[ { "path": ".gitignore", "content": "## MATLAB files to ignore ##\n\n# Windows default autosave extension\n*.asv\n\n# OSX / *nix default autosave extension\n*.m~\n\n# Compiled MEX binaries (all platforms)\n*.mex*\n\n# Packaged app and toolbox files\n*.mlappinstall\n*.mltbx\n\n# Generated helpsearch folders\nhelpsearch*/\n\n# Simulink code generation folders\nslprj/\nsccprj/\n\n# Matlab code generation folders\ncodegen/\n\n# Simulink autosave extension\n*.autosave\n\n# Simulink cache files\n*.slxc\n\n# Octave session info\noctave-workspace\n" }, { "path": "LICENSE", "content": "MIT License\n\nCopyright (c) 2021 Jun Zeng\n\nPermission is hereby granted, free of charge, to any person obtaining a copy\nof this software and associated documentation files (the \"Software\"), to deal\nin the Software without restriction, including without limitation the rights\nto use, copy, modify, merge, publish, distribute, sublicense, and/or sell\ncopies of the Software, and to permit persons to whom the Software is\nfurnished to do so, subject to the following conditions:\n\nThe above copyright notice and this permission notice shall be included in all\ncopies or substantial portions of the Software.\n\nTHE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\nIMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\nFITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\nAUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\nLIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\nOUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE\nSOFTWARE.\n" }, { "path": "README.md", "content": "**Status**: The implementation code for corresponding papers will be merged here and new papers will be added in an inverse order of submission.\n\n### Introduction\n\nIn this repository, a collection of our work is presented where nonlinear model predictive control (NMPC) with control Lyapunov functions (CLFs) and control barrier functions (CBFs) are applied.\n\n### Dependencies\nThe packages needed for running the code are [Yalmip](https://yalmip.github.io/) and [IPOPT](https://projects.coin-or.org/Ipopt/wiki/MatlabInterface).\n\nWe also provide the zipped version of precompiled .mex files for IPOPT in the folder `packages` in case you don't have it. Unzip the file based on the operating system and add those .mex files into your MATLAB path.\n\n### Citing\n\n#### Theoretical Publications\n\nIf you find this project useful in your work, please consider citing following work:\n\n* S. Liu, J. Zeng, K. Sreenath and C. Belta. \"Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions.\" *2023 IEEE American Control Conference (ACC)*. [[arXiv]](https://arxiv.org/abs/2210.04361) [[Docs]](matlab/acc2023/README.md) [[Code]](matlab/acc2023) [[BibTex]](bibtex/acc2023_impc_dhocbf.md)\n\n* A. Thirugnanam, J. Zeng, K. Sreenath. \"Safety-Critical Control and Planning for Obstacle Avoidance between Polytopes with Control Barrier Functions.\" *2022 IEEE International Conference on Robotics and Automation (ICRA)*. [[IEEE]](https://ieeexplore.ieee.org/document/9812334) [[arXiv]](https://arxiv.org/abs/2109.12313) [[Video]](https://youtu.be/wucophROPRY) [[BibTex]](bibtex/icra2022_nmpc_dcbf_polytope.md)\n\n* J. Zeng, Z. Li, K. Sreenath. \"Enhancing Feasibility and Safety of Nonlinear Model Predictive Control with Discrete-Time Control Barrier Functions.\" *2021 IEEE Conference on Decision and Control (CDC)*. [[IEEE]](https://ieeexplore.ieee.org/document/9683174) [[arXiv]](https://arxiv.org/abs/2105.10596) [[Docs]](matlab/cdc2021/README.md) [[Code]](matlab/cdc2021) [[BibTex]](bibtex/cdc2022_nmpc_dcbf_feasibility.md)\n\n* J. Zeng, B. Zhang and K. Sreenath. \"Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function.\" *2021 IEEE American Control Conference (ACC)*. [[IEEE]](https://ieeexplore.ieee.org/document/9483029) [[arXiv]](https://arxiv.org/abs/2007.11718) [[Docs]](matlab/acc2021/README.md) [[Code]](matlab/acc2021) [[BibTex]](bibtex/acc2021_nmpc_dcbf.md)\n\n#### Applicational Publications\n\n* Z. Li, J. Zeng, A. Thirugnanam, K. Sreenath. \"Bridging Model-based Safety and Model-free Reinforcement Learning through System Identification of Low Dimensional Linear Models.\" *2022 Proceedings of Robotics: Science and Systems (RSS)*. [[RSS]](http://www.roboticsproceedings.org/rss18/p033.html) [[arXiv]](https://arxiv.org/abs/2205.05787) [[BibTex]](bibtex/rss2022_nmpc_dcbf_legged_robots.md) [[Webpage]](https://sites.google.com/berkeley.edu/rl-sysid-rss2022/home)\n" }, { "path": "bibtex/acc2021_nmpc_dcbf.md", "content": "```\n@inproceedings{zeng2021mpccbf,\n title={Safety-critical model predictive control with discrete-time control barrier function},\n author={Zeng, Jun and Zhang, Bike and Sreenath, Koushil},\n booktitle={2021 American Control Conference (ACC)},\n year={2021},\n volume={},\n number={},\n pages={3882-3889}\n}\n```" }, { "path": "bibtex/acc2023_impc_dhocbf.md", "content": "```\n@inproceedings{liu2023iterative,\n title={Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions},\n author={Liu, Shuo and Zeng, Jun and Sreenath, Koushil and Belta, Calin A},\n booktitle={2023 American Control Conference (ACC)},\n year={2023}\n}\n```" }, { "path": "bibtex/cdc2022_nmpc_dcbf_feasibility.md", "content": "```\n@inproceedings{zeng2021mpccbf-feasibility,\n title={Enhancing feasibility and safety of nonlinear model predictive control with discrete-time control barrier functions},\n author={Zeng, Jun and Li, Zhongyu and Sreenath, Koushil},\n booktitle={2021 Conference on Decision and Control (CDC)},\n year={2021},\n volume={},\n number={},\n pages={6137-6144}\n}\n``` " }, { "path": "bibtex/icra2022_nmpc_dcbf_polytope.md", "content": "```\n@inproceedings{thirugnanam2021safetycritical,\n author={Thirugnanam, Akshay and Zeng, Jun and Sreenath, Koushil},\n booktitle={2022 International Conference on Robotics and Automation (ICRA)}, \n title={Safety-Critical Control and Planning for Obstacle Avoidance between Polytopes with Control Barrier Functions}, \n year={2022},\n volume={},\n number={},\n pages={286-292},\n}\n```" }, { "path": "bibtex/rss2022_nmpc_dcbf_legged_robots.md", "content": "```\n@inproceedings{li2022bridging, \n author = {Li, Zhongyu and Zeng, Jun and Thirugnanam, Akshay and Sreenath, Koushil}, \n title = {{Bridging Model-based Safety and Model-free Reinforcement Learning through System Identification of Low Dimensional Linear Models}}, \n booktitle = {Proceedings of Robotics: Science and Systems}, \n year = {2022}, \n address = {New York City, NY, USA}, \n month = {June}, \n}\n```" }, { "path": "matlab/acc2021/DCLFDCBF.m", "content": "classdef DCLFDCBF < handle\n properties\n system\n params\n x0\n x_curr\n time_curr = 0.0\n xlog = []\n ulog = []\n u_cost = 0\n obs\n end\n \n methods\n function self = DCLFDCBF(x0, system, params)\n % Define DCLF-DCBF controller\n self.x0 = x0;\n self.x_curr = x0;\n self.system = system;\n self.params = params;\n end\n \n function sim(self, time)\n % Simulate the system until a given time\n xk = self.x_curr;\n while self.time_curr <= time\n % Solve DCLF-DCBF\n uk = self.solveDCLFDCBF();\n xk = self.system.A * xk + self.system.B * uk;\n % update system\n self.x_curr = xk;\n self.time_curr = self.time_curr + self.system.dt;\n self.xlog = [self.xlog, xk];\n self.ulog = [self.ulog, uk];\n self.u_cost = self.u_cost + uk'*uk*self.system.dt;\n end\n end\n \n function uopt = solveDCLFDCBF(self)\n % Solve DCLF-DCBF\n x = self.x_curr;\n u = sdpvar(2,1);\n delta = sdpvar(1,1);\n constraints = [];\n % add DCLF constraints\n x_next = self.system.A * x + self.system.B * u;\n v = x' * self.params.P * x;\n v_next = x_next' * self.params.P * x_next;\n constraints = [constraints; v_next - v + self.params.alpha * v <= delta];\n % add DCBF constraints\n pos = self.obs.pos;\n r = self.obs.r;\n b = (x(1:2)-pos)'*(x(1:2)-pos) - r^2;\n b_next = (x_next(1:2)-pos)'*(x_next(1:2)-pos) - r^2;\n constraints = [constraints; b_next - b + self.params.gamma*b >= 0];\n % input constraints\n constraints = [constraints; self.system.ul <= u <= self.system.uu];\n % cost\n cost = u'*self.params.H*u + delta'*self.params.p*delta;\n % solve optimization\n ops = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, ops);\n if diagnostics.problem == 0\n feas = true;\n uopt = value(u);\n else\n feas = false;\n uopt = [];\n end\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function plot(self,figure_name)\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot trajectory\n plot(self.xlog(1,:), self.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0,'MarkerSize',4);\n % plot obstacle\n pos = self.obs.pos;\n r = self.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980], 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(self.x0(1), self.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'DCLF-DCBF'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf,strcat('figures/',figure_name), '-depsc');\n end\n end\nend" }, { "path": "matlab/acc2021/MPCCBF.m", "content": "classdef MPCCBF < handle\n % MPC with distance constraints\n properties\n system\n params\n x0\n x_curr\n time_curr = 0.0\n xlog = []\n ulog = []\n distlog = []\n solvertime = []\n xopenloop = {}\n uopenloop = {}\n u_cost = 0\n obs\n end\n methods\n function self = MPCCBF(x0, system, params)\n % Define MPC_CBF controller\n self.x0 = x0;\n self.x_curr = x0;\n self.system = system;\n self.params = params;\n end\n \n function sim(self, time)\n % Simulate the system until a given time\n xk = self.x_curr;\n while self.time_curr <= time\n % Solve CFTOC\n [~, uk] = self.solveMPCCBF(self.x_curr);\n xk = self.system.A * xk + self.system.B * uk;\n % update system\n self.x_curr = xk;\n self.time_curr = self.time_curr + self.system.dt;\n self.xlog = [self.xlog, xk];\n self.ulog = [self.ulog, uk];\n self.u_cost = self.u_cost + uk'*uk*self.system.dt;\n end\n end\n \n function [xopt, uopt] = solveMPCCBF(self, xk)\n % Solve MPC-CBF\n [feas, x, u, J] = self.solve_cftoc(xk);\n if ~feas\n xopt = [];\n uopt = [];\n return\n else\n xopt = x(:,2);\n uopt = u(:,1);\n end\n end\n \n function [feas, xopt, uopt, Jopt] = solve_cftoc(self, xk)\n % Solve CFTOC\n % extract variables\n N = self.params.N;\n % define variables and cost\n x = sdpvar(4, N+1);\n u = sdpvar(2, N);\n constraints = [];\n cost = 0;\n % initial constraint\n constraints = [constraints; x(:,1) == xk];\n % add constraints and costs\n for i = 1:N\n constraints = [constraints;\n self.system.xl <= x(:,i) <= self.system.xu;\n self.system.ul <= u(:,i) <= self.system.uu\n x(:,i+1) == self.system.A * x(:,i) + self.system.B * u(:,i)];\n cost = cost + x(:,i)'*self.params.Q*x(:,i) + u(:,i)'*self.params.R*u(:,i);\n end\n % add CBF constraints\n for i = 1:N\n pos = self.obs.pos;\n r = self.obs.r ;\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos)) - r^2;\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos)) - r^2;\n constraints = [constraints; b_next - b + self.params.gamma * b >= 0];\n end\n % add terminal cost\n cost = cost + x(:,N+1)'*self.params.P*x(:,N+1);\n ops = sdpsettings('solver','ipopt','verbose',0);\n % solve optimization\n diagnostics = optimize(constraints, cost, ops);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt = [];\n uopt = [];\n Jopt = [];\n end\n self.distlog = [self.distlog, sqrt((xk(1:2)-pos)'*(xk(1:2)-pos)-r^2)];\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\n self.solvertime = [self.solvertime, diagnostics.solvertime];\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function plot(self, figure_name)\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot closed-loop trajectory\n plot(self.xlog(1,:), self.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0, 'MarkerSize', 4);\n % plot open-loop trajectory\n for i = 1:size(self.xopenloop, 2)\n plot(self.xopenloop{i}(1,:), self.xopenloop{i}(2,:),...\n 'k*-.', 'LineWidth', 0.5,'MarkerSize',0.5)\n end\n % plot obstacle\n pos = self.obs.pos;\n r = self.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980],...\n 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980],...\n 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(self.x0(1), self.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'MPC-CBF'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf,strcat('figures/',figure_name), '-depsc');\n end\n end\nend" }, { "path": "matlab/acc2021/MPCDC.m", "content": "classdef MPCDC < handle\n % MPC with distance constraints\n properties\n system\n params\n x0\n x_curr\n time_curr = 0.0\n xlog = []\n ulog = []\n distlog = []\n solvertime = []\n xopenloop = {}\n uopenloop = {}\n u_cost = 0\n obs\n end\n methods\n function self = MPCDC(x0, system, params)\n % Define MPC_DC controller\n self.x0 = x0;\n self.x_curr = x0;\n self.system = system;\n self.params = params;\n end\n \n function sim(self, time)\n % Simulate the system until a given time\n xk = self.x_curr;\n while self.time_curr <= time\n % Solve CFTOC\n [~, uk] = self.solveMPCDC(self.x_curr);\n xk = self.system.A * xk + self.system.B * uk;\n % update system\n self.x_curr = xk;\n self.time_curr = self.time_curr + self.system.dt;\n self.xlog = [self.xlog, xk];\n self.ulog = [self.ulog, uk];\n self.u_cost = self.u_cost + uk'*uk*self.system.dt;\n end\n end\n \n function [xopt, uopt] = solveMPCDC(self, xk)\n % Solve MPC-DC\n [feas, x, u, J] = self.solve_cftoc(xk);\n if ~feas\n xopt = [];\n uopt = [];\n return\n else\n xopt = x(:,2);\n uopt = u(:,1);\n end\n end\n \n function [feas, xopt, uopt, Jopt] = solve_cftoc(self, xk)\n % Solve CFTOC\n % extract variables\n N = self.params.N;\n % define variables and cost\n x = sdpvar(4, N+1);\n u = sdpvar(2, N);\n constraints = [];\n cost = 0;\n % initial constraint\n constraints = [constraints; x(:,1) == xk];\n % add constraints and costs\n for i = 1:N\n constraints = [constraints;\n self.system.xl <= x(:,i) <= self.system.xu;\n self.system.ul <= u(:,i) <= self.system.uu\n x(:,i+1) == self.system.A * x(:,i) + self.system.B * u(:,i)];\n cost = cost + x(:,i)'*self.params.Q*x(:,i) + u(:,i)'*self.params.R*u(:,i);\n end\n % add CBF constraints\n for i = 1:N\n pos = self.obs.pos;\n r = self.obs.r ;\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos)) - r^2;\n constraints = [constraints; b >= 0];\n end\n % add terminal cost\n cost = cost + x(:,N+1)'*self.params.P*x(:,N+1);\n ops = sdpsettings('solver','ipopt','verbose',0);\n % solve optimization\n diagnostics = optimize(constraints, cost, ops);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt = [];\n uopt = [];\n Jopt = [];\n end\n self.distlog = [self.distlog, sqrt((xk(1:2)-pos)'*(xk(1:2)-pos)-r^2)];\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\n self.solvertime = [self.solvertime, diagnostics.solvertime];\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function plot(self, figure_name)\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot closed-loop trajectory\n plot(self.xlog(1,:), self.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0, 'MarkerSize', 4);\n % plot open-loop trajectory\n for i = 1:size(self.xopenloop, 2)\n plot(self.xopenloop{i}(1,:), self.xopenloop{i}(2,:),...\n 'k*-.', 'LineWidth', 0.5,'MarkerSize',0.5)\n end\n % plot obstacle\n pos = self.obs.pos;\n r = self.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980],...\n 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980],...\n 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(self.x0(1), self.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'MPC-DC'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf,strcat('figures/',figure_name), '-depsc');\n end\n end\nend" }, { "path": "matlab/acc2021/README.md", "content": "## MPC-CBF-ACC2021\nWe propose a control framework which unifies the model predictive control and control barrier functions, where terminal cost function serves as control Lyapunov functions for stability. This is the reference implementation of our paper:\n\nIf you find this project useful in your work, please consider citing following work:\n```\n@inproceedings{zeng2021mpccbf,\n title={Safety-critical model predictive control with discrete-time control barrier function},\n author={Zeng, Jun and Zhang, Bike and Sreenath, Koushil},\n booktitle={2021 American Control Conference (ACC)},\n year={2021}\n}\n```\n\n### Instructions\nThe 2D double integrator is assigned to reach the target position at origin while avoiding obstacles. We have three classes for different controllers: `DCLFDCBF.m` (DCLF-DCBF), `MPCCBF.m` (MPC-CBF) and `MPCDC` (MPC-DC), respectively.\n\nMoreover, to illustrate the performance among them, we have:\n* `test.m`: Run DCLF-DCBF/MPC-CBF/MPC-DC respectively.\n* `testGamma.m`: Run analysis for different hyperparameter $\\gamma$.\n* `testHorizon.m`: Run analysis for different horizon.\n* `testBenchmark.m`: Run analysis for some benchmark." }, { "path": "matlab/acc2021/figures/.gitignore", "content": "*.eps\n" }, { "path": "matlab/acc2021/test.m", "content": "close all\nclear\n\n%% General Flags\nrun_dclf_dcbf = true;\ndisplay_dclf_dcbf = true;\nrun_mpc_cbf_one = true;\ndisplay_mpc_cbf_one = true;\nrun_mpc_cbf_multiple = true;\nrun_mpc_cbf_multiple = true;\nrun_mpc_dc = true;\ndisplay_mpc_dc = true;\n\n%% Setup and Parameters\nx0 = [-5; -5; 0; 0];\ntime_total = 30.0;\ndt = 0.2;\nP = 100*eye(4);\nQ = 10*eye(4);\nR = eye(2);\nN = 8;\nxmin = [-5; -5; -5; -5];\nxmax = [5; 5; 5; 5];\numin = [-1; -1];\numax = [1; 1];\n\n%% Discrete-time double integrator 2D\nsystem.dt = dt;\nsystem.A = [1 0 dt 0;\n 0 1 0 dt;\n 0 0 1 0;\n 0 0 0 1];\nsystem.B = [0.5*dt^2 0;\n 0 0.5*dt^2;\n dt 0;\n 0 dt];\nsystem.xl = xmin;\nsystem.xu = xmax;\nsystem.ul = umin;\nsystem.uu = umax;\n\n%% DCLF-DCBF parameters\np = 1e3;\nparams_dclf_dcbf.P = P;\nparams_dclf_dcbf.H = R;\nparams_dclf_dcbf.p = p;\nparams_dclf_dcbf.alpha = 1.0;\nparams_dclf_dcbf.gamma = 0.4;\n\n%% MPC-CBF parameters\nparams_mpc_cbf.Q = Q;\nparams_mpc_cbf.R = R;\nparams_mpc_cbf.P = P;\nparams_mpc_cbf.N = N;\nparams_mpc_cbf.gamma = 0.4;\n\n%% MPC-DC parameters\nparams_mpc_dc.Q = Q;\nparams_mpc_dc.R = R;\nparams_mpc_dc.P = P;\nparams_mpc_dc.N = N;\n\n%% Obstacle\nobs.pos = [-2; -2.25];\nobs.r = 1.5;\n\n%% Simulate DCLF-DCBF\nif run_dclf_dcbf\n fprintf('Run DCLF-DCBF\\n');\n controller_dclf_dcbf = DCLFDCBF(x0, system, params_dclf_dcbf);\n controller_dclf_dcbf.obs = obs;\n controller_dclf_dcbf.sim(time_total);\nend\n\n%% Display DCLF-DCBF simulation\nif display_dclf_dcbf\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot trajectory\n plot(controller_dclf_dcbf.xlog(1,:), controller_dclf_dcbf.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0,'MarkerSize',4);\n % plot obstacle\n pos = controller_dclf_dcbf.obs.pos;\n r = controller_dclf_dcbf.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980], 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(controller_dclf_dcbf.x0(1), controller_dclf_dcbf.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'DCLF-DCBF'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf,'figures/dclf-dcbf-avoidance.eps', '-depsc');\n print(gcf,'figures/dclf-dcbf-avoidance.png', '-dpng', '-r800');\nend\n\n%% Simulate MPC-CBF with N=1;\nparams_mpc_cbf.N = 1;\nif run_mpc_cbf_one\n fprintf('Run MPC-CBF\\n');\n controller_mpc_cbf_one = MPCCBF(x0, system, params_mpc_cbf);\n controller_mpc_cbf_one.obs = obs;\n controller_mpc_cbf_one.sim(time_total);\nend\n\n%% Display MPC-CBF simulation with N=1\nif display_mpc_cbf_one\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot trajectory\n plot(controller_mpc_cbf_one.xlog(1,:), controller_mpc_cbf_one.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0,'MarkerSize',4);\n % plot obstacle\n pos = controller_mpc_cbf_one.obs.pos;\n r = controller_mpc_cbf_one.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980], 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(controller_mpc_cbf_one.x0(1), controller_mpc_cbf_one.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'MPC-CBF ($N=1$)'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf, 'figures/mpc-cbf-avoidance-one-step.eps', '-depsc');\n print(gcf, 'figures/mpc-cbf-avoidance-one-step.png', '-dpng', '-r800');\nend\n\n%% Simulate MPC-CBF with other N\nparams_mpc_cbf.N = 8;\nif run_mpc_cbf_one\n fprintf('Run MPC-CBF\\n');\n controller_mpc_cbf_multiple = MPCCBF(x0, system, params_mpc_cbf);\n controller_mpc_cbf_multiple.obs = obs;\n controller_mpc_cbf_multiple.sim(time_total);\nend\n\n%% Display MPC-CBF simulation with other N\nif display_mpc_cbf_one\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot trajectory\n plot(controller_mpc_cbf_multiple.xlog(1,:), controller_mpc_cbf_multiple.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0,'MarkerSize',4);\n % plot obstacle\n pos = controller_mpc_cbf_multiple.obs.pos;\n r = controller_mpc_cbf_multiple.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980], 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(controller_mpc_cbf_multiple.x0(1), controller_mpc_cbf_multiple.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'MPC-CBF ($N=8$)'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf, 'figures/mpc-cbf-avoidance-several-steps.eps', '-depsc');\n print(gcf, 'figures/mpc-cbf-avoidance-several-steps.png', '-dpng', '-r800');\nend\n\n%% Simulate MPC-DC\nif run_mpc_dc\n fprintf('Run MPC-DC\\n');\n controller_mpc_dc = MPCDC(x0, system, params_mpc_dc);\n controller_mpc_dc.obs = obs;\n controller_mpc_dc.sim(time_total);\nend\n\n%% Display MPC-DC simulation\nif display_mpc_dc\n % Plot simulation\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n % plot trajectory\n plot(controller_mpc_dc.xlog(1,:), controller_mpc_dc.xlog(2,:), 'ko-',...\n 'LineWidth', 1.0,'MarkerSize',4);\n % plot obstacle\n pos = controller_mpc_dc.obs.pos;\n r = controller_mpc_dc.obs.r;\n th = linspace(0,2*pi*100);\n x = cos(th) ; y = sin(th) ;\n plot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 2);\n plot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980], 'MarkerSize', 5, 'LineWidth', 2);\n % plot target position\n plot(controller_mpc_dc.x0(1), controller_mpc_dc.x0(2), 'db', 'LineWidth', 1);\n plot(0.0, 0.0, 'dr', 'LineWidth', 1);\n h=get(gca,'Children');\n h_legend = legend(h([end]), {'MPC-DC ($N=8$)'}, 'Location', 'SouthEast');\n set(h_legend, 'Interpreter','latex');\n set(gca,'LineWidth', 0.2, 'FontSize', 15);\n grid on\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$y (m)$','interpreter','latex','FontSize',20);\n xticks(-5:0);\n yticks(-5:0);\n xlim([-5,0.2]);\n ylim([-5,0.2]);\n print(gcf, 'figures/mpc-dc-avoidance.eps', '-depsc');\n print(gcf, 'figures/mpc-dc-avoidance.png', '-dpng', '-r800');\nend" }, { "path": "matlab/acc2021/testBenchmark.m", "content": "close all\nclear\n\n%% Setup and Parameters\nx0 = [-5; -5; 0; 0];\ntime_total = 20.0;\ndt = 0.2;\nP = 100*eye(4);\nQ = 10*eye(4);\nR = eye(2);\nN = 8;\nxmin = [-5; -5; -5; -5];\nxmax = [5; 5; 5; 5];\numin = [-1; -1];\numax = [1; 1];\n\n%% Discrete-time double integrator 2D\nsystem.dt = dt;\nsystem.A = [1 0 dt 0;\n 0 1 0 dt;\n 0 0 1 0;\n 0 0 0 1];\nsystem.B = [0.5*dt^2 0;\n 0 0.5*dt^2;\n dt 0;\n 0 dt];\nsystem.xl = xmin;\nsystem.xu = xmax;\nsystem.ul = umin;\nsystem.uu = umax;\n\n%% MPC-CBF parameters\nparams.Q = Q;\nparams.R = R;\nparams.P = P;\nparams.N = N;\nparams.gamma = 0.5;\n\n%% Obstacle\nobs.pos = [-2; -2.25];\nobs.r = 1.5;\n\n%% Simulate MPC-CBF\ngamma_list = linspace(0.1, 0.5, 5);\ncontroller_mpc_cbf_list = {};\nfor i = 1:size(gamma_list, 2)\n new_params = params;\n new_params.N = 5;\n new_params.gamma = gamma_list(i);\n controller_mpc_cbf = MPCCBF(x0, system, new_params);\n controller_mpc_cbf.obs = obs;\n controller_mpc_cbf.sim(time_total);\n controller_mpc_cbf_list{i} = controller_mpc_cbf;\nend\n\n%% Computational time benchmark\nfor i = 1:size(gamma_list, 2)\n controller_mpc_cbf = controller_mpc_cbf_list{i};\n fprintf('Computational time for MPC-CBF3: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_cbf.solvertime),...\n std(controller_mpc_cbf.solvertime),...\n min(controller_mpc_cbf.solvertime),...\n max(controller_mpc_cbf.solvertime),...\n controller_mpc_cbf.u_cost,...\n min(controller_mpc_cbf.distlog)]);\nend" }, { "path": "matlab/acc2021/testGamma.m", "content": "close all\nclear\n\n%% Setup and Parameters\nx0 = [-5; -5; 0; 0];\ntime_total = 20.0;\ndt = 0.2;\nP = 100*eye(4);\nQ = 10*eye(4);\nR = eye(2);\nN = 8;\nxmin = [-5; -5; -5; -5];\nxmax = [5; 5; 5; 5];\numin = [-1; -1];\numax = [1; 1];\n\n%% Discrete-time double integrator 2D\nsystem.dt = dt;\nsystem.A = [1 0 dt 0;\n 0 1 0 dt;\n 0 0 1 0;\n 0 0 0 1];\nsystem.B = [0.5*dt^2 0;\n 0 0.5*dt^2;\n dt 0;\n 0 dt];\nsystem.xl = xmin;\nsystem.xu = xmax;\nsystem.ul = umin;\nsystem.uu = umax;\n\n%% MPC-CBF parameters\nparams_mpc_cbf.Q = Q;\nparams_mpc_cbf.R = R;\nparams_mpc_cbf.P = P;\nparams_mpc_cbf.N = N;\nparams_mpc_cbf.gamma = 0.5;\n\n%% Obstacle\nobs.pos = [-2; -2.25];\nobs.r = 1.5;\n\n%% Simulate MPC-CBF\ngamma_list = [0.1, 0.2, 0.3, 1.0]; \nfor ind = 1:size(gamma_list, 2)\n fprintf('Run MPC-CBF with gamma %f\\n', gamma_list(ind));\n params_mpc_cbf.gamma = gamma_list(ind);\n controller_mpc_cbf_list{ind} = MPCCBF(x0, system, params_mpc_cbf);\n controller_mpc_cbf_list{ind}.obs = obs;\n controller_mpc_cbf_list{ind}.sim(time_total);\nend\n\n%% Simulate MPC-DC\nparams_mpc_dc = params_mpc_cbf;\ncontroller_mpc_dc = MPCDC(x0, system, params_mpc_cbf);\ncontroller_mpc_dc.obs = obs;\ncontroller_mpc_dc.sim(time_total);\n\n%% Visualization benchmark\nfigure('Renderer', 'painters', 'Position', [0 0 400 400]);\nset(gca,'LooseInset',get(gca,'TightInset'));\nhold on\nfor ind = 1:size(gamma_list, 2)\n plot(controller_mpc_cbf_list{ind}.xlog(1,:), controller_mpc_cbf_list{ind}.xlog(2,:),...\n '-', 'LineWidth', 1.0, 'MarkerSize', 4);\nend\nplot(controller_mpc_dc.xlog(1,:), controller_mpc_dc.xlog(2,:),...\n 'k--', 'LineWidth', 1.0, 'MarkerSize', 4);\n% plot obstacle\npos = obs.pos;\nr = obs.r;\nth = linspace(0,2*pi*100);\nx = cos(th) ; y = sin(th) ;\nplot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980],...\n 'LineWidth', 2);\nplot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980],...\n 'MarkerSize', 5, 'LineWidth', 2);\n% plot target position\nplot(x0(1), x0(2), 'db', 'LineWidth', 1);\nplot(0.0, 0.0, 'dr', 'LineWidth', 1);\nh=get(gca,'Children');\nh_legend = legend(h([end, end-1, end-2, end-3, end-4]),...\n {'MPC-CBF ($\\gamma=0.1$)', 'MPC-CBF ($\\gamma=0.2$)',...\n 'MPC-CBF ($\\gamma=0.3$)', 'MPC-CBF ($\\gamma=1.0$)', 'MPC-DC'}, 'Location', 'SouthEast');\nset(h_legend, 'Interpreter','latex');\nset(gca,'LineWidth', 0.2, 'FontSize', 15);\ngrid on\nxlabel('$x (m)$','interpreter','latex','FontSize',20);\nylabel('$y (m)$','interpreter','latex','FontSize',20);\nxticks(-5:1);\nyticks(-5:1);\nxlim([-5,0.2]);\nylim([-5,0.2]);\nprint(gcf,'figures/benchmark-gamma.eps', '-depsc');\nprint(gcf,'figures/benchmark-gamma.png', '-dpng', '-r800');" }, { "path": "matlab/acc2021/testHorizon.m", "content": "close all\nclear\n\n%% Setup and Parameters\nx0 = [-5; -5; 0; 0];\ntime_total = 20.0;\ndt = 0.2;\nP = 100*eye(4);\nQ = 10*eye(4);\nR = eye(2);\nN = 8;\nxmin = [-5; -5; -5; -5];\nxmax = [5; 5; 5; 5];\numin = [-1; -1];\numax = [1; 1];\n\n%% Discrete-time double integrator 2D\nsystem.dt = dt;\nsystem.A = [1 0 dt 0;\n 0 1 0 dt;\n 0 0 1 0;\n 0 0 0 1];\nsystem.B = [0.5*dt^2 0;\n 0 0.5*dt^2;\n dt 0;\n 0 dt];\nsystem.xl = xmin;\nsystem.xu = xmax;\nsystem.ul = umin;\nsystem.uu = umax;\n\n%% MPC-CBF parameters\nparams.Q = Q;\nparams.R = R;\nparams.P = P;\nparams.N = N;\nparams.gamma = 0.5;\n\n%% Obstacle\nobs.pos = [-2; -2.25];\nobs.r = 1.5;\n\n%% Simulate MPC-CBF\nparams.N = 5;\nparams.gamma = 0.25;\ncontroller_mpc_cbf_1 = MPCCBF(x0, system, params);\ncontroller_mpc_cbf_1.obs = obs;\ncontroller_mpc_cbf_1.sim(time_total);\n\n%% Simulate MPC-CBF with lower gamma\nparams.gamma = 0.20;\ncontroller_mpc_cbf_2 = MPCCBF(x0, system, params);\ncontroller_mpc_cbf_2.obs = obs;\ncontroller_mpc_cbf_2.sim(time_total);\n\nparams.gamma = 0.15;\ncontroller_mpc_cbf_3 = MPC_CBF(x0, system, params);\ncontroller_mpc_cbf_3.obs = obs;\ncontroller_mpc_cbf_3.sim(time_total);\n\n%% Simulate MPC-DC\n\n% The problem is infeasible at N=5\n% params.N = 5;\n% controller_mpc_dc_0 = MPCDC(x0, system, params);\n% controller_mpc_dc_0.obs = obs;\n% controller_mpc_dc_0.sim(time_total);\n\nparams.N = 7;\ncontroller_mpc_dc_1 = MPCDC(x0, system, params);\ncontroller_mpc_dc_1.obs = obs;\ncontroller_mpc_dc_1.sim(time_total);\n\nparams.N = 15;\ncontroller_mpc_dc_2 = MPCDC(x0, system, params);\ncontroller_mpc_dc_2.obs = obs;\ncontroller_mpc_dc_2.sim(time_total);\n\n%%\nparams.N = 30;\ncontroller_mpc_dc_3 = MPCDC(x0, system, params);\ncontroller_mpc_dc_3.obs = obs;\ncontroller_mpc_dc_3.sim(time_total);\n\n%% Visualization benchmark\nfigure('Renderer', 'painters', 'Position', [0 0 400 400]);\nset(gca,'LooseInset',get(gca,'TightInset'));\nhold on\nplot(controller_mpc_cbf_3.xlog(1,:), controller_mpc_cbf_3.xlog(2,:),...\n '-', 'Color', [0, 0.4470, 0.7410], 'LineWidth', 1.0, 'MarkerSize', 4);\nplot(controller_mpc_cbf_2.xlog(1,:), controller_mpc_cbf_2.xlog(2,:),...\n '-', 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 1.0, 'MarkerSize', 4);\nplot(controller_mpc_cbf_1.xlog(1,:), controller_mpc_cbf_1.xlog(2,:),...\n '-', 'Color', [0.9290, 0.6940, 0.1250], 'LineWidth', 1.0, 'MarkerSize', 4);\nplot(controller_mpc_dc_1.xlog(1,:), controller_mpc_dc_1.xlog(2,:),...\n '--', 'Color', [0.4940, 0.1840, 0.5560], 'LineWidth', 1.0, 'MarkerSize', 4);\nplot(controller_mpc_dc_2.xlog(1,:), controller_mpc_dc_2.xlog(2,:),...\n '--', 'Color', [0.4660, 0.6740, 0.1880], 'LineWidth', 1.0, 'MarkerSize', 4);\nplot(controller_mpc_dc_3.xlog(1,:), controller_mpc_dc_3.xlog(2,:),...\n '--', 'Color', [0.3010, 0.7450, 0.9330], 'LineWidth', 1.0, 'MarkerSize', 4);\n% plot obstacle\npos = obs.pos;\nr = obs.r;\nth = linspace(0,2*pi*100);\nx = cos(th) ; y = sin(th) ;\nplot(pos(1) + r*x, pos(2) + r*y, 'Color', [0.8500, 0.3250, 0.0980],...\n 'LineWidth', 2);\nplot(pos(1), pos(2), 'Color', [0.8500, 0.3250, 0.0980],...\n 'MarkerSize', 5, 'LineWidth', 2);\n% plot target position\nplot(x0(1), x0(2), 'db', 'LineWidth', 1);\nplot(0.0, 0.0, 'dr', 'LineWidth', 1);\nh=get(gca,'Children');\nh_legend = legend(h([end, end-1, end-2, end-3, end-4, end-5]),...\n {'MPC-CBF ($N = 5, \\gamma = 0.15$)', 'MPC-CBF ($N = 5, \\gamma = 0.20$)',...\n 'MPC-CBF ($N = 5, \\gamma = 0.25$)', 'MPC-DC ($N = 7$)', 'MPC-DC ($N = 15$)',...\n 'MPC-DC ($N = 30$)'}, 'Location', 'SouthEast');\nset(h_legend, 'Interpreter','latex', 'FontSize', 10);\nset(gca,'LineWidth', 0.2, 'FontSize', 15);\ngrid on\nxlabel('$x (m)$','interpreter','latex','FontSize',20);\nylabel('$y (m)$','interpreter','latex','FontSize',20);\nxticks(-5:0);\nyticks(-5:0);\nxlim([-5,0.2]);\nylim([-5,0.2]);\nprint(gcf,'figures/benchmark-horizon.eps', '-depsc');\nprint(gcf,'figures/benchmark-horizon.png', '-dpng', '-r800');\n\n%% Computational time benchmark\nfigure('Renderer', 'painters', 'Position', [0 0 800 400]);\nxlabel('Computational time (s)', 'interpreter', 'latex', 'FontSize', 16);\nylabel('Percentage', 'interpreter', 'latex', 'FontSize', 16);\nhold on\n% [N_mpc_3, edges_mpc_3] = histcounts(controller_mpc_cbf_3.solvertime, 10, 'Normalization', 'probability');\n% plot(edges_mpc_3(1:end-1), N_mpc_3, '-', 'Color', [0, 0.4470, 0.7410], 'LineWidth', 1);\n% [N_mpc_2, edges_mpc_2] = histcounts(controller_mpc_cbf_2.solvertime, 10, 'Normalization', 'probability');\n% plot(edges_mpc_2(1:end-1), N_mpc_2, '-', 'Color', [0.8500, 0.3250, 0.0980], 'LineWidth', 1);\n[N_mpc_1, edges_mpc_1] = histcounts(controller_mpc_cbf_1.solvertime, 10, 'Normalization', 'probability');\nplot(edges_mpc_1(1:end-1), N_mpc_1, '-', 'Color', [0.9290, 0.6940, 0.1250], 'LineWidth', 1);\n[N_dc_1, edges_dc_1] = histcounts(controller_mpc_dc_1.solvertime, 10, 'Normalization', 'probability');\nplot(edges_dc_1(1:end-1), N_dc_1, '--', 'Color', [0.4940, 0.1840, 0.5560], 'LineWidth', 1);\n[N_dc_2, edges_dc_2] = histcounts(controller_mpc_dc_2.solvertime, 10, 'Normalization', 'probability');\nplot(edges_dc_2(1:end-1), N_dc_2, '--', 'Color', [0.4660, 0.6740, 0.1880], 'LineWidth', 1);\n[N_dc_3, edges_dc_3] = histcounts(controller_mpc_dc_3.solvertime, 10, 'Normalization', 'probability');\nplot(edges_dc_3(1:end-1), N_dc_3, '--', 'Color', [0.3010, 0.7450, 0.9330], 'LineWidth', 1);\nh=get(gca,'Children');\nh_legend = legend(h([end, end-1, end-2, end-3]),...\n {'MPC-CBF ($N = 5, \\gamma = 0.25$)',...\n 'MPC-DC ($N = 7$)',...\n 'MPC-DC ($N = 15$)',...\n 'MPC-DC ($N = 30$)'}, 'Location', 'NorthEast');\nset(h_legend, 'Interpreter','latex');\nset(gca,'LineWidth', 0.2, 'FontSize', 15);\nprint(gcf,'figures/benchmark-horizon-computational-time.eps', '-depsc');\nprint(gcf,'figures/benchmark-horizon-computational-time.png', '-dpng', '-r800');\n%% Computational time table\nfprintf('Computational time for MPC-CBF3: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_cbf_3.solvertime),...\n std(controller_mpc_cbf_3.solvertime),...\n min(controller_mpc_cbf_3.solvertime),...\n max(controller_mpc_cbf_3.solvertime),...\n controller_mpc_cbf_3.u_cost,...\n min(controller_mpc_cbf_3.distlog)]);\n\nfprintf('Computational time for MPC-CBF2: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_cbf_2.solvertime),...\n std(controller_mpc_cbf_2.solvertime),...\n min(controller_mpc_cbf_2.solvertime),...\n max(controller_mpc_cbf_2.solvertime),...\n controller_mpc_cbf_2.u_cost,...\n min(controller_mpc_cbf_2.distlog)]);\n\nfprintf('Computational time for MPC-CBF1: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_cbf_1.solvertime),...\n std(controller_mpc_cbf_1.solvertime),...\n min(controller_mpc_cbf_1.solvertime),...\n max(controller_mpc_cbf_1.solvertime),...\n controller_mpc_cbf_1.u_cost,...\n min(controller_mpc_cbf_1.distlog)]);\n\n% fprintf('Computational time for MPC-DC0: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n% [mean(controller_mpc_dc_0.solvertime),...\n% std(controller_mpc_dc_0.solvertime),...\n% min(controller_mpc_dc_0.solvertime),...\n% max(controller_mpc_dc_0.solvertime),...\n% controller_mpc_dc_0.u_cost,...\n% min(controller_mpc_dc_0.distlog)]);\n\nfprintf('Computational time for MPC-DC1: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_dc_1.solvertime),...\n std(controller_mpc_dc_1.solvertime),...\n min(controller_mpc_dc_1.solvertime),...\n max(controller_mpc_dc_1.solvertime),...\n controller_mpc_dc_1.u_cost,...\n min(controller_mpc_dc_1.distlog)]);\n\nfprintf('Computational time for MPC-DC2: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_dc_2.solvertime),...\n std(controller_mpc_dc_2.solvertime),...\n min(controller_mpc_dc_2.solvertime),...\n max(controller_mpc_dc_2.solvertime),...\n controller_mpc_dc_2.u_cost,...\n min(controller_mpc_dc_2.distlog)]);\n\nfprintf('Computational time for MPC-DC3: mean %.3f, std %.3f, min %.3f, max %.3f, input cost %.3f, min dist %f\\n',...\n [mean(controller_mpc_dc_3.solvertime),...\n std(controller_mpc_dc_3.solvertime),...\n min(controller_mpc_dc_3.solvertime),...\n max(controller_mpc_dc_3.solvertime),...\n controller_mpc_dc_3.u_cost,...\n min(controller_mpc_dc_3.distlog)]);" }, { "path": "matlab/acc2023/README.md", "content": "# Iterative-MPC-DHOCBF\nMatlab code for the paper *\"Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions\"*, accepted by IEEE American Control Conference (ACC) 2023, Authors: *Shuo Liu, Jun Zeng, Koushil Sreenath and Calin Belta* [[PDF]](https://arxiv.org/pdf/2210.04361.pdf)\n\nIn this paper, we propose a framework that solves the safety critical MPC problem in an iterative optimization, which is applicable for any relative-degree control barrier functions. In the proposed formulation, the nonlinear system dynamics as well as the safety constraints modeled as discrete-time high order control barrier functions (DHOCBF) are linearized at each time step. Our formulation is generally valid for any control barrier function with an arbitrary relative-degree. The advantages of fast computational performance with safety guarantee are analyzed and validated with numerical results.\n\n## Citing\nIf you find this project useful in your work, please consider citing following work:\n```\n@inproceedings{liu2023iterative,\n title={Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions},\n author={Liu, Shuo and Zeng, Jun and Sreenath, Koushil and Belta, Calin A},\n booktitle={2023 American Control Conference (ACC)},\n year={2023}\n}\n```\n\n## Instruction\nThere are two subfolders `closedloop_performance` and `benchmark`. `closedloop_performance` contains all information to generate figure 2, 3, 4 in paper and `benchmark` conatins code to generate data for table 1,2 in paper.\n\n## Subfolder `closedloop_performance`\n\n### Code Descriptions\n* `Closedloop_Trajectories_Hyperparameters.m` includes codes to generate necessary data for figure 2-a and figure 3; *\"Iterative_Convergence\"* includes codes to generate necessary data for figure 2-b,c,d.\n* `Maximum_Iterations.m` includes codes to generate necessary data for figure 4. \n* `NMPCDCBF1.m` and `NMPCDCBF2.m` include codes related to NMPC-DHOCBF with `mcbf=1` and `mcbf=2` respectively and `FigureGenerate.m` includes codes to transfer the data into figures in paper.\n\n### Usage\n* Run `Closedloop_Trajectories_Hyperparameters.m`, `Iterative_Convergence.m` and `Maximum_Iterations.m` and the data files are generated as MATLAB mat files. \n* Run `FigureGenerate.m` to generate all figures in paper, which will be saved in folder `closedloop_performance/figures`.\n\n## Subfolder `benchmark`\n\n### Code Descriptions\nThere are four folders including the codes to generate necessary dada for table 1,2 in paper. \n* `gamma1_0p4gamma2_0p4` includes the codes for iMPC-DHOCBF and NMPC-DHOCBF with decay rate parameters `gamma1=0.4, gamma2=0.4, mcbf=2`.\n* `gamma1_0p4` includes the codes for iMPC-DHOCBF and NMPC-DHOCBF with decay rate parameters `gamma1=0.4, mcbf=1`.\n* `gamma1_0p6gamma2_0p6` includes the codes for iMPC-DHOCBF and NMPC-DHOCBF with decay rate parameters `gamma1=0.6, gamma2=0.6, mcbf=2`.\n* `gamma1_0p6` includes the codes for iMPC-DHOCBF and NMPC-DHOCBF with decay rate parameters `gamma1=0.6, mcbf=1`.\n### Usage\n* Run `InitialState.m` first to generate 1000 random initial states in the mat file `InitialStateData.mat`, then copy it to each folder.\n* In each folder, run the file `test_comprehensive.m` and you will see corresponding data files generated as MATLAB mat files `feasibility_N.mat` and `timecom_N.mat` which includes the information about infeasible rate and mean/variance of computing time (stored in matrices `nmpcdata` and `impcdata`) in paper from generating one time-step trajectories for iMPC-DHOCBF and NMPC-DHOCBF.\n\n### Warnings\n* It may take a long time to run the file `test_comprehensive.m` (several hours to a day depending on your computer when the number of horizon reaches large). Instead, if you want to run the file for different number of horizon parallelly to speed up the running process, in folder `test_each_horizon.m`, there are six files called `test_N4.m`, `test_N8.m`, `test_N12.m`, `test_N16.m`, `test_N20.m`,`test_N24.m` which correspond to the number of horizon 4, 8, 12, 16, 20, 24 for iMPC-DHOCBF and NMPC-DHOCBF. By copying `InitialStateData.mat` to this folder then you can run these 6 files parallely, in order to generate same MATLAB mat files `feasibility_N.mat` and `timecom_N.mat` which includes the information about infeasible rate and mean/variance of computing time in paper from generating one time-step trajectories for iMPC-DHOCBF and NMPC-DHOCBF. You should run the file `tabledata.m` next and all useful data in table are stored in matrices `nmpcdata` and `impcdata`.\n\n## Post Analysis\n1. For comparison of capability of generating safe optimal trajectories for different numbers of horizon and hyperparameters, please see the figure below:\n![image](https://github.com/ShockLeo/Iterative-MPC-DHOCBF/blob/main/closedloop_performance/performance2.png)\n2. For comparison of infeasible rate and mean/variance of computing time from generating one time-step trajectories, please see the figure below:\n![image](https://github.com/ShockLeo/Iterative-MPC-DHOCBF/blob/main/benchmark/performance1.png) \nFor other figures please refer to the paper.\n\n## Dependencies\nThe packages needed for running the code are [Yalmip](https://yalmip.github.io/) and [IPOPT](https://github.com/coin-or/Ipopt) for NMPC-DHOCBF and [OSQP](https://github.com/osqp/osqp) for iMPC-DHOCBF.\n" }, { "path": "matlab/acc2023/benchmark/InitialState.m", "content": "samplen=1000;%Total number of initial states\r\nInitialMat=zeros(4,samplen);%Store initialized x, y, theta, v into each row by sequence\r\nfor i=1:samplen\r\nrr=0;\r\nwhile rr<=1\r\nxini1=(-10+rand(1)*(20));%-10~10\r\nyini1=(-10+rand(1)*(20));%-10~10\r\nrr = (xini1)^2+(yini1)^2;%Initialize location states which are inside safe region\r\nend\r\nthetaini1=(-10+rand(1)*(20));%-10~10\r\nvini1=(-10+rand(1)*(20));%-10~10\r\nInitialMat(1,i)=xini1;\r\nInitialMat(2,i)=yini1;\r\nInitialMat(3,i)=thetaini1;\r\nInitialMat(4,i)=vini1;\r\nend\r\nsave('InitialStateData','InitialMat');" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/NMPCDCBF1.m", "content": "classdef NMPCDCBF1 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF1(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);\r\n if tt == -1 %if infeasiblility happens, stop\r\n self.tt = tt;\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;%record computing time for each time-step\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n constraints = [constraints; b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=1\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_comprehensive.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=ii*4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nsave(filenm1,'nmpcplot1','impcplot1');\r\nsave(filenm2,'nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/NMPCDCBF1.m", "content": "classdef NMPCDCBF1 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF1(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);\r\n if tt == -1 %if infeasiblility happens, stop\r\n self.tt = tt;\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;%record computing time for each time-step\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n constraints = [constraints; b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=1\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/tabledata.m", "content": "nmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\ni=ii*4;\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nload(filenm1);\r\nload(filenm2);\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N12.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=12;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom12','nmpcplot1','impcplot1');\r\nsave('feasibility12','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-12\\n');\r\n controller_nmpc_dcbf_multiple12 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple12.obs = obs;\r\n controller_nmpc_dcbf_multiple12.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple12.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N16.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=16;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom16','nmpcplot1','impcplot1');\r\nsave('feasibility16','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-16\\n');\r\n controller_nmpc_dcbf_multiple16 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple16.obs = obs;\r\n controller_nmpc_dcbf_multiple16.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple16.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N20.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=20;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom20','nmpcplot1','impcplot1');\r\nsave('feasibility20','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-20\\n');\r\n controller_nmpc_dcbf_multiple20 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple20.obs = obs;\r\n controller_nmpc_dcbf_multiple20.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple20.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N24.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=24;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom24','nmpcplot1','impcplot1');\r\nsave('feasibility24','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-24\\n');\r\n controller_nmpc_dcbf_multiple24 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple24.obs = obs;\r\n controller_nmpc_dcbf_multiple24.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple24.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N4.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom4','nmpcplot1','impcplot1');\r\nsave('feasibility4','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6/test_each_horizon/test_N8.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=8;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom8','nmpcplot1','impcplot1');\r\nsave('feasibility8','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-8\\n');\r\n controller_nmpc_dcbf_multiple8 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple8.obs = obs;\r\n controller_nmpc_dcbf_multiple8.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple8.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/NMPCDCBF2.m", "content": "classdef NMPCDCBF2 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF2(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);%if infeasiblility happens, stop\r\n if tt == -1\r\n self.tt = tt;%record computing time for each time-step\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n b_next_next = (x([1:2],i+2)-pos)'*((x([1:2],i+2)-pos))-r^2; \r\n b1 = (b_next - b)/self.system.dt + self.params.gamma1/self.system.dt * (b);\r\n b1_next = (b_next_next - b_next)/self.system.dt + self.params.gamma1/self.system.dt * (b_next);\r\n constraints = [constraints; b1_next >=u(4,i)*(1-self.params.gamma2)*b1;b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=2\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_comprehensive.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=ii*4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nsave(filenm1,'nmpcplot1','impcplot1');\r\nsave(filenm2,'nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/NMPCDCBF2.m", "content": "classdef NMPCDCBF2 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF2(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);%if infeasiblility happens, stop\r\n if tt == -1\r\n self.tt = tt;%record computing time for each time-step\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n b_next_next = (x([1:2],i+2)-pos)'*((x([1:2],i+2)-pos))-r^2; \r\n b1 = (b_next - b)/self.system.dt + self.params.gamma1/self.system.dt * (b);\r\n b1_next = (b_next_next - b_next)/self.system.dt + self.params.gamma1/self.system.dt * (b_next);\r\n constraints = [constraints; b1_next >=u(4,i)*(1-self.params.gamma2)*b1;b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=2\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/tabledata.m", "content": "nmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\ni=ii*4;\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nload(filenm1);\r\nload(filenm2);\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N12.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=12;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom12','nmpcplot1','impcplot1');\r\nsave('feasibility12','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-12\\n');\r\n controller_nmpc_dcbf_multiple12 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple12.obs = obs;\r\n controller_nmpc_dcbf_multiple12.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple12.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N16.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=16;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom16','nmpcplot1','impcplot1');\r\nsave('feasibility16','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-16\\n');\r\n controller_nmpc_dcbf_multiple16 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple16.obs = obs;\r\n controller_nmpc_dcbf_multiple16.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple16.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N20.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=20;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom20','nmpcplot1','impcplot1');\r\nsave('feasibility20','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-20\\n');\r\n controller_nmpc_dcbf_multiple20 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple20.obs = obs;\r\n controller_nmpc_dcbf_multiple20.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple20.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N24.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=24;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom24','nmpcplot1','impcplot1');\r\nsave('feasibility24','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-24\\n');\r\n controller_nmpc_dcbf_multiple24 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple24.obs = obs;\r\n controller_nmpc_dcbf_multiple24.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple24.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N4.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom4','nmpcplot1','impcplot1');\r\nsave('feasibility4','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1-0p6gamma2-0p6/test_each_horizon/test_N8.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=8;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p6gamma2_0p6\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom8','nmpcplot1','impcplot1');\r\nsave('feasibility8','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-8\\n');\r\n controller_nmpc_dcbf_multiple8 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple8.obs = obs;\r\n controller_nmpc_dcbf_multiple8.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple8.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 6;\r\ngamma2 = 6;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/NMPCDCBF1.m", "content": "classdef NMPCDCBF1 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF1(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);\r\n if tt == -1 %if infeasiblility happens, stop\r\n self.tt = tt;\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;%record computing time for each time-step\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n constraints = [constraints; b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=1\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_comprehensive.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=ii*4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nsave(filenm1,'nmpcplot1','impcplot1');\r\nsave(filenm2,'nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/NMPCDCBF1.m", "content": "classdef NMPCDCBF1 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF1(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);\r\n if tt == -1 %if infeasiblility happens, stop\r\n self.tt = tt;\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;%record computing time for each time-step\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n constraints = [constraints; b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=1\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/tabledata.m", "content": "nmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\ni=ii*4;\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nload(filenm1);\r\nload(filenm2);\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1; \r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N12.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=12;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom12','nmpcplot1','impcplot1');\r\nsave('feasibility12','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-12\\n');\r\n controller_nmpc_dcbf_multiple12 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple12.obs = obs;\r\n controller_nmpc_dcbf_multiple12.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple12.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N16.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=16;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom16','nmpcplot1','impcplot1');\r\nsave('feasibility16','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-16\\n');\r\n controller_nmpc_dcbf_multiple16 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple16.obs = obs;\r\n controller_nmpc_dcbf_multiple16.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple16.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N20.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=20;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom20','nmpcplot1','impcplot1');\r\nsave('feasibility20','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-20\\n');\r\n controller_nmpc_dcbf_multiple20 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple20.obs = obs;\r\n controller_nmpc_dcbf_multiple20.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple20.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N24.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=24;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom24','nmpcplot1','impcplot1');\r\nsave('feasibility24','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-24\\n');\r\n controller_nmpc_dcbf_multiple24 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple24.obs = obs;\r\n controller_nmpc_dcbf_multiple24.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple24.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N4.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom4','nmpcplot1','impcplot1');\r\nsave('feasibility4','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4/test_each_horizon/test_N8.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=8;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom8','nmpcplot1','impcplot1');\r\nsave('feasibility8','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-8\\n');\r\n controller_nmpc_dcbf_multiple8 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple8.obs = obs;\r\n controller_nmpc_dcbf_multiple8.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple8.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/NMPCDCBF2.m", "content": "classdef NMPCDCBF2 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF2(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);%if infeasiblility happens, stop\r\n if tt == -1\r\n self.tt = tt;%record computing time for each time-step\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n b_next_next = (x([1:2],i+2)-pos)'*((x([1:2],i+2)-pos))-r^2; \r\n b1 = (b_next - b)/self.system.dt + self.params.gamma1/self.system.dt * (b);\r\n b1_next = (b_next_next - b_next)/self.system.dt + self.params.gamma1/self.system.dt * (b_next);\r\n constraints = [constraints; b1_next >=u(4,i)*(1-self.params.gamma2)*b1;b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=2\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_comprehensive.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=ii*4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nsave(filenm1,'nmpcplot1','impcplot1');\r\nsave(filenm2,'nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/NMPCDCBF2.m", "content": "classdef NMPCDCBF2 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF2(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);%if infeasiblility happens, stop\r\n if tt == -1\r\n self.tt = tt;%record computing time for each time-step\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n b_next_next = (x([1:2],i+2)-pos)'*((x([1:2],i+2)-pos))-r^2; \r\n b1 = (b_next - b)/self.system.dt + self.params.gamma1/self.system.dt * (b);\r\n b1_next = (b_next_next - b_next)/self.system.dt + self.params.gamma1/self.system.dt * (b_next);\r\n constraints = [constraints; b1_next >=u(4,i)*(1-self.params.gamma2)*b1;b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=2\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/tabledata.m", "content": "nmpcdata=zeros(3,6);%Store data in matrix for nmpc and impc, first row: mean of computing time; second row: standard deviation of computing time; third row: infeasible rate\r\nimpcdata=zeros(3,6);\r\nfor ii = 1:6\r\ni=ii*4;\r\nfilenm1 = ['timecom' num2str(i) '.mat'];\r\nfilenm2 = ['feasibility' num2str(i) '.mat'];\r\nload(filenm1);\r\nload(filenm2);\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\nnmpcdata(1,ii)=mu1;\r\nimpcdata(1,ii)=mu2;\r\nnmpcdata(2,ii)=sigma1;\r\nimpcdata(2,ii)=sigma2;\r\nnmpcdata(3,ii)=nmpcinf;\r\nimpcdata(3,ii)=impcinf;\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N12.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=12;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom12','nmpcplot1','impcplot1');\r\nsave('feasibility12','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-12\\n');\r\n controller_nmpc_dcbf_multiple12 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple12.obs = obs;\r\n controller_nmpc_dcbf_multiple12.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple12.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N16.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=16;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom16','nmpcplot1','impcplot1');\r\nsave('feasibility16','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-16\\n');\r\n controller_nmpc_dcbf_multiple16 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple16.obs = obs;\r\n controller_nmpc_dcbf_multiple16.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple16.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N20.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=20;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom20','nmpcplot1','impcplot1');\r\nsave('feasibility20','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-20\\n');\r\n controller_nmpc_dcbf_multiple20 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple20.obs = obs;\r\n controller_nmpc_dcbf_multiple20.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple20.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N24.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=24;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom24','nmpcplot1','impcplot1');\r\nsave('feasibility24','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-24\\n');\r\n controller_nmpc_dcbf_multiple24 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple24.obs = obs;\r\n controller_nmpc_dcbf_multiple24.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple24.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N4.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=4;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom4','nmpcplot1','impcplot1');\r\nsave('feasibility4','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-4\\n');\r\n controller_nmpc_dcbf_multiple4 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple4.obs = obs;\r\n controller_nmpc_dcbf_multiple4.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple4.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/benchmark/gamma1_0p4gamma2_0p4/test_each_horizon/test_N8.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------------------------------------------------------\r\nnb = 1000;%Total number of initial states\r\nnmpcd=zeros(1,nb);%Store computing time for one time-step for NMPC-DCBF\r\nimpcd=zeros(1,nb);%Store computing time for one time-step for iMPC-DCBF\r\nnmpcinf=zeros(1,1);%Store infeasible rate for one time-step for NMPC-DCBF\r\nimpcinf=zeros(1,1);%Store infeasible rate for one time-step for iMPC-DCBF\r\ni=8;%i is number of horozon\r\nts = 1;%Total time steps\r\nload(gamma1_0p4gamma2_0p4\\InitialStateData');%Load the generated 1000 initial states\r\n[a, b, c, d]=compare(i,ts,nb,InitialMat);%Involves NMPC-DCBF and iMPC-DCBF\r\nnmpcd(1,:)=a;\r\nimpcd(1,:)=b;\r\nnmpcinf(1,1)=c;\r\nimpcinf(1,1)=d;\r\nnmpcplot1=nmpcd(1,:);\r\nnmpcplot1(nmpcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\nimpcplot1=impcd(1,:);\r\nimpcplot1(impcplot1==-1)=[];%Eliminate infeasible one time-step trajectories\r\n\r\nsave('timecom8','nmpcplot1','impcplot1');\r\nsave('feasibility8','nmpcinf','impcinf');\r\n\r\ndistnmpc = fitdist(nmpcplot1','Normal');\r\ndistimpc = fitdist(impcplot1','Normal');\r\nmu1=distnmpc.mu;%Get mean of sample of computing time for NMPC-DCBF\r\nmu2=distimpc.mu;%Get mean of sample of computing time for iMPC-DCBF\r\nsigma1=distnmpc.sigma;%Get variance of sample of computing time for NMPC-DCBF\r\nsigma2=distimpc.sigma;%Get variance of sample of computing time for iMPC-DCBF\r\n\r\n\r\n%Initialize atmosphere parameters\r\nfunction [tnmpc, timpc, ratiotnmpc, ratiotimpc]=compare(N11,ttsim,samplen,InitialMat)\r\ntnmpc=[];\r\ntimpc=[];\r\nfor i=1:samplen\r\nN1=N11;\r\ntsim=ttsim;\r\nxini1=InitialMat(1,i);\r\nyini1=InitialMat(2,i);\r\nthetaini1=InitialMat(3,i);\r\nvini1=InitialMat(4,i);\r\nx01=[xini1;yini1;thetaini1;vini1];\r\nx02=[xini1 yini1 thetaini1 vini1];\r\nt1 = nmpcdcbf(x01, N1, tsim);\r\nt2 = impcdcbf(x02, N1, tsim);\r\ntindex=[N11 i];\r\ndisp(tindex);\r\ntnmpc=[tnmpc t1];%Computing time for NMPC-DCBF\r\ntimpc=[timpc t2];%Computing time for iMPC-DCBF\r\nend\r\nnnmpc1 = length(tnmpc);\r\nnimpc1 = length(timpc);\r\ntnmpcs = tnmpc;\r\ntimpcs = timpc;\r\ntnmpcs(tnmpcs==-1)=[];\r\ntimpcs(timpcs==-1)=[];\r\nnnmpc2 = length(tnmpcs);\r\nnimpc2 = length(timpcs);\r\nratiotnmpc = (nnmpc1-nnmpc2)/nnmpc1;%Infeasible rate for NMPC-DCBF\r\nratiotimpc = (nimpc1-nimpc2)/nimpc1;%Infeasible rate for iMPC-DCBF\r\nend\r\n\r\n\r\n\r\n\r\nfunction t1 = nmpcdcbf(x00, N1, ttsim)\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = x00;\r\ndt = 0.1;\r\ntime_total = ttsim *dt;\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time unicycle model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% NMPC-DCBF parameters\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n\r\n\r\n%% Simulate NMPC-DCBF with other N\r\nparams_nmpc_dcbf.N = N1;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-8\\n');\r\n controller_nmpc_dcbf_multiple8 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple8.obs = obs;\r\n controller_nmpc_dcbf_multiple8.sim(time_total);\r\nend\r\nt1=controller_nmpc_dcbf_multiple8.tt;\r\nend\r\n\r\n\r\nfunction t2=impcdcbf(x00, N1, ttsim)\r\nt2=0;\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = ttsim;\r\n\r\ntic;\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = x00;\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nt2=t2+toc;\r\nreturn\r\n\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n% error('OSQP did not solve the problem!')\r\n t2=-1;\r\n return\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n testx0 = [testx0 x0];\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/Closedloop_Trajectories_Hyperparameters.m", "content": "close all\r\nclear\r\n\r\n%------------------------------------------NMPC-DCBF---------------------------------------------\r\n\r\n%% General Flags\r\nrun_nmpc_dcbf_one = true;\r\nrun_nmpc_dcbf_multiple = true;\r\n\r\n%% Setup and Parameters\r\nx0 = [-3; 0; 0; 0];%Initial Condition\r\ntime_total = 4.5;%Time for the total steps, equal to tsim\r\ndt = 0.1;%One time-step\r\nP = zeros(4,4);%Weight matrix P\r\nP(1,1) = 10; P(2,2) = 10; P(3,3) = 10;P(4,4) = 10;\r\nQ = zeros(4,4);%Weight matrix Q\r\nQ(1,1) = 10; Q(2,2) = 10; Q(3,3) = 10;Q(4,4) = 10;\r\nR = zeros(4,4);%Weight matrix R\r\nR(1,1) = 1; R(2,2) = 1;R(3,3) = 1000;R(4,4) = 1000;\r\n%Variables range as below\r\nxmin = [-10; -10; -10;-10];\r\nxmax = [10; 10; 10;10];\r\numin = [-7; -5;-inf;-inf];\r\numax = [7; 5;inf;inf];\r\n\r\n%% Discrete-time Unicycle Model\r\nsystem.dt = dt;\r\nsystem.A = [1 0 0 0;\r\n 0 1 0 0;\r\n 0 0 1 0;\r\n 0 0 0 1];\r\nsystem.B = [x0(4,1)*cos(x0(3,1))*dt;\r\n x0(4,1)*sin(x0(3,1))*dt;\r\n 0;\r\n 0];\r\nsystem.C =[0 0 0 0;\r\n 0 0 0 0;\r\n 1*dt 0 0 0;\r\n 0 1*dt 0 0];\r\nsystem.xl = xmin;\r\nsystem.xu = xmax;\r\nsystem.ul = umin;\r\nsystem.uu = umax;\r\n\r\n%% Obstacle\r\nobs.pos1 = [0; 0];\r\nobs.r1 = 1;\r\n\r\n%% NMPC-DCBF parameters set 1\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n%% Simulate NMPC-DCBF with Number of Horizon N\r\nparams_nmpc_dcbf.N = 24;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-11\\n');\r\n controller_nmpc_dcbf_multiple11 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=1\r\n controller_nmpc_dcbf_multiple11.obs = obs;\r\n controller_nmpc_dcbf_multiple11.sim(time_total);\r\nend\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-12\\n');\r\n controller_nmpc_dcbf_multiple12 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple12.obs = obs;\r\n controller_nmpc_dcbf_multiple12.sim(time_total);\r\nend\r\n\r\n%% NMPC-DCBF parameters set 2\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;%\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n%% Simulate NMPC-DCBF with Number of Horizon N\r\nparams_nmpc_dcbf.N = 24;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-21\\n');\r\n controller_nmpc_dcbf_multiple21 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=1\r\n controller_nmpc_dcbf_multiple21.obs = obs;\r\n controller_nmpc_dcbf_multiple21.sim(time_total);\r\nend\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-22\\n');\r\n controller_nmpc_dcbf_multiple22 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple22.obs = obs;\r\n controller_nmpc_dcbf_multiple22.sim(time_total);\r\nend\r\n\r\n%% NMPC-DCBF parameters set 3\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.4;%\r\nparams_nmpc_dcbf.gamma2 = 0.4;\r\n%% Simulate NMPC-DCBF with Number of Horizon N\r\nparams_nmpc_dcbf.N = 16;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-31\\n');\r\n controller_nmpc_dcbf_multiple31 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=1\r\n controller_nmpc_dcbf_multiple31.obs = obs;\r\n controller_nmpc_dcbf_multiple31.sim(time_total);\r\nend\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-32\\n');\r\n controller_nmpc_dcbf_multiple32 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple32.obs = obs;\r\n controller_nmpc_dcbf_multiple32.sim(time_total);\r\nend\r\n\r\n%% NMPC-DCBF parameters set 4\r\nparams_nmpc_dcbf.Q = Q;\r\nparams_nmpc_dcbf.R = R;\r\nparams_nmpc_dcbf.P = P;\r\nparams_nmpc_dcbf.gamma1 = 0.6;%\r\nparams_nmpc_dcbf.gamma2 = 0.6;\r\n%% Simulate NMPC-DCBF with Number of Horizon N\r\nparams_nmpc_dcbf.N = 16;\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-41\\n');\r\n controller_nmpc_dcbf_multiple41 = NMPCDCBF1(x0, system, params_nmpc_dcbf);%Highest order mcbf=1\r\n controller_nmpc_dcbf_multiple41.obs = obs;\r\n controller_nmpc_dcbf_multiple41.sim(time_total);\r\nend\r\nif run_nmpc_dcbf_one\r\n fprintf('Run NMPC-DCBF-42\\n');\r\n controller_nmpc_dcbf_multiple42 = NMPCDCBF2(x0, system, params_nmpc_dcbf);%Highest order mcbf=2\r\n controller_nmpc_dcbf_multiple42.obs = obs;\r\n controller_nmpc_dcbf_multiple42.sim(time_total);\r\nend\r\n%% Initialize atmosphere parameters\r\nimpc1 = impcdcbf2(4,4,24);%Highest order mcbf=2\r\nimpc2 = impcdcbf2(6,6,24);\r\nimpc3 = impcdcbf2(4,4,16);\r\nimpc4 = impcdcbf2(6,6,16);\r\nimpc5 = impcdcbf1(4,24);%Highest order mcbf=1\r\nimpc6 = impcdcbf1(6,24);\r\n\r\n\r\nsave('impc_N24_Gamma4_4','impc1');%impc-solid lines\r\nsave('impc_N24_Gamma6_6','impc2');\r\nsave('impc_N16_Gamma4_4','impc3');\r\nsave('impc_N16_Gamma6_6','impc4');\r\nsave('impc_N24_Gamma4','impc5');\r\nsave('impc_N24_Gamma6','impc6');\r\n\r\ncontroller_nmpc_dcbf_multiple1=controller_nmpc_dcbf_multiple12.xlog;%Highest order mcbf=2\r\ncontroller_nmpc_dcbf_multiple2=controller_nmpc_dcbf_multiple22.xlog;\r\ncontroller_nmpc_dcbf_multiple3=controller_nmpc_dcbf_multiple32.xlog;\r\ncontroller_nmpc_dcbf_multiple4=controller_nmpc_dcbf_multiple42.xlog;\r\ncontroller_nmpc_dcbf_multiple5=controller_nmpc_dcbf_multiple11.xlog;%Highest order mcbf=1\r\ncontroller_nmpc_dcbf_multiple6=controller_nmpc_dcbf_multiple21.xlog;\r\n\r\nsave('nmpc_N24_Gamma4_4','controller_mpc_cbf_multiple1');%nmpc-dashed lines\r\nsave('nmpc_N24_Gamma6_6','controller_mpc_cbf_multiple2');\r\nsave('nmpc_N16_Gamma4_4','controller_mpc_cbf_multiple3');\r\nsave('nmpc_N16_Gamma6_6','controller_mpc_cbf_multiple4');\r\nsave('nmpc_N24_Gamma4','controller_mpc_cbf_multiple5');\r\nsave('nmpc_N24_Gamma6','controller_mpc_cbf_multiple6');\r\n\r\n\r\n%--------------------------------------iMPC-DCBF------------------------------------------------\r\nfunction impc=impcdcbf2(gamma1, gamma2, N1)\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = gamma1;\r\ngamma2 = gamma2;\r\nnsim = 45;%Total time steps, corresponding to time_total in NMPC-DCBF\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = [-3 0 0 0];\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);%First order CBF constraints\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);%Second order CBF constraints\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\n\r\nA = [Aeq; Aineq; Acbf; A2cbf];\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step\r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\nfunction impc=impcdcbf1(gamma1, N1)\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = N1;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = gamma1;\r\nnsim = 45;%Total time steps, corresponding to time_total in NMPC-DCBF\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = [-3 0 0 0];\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);%First order CBF constraints\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf];\r\nl = [leq; lineq; lcbf];\r\nu = [ueq; uineq; ucbf];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)%Iterations for the first time-step \r\n storagex = ctrlx;%store the state variables and inputs in order to compare them with the next iteration\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n \r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf];\r\n l = [leq; lineq; lcbf];\r\n u = [ueq; uineq; ucbf];\r\n\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break \r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)%Iterations for the left time steps\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf];\r\n l = [leq; lineq; lcbf];\r\n u = [ueq; uineq; ucbf];\r\n\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break \r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/FigureGenerate.m", "content": "close all\r\nclear\r\n\r\n%--------------------------------------------------------------------------------------\r\nnsim=100;%Total time-steps for the trajectory\r\nxo = 0;%Obstacle parameters\r\nyo = 0;\r\nr = 1;\r\nN=24;%Number of iteration\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\trajectory');%For the location address, please change it later based on where you download the files\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\iteration_x');\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\iteration_y');\r\nfigure(1)%Figure 2-a of paper\r\nset(0,'defaultfigurecolor','w'); \r\nset(gca,'LooseInset',get(gca,'TightInset'));\r\nhold on;\r\nth = linspace(0,2*pi*100);\r\nx = cos(th) ; y = sin(th) ;\r\nplot(xo + r*x, yo + r*y, 'k-', 'LineWidth', 2);\r\nfill(xo + r*x, yo + r*y, [1 0.5 0]);\r\naxis square\r\nplot(impctra(1,1), impctra(2,1), 'db', 'LineWidth', 1);\r\nplot(3, 0, 'dr', 'LineWidth', 1);\r\nh = plot(impctra(1,:), impctra(2,:), 'ko-',...\r\n 'LineWidth', 1.0,'MarkerSize',4);\r\nh1 = plot(Itera_x(1,:), Itera_y(1,:),'c', 'LineWidth', 1);%\r\nh2 =plot(Itera_x(3,:), Itera_y(3,:),'g', 'LineWidth', 1);\r\nh3 =plot(Itera_x(5,:), Itera_y(5,:),'m', 'LineWidth', 1);\r\nh4 =plot(Itera_x(10,:), Itera_y(10,:),'r', 'LineWidth', 1);\r\nh5 =plot(Itera_x(32,:), Itera_y(32,:),'b', 'LineWidth', 1);\r\n\r\nh_legend = legend([h,h1,h2,h3,h4,h5], {'Closed-loop ($t_{\\mathrm{sim}}=100$)','Open-loop ($j=1$)','Open-loop ($j=3$)','Open-loop ($j=5$)',...\r\n 'Open-loop ($j=10$)','Open-loop ($j=32$)'}, 'Location', 'SouthEast');\r\nset(h_legend, 'Interpreter','latex');\r\nset(gca,'LineWidth', 0.2, 'FontSize', 19);\r\ngrid on\r\naxis equal\r\nxlim([-3,3]);\r\nylim([-2,2]);\r\nxlabel('$x(m)$','interpreter','latex','FontSize',20);\r\nylabel('$y(m)$','interpreter','latex','FontSize',20);\r\ntext('Interpreter','latex','String','$t=6$','Position',[-2.45 0.9],'FontSize',18);\r\nannotation('arrow',[0.21 0.21],[0.70 0.60],'LineStyle', '-','color',[0 0 0]);\r\nprint(gcf,'figures/openloop-snapshots', '-depsc');\r\nprint(gcf,'figures/openloop-snapshots.png', '-dpng', '-r800');\r\n\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N24_Gamma4_4');%impc-24-4-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N24_Gamma6_6');%impc-24-6-6\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N16_Gamma4_4');%impc-16-4-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N16_Gamma6_6');%impc-16-6-6\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N24_Gamma4_4');%nmpc-24-6-6\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N24_Gamma6_6');%nmpc-24-4-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N16_Gamma4_4');%nmpc-16-6-6\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N16_Gamma6_6');%nmpc-16-4-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N24_Gamma4');%impc-24-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\impc_N24_Gamma6');%impc-24-6\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N24_Gamma4');%nmpc-24-4\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\nmpc_N24_Gamma6');%nmpc-24-6\r\n\r\nfigure(2)%Figure 2-b of paper\r\n%Draw the figure\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nxr = [3;0.01;0;0];\r\nset(0,'defaultfigurecolor','w'); \r\nset(gca,'LooseInset',get(gca,'TightInset'));\r\nhold on;\r\nth = linspace(0,2*pi*100);\r\nx = cos(th) ; y = sin(th) ;\r\nplot(xo + r*x, yo + r*y, 'k-', 'LineWidth', 2);\r\nfill(xo + r*x, yo + r*y, [1 0.5 0]);\r\naxis square\r\nplot(impc1(1,1), impc2(2,1), 'db', 'LineWidth', 1);\r\nplot(xr(1), xr(2), 'dr', 'LineWidth', 1);\r\nh1 = plot(impc1(1,:), impc1(2,:), 'k',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh2 = plot(impc2(1,:), impc2(2,:), 'r',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh3 = plot(impc3(1,:), impc3(2,:), 'b',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh4 = plot(impc4(1,:), impc4(2,:), 'm',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\n\r\nh_legend1 = legend([h1,h2,h3,h4], {'$N=24,\\gamma_{1}=0.4,\\gamma_{2}=0.4$','$N=24,\\gamma_{1}=0.6,\\gamma_{2}=0.6$','$N=16,\\gamma_{1}=0.4,\\gamma_{2}=0.4$','$N=16,\\gamma_{1}=0.6,\\gamma_{2}=0.6$'}, 'Location', 'SouthEast');\r\nset(h_legend1, 'Interpreter','latex');\r\nset(gca,'LineWidth', 0.2, 'FontSize', 18);\r\ngrid on\r\naxis equal\r\nxlim([-3,3]);\r\nylim([-2,2]);\r\nxlabel('$x(m)$','interpreter','latex','FontSize',20);\r\nylabel('$y(m)$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/closedloop-snapshots1', '-depsc');\r\nprint(gcf,'figures/closedloop-snapshots1.png', '-dpng', '-r800');\r\n\r\nfigure(3)%Figure 2-c of paper\r\n% Draw the figure\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nxr = [3;0.01;0;0];\r\nset(0,'defaultfigurecolor','w'); \r\nset(gca,'LooseInset',get(gca,'TightInset'));\r\nhold on;\r\nth = linspace(0,2*pi*100);\r\nx = cos(th) ; y = sin(th) ;\r\nplot(xo + r*x, yo + r*y, 'k-', 'LineWidth', 2);\r\nfill(xo + r*x, yo + r*y, [1 0.5 0]);\r\naxis square\r\nplot(impc1(1,1), impc2(2,1), 'db', 'LineWidth', 1);\r\nplot(xr(1), xr(2), 'dr', 'LineWidth', 1);\r\nh5 = plot(controller_nmpc_dcbf_multiple1(1,:), controller_nmpc_dcbf_multiple1(2,:), 'k--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh6 = plot(controller_nmpc_dcbf_multiple2(1,:), controller_nmpc_dcbf_multiple2(2,:), 'r--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh7 = plot(controller_nmpc_dcbf_multiple3(1,:), controller_nmpc_dcbf_multiple3(2,:), 'b--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh8 = plot(controller_nmpc_dcbf_multiple4(1,:), controller_nmpc_dcbf_multiple4(2,:), 'm--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh_legend2 = legend([h5,h6,h7,h8], {'$N=24,\\gamma_{1}=0.4,\\gamma_{2}=0.4$','$N=24,\\gamma_{1}=0.6,\\gamma_{2}=0.6$','$N=16,\\gamma_{1}=0.4,\\gamma_{2}=0.4$','$N=16,\\gamma_{1}=0.6,\\gamma_{2}=0.6$'}, 'Location', 'SouthEast');\r\nset(h_legend2, 'Interpreter','latex');\r\nset(gca,'LineWidth', 0.2, 'FontSize', 16);\r\ngrid on\r\naxis equal\r\nxlim([-3,3]);\r\nylim([-2,2]);\r\nxlabel('$x(m)$','interpreter','latex','FontSize',20);\r\nylabel('$y(m)$','interpreter','latex','FontSize',20);\r\ntext('Interpreter','latex','String','$t=13$','Position',[-2 -1.1],'FontSize',18);\r\nannotation('arrow',[0.37 0.47],[0.33 0.38],'LineStyle', '-','color',[0 0 0]);\r\ntext('Interpreter','latex','String','$t=33$','Position',[1 0.1],'FontSize',18);\r\nannotation('arrow',[0.78 0.88],[0.53 0.48],'LineStyle', '-','color',[0 0 0]);\r\nprint(gcf,'figures/closedloop-snapshots2', '-depsc');\r\nprint(gcf,'figures/closedloop-snapshots2.png', '-dpng', '-r800');\r\n\r\nfigure(4)%Figure 2-d of paper\r\n% Draw the figure\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nxr = [3;0.01;0;0];\r\nset(0,'defaultfigurecolor','w'); \r\nset(gca,'FontSize',20)\r\nset(gca,'LooseInset',get(gca,'TightInset'));\r\nhold on;\r\nth = linspace(0,2*pi*100);\r\nx = cos(th) ; y = sin(th) ;\r\nplot(xo + r*x, yo + r*y, 'k-', 'LineWidth', 2);\r\nfill(xo + r*x, yo + r*y, [1 0.5 0]);\r\naxis square\r\nplot(impc1(1,1), impc2(2,1), 'db', 'LineWidth', 1);\r\nplot(xr(1), xr(2), 'dr', 'LineWidth', 1);\r\nh9 = plot(impc5(1,1:46), impc5(2,1:46), 'k',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh10 = plot(impc6(1,1:46), impc6(2,1:46), 'r',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh11 = plot(controller_nmpc_dcbf_multiple5(1,:), controller_nmpc_dcbf_multiple5(2,:), 'b--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh12 = plot(controller_nmpc_dcbf_multiple6(1,:), controller_nmpc_dcbf_multiple6(2,:), 'm--',...\r\n 'LineWidth', 2.0,'MarkerSize',4);\r\nh_legend3 = legend([h9,h10,h11,h12], {'$N=24,\\gamma_{1}=0.4$','$N=24,\\gamma_{1}=0.6$','$N=24,\\gamma_{1}=0.4$','$N=24,\\gamma_{1}=0.6$'}, 'Location', 'SouthEast');\r\nset(h_legend3, 'Interpreter','latex');\r\nset(gca,'LineWidth', 0.2, 'FontSize', 18);\r\ngrid on\r\naxis equal\r\nxlim([-3,3]);\r\nylim([-2,2]);\r\nxlabel('$x(m)$','interpreter','latex','FontSize',20);\r\nylabel('$y(m)$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/closedloop-snapshots3', '-depsc');\r\nprint(gcf,'figures/closedloop-snapshots3.png', '-dpng', '-r800');\r\n\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\jtconv_Gamma4_4');\r\nfigure(5)%Figure 4-a of paper\r\nkk = 0:1:(nsim-1);\r\nhk = plot(kk, kk44(1,:), 'ko-',...\r\n 'LineWidth', 1.0,'MarkerSize',4);\r\nxlim([0,99]);\r\nylim([0,1000]);\r\nset(gca,'FontSize',18)\r\nxlabel('$t$','interpreter','latex','FontSize',20);\r\nylabel('$j_{t,\\mathrm{conv}}$','interpreter','latex','FontSize',20);\r\nset(gca, 'YScale', 'log')\r\nprint(gcf,'figures/J-converge-1', '-depsc');\r\nprint(gcf,'figures/J-converge-1.png', '-dpng', '-r400');\r\n\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\jtconv_Gamma4_6'); \r\n\r\nfigure(6)%Figure 4-b of paper\r\nkk = 0:1:(nsim-1);\r\nhk = plot(kk, kk46(1,:), 'ko-',...\r\n 'LineWidth', 1.0,'MarkerSize',4);\r\nxlim([0,99]);\r\nylim([0,1000]);\r\nxlabel('$t$','interpreter','latex','FontSize',20);\r\nylabel('$j_{t,\\mathrm{conv}}$','interpreter','latex','FontSize',20);\r\nset(gca,'FontSize',18)\r\nset(gca, 'YScale', 'log')\r\nprint(gcf,'figures/J-converge-2', '-depsc');\r\nprint(gcf,'figures/J-converge-2.png', '-dpng', '-r400');\r\n\r\n\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\jtconv_Gamma6_4');\r\n\r\nfigure(7)%Figure 4-c of paper\r\nkk = 0:1:(nsim-1);\r\nhk = plot(kk, kk64(1,:), 'ko-',...\r\n 'LineWidth', 1.0,'MarkerSize',4);\r\nxlim([0,99]);\r\nxlabel('$t$','interpreter','latex','FontSize',20);\r\nylabel('$j_{t,\\mathrm{conv}}$','interpreter','latex','FontSize',20);\r\nset(gca,'FontSize',18)\r\nset(gca, 'YScale', 'log')\r\nprint(gcf,'figures/J-converge-3', '-depsc');\r\nprint(gcf,'figures/J-converge-3.png', '-dpng', '-r400');\r\n\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\jtconv_Gamma6_6');\r\n\r\nfigure(8)%Figure 4-d of paper\r\nkk = 0:1:(nsim-1);\r\nhk = plot(kk, kk66(1,:), 'ko-',...\r\n 'LineWidth', 1.0,'MarkerSize',4);\r\nxlim([0,99]);\r\nxlabel('$t$','interpreter','latex','FontSize',20);\r\nylabel('$j_{t,\\mathrm{conv}}$','interpreter','latex','FontSize',20);\r\nset(gca,'FontSize',18)\r\nset(gca, 'YScale', 'log')\r\nprint(gcf,'figures/J-converge-4', '-depsc');\r\nprint(gcf,'figures/J-converge-4.png', '-dpng', '-r400');\r\n\r\n \r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\iteration_theta');\r\nload('D:\\onedri\\OneDrive\\Desktop\\shuo\\Github\\Figure2_3_4\\iteration_v');\r\n\r\nfigure(9)%Figure 3-a of paper\r\nkk1 = 0:1:N;\r\nhh = plot(kk1, Itera_x(32,:),'r', 'LineWidth', 1);\r\nhh1 = plot(kk1, Itera_x(1,:),'r', 'LineWidth', 1);\r\nhold on \r\nfor i=2:1:32\r\nif i<=20\r\nhh1= plot(kk1, Itera_x(i,:),'r', 'LineWidth', 1);\r\nelse\r\nhh=plot(kk1, Itera_x(i,:),'b', 'LineWidth', 2);\r\nhold on \r\nxlim([0,24]);\r\nend\r\nend\r\nhh_legend = legend([hh,hh1], {'Converged ($j=32$)','Converging ($1\\le j<32$)'}, 'Location', 'SouthEast');\r\nset(hh_legend, 'Interpreter','latex','FontSize',18);\r\ntext('Interpreter','latex','String','$j=1$','Position',[19 0.55],'FontSize',18);\r\nannotation('arrow',[0.75 0.7],[0.62 0.71],'LineStyle', '-','color',[0 0 0]);\r\ntext('Interpreter','latex','String','$j=2$','Position',[5.3 0.6],'FontSize',18);\r\nannotation('arrow',[0.3 0.4],[0.6 0.55],'LineStyle', '-','color',[0 0 0]);\r\nset(gca,'FontSize',18)\r\nxlabel('$k$','interpreter','latex','FontSize',20);\r\nylabel('$x(m)$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/state-converge-1', '-depsc');\r\nprint(gcf,'figures/state-converge-1.png', '-dpng', '-r400');\r\n \r\nfigure(10)%Figure 3-b of paper\r\nkk1 = 0:1:N;\r\nhh = plot(kk1, Itera_y(32,:),'r', 'LineWidth', 1);\r\nhh1 = plot(kk1, Itera_y(1,:),'r', 'LineWidth', 1);\r\nhold on \r\nfor i=2:1:32\r\nif i<=20\r\nhh1= plot(kk1, Itera_y(i,:),'r', 'LineWidth', 1);\r\nelse\r\nhh=plot(kk1, Itera_y(i,:),'b', 'LineWidth', 2);\r\nhold on \r\nxlim([0,24]);\r\nend\r\nend\r\nhh_legend = legend([hh,hh1], {'Converged ($j=32$)','Converging ($1\\le j<32$)'}, 'Location', 'SouthEast');\r\nset(hh_legend, 'Interpreter','latex','FontSize',18);\r\ntext('Interpreter','latex','String','$j=1$','Position',[16.5 1],'FontSize',18);\r\nannotation('arrow',[0.7 0.75],[0.75 0.66],'LineStyle', '-','color',[0 0 0]);\r\ntext('Interpreter','latex','String','$j=2$','Position',[1.9 0.90],'FontSize',18);\r\nannotation('arrow',[0.2 0.27],[0.68 0.6],'LineStyle', '-','color',[0 0 0]);\r\nset(gca,'FontSize',18)\r\nxlabel('$k$','interpreter','latex','FontSize',20);\r\nylabel('$y(m)$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/state-converge-2', '-depsc');\r\nprint(gcf,'figures/state-converge-2.png', '-dpng', '-r400');\r\n\r\n\r\nfigure(11)%Figure 3-c of paper\r\nkk1 = 0:1:N;\r\nhh = plot(kk1, Itera_theta(32,:),'r', 'LineWidth', 1);\r\nhh1 = plot(kk1, Itera_theta(1,:),'r', 'LineWidth', 1);\r\nhold on \r\nfor i=2:1:32\r\nif i<=20\r\nhh1= plot(kk1, Itera_theta(i,:),'r', 'LineWidth', 1);\r\nelse\r\nhh=plot(kk1, Itera_theta(i,:),'b', 'LineWidth', 2);\r\nhold on \r\nxlim([0,24]);\r\nend\r\nend\r\nhh_legend = legend([hh,hh1], {'Converged ($j=32$)','Converging ($1\\le j<32$)'}, 'Location', 'SouthEast');\r\nset(hh_legend, 'Interpreter','latex','FontSize',18);\r\ntext('Interpreter','latex','String','$j=1$','Position',[13.5 0],'FontSize',18);\r\nannotation('arrow',[0.58 0.48],[0.55 0.5],'LineStyle', '-','color',[0 0 0]);\r\ntext('Interpreter','latex','String','$j=2$','Position',[5.4 -0.17],'FontSize',18);\r\nannotation('arrow',[0.37 0.44],[0.43 0.35],'LineStyle', '-','color',[0 0 0]);\r\nset(gca,'FontSize',18)\r\nxlabel('$k$','interpreter','latex','FontSize',20);\r\nylabel('$\\theta(rad)$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/state-converge-3', '-depsc');\r\nprint(gcf,'figures/state-converge-3.png', '-dpng', '-r400');\r\n\r\n \r\nfigure(12)%Figure 3-d of paper\r\nkk1 = 0:1:N;\r\nhh = plot(kk1, Itera_v(32,:),'r', 'LineWidth', 1);\r\nhh1 = plot(kk1, Itera_v(1,:),'r', 'LineWidth', 1);\r\nhold on \r\nfor i=2:1:32\r\nif i<=20\r\nhh1= plot(kk1, Itera_v(i,:),'r', 'LineWidth', 1);\r\nelse\r\nhh=plot(kk1, Itera_v(i,:),'b', 'LineWidth', 2);\r\nhold on \r\nxlim([0,24]);\r\nend\r\nend\r\nhh_legend = legend([hh,hh1], {'Converged ($j=32$)','Converging ($1\\le j<32$)'}, 'Location', 'SouthEast');\r\nset(hh_legend, 'Interpreter','latex','FontSize',18);\r\ntext('Interpreter','latex','String','$j=1$','Position',[17.5 1.7],'FontSize',18);\r\nannotation('arrow',[0.74 0.64],[0.55 0.5],'LineStyle', '-','color',[0 0 0]);\r\ntext('Interpreter','latex','String','$j=2$','Position',[8.5 2.7],'FontSize',18);\r\nannotation('arrow',[0.42 0.32],[0.83 0.88],'LineStyle', '-','color',[0 0 0]);\r\nset(gca,'FontSize',18)\r\nxlabel('$k$','interpreter','latex','FontSize',20);\r\nylabel('$v(m\\cdot s^{-1})$','interpreter','latex','FontSize',20);\r\nprint(gcf,'figures/state-converge-4', '-depsc');\r\nprint(gcf,'figures/state-converge-4.png', '-dpng', '-r400');\r\n" }, { "path": "matlab/acc2023/closedloop_performance/Iterative_Convergence.m", "content": "close all\r\nclear\r\n\r\n%--------------------------------------iMPC-DCBF------------------------------------------------\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = 24;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = 4;\r\ngamma2 = 4;\r\nnsim = 100;%Total time steps\r\nItera_x = [];%This is to store iterative convergence of location x\r\nItera_y = [];%This is to store iterative convergence of location y\r\nItera_theta = [];%This is to store iterative convergence of orientation theta\r\nItera_v = [];%This is to store iterative convergence of speed v\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = [-3 0 0 0];\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n if i == 6 %Get several open-loop trajectories at different iterations predicted at t = 6\r\n Itera_x = [Itera_x; ctrlx(1:4:(N*nx)+1)'];\r\n Itera_y = [Itera_y; ctrlx(2:4:(N*nx)+2)'];\r\n Itera_theta = [Itera_theta; ctrlx(3:4:(N*nx)+3)'];\r\n Itera_v = [Itera_v; ctrlx(4:4:(N*nx)+4)'];\r\n end\r\n x0 = ctrlx(1:nx);\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n break\r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\n\r\nsave('trajectory','impctra');%Close-loop 2D-trajectory where tsim=100\r\nsave('iteration_x','Itera_x');\r\nsave('iteration_y','Itera_y');\r\nsave('iteration_theta','Itera_theta');\r\nsave('iteration_v','Itera_v');\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/Maximum_Iterations.m", "content": "close all\r\nclear\r\n\r\n%--------------------------------------iMPC-DCBF------------------------------------------------\r\nkk44=getkk(4,4);\r\nkk46=getkk(4,6);\r\nkk64=getkk(6,4);\r\nkk66=getkk(6,6);\r\nsave('jtconv_Gamma4_4','kk44');%44 corrsponds to gamma1=0.4 and gamma2=0.4 in paper\r\nsave('jtconv_Gamma4_6','kk46');%46 corrsponds to gamma1=0.4 and gamma2=0.6 in paper\r\nsave('jtconv_Gamma6_4','kk64');%64 corrsponds to gamma1=0.6 and gamma2=0.4 in paper\r\nsave('jtconv_Gamma6_6','kk66');%66 corrsponds to gamma1=0.6 and gamma2=0.6 in paper\r\n\r\nfunction kk = getkk(gamma1,gamma2)\r\nxo = 0;\r\nyo = 0;\r\nr = 1;\r\nN = 24;%Number of horizon\r\nK1 = 1000;%Maximum iteration times, jmax\r\nK = 1000;%Maximum iteration times, jmax\r\ndt = 0.1;\r\ngamma1 = gamma1;\r\ngamma2 = gamma2;\r\nnsim = 100;%Total time steps\r\nkk = [];%This is for you to see the convergence interation jt,conv for every time step\r\n\r\n% Constraints\r\n\r\numin = [-7; -5; -inf; -inf;];\r\numax = [7; 5; Inf; Inf];\r\nxmin = [-10; -10; -10; -10];\r\nxmax = [ 10; 10; 10; 10];\r\n\r\n% Objective function\r\nQ = diag([10 10 10 10]);\r\nQN = Q;\r\nR = 1 * eye(4);\r\nR(3,3) = 1000;\r\nR(4,4) = 1000;\r\n% Initial and reference states\r\nx0 = [-3 0 0 0];\r\nxr = [3; 0.01; 0; 0];\r\nur = [0; 0; 1; 1];\r\n\r\n% Dynamic system initialization\r\nBB = [0 0 0 0;0 0 0 0;dt 0 0 0;0 dt 0 0];\r\n\r\n% Convex MPC frame\r\n[nx, nu] = size(BB);\r\n% Initialize states set (x00,x01,,,x0N）\r\nx_0 = [];\r\nx0 = x0';\r\nu_0 = zeros(nu, N);\r\nu0 = zeros(nu, 1);% we may need to change this for a better warm-up start\r\nimpc = zeros(nx, nsim + 1);\r\nimpc(:, 1) = x0;\r\nx0new = x0;\r\nfor i=1 : (N+1)\r\n x_0 = [x_0 x0new];\r\n x0new = [dt*x0new(4)*cos(x0new(3))+x0new(1);dt*x0new(4)*sin(x0new(3))+x0new(2);dt*u0(1)+x0new(3);dt*u0(2)+x0new(4)];\r\nend\r\n\r\nabc = tangentline(x_0, r);\r\nAcbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\nlcbf = -abc(3, :);\r\nlcbf(1) = [];\r\nlcbf = lcbf';\r\nucbf = inf * ones(N, 1);\r\nA2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\nlcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\nucbf2 = inf * ones(N-1, 1);\r\n\r\n% Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n% - quadratic objective\r\nP = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n% - linear objective\r\nq = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n% - linear dynamics\r\nAx = getax(x_0, dt, N, nx);\r\nBu = kron([sparse(1, N); speye(N)], BB);\r\nCx = getcx(x_0, u_0, dt, N, nx);\r\nAeq = [Ax, Bu];\r\nleq = [-x0; -Cx];\r\nueq = leq;\r\n% - input and state constraints\r\nAineq = speye((N+1)*nx + N*nu);\r\nlineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\nuineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n% - OSQP constraints\r\nA = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\nl = [leq; lineq; lcbf; lcbf2];\r\nu = [ueq; uineq; ucbf; ucbf2];\r\n\r\n% Create an OSQP object\r\nprob = osqp;\r\n% Setup workspace\r\nprob.setup(P, q, A, l, u, 'warm_start', true);\r\n% Solve\r\nres = prob.solve();\r\n\r\n% Check solver status\r\nif ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\nend\r\nctrlx = res.x(1:(N+1)*nx);\r\nctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n\r\nfor j = 1 : (K1-1)\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n kk = [kk j];%If convergence criterion is satisfied, stop and record jt,conv\r\n break\r\n elseif j == K1-1\r\n kk = [kk j];%If convergence criterion is not satisfied, stop at jmax and record it as jt,conv \r\n end\r\nend % Move for the first step\r\nctrl = ctrlu(1:nu);\r\nx0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\nctrlu = rewrite(ctrlu, nu, N);\r\nx_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\nu_0 = transu(ctrlu, nu, N);\r\nimpc(:, 2) = x0;\r\nstoragex = ctrlx;\r\nstorageu = ctrlu;\r\n\r\nfor i = 1 : (nsim-1)\r\n\r\n for j = 1 : K\r\n abc = tangentline(x_0, r);\r\n Acbf = cbfmat(abc, nx, nu, dt, gamma1, x0);\r\n lcbf = -abc(3, :);\r\n lcbf(1) = [];\r\n lcbf = lcbf';\r\n ucbf = inf * ones(N, 1);\r\n A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0);\r\n lcbf2 = getlcbf2(abc, dt, gamma1, gamma2);\r\n ucbf2 = inf * ones(N-1, 1);\r\n % Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))\r\n % - quadratic objective\r\n P = blkdiag( kron(speye(N), Q), QN, kron(speye(N), R) );\r\n % - linear objective\r\n q = [repmat(-Q*xr, N, 1); -QN*xr; repmat(-R*ur, N, 1)];\r\n % - linear dynamics\r\n Ax = getax(x_0, dt, N, nx);\r\n Bu = kron([sparse(1, N); speye(N)], BB);\r\n Cx = getcx(x_0, u_0, dt, N, nx);\r\n Aeq = [Ax, Bu];\r\n leq = [-x0; -Cx];\r\n ueq = leq;\r\n % - input and state constraints\r\n Aineq = speye((N+1)*nx + N*nu);\r\n lineq = [repmat(xmin, N+1, 1); repmat(umin, N, 1)];\r\n uineq = [repmat(xmax, N+1, 1); repmat(umax, N, 1)];\r\n\r\n % - OSQP constraints\r\n A = [Aeq; Aineq; Acbf; A2cbf];%Highest order mcbf=2\r\n l = [leq; lineq; lcbf; lcbf2];\r\n u = [ueq; uineq; ucbf; ucbf2];\r\n % Create an OSQP object\r\n prob = osqp;\r\n % Setup workspace\r\n prob.setup(P, q, A, l, u, 'warm_start', true);\r\n % Solve\r\n res = prob.solve();\r\n\r\n % Check solver status\r\n if ~strcmp(res.info.status, 'solved')\r\n error('OSQP did not solve the problem!')\r\n end\r\n ctrlx = res.x(1:(N+1)*nx);\r\n ctrlu = res.x((N+1)*nx+1:(N+1)*nx+N*nu);\r\n x0 = ctrlx(1:nx);\r\n x_0 = trans(x0, ctrlx, nx, N+1);\r\n u_0 = transu(ctrlu, nu, N);\r\n testx = (storagex - ctrlx)'*(storagex - ctrlx);\r\n testu = (storageu - ctrlu)'*(storageu - ctrlu);\r\n test = (testx)/(storagex'*storagex);\r\n if (test)^(0.5)<=10^(-2)&& (testx/((N+1)*nx))^(0.5)<=10^(-4)%Convergence criterion\r\n kk = [kk j];%If convergence criterion is satisfied, stop and record jt,conv\r\n break\r\n elseif j == K\r\n kk = [kk j];%If convergence criterion is not satisfied, stop at jmax and record it as jt,conv \r\n end\r\n storagex = ctrlx;\r\n storageu = ctrlu;\r\n end\r\n ctrl = ctrlu(1:nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*ctrl(1)+x0(3);dt*ctrl(2)+x0(4)];\r\n ctrlu = rewrite(ctrlu, nu, N);\r\n x_0 = newinit(x0, ctrlu, nx, nu, N, dt);\r\n u_0 = transu(ctrlu, nu, N);\r\n impc(:, i+2) = x0;\r\nend\r\nend\r\n\r\n% Linerize the CBF constraints (get a, b, c for lines)\r\nfunction abc = tangentline(xy, r)% x and y from initialize states set, abc are coeeficients for linear equation a*x+b*y+c=0\r\n[xx, ~] = size(xy);%xx=2,yy=N+1\r\nxy(xx,:) = []; % this part should be changed for other case\r\nxy((xx-1),:) = []; % this part should be changed other case\r\n[xx, yy] = size(xy);%xx=2,yy=N+1\r\nxyjiao = zeros(xx, yy);%intersection points\r\nfor i = 1 : xx\r\n for j = 1 : yy\r\n xyjiao(i, j) = r * xy(i, j) * (1 / (xy(:, j)' * xy(:, j)))^(0.5);%calculate coordinates of intersection points\r\n end\r\nend\r\ncc = -r^2 * ones(1, yy);\r\nabc = [xyjiao; cc];\r\nend\r\n\r\n% Get CBF constraints matrix \r\nfunction Acbf = cbfmat(abc, nx, nu, dt, gamma, x0)\r\n[~, yy] = size(abc);\r\nAcbfx = zeros((yy-1), yy*nx);\r\nAcbfu = zeros((yy-1), (yy-1)*nu);\r\nfor i = 1 : (yy-1)\r\n Acbfx(i, (i*nx)+1) = abc(1, (i+1));\r\n Acbfx(i, (i*nx)+2) = abc(2, (i+1));\r\nend\r\nfor i = 1 : (yy-1)\r\n Acbfu(i, ((i-1)*nu+3)) = - (1 - dt * gamma)^(i) * (abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nAcbf = [Acbfx Acbfu];\r\nend\r\n\r\n% Transfer vector x into matrix \r\nfunction res = trans(x0, vector, nxx, nyy)%nxx=nx,nyy=N+1\r\n res = zeros(nxx, nyy);\r\n res(:,1) = x0;\r\n for i = 1 : (nyy -1)\r\n res(:,i+1)= vector(((i)*nxx+1):(i+1)*nxx);\r\n end \r\nend\r\n\r\n% Transfer vector u into matrix \r\nfunction resu = transu(vector, nxx, nyy)%nxx=nu,nyy=N\r\n resu = zeros(nxx, nyy);\r\n for i = 1 : (nyy)\r\n resu(:,i)= vector(((i-1)*nxx+1):(i)*nxx);\r\n end \r\nend\r\n\r\n% Rewrite u vector\r\nfunction reu = rewrite(vector, nu, N)\r\nappend = vector((N-1)*nu+1:N*nu);\r\nvector(1:nu) = [];\r\nreu = [vector;append];\r\nend\r\n\r\n% Get new x_0\r\nfunction x_0 = newinit(x0, ctrlu, nx, nu, N, dt)\r\n x_0 = zeros(nx, N+1);\r\n x_0(:, 1) = x0;\r\n for i=1 : N\r\n u0 = ctrlu((i-1)*nu+1:i*nu);\r\n x0 = [dt*x0(4)*cos(x0(3))+x0(1);dt*x0(4)*sin(x0(3))+x0(2);dt*u0(1)+x0(3);dt*u0(2)+x0(4)];\r\n x_0(:, i + 1) = x0;\r\n end\r\nend\r\n\r\n% Get AA matrix\r\nfunction AA = getaa(x0, dt)\r\n AA = [1 0 -x0(4)*sin(x0(3))*dt cos(x0(3))*dt;0 1 x0(4)*cos(x0(3))*dt sin(x0(3))*dt;0 0 1 0;0 0 0 1];\r\nend\r\n% Get CC matrix\r\nfunction CC = getcc(x0, x1, u0, dt)\r\n CC = [x0(4)*sin(x0(3))*x0(3)*dt-x0(4)*cos(x0(3))*dt+x1(1)-x0(1);-x0(4)*cos(x0(3))*x0(3)*dt-x0(4)*sin(x0(3))*dt+x1(2)-x0(2);-u0(1)*dt+x1(3)-x0(3);-u0(2)*dt+x1(4)-x0(4)];\r\nend\r\n% Get Ax matrix\r\nfunction Ax = getax(x_0, dt, N, nx)\r\nx0 = x_0(:,1);\r\nAA = getaa(x0, dt);\r\nAx = kron(speye(N+1), -speye(nx)) + kron(sparse(diag(ones(N, 1), -1)), AA);\r\nfor i = 1 : (N-1)\r\n x0 = x_0(:,i+1);\r\n AA = getaa(x0, dt);\r\n Ax(nx*(i+1)+1:nx*(i+1)+nx,nx*i+1:nx*i+nx) = AA;\r\nend\r\nend\r\n% Get Cx matrix\r\nfunction Cx = getcx(x_0, u_0, dt, N, nx)\r\nCx = zeros(N*nx, 1);\r\nfor i = 1 : N\r\n u0 = u_0(:,i);\r\n x0 = x_0(:,i);\r\n x1 = x_0(:,i+1);\r\n CC = getcc(x0, x1, u0, dt);\r\n Cx((i-1)*nx+1:(i-1)*nx+nx) = CC;\r\nend\r\nend\r\n% Get A2cbf\r\nfunction A2cbf = cbfmat2(abc, nx, nu, dt, gamma1, gamma2, x0)\r\n[~, yy] = size(abc);\r\nAcbfx2 = zeros((yy-2), yy*nx);\r\nAcbfx22 = zeros((yy-2), yy*nx);\r\nAcbfu2 = zeros((yy-2), (yy-1)*nu);\r\nfor i = 1 : (yy-2)\r\n Acbfx2(i, (i*nx)+1) = (gamma1-1/dt)*abc(1, (i+1));\r\n Acbfx2(i, (i*nx)+2) = (gamma1-1/dt)*abc(2, (i+1));\r\n Acbfx2(i, ((i+1)*nx)+1) = (1/dt)*abc(1, (i+2));\r\n Acbfx2(i, ((i+1)*nx)+2) = (1/dt)*abc(2, (i+2));\r\nend\r\nfor i = 1 : (yy-2)\r\n Acbfx22(i, (1*nx)+1) = -(1-dt*gamma2)^(i)/dt*abc(1, (1+1));\r\n Acbfx22(i, (1*nx)+2) = -(1-dt*gamma2)^(i)/dt*abc(2, (1+1));\r\nend\r\nAcbfx2 = Acbfx2 + Acbfx22;\r\nfor i = 1 : (yy-2)\r\n Acbfu2(i, ((i-1)*nu+4)) = - (1 - dt * gamma2)^(i)*(gamma1-1/dt)*(abc(1, 1) * x0(1, 1) + abc(2, 1) * x0(2, 1) + abc(3, 1));\r\nend\r\nA2cbf = [Acbfx2 Acbfu2];\r\nend\r\n% Get lcbf2\r\nfunction lcbf2 = getlcbf2(abc, dt, gamma1, gamma2)\r\n[~, yy] = size(abc);\r\nlcbf2 = zeros((yy-2),1);\r\nfor i = 1 : (yy-2)\r\n lcbf2(i, 1) = -abc(3, (i+2))/dt-(gamma1-1/dt)*abc(3, (i+1))+(1 - dt * gamma2)^(i)*abc(3, (1+1))/dt;\r\nend\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/NMPCDCBF1.m", "content": "classdef NMPCDCBF1 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF1(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);\r\n if tt == -1 %if infeasiblility happens, stop\r\n self.tt = tt;\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;%record computing time for each time-step\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n constraints = [constraints; b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=1\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/NMPCDCBF2.m", "content": "classdef NMPCDCBF2 < handle\r\n % MPC with distance constraints\r\n properties\r\n system\r\n params\r\n x0\r\n x_curr\r\n time_curr = 0.0\r\n xlog = []\r\n ulog = []\r\n tt = 0\r\n distlog = []\r\n solvertime = []\r\n xopenloop = {}\r\n uopenloop = {}\r\n u_cost = 0\r\n obs\r\n end\r\n methods\r\n function self = NMPCDCBF2(x0, system, params)\r\n % Define MPC_CBF controller\r\n self.x0 = x0;\r\n self.x_curr = x0;\r\n self.system = system;\r\n self.params = params;\r\n end\r\n \r\n function sim(self, time)\r\n % Simulate the system until a given time\r\n tic;\r\n xk = self.x_curr;\r\n while self.time_curr < time\r\n [~, uk, tt] = self.solveNMPCDCBF1(self.x_curr);%if infeasiblility happens, stop\r\n if tt == -1\r\n self.tt = tt;%record computing time for each time-step\r\n return\r\n end\r\n xk = self.system.A * xk + [xk(4,1) * cos(xk(3,1))*self.system.dt;xk(4,1) * sin(xk(3,1))*self.system.dt;0;0] + self.system.C * uk;\r\n % update system\r\n self.x_curr = xk;\r\n self.time_curr = self.time_curr + self.system.dt;\r\n self.xlog = [self.xlog, xk];\r\n self.ulog = [self.ulog, uk];\r\n self.u_cost = self.u_cost + uk'*uk;\r\n tt = tt + toc;\r\n self.tt = tt;\r\n return\r\n end\r\n \r\n end\r\n \r\n function [xopt, uopt,tt] = solveNMPCDCBF1(self, xk)\r\n % Solve NMPC-DCBF\r\n [feas, x, u, J] = self.solve_cftoc1(xk);\r\n if ~feas\r\n xopt = [];\r\n uopt = [];\r\n tt = -1;\r\n return\r\n else\r\n xopt = x(:,2);\r\n uopt = u(:,1);\r\n tt = 0;\r\n end\r\n end\r\n \r\n function [feas, xopt, uopt, Jopt] = solve_cftoc1(self, xk)\r\n % Solve CFTOC\r\n % extract variables\r\n N = self.params.N;\r\n % define variables and cost\r\n x = sdpvar(4, N+1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n u = sdpvar(4, N);\r\n constraints = [];\r\n cost = 0;\r\n % initial constraint\r\n constraints = [constraints; x(:,1) == xk];\r\n % add constraints and costs\r\n AA=self.system.A;\r\n BB=self.system.B;\r\n CC=self.system.C;\r\n constraints = [constraints;\r\n self.system.xl <= x(:,1) <= self.system.xu;\r\n self.system.ul <= u(:,1) <= self.system.uu;\r\n x(:,2) == AA * x(:,1) + [x(4,1)*cos(x(3,1))*self.system.dt;x(4,1)*sin(x(3,1))*self.system.dt;0; 0] + CC * u(:,1)];\r\n cost = cost + (x(:,1)-[3;0.01;0;0])'*self.params.Q*(x(:,1)-[3;0.01;0;0]) + (u(:,1)-[0;0;1;1])'*self.params.R*(u(:,1)-[0;0;1;1]); % self.params.S*(w(:,1)-1)^2;\r\n for i = 2:1:N\r\n constraints = [constraints;\r\n self.system.xl <= x(:,i) <= self.system.xu;\r\n self.system.ul <= u(:,i) <= self.system.uu;\r\n x(:,i+1) == AA * x(:,i) + [x(4,i)*cos(x(3,i))*self.system.dt;x(4,i)*sin(x(3,i))*self.system.dt;0; 0] + CC * u(:,i)];\r\n cost = cost + (x(:,i)-[3;0.01;0;0])'*self.params.Q*(x(:,i)-[3;0.01;0;0]) + (u(:,i)-[0;0;1;1])'*self.params.R*(u(:,i)-[0;0;1;1]);% self.params.S*(w(:,i)-1)^2;\r\n end\r\n % add CBF constraints\r\n for i = 1:N-1\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n b = (x([1:2],i)-pos)'*((x([1:2],i)-pos))-r^2;\r\n b_next = (x([1:2],i+1)-pos)'*((x([1:2],i+1)-pos))-r^2;\r\n b_next_next = (x([1:2],i+2)-pos)'*((x([1:2],i+2)-pos))-r^2; \r\n b1 = (b_next - b)/self.system.dt + self.params.gamma1/self.system.dt * (b);\r\n b1_next = (b_next_next - b_next)/self.system.dt + self.params.gamma1/self.system.dt * (b_next);\r\n constraints = [constraints; b1_next >=u(4,i)*(1-self.params.gamma2)*b1;b_next >= u(3,i)*(1-self.params.gamma1)*b];%Highest order mcbf=2\r\n end\r\n % add terminal cost\r\n cost = cost + (x(:,N+1)-[3;0.01;0;0])'*self.params.P*(x(:,N+1)-[3;0.01;0;0]);\r\n ops = sdpsettings('solver','ipopt','verbose',0);\r\n % solve optimization\r\n diagnostics = optimize(constraints, cost, ops);\r\n if diagnostics.problem == 0\r\n feas = true;\r\n xopt = value(x);\r\n uopt = value(u);\r\n Jopt = value(cost);\r\n else\r\n feas = false;\r\n xopt = [];\r\n uopt = [];\r\n Jopt = [];\r\n end\r\n pos = self.obs.pos1;\r\n r = self.obs.r1 ;\r\n self.distlog = [self.distlog, (r^2-(xk(1:2)-pos)'*(xk(1:2)-pos))];\r\n self.xopenloop{size(self.xopenloop,2)+1} = xopt;\r\n self.uopenloop{size(self.uopenloop,2)+1} = uopt;\r\n self.solvertime = [self.solvertime, diagnostics.solvertime];\r\n fprintf('solver time: %f\\n', diagnostics.solvertime);\r\n end\r\n end\r\nend" }, { "path": "matlab/acc2023/closedloop_performance/figures/J-converge-1.eps", "content": "%!PS-Adobe-3.0 EPSF-3.0\r\n%%Creator: (MATLAB, The Mathworks, Inc. 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License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/BeginEPSF { %def\r\n/b4_Inc_state save def % Save state for cleanup\r\n/dict_count countdictstack def % Count objects on dict stack\r\n/op_count count 1 sub def % Count objects on operand stack\r\nuserdict begin % Push userdict on dict stack\r\n/showpage { } def % Redefine showpage, { } = null proc\r\n0 setgray 0 setlinecap % Prepare graphics state\r\n1 setlinewidth 0 setlinejoin\r\n10 setmiterlimit [ ] 0 setdash newpath\r\n/languagelevel where % If level not equal to 1 then\r\n{pop languagelevel % set strokeadjust and\r\n1 ne % overprint to their defaults.\r\n{false setstrokeadjust false setoverprint\r\n} if\r\n} if\r\n} bd\r\n/EndEPSF { %def\r\ncount op_count sub {pop} repeat % Clean up stacks\r\ncountdictstack dict_count sub {end} repeat\r\nb4_Inc_state restore\r\n} bd\r\n%%EndResource\r\n%FOPBeginFontDict\r\n%%IncludeResource: font Courier-Oblique\r\n%%IncludeResource: font Courier-BoldOblique\r\n%%IncludeResource: font Courier-Bold\r\n%%IncludeResource: font ZapfDingbats\r\n%%IncludeResource: font Symbol\r\n%%IncludeResource: font Helvetica\r\n%%IncludeResource: font Helvetica-Oblique\r\n%%IncludeResource: font Helvetica-Bold\r\n%%IncludeResource: font Helvetica-BoldOblique\r\n%%IncludeResource: font Times-Roman\r\n%%IncludeResource: font Times-Italic\r\n%%IncludeResource: font Times-Bold\r\n%%IncludeResource: font Times-BoldItalic\r\n%%IncludeResource: font Courier\r\n%FOPEndFontDict\r\n%%BeginResource: encoding WinAnsiEncoding\r\n/WinAnsiEncoding [\r\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /space /exclam /quotedbl\n/numbersign /dollar /percent /ampersand /quotesingle\n/parenleft /parenright /asterisk /plus /comma\n/hyphen /period /slash /zero /one\n/two /three /four /five /six\n/seven /eight /nine /colon /semicolon\n/less /equal /greater /question /at\n/A /B /C /D /E\n/F /G /H /I /J\n/K /L /M /N /O\n/P /Q /R /S /T\n/U /V /W /X /Y\n/Z /bracketleft /backslash /bracketright /asciicircum\n/underscore /quoteleft /a /b /c\n/d /e /f /g /h\n/i /j /k /l /m\n/n /o /p /q /r\n/s /t /u /v /w\n/x /y /z /braceleft /bar\n/braceright /asciitilde /bullet /Euro /bullet\n/quotesinglbase /florin /quotedblbase /ellipsis /dagger\n/daggerdbl /circumflex /perthousand /Scaron /guilsinglleft\n/OE /bullet /Zcaron /bullet /bullet\n/quoteleft /quoteright /quotedblleft /quotedblright /bullet\n/endash /emdash /asciitilde /trademark /scaron\n/guilsinglright /oe /bullet /zcaron /Ydieresis\n/space /exclamdown /cent /sterling /currency\n/yen /brokenbar /section /dieresis /copyright\n/ordfeminine /guillemotleft /logicalnot /sfthyphen /registered\n/macron /degree /plusminus /twosuperior /threesuperior\n/acute /mu /paragraph /middot /cedilla\n/onesuperior /ordmasculine /guillemotright /onequarter /onehalf\n/threequarters /questiondown /Agrave /Aacute /Acircumflex\n/Atilde /Adieresis /Aring /AE /Ccedilla\n/Egrave /Eacute /Ecircumflex /Edieresis /Igrave\n/Iacute /Icircumflex /Idieresis /Eth /Ntilde\n/Ograve /Oacute /Ocircumflex /Otilde /Odieresis\n/multiply /Oslash /Ugrave /Uacute /Ucircumflex\n/Udieresis /Yacute /Thorn /germandbls /agrave\n/aacute /acircumflex /atilde /adieresis /aring\n/ae /ccedilla /egrave /eacute /ecircumflex\n/edieresis /igrave /iacute /icircumflex /idieresis\n/eth /ntilde /ograve /oacute /ocircumflex\n/otilde /odieresis /divide /oslash /ugrave\n/uacute /ucircumflex /udieresis /yacute /thorn\n/ydieresis\n] def\r\n%%EndResource\r\n%FOPBeginFontReencode\r\n/Courier-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Oblique exch definefont pop\r\n/Courier-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-BoldOblique exch definefont pop\r\n/Courier-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Bold exch definefont pop\r\n/Helvetica findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica exch definefont pop\r\n/Helvetica-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Oblique exch definefont pop\r\n/Helvetica-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Bold exch definefont pop\r\n/Helvetica-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-BoldOblique exch definefont pop\r\n/Times-Roman findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Roman exch definefont pop\r\n/Times-Italic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Italic exch definefont pop\r\n/Times-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Bold exch definefont pop\r\n/Times-BoldItalic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-BoldItalic exch definefont pop\r\n/Courier findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier exch definefont pop\r\n%FOPEndFontReencode\r\n%%EndProlog\r\n%%Page: 1 1\r\n%%PageBoundingBox: 0 0 420 315\r\n%%BeginPageSetup\r\n[1 0 0 -1 0 315] CT\r\n%%EndPageSetup\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n163 698 M\n1014 698 L\n1014 63 L\n163 63 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 698 M\n1014 698 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 63 M\n1014 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 698 M\n163 689.49 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n334.919 698 M\n334.919 689.49 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n506.838 698 M\n506.838 689.49 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n678.758 698 M\n678.758 689.49 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n850.677 698 M\n850.677 689.49 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 63 M\n163 71.51 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n334.919 63 M\n334.919 71.51 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n506.838 63 M\n506.838 71.51 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n678.758 63 M\n678.758 71.51 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n850.677 63 M\n850.677 71.51 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 61.125 267.35001] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(0) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 125.5947 267.35001] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(20) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 190.06439 267.35001] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(40) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 254.53409 267.35001] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(60) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 319.00378 267.35001] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(80) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.68766 290.84999] CT\r\n0.149 GC\r\nN\r\n-6.348 40.503 M\n-6.348 39.722 -6.192 39.034 QT\n-2.364 23.815 L\n-7.927 23.815 L\n-8.458 23.815 -8.458 23.112 QT\n-8.255 21.956 -7.77 21.956 QT\n-1.895 21.956 L\n0.23 13.3 L\n0.433 12.612 1.058 12.12 QT\n1.683 11.628 2.448 11.628 QT\n3.136 11.628 3.589 12.026 QT\n4.042 12.425 4.042 13.097 QT\n4.042 13.253 4.027 13.347 QT\n4.011 13.44 3.98 13.534 QT\n1.855 21.956 L\n7.323 21.956 L\n7.87 21.956 7.87 22.644 QT\n7.839 22.784 7.761 23.089 QT\n7.683 23.394 7.55 23.604 QT\n7.417 23.815 7.167 23.815 QT\n1.402 23.815 L\n-2.411 39.128 L\n-2.802 40.628 -2.802 41.722 QT\n-2.802 44.003 -1.239 44.003 QT\n1.089 44.003 2.886 41.815 QT\n4.683 39.628 5.636 37.003 QT\n5.839 36.706 6.058 36.706 QT\n6.698 36.706 L\n6.902 36.706 7.034 36.847 QT\n7.167 36.987 7.167 37.159 QT\n7.167 37.269 7.12 37.315 QT\n5.948 40.534 3.722 42.964 QT\n1.495 45.394 -1.38 45.394 QT\n-3.473 45.394 -4.911 44.026 QT\n-6.348 42.659 -6.348 40.503 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 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setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 274.667 M\n1005.49 274.667 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 63 M\n1005.49 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 698 M\n167.255 698 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 634.282 M\n167.255 634.282 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 597.009 M\n167.255 597.009 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 570.564 M\n167.255 570.564 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 550.051 M\n167.255 550.051 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n163 533.291 M\n167.255 533.291 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L\n670.162 634.282 L\n678.758 634.282 L\n687.354 634.282 L\n695.949 634.282 L\n704.545 634.282 L\n713.141 634.282 L\n721.737 634.282 L\n730.333 634.282 L\n738.929 634.282 L\n747.525 634.282 L\n756.121 634.282 L\n764.717 634.282 L\n773.313 634.282 L\n781.909 634.282 L\n790.505 634.282 L\n799.101 634.282 L\n807.697 634.282 L\n816.293 634.282 L\n824.889 698 L\n833.485 634.282 L\n842.081 698 L\n850.677 634.282 L\n859.273 698 L\n867.869 634.282 L\n876.465 698 L\n885.061 698 L\n893.657 634.282 L\n902.253 698 L\n910.849 698 L\n919.444 698 L\n928.04 634.282 L\n936.636 698 L\n945.232 698 L\n953.828 698 L\n962.424 698 L\n971.02 698 L\n979.616 698 L\n988.212 698 L\n996.808 698 L\n1005.404 634.282 L\n1014 698 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 61.125 125.54224] CT\r\nN\r\n0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 64.34849 124.24125] CT\r\nN\r\n/f-1792389198{0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp}def\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 67.57197 166.17297] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 70.79545 150.78851] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 74.01894 160.24893] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 77.24242 155.19519] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 80.46591 142.27872] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 83.68939 173.33075] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 86.91288 160.24893] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 90.13636 166.17297] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 93.35985 166.17297] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 96.58333 155.19519] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 99.80682 158.48075] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 103.0303 138.21849] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 106.25378 143.37316] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 109.47727 190.06723] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 112.70075 186.007] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 115.92424 199.98425] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 119.14772 199.98425] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 122.37122 206.26925] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 125.5947 206.26925] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 128.81819 213.96149] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 132.04167 213.96149] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 135.26515 213.96149] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 138.48864 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 141.71212 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 144.9356 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 148.15909 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 151.38257 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 154.60606 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 157.82954 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 161.05302 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 164.27652 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 167.50001 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 170.72348 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 173.94697 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.17046 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 180.39394 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 183.61742 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 186.84091 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 190.06439 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 193.28787 223.8785] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 196.51138 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 199.73486 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 202.95834 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 206.18182 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 209.4053 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 212.6288 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 215.85228 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 219.07576 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.29924 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 225.52272 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 228.7462 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 231.9697 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 235.19318 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 238.41666 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 241.64014 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 244.86362 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 248.08713 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 251.31061 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 254.53409 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 257.75759 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 260.98105 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 264.20455 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 267.42803 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 270.65151 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 273.87502 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 277.09847 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 280.32195 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 283.54546 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.76894 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 289.99244 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 293.21592 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 296.43938 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 299.66288 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 302.88636 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 306.10984 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 309.33334 261.75] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 312.55682 237.85574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 315.7803 261.75] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 319.00378 237.85574] 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CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 325.38636 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 328.61364 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 331.84092 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 335.06818 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 338.29546 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 341.52274 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 344.74999 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 347.97727 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 351.20455 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 354.43181 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 357.65909 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 360.88637 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 364.11365 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 367.3409 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 370.56818 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 373.79544 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 377.02272 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 380.25 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\n%%Trailer\r\n%%Pages: 1\r\n%%EOF\r\n" }, { "path": "matlab/acc2023/closedloop_performance/figures/J-converge-3.eps", "content": "%!PS-Adobe-3.0 EPSF-3.0\r\n%%Creator: (MATLAB, The Mathworks, Inc. Version 9.10.0.1649659 \\(R2021a\\) Update 1. Operating System: Windows 10)\r\n%%Title: figures/J-converge-3.eps\r\n%%CreationDate: 2023-01-31T03:11:56\r\n%%Pages: (atend)\r\n%%BoundingBox: 3 6 383 301\r\n%%LanguageLevel: 3\r\n%%EndComments\r\n%%BeginProlog\r\n%%BeginResource: procset (Apache XML Graphics Std ProcSet) 1.2 0\r\n%%Version: 1.2 0\r\n%%Copyright: (Copyright 2001-2003,2010 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/bd{bind def}bind def\r\n/ld{load def}bd\r\n/GR/grestore ld\r\n/GS/gsave ld\r\n/RM/rmoveto ld\r\n/C/curveto ld\r\n/t/show ld\r\n/L/lineto ld\r\n/ML/setmiterlimit ld\r\n/CT/concat ld\r\n/f/fill ld\r\n/N/newpath ld\r\n/S/stroke ld\r\n/CC/setcmykcolor ld\r\n/A/ashow ld\r\n/cp/closepath ld\r\n/RC/setrgbcolor ld\r\n/LJ/setlinejoin ld\r\n/GC/setgray ld\r\n/LW/setlinewidth ld\r\n/M/moveto ld\r\n/re {4 2 roll M\r\n1 index 0 rlineto\r\n0 exch rlineto\r\nneg 0 rlineto\r\ncp } bd\r\n/_ctm matrix def\r\n/_tm matrix def\r\n/BT { _ctm currentmatrix pop matrix _tm copy pop 0 0 moveto } bd\r\n/ET { _ctm setmatrix } bd\r\n/iTm { _ctm setmatrix _tm concat } bd\r\n/Tm { _tm astore pop iTm 0 0 moveto } bd\r\n/ux 0.0 def\r\n/uy 0.0 def\r\n/F {\r\n /Tp exch def\r\n /Tf exch def\r\n Tf findfont Tp scalefont setfont\r\n /cf Tf def /cs Tp def\r\n} bd\r\n/ULS {currentpoint /uy exch def /ux exch def} bd\r\n/ULE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add moveto Tcx uy To add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/OLE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs add moveto Tcx uy To add cs add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/SOE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/QT {\r\n/Y22 exch store\r\n/X22 exch store\r\n/Y21 exch store\r\n/X21 exch store\r\ncurrentpoint\r\n/Y21 load 2 mul add 3 div exch\r\n/X21 load 2 mul add 3 div exch\r\n/X21 load 2 mul /X22 load add 3 div\r\n/Y21 load 2 mul /Y22 load add 3 div\r\n/X22 load /Y22 load curveto\r\n} bd\r\n/SSPD {\r\ndup length /d exch dict def\r\n{\r\n/v exch def\r\n/k exch def\r\ncurrentpagedevice k known {\r\n/cpdv currentpagedevice k get def\r\nv cpdv ne {\r\n/upd false def\r\n/nullv v type /nulltype eq def\r\n/nullcpdv cpdv type /nulltype eq def\r\nnullv nullcpdv or\r\n{\r\n/upd true def\r\n} {\r\n/sametype v type cpdv type eq def\r\nsametype {\r\nv type /arraytype eq {\r\n/vlen v length def\r\n/cpdvlen cpdv length def\r\nvlen cpdvlen eq {\r\n0 1 vlen 1 sub {\r\n/i exch def\r\n/obj v i get def\r\n/cpdobj cpdv i get def\r\nobj cpdobj ne {\r\n/upd true def\r\nexit\r\n} if\r\n} for\r\n} {\r\n/upd true def\r\n} ifelse\r\n} {\r\nv type /dicttype eq {\r\nv {\r\n/dv exch def\r\n/dk exch def\r\n/cpddv cpdv dk get def\r\ndv cpddv ne {\r\n/upd true def\r\nexit\r\n} if\r\n} forall\r\n} {\r\n/upd true def\r\n} ifelse\r\n} ifelse\r\n} if\r\n} ifelse\r\nupd true eq {\r\nd k v put\r\n} if\r\n} if\r\n} if\r\n} forall\r\nd length 0 gt {\r\nd setpagedevice\r\n} if\r\n} bd\r\n/RE { % /NewFontName [NewEncodingArray] /FontName RE -\r\n findfont dup length dict begin\r\n {\r\n 1 index /FID ne\r\n {def} {pop pop} ifelse\r\n } forall\r\n /Encoding exch def\r\n /FontName 1 index def\r\n currentdict definefont pop\r\n end\r\n} bind def\r\n%%EndResource\r\n%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0\r\n%%Version: 1.0 0\r\n%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/BeginEPSF { %def\r\n/b4_Inc_state save def % Save state for cleanup\r\n/dict_count countdictstack def % Count objects on dict stack\r\n/op_count count 1 sub def % Count objects on operand stack\r\nuserdict begin % Push userdict on dict stack\r\n/showpage { } def % Redefine showpage, { } = null proc\r\n0 setgray 0 setlinecap % Prepare graphics state\r\n1 setlinewidth 0 setlinejoin\r\n10 setmiterlimit [ ] 0 setdash newpath\r\n/languagelevel where % If level not equal to 1 then\r\n{pop languagelevel % set strokeadjust and\r\n1 ne % overprint to their defaults.\r\n{false setstrokeadjust false setoverprint\r\n} if\r\n} if\r\n} bd\r\n/EndEPSF { %def\r\ncount op_count sub {pop} repeat % Clean up stacks\r\ncountdictstack dict_count sub {end} repeat\r\nb4_Inc_state restore\r\n} bd\r\n%%EndResource\r\n%FOPBeginFontDict\r\n%%IncludeResource: font Courier-Oblique\r\n%%IncludeResource: font Courier-BoldOblique\r\n%%IncludeResource: font 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/one\n/two /three /four /five /six\n/seven /eight /nine /colon /semicolon\n/less /equal /greater /question /at\n/A /B /C /D /E\n/F /G /H /I /J\n/K /L /M /N /O\n/P /Q /R /S /T\n/U /V /W /X /Y\n/Z /bracketleft /backslash /bracketright /asciicircum\n/underscore /quoteleft /a /b /c\n/d /e /f /g /h\n/i /j /k /l /m\n/n /o /p /q /r\n/s /t /u /v /w\n/x /y /z /braceleft /bar\n/braceright /asciitilde /bullet /Euro /bullet\n/quotesinglbase /florin /quotedblbase /ellipsis /dagger\n/daggerdbl /circumflex /perthousand /Scaron /guilsinglleft\n/OE /bullet /Zcaron /bullet /bullet\n/quoteleft /quoteright /quotedblleft /quotedblright /bullet\n/endash /emdash /asciitilde /trademark /scaron\n/guilsinglright /oe /bullet /zcaron /Ydieresis\n/space /exclamdown /cent /sterling /currency\n/yen /brokenbar /section /dieresis /copyright\n/ordfeminine /guillemotleft /logicalnot /sfthyphen /registered\n/macron /degree /plusminus /twosuperior /threesuperior\n/acute /mu /paragraph /middot /cedilla\n/onesuperior /ordmasculine /guillemotright /onequarter /onehalf\n/threequarters /questiondown /Agrave /Aacute /Acircumflex\n/Atilde /Adieresis /Aring /AE /Ccedilla\n/Egrave /Eacute /Ecircumflex /Edieresis /Igrave\n/Iacute /Icircumflex /Idieresis /Eth /Ntilde\n/Ograve /Oacute /Ocircumflex /Otilde /Odieresis\n/multiply /Oslash /Ugrave /Uacute /Ucircumflex\n/Udieresis /Yacute /Thorn /germandbls /agrave\n/aacute /acircumflex /atilde /adieresis /aring\n/ae /ccedilla /egrave /eacute /ecircumflex\n/edieresis /igrave /iacute /icircumflex /idieresis\n/eth /ntilde /ograve /oacute /ocircumflex\n/otilde /odieresis /divide /oslash /ugrave\n/uacute /ucircumflex /udieresis /yacute /thorn\n/ydieresis\n] def\r\n%%EndResource\r\n%FOPBeginFontReencode\r\n/Courier-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Oblique exch definefont pop\r\n/Courier-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-BoldOblique exch definefont pop\r\n/Courier-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Bold exch definefont pop\r\n/Helvetica findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica exch definefont pop\r\n/Helvetica-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Oblique exch definefont pop\r\n/Helvetica-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Bold exch definefont pop\r\n/Helvetica-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-BoldOblique exch definefont pop\r\n/Times-Roman findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Roman exch definefont pop\r\n/Times-Italic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Italic exch definefont pop\r\n/Times-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Bold exch definefont pop\r\n/Times-BoldItalic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-BoldItalic exch definefont pop\r\n/Courier findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier exch definefont pop\r\n%FOPEndFontReencode\r\n%%EndProlog\r\n%%Page: 1 1\r\n%%PageBoundingBox: 0 0 420 315\r\n%%BeginPageSetup\r\n[1 0 0 -1 0 315] CT\r\n%%EndPageSetup\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n162 699 M\n1014 699 L\n1014 63 L\n162 63 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 699 M\n1014 699 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 63 M\n1014 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 699 M\n162 690.48 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n334.121 699 M\n334.121 690.48 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n506.242 699 M\n506.242 690.48 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n678.364 699 M\n678.364 690.48 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n850.485 699 M\n850.485 690.48 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 63 M\n162 71.52 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n334.121 63 M\n334.121 71.52 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n506.242 63 M\n506.242 71.52 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n678.364 63 M\n678.364 71.52 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n850.485 63 M\n850.485 71.52 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 60.75 267.72501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(0) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 125.29546 267.72501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(20) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 189.84091 267.72501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(40) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 254.38637 267.72501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(60) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 318.93182 267.72501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-27 49 moveto \r\n1 -1 scale\r\n(80) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.50016 291.22499] CT\r\n0.149 GC\r\nN\r\n-6.252 40.055 M\n-6.252 39.274 -6.096 38.586 QT\n-2.268 23.367 L\n-7.83 23.367 L\n-8.361 23.367 -8.361 22.664 QT\n-8.158 21.508 -7.674 21.508 QT\n-1.799 21.508 L\n0.326 12.852 L\n0.529 12.164 1.154 11.672 QT\n1.779 11.18 2.545 11.18 QT\n3.232 11.18 3.685 11.578 QT\n4.139 11.977 4.139 12.649 QT\n4.139 12.805 4.123 12.899 QT\n4.107 12.992 4.076 13.086 QT\n1.951 21.508 L\n7.42 21.508 L\n7.967 21.508 7.967 22.196 QT\n7.935 22.336 7.857 22.641 QT\n7.779 22.946 7.646 23.156 QT\n7.514 23.367 7.264 23.367 QT\n1.498 23.367 L\n-2.315 38.68 L\n-2.705 40.18 -2.705 41.274 QT\n-2.705 43.555 -1.143 43.555 QT\n1.185 43.555 2.982 41.367 QT\n4.779 39.18 5.732 36.555 QT\n5.935 36.258 6.154 36.258 QT\n6.795 36.258 L\n6.998 36.258 7.131 36.399 QT\n7.264 36.539 7.264 36.711 QT\n7.264 36.821 7.217 36.867 QT\n6.045 40.086 3.818 42.516 QT\n1.592 44.946 -1.283 44.946 QT\n-3.377 44.946 -4.815 43.578 QT\n-6.252 42.211 -6.252 40.055 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 699 M\n162 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 699 M\n1014 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 699 M\n170.52 699 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 487 M\n170.52 487 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 275 M\n170.52 275 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 63 M\n170.52 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 699 M\n1005.48 699 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 487 M\n1005.48 487 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 275 M\n1005.48 275 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n1014 63 M\n1005.48 63 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 699 M\n166.26 699 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 635.182 M\n166.26 635.182 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 597.85 M\n166.26 597.85 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 571.363 M\n166.26 571.363 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 550.818 M\n166.26 550.818 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 534.032 M\n166.26 534.032 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 519.839 M\n166.26 519.839 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 507.545 M\n166.26 507.545 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 496.701 M\n166.26 496.701 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 487 M\n166.26 487 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 423.182 M\n166.26 423.182 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 385.85 M\n166.26 385.85 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 359.363 M\n166.26 359.363 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 338.818 M\n166.26 338.818 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 322.032 M\n166.26 322.032 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 307.839 M\n166.26 307.839 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 295.545 M\n166.26 295.545 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 284.701 M\n166.26 284.701 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 275 M\n166.26 275 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 211.182 M\n166.26 211.182 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n1.333 LW\r\nN\r\n162 173.85 M\n166.26 173.85 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 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L\n712.788 635.182 L\n721.394 635.182 L\n730 635.182 L\n738.606 635.182 L\n747.212 635.182 L\n755.818 635.182 L\n764.424 635.182 L\n773.03 635.182 L\n781.636 635.182 L\n790.242 635.182 L\n798.848 635.182 L\n807.455 635.182 L\n816.061 635.182 L\n824.667 635.182 L\n833.273 699 L\n841.879 635.182 L\n850.485 699 L\n859.091 635.182 L\n867.697 699 L\n876.303 635.182 L\n884.909 699 L\n893.515 699 L\n902.121 635.182 L\n910.727 699 L\n919.333 699 L\n927.939 699 L\n936.545 699 L\n945.151 635.182 L\n953.758 699 L\n962.364 699 L\n970.97 699 L\n979.576 699 L\n988.182 699 L\n996.788 699 L\n1005.394 699 L\n1014 699 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 60.75 147.07593] CT\r\nN\r\n0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 63.97727 132.26426] CT\r\nN\r\n/f-1792389198{0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp}def\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 67.20455 150.98877] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 70.43182 147.07593] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 73.65909 164.30431] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 76.88636 144.69386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 80.11364 157.00857] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 83.34091 66.36436] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 86.56818 149.63462] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 89.79545 150.98877] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 93.02273 168.62574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 96.25 160.46409] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 99.47728 157.00857] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 102.70454 186.26272] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 105.93181 179.33429] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 109.15909 168.62574] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 112.38636 206.55688] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 115.61364 194.93971] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 118.84091 200.26197] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 122.06819 206.55688] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 125.29546 206.55688] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 128.52272 206.55688] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 131.75 214.26123] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 134.97727 214.26123] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 138.20455 214.26123] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 141.43182 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 144.65909 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 147.88637 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 151.11364 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 154.3409 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 157.56818 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 160.79545 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 164.02273 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 167.25 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 170.47727 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 173.70455 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 176.93182 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 180.15908 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 183.38636 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 186.61364 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 189.84091 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 193.06819 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 196.29545 224.19386] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 199.52273 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 202.75001 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 205.97729 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 209.20454 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 212.43182 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 215.6591 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 218.88636 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.11364 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 225.3409 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 228.56818 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 231.79546 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 235.02274 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 238.25002 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 241.47727 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 244.70455 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 247.93181 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 251.15909 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 254.38637 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 257.61365 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 260.8409 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 264.06818 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 267.29546 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 270.52272 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 273.75 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 276.97726 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 280.20454 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 283.43182 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.6591 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 289.88638 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 293.11363 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 296.34089 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 299.56817 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 302.79545 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 306.02273 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 309.25001 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 312.47729 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 315.70454 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 318.93182 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 322.15908 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 325.38636 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 328.61364 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 331.84092 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 335.06818 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 338.29546 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 341.52274 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 344.74999 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 347.97727 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 351.20455 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 354.43181 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 357.65909 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 360.88637 262.125] 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CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 325.38636 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 328.61364 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 331.84092 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 335.06818 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 338.29546 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 341.52274 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 344.74999 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 347.97727 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 351.20455 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 354.43181 238.19312] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 357.65909 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 360.88637 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 364.11365 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 367.3409 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 370.56818 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 373.79544 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 377.02272 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 380.25 262.125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\n%%Trailer\r\n%%Pages: 1\r\n%%EOF\r\n" }, { "path": "matlab/acc2023/closedloop_performance/figures/closedloop-snapshots1.eps", "content": "%!PS-Adobe-3.0 EPSF-3.0\r\n%%Creator: (MATLAB, The Mathworks, Inc. Version 9.10.0.1649659 \\(R2021a\\) Update 1. Operating System: Windows 10)\r\n%%Title: figures/closedloop-snapshots1.eps\r\n%%CreationDate: 2023-01-31T03:11:33\r\n%%Pages: (atend)\r\n%%BoundingBox: 1 1 418 301\r\n%%LanguageLevel: 3\r\n%%EndComments\r\n%%BeginProlog\r\n%%BeginResource: procset (Apache XML Graphics Std ProcSet) 1.2 0\r\n%%Version: 1.2 0\r\n%%Copyright: (Copyright 2001-2003,2010 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/bd{bind def}bind def\r\n/ld{load def}bd\r\n/GR/grestore ld\r\n/GS/gsave ld\r\n/RM/rmoveto ld\r\n/C/curveto ld\r\n/t/show ld\r\n/L/lineto ld\r\n/ML/setmiterlimit ld\r\n/CT/concat ld\r\n/f/fill ld\r\n/N/newpath ld\r\n/S/stroke ld\r\n/CC/setcmykcolor ld\r\n/A/ashow ld\r\n/cp/closepath ld\r\n/RC/setrgbcolor ld\r\n/LJ/setlinejoin ld\r\n/GC/setgray ld\r\n/LW/setlinewidth ld\r\n/M/moveto ld\r\n/re {4 2 roll M\r\n1 index 0 rlineto\r\n0 exch rlineto\r\nneg 0 rlineto\r\ncp } bd\r\n/_ctm matrix def\r\n/_tm matrix def\r\n/BT { _ctm currentmatrix pop matrix _tm copy pop 0 0 moveto } bd\r\n/ET { _ctm setmatrix } bd\r\n/iTm { _ctm setmatrix _tm concat } bd\r\n/Tm { _tm astore pop iTm 0 0 moveto } bd\r\n/ux 0.0 def\r\n/uy 0.0 def\r\n/F {\r\n /Tp exch def\r\n /Tf exch def\r\n Tf findfont Tp scalefont setfont\r\n /cf Tf def /cs Tp def\r\n} bd\r\n/ULS {currentpoint /uy exch def /ux exch def} bd\r\n/ULE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add moveto Tcx uy To add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/OLE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs add moveto Tcx uy To add cs add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/SOE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/QT {\r\n/Y22 exch store\r\n/X22 exch store\r\n/Y21 exch store\r\n/X21 exch store\r\ncurrentpoint\r\n/Y21 load 2 mul add 3 div exch\r\n/X21 load 2 mul add 3 div exch\r\n/X21 load 2 mul /X22 load add 3 div\r\n/Y21 load 2 mul /Y22 load add 3 div\r\n/X22 load /Y22 load curveto\r\n} bd\r\n/SSPD {\r\ndup length /d exch dict def\r\n{\r\n/v exch def\r\n/k exch def\r\ncurrentpagedevice k known {\r\n/cpdv currentpagedevice k get def\r\nv cpdv ne {\r\n/upd false def\r\n/nullv v type /nulltype eq def\r\n/nullcpdv cpdv type /nulltype eq def\r\nnullv nullcpdv or\r\n{\r\n/upd true def\r\n} {\r\n/sametype v type cpdv type eq def\r\nsametype {\r\nv type /arraytype eq {\r\n/vlen v length def\r\n/cpdvlen cpdv length def\r\nvlen cpdvlen eq {\r\n0 1 vlen 1 sub {\r\n/i exch def\r\n/obj v i get def\r\n/cpdobj cpdv i get def\r\nobj cpdobj ne {\r\n/upd true def\r\nexit\r\n} if\r\n} for\r\n} {\r\n/upd true def\r\n} ifelse\r\n} {\r\nv type /dicttype eq {\r\nv {\r\n/dv exch def\r\n/dk exch def\r\n/cpddv cpdv dk get def\r\ndv cpddv ne {\r\n/upd true def\r\nexit\r\n} if\r\n} forall\r\n} {\r\n/upd true def\r\n} ifelse\r\n} ifelse\r\n} if\r\n} ifelse\r\nupd true eq {\r\nd k v put\r\n} if\r\n} if\r\n} if\r\n} forall\r\nd length 0 gt {\r\nd setpagedevice\r\n} if\r\n} bd\r\n/RE { % /NewFontName [NewEncodingArray] /FontName RE -\r\n findfont dup length dict begin\r\n {\r\n 1 index /FID ne\r\n {def} {pop pop} ifelse\r\n } forall\r\n /Encoding exch def\r\n /FontName 1 index def\r\n currentdict definefont pop\r\n end\r\n} bind def\r\n%%EndResource\r\n%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0\r\n%%Version: 1.0 0\r\n%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. 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L\n138.091 376.924 L\n153.947 373.686 L\n177.179 366.646 L\n206.631 356.87 L\n241.756 345.523 L\n281.452 332.505 L\n323.644 317.47 L\n365.631 299.287 L\n406.13 278.131 L\n445.652 256.089 L\n485.316 235.728 L\n525.911 219.641 L\n567.219 209.885 L\n608.265 208.904 L\n646.783 213.683 L\n682.712 222.471 L\n716.211 232.457 L\n747.466 242.446 L\n776.582 251.967 L\n803.645 260.829 L\n828.74 268.96 L\n851.963 276.347 L\n873.415 283.005 L\n893.2 288.966 L\n911.422 294.271 L\n928.185 298.965 L\n943.587 303.097 L\n957.725 306.715 L\n970.69 309.867 L\n982.567 312.6 L\n993.436 314.957 L\n1003.374 316.983 L\n1012.451 318.715 L\n1020.735 320.191 L\n1028.287 321.442 L\n1035.166 322.5 L\n1041.427 323.391 L\n1047.119 324.138 L\n1052.29 324.764 L\n1056.985 325.286 L\n1061.243 325.72 L\n1065.104 326.08 L\n1068.6 326.378 L\n1071.766 326.625 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 1 RC\r\n1 LJ\r\n5.333 LW\r\nN\r\n130 377 M\n130 377 L\n138.083 376.795 L\n154.099 379.095 L\n176.473 384.836 L\n204.294 394.103 L\n237.234 406.051 L\n276.694 400.199 L\n312.148 385.834 L\n345.322 368.539 L\n376.091 349.449 L\n408.142 327.27 L\n443.765 301.354 L\n480.264 272.516 L\n517.332 243.227 L\n559.815 223.586 L\n604.621 215.437 L\n648.047 218.959 L\n688.716 228.376 L\n726 239.916 L\n760.857 251.891 L\n793.331 263.567 L\n823.501 274.581 L\n851.446 284.771 L\n877.239 294.073 L\n900.977 302.464 L\n922.773 309.952 L\n942.746 316.568 L\n961.014 322.36 L\n977.695 327.383 L\n992.9 331.702 L\n1006.701 335.386 L\n1019.232 338.499 L\n1030.591 341.108 L\n1040.872 343.278 L\n1050.162 345.069 L\n1058.545 346.539 L\n1066.06 347.73 L\n1072.755 348.685 L\n1078.684 349.441 L\n1083.905 350.033 L\n1088.48 350.493 L\n1092.472 350.847 L\n1095.938 351.118 L\n1098.935 351.322 L\n1101.039 351.444 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n1080 679 M\n1080 461 L\n494 461 L\n494 679 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n2.016 10.368 M\n1.594 10.368 1.594 9.805 QT\n1.609 9.696 1.68 9.43 QT\n1.75 9.165 1.859 9.008 QT\n1.969 8.852 2.141 8.852 QT\n6.25 8.852 6.922 6.227 QT\n12.766 -17.288 L\n11.563 -17.492 8.984 -17.492 QT\n8.563 -17.492 8.563 -18.054 QT\n8.594 -18.163 8.656 -18.429 QT\n8.719 -18.695 8.828 -18.851 QT\n8.938 -19.007 9.109 -19.007 QT\n16.547 -19.007 L\n16.859 -19.007 16.938 -18.726 QT\n26.5 4.008 L\n31.266 -15.038 L\n31.344 -15.507 31.344 -15.695 QT\n31.344 -17.492 28.031 -17.492 QT\n27.609 -17.492 27.609 -18.054 QT\n27.75 -18.601 27.836 -18.804 QT\n27.922 -19.007 28.344 -19.007 QT\n37.547 -19.007 L\n37.969 -19.007 37.969 -18.445 QT\n37.938 -18.335 37.875 -18.07 QT\n37.813 -17.804 37.695 -17.648 QT\n37.578 -17.492 37.422 -17.492 QT\n33.297 -17.492 32.625 -14.867 QT\n26.453 9.993 L\n26.313 10.368 26.016 10.368 QT\n25.484 10.368 L\n25.203 10.368 25.109 10.071 QT\n14.156 -15.945 L\n14.063 -16.21 L\n13.984 -16.304 13.984 -16.335 QT\n8.297 6.415 L\n8.25 6.524 8.227 6.68 QT\n8.203 6.837 8.172 7.055 QT\n8.172 8.165 9.133 8.508 QT\n10.094 8.852 11.531 8.852 QT\n11.953 8.852 11.953 9.43 QT\n11.797 10.008 11.695 10.188 QT\n11.594 10.368 11.234 10.368 QT\n2.016 10.368 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n54.654 4.649 M\n54.295 4.649 54.068 4.383 QT\n53.841 4.118 53.841 3.79 QT\n53.841 3.446 54.068 3.188 QT\n54.295 2.93 54.654 2.93 QT\n81.654 2.93 L\n81.966 2.93 82.201 3.188 QT\n82.435 3.446 82.435 3.79 QT\n82.435 4.118 82.201 4.383 QT\n81.966 4.649 81.654 4.649 QT\n54.654 4.649 L\ncp\n54.654 -3.695 M\n54.295 -3.695 54.068 -3.952 QT\n53.841 -4.21 53.841 -4.554 QT\n53.841 -4.882 54.068 -5.156 QT\n54.295 -5.429 54.654 -5.429 QT\n81.654 -5.429 L\n81.966 -5.429 82.201 -5.156 QT\n82.435 -4.882 82.435 -4.554 QT\n82.435 -4.21 82.201 -3.952 QT\n81.966 -3.695 81.654 -3.695 QT\n54.654 -3.695 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n99.16 10.368 M\n99.16 9.212 L\n99.16 9.102 99.238 8.977 QT\n105.926 1.587 L\n107.441 -0.054 108.379 -1.163 QT\n109.316 -2.273 110.246 -3.718 QT\n111.176 -5.163 111.707 -6.671 QT\n112.238 -8.179 112.238 -9.851 QT\n112.238 -11.617 111.59 -13.226 QT\n110.941 -14.835 109.652 -15.796 QT\n108.363 -16.757 106.535 -16.757 QT\n104.66 -16.757 103.168 -15.632 QT\n101.676 -14.507 101.066 -12.726 QT\n101.238 -12.773 101.535 -12.773 QT\n102.504 -12.773 103.183 -12.124 QT\n103.863 -11.476 103.863 -10.445 QT\n103.863 -9.46 103.183 -8.773 QT\n102.504 -8.085 101.535 -8.085 QT\n100.519 -8.085 99.84 -8.788 QT\n99.16 -9.492 99.16 -10.445 QT\n99.16 -12.054 99.769 -13.476 QT\n100.379 -14.898 101.519 -15.999 QT\n102.66 -17.101 104.105 -17.687 QT\n105.551 -18.273 107.16 -18.273 QT\n109.613 -18.273 111.738 -17.234 QT\n113.863 -16.195 115.098 -14.296 QT\n116.332 -12.398 116.332 -9.851 QT\n116.332 -7.976 115.512 -6.296 QT\n114.691 -4.617 113.418 -3.242 QT\n112.144 -1.867 110.144 -0.124 QT\n108.144 1.618 107.519 2.196 QT\n102.644 6.883 L\n106.785 6.883 L\n109.832 6.883 111.879 6.829 QT\n113.926 6.774 114.051 6.665 QT\n114.551 6.133 115.082 2.712 QT\n116.332 2.712 L\n115.113 10.368 L\n99.16 10.368 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n119.823 3.274 M\n119.823 1.758 L\n133.104 -18.054 L\n133.26 -18.273 133.541 -18.273 QT\n134.182 -18.273 L\n134.666 -18.273 134.666 -17.788 QT\n134.666 1.758 L\n138.885 1.758 L\n138.885 3.274 L\n134.666 3.274 L\n134.666 7.493 L\n134.666 8.368 135.924 8.61 QT\n137.182 8.852 138.838 8.852 QT\n138.838 10.368 L\n126.994 10.368 L\n126.994 8.852 L\n128.651 8.852 129.916 8.61 QT\n131.182 8.368 131.182 7.493 QT\n131.182 3.274 L\n119.823 3.274 L\ncp\n121.244 1.758 M\n131.432 1.758 L\n131.432 -13.46 L\n121.244 1.758 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n144.485 18.118 M\n144.485 17.93 144.657 17.758 QT\n146.204 16.274 147.063 14.321 QT\n147.923 12.368 147.923 10.196 QT\n147.923 9.68 L\n147.235 10.368 146.204 10.368 QT\n145.22 10.368 144.524 9.673 QT\n143.829 8.977 143.829 7.993 QT\n143.829 6.993 144.524 6.321 QT\n145.22 5.649 146.204 5.649 QT\n147.735 5.649 148.384 7.063 QT\n149.032 8.477 149.032 10.196 QT\n149.032 12.587 148.079 14.743 QT\n147.126 16.899 145.391 18.618 QT\n145.22 18.696 145.11 18.696 QT\n144.907 18.696 144.696 18.508 QT\n144.485 18.321 144.485 18.118 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n172.107 18.79 M\n172.107 17.821 172.537 15.977 QT\n172.966 14.133 173.583 12.04 QT\n174.201 9.946 174.638 8.602 QT\n174.81 6.462 174.81 5.368 QT\n174.81 2.602 174.263 0.219 QT\n173.716 -2.163 172.208 -3.773 QT\n170.701 -5.382 167.982 -5.382 QT\n166.56 -5.382 165.138 -4.773 QT\n163.716 -4.163 162.701 -3.07 QT\n161.685 -1.976 161.31 -0.585 QT\n161.232 -0.335 160.966 -0.335 QT\n160.451 -0.335 L\n160.107 -0.335 160.107 -0.726 QT\n160.107 -0.867 L\n160.591 -2.804 161.818 -4.593 QT\n163.044 -6.382 164.826 -7.507 QT\n166.607 -8.632 168.576 -8.632 QT\n170.591 -8.632 172.06 -7.265 QT\n173.529 -5.898 174.412 -3.804 QT\n175.294 -1.71 175.685 0.43 QT\n176.076 2.571 176.076 4.462 QT\n178.232 -1.413 180.919 -6.663 QT\n181.576 -8.038 L\n181.701 -8.179 181.857 -8.179 QT\n182.388 -8.179 L\n182.732 -8.179 182.732 -7.773 QT\n182.732 -7.695 182.638 -7.523 QT\n181.998 -6.179 L\n180.123 -2.46 178.56 1.36 QT\n176.998 5.18 175.888 8.727 QT\n175.716 10.18 175.24 12.696 QT\n174.763 15.212 174.091 17.391 QT\n173.419 19.571 172.748 19.649 QT\n172.107 19.649 172.107 18.79 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n184.567 16.848 M\n184.567 15.785 L\n188.317 15.785 188.317 14.848 QT\n188.317 -0.902 L\n186.77 -0.152 184.395 -0.152 QT\n184.395 -1.215 L\n188.067 -1.215 189.942 -3.137 QT\n190.363 -3.137 L\n190.473 -3.137 190.567 -3.051 QT\n190.66 -2.965 190.66 -2.871 QT\n190.66 14.848 L\n190.66 15.785 194.41 15.785 QT\n194.41 16.848 L\n184.567 16.848 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n214.299 4.649 M\n213.94 4.649 213.714 4.383 QT\n213.487 4.118 213.487 3.79 QT\n213.487 3.446 213.714 3.188 QT\n213.94 2.93 214.299 2.93 QT\n241.299 2.93 L\n241.612 2.93 241.846 3.188 QT\n242.081 3.446 242.081 3.79 QT\n242.081 4.118 241.846 4.383 QT\n241.612 4.649 241.299 4.649 QT\n214.299 4.649 L\ncp\n214.299 -3.695 M\n213.94 -3.695 213.714 -3.952 QT\n213.487 -4.21 213.487 -4.554 QT\n213.487 -4.882 213.714 -5.156 QT\n213.94 -5.429 214.299 -5.429 QT\n241.299 -5.429 L\n241.612 -5.429 241.846 -5.156 QT\n242.081 -4.882 242.081 -4.554 QT\n242.081 -4.21 241.846 -3.952 QT\n241.612 -3.695 241.299 -3.695 QT\n214.299 -3.695 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n267.415 11.305 M\n262.149 11.305 260.251 6.969 QT\n258.352 2.633 258.352 -3.335 QT\n258.352 -7.085 259.032 -10.382 QT\n259.712 -13.679 261.735 -15.976 QT\n263.759 -18.273 267.415 -18.273 QT\n270.243 -18.273 272.056 -16.89 QT\n273.868 -15.507 274.806 -13.312 QT\n275.743 -11.117 276.095 -8.609 QT\n276.446 -6.101 276.446 -3.335 QT\n276.446 0.352 275.759 3.579 QT\n275.071 6.805 273.079 9.055 QT\n271.087 11.305 267.415 11.305 QT\ncp\n267.415 10.196 M\n269.806 10.196 270.985 7.743 QT\n272.165 5.29 272.438 2.305 QT\n272.712 -0.679 272.712 -4.038 QT\n272.712 -7.273 272.438 -9.999 QT\n272.165 -12.726 271.001 -14.945 QT\n269.837 -17.163 267.415 -17.163 QT\n264.977 -17.163 263.806 -14.937 QT\n262.634 -12.71 262.36 -9.992 QT\n262.087 -7.273 262.087 -4.038 QT\n262.087 -1.648 262.196 0.477 QT\n262.306 2.602 262.813 4.86 QT\n263.321 7.118 264.446 8.657 QT\n265.571 10.196 267.415 10.196 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n281.874 7.993 M\n281.874 7.024 282.593 6.337 QT\n283.312 5.649 284.249 5.649 QT\n284.843 5.649 285.406 5.962 QT\n285.968 6.274 286.281 6.844 QT\n286.593 7.415 286.593 7.993 QT\n286.593 8.946 285.906 9.657 QT\n285.218 10.368 284.249 10.368 QT\n283.312 10.368 282.593 9.657 QT\n281.874 8.946 281.874 7.993 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n291.468 3.274 M\n291.468 1.758 L\n304.749 -18.054 L\n304.906 -18.273 305.187 -18.273 QT\n305.828 -18.273 L\n306.312 -18.273 306.312 -17.788 QT\n306.312 1.758 L\n310.531 1.758 L\n310.531 3.274 L\n306.312 3.274 L\n306.312 7.493 L\n306.312 8.368 307.57 8.61 QT\n308.828 8.852 310.484 8.852 QT\n310.484 10.368 L\n298.64 10.368 L\n298.64 8.852 L\n300.296 8.852 301.562 8.61 QT\n302.828 8.368 302.828 7.493 QT\n302.828 3.274 L\n291.468 3.274 L\ncp\n292.89 1.758 M\n303.078 1.758 L\n303.078 -13.46 L\n292.89 1.758 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n316.131 18.118 M\n316.131 17.93 316.303 17.758 QT\n317.849 16.274 318.709 14.321 QT\n319.568 12.368 319.568 10.196 QT\n319.568 9.68 L\n318.881 10.368 317.849 10.368 QT\n316.865 10.368 316.17 9.673 QT\n315.474 8.977 315.474 7.993 QT\n315.474 6.993 316.17 6.321 QT\n316.865 5.649 317.849 5.649 QT\n319.381 5.649 320.029 7.063 QT\n320.678 8.477 320.678 10.196 QT\n320.678 12.587 319.724 14.743 QT\n318.771 16.899 317.037 18.618 QT\n316.865 18.696 316.756 18.696 QT\n316.553 18.696 316.342 18.508 QT\n316.131 18.321 316.131 18.118 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 184.0765] CT\r\nN\r\n343.752 18.79 M\n343.752 17.821 344.182 15.977 QT\n344.612 14.133 345.229 12.04 QT\n345.846 9.946 346.284 8.602 QT\n346.456 6.462 346.456 5.368 QT\n346.456 2.602 345.909 0.219 QT\n345.362 -2.163 343.854 -3.773 QT\n342.346 -5.382 339.627 -5.382 QT\n338.206 -5.382 336.784 -4.773 QT\n335.362 -4.163 334.346 -3.07 QT\n333.331 -1.976 332.956 -0.585 QT\n332.877 -0.335 332.612 -0.335 QT\n332.096 -0.335 L\n331.752 -0.335 331.752 -0.726 QT\n331.752 -0.867 L\n332.237 -2.804 333.463 -4.593 QT\n334.69 -6.382 336.471 -7.507 QT\n338.252 -8.632 340.221 -8.632 QT\n342.237 -8.632 343.706 -7.265 QT\n345.174 -5.898 346.057 -3.804 QT\n346.94 -1.71 347.331 0.43 QT\n347.721 2.571 347.721 4.462 QT\n349.877 -1.413 352.565 -6.663 QT\n353.221 -8.038 L\n353.346 -8.179 353.502 -8.179 QT\n354.034 -8.179 L\n354.377 -8.179 354.377 -7.773 QT\n354.377 -7.695 354.284 -7.523 QT\n353.643 -6.179 L\n351.768 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267.415 10.196 QT\ncp}def\r\nf2141106481\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f-689209901{281.874 7.993 M\n281.874 7.024 282.593 6.337 QT\n283.312 5.649 284.249 5.649 QT\n284.843 5.649 285.406 5.962 QT\n285.968 6.274 286.281 6.844 QT\n286.593 7.415 286.593 7.993 QT\n286.593 8.946 285.906 9.657 QT\n285.218 10.368 284.249 10.368 QT\n283.312 10.368 282.593 9.657 QT\n281.874 8.946 281.874 7.993 QT\ncp}def\r\nf-689209901\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f-830940872{301.015 11.305 M\n298.343 11.305 296.562 9.891 QT\n294.781 8.477 293.804 6.219 QT\n292.828 3.962 292.453 1.485 QT\n292.078 -0.992 292.078 -3.538 QT\n292.078 -6.929 293.398 -10.351 QT\n294.718 -13.773 297.288 -16.023 QT\n299.859 -18.273 303.39 -18.273 QT\n304.859 -18.273 306.132 -17.718 QT\n307.406 -17.163 308.124 -16.077 QT\n308.843 -14.992 308.843 -13.46 QT\n308.843 -12.585 308.249 -11.984 QT\n307.656 -11.382 306.765 -11.382 QT\n305.921 -11.382 305.32 -11.992 QT\n304.718 -12.601 304.718 -13.46 QT\n304.718 -14.304 305.32 -14.913 QT\n305.921 -15.523 306.765 -15.523 QT\n306.999 -15.523 L\n306.453 -16.304 305.453 -16.671 QT\n304.453 -17.038 303.39 -17.038 QT\n302.093 -17.038 300.984 -16.468 QT\n299.874 -15.898 298.999 -14.929 QT\n298.124 -13.96 297.531 -12.796 QT\n296.937 -11.632 296.617 -10.14 QT\n296.296 -8.648 296.21 -7.351 QT\n296.124 -6.054 296.124 -4.085 QT\n296.874 -5.835 298.265 -6.96 QT\n299.656 -8.085 301.39 -8.085 QT\n303.296 -8.085 304.874 -7.312 QT\n306.453 -6.538 307.585 -5.163 QT\n308.718 -3.788 309.32 -2.023 QT\n309.921 -0.257 309.921 1.555 QT\n309.921 4.071 308.796 6.344 QT\n307.671 8.618 305.632 9.962 QT\n303.593 11.305 301.015 11.305 QT\ncp\n301.015 9.946 M\n302.671 9.946 303.679 9.188 QT\n304.687 8.43 305.163 7.18 QT\n305.64 5.93 305.749 4.665 QT\n305.859 3.399 305.859 1.555 QT\n305.859 -0.882 305.632 -2.609 QT\n305.406 -4.335 304.374 -5.648 QT\n303.343 -6.96 301.218 -6.96 QT\n299.484 -6.96 298.359 -5.781 QT\n297.234 -4.601 296.718 -2.804 QT\n296.203 -1.007 296.203 0.649 QT\n296.203 1.212 296.249 1.508 QT\n296.249 1.571 296.242 1.61 QT\n296.234 1.649 296.203 1.712 QT\n296.203 3.571 296.585 5.454 QT\n296.968 7.337 298.038 8.641 QT\n299.109 9.946 301.015 9.946 QT\ncp}def\r\nf-830940872\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f1312883481{316.131 18.118 M\n316.131 17.93 316.303 17.758 QT\n317.849 16.274 318.709 14.321 QT\n319.568 12.368 319.568 10.196 QT\n319.568 9.68 L\n318.881 10.368 317.849 10.368 QT\n316.865 10.368 316.17 9.673 QT\n315.474 8.977 315.474 7.993 QT\n315.474 6.993 316.17 6.321 QT\n316.865 5.649 317.849 5.649 QT\n319.381 5.649 320.029 7.063 QT\n320.678 8.477 320.678 10.196 QT\n320.678 12.587 319.724 14.743 QT\n318.771 16.899 317.037 18.618 QT\n316.865 18.696 316.756 18.696 QT\n316.553 18.696 316.342 18.508 QT\n316.131 18.321 316.131 18.118 QT\ncp}def\r\nf1312883481\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f1797501460{343.752 18.79 M\n343.752 17.821 344.182 15.977 QT\n344.612 14.133 345.229 12.04 QT\n345.846 9.946 346.284 8.602 QT\n346.456 6.462 346.456 5.368 QT\n346.456 2.602 345.909 0.219 QT\n345.362 -2.163 343.854 -3.773 QT\n342.346 -5.382 339.627 -5.382 QT\n338.206 -5.382 336.784 -4.773 QT\n335.362 -4.163 334.346 -3.07 QT\n333.331 -1.976 332.956 -0.585 QT\n332.877 -0.335 332.612 -0.335 QT\n332.096 -0.335 L\n331.752 -0.335 331.752 -0.726 QT\n331.752 -0.867 L\n332.237 -2.804 333.463 -4.593 QT\n334.69 -6.382 336.471 -7.507 QT\n338.252 -8.632 340.221 -8.632 QT\n342.237 -8.632 343.706 -7.265 QT\n345.174 -5.898 346.057 -3.804 QT\n346.94 -1.71 347.331 0.43 QT\n347.721 2.571 347.721 4.462 QT\n349.877 -1.413 352.565 -6.663 QT\n353.221 -8.038 L\n353.346 -8.179 353.502 -8.179 QT\n354.034 -8.179 L\n354.377 -8.179 354.377 -7.773 QT\n354.377 -7.695 354.284 -7.523 QT\n353.643 -6.179 L\n351.768 -2.46 350.206 1.36 QT\n348.643 5.18 347.534 8.727 QT\n347.362 10.18 346.885 12.696 QT\n346.409 15.212 345.737 17.391 QT\n345.065 19.571 344.393 19.649 QT\n343.752 19.649 343.752 18.79 QT\ncp}def\r\nf1797501460\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f1851169393{354.931 16.848 M\n354.931 16.035 L\n354.931 15.973 354.978 15.879 QT\n359.65 10.723 L\n360.697 9.582 361.353 8.809 QT\n362.009 8.035 362.657 7.02 QT\n363.306 6.004 363.681 4.957 QT\n364.056 3.91 364.056 2.738 QT\n364.056 1.504 363.603 0.387 QT\n363.15 -0.73 362.243 -1.402 QT\n361.337 -2.074 360.072 -2.074 QT\n358.759 -2.074 357.72 -1.293 QT\n356.681 -0.512 356.259 0.738 QT\n356.368 0.707 356.587 0.707 QT\n357.259 0.707 357.736 1.16 QT\n358.212 1.613 358.212 2.332 QT\n358.212 3.02 357.736 3.496 QT\n357.259 3.973 356.587 3.973 QT\n355.884 3.973 355.407 3.481 QT\n354.931 2.988 354.931 2.332 QT\n354.931 1.207 355.353 0.215 QT\n355.775 -0.777 356.572 -1.543 QT\n357.368 -2.308 358.376 -2.723 QT\n359.384 -3.137 360.509 -3.137 QT\n362.228 -3.137 363.704 -2.41 QT\n365.181 -1.683 366.048 -0.355 QT\n366.915 0.973 366.915 2.738 QT\n366.915 4.051 366.337 5.223 QT\n365.759 6.395 364.868 7.348 QT\n363.978 8.301 362.587 9.52 QT\n361.197 10.738 360.759 11.145 QT\n357.353 14.41 L\n360.243 14.41 L\n362.368 14.41 363.798 14.379 QT\n365.228 14.348 365.306 14.27 QT\n365.665 13.895 366.025 11.504 QT\n366.915 11.504 L\n366.056 16.848 L\n354.931 16.848 L\ncp}def\r\nf1851169393\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f464326539{385.945 4.649 M\n385.586 4.649 385.359 4.383 QT\n385.133 4.118 385.133 3.79 QT\n385.133 3.446 385.359 3.188 QT\n385.586 2.93 385.945 2.93 QT\n412.945 2.93 L\n413.258 2.93 413.492 3.188 QT\n413.726 3.446 413.726 3.79 QT\n413.726 4.118 413.492 4.383 QT\n413.258 4.649 412.945 4.649 QT\n385.945 4.649 L\ncp\n385.945 -3.695 M\n385.586 -3.695 385.359 -3.952 QT\n385.133 -4.21 385.133 -4.554 QT\n385.133 -4.882 385.359 -5.156 QT\n385.586 -5.429 385.945 -5.429 QT\n412.945 -5.429 L\n413.258 -5.429 413.492 -5.156 QT\n413.726 -4.882 413.726 -4.554 QT\n413.726 -4.21 413.492 -3.952 QT\n413.258 -3.695 412.945 -3.695 QT\n385.945 -3.695 L\ncp}def\r\nf464326539\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f769333717{439.06 11.305 M\n433.795 11.305 431.896 6.969 QT\n429.998 2.633 429.998 -3.335 QT\n429.998 -7.085 430.678 -10.382 QT\n431.357 -13.679 433.381 -15.976 QT\n435.404 -18.273 439.06 -18.273 QT\n441.889 -18.273 443.701 -16.89 QT\n445.514 -15.507 446.451 -13.312 QT\n447.389 -11.117 447.74 -8.609 QT\n448.092 -6.101 448.092 -3.335 QT\n448.092 0.352 447.404 3.579 QT\n446.717 6.805 444.724 9.055 QT\n442.732 11.305 439.06 11.305 QT\ncp\n439.06 10.196 M\n441.451 10.196 442.631 7.743 QT\n443.81 5.29 444.084 2.305 QT\n444.357 -0.679 444.357 -4.038 QT\n444.357 -7.273 444.084 -9.999 QT\n443.81 -12.726 442.646 -14.945 QT\n441.482 -17.163 439.06 -17.163 QT\n436.623 -17.163 435.451 -14.937 QT\n434.279 -12.71 434.006 -9.992 QT\n433.732 -7.273 433.732 -4.038 QT\n433.732 -1.648 433.842 0.477 QT\n433.951 2.602 434.459 4.86 QT\n434.967 7.118 436.092 8.657 QT\n437.217 10.196 439.06 10.196 QT\ncp}def\r\nf769333717\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f187970572{453.52 7.993 M\n453.52 7.024 454.239 6.337 QT\n454.957 5.649 455.895 5.649 QT\n456.489 5.649 457.051 5.962 QT\n457.614 6.274 457.926 6.844 QT\n458.239 7.415 458.239 7.993 QT\n458.239 8.946 457.551 9.657 QT\n456.864 10.368 455.895 10.368 QT\n454.957 10.368 454.239 9.657 QT\n453.52 8.946 453.52 7.993 QT\ncp}def\r\nf187970572\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 203.85883] CT\r\nN\r\n/f-1662342170{472.661 11.305 M\n469.989 11.305 468.207 9.891 QT\n466.426 8.477 465.45 6.219 QT\n464.473 3.962 464.098 1.485 QT\n463.723 -0.992 463.723 -3.538 QT\n463.723 -6.929 465.043 -10.351 QT\n466.364 -13.773 468.934 -16.023 QT\n471.504 -18.273 475.036 -18.273 QT\n476.504 -18.273 477.778 -17.718 QT\n479.051 -17.163 479.77 -16.077 QT\n480.489 -14.992 480.489 -13.46 QT\n480.489 -12.585 479.895 -11.984 QT\n479.301 -11.382 478.411 -11.382 QT\n477.567 -11.382 476.965 -11.992 QT\n476.364 -12.601 476.364 -13.46 QT\n476.364 -14.304 476.965 -14.913 QT\n477.567 -15.523 478.411 -15.523 QT\n478.645 -15.523 L\n478.098 -16.304 477.098 -16.671 QT\n476.098 -17.038 475.036 -17.038 QT\n473.739 -17.038 472.629 -16.468 QT\n471.52 -15.898 470.645 -14.929 QT\n469.77 -13.96 469.176 -12.796 QT\n468.582 -11.632 468.262 -10.14 QT\n467.942 -8.648 467.856 -7.351 QT\n467.77 -6.054 467.77 -4.085 QT\n468.52 -5.835 469.911 -6.96 QT\n471.301 -8.085 473.036 -8.085 QT\n474.942 -8.085 476.52 -7.312 QT\n478.098 -6.538 479.231 -5.163 QT\n480.364 -3.788 480.965 -2.023 QT\n481.567 -0.257 481.567 1.555 QT\n481.567 4.071 480.442 6.344 QT\n479.317 8.618 477.278 9.962 QT\n475.239 11.305 472.661 11.305 QT\ncp\n472.661 9.946 M\n474.317 9.946 475.325 9.188 QT\n476.332 8.43 476.809 7.18 QT\n477.286 5.93 477.395 4.665 QT\n477.504 3.399 477.504 1.555 QT\n477.504 -0.882 477.278 -2.609 QT\n477.051 -4.335 476.02 -5.648 QT\n474.989 -6.96 472.864 -6.96 QT\n471.129 -6.96 470.004 -5.781 QT\n468.879 -4.601 468.364 -2.804 QT\n467.848 -1.007 467.848 0.649 QT\n467.848 1.212 467.895 1.508 QT\n467.895 1.571 467.887 1.61 QT\n467.879 1.649 467.848 1.712 QT\n467.848 3.571 468.231 5.454 QT\n468.614 7.337 469.684 8.641 QT\n470.754 9.946 472.661 9.946 QT\ncp}def\r\nf-1662342170\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n5.333 LW\r\nN\r\n502.007 543.624 M\n582.074 543.624 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-1698885829\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-1758447675\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\n/f1717920742{101.004 10.368 M\n101.004 8.852 L\n106.379 8.852 106.379 7.493 QT\n106.379 -15.085 L\n104.16 -14.007 100.754 -14.007 QT\n100.754 -15.523 L\n106.019 -15.523 108.707 -18.273 QT\n109.316 -18.273 L\n109.473 -18.273 109.605 -18.156 QT\n109.738 -18.038 109.738 -17.898 QT\n109.738 7.493 L\n109.738 8.852 115.113 8.852 QT\n115.113 10.368 L\n101.004 10.368 L\ncp}def\r\nf1717920742\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\n/f-1038411071{129.369 11.305 M\n126.698 11.305 124.916 9.891 QT\n123.135 8.477 122.159 6.219 QT\n121.182 3.962 120.807 1.485 QT\n120.432 -0.992 120.432 -3.538 QT\n120.432 -6.929 121.752 -10.351 QT\n123.073 -13.773 125.643 -16.023 QT\n128.213 -18.273 131.744 -18.273 QT\n133.213 -18.273 134.487 -17.718 QT\n135.76 -17.163 136.479 -16.077 QT\n137.198 -14.992 137.198 -13.46 QT\n137.198 -12.585 136.604 -11.984 QT\n136.01 -11.382 135.119 -11.382 QT\n134.276 -11.382 133.674 -11.992 QT\n133.073 -12.601 133.073 -13.46 QT\n133.073 -14.304 133.674 -14.913 QT\n134.276 -15.523 135.119 -15.523 QT\n135.354 -15.523 L\n134.807 -16.304 133.807 -16.671 QT\n132.807 -17.038 131.744 -17.038 QT\n130.448 -17.038 129.338 -16.468 QT\n128.229 -15.898 127.354 -14.929 QT\n126.479 -13.96 125.885 -12.796 QT\n125.291 -11.632 124.971 -10.14 QT\n124.651 -8.648 124.565 -7.351 QT\n124.479 -6.054 124.479 -4.085 QT\n125.229 -5.835 126.619 -6.96 QT\n128.01 -8.085 129.744 -8.085 QT\n131.651 -8.085 133.229 -7.312 QT\n134.807 -6.538 135.94 -5.163 QT\n137.073 -3.788 137.674 -2.023 QT\n138.276 -0.257 138.276 1.555 QT\n138.276 4.071 137.151 6.344 QT\n136.026 8.618 133.987 9.962 QT\n131.948 11.305 129.369 11.305 QT\ncp\n129.369 9.946 M\n131.026 9.946 132.034 9.188 QT\n133.041 8.43 133.518 7.18 QT\n133.994 5.93 134.104 4.665 QT\n134.213 3.399 134.213 1.555 QT\n134.213 -0.882 133.987 -2.609 QT\n133.76 -4.335 132.729 -5.648 QT\n131.698 -6.96 129.573 -6.96 QT\n127.838 -6.96 126.713 -5.781 QT\n125.588 -4.601 125.073 -2.804 QT\n124.557 -1.007 124.557 0.649 QT\n124.557 1.212 124.604 1.508 QT\n124.604 1.571 124.596 1.61 QT\n124.588 1.649 124.557 1.712 QT\n124.557 3.571 124.94 5.454 QT\n125.323 7.337 126.393 8.641 QT\n127.463 9.946 129.369 9.946 QT\ncp}def\r\nf-1038411071\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-1779188984\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf2034987691\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-173962665\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-988291271\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf2141106481\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\nf-689209901\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 220.52946 223.64117] CT\r\nN\r\n/f2096529930{291.468 3.274 M\n291.468 1.758 L\n304.749 -18.054 L\n304.906 -18.273 305.187 -18.273 QT\n305.828 -18.273 L\n306.312 -18.273 306.312 -17.788 QT\n306.312 1.758 L\n310.531 1.758 L\n310.531 3.274 L\n306.312 3.274 L\n306.312 7.493 L\n306.312 8.368 307.57 8.61 QT\n308.828 8.852 310.484 8.852 QT\n310.484 10.368 L\n298.64 10.368 L\n298.64 8.852 L\n300.296 8.852 301.562 8.61 QT\n302.828 8.368 302.828 7.493 QT\n302.828 3.274 L\n291.468 3.274 L\ncp\n292.89 1.758 M\n303.078 1.758 L\n303.078 -13.46 L\n292.89 1.758 L\ncp}def\r\nf2096529930\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 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QT\n119.19 -16.097 L\n119.611 -16.097 119.611 -15.659 QT\n119.611 1.606 L\n123.346 1.606 L\n123.346 2.95 L\n119.611 2.95 L\n119.611 6.669 L\n119.611 7.45 120.729 7.661 QT\n121.846 7.872 123.315 7.872 QT\n123.315 9.216 L\n112.846 9.216 L\n112.846 7.872 L\n114.315 7.872 115.424 7.661 QT\n116.533 7.45 116.533 6.669 QT\n116.533 2.95 L\n106.502 2.95 L\ncp\n107.752 1.606 M\n116.752 1.606 L\n116.752 -11.847 L\n107.752 1.606 L\ncp}def\r\nf-819923682\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1106082528{128.405 16.06 M\n128.405 15.903 128.561 15.747 QT\n129.921 14.435 130.686 12.708 QT\n131.452 10.981 131.452 9.06 QT\n131.452 8.606 L\n130.843 9.216 129.921 9.216 QT\n129.061 9.216 128.444 8.606 QT\n127.827 7.997 127.827 7.122 QT\n127.827 6.231 128.444 5.638 QT\n129.061 5.044 129.921 5.044 QT\n131.28 5.044 131.858 6.294 QT\n132.436 7.544 132.436 9.06 QT\n132.436 11.185 131.593 13.083 QT\n130.749 14.981 129.202 16.513 QT\n129.061 16.575 128.968 16.575 QT\n128.78 16.575 128.593 16.411 QT\n128.405 16.247 128.405 16.06 QT\ncp}def\r\nf-1106082528\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-2022938735{152.909 16.653 M\n152.909 15.81 153.292 14.177 QT\n153.675 12.544 154.222 10.692 QT\n154.769 8.841 155.159 7.653 QT\n155.3 5.763 155.3 4.794 QT\n155.3 2.356 154.823 0.247 QT\n154.347 -1.862 153.011 -3.284 QT\n151.675 -4.706 149.284 -4.706 QT\n148.019 -4.706 146.761 -4.167 QT\n145.503 -3.628 144.605 -2.659 QT\n143.706 -1.69 143.378 -0.472 QT\n143.3 -0.253 143.081 -0.253 QT\n142.612 -0.253 L\n142.315 -0.253 142.315 -0.581 QT\n142.315 -0.706 L\n142.753 -2.425 143.831 -4.011 QT\n144.909 -5.597 146.48 -6.589 QT\n148.05 -7.581 149.8 -7.581 QT\n151.581 -7.581 152.878 -6.37 QT\n154.175 -5.159 154.956 -3.308 QT\n155.737 -1.456 156.081 0.435 QT\n156.425 2.325 156.425 3.997 QT\n158.331 -1.19 160.706 -5.831 QT\n161.284 -7.05 L\n161.394 -7.175 161.534 -7.175 QT\n162.003 -7.175 L\n162.3 -7.175 162.3 -6.815 QT\n162.3 -6.737 162.222 -6.597 QT\n161.659 -5.409 L\n160.003 -2.128 158.62 1.255 QT\n157.237 4.638 156.253 7.763 QT\n156.112 9.044 155.69 11.271 QT\n155.269 13.497 154.675 15.419 QT\n154.081 17.341 153.487 17.419 QT\n152.909 17.419 152.909 16.653 QT\ncp}def\r\nf-2022938735\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1608834715{164.087 14.976 M\n164.087 14.023 L\n167.462 14.023 167.462 13.163 QT\n167.462 -1.009 L\n166.071 -0.337 163.931 -0.337 QT\n163.931 -1.274 L\n167.243 -1.274 168.931 -3.009 QT\n169.306 -3.009 L\n169.399 -3.009 169.485 -2.938 QT\n169.571 -2.868 169.571 -2.774 QT\n169.571 13.163 L\n169.571 14.023 172.946 14.023 QT\n172.946 14.976 L\n164.087 14.976 L\ncp}def\r\nf-1608834715\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1300927600{190.473 4.169 M\n190.16 4.169 189.957 3.927 QT\n189.754 3.685 189.754 3.403 QT\n189.754 3.091 189.957 2.872 QT\n190.16 2.653 190.473 2.653 QT\n214.332 2.653 L\n214.613 2.653 214.816 2.872 QT\n215.02 3.091 215.02 3.403 QT\n215.02 3.685 214.816 3.927 QT\n214.613 4.169 214.332 4.169 QT\n190.473 4.169 L\ncp\n190.473 -3.222 M\n190.16 -3.222 189.957 -3.44 QT\n189.754 -3.659 189.754 -3.972 QT\n189.754 -4.253 189.957 -4.495 QT\n190.16 -4.737 190.473 -4.737 QT\n214.332 -4.737 L\n214.613 -4.737 214.816 -4.495 QT\n215.02 -4.253 215.02 -3.972 QT\n215.02 -3.659 214.816 -3.44 QT\n214.613 -3.222 214.332 -3.222 QT\n190.473 -3.222 L\ncp}def\r\nf-1300927600\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1997795938{237.647 10.044 M\n232.99 10.044 231.311 6.216 QT\n229.631 2.388 229.631 -2.894 QT\n229.631 -6.206 230.232 -9.12 QT\n230.834 -12.034 232.623 -14.065 QT\n234.412 -16.097 237.647 -16.097 QT\n240.147 -16.097 241.748 -14.87 QT\n243.35 -13.644 244.178 -11.706 QT\n245.006 -9.769 245.318 -7.55 QT\n245.631 -5.331 245.631 -2.894 QT\n245.631 0.372 245.022 3.216 QT\n244.412 6.06 242.654 8.052 QT\n240.897 10.044 237.647 10.044 QT\ncp\n237.647 9.06 M\n239.756 9.06 240.795 6.896 QT\n241.834 4.731 242.076 2.091 QT\n242.318 -0.55 242.318 -3.519 QT\n242.318 -6.378 242.076 -8.784 QT\n241.834 -11.19 240.811 -13.151 QT\n239.787 -15.112 237.647 -15.112 QT\n235.49 -15.112 234.451 -13.144 QT\n233.412 -11.175 233.17 -8.776 QT\n232.928 -6.378 232.928 -3.519 QT\n232.928 -1.394 233.029 0.481 QT\n233.131 2.356 233.576 4.349 QT\n234.022 6.341 235.022 7.7 QT\n236.022 9.06 237.647 9.06 QT\ncp}def\r\nf-1997795938\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f1065174994{250.534 7.122 M\n250.534 6.263 251.167 5.653 QT\n251.8 5.044 252.628 5.044 QT\n253.159 5.044 253.659 5.317 QT\n254.159 5.591 254.433 6.099 QT\n254.706 6.606 254.706 7.122 QT\n254.706 7.95 254.097 8.583 QT\n253.487 9.216 252.628 9.216 QT\n251.8 9.216 251.167 8.583 QT\n250.534 7.95 250.534 7.122 QT\ncp}def\r\nf1065174994\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f1247873414{259.076 2.95 M\n259.076 1.606 L\n270.81 -15.909 L\n270.951 -16.097 271.201 -16.097 QT\n271.763 -16.097 L\n272.185 -16.097 272.185 -15.659 QT\n272.185 1.606 L\n275.92 1.606 L\n275.92 2.95 L\n272.185 2.95 L\n272.185 6.669 L\n272.185 7.45 273.302 7.661 QT\n274.42 7.872 275.888 7.872 QT\n275.888 9.216 L\n265.42 9.216 L\n265.42 7.872 L\n266.888 7.872 267.998 7.661 QT\n269.107 7.45 269.107 6.669 QT\n269.107 2.95 L\n259.076 2.95 L\ncp\n260.326 1.606 M\n269.326 1.606 L\n269.326 -11.847 L\n260.326 1.606 L\ncp}def\r\nf1247873414\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1867049319{280.979 16.06 M\n280.979 15.903 281.135 15.747 QT\n282.495 14.435 283.26 12.708 QT\n284.026 10.981 284.026 9.06 QT\n284.026 8.606 L\n283.416 9.216 282.495 9.216 QT\n281.635 9.216 281.018 8.606 QT\n280.401 7.997 280.401 7.122 QT\n280.401 6.231 281.018 5.638 QT\n281.635 5.044 282.495 5.044 QT\n283.854 5.044 284.432 6.294 QT\n285.01 7.544 285.01 9.06 QT\n285.01 11.185 284.166 13.083 QT\n283.323 14.981 281.776 16.513 QT\n281.635 16.575 281.541 16.575 QT\n281.354 16.575 281.166 16.411 QT\n280.979 16.247 280.979 16.06 QT\ncp}def\r\nf-1867049319\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1823982240{305.483 16.653 M\n305.483 15.81 305.866 14.177 QT\n306.249 12.544 306.796 10.692 QT\n307.342 8.841 307.733 7.653 QT\n307.874 5.763 307.874 4.794 QT\n307.874 2.356 307.397 0.247 QT\n306.921 -1.862 305.585 -3.284 QT\n304.249 -4.706 301.858 -4.706 QT\n300.592 -4.706 299.335 -4.167 QT\n298.077 -3.628 297.178 -2.659 QT\n296.28 -1.69 295.952 -0.472 QT\n295.874 -0.253 295.655 -0.253 QT\n295.186 -0.253 L\n294.889 -0.253 294.889 -0.581 QT\n294.889 -0.706 L\n295.327 -2.425 296.405 -4.011 QT\n297.483 -5.597 299.053 -6.589 QT\n300.624 -7.581 302.374 -7.581 QT\n304.155 -7.581 305.452 -6.37 QT\n306.749 -5.159 307.53 -3.308 QT\n308.311 -1.456 308.655 0.435 QT\n308.999 2.325 308.999 3.997 QT\n310.905 -1.19 313.28 -5.831 QT\n313.858 -7.05 L\n313.967 -7.175 314.108 -7.175 QT\n314.577 -7.175 L\n314.874 -7.175 314.874 -6.815 QT\n314.874 -6.737 314.796 -6.597 QT\n314.233 -5.409 L\n312.577 -2.128 311.194 1.255 QT\n309.811 4.638 308.827 7.763 QT\n308.686 9.044 308.264 11.271 QT\n307.842 13.497 307.249 15.419 QT\n306.655 17.341 306.061 17.419 QT\n305.483 17.419 305.483 16.653 QT\ncp}def\r\nf-1823982240\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1287452466{315.505 14.976 M\n315.505 14.257 L\n315.505 14.179 315.551 14.101 QT\n319.755 9.46 L\n320.708 8.429 321.294 7.734 QT\n321.88 7.038 322.465 6.132 QT\n323.051 5.226 323.387 4.28 QT\n323.723 3.335 323.723 2.273 QT\n323.723 1.179 323.309 0.163 QT\n322.895 -0.852 322.09 -1.454 QT\n321.286 -2.055 320.13 -2.055 QT\n318.958 -2.055 318.02 -1.352 QT\n317.083 -0.649 316.708 0.476 QT\n316.817 0.445 316.989 0.445 QT\n317.598 0.445 318.028 0.851 QT\n318.458 1.257 318.458 1.913 QT\n318.458 2.523 318.028 2.952 QT\n317.598 3.382 316.989 3.382 QT\n316.364 3.382 315.934 2.945 QT\n315.505 2.507 315.505 1.913 QT\n315.505 0.898 315.887 0.007 QT\n316.27 -0.884 316.989 -1.579 QT\n317.708 -2.274 318.614 -2.641 QT\n319.52 -3.009 320.536 -3.009 QT\n322.067 -3.009 323.403 -2.352 QT\n324.739 -1.696 325.512 -0.509 QT\n326.286 0.679 326.286 2.273 QT\n326.286 3.46 325.77 4.515 QT\n325.255 5.57 324.458 6.429 QT\n323.661 7.288 322.403 8.382 QT\n321.145 9.476 320.755 9.851 QT\n317.692 12.788 L\n320.286 12.788 L\n322.208 12.788 323.489 12.757 QT\n324.77 12.726 324.848 12.648 QT\n325.176 12.32 325.505 10.163 QT\n326.286 10.163 L\n325.52 14.976 L\n315.505 14.976 L\ncp}def\r\nf-1287452466\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-172155240{343.047 4.169 M\n342.734 4.169 342.531 3.927 QT\n342.328 3.685 342.328 3.403 QT\n342.328 3.091 342.531 2.872 QT\n342.734 2.653 343.047 2.653 QT\n366.906 2.653 L\n367.187 2.653 367.39 2.872 QT\n367.593 3.091 367.593 3.403 QT\n367.593 3.685 367.39 3.927 QT\n367.187 4.169 366.906 4.169 QT\n343.047 4.169 L\ncp\n343.047 -3.222 M\n342.734 -3.222 342.531 -3.44 QT\n342.328 -3.659 342.328 -3.972 QT\n342.328 -4.253 342.531 -4.495 QT\n342.734 -4.737 343.047 -4.737 QT\n366.906 -4.737 L\n367.187 -4.737 367.39 -4.495 QT\n367.593 -4.253 367.593 -3.972 QT\n367.593 -3.659 367.39 -3.44 QT\n367.187 -3.222 366.906 -3.222 QT\n343.047 -3.222 L\ncp}def\r\nf-172155240\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1794144311{390.22 10.044 M\n385.564 10.044 383.884 6.216 QT\n382.205 2.388 382.205 -2.894 QT\n382.205 -6.206 382.806 -9.12 QT\n383.408 -12.034 385.197 -14.065 QT\n386.986 -16.097 390.22 -16.097 QT\n392.72 -16.097 394.322 -14.87 QT\n395.923 -13.644 396.752 -11.706 QT\n397.58 -9.769 397.892 -7.55 QT\n398.205 -5.331 398.205 -2.894 QT\n398.205 0.372 397.595 3.216 QT\n396.986 6.06 395.228 8.052 QT\n393.47 10.044 390.22 10.044 QT\ncp\n390.22 9.06 M\n392.33 9.06 393.369 6.896 QT\n394.408 4.731 394.65 2.091 QT\n394.892 -0.55 394.892 -3.519 QT\n394.892 -6.378 394.65 -8.784 QT\n394.408 -11.19 393.384 -13.151 QT\n392.361 -15.112 390.22 -15.112 QT\n388.064 -15.112 387.025 -13.144 QT\n385.986 -11.175 385.744 -8.776 QT\n385.502 -6.378 385.502 -3.519 QT\n385.502 -1.394 385.603 0.481 QT\n385.705 2.356 386.15 4.349 QT\n386.595 6.341 387.595 7.7 QT\n388.595 9.06 390.22 9.06 QT\ncp}def\r\nf-1794144311\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f-1192358335{403.108 7.122 M\n403.108 6.263 403.741 5.653 QT\n404.374 5.044 405.202 5.044 QT\n405.733 5.044 406.233 5.317 QT\n406.733 5.591 407.006 6.099 QT\n407.28 6.606 407.28 7.122 QT\n407.28 7.95 406.67 8.583 QT\n406.061 9.216 405.202 9.216 QT\n404.374 9.216 403.741 8.583 QT\n403.108 7.95 403.108 7.122 QT\ncp}def\r\nf-1192358335\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 193.8734] CT\r\nN\r\n/f947843481{411.65 2.95 M\n411.65 1.606 L\n423.384 -15.909 L\n423.525 -16.097 423.775 -16.097 QT\n424.337 -16.097 L\n424.759 -16.097 424.759 -15.659 QT\n424.759 1.606 L\n428.493 1.606 L\n428.493 2.95 L\n424.759 2.95 L\n424.759 6.669 L\n424.759 7.45 425.876 7.661 QT\n426.993 7.872 428.462 7.872 QT\n428.462 9.216 L\n417.993 9.216 L\n417.993 7.872 L\n419.462 7.872 420.571 7.661 QT\n421.681 7.45 421.681 6.669 QT\n421.681 2.95 L\n411.65 2.95 L\ncp\n412.9 1.606 M\n421.9 1.606 L\n421.9 -11.847 L\n412.9 1.606 L\ncp}def\r\nf947843481\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n[20 20] 0 setdash\r\n2 LJ\r\n5.333 LW\r\nN\r\n556.003 516.996 M\n636.035 516.996 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf574533680\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf735976676\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf1533900297\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-819923682\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1106082528\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-2022938735\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1608834715\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1300927600\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1997795938\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf1065174994\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\n/f-1449850694{267.513 10.044 M\n265.154 10.044 263.576 8.794 QT\n261.998 7.544 261.138 5.552 QT\n260.279 3.56 259.943 1.372 QT\n259.607 -0.815 259.607 -3.065 QT\n259.607 -6.081 260.779 -9.104 QT\n261.951 -12.128 264.224 -14.112 QT\n266.498 -16.097 269.607 -16.097 QT\n270.904 -16.097 272.029 -15.604 QT\n273.154 -15.112 273.795 -14.159 QT\n274.435 -13.206 274.435 -11.847 QT\n274.435 -11.065 273.904 -10.534 QT\n273.373 -10.003 272.591 -10.003 QT\n271.857 -10.003 271.318 -10.542 QT\n270.779 -11.081 270.779 -11.847 QT\n270.779 -12.581 271.318 -13.12 QT\n271.857 -13.659 272.591 -13.659 QT\n272.795 -13.659 L\n272.326 -14.347 271.443 -14.675 QT\n270.56 -15.003 269.607 -15.003 QT\n268.466 -15.003 267.49 -14.503 QT\n266.513 -14.003 265.732 -13.144 QT\n264.951 -12.284 264.435 -11.261 QT\n263.92 -10.237 263.63 -8.917 QT\n263.341 -7.597 263.263 -6.448 QT\n263.185 -5.3 263.185 -3.55 QT\n263.857 -5.112 265.084 -6.104 QT\n266.31 -7.097 267.841 -7.097 QT\n269.529 -7.097 270.927 -6.409 QT\n272.326 -5.722 273.326 -4.503 QT\n274.326 -3.284 274.857 -1.729 QT\n275.388 -0.175 275.388 1.419 QT\n275.388 3.653 274.388 5.661 QT\n273.388 7.669 271.591 8.856 QT\n269.795 10.044 267.513 10.044 QT\ncp\n267.513 8.841 M\n268.982 8.841 269.873 8.177 QT\n270.763 7.513 271.177 6.403 QT\n271.591 5.294 271.693 4.177 QT\n271.795 3.06 271.795 1.419 QT\n271.795 -0.737 271.591 -2.253 QT\n271.388 -3.769 270.482 -4.933 QT\n269.576 -6.097 267.701 -6.097 QT\n266.154 -6.097 265.162 -5.058 QT\n264.17 -4.019 263.716 -2.433 QT\n263.263 -0.847 263.263 0.622 QT\n263.263 1.122 263.295 1.388 QT\n263.295 1.435 263.287 1.474 QT\n263.279 1.513 263.263 1.575 QT\n263.263 3.2 263.599 4.872 QT\n263.935 6.544 264.88 7.692 QT\n265.826 8.841 267.513 8.841 QT\ncp}def\r\nf-1449850694\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1867049319\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1823982240\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1287452466\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-172155240\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1794144311\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\nf-1192358335\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 240.76412 211.49947] CT\r\nN\r\n/f1069079805{420.087 10.044 M\n417.728 10.044 416.15 8.794 QT\n414.571 7.544 413.712 5.552 QT\n412.853 3.56 412.517 1.372 QT\n412.181 -0.815 412.181 -3.065 QT\n412.181 -6.081 413.353 -9.104 QT\n414.525 -12.128 416.798 -14.112 QT\n419.071 -16.097 422.181 -16.097 QT\n423.478 -16.097 424.603 -15.604 QT\n425.728 -15.112 426.368 -14.159 QT\n427.009 -13.206 427.009 -11.847 QT\n427.009 -11.065 426.478 -10.534 QT\n425.946 -10.003 425.165 -10.003 QT\n424.431 -10.003 423.892 -10.542 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License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/bd{bind def}bind def\r\n/ld{load def}bd\r\n/GR/grestore ld\r\n/GS/gsave ld\r\n/RM/rmoveto ld\r\n/C/curveto ld\r\n/t/show ld\r\n/L/lineto ld\r\n/ML/setmiterlimit ld\r\n/CT/concat ld\r\n/f/fill ld\r\n/N/newpath ld\r\n/S/stroke ld\r\n/CC/setcmykcolor ld\r\n/A/ashow ld\r\n/cp/closepath ld\r\n/RC/setrgbcolor ld\r\n/LJ/setlinejoin ld\r\n/GC/setgray ld\r\n/LW/setlinewidth ld\r\n/M/moveto ld\r\n/re {4 2 roll M\r\n1 index 0 rlineto\r\n0 exch rlineto\r\nneg 0 rlineto\r\ncp } bd\r\n/_ctm matrix def\r\n/_tm matrix def\r\n/BT { _ctm currentmatrix pop matrix _tm copy pop 0 0 moveto } bd\r\n/ET { _ctm setmatrix } bd\r\n/iTm { _ctm setmatrix _tm concat } bd\r\n/Tm { _tm astore pop iTm 0 0 moveto } bd\r\n/ux 0.0 def\r\n/uy 0.0 def\r\n/F {\r\n /Tp exch def\r\n /Tf exch def\r\n Tf findfont Tp scalefont setfont\r\n /cf Tf def /cs Tp def\r\n} bd\r\n/ULS {currentpoint /uy exch def /ux exch def} bd\r\n/ULE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add moveto Tcx uy To add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/OLE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs add moveto Tcx uy To add cs add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/SOE {\r\n /Tcx currentpoint pop def\r\n gsave\r\n newpath\r\n cf findfont cs scalefont dup\r\n /FontMatrix get 0 get /Ts exch def /FontInfo get dup\r\n /UnderlinePosition get Ts mul /To exch def\r\n /UnderlineThickness get Ts mul /Tt exch def\r\n ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto\r\n Tt setlinewidth stroke\r\n grestore\r\n} bd\r\n/QT {\r\n/Y22 exch store\r\n/X22 exch store\r\n/Y21 exch store\r\n/X21 exch store\r\ncurrentpoint\r\n/Y21 load 2 mul add 3 div exch\r\n/X21 load 2 mul add 3 div exch\r\n/X21 load 2 mul /X22 load add 3 div\r\n/Y21 load 2 mul /Y22 load add 3 div\r\n/X22 load /Y22 load curveto\r\n} bd\r\n/SSPD {\r\ndup length /d exch dict def\r\n{\r\n/v exch def\r\n/k exch def\r\ncurrentpagedevice k known {\r\n/cpdv currentpagedevice k get def\r\nv cpdv ne {\r\n/upd false def\r\n/nullv v type /nulltype eq def\r\n/nullcpdv cpdv type /nulltype eq def\r\nnullv nullcpdv or\r\n{\r\n/upd true def\r\n} {\r\n/sametype v type cpdv type eq def\r\nsametype {\r\nv type /arraytype eq {\r\n/vlen v length def\r\n/cpdvlen cpdv length def\r\nvlen cpdvlen eq {\r\n0 1 vlen 1 sub {\r\n/i exch def\r\n/obj v i get def\r\n/cpdobj cpdv i get def\r\nobj cpdobj ne {\r\n/upd true def\r\nexit\r\n} if\r\n} for\r\n} {\r\n/upd true def\r\n} ifelse\r\n} {\r\nv type /dicttype eq {\r\nv {\r\n/dv exch def\r\n/dk exch def\r\n/cpddv cpdv dk get def\r\ndv cpddv ne {\r\n/upd true def\r\nexit\r\n} if\r\n} forall\r\n} {\r\n/upd true def\r\n} ifelse\r\n} ifelse\r\n} if\r\n} ifelse\r\nupd true eq {\r\nd k v put\r\n} if\r\n} if\r\n} if\r\n} forall\r\nd length 0 gt {\r\nd setpagedevice\r\n} if\r\n} bd\r\n/RE { % /NewFontName [NewEncodingArray] /FontName RE -\r\n findfont dup length dict begin\r\n {\r\n 1 index /FID ne\r\n {def} {pop pop} ifelse\r\n } forall\r\n /Encoding exch def\r\n /FontName 1 index def\r\n currentdict definefont pop\r\n end\r\n} bind def\r\n%%EndResource\r\n%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0\r\n%%Version: 1.0 0\r\n%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)\r\n/BeginEPSF { %def\r\n/b4_Inc_state save def % Save state for cleanup\r\n/dict_count countdictstack def % Count objects on dict stack\r\n/op_count count 1 sub def % Count objects on operand stack\r\nuserdict begin % Push userdict on dict stack\r\n/showpage { } def % Redefine showpage, { } = null proc\r\n0 setgray 0 setlinecap % Prepare graphics state\r\n1 setlinewidth 0 setlinejoin\r\n10 setmiterlimit [ ] 0 setdash newpath\r\n/languagelevel where % If level not equal to 1 then\r\n{pop languagelevel % set strokeadjust and\r\n1 ne % overprint to their defaults.\r\n{false setstrokeadjust false setoverprint\r\n} if\r\n} if\r\n} bd\r\n/EndEPSF { %def\r\ncount op_count sub {pop} repeat % Clean up stacks\r\ncountdictstack dict_count sub {end} repeat\r\nb4_Inc_state restore\r\n} bd\r\n%%EndResource\r\n%FOPBeginFontDict\r\n%%IncludeResource: font Courier-Oblique\r\n%%IncludeResource: font Courier-BoldOblique\r\n%%IncludeResource: font Courier-Bold\r\n%%IncludeResource: font ZapfDingbats\r\n%%IncludeResource: font Symbol\r\n%%IncludeResource: font Helvetica\r\n%%IncludeResource: font Helvetica-Oblique\r\n%%IncludeResource: font Helvetica-Bold\r\n%%IncludeResource: font Helvetica-BoldOblique\r\n%%IncludeResource: font Times-Roman\r\n%%IncludeResource: font Times-Italic\r\n%%IncludeResource: font Times-Bold\r\n%%IncludeResource: font Times-BoldItalic\r\n%%IncludeResource: font Courier\r\n%FOPEndFontDict\r\n%%BeginResource: encoding WinAnsiEncoding\r\n/WinAnsiEncoding [\r\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /.notdef /.notdef /.notdef\n/.notdef /.notdef /space /exclam /quotedbl\n/numbersign /dollar /percent /ampersand /quotesingle\n/parenleft /parenright /asterisk /plus /comma\n/hyphen /period /slash /zero /one\n/two /three /four /five /six\n/seven /eight /nine /colon /semicolon\n/less /equal /greater /question /at\n/A /B /C /D /E\n/F /G /H /I /J\n/K /L /M /N /O\n/P /Q /R /S /T\n/U /V /W /X /Y\n/Z /bracketleft /backslash /bracketright /asciicircum\n/underscore /quoteleft /a /b /c\n/d /e /f /g /h\n/i /j /k /l /m\n/n /o /p /q /r\n/s /t /u /v /w\n/x /y /z /braceleft /bar\n/braceright /asciitilde /bullet /Euro /bullet\n/quotesinglbase /florin /quotedblbase /ellipsis /dagger\n/daggerdbl /circumflex /perthousand /Scaron /guilsinglleft\n/OE /bullet /Zcaron /bullet /bullet\n/quoteleft /quoteright /quotedblleft /quotedblright /bullet\n/endash /emdash /asciitilde /trademark /scaron\n/guilsinglright /oe /bullet /zcaron /Ydieresis\n/space /exclamdown /cent /sterling /currency\n/yen /brokenbar /section /dieresis /copyright\n/ordfeminine /guillemotleft /logicalnot /sfthyphen /registered\n/macron /degree /plusminus /twosuperior /threesuperior\n/acute /mu /paragraph /middot /cedilla\n/onesuperior /ordmasculine /guillemotright /onequarter /onehalf\n/threequarters /questiondown /Agrave /Aacute /Acircumflex\n/Atilde /Adieresis /Aring /AE /Ccedilla\n/Egrave /Eacute /Ecircumflex /Edieresis /Igrave\n/Iacute /Icircumflex /Idieresis /Eth /Ntilde\n/Ograve /Oacute /Ocircumflex /Otilde /Odieresis\n/multiply /Oslash /Ugrave /Uacute /Ucircumflex\n/Udieresis /Yacute /Thorn /germandbls /agrave\n/aacute /acircumflex /atilde /adieresis /aring\n/ae /ccedilla /egrave /eacute /ecircumflex\n/edieresis /igrave /iacute /icircumflex /idieresis\n/eth /ntilde /ograve /oacute /ocircumflex\n/otilde /odieresis /divide /oslash /ugrave\n/uacute /ucircumflex /udieresis /yacute /thorn\n/ydieresis\n] def\r\n%%EndResource\r\n%FOPBeginFontReencode\r\n/Courier-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Oblique exch definefont pop\r\n/Courier-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-BoldOblique exch definefont pop\r\n/Courier-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier-Bold exch definefont pop\r\n/Helvetica findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica exch definefont pop\r\n/Helvetica-Oblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Oblique exch definefont pop\r\n/Helvetica-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-Bold exch definefont pop\r\n/Helvetica-BoldOblique findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Helvetica-BoldOblique exch definefont pop\r\n/Times-Roman findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Roman exch definefont pop\r\n/Times-Italic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Italic exch definefont pop\r\n/Times-Bold findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-Bold exch definefont pop\r\n/Times-BoldItalic findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Times-BoldItalic exch definefont pop\r\n/Courier findfont\r\ndup length dict begin\r\n {1 index /FID ne {def} {pop pop} ifelse} forall\r\n /Encoding WinAnsiEncoding def\r\n currentdict\r\nend\r\n/Courier exch definefont pop\r\n%FOPEndFontReencode\r\n%%EndProlog\r\n%%Page: 1 1\r\n%%PageBoundingBox: 0 0 420 315\r\n%%BeginPageSetup\r\n[1 0 0 -1 0 315] CT\r\n%%EndPageSetup\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n0 0 1120 840 re\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n130 703 M\n1105 703 L\n1105 53 L\n130 53 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n130 703 M\n130 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n292.5 703 M\n292.5 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n455 703 M\n455 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n617.5 703 M\n617.5 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n780 703 M\n780 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n942.5 703 M\n942.5 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 703 M\n1105 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 703 M\n130 703 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 540.5 M\n130 540.5 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 378 M\n130 378 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 215.5 M\n130 215.5 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.873 GC\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 53 M\n130 53 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n130 703 M\n1105 703 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n130 703 M\n130 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n292.5 703 M\n292.5 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n455 703 M\n455 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n617.5 703 M\n617.5 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n780 703 M\n780 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n942.5 703 M\n942.5 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0.149 GC\r\n2 setlinecap\r\n1 LJ\r\n0.533 LW\r\nN\r\n1105 703 M\n1105 693.25 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 48.75 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-21.5 49 moveto \r\n1 -1 scale\r\n(-3) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 109.6875 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-21.5 49 moveto \r\n1 -1 scale\r\n(-2) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 170.625 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-21.5 49 moveto \r\n1 -1 scale\r\n(-1) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 231.5625 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(0) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 292.5 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(1) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 353.4375 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(2) t \r\nGR\r\nGR\r\nGS\r\n[0.375 0 0 0.375 414.375 269.22501] CT\r\n0.149 GC\r\n/Helvetica 48 F\r\nGS\r\n[1 0 0 1 0 0] CT\r\n-13.5 49 moveto \r\n1 -1 scale\r\n(3) t 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L\n584.219 537.055 L\n594.374 538.846 L\n604.622 539.989 L\n614.922 540.48 L\n625.232 540.316 L\n635.511 539.499 L\n645.718 538.031 L\n655.811 535.919 L\n665.75 533.172 L\n675.494 529.799 L\n685.005 525.815 L\n694.244 521.236 L\n703.174 516.08 L\n711.759 510.369 L\n719.965 504.124 L\n727.758 497.371 L\n735.107 490.138 L\n741.982 482.453 L\n748.356 474.348 L\n754.204 465.854 L\n759.501 457.007 L\n764.226 447.842 L\n768.36 438.395 L\n771.887 428.705 L\n774.792 418.812 L\n777.063 408.753 L\n778.693 398.571 L\n779.673 388.306 L\n780 378 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0.502 0 RC\r\nN\r\n489.766 478.451 M\n496.395 486.35 L\n503.51 493.813 L\n511.085 500.809 L\n519.088 507.311 L\n527.488 513.293 L\n536.25 518.729 L\n545.339 523.599 L\n554.719 527.883 L\n564.351 531.563 L\n574.198 534.624 L\n584.219 537.055 L\n594.374 538.846 L\n604.622 539.989 L\n614.922 540.48 L\n625.232 540.316 L\n635.511 539.499 L\n645.718 538.031 L\n655.811 535.919 L\n665.75 533.172 L\n675.494 529.799 L\n685.005 525.815 L\n694.244 521.236 L\n703.174 516.08 L\n711.759 510.369 L\n719.965 504.124 L\n727.758 497.371 L\n735.107 490.138 L\n741.982 482.453 L\n748.356 474.348 L\n754.204 465.854 L\n759.501 457.007 L\n764.226 447.842 L\n768.36 438.395 L\n771.887 428.705 L\n774.792 418.812 L\n777.063 408.753 L\n778.693 398.571 L\n779.673 388.306 L\n780 378 L\n779.673 367.694 L\n778.693 357.429 L\n777.063 347.247 L\n774.792 337.188 L\n771.887 327.295 L\n768.36 317.605 L\n764.226 308.158 L\n759.501 298.993 L\n754.204 290.146 L\n748.356 281.652 L\n741.982 273.547 L\n735.107 265.862 L\n727.758 258.629 L\n719.965 251.876 L\n711.759 245.631 L\n703.174 239.92 L\n694.244 234.764 L\n685.005 230.185 L\n675.494 226.201 L\n665.75 222.828 L\n655.811 220.081 L\n645.718 217.969 L\n635.511 216.501 L\n625.232 215.684 L\n614.922 215.52 L\n604.622 216.011 L\n594.374 217.154 L\n584.219 218.945 L\n574.198 221.376 L\n564.351 224.437 L\n554.719 228.117 L\n545.339 232.401 L\n536.25 237.271 L\n527.488 242.707 L\n519.088 248.689 L\n511.085 255.191 L\n503.51 262.187 L\n496.395 269.65 L\n489.766 277.549 L\n483.653 285.853 L\n478.078 294.527 L\n473.064 303.538 L\n468.632 312.849 L\n464.8 322.422 L\n461.582 332.218 L\n458.993 342.2 L\n457.041 352.325 L\n455.736 362.553 L\n455.082 372.844 L\n455.082 383.156 L\n455.736 393.447 L\n457.041 403.675 L\n458.993 413.8 L\n461.582 423.782 L\n464.8 433.578 L\n468.632 443.151 L\n473.064 452.462 L\n478.078 461.473 L\n483.653 470.147 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n2 setlinecap\r\n10.0 ML\r\n1.333 LW\r\nN\r\n780 378 M\n779.673 367.694 L\n778.693 357.429 L\n777.063 347.247 L\n774.792 337.188 L\n771.887 327.295 L\n768.36 317.605 L\n764.226 308.158 L\n759.501 298.993 L\n754.204 290.146 L\n748.356 281.652 L\n741.982 273.547 L\n735.107 265.862 L\n727.758 258.629 L\n719.965 251.876 L\n711.759 245.631 L\n703.174 239.92 L\n694.244 234.764 L\n685.005 230.185 L\n675.494 226.201 L\n665.75 222.828 L\n655.811 220.081 L\n645.718 217.969 L\n635.511 216.501 L\n625.232 215.684 L\n614.922 215.52 L\n604.622 216.011 L\n594.374 217.154 L\n584.219 218.945 L\n574.198 221.376 L\n564.351 224.437 L\n554.719 228.117 L\n545.339 232.401 L\n536.25 237.271 L\n527.488 242.707 L\n519.088 248.689 L\n511.085 255.191 L\n503.51 262.187 L\n496.395 269.65 L\n489.766 277.549 L\n483.653 285.853 L\n478.078 294.527 L\n473.064 303.538 L\n468.632 312.849 L\n464.8 322.422 L\n461.582 332.218 L\n458.993 342.2 L\n457.041 352.325 L\n455.736 362.553 L\n455.082 372.844 L\n455.082 383.156 L\n455.736 393.447 L\n457.041 403.675 L\n458.993 413.8 L\n461.582 423.782 L\n464.8 433.578 L\n468.632 443.151 L\n473.064 452.462 L\n478.078 461.473 L\n483.653 470.147 L\n489.766 478.451 L\n496.395 486.35 L\n503.51 493.813 L\n511.085 500.809 L\n519.088 507.311 L\n527.488 513.293 L\n536.25 518.729 L\n545.339 523.599 L\n554.719 527.883 L\n564.351 531.563 L\n574.198 534.624 L\n584.219 537.055 L\n594.374 538.846 L\n604.622 539.989 L\n614.922 540.48 L\n625.232 540.316 L\n635.511 539.499 L\n645.718 538.031 L\n655.811 535.919 L\n665.75 533.172 L\n675.494 529.799 L\n685.005 525.815 L\n694.244 521.236 L\n703.174 516.08 L\n711.759 510.369 L\n719.965 504.124 L\n727.758 497.371 L\n735.107 490.138 L\n741.982 482.453 L\n748.356 474.348 L\n754.204 465.854 L\n759.501 457.007 L\n764.226 447.842 L\n768.36 438.395 L\n771.887 428.705 L\n774.792 418.812 L\n777.063 408.753 L\n778.693 398.571 L\n779.673 388.306 L\n780 378 L\n780 378 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 48.75 141.75] CT\r\n0 0 1 RC\r\n10.0 ML\r\n2.667 LW\r\nN\r\n-8 0 M\n0 10.64 L\n8 0 L\n0 -10.64 L\ncp\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 414.375 141.14063] CT\r\n1 0 0 RC\r\n10.0 ML\r\n2.667 LW\r\nN\r\n-8 0 M\n0 10.64 L\n8 0 L\n0 -10.64 L\ncp\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 LJ\r\n5.333 LW\r\nN\r\n130 378 M\n130 378 L\n138.118 377.512 L\n154.199 380.002 L\n177.031 386.679 L\n206.063 396.743 L\n238.228 408.147 L\n271.882 420.187 L\n305.911 431.631 L\n340.589 444.026 L\n377.263 458.484 L\n418.522 475.762 L\n460.353 496.861 L\n501.749 519.954 L\n545.468 537.374 L\n590.794 546.087 L\n635.044 543.053 L\n676.095 532.99 L\n714.002 520.799 L\n749.653 508.197 L\n783.025 495.889 L\n814.135 484.245 L\n843.009 473.429 L\n869.735 463.526 L\n894.378 454.558 L\n917.039 446.523 L\n937.829 439.395 L\n956.865 433.132 L\n974.263 427.679 L\n990.135 422.974 L\n1004.556 418.946 L\n1017.658 415.531 L\n1029.543 412.66 L\n1040.305 410.265 L\n1050.034 408.282 L\n1058.816 406.651 L\n1066.73 405.32 L\n1073.816 404.245 L\n1080.121 403.387 L\n1085.698 402.711 L\n1090.605 402.182 L\n1094.904 401.772 L\n1098.652 401.458 L\n1101.905 401.219 L\n1104.717 401.039 L\n1105.039 401.021 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n5.333 LW\r\nN\r\n130 378 M\n130 378 L\n138.116 377.794 L\n154.198 380.103 L\n176.665 385.868 L\n204.6 395.173 L\n237.676 407.171 L\n277.298 401.294 L\n312.899 386.87 L\n346.209 369.504 L\n377.105 350.335 L\n409.288 328.065 L\n445.058 302.043 L\n481.707 273.085 L\n518.928 243.676 L\n561.586 223.954 L\n606.576 215.771 L\n650.181 219.308 L\n691.018 228.764 L\n728.456 240.351 L\n763.455 252.376 L\n796.063 264.1 L\n826.358 275.159 L\n854.418 285.391 L\n880.317 294.732 L\n904.153 303.157 L\n926.039 310.676 L\n946.094 317.319 L\n964.437 323.135 L\n981.187 328.179 L\n996.454 332.515 L\n1010.312 336.215 L\n1022.895 339.34 L\n1034.301 341.96 L\n1044.624 344.139 L\n1053.953 345.938 L\n1062.37 347.414 L\n1069.917 348.61 L\n1076.639 349.568 L\n1082.593 350.327 L\n1087.835 350.922 L\n1092.429 351.384 L\n1096.437 351.74 L\n1099.917 352.011 L\n1102.926 352.217 L\n1105.039 352.339 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0 0 1 RC\r\n[20 20] 0 setdash\r\n2 LJ\r\n5.333 LW\r\nN\r\n130 378 M\n138.086 377.201 L\n154.269 378.675 L\n178.439 375.519 L\n210.533 380.64 L\n249.598 391.788 L\n296.236 379.618 L\n343.115 356.658 L\n387.713 327.195 L\n428.471 293.148 L\n469.021 260.132 L\n512.422 233.248 L\n558.649 215.967 L\n605.51 209.958 L\n650.067 215.121 L\n690.887 225.864 L\n728.369 238.034 L\n762.869 250.109 L\n794.608 261.51 L\n823.756 272.02 L\n850.473 281.571 L\n874.914 290.162 L\n897.237 297.826 L\n917.597 304.612 L\n936.141 310.579 L\n953.012 315.789 L\n968.343 320.308 L\n982.258 324.203 L\n994.873 327.537 L\n1006.296 330.375 L\n1016.626 332.776 L\n1025.955 334.794 L\n1034.37 336.482 L\n1041.949 337.886 L\n1048.767 339.048 L\n1054.892 340.004 L\n1060.387 340.788 L\n1065.312 341.428 L\n1069.719 341.949 L\n1073.658 342.37 L\n1077.177 342.71 L\n1080.315 342.983 L\n1083.112 343.202 L\n1085.603 343.378 L\n1087.818 343.518 L\n1089.788 343.629 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 1 RC\r\n[20 20] 0 setdash\r\n2 LJ\r\n5.333 LW\r\nN\r\n130 378 M\n138.089 377.233 L\n154.129 374.633 L\n177.997 369.683 L\n209.58 362.02 L\n248.741 351.211 L\n295.28 336.697 L\n345.413 319.032 L\n396.394 298.632 L\n446.211 275.761 L\n493.253 250.448 L\n539.654 228.164 L\n587.188 215.744 L\n633.671 215.272 L\n676.416 224.182 L\n715.443 235.836 L\n751.231 247.868 L\n784.088 259.401 L\n814.23 270.105 L\n841.836 279.867 L\n867.079 288.67 L\n890.127 296.538 L\n911.141 303.516 L\n930.278 309.661 L\n947.687 315.035 L\n963.506 319.705 L\n977.864 323.736 L\n990.882 327.193 L\n1002.672 330.14 L\n1013.336 332.638 L\n1022.97 334.741 L\n1031.662 336.503 L\n1039.495 337.971 L\n1046.544 339.188 L\n1052.879 340.191 L\n1058.566 341.015 L\n1063.664 341.689 L\n1068.23 342.237 L\n1072.314 342.682 L\n1075.963 343.041 L\n1079.22 343.331 L\n1082.125 343.564 L\n1084.713 343.75 L\n1087.017 343.899 L\n1089.067 344.017 L\n1090.889 344.112 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 GC\r\nN\r\n1084 682 M\n1084 463 L\n670 463 L\n670 682 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 184.87788] CT\r\nN\r\n2.016 10.368 M\n1.594 10.368 1.594 9.805 QT\n1.609 9.696 1.68 9.43 QT\n1.75 9.165 1.859 9.008 QT\n1.969 8.852 2.141 8.852 QT\n6.25 8.852 6.922 6.227 QT\n12.766 -17.288 L\n11.563 -17.492 8.984 -17.492 QT\n8.563 -17.492 8.563 -18.054 QT\n8.594 -18.163 8.656 -18.429 QT\n8.719 -18.695 8.828 -18.851 QT\n8.938 -19.007 9.109 -19.007 QT\n16.547 -19.007 L\n16.859 -19.007 16.938 -18.726 QT\n26.5 4.008 L\n31.266 -15.038 L\n31.344 -15.507 31.344 -15.695 QT\n31.344 -17.492 28.031 -17.492 QT\n27.609 -17.492 27.609 -18.054 QT\n27.75 -18.601 27.836 -18.804 QT\n27.922 -19.007 28.344 -19.007 QT\n37.547 -19.007 L\n37.969 -19.007 37.969 -18.445 QT\n37.938 -18.335 37.875 -18.07 QT\n37.813 -17.804 37.695 -17.648 QT\n37.578 -17.492 37.422 -17.492 QT\n33.297 -17.492 32.625 -14.867 QT\n26.453 9.993 L\n26.313 10.368 26.016 10.368 QT\n25.484 10.368 L\n25.203 10.368 25.109 10.071 QT\n14.156 -15.945 L\n14.063 -16.21 L\n13.984 -16.304 13.984 -16.335 QT\n8.297 6.415 L\n8.25 6.524 8.227 6.68 QT\n8.203 6.837 8.172 7.055 QT\n8.172 8.165 9.133 8.508 QT\n10.094 8.852 11.531 8.852 QT\n11.953 8.852 11.953 9.43 QT\n11.797 10.008 11.695 10.188 QT\n11.594 10.368 11.234 10.368 QT\n2.016 10.368 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 184.87788] CT\r\nN\r\n54.654 4.649 M\n54.295 4.649 54.068 4.383 QT\n53.841 4.118 53.841 3.79 QT\n53.841 3.446 54.068 3.188 QT\n54.295 2.93 54.654 2.93 QT\n81.654 2.93 L\n81.966 2.93 82.201 3.188 QT\n82.435 3.446 82.435 3.79 QT\n82.435 4.118 82.201 4.383 QT\n81.966 4.649 81.654 4.649 QT\n54.654 4.649 L\ncp\n54.654 -3.695 M\n54.295 -3.695 54.068 -3.952 QT\n53.841 -4.21 53.841 -4.554 QT\n53.841 -4.882 54.068 -5.156 QT\n54.295 -5.429 54.654 -5.429 QT\n81.654 -5.429 L\n81.966 -5.429 82.201 -5.156 QT\n82.435 -4.882 82.435 -4.554 QT\n82.435 -4.21 82.201 -3.952 QT\n81.966 -3.695 81.654 -3.695 QT\n54.654 -3.695 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 184.87788] CT\r\nN\r\n99.16 10.368 M\n99.16 9.212 L\n99.16 9.102 99.238 8.977 QT\n105.926 1.587 L\n107.441 -0.054 108.379 -1.163 QT\n109.316 -2.273 110.246 -3.718 QT\n111.176 -5.163 111.707 -6.671 QT\n112.238 -8.179 112.238 -9.851 QT\n112.238 -11.617 111.59 -13.226 QT\n110.941 -14.835 109.652 -15.796 QT\n108.363 -16.757 106.535 -16.757 QT\n104.66 -16.757 103.168 -15.632 QT\n101.676 -14.507 101.066 -12.726 QT\n101.238 -12.773 101.535 -12.773 QT\n102.504 -12.773 103.183 -12.124 QT\n103.863 -11.476 103.863 -10.445 QT\n103.863 -9.46 103.183 -8.773 QT\n102.504 -8.085 101.535 -8.085 QT\n100.519 -8.085 99.84 -8.788 QT\n99.16 -9.492 99.16 -10.445 QT\n99.16 -12.054 99.769 -13.476 QT\n100.379 -14.898 101.519 -15.999 QT\n102.66 -17.101 104.105 -17.687 QT\n105.551 -18.273 107.16 -18.273 QT\n109.613 -18.273 111.738 -17.234 QT\n113.863 -16.195 115.098 -14.296 QT\n116.332 -12.398 116.332 -9.851 QT\n116.332 -7.976 115.512 -6.296 QT\n114.691 -4.617 113.418 -3.242 QT\n112.144 -1.867 110.144 -0.124 QT\n108.144 1.618 107.519 2.196 QT\n102.644 6.883 L\n106.785 6.883 L\n109.832 6.883 111.879 6.829 QT\n113.926 6.774 114.051 6.665 QT\n114.551 6.133 115.082 2.712 QT\n116.332 2.712 L\n115.113 10.368 L\n99.16 10.368 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 184.87788] CT\r\nN\r\n119.823 3.274 M\n119.823 1.758 L\n133.104 -18.054 L\n133.26 -18.273 133.541 -18.273 QT\n134.182 -18.273 L\n134.666 -18.273 134.666 -17.788 QT\n134.666 1.758 L\n138.885 1.758 L\n138.885 3.274 L\n134.666 3.274 L\n134.666 7.493 L\n134.666 8.368 135.924 8.61 QT\n137.182 8.852 138.838 8.852 QT\n138.838 10.368 L\n126.994 10.368 L\n126.994 8.852 L\n128.651 8.852 129.916 8.61 QT\n131.182 8.368 131.182 7.493 QT\n131.182 3.274 L\n119.823 3.274 L\ncp\n121.244 1.758 M\n131.432 1.758 L\n131.432 -13.46 L\n121.244 1.758 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 184.87788] CT\r\nN\r\n144.485 18.118 M\n144.485 17.93 144.657 17.758 QT\n146.204 16.274 147.063 14.321 QT\n147.923 12.368 147.923 10.196 QT\n147.923 9.68 L\n147.235 10.368 146.204 10.368 QT\n145.22 10.368 144.524 9.673 QT\n143.829 8.977 143.829 7.993 QT\n143.829 6.993 144.524 6.321 QT\n145.22 5.649 146.204 5.649 QT\n147.735 5.649 148.384 7.063 QT\n149.032 8.477 149.032 10.196 QT\n149.032 12.587 148.079 14.743 QT\n147.126 16.899 145.391 18.618 QT\n145.22 18.696 145.11 18.696 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172.748 19.649 QT\n172.107 19.649 172.107 18.79 QT\ncp}def\r\nf2034987691\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 204.75096] CT\r\nN\r\n/f-173962665{184.567 16.848 M\n184.567 15.785 L\n188.317 15.785 188.317 14.848 QT\n188.317 -0.902 L\n186.77 -0.152 184.395 -0.152 QT\n184.395 -1.215 L\n188.067 -1.215 189.942 -3.137 QT\n190.363 -3.137 L\n190.473 -3.137 190.567 -3.051 QT\n190.66 -2.965 190.66 -2.871 QT\n190.66 14.848 L\n190.66 15.785 194.41 15.785 QT\n194.41 16.848 L\n184.567 16.848 L\ncp}def\r\nf-173962665\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 286.51149 204.75096] CT\r\nN\r\n/f-988291271{214.299 4.649 M\n213.94 4.649 213.714 4.383 QT\n213.487 4.118 213.487 3.79 QT\n213.487 3.446 213.714 3.188 QT\n213.94 2.93 214.299 2.93 QT\n241.299 2.93 L\n241.612 2.93 241.846 3.188 QT\n242.081 3.446 242.081 3.79 QT\n242.081 4.118 241.846 4.383 QT\n241.612 4.649 241.299 4.649 QT\n214.299 4.649 L\ncp\n214.299 -3.695 M\n213.94 -3.695 213.714 -3.952 QT\n213.487 -4.21 213.487 -4.554 QT\n213.487 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RC\r\n10.0 ML\r\n2.667 LW\r\nN\r\n-8 0 M\n0 10.64 L\n8 0 L\n0 -10.64 L\ncp\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 LJ\r\n2.667 LW\r\nN\r\n133 375.5 M\n133 375.5 L\n140.853 373.654 L\n156.077 368.315 L\n177.116 359.28 L\n204.989 347.858 L\n236.14 335.214 L\n271.673 322.137 L\n313.271 306.838 L\n357.777 288.674 L\n402.46 267.286 L\n446.991 244.907 L\n492.11 224.784 L\n538.24 210.247 L\n584.617 203.594 L\n629.489 206.163 L\n670.815 216.322 L\n708.788 228.65 L\n743.987 241.064 L\n776.552 252.895 L\n806.595 263.881 L\n834.23 273.927 L\n859.584 283.014 L\n882.795 291.162 L\n904.005 298.412 L\n923.355 304.815 L\n940.983 310.433 L\n957.022 315.328 L\n971.596 319.564 L\n984.823 323.206 L\n996.812 326.317 L\n1007.665 328.959 L\n1017.476 331.188 L\n1026.333 333.058 L\n1034.318 334.619 L\n1041.506 335.914 L\n1047.969 336.983 L\n1053.772 337.862 L\n1058.975 338.58 L\n1063.635 339.166 L\n1067.803 339.641 L\n1071.526 340.025 L\n1074.849 340.334 L\n1077.812 340.583 L\n1080.451 340.782 L\n1082.799 340.94 L\n1084.887 341.067 L\n1086.742 341.167 L\n1088.389 341.247 L\n1089.85 341.31 L\n1091.145 341.36 L\n1092.293 341.399 L\n1093.309 341.43 L\n1094.209 341.454 L\n1095.005 341.473 L\n1095.709 341.488 L\n1096.332 341.5 L\n1096.882 341.509 L\n1097.368 341.517 L\n1097.797 341.522 L\n1098.176 341.527 L\n1098.511 341.53 L\n1098.806 341.533 L\n1099.066 341.535 L\n1099.296 341.536 L\n1099.499 341.538 L\n1099.677 341.539 L\n1099.835 341.539 L\n1099.974 341.54 L\n1100.096 341.54 L\n1100.204 341.541 L\n1100.299 341.541 L\n1100.383 341.541 L\n1100.456 341.542 L\n1100.521 341.542 L\n1100.579 341.542 L\n1100.629 341.542 L\n1100.673 341.542 L\n1100.713 341.542 L\n1100.747 341.542 L\n1100.777 341.542 L\n1100.804 341.542 L\n1100.827 341.542 L\n1100.848 341.542 L\n1100.866 341.542 L\n1100.882 341.542 L\n1100.896 341.542 L\n1100.909 341.542 L\n1100.92 341.542 L\n1100.929 341.542 L\n1100.938 341.542 L\n1100.945 341.542 L\n1100.952 341.542 L\n1100.958 341.542 L\n1100.963 341.542 L\n1100.967 341.542 L\n1100.971 341.542 L\n1100.974 341.542 L\n1100.978 341.542 L\n1100.98 341.542 L\n1100.983 341.542 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 49.875 140.8125] CT\r\nN\r\n0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 49.875 140.8125] CT\r\nN\r\n/f-1792389198{0 -4 M\n2.209 -4 4 -2.209 4 0 C\n4 0 L\n4 2.209 2.209 4 0 4 C\n-2.209 4 -4 2.209 -4 0 C\n-4 -2.209 -2.209 -4 0 -4 C\ncp\n0 -6.667 M\n-3.682 -6.667 -6.667 -3.682 -6.667 0 C\n-6.667 3.682 -3.682 6.667 0 6.667 C\n3.682 6.667 6.667 3.682 6.667 0 C\n6.667 0 L\n6.667 -3.682 3.682 -6.667 0 -6.667 C\ncp}def\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 52.81999 140.12026] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 58.52871 138.11801] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 66.41841 134.73005] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 76.87071 130.44685] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 88.5524 125.70506] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 101.87727 120.80127] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 117.47675 115.06416] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 134.16656 108.25274] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 150.92249 100.23207] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 167.62165 91.83996] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 184.54131 84.29418] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 201.84002 78.84247] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 219.23151 76.34766] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 236.05854 77.31118] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 251.55547 81.12088] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 265.7956 85.74387] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 278.99494 90.39893] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 291.20707 94.83567] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 302.47327 98.95523] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 312.83629 102.72255] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 322.34404 106.13036] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 331.04816 109.18579] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 339.00179 111.90435] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 346.25805 114.30575] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 352.86861 116.41246] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 358.88317 118.24794] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 364.34857 119.83645] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 369.30874 121.20222] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 373.80464 122.369] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 377.87441 123.35958] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 381.5535 124.19552] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 384.87488 124.89687] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 387.8692 125.48203] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 390.56493 125.96765] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 392.98851 126.36861] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 395.16454 126.69811] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 397.11571 126.96764] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 398.86304 127.18721] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 400.42598 127.36535] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 401.82234 127.50936] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 403.06856 127.62537] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 404.1796 127.71852] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 405.16919 127.79311] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 406.04979 127.85264] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 406.8327 127.90006] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 407.52827 127.93774] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 408.1458 127.96762] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 408.69365 127.99125] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 409.17938 128.00993] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 409.60977 128.02467] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 409.991 128.03626] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 410.32841 128.04539] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 410.62697 128.05256] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 410.89101 128.05817] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.12442 128.06258] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.33073 128.06602] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.51297 128.06873] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.67397 128.07083] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.81606 128.07248] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 411.94148 128.07377] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.05222 128.07477] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.14986 128.07556] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.23605 128.07616] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.312 128.07663] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.37901 128.07701] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.43806 128.0773] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.49016 128.07751] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.53603 128.07769] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.57649 128.07782] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.61211 128.07793] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.64351 128.07801] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.67116 128.07808] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.69551 128.07812] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.71698 128.07817] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.73589 128.07819] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.75255 128.07821] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.7672 128.07824] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.78011 128.07825] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.79146 128.07826] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.80148 128.07827] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.81027 128.07827] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.81801 128.07828] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.82483 128.07828] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.83083 128.07828] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.83614 128.07828] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.84081 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.84488 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.8485 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.85165 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.85445 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.85692 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.85907 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86099 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86269 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86415 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86543 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86658 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.86758 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 412.8685 128.07829] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0 1 1 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n236.14 335.214 M\n271.673 322.137 L\n312.803 311.451 L\n355.174 299.406 L\n395.525 283.346 L\n432.849 263.393 L\n468.87 243.117 L\n504.988 224.901 L\n542.672 209.148 L\n582.056 197.854 L\n620.712 193.807 L\n657.157 196.677 L\n690.978 202.91 L\n722.125 210.358 L\n750.608 217.769 L\n776.442 224.463 L\n799.648 230.122 L\n820.266 234.665 L\n838.367 238.154 L\n854.06 240.731 L\n867.496 242.574 L\n878.873 243.86 L\n888.439 244.753 L\n896.503 245.387 L\n903.45 245.87 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0 1 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n236.14 335.214 M\n271.673 322.137 L\n315.616 308.856 L\n361.543 291.881 L\n404.426 270.071 L\n446.378 246.49 L\n489.255 225.336 L\n533.735 209.818 L\n581.433 201.703 L\n631.217 204.13 L\n676.949 213.181 L\n717.344 223.125 L\n752.328 231.868 L\n782.682 239.204 L\n809.306 245.424 L\n832.808 250.718 L\n853.53 255.146 L\n871.684 258.739 L\n887.445 261.554 L\n900.988 263.682 L\n912.504 265.236 L\n922.204 266.342 L\n930.329 267.119 L\n937.162 267.676 L\n943.042 268.103 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 1 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n236.14 335.214 M\n271.673 322.137 L\n315.617 307.716 L\n361.55 290.154 L\n404.689 268.928 L\n446.673 246.088 L\n489.275 225.469 L\n533.291 210.192 L\n580.499 202.03 L\n629.908 204.13 L\n675.693 213.504 L\n716.615 225.091 L\n752.132 235.683 L\n782.649 244.373 L\n809.129 251.37 L\n832.382 257.032 L\n852.878 261.6 L\n870.863 265.224 L\n886.5 268.023 L\n899.951 270.117 L\n911.393 271.637 L\n921.033 272.714 L\n929.108 273.468 L\n935.899 274.009 L\n941.743 274.423 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n236.14 335.214 M\n271.673 322.137 L\n315.498 306.547 L\n361.646 288.526 L\n405.193 267.353 L\n447.589 244.666 L\n490.561 224.286 L\n534.864 209.355 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33.369 4.176 QT\n33.369 3.785 33.627 3.504 QT\n33.885 3.223 34.276 3.223 QT\n64.416 3.223 L\n64.776 3.223 65.034 3.504 QT\n65.291 3.785 65.291 4.176 QT\n65.291 4.535 65.034 4.84 QT\n64.776 5.145 64.416 5.145 QT\n34.276 5.145 L\ncp\n34.276 -4.183 M\n33.885 -4.183 33.627 -4.465 QT\n33.369 -4.746 33.369 -5.152 QT\n33.369 -5.496 33.627 -5.8 QT\n33.885 -6.105 34.276 -6.105 QT\n64.416 -6.105 L\n64.776 -6.105 65.034 -5.8 QT\n65.291 -5.496 65.291 -5.152 QT\n65.291 -4.746 65.034 -4.465 QT\n64.776 -4.183 64.416 -4.183 QT\n34.276 -4.183 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 83.15 86.3625] CT\r\nN\r\n93.333 12.582 M\n90.364 12.582 88.372 10.996 QT\n86.38 9.41 85.286 6.887 QT\n84.192 4.364 83.77 1.598 QT\n83.348 -1.168 83.348 -3.996 QT\n83.348 -7.793 84.825 -11.613 QT\n86.302 -15.433 89.177 -17.941 QT\n92.052 -20.449 95.989 -20.449 QT\n97.63 -20.449 99.044 -19.832 QT\n100.458 -19.215 101.27 -18.004 QT\n102.083 -16.793 102.083 -15.09 QT\n102.083 -14.105 101.411 -13.433 QT\n100.739 -12.761 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9.202 QT\n129.513 9.319 130.982 9.319 QT\n130.982 10.944 L\n124.044 11.459 L\n124.044 8.569 L\n122.841 9.928 121.208 10.694 QT\n119.575 11.459 117.888 11.459 QT\ncp\n113.185 6.944 M\n113.982 8.444 115.294 9.342 QT\n116.607 10.241 118.153 10.241 QT\n120.06 10.241 121.653 9.147 QT\n123.247 8.053 124.044 6.288 QT\n124.044 -4.728 L\n123.497 -5.744 122.677 -6.541 QT\n121.857 -7.337 120.833 -7.767 QT\n119.81 -8.197 118.669 -8.197 QT\n116.263 -8.197 114.802 -6.837 QT\n113.341 -5.478 112.763 -3.369 QT\n112.185 -1.259 112.185 1.053 QT\n112.185 2.897 112.372 4.272 QT\n112.56 5.647 113.185 6.944 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n132.503 2.428 M\n132.503 -0.416 L\n144.674 -0.416 L\n144.674 2.428 L\n132.503 2.428 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n148.609 10.944 M\n148.609 9.319 L\n150.171 9.319 151.187 9.077 QT\n152.202 8.834 152.202 7.866 QT\n152.202 -16.275 L\n152.202 -17.509 151.827 -18.064 QT\n151.452 -18.619 150.757 -18.744 QT\n150.062 -18.869 148.609 -18.869 QT\n148.609 -20.478 L\n155.437 -20.978 L\n155.437 7.866 L\n155.437 8.834 156.445 9.077 QT\n157.452 9.319 158.999 9.319 QT\n158.999 10.944 L\n148.609 10.944 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n171.354 11.459 M\n168.588 11.459 166.229 10.053 QT\n163.869 8.647 162.502 6.288 QT\n161.135 3.928 161.135 1.147 QT\n161.135 -0.962 161.885 -2.916 QT\n162.635 -4.869 164.041 -6.408 QT\n165.447 -7.947 167.314 -8.814 QT\n169.182 -9.681 171.354 -9.681 QT\n174.182 -9.681 176.51 -8.181 QT\n178.838 -6.681 180.182 -4.181 QT\n181.525 -1.681 181.525 1.147 QT\n181.525 3.913 180.158 6.28 QT\n178.791 8.647 176.439 10.053 QT\n174.088 11.459 171.354 11.459 QT\ncp\n171.354 10.116 M\n175.041 10.116 176.275 7.444 QT\n177.51 4.772 177.51 0.631 QT\n177.51 -1.681 177.26 -3.197 QT\n177.01 -4.712 176.182 -5.947 QT\n175.666 -6.712 174.869 -7.283 QT\n174.072 -7.853 173.182 -8.158 QT\n172.291 -8.462 171.354 -8.462 QT\n169.916 -8.462 168.627 -7.814 QT\n167.338 -7.166 166.479 -5.947 QT\n165.619 -4.65 165.385 -3.087 QT\n165.15 -1.525 165.15 0.631 QT\n165.15 3.209 165.604 5.272 QT\n166.057 7.334 167.416 8.725 QT\n168.775 10.116 171.354 10.116 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n195.42 11.459 M\n192.655 11.459 190.295 10.053 QT\n187.936 8.647 186.569 6.288 QT\n185.202 3.928 185.202 1.147 QT\n185.202 -0.962 185.952 -2.916 QT\n186.702 -4.869 188.108 -6.408 QT\n189.514 -7.947 191.381 -8.814 QT\n193.248 -9.681 195.42 -9.681 QT\n198.248 -9.681 200.577 -8.181 QT\n202.905 -6.681 204.248 -4.181 QT\n205.592 -1.681 205.592 1.147 QT\n205.592 3.913 204.225 6.28 QT\n202.858 8.647 200.506 10.053 QT\n198.155 11.459 195.42 11.459 QT\ncp\n195.42 10.116 M\n199.108 10.116 200.342 7.444 QT\n201.577 4.772 201.577 0.631 QT\n201.577 -1.681 201.327 -3.197 QT\n201.077 -4.712 200.248 -5.947 QT\n199.733 -6.712 198.936 -7.283 QT\n198.139 -7.853 197.248 -8.158 QT\n196.358 -8.462 195.42 -8.462 QT\n193.983 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9.178 QT\n218.533 10.241 220.236 10.241 QT\n221.83 10.241 223.056 9.389 QT\n224.283 8.538 225.119 7.1 QT\n225.955 5.663 226.345 4.077 QT\n226.736 2.491 226.736 1.022 QT\n226.736 -0.822 226.072 -2.97 QT\n225.408 -5.119 224.049 -6.587 QT\n222.689 -8.056 220.752 -8.056 QT\n218.892 -8.056 217.33 -7.103 QT\n215.767 -6.15 214.845 -4.509 QT\n214.845 6.475 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n261.519 22.35 M\n258.957 20.334 257.105 17.717 QT\n255.254 15.1 254.074 12.131 QT\n252.894 9.163 252.308 5.928 QT\n251.723 2.694 251.723 -0.556 QT\n251.723 -3.853 252.308 -7.087 QT\n252.894 -10.322 254.098 -13.314 QT\n255.301 -16.306 257.16 -18.908 QT\n259.019 -21.509 261.519 -23.462 QT\n261.519 -23.556 261.738 -23.556 QT\n262.176 -23.556 L\n262.301 -23.556 262.418 -23.431 QT\n262.535 -23.306 262.535 -23.15 QT\n262.535 -22.947 262.441 -22.853 QT\n260.191 -20.666 258.699 -18.15 QT\n257.207 -15.634 256.293 -12.791 QT\n255.379 -9.947 254.98 -6.9 QT\n254.582 -3.853 254.582 -0.556 QT\n254.582 14.069 262.394 21.663 QT\n262.535 21.788 262.535 22.038 QT\n262.535 22.147 262.41 22.295 QT\n262.285 22.444 262.176 22.444 QT\n261.738 22.444 L\n261.519 22.444 261.519 22.35 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n267.847 7.209 M\n267.847 6.538 267.972 5.928 QT\n271.3 -7.275 L\n266.472 -7.275 L\n266.003 -7.275 266.003 -7.884 QT\n266.175 -8.884 266.612 -8.884 QT\n271.706 -8.884 L\n273.55 -16.384 L\n273.722 -16.994 274.261 -17.423 QT\n274.8 -17.853 275.472 -17.853 QT\n276.065 -17.853 276.456 -17.501 QT\n276.847 -17.15 276.847 -16.572 QT\n276.847 -16.431 276.839 -16.353 QT\n276.831 -16.275 276.8 -16.197 QT\n274.956 -8.884 L\n279.706 -8.884 L\n280.175 -8.884 280.175 -8.291 QT\n280.143 -8.166 280.081 -7.9 QT\n280.018 -7.634 279.901 -7.455 QT\n279.784 -7.275 279.565 -7.275 QT\n274.55 -7.275 L\n271.253 6.022 L\n270.925 7.334 270.925 8.272 QT\n270.925 10.241 272.268 10.241 QT\n274.284 10.241 275.847 8.342 QT\n277.409 6.444 278.237 4.178 QT\n278.425 3.913 278.597 3.913 QT\n279.159 3.913 L\n279.347 3.913 279.456 4.038 QT\n279.565 4.163 279.565 4.319 QT\n279.565 4.413 279.518 4.459 QT\n278.503 7.241 276.573 9.35 QT\n274.643 11.459 272.159 11.459 QT\n270.331 11.459 269.089 10.272 QT\n267.847 9.084 267.847 7.209 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n282.516 17.878 M\n282.516 12.659 L\n282.516 12.409 282.798 12.409 QT\n283.188 12.409 L\n283.376 12.409 283.438 12.659 QT\n284.329 17.299 287.751 17.299 QT\n289.266 17.299 290.29 16.612 QT\n291.313 15.924 291.313 14.487 QT\n291.313 13.456 290.516 12.729 QT\n289.72 12.003 288.626 11.737 QT\n286.485 11.315 L\n285.407 11.081 284.524 10.596 QT\n283.641 10.112 283.079 9.307 QT\n282.516 8.503 282.516 7.44 QT\n282.516 6.034 283.259 5.135 QT\n284.001 4.237 285.188 3.838 QT\n286.376 3.44 287.751 3.44 QT\n289.391 3.44 290.61 4.315 QT\n291.532 3.518 L\n291.532 3.44 291.688 3.44 QT\n291.923 3.44 L\n292.016 3.44 292.095 3.526 QT\n292.173 3.612 292.173 3.706 QT\n292.173 7.893 L\n292.173 8.19 291.923 8.19 QT\n291.532 8.19 L\n291.251 8.19 291.251 7.893 QT\n291.251 6.221 290.321 5.206 QT\n289.391 4.19 287.72 4.19 QT\n286.282 4.19 285.227 4.721 QT\n284.173 5.253 284.173 6.549 QT\n284.173 7.44 284.93 8.01 QT\n285.688 8.581 286.704 8.831 QT\n288.876 9.237 L\n289.97 9.487 290.915 10.081 QT\n291.86 10.674 292.415 11.581 QT\n292.97 12.487 292.97 13.628 QT\n292.97 14.784 292.571 15.635 QT\n292.173 16.487 291.462 17.049 QT\n290.751 17.612 289.782 17.878 QT\n288.813 18.143 287.751 18.143 QT\n285.751 18.143 284.329 16.799 QT\n283.157 18.065 L\n283.157 18.143 282.985 18.143 QT\n282.798 18.143 L\n282.516 18.143 282.516 17.878 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n295.029 17.784 M\n295.029 16.659 L\n296.123 16.659 296.826 16.487 QT\n297.529 16.315 297.529 15.643 QT\n297.529 6.909 L\n297.529 5.674 297.052 5.393 QT\n296.576 5.112 295.17 5.112 QT\n295.17 3.987 L\n299.779 3.643 L\n299.779 15.643 L\n299.779 16.315 300.388 16.487 QT\n300.998 16.659 302.013 16.659 QT\n302.013 17.784 L\n295.029 17.784 L\ncp\n296.388 -1.857 M\n296.388 -2.56 296.92 -3.091 QT\n297.451 -3.622 298.138 -3.622 QT\n298.591 -3.622 299.013 -3.388 QT\n299.435 -3.154 299.67 -2.732 QT\n299.904 -2.31 299.904 -1.857 QT\n299.904 -1.169 299.373 -0.638 QT\n298.841 -0.107 298.138 -0.107 QT\n297.451 -0.107 296.92 -0.638 QT\n296.388 -1.169 296.388 -1.857 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n303.864 17.784 M\n303.864 16.659 L\n304.958 16.659 305.661 16.487 QT\n306.364 16.315 306.364 15.643 QT\n306.364 6.909 L\n306.364 6.049 306.107 5.667 QT\n305.849 5.284 305.364 5.198 QT\n304.88 5.112 303.864 5.112 QT\n303.864 3.987 L\n308.505 3.643 L\n308.505 6.768 L\n309.146 5.393 310.403 4.518 QT\n311.661 3.643 313.146 3.643 QT\n316.833 3.643 317.474 6.643 QT\n318.114 5.299 319.349 4.471 QT\n320.583 3.643 322.052 3.643 QT\n323.505 3.643 324.497 4.112 QT\n325.489 4.581 325.989 5.542 QT\n326.489 6.503 326.489 7.956 QT\n326.489 15.643 L\n326.489 16.315 327.2 16.487 QT\n327.911 16.659 328.989 16.659 QT\n328.989 17.784 L\n321.661 17.784 L\n321.661 16.659 L\n322.755 16.659 323.458 16.487 QT\n324.161 16.315 324.161 15.643 QT\n324.161 8.049 L\n324.161 6.44 323.708 5.456 QT\n323.255 4.471 321.849 4.471 QT\n320.005 4.471 318.802 5.956 QT\n317.599 7.44 317.599 9.331 QT\n317.599 15.643 L\n317.599 16.315 318.302 16.487 QT\n319.005 16.659 320.099 16.659 QT\n320.099 17.784 L\n312.771 17.784 L\n312.771 16.659 L\n313.864 16.659 314.567 16.487 QT\n315.271 16.315 315.271 15.643 QT\n315.271 8.049 L\n315.271 6.487 314.817 5.479 QT\n314.364 4.471 312.958 4.471 QT\n311.099 4.471 309.903 5.956 QT\n308.708 7.44 308.708 9.331 QT\n308.708 15.643 L\n308.708 16.315 309.411 16.487 QT\n310.114 16.659 311.192 16.659 QT\n311.192 17.784 L\n303.864 17.784 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n347.911 4.834 M\n347.536 4.834 347.286 4.545 QT\n347.036 4.256 347.036 3.913 QT\n347.036 3.538 347.286 3.264 QT\n347.536 2.991 347.911 2.991 QT\n376.801 2.991 L\n377.145 2.991 377.387 3.264 QT\n377.629 3.538 377.629 3.913 QT\n377.629 4.256 377.387 4.545 QT\n377.145 4.834 376.801 4.834 QT\n347.911 4.834 L\ncp\n347.911 -4.103 M\n347.536 -4.103 347.286 -4.376 QT\n347.036 -4.65 347.036 -5.025 QT\n347.036 -5.369 347.286 -5.658 QT\n347.536 -5.947 347.911 -5.947 QT\n376.801 -5.947 L\n377.145 -5.947 377.387 -5.658 QT\n377.629 -5.369 377.629 -5.025 QT\n377.629 -4.65 377.387 -4.376 QT\n377.145 -4.103 376.801 -4.103 QT\n347.911 -4.103 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n396.856 10.944 M\n396.856 9.319 L\n402.606 9.319 402.606 7.866 QT\n402.606 -16.275 L\n400.231 -15.134 396.591 -15.134 QT\n396.591 -16.744 L\n402.231 -16.744 405.106 -19.697 QT\n405.747 -19.697 L\n405.903 -19.697 406.052 -19.572 QT\n406.2 -19.447 406.2 -19.291 QT\n406.2 7.866 L\n406.2 9.319 411.95 9.319 QT\n411.95 10.944 L\n396.856 10.944 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0.375 222.39173 137.37397] CT\r\nN\r\n449.691 11.959 M\n444.05 11.959 442.019 7.319 QT\n439.988 2.678 439.988 -3.728 QT\n439.988 -7.728 440.714 -11.251 QT\n441.441 -14.775 443.613 -17.236 QT\n445.785 -19.697 449.691 -19.697 QT\n452.722 -19.697 454.652 -18.212 QT\n456.582 -16.728 457.597 -14.384 QT\n458.613 -12.041 458.98 -9.353 QT\n459.347 -6.666 459.347 -3.728 QT\n459.347 0.225 458.621 3.678 QT\n457.894 7.131 455.761 9.545 QT\n453.628 11.959 449.691 11.959 QT\ncp\n449.691 10.756 M\n452.253 10.756 453.511 8.131 QT\n454.769 5.506 455.058 2.319 QT\n455.347 -0.869 455.347 -4.462 QT\n455.347 -7.931 455.058 -10.845 QT\n454.769 -13.759 453.519 -16.134 QT\n452.269 -18.509 449.691 -18.509 QT\n447.082 -18.509 445.824 -16.126 QT\n444.566 -13.744 444.277 -10.837 QT\n443.988 -7.931 443.988 -4.462 QT\n443.988 -1.9 444.113 0.366 QT\n444.238 2.631 444.777 5.045 QT\n445.316 7.459 446.519 9.108 QT\n447.722 10.756 449.691 10.756 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 137.37397] CT\r\nN\r\n463.975 22.444 M\n463.569 22.444 463.569 22.038 QT\n463.569 21.834 463.663 21.741 QT\n471.522 14.069 471.522 -0.556 QT\n471.522 -15.181 463.757 -22.775 QT\n463.569 -22.884 463.569 -23.15 QT\n463.569 -23.306 463.694 -23.431 QT\n463.819 -23.556 463.975 -23.556 QT\n464.413 -23.556 L\n464.538 -23.556 464.632 -23.462 QT\n467.928 -20.869 470.132 -17.134 QT\n472.335 -13.4 473.358 -9.181 QT\n474.382 -4.962 474.382 -0.556 QT\n474.382 2.694 473.827 5.85 QT\n473.272 9.006 472.077 12.077 QT\n470.882 15.147 469.038 17.741 QT\n467.194 20.334 464.632 22.35 QT\n464.538 22.444 464.413 22.444 QT\n463.975 22.444 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 LJ\r\n2.667 LW\r\nN\r\n507.004 366.331 M\n587.042 366.331 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 205.13354 137.37397] CT\r\nN\r\nf-1792389198\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 158.24938] CT\r\nN\r\n17.906 11.959 M\n13.516 11.959 10.023 9.631 QT\n6.531 7.303 4.555 3.452 QT\n2.578 -0.4 2.578 -4.65 QT\n2.578 -7.791 3.719 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87.302 7.866 QT\n87.302 -4.697 L\n87.302 -5.931 86.927 -6.478 QT\n86.552 -7.025 85.857 -7.15 QT\n85.162 -7.275 83.709 -7.275 QT\n83.709 -8.884 L\n90.38 -9.384 L\n90.38 -4.884 L\n91.302 -6.869 93.107 -8.126 QT\n94.912 -9.384 97.052 -9.384 QT\n100.24 -9.384 101.841 -7.853 QT\n103.443 -6.322 103.443 -3.181 QT\n103.443 7.866 L\n103.443 8.834 104.459 9.077 QT\n105.474 9.319 107.037 9.319 QT\n107.037 10.944 L\n96.505 10.944 L\n96.505 9.319 L\n98.084 9.319 99.091 9.077 QT\n100.099 8.834 100.099 7.866 QT\n100.099 -3.056 L\n100.099 -5.291 99.451 -6.744 QT\n98.802 -8.197 96.771 -8.197 QT\n94.099 -8.197 92.38 -6.064 QT\n90.662 -3.931 90.662 -1.212 QT\n90.662 7.866 L\n90.662 8.834 91.677 9.077 QT\n92.693 9.319 94.24 9.319 QT\n94.24 10.944 L\n83.709 10.944 L\ncp}def\r\nf-1076963521\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f-11361369{108.183 2.428 M\n108.183 -0.416 L\n120.354 -0.416 L\n120.354 2.428 L\n108.183 2.428 L\ncp}def\r\nf-11361369\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f339891938{124.289 10.944 M\n124.289 9.319 L\n125.851 9.319 126.867 9.077 QT\n127.883 8.834 127.883 7.866 QT\n127.883 -16.275 L\n127.883 -17.509 127.508 -18.064 QT\n127.133 -18.619 126.437 -18.744 QT\n125.742 -18.869 124.289 -18.869 QT\n124.289 -20.478 L\n131.117 -20.978 L\n131.117 7.866 L\n131.117 8.834 132.125 9.077 QT\n133.133 9.319 134.679 9.319 QT\n134.679 10.944 L\n124.289 10.944 L\ncp}def\r\nf339891938\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f-2132429885{147.034 11.459 M\n144.268 11.459 141.909 10.053 QT\n139.549 8.647 138.182 6.288 QT\n136.815 3.928 136.815 1.147 QT\n136.815 -0.962 137.565 -2.916 QT\n138.315 -4.869 139.721 -6.408 QT\n141.127 -7.947 142.995 -8.814 QT\n144.862 -9.681 147.034 -9.681 QT\n149.862 -9.681 152.19 -8.181 QT\n154.518 -6.681 155.862 -4.181 QT\n157.206 -1.681 157.206 1.147 QT\n157.206 3.913 155.838 6.28 QT\n154.471 8.647 152.12 10.053 QT\n149.768 11.459 147.034 11.459 QT\ncp\n147.034 10.116 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171.1 11.459 QT\ncp\n171.1 10.116 M\n174.788 10.116 176.022 7.444 QT\n177.257 4.772 177.257 0.631 QT\n177.257 -1.681 177.007 -3.197 QT\n176.757 -4.712 175.929 -5.947 QT\n175.413 -6.712 174.616 -7.283 QT\n173.819 -7.853 172.929 -8.158 QT\n172.038 -8.462 171.1 -8.462 QT\n169.663 -8.462 168.374 -7.814 QT\n167.085 -7.166 166.225 -5.947 QT\n165.366 -4.65 165.132 -3.087 QT\n164.897 -1.525 164.897 0.631 QT\n164.897 3.209 165.35 5.272 QT\n165.804 7.334 167.163 8.725 QT\n168.522 10.116 171.1 10.116 QT\ncp}def\r\nf1863324703\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f381148119{183.588 19.866 M\n183.588 18.272 L\n185.166 18.272 186.174 17.999 QT\n187.182 17.725 187.182 16.788 QT\n187.182 -5.384 L\n187.182 -6.65 186.283 -6.962 QT\n185.385 -7.275 183.588 -7.275 QT\n183.588 -8.884 L\n190.4 -9.384 L\n190.4 -6.525 L\n191.65 -7.931 193.322 -8.658 QT\n194.994 -9.384 196.885 -9.384 QT\n199.604 -9.384 201.783 -7.916 QT\n203.963 -6.447 205.19 -4.048 QT\n206.416 -1.65 206.416 1.022 QT\n206.416 3.803 205.057 6.202 QT\n203.697 8.6 201.346 10.03 QT\n198.994 11.459 196.197 11.459 QT\n192.838 11.459 190.525 8.741 QT\n190.525 16.788 L\n190.525 17.725 191.549 17.999 QT\n192.572 18.272 194.119 18.272 QT\n194.119 19.866 L\n183.588 19.866 L\ncp\n190.525 6.475 M\n191.338 8.116 192.775 9.178 QT\n194.213 10.241 195.916 10.241 QT\n197.51 10.241 198.736 9.389 QT\n199.963 8.538 200.799 7.1 QT\n201.635 5.663 202.025 4.077 QT\n202.416 2.491 202.416 1.022 QT\n202.416 -0.822 201.752 -2.97 QT\n201.088 -5.119 199.729 -6.587 QT\n198.369 -8.056 196.432 -8.056 QT\n194.572 -8.056 193.01 -7.103 QT\n191.447 -6.15 190.525 -4.509 QT\n190.525 6.475 L\ncp}def\r\nf381148119\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f350766748{237.2 22.35 M\n234.637 20.334 232.785 17.717 QT\n230.934 15.1 229.754 12.131 QT\n228.575 9.163 227.989 5.928 QT\n227.403 2.694 227.403 -0.556 QT\n227.403 -3.853 227.989 -7.087 QT\n228.575 -10.322 229.778 -13.314 QT\n230.981 -16.306 232.84 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-2.541 342.793 -0.783 QT\n343.887 0.975 343.887 2.991 QT\n343.887 5.538 342.488 7.592 QT\n341.09 9.647 338.824 10.803 QT\n336.559 11.959 334.043 11.959 QT\n331.887 11.959 329.715 11.139 QT\n327.543 10.319 326.168 8.678 QT\n324.793 7.038 324.793 4.741 QT\n324.793 3.6 325.551 2.834 QT\n326.309 2.069 327.465 2.069 QT\n328.199 2.069 328.816 2.42 QT\n329.434 2.772 329.785 3.397 QT\n330.137 4.022 330.137 4.741 QT\n330.137 5.866 329.348 6.631 QT\n328.559 7.397 327.465 7.397 QT\n327.23 7.397 L\ncp}def\r\nf511004193\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 222.39173 179.1248] CT\r\nN\r\n/f-1239821374{348.64 22.444 M\n348.234 22.444 348.234 22.038 QT\n348.234 21.834 348.327 21.741 QT\n356.187 14.069 356.187 -0.556 QT\n356.187 -15.181 348.421 -22.775 QT\n348.234 -22.884 348.234 -23.15 QT\n348.234 -23.306 348.359 -23.431 QT\n348.484 -23.556 348.64 -23.556 QT\n349.077 -23.556 L\n349.202 -23.556 349.296 -23.462 QT\n352.593 -20.869 354.796 -17.134 QT\n356.999 -13.4 358.023 -9.181 QT\n359.046 -4.962 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0 0.375 0 0] CT\r\n0 0 1 RC\r\n1 LJ\r\n5.333 LW\r\nN\r\n146 630.341 M\n182.167 607.032 L\n218.333 578.274 L\n254.5 548.034 L\n290.667 519.525 L\n326.833 491.782 L\n363 463.666 L\n399.167 434.673 L\n435.333 403.658 L\n471.5 371.322 L\n507.667 341.392 L\n543.833 314.617 L\n580 291.452 L\n616.167 271.678 L\n652.333 254.582 L\n688.5 239.544 L\n724.667 226.227 L\n760.833 214.491 L\n797 204.256 L\n833.167 195.44 L\n869.333 187.936 L\n905.5 181.614 L\n941.667 176.319 L\n977.833 171.866 L\n1014 168.036 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n0 0 1 RC\r\n1 LJ\r\n5.333 LW\r\nN\r\n146 630.341 M\n182.167 607.032 L\n218.333 578.274 L\n254.5 548.035 L\n290.667 519.527 L\n326.833 491.786 L\n363 463.672 L\n399.167 434.68 L\n435.333 403.665 L\n471.5 371.329 L\n507.667 341.397 L\n543.833 314.617 L\n580 291.448 L\n616.167 271.671 L\n652.333 254.573 L\n688.5 239.534 L\n724.667 226.217 L\n760.833 214.481 L\n797 204.247 L\n833.167 195.431 L\n869.333 187.928 L\n905.5 181.606 L\n941.667 176.311 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L\n14.156 -5.433 14.156 -6.386 QT\n14.156 -8.449 12.734 -8.449 QT\n10.625 -8.449 9.039 -6.543 QT\n7.453 -4.636 6.469 -2.121 QT\n6.375 -1.824 6.094 -1.824 QT\n5.531 -1.824 L\n5.141 -1.824 5.141 -2.261 QT\n5.141 -2.402 L\n6.219 -5.293 8.211 -7.496 QT\n10.203 -9.699 12.828 -9.699 QT\n14.813 -9.699 16.094 -8.457 QT\n17.375 -7.215 17.375 -5.293 QT\n17.375 -4.605 17.234 -3.933 QT\n12.688 14.317 L\n12.188 16.254 10.805 17.871 QT\n9.422 19.489 7.5 20.426 QT\n5.578 21.364 3.563 21.364 QT\n1.953 21.364 0.656 20.59 QT\n-0.641 19.817 -0.641 18.348 QT\ncp\n14.563 -17.574 M\n14.563 -18.621 15.383 -19.418 QT\n16.203 -20.215 17.234 -20.215 QT\n18.047 -20.215 18.578 -19.715 QT\n19.109 -19.215 19.109 -18.433 QT\n19.109 -17.355 18.25 -16.55 QT\n17.391 -15.746 16.359 -15.746 QT\n15.594 -15.746 15.078 -16.285 QT\n14.563 -16.824 14.563 -17.574 QT\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 312.43751 120.85937] CT\r\nN\r\n39.457 5.145 M\n39.066 5.145 38.808 4.84 QT\n38.55 4.535 38.55 4.176 QT\n38.55 3.785 38.808 3.504 QT\n39.066 3.223 39.457 3.223 QT\n69.597 3.223 L\n69.957 3.223 70.215 3.504 QT\n70.472 3.785 70.472 4.176 QT\n70.472 4.535 70.215 4.84 QT\n69.957 5.145 69.597 5.145 QT\n39.457 5.145 L\ncp\n39.457 -4.183 M\n39.066 -4.183 38.808 -4.465 QT\n38.55 -4.746 38.55 -5.152 QT\n38.55 -5.496 38.808 -5.8 QT\n39.066 -6.105 39.457 -6.105 QT\n69.597 -6.105 L\n69.957 -6.105 70.215 -5.8 QT\n70.472 -5.496 70.472 -5.152 QT\n70.472 -4.746 70.215 -4.465 QT\n69.957 -4.183 69.597 -4.183 QT\n39.457 -4.183 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 312.43751 120.85937] CT\r\nN\r\n90.967 11.52 M\n90.967 9.832 L\n96.967 9.832 96.967 8.301 QT\n96.967 -16.886 L\n94.483 -15.699 90.686 -15.699 QT\n90.686 -17.386 L\n96.576 -17.386 99.576 -20.449 QT\n100.248 -20.449 L\n100.42 -20.449 100.569 -20.324 QT\n100.717 -20.199 100.717 -20.027 QT\n100.717 8.301 L\n100.717 9.832 106.717 9.832 QT\n106.717 11.52 L\n90.967 11.52 L\ncp\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 126.63126 118.875] CT\r\nN\r\n/f946251025{-0.641 18.348 M\n-0.641 17.176 0.172 16.317 QT\n0.984 15.457 2.156 15.457 QT\n2.938 15.457 3.461 15.942 QT\n3.984 16.426 3.984 17.192 QT\n3.984 18.114 3.414 18.848 QT\n2.844 19.582 1.969 19.817 QT\n2.813 20.129 3.656 20.129 QT\n5.703 20.129 7.203 18.239 QT\n8.703 16.348 9.313 14.004 QT\n13.828 -4.043 L\n14.156 -5.433 14.156 -6.386 QT\n14.156 -8.449 12.734 -8.449 QT\n10.625 -8.449 9.039 -6.543 QT\n7.453 -4.636 6.469 -2.121 QT\n6.375 -1.824 6.094 -1.824 QT\n5.531 -1.824 L\n5.141 -1.824 5.141 -2.261 QT\n5.141 -2.402 L\n6.219 -5.293 8.211 -7.496 QT\n10.203 -9.699 12.828 -9.699 QT\n14.813 -9.699 16.094 -8.457 QT\n17.375 -7.215 17.375 -5.293 QT\n17.375 -4.605 17.234 -3.933 QT\n12.688 14.317 L\n12.188 16.254 10.805 17.871 QT\n9.422 19.489 7.5 20.426 QT\n5.578 21.364 3.563 21.364 QT\n1.953 21.364 0.656 20.59 QT\n-0.641 19.817 -0.641 18.348 QT\ncp\n14.563 -17.574 M\n14.563 -18.621 15.383 -19.418 QT\n16.203 -20.215 17.234 -20.215 QT\n18.047 -20.215 18.578 -19.715 QT\n19.109 -19.215 19.109 -18.433 QT\n19.109 -17.355 18.25 -16.55 QT\n17.391 -15.746 16.359 -15.746 QT\n15.594 -15.746 15.078 -16.285 QT\n14.563 -16.824 14.563 -17.574 QT\ncp}def\r\nf946251025\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 126.63126 118.875] CT\r\nN\r\n/f-1792154824{39.457 5.145 M\n39.066 5.145 38.808 4.84 QT\n38.55 4.535 38.55 4.176 QT\n38.55 3.785 38.808 3.504 QT\n39.066 3.223 39.457 3.223 QT\n69.597 3.223 L\n69.957 3.223 70.215 3.504 QT\n70.472 3.785 70.472 4.176 QT\n70.472 4.535 70.215 4.84 QT\n69.957 5.145 69.597 5.145 QT\n39.457 5.145 L\ncp\n39.457 -4.183 M\n39.066 -4.183 38.808 -4.465 QT\n38.55 -4.746 38.55 -5.152 QT\n38.55 -5.496 38.808 -5.8 QT\n39.066 -6.105 39.457 -6.105 QT\n69.597 -6.105 L\n69.957 -6.105 70.215 -5.8 QT\n70.472 -5.496 70.472 -5.152 QT\n70.472 -4.746 70.215 -4.465 QT\n69.957 -4.183 69.597 -4.183 QT\n39.457 -4.183 L\ncp}def\r\nf-1792154824\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 126.63126 118.875] CT\r\nN\r\n/f1600933324{88.904 11.52 M\n88.904 10.223 L\n88.904 10.114 88.998 9.973 QT\n96.451 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126.547 5.66 QT\n126.594 5.489 126.719 5.364 QT\n126.844 5.239 127.016 5.239 QT\n127.438 5.239 L\n127.922 5.239 127.922 5.848 QT\n127.25 8.567 125 10.317 QT\n122.75 12.067 119.953 12.067 QT\ncp\n113.578 -0.761 M\n124.531 -0.761 L\n124.531 -2.574 124.024 -4.425 QT\n123.516 -6.277 122.344 -7.504 QT\n121.172 -8.73 119.281 -8.73 QT\n116.563 -8.73 115.071 -6.191 QT\n113.578 -3.652 113.578 -0.761 QT\ncp}def\r\nf1730672616\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1581324804{130.584 11.52 M\n130.584 9.832 L\n132.224 9.832 133.279 9.575 QT\n134.334 9.317 134.334 8.301 QT\n134.334 -4.793 L\n134.334 -6.09 133.943 -6.66 QT\n133.552 -7.23 132.826 -7.363 QT\n132.099 -7.496 130.584 -7.496 QT\n130.584 -9.183 L\n137.443 -9.699 L\n137.443 -5.011 L\n138.224 -7.09 139.654 -8.394 QT\n141.084 -9.699 143.115 -9.699 QT\n144.552 -9.699 145.677 -8.855 QT\n146.802 -8.011 146.802 -6.621 QT\n146.802 -5.761 146.177 -5.113 QT\n145.552 -4.465 144.646 -4.465 QT\n143.755 -4.465 143.123 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46.667 10.645 QT\ncp}def\r\nf-518588683\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f763322885{60.104 11.52 M\n60.104 9.832 L\n61.745 9.832 62.8 9.575 QT\n63.854 9.317 63.854 8.301 QT\n63.854 -4.793 L\n63.854 -6.09 63.464 -6.66 QT\n63.073 -7.23 62.346 -7.363 QT\n61.62 -7.496 60.104 -7.496 QT\n60.104 -9.183 L\n67.057 -9.699 L\n67.057 -5.011 L\n68.026 -7.074 69.909 -8.386 QT\n71.792 -9.699 74.026 -9.699 QT\n77.354 -9.699 79.026 -8.105 QT\n80.698 -6.511 80.698 -3.23 QT\n80.698 8.301 L\n80.698 9.317 81.753 9.575 QT\n82.807 9.832 84.448 9.832 QT\n84.448 11.52 L\n73.464 11.52 L\n73.464 9.832 L\n75.104 9.832 76.159 9.575 QT\n77.214 9.317 77.214 8.301 QT\n77.214 -3.09 L\n77.214 -5.433 76.534 -6.941 QT\n75.854 -8.449 73.745 -8.449 QT\n70.948 -8.449 69.159 -6.222 QT\n67.37 -3.996 67.37 -1.168 QT\n67.37 8.301 L\n67.37 9.317 68.425 9.575 QT\n69.479 9.832 71.089 9.832 QT\n71.089 11.52 L\n60.104 11.52 L\ncp}def\r\nf763322885\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1813278697{95.719 11.52 M\n88.719 -6.105 L\n88.266 -6.996 87.344 -7.246 QT\n86.422 -7.496 84.922 -7.496 QT\n84.922 -9.183 L\n95 -9.183 L\n95 -7.496 L\n92.297 -7.496 92.297 -6.34 QT\n92.297 -6.152 92.344 -6.058 QT\n97.735 7.489 L\n102.594 -4.746 L\n102.735 -5.121 102.735 -5.527 QT\n102.735 -6.433 102.039 -6.965 QT\n101.344 -7.496 100.406 -7.496 QT\n100.406 -9.183 L\n108.375 -9.183 L\n108.375 -7.496 L\n106.906 -7.496 105.789 -6.816 QT\n104.672 -6.136 104.063 -4.793 QT\n97.594 11.52 L\n97.406 12.067 96.828 12.067 QT\n96.469 12.067 L\n95.891 12.067 95.719 11.52 QT\ncp}def\r\nf1813278697\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1730672616{119.953 12.067 M\n117.031 12.067 114.578 10.528 QT\n112.125 8.989 110.735 6.418 QT\n109.344 3.848 109.344 0.989 QT\n109.344 -1.824 110.617 -4.355 QT\n111.891 -6.886 114.18 -8.441 QT\n116.469 -9.996 119.281 -9.996 QT\n121.485 -9.996 123.11 -9.261 QT\n124.735 -8.527 125.789 -7.215 QT\n126.844 -5.902 127.383 -4.121 QT\n127.922 -2.34 127.922 -0.199 QT\n127.922 0.426 127.438 0.426 QT\n113.531 0.426 L\n113.531 0.942 L\n113.531 4.926 115.141 7.785 QT\n116.75 10.645 120.375 10.645 QT\n121.86 10.645 123.11 9.989 QT\n124.36 9.332 125.289 8.16 QT\n126.219 6.989 126.547 5.66 QT\n126.594 5.489 126.719 5.364 QT\n126.844 5.239 127.016 5.239 QT\n127.438 5.239 L\n127.922 5.239 127.922 5.848 QT\n127.25 8.567 125 10.317 QT\n122.75 12.067 119.953 12.067 QT\ncp\n113.578 -0.761 M\n124.531 -0.761 L\n124.531 -2.574 124.024 -4.425 QT\n123.516 -6.277 122.344 -7.504 QT\n121.172 -8.73 119.281 -8.73 QT\n116.563 -8.73 115.071 -6.191 QT\n113.578 -3.652 113.578 -0.761 QT\ncp}def\r\nf1730672616\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1581324804{130.584 11.52 M\n130.584 9.832 L\n132.224 9.832 133.279 9.575 QT\n134.334 9.317 134.334 8.301 QT\n134.334 -4.793 L\n134.334 -6.09 133.943 -6.66 QT\n133.552 -7.23 132.826 -7.363 QT\n132.099 -7.496 130.584 -7.496 QT\n130.584 -9.183 L\n137.443 -9.699 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156.641 19.598 QT\n158.493 20.129 160.134 20.129 QT\n161.759 20.129 163.61 19.598 QT\n165.462 19.067 166.759 17.981 QT\n168.055 16.895 168.055 15.27 QT\n168.055 12.77 165.759 12.028 QT\n163.462 11.285 160.18 11.285 QT\n156.243 11.285 L\n155.149 11.285 154.22 11.817 QT\n153.29 12.348 152.727 13.293 QT\n152.165 14.239 152.165 15.27 QT\ncp\n158.805 3.176 M\n162.884 3.176 162.884 -2.636 QT\n162.884 -5.152 162.016 -6.777 QT\n161.149 -8.402 158.805 -8.402 QT\n156.462 -8.402 155.595 -6.777 QT\n154.727 -5.152 154.727 -2.636 QT\n154.727 -1.043 155.055 0.246 QT\n155.384 1.535 156.274 2.356 QT\n157.165 3.176 158.805 3.176 QT\ncp}def\r\nf-1655348641\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f402514587{184.087 12.067 M\n181.165 12.067 178.712 10.528 QT\n176.259 8.989 174.868 6.418 QT\n173.477 3.848 173.477 0.989 QT\n173.477 -1.824 174.751 -4.355 QT\n176.024 -6.886 178.313 -8.441 QT\n180.602 -9.996 183.415 -9.996 QT\n185.618 -9.996 187.243 -9.261 QT\n188.868 -8.527 189.923 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467.549 L\n799 499.15 L\n834.833 528.387 L\n870.667 554.779 L\n906.5 577.8 L\n942.333 596.791 L\n978.167 610.865 L\n1014 618.777 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 128.386 L\n225.667 79.354 L\n261.5 63 L\n297.333 69.932 L\n333.167 86.646 L\n369 109.644 L\n404.833 139.632 L\n440.667 175.094 L\n476.5 214.238 L\n512.333 253.648 L\n548.167 291.663 L\n584 329.125 L\n619.833 365.829 L\n655.667 401.383 L\n691.5 435.442 L\n727.333 467.74 L\n763.167 498.059 L\n799 526.19 L\n834.833 551.907 L\n870.667 574.932 L\n906.5 594.896 L\n942.333 611.289 L\n978.167 623.39 L\n1014 630.17 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 146.994 L\n225.667 111.335 L\n261.5 98.378 L\n297.333 98.454 L\n333.167 107.935 L\n369 125.267 L\n404.833 149.921 L\n440.667 179.109 L\n476.5 211.943 L\n512.333 247.772 L\n548.167 285.489 L\n584 323.745 L\n619.833 361.615 L\n655.667 398.417 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289.351 L\n584 327.245 L\n619.833 364.732 L\n655.667 401.143 L\n691.5 435.994 L\n727.333 468.93 L\n763.167 499.688 L\n799 528.059 L\n834.833 553.842 L\n870.667 576.799 L\n906.5 596.608 L\n942.333 612.807 L\n978.167 624.724 L\n1014 631.384 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 141.18 L\n225.667 101.803 L\n261.5 92.381 L\n297.333 96.853 L\n333.167 109.04 L\n369 127.944 L\n404.833 153.838 L\n440.667 184.594 L\n476.5 219.217 L\n512.333 256.221 L\n548.167 294.174 L\n584 331.972 L\n619.833 368.893 L\n655.667 404.477 L\n691.5 438.417 L\n727.333 470.496 L\n763.167 500.538 L\n799 528.371 L\n834.833 553.792 L\n870.667 576.538 L\n906.5 596.254 L\n942.333 612.439 L\n978.167 624.383 L\n1014 631.074 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 144.482 L\n225.667 107.382 L\n261.5 98.071 L\n297.333 102.175 L\n333.167 113.86 L\n369 131.999 L\n404.833 156.66 L\n440.667 185.449 L\n476.5 218.071 L\n512.333 253.87 L\n548.167 291.353 L\n584 329.228 L\n619.833 366.581 L\n655.667 402.775 L\n691.5 437.367 L\n727.333 470.037 L\n763.167 500.553 L\n799 528.718 L\n834.833 554.336 L\n870.667 577.168 L\n906.5 596.887 L\n942.333 613.026 L\n978.167 624.907 L\n1014 631.549 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 142.164 L\n225.667 103.471 L\n261.5 94.045 L\n297.333 98.427 L\n333.167 110.523 L\n369 129.294 L\n404.833 154.967 L\n440.667 185.359 L\n476.5 219.584 L\n512.333 256.322 L\n548.167 294.15 L\n584 331.934 L\n619.833 368.907 L\n655.667 404.569 L\n691.5 438.59 L\n727.333 470.739 L\n763.167 500.832 L\n799 528.692 L\n834.833 554.118 L\n870.667 576.851 L\n906.5 596.54 L\n942.333 612.693 L\n978.167 624.605 L\n1014 631.276 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 144.033 L\n225.667 106.633 L\n261.5 97.411 L\n297.333 101.668 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143.764 L\n225.667 106.182 L\n261.5 96.966 L\n297.333 101.279 L\n333.167 113.185 L\n369 131.594 L\n404.833 156.632 L\n440.667 185.981 L\n476.5 219.129 L\n512.333 255.223 L\n548.167 292.811 L\n584 330.646 L\n619.833 367.853 L\n655.667 403.836 L\n691.5 438.192 L\n727.333 470.638 L\n763.167 500.963 L\n799 528.979 L\n834.833 554.49 L\n870.667 577.252 L\n906.5 596.929 L\n942.333 613.047 L\n978.167 624.92 L\n1014 631.561 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 142.925 L\n225.667 104.76 L\n261.5 95.422 L\n297.333 99.782 L\n333.167 111.813 L\n369 130.449 L\n404.833 155.876 L\n440.667 185.856 L\n476.5 219.642 L\n512.333 256.106 L\n548.167 293.823 L\n584 331.616 L\n619.833 368.668 L\n655.667 404.441 L\n691.5 438.578 L\n727.333 470.828 L\n763.167 500.997 L\n799 528.905 L\n834.833 554.352 L\n870.667 577.085 L\n906.5 596.759 L\n942.333 612.888 L\n978.167 624.778 L\n1014 631.433 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 143.597 L\n225.667 105.9 L\n261.5 96.673 L\n297.333 101.007 L\n333.167 112.948 L\n369 131.411 L\n404.833 156.534 L\n440.667 186.017 L\n476.5 219.297 L\n512.333 255.467 L\n548.167 293.08 L\n584 330.905 L\n619.833 368.078 L\n655.667 404.013 L\n691.5 438.318 L\n727.333 470.717 L\n763.167 501.004 L\n799 528.993 L\n834.833 554.486 L\n870.667 577.238 L\n906.5 596.911 L\n942.333 613.028 L\n978.167 624.902 L\n1014 631.546 L\r\nS\r\nGR\r\nGS\r\n[0.375 0 0 0.375 0 0] CT\r\n1 0 0 RC\r\n1 LJ\r\n2.667 LW\r\nN\r\n154 224.145 M\n189.833 143.08 L\n225.667 105.023 L\n261.5 95.711 L\n297.333 100.069 L\n333.167 112.083 L\n369 130.685 L\n404.833 156.05 L\n440.667 185.927 L\n476.5 219.606 L\n512.333 256.006 L\n548.167 293.7 L\n584 331.498 L\n619.833 368.573 L\n655.667 404.378 L\n691.5 438.546 L\n727.333 470.825 L\n763.167 501.016 L\n799 528.939 L\n834.833 554.394 L\n870.667 577.128 L\n906.5 596.8 L\n942.333 612.926 L\n978.167 624.811 L\n1014 631.463 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12.067 QT\ncp\n46.667 10.645 M\n50.51 10.645 51.8 7.856 QT\n53.089 5.067 53.089 0.754 QT\n53.089 -1.652 52.831 -3.238 QT\n52.573 -4.824 51.714 -6.105 QT\n51.167 -6.902 50.339 -7.504 QT\n49.51 -8.105 48.581 -8.418 QT\n47.651 -8.73 46.667 -8.73 QT\n45.167 -8.73 43.823 -8.05 QT\n42.479 -7.371 41.589 -6.105 QT\n40.698 -4.746 40.448 -3.121 QT\n40.198 -1.496 40.198 0.754 QT\n40.198 3.457 40.667 5.598 QT\n41.135 7.739 42.557 9.192 QT\n43.979 10.645 46.667 10.645 QT\ncp}def\r\nf-518588683\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f763322885{60.104 11.52 M\n60.104 9.832 L\n61.745 9.832 62.8 9.575 QT\n63.854 9.317 63.854 8.301 QT\n63.854 -4.793 L\n63.854 -6.09 63.464 -6.66 QT\n63.073 -7.23 62.346 -7.363 QT\n61.62 -7.496 60.104 -7.496 QT\n60.104 -9.183 L\n67.057 -9.699 L\n67.057 -5.011 L\n68.026 -7.074 69.909 -8.386 QT\n71.792 -9.699 74.026 -9.699 QT\n77.354 -9.699 79.026 -8.105 QT\n80.698 -6.511 80.698 -3.23 QT\n80.698 8.301 L\n80.698 9.317 81.753 9.575 QT\n82.807 9.832 84.448 9.832 QT\n84.448 11.52 L\n73.464 11.52 L\n73.464 9.832 L\n75.104 9.832 76.159 9.575 QT\n77.214 9.317 77.214 8.301 QT\n77.214 -3.09 L\n77.214 -5.433 76.534 -6.941 QT\n75.854 -8.449 73.745 -8.449 QT\n70.948 -8.449 69.159 -6.222 QT\n67.37 -3.996 67.37 -1.168 QT\n67.37 8.301 L\n67.37 9.317 68.425 9.575 QT\n69.479 9.832 71.089 9.832 QT\n71.089 11.52 L\n60.104 11.52 L\ncp}def\r\nf763322885\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1813278697{95.719 11.52 M\n88.719 -6.105 L\n88.266 -6.996 87.344 -7.246 QT\n86.422 -7.496 84.922 -7.496 QT\n84.922 -9.183 L\n95 -9.183 L\n95 -7.496 L\n92.297 -7.496 92.297 -6.34 QT\n92.297 -6.152 92.344 -6.058 QT\n97.735 7.489 L\n102.594 -4.746 L\n102.735 -5.121 102.735 -5.527 QT\n102.735 -6.433 102.039 -6.965 QT\n101.344 -7.496 100.406 -7.496 QT\n100.406 -9.183 L\n108.375 -9.183 L\n108.375 -7.496 L\n106.906 -7.496 105.789 -6.816 QT\n104.672 -6.136 104.063 -4.793 QT\n97.594 11.52 L\n97.406 12.067 96.828 12.067 QT\n96.469 12.067 L\n95.891 12.067 95.719 11.52 QT\ncp}def\r\nf1813278697\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f1730672616{119.953 12.067 M\n117.031 12.067 114.578 10.528 QT\n112.125 8.989 110.735 6.418 QT\n109.344 3.848 109.344 0.989 QT\n109.344 -1.824 110.617 -4.355 QT\n111.891 -6.886 114.18 -8.441 QT\n116.469 -9.996 119.281 -9.996 QT\n121.485 -9.996 123.11 -9.261 QT\n124.735 -8.527 125.789 -7.215 QT\n126.844 -5.902 127.383 -4.121 QT\n127.922 -2.34 127.922 -0.199 QT\n127.922 0.426 127.438 0.426 QT\n113.531 0.426 L\n113.531 0.942 L\n113.531 4.926 115.141 7.785 QT\n116.75 10.645 120.375 10.645 QT\n121.86 10.645 123.11 9.989 QT\n124.36 9.332 125.289 8.16 QT\n126.219 6.989 126.547 5.66 QT\n126.594 5.489 126.719 5.364 QT\n126.844 5.239 127.016 5.239 QT\n127.438 5.239 L\n127.922 5.239 127.922 5.848 QT\n127.25 8.567 125 10.317 QT\n122.75 12.067 119.953 12.067 QT\ncp\n113.578 -0.761 M\n124.531 -0.761 L\n124.531 -2.574 124.024 -4.425 QT\n123.516 -6.277 122.344 -7.504 QT\n121.172 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QT\n154.727 -1.043 155.055 0.246 QT\n155.384 1.535 156.274 2.356 QT\n157.165 3.176 158.805 3.176 QT\ncp}def\r\nf-1655348641\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f402514587{184.087 12.067 M\n181.165 12.067 178.712 10.528 QT\n176.259 8.989 174.868 6.418 QT\n173.477 3.848 173.477 0.989 QT\n173.477 -1.824 174.751 -4.355 QT\n176.024 -6.886 178.313 -8.441 QT\n180.602 -9.996 183.415 -9.996 QT\n185.618 -9.996 187.243 -9.261 QT\n188.868 -8.527 189.923 -7.215 QT\n190.977 -5.902 191.516 -4.121 QT\n192.056 -2.34 192.056 -0.199 QT\n192.056 0.426 191.571 0.426 QT\n177.665 0.426 L\n177.665 0.942 L\n177.665 4.926 179.274 7.785 QT\n180.884 10.645 184.509 10.645 QT\n185.993 10.645 187.243 9.989 QT\n188.493 9.332 189.423 8.16 QT\n190.352 6.989 190.681 5.66 QT\n190.727 5.489 190.852 5.364 QT\n190.977 5.239 191.149 5.239 QT\n191.571 5.239 L\n192.056 5.239 192.056 5.848 QT\n191.384 8.567 189.134 10.317 QT\n186.884 12.067 184.087 12.067 QT\ncp\n177.712 -0.761 M\n188.665 -0.761 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QT\n244.618 -10.277 244.196 -7.105 QT\n243.774 -3.933 243.774 -0.48 QT\n243.774 14.785 251.931 22.707 QT\n252.071 22.848 252.071 23.098 QT\n252.071 23.223 251.946 23.371 QT\n251.821 23.52 251.696 23.52 QT\n251.259 23.52 L\n251.024 23.52 251.024 23.426 QT\ncp}def\r\nf-583193730\r\nf\r\nGR\r\nGS\r\n[0.375 0 0 0.375 177.3761 219.6552] CT\r\nN\r\n/f-129920944{254.16 18.348 M\n254.16 17.176 254.972 16.317 QT\n255.785 15.457 256.957 15.457 QT\n257.738 15.457 258.261 15.942 QT\n258.785 16.426 258.785 17.192 QT\n258.785 18.114 258.214 18.848 QT\n257.644 19.582 256.769 19.817 QT\n257.613 20.129 258.457 20.129 QT\n260.504 20.129 262.004 18.239 QT\n263.504 16.348 264.113 14.004 QT\n268.629 -4.043 L\n268.957 -5.433 268.957 -6.386 QT\n268.957 -8.449 267.535 -8.449 QT\n265.425 -8.449 263.839 -6.543 QT\n262.254 -4.636 261.269 -2.121 QT\n261.175 -1.824 260.894 -1.824 QT\n260.332 -1.824 L\n259.941 -1.824 259.941 -2.261 QT\n259.941 -2.402 L\n261.019 -5.293 263.011 -7.496 QT\n265.004 -9.699 267.629 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self.systemParam.B * uk;\n self.xcurr = xk;\n self.timecurr = self.timecurr + self.systemParam.timestep;\n self.tlog = [self.tlog, self.timecurr];\n self.xlog = [self.xlog, xk];\n self.ulog = [self.ulog, uk];\n end\n end\n \n function [feas, xopt, uopt, Jopt] = solve(self)\n if strcmp(self.optType,'dclfdcbf')\n [feas, xopt, uopt, Jopt] = solveDCLFDCBF(self);\n elseif strcmp(self.optType,'nmpcdclfdcbf')\n [feas, xopt, uopt, Jopt] = solveNMPCDCLFDCBF(self);\n elseif strcmp(self.optType,'mpcdcbf')\n [feas, xopt, uopt, Jopt] = solveMPCDCBF(self);\n elseif strcmp(self.optType,'mpcgcbf')\n [feas, xopt, uopt, Jopt] = solveMPCGCBF(self);\n elseif strcmp(self.optType,'nmpcdcbf')\n [feas, xopt, uopt, Jopt] = solveNMPCDCBF(self);\n else\n disp('optimal control policy undefined');\n end\n end\n \n function [feas, xopt, uopt, Jopt] = solveDCLFDCBF(self)\n x = sdpvar(self.xdim, 2);\n u = sdpvar(self.udim, 1);\n s = sdpvar(1,1);\n cost = 0;\n constraints = [];\n % initial condition\n constraints = [constraints; x(:,1) == self.xcurr];\n % dynamics\n constraints = [constraints; x(:,2) == self.systemParam.A * x(:,1) + self.systemParam.B * u(:,1)];\n % DCLF constraint\n v = x(:,1)' * self.optParam.P * x(:,1);\n vnext = x(:,2)' * self.optParam.P * x(:,2);\n constraints = [constraints; vnext - v + self.optParam.alpha * v <= s];\n % DCBF constraint\n r = 1;\n b = x(:,1)' * x(:,1) - r^2;\n bnext = x(:,2)' * x(:,2) - r^2;\n constraints = [constraints; bnext - b + self.optParam.gamma * b >= 0];\n % input constraint\n constraints = [constraints; self.systemParam.ul <= u <= self.systemParam.uu];\n % cost\n cost = cost + u' * self.optParam.uWeight * u + s' * self.optParam.sWeight * s;\n opt_settings = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, opt_settings);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt= [];\n uopt = [];\n Jopt = value(cost);\n end\n self.solvertime = diagnostics.solvertime;\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function [feas, xopt, uopt, Jopt] = solveNMPCDCLFDCBF(self)\n x = sdpvar(self.xdim, self.optParam.horizon + 1);\n u = sdpvar(self.udim, self.optParam.horizon);\n s = sdpvar(1,self.optParam.horizonCLF);\n omega = sdpvar(1,self.optParam.horizonCBF);\n cost = 0;\n constraints = [];\n % initial condition\n constraints = [constraints; x(:,1) == self.xcurr];\n % MPC cost and constraints\n for i = 1:self.optParam.horizon\n constraints = [constraints;\n self.systemParam.ul <= u(:,i) <= self.systemParam.uu;\n x(:,i+1) == self.systemParam.A * x(:,i) + self.systemParam.B * u(:,i)];\n cost = cost + x(:,i)'*self.optParam.xWeight*x(:,i);\n cost = cost + u(:,i)'*self.optParam.uWeight*u(:,i);\n end\n cost = cost + x(:,self.optParam.horizon)' * self.optParam.P * x(:,self.optParam.horizon);\n % CLF constraints\n for i = 1:self.optParam.horizonCLF\n v = x(:,1)' * self.optParam.P * x(:,1);\n vnext = x(:,2)' * self.optParam.P * x(:,2);\n constraints = [constraints; vnext - v + self.optParam.alpha * v <= s(i)];\n cost = cost + s(i)' * self.optParam.sWeight * s(i);\n end\n % CBF constraints\n for i = 1:self.optParam.horizonCBF\n r = 1;\n b = x(:,1)' * x(:,1) - r^2;\n bnext = x(:,2)' * x(:,2) - r^2;\n constraints = [constraints; bnext >= omega(i) * (1 - self.optParam.gamma) * b];\n constraints = [constraints; omega(i) >= 0];\n assign(omega(i), 1);\n cost = cost + self.optParam.omegaWeight * (omega(i) - 1)^2;\n end\n opt_settings = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, opt_settings);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt= [];\n uopt = [];\n Jopt = value(cost);\n end\n self.solvertime = diagnostics.solvertime;\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function [feas, xopt, uopt, Jopt] = solveMPCDCBF(self)\n x = sdpvar(self.xdim, self.optParam.horizon + 1);\n u = sdpvar(self.udim, self.optParam.horizon);\n cost = 0;\n constraints = [];\n % initial condition\n constraints = [constraints; x(:,1) == self.xcurr];\n % MPC cost and constraints\n for i = 1:self.optParam.horizon\n constraints = [constraints;\n self.systemParam.ul <= u(:,i) <= self.systemParam.uu;\n x(:,i+1) == self.systemParam.A * x(:,i) + self.systemParam.B * u(:,i)];\n cost = cost + x(:,i)'*self.optParam.xWeight*x(:,i);\n cost = cost + u(:,i)'*self.optParam.uWeight*u(:,i);\n end\n cost = cost + x(:,self.optParam.horizon)' * self.optParam.P * x(:,self.optParam.horizon);\n % CBF constraints\n for i = 1:self.optParam.horizon\n r = 0;\n b = r - x(1,i);\n bnext = r - x(1,i+1);\n constraints = [constraints; bnext - b + self.optParam.gamma * b >= 0];\n end\n opt_settings = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, opt_settings);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt= [];\n uopt = [];\n Jopt = value(cost);\n end\n self.solvertime = diagnostics.solvertime;\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function [feas, xopt, uopt, Jopt] = solveMPCGCBF(self)\n x = sdpvar(self.xdim, self.optParam.horizon + 1);\n u = sdpvar(self.udim, self.optParam.horizon);\n cost = 0;\n constraints = [];\n % initial condition\n constraints = [constraints; x(:,1) == self.xcurr];\n % MPC cost and constraints\n for i = 1:self.optParam.horizon\n constraints = [constraints;\n self.systemParam.ul <= u(:,i) <= self.systemParam.uu;\n x(:,i+1) == self.systemParam.A * x(:,i) + self.systemParam.B * u(:,i)];\n cost = cost + x(:,i)'*self.optParam.xWeight*x(:,i);\n cost = cost + u(:,i)'*self.optParam.uWeight*u(:,i);\n end\n cost = cost + x(:,self.optParam.horizon)' * self.optParam.P * x(:,self.optParam.horizon);\n % GCBF constraints\n num_HO = 3;\n r = 0;\n b = r - x(1,1);\n bnext = r - x(1,1+num_HO);\n constraints = [constraints; bnext >= (1 - self.optParam.gamma)^num_HO * b]; \n opt_settings = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, opt_settings);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt= [];\n uopt = [];\n Jopt = value(cost);\n end\n self.solvertime = diagnostics.solvertime;\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n \n function [feas, xopt, uopt, Jopt] = solveNMPCDCBF(self)\n x = sdpvar(self.xdim, self.optParam.horizon + 1);\n u = sdpvar(self.udim, self.optParam.horizon);\n omega = sdpvar(1,self.optParam.horizonCBF);\n cost = 0;\n constraints = [];\n % initial condition\n constraints = [constraints; x(:,1) == self.xcurr];\n % MPC cost and constraints\n for i = 1:self.optParam.horizon\n constraints = [constraints;\n self.systemParam.ul <= u(:,i) <= self.systemParam.uu;\n x(:,i+1) == self.systemParam.A * x(:,i) + self.systemParam.B * u(:,i)];\n cost = cost + x(:,i)'*self.optParam.xWeight*x(:,i);\n cost = cost + u(:,i)'*self.optParam.uWeight*u(:,i);\n end\n cost = cost + x(:,self.optParam.horizon)' * self.optParam.P * x(:,self.optParam.horizon);\n % CBF constraints\n for i = 1:self.optParam.horizonCBF\n r = 0;\n b = r - x(1,i);\n bnext = r - x(1,i+1);\n constraints = [constraints; bnext >= omega(i) * (1 - self.optParam.gamma) * b];\n constraints = [constraints; omega(i) >= 0];\n assign(omega(i), 1);\n cost = cost + self.optParam.omegaWeight * (omega(i) - 1)^2;\n end\n opt_settings = sdpsettings('solver','ipopt','verbose',0);\n diagnostics = optimize(constraints, cost, opt_settings);\n if diagnostics.problem == 0\n feas = true;\n xopt = value(x);\n uopt = value(u);\n Jopt = value(cost);\n else\n feas = false;\n xopt= [];\n uopt = [];\n Jopt = value(cost);\n end\n self.solvertime = diagnostics.solvertime;\n fprintf('solver time: %f\\n', diagnostics.solvertime);\n end\n end\nend\n\n" }, { "path": "matlab/cdc2021/ParamDCLFDCBF.m", "content": "classdef ParamDCLFDCBF < handle\n properties\n alpha\n gamma\n P % weight for Lyapunov function\n uWeight\n sWeight\n end\n methods\n function self = ParamDCLFDCBF(alpha, gamma, P, u_weight, s_weight)\n self.alpha = alpha;\n self.gamma = gamma;\n self.P = P;\n self.uWeight = u_weight;\n self.sWeight = s_weight;\n end\n end\nend" }, { "path": "matlab/cdc2021/ParamMPCDCBF.m", "content": "classdef ParamMPCDCBF < handle\n properties\n horizon\n gamma\n P % weight for terminal cost (CLF)\n xWeight\n uWeight\n end\n methods\n function self = ParamMPCDCBF(horizon, gamma, P, x_weight, u_weight)\n self.horizon = horizon;\n self.gamma = gamma;\n self.P = P;\n self.xWeight = x_weight;\n self.uWeight = u_weight;\n end\n end\nend" }, { "path": "matlab/cdc2021/ParamMPCGCBF.m", "content": "classdef ParamMPCGCBF < handle\n properties\n horizon\n gamma\n P % weight for terminal cost (CLF)\n xWeight\n uWeight\n end\n methods\n function self = ParamMPCGCBF(horizon, gamma, P, x_weight, u_weight)\n self.horizon = horizon;\n self.gamma = gamma;\n self.P = P;\n self.xWeight = x_weight;\n self.uWeight = u_weight;\n end\n end\nend" }, { "path": "matlab/cdc2021/ParamNMPCDCBF.m", "content": "classdef ParamNMPCDCBF < handle\n properties\n horizon\n horizonCBF\n gamma\n P % weight for terminal cost (CLF)\n xWeight\n uWeight\n omegaWeight\n end\n methods\n function self = ParamNMPCDCBF(horizon, horizon_cbf, gamma, P, x_weight, u_weight, omega_weight)\n self.horizon = horizon;\n self.horizonCBF = horizon_cbf;\n self.gamma = gamma;\n self.P = P;\n self.xWeight = x_weight;\n self.uWeight = u_weight;\n self.omegaWeight = omega_weight;\n end\n end\nend" }, { "path": "matlab/cdc2021/ParamNMPCDCLFDCBF.m", "content": "classdef ParamNMPCDCLFDCBF < handle\n properties\n horizon\n horizonCLF\n horizonCBF\n alpha\n gamma\n P % weight for Lyapunov function\n xWeight\n uWeight\n sWeight\n omegaWeight\n end\n methods\n function self = ParamNMPCDCLFDCBF(horizon, horizon_clf, horizon_cbf, alpha, gamma, P, x_weight, u_weight, s_weight, omega_weight)\n self.horizon = horizon;\n self.horizonCLF = horizon_clf;\n self.horizonCBF = horizon_cbf;\n self.alpha = alpha;\n self.gamma = gamma;\n self.P = P;\n self.xWeight = x_weight;\n self.uWeight = u_weight;\n self.sWeight = s_weight;\n self.omegaWeight = omega_weight;\n end\n end\nend" }, { "path": "matlab/cdc2021/README.md", "content": "## NMPC-DCLF-DCBF-CDC2021\nIn this paper, we discuss the feasibility, safety and complexity about existing optimal control laws using control barrier functions. This is the reference implementation of our paper.\n\nIf you find this code useful in your work, please consider citing:\n```\n@inproceedings{zeng2021nmpc-dclf-dcbf,\n title={Enhancing Feasibility and Safety of Nonlinear Model Predictive Control with Discrete-Time Control Barrier Functions},\n author={Zeng, Jun and Li, Zhongyu and Sreenath, Koushil},\n booktitle={2021 IEEE Conference on Decision and Control (CDC)},\n year={2021}\n}\n```\n\n### Instructions\nSeveral test files are created to reproduce the results in the paper:\n* `testFeasibilityMPCDCBF.m` compares the feasibility performance between MPC-DCBF and NMPC-DCBF with different hyperparameters.\n* `testFeasibilityMPCGCBF.m` compares the feasibility performance between MPC-GCBF and NMPC-DCBF with different hyperparameters.\n* `testFeasibilityDCLFDCBF.m` compares the feasibility performance between DCLF-DCBF and NMPC-DCLF-DCBF.\n* `testSafety.m` compares the safety performance between MPC-CBF, MPC-GCBF and NMPC-DCBF.\n\nNotice that running either `testFeasibilityMPCDCBF.m` or `testFeasibilityMPCGCBF.m` or `testFeasibilityDCLFDCBF.m` will take more than 4 hours due to the weak memory management using matlab together with excessive Yalmip calling. To check results or have a quick visualization, feel free to load data directly from the `/data`." }, { "path": "matlab/cdc2021/test.m", "content": "close all\nclear\n\n% system setup and initial condition\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [-2;0;1];\nt0 = 0.0;\ntotal_time = 1.0;\n\n% problem setup\nsimulator = CBFDT(system_param, x0, t0);\n\nparam_mpcdcbf = ParamMPCDCBF(6, 0.1, 10 * eye(3), 10.0 * eye(3), 1.0);\nsimulator.setOpt('mpcdcbf', param_mpcdcbf);\nsimulator.sim(total_time);\n\nparam_mpcgcbf = ParamMPCGCBF(6, 0.1, 10 * eye(3), 10.0 * eye(3), 1.0);\nsimulator.setOpt('mpcgcbf', param_mpcgcbf);\nsimulator.sim(total_time);\n\nparam_nmpcdcbf = ParamNMPCDCBF(6, 6, 0.1, 10 * eye(3), 10.0 * eye(3), 1.0, 10.0);\nsimulator.setOpt('nmpcdcbf', param_nmpcdcbf);\nsimulator.sim(total_time);\n\nparam_dclfdcbf = ParamDCLFDCBF(0.1, 0.1, 10 * eye(3), 1.0, 10.0);\nsimulator.setOpt('dclfdcbf', param_dclfdcbf);\nsimulator.sim(total_time);\n\nparam_nmpcdclfdcbf = ParamNMPCDCLFDCBF(6, 6, 6, 0.1, 0.1, 10.0 * eye(3), 10.0 * eye(3), 1.0, 10.0, 10.0);\nsimulator.setOpt('nmpcdclfdcbf', param_nmpcdclfdcbf);\nsimulator.sim(total_time);" }, { "path": "matlab/cdc2021/testComplexity.m", "content": "clc\nclose all\nclear\n% This file tests computational time for various approaches\n\n%% System setup\n\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [0;0;0]; % this value will be overrided\nt0 = 0.0;\n\n%% Sampling states\n\nx1list = linspace(-2,0,11);\nx2list = linspace(0,2,11);\nx3list = linspace(0,2,11);\n\n%% MPC-CBF (N=10)\nsolvertime_list1 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_mpcdcbf = ParamMPCDCBF(10, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0);\nsimulator_dt.setOpt('mpcdcbf', param_mpcdcbf);\nsolvertime_list1 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with MPC-DCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list1 = [solvertime_list1, simulator_dt.solvertime];\n end\n end\n end\nend\n\n%% MPC-GCBF (N=10)\nsolvertime_list2 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_mpcgcbf = ParamMPCGCBF(10, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0);\nsimulator_dt.setOpt('mpcgcbf', param_mpcgcbf);\nsolvertime_list2 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with MPC-GCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list2 = [solvertime_list2, simulator_dt.solvertime];\n end\n end\n end\nend\n\n%% NMPC-DCBF (N=10, MCBF=10)\nsolvertime_list3 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_nmpcdcbf = ParamNMPCDCBF(10, 10, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\nsimulator_dt.setOpt('nmpcdcbf', param_nmpcdcbf);\nsolvertime_list3 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with NMPC-DCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list3 = [solvertime_list3, simulator_dt.solvertime];\n end\n end\n end\nend\n\n%% NMPC-DCBF (N=10, MCBF=8)\nsolvertime_list4 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_nmpcdcbf = ParamNMPCDCBF(10, 8, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\nsimulator_dt.setOpt('nmpcdcbf', param_nmpcdcbf);\nsolvertime_list4 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with NMPC-DCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list4 = [solvertime_list4, simulator_dt.solvertime];\n end\n end\n end\nend\n\n%% NMPC-DCBF (N=10, MCBF=5)\nsolvertime_list5 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_nmpcdcbf = ParamNMPCDCBF(10, 5, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\nsimulator_dt.setOpt('nmpcdcbf', param_nmpcdcbf);\nsolvertime_list5 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with NMPC-DCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list5 = [solvertime_list5, simulator_dt.solvertime];\n end\n end\n end\nend\n\n%% NMPC-DCBF (N=10, MCBF=3)\nsolvertime_list6 = [];\nsimulator_dt = CBFDT(system_param, x0, t0);\nparam_nmpcdcbf = ParamNMPCDCBF(10, 3, 0.2, 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\nsimulator_dt.setOpt('nmpcdcbf', param_nmpcdcbf);\nsolvertime_list6 = [];\nfor k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with NMPC-DCBF\n simulator_dt.xcurr = xfeas;\n [feas, ~, ~, ~] = simulator_dt.solve;\n if feas == 1\n solvertime_list6 = [solvertime_list6, simulator_dt.solvertime];\n end\n end\n end\nend" }, { "path": "matlab/cdc2021/testFeasibilityDCLFDCBF.m", "content": "clc\nclose all\nclear\n% This file tests feasibility between DCLF-DCBF and NMPC-DCLF-DCBF.\n\n%% System setup\n\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [0;0;0]; % this value will be overrided\nt0 = 0.0;\n\ngammalist = [0.05, 0.1, 0.15, 0.2];\nfor gammaindex = 1:length(gammalist)\n %% Problem setup with different hyperparameters\n \n % MPC-GCBF simulator\n simulator_dclfdcbf = CBFDT(system_param, x0, t0);\n param_dclfdcbf = ParamDCLFDCBF(0.1, gammalist(gammaindex), 10 * eye(3), 1.0, 10.0);\n simulator_dclfdcbf.setOpt('dclfdcbf', param_dclfdcbf);\n % NMPC-DCBF simulator\n simulator_nmpcdclfdcbf = CBFDT(system_param, x0, t0);\n param_nmpcdclfdcbf = ParamNMPCDCLFDCBF(8, 8, 8, 0.1, gammalist(gammaindex), 10.0 * eye(3), 10.0 * eye(3), 1.0, 10.0, 10.0);\n simulator_nmpcdclfdcbf.setOpt('nmpcdclfdcbf', param_nmpcdclfdcbf);\n % iterate over states\n x1list = linspace(-2,0,11);\n x2list = linspace(0,2,11);\n x3list = linspace(0,2,11);\n % data collection\n feas_points_dclfdcbf = [];\n feas_points_nmpcdclfdcbf = [];\n infeas_points_all = [];\n \n %% Test feasibility with sampling states among controllers\n \n for k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % exclude the point if it's always unsafe\n if xfeas' * xfeas <= 1\n continue;\n end\n % solve the problem with MPC-GCBF\n simulator_dclfdcbf.xcurr = xfeas;\n [feas_dclfdcbf, ~, ~, ~] = simulator_dclfdcbf.solve;\n % solve the problem with NMPC-DCLF-DCBF\n simulator_nmpcdclfdcbf.xcurr = xfeas;\n [feas_nmpcdclfdcbf, ~, ~, ~] = simulator_nmpcdclfdcbf.solve;\n if feas_dclfdcbf == 1\n feas_points_dclfdcbf = [feas_points_dclfdcbf, xfeas];\n end\n if feas_nmpcdclfdcbf == 1\n feas_points_nmpcdclfdcbf = [feas_points_nmpcdclfdcbf, xfeas];\n end\n if feas_dclfdcbf == 0 && feas_nmpcdclfdcbf == 0\n infeas_points_all = [infeas_points_all, xfeas];\n end\n end\n end\n end\n \n %% Plotting\n close all\n outer_color = [0.3010, 0.7450, 0.9330];\n inner_color = [1, 0, 0];\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n scatter3(feas_points_dclfdcbf(1,:),feas_points_dclfdcbf(2,:),feas_points_dclfdcbf(3,:),...\n 5, 'o', 'LineWidth', 1.0, 'MarkerEdgeColor', inner_color, 'MarkerFaceColor', inner_color);\n scatter3(feas_points_nmpcdclfdcbf(1,:),feas_points_nmpcdclfdcbf(2,:),feas_points_nmpcdclfdcbf(3,:),...\n 20, 'o', 'LineWidth', 1.2, 'MarkerEdgeColor', outer_color);\n r = 1.0;\n [x,y,z] = sphere(50);\n x = x*r; y = y*r; z = z*r;\n lightGrey = 0.8*[1 1 1]; % It looks better if the lines are lighter\n ball_surface = surface(x,y,z, 'FaceColor', 'none', 'EdgeColor', [0.75, 0.75, 0], 'facealpha', 0.5);\n axis equal\n if gammaindex == 1\n h=get(gca,'Children');\n h_legend = legend(h([end, end-1]),...\n {'DCLF-DCBF', 'NMPC-DCLF-DCBF'}, 'Location', 'NorthEast');\n set(h_legend, 'Interpreter','latex');\n end\n set(gca,'LineWidth', 1.0, 'FontSize', 15);\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$v (m/s)$','interpreter','latex','FontSize',20);\n zlabel('$a (m/s^2)$','interpreter','latex','FontSize',20);\n xlim([-2, 0]);\n ylim([0, 2]);\n zlim([0, 2]);\n view(160, 4);\n grid on;\n figurename_eps = \"feasibility-dclfdcbf\" + gammaindex + \".eps\";\n figurename_png = \"feasibility-dclfdcbf\" + gammaindex + \".png\";\n dataname = \"feasibility-dclfdcbf\" + gammaindex + \".mat\";\n % save data and generate figures\n print(gcf,strcat('figures/',figurename_eps), '-depsc');\n print(gcf,strcat('figures/',figurename_png), '-dpng', '-r800');\n save(strcat('data/',dataname));\nend" }, { "path": "matlab/cdc2021/testFeasibilityMPCDCBF.m", "content": "clc\nclose all\nclear\n% This file tests feasibility between MPC-DCBF and NMPC-DCBF.\n\n%% System setup\n\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [0;0;0]; % this value will be overrided\nt0 = 0.0;\n\ngammalist = [0.05, 0.1, 0.15, 0.2];\nfor gammaindex = 1:length(gammalist)\n %% Problem setup with different hyperparameters\n \n % MPC-DCBF simulator\n simulator_mpcdcbf = CBFDT(system_param, x0, t0);\n param_mpcdcbf = ParamMPCDCBF(8, gammalist(gammaindex), 10.0*eye(3), 10.0*eye(3), 1.0);\n simulator_mpcdcbf.setOpt('mpcdcbf', param_mpcdcbf);\n % NMPC-DCBF simulator\n simulator_nmpcdcbf = CBFDT(system_param, x0, t0);\n param_nmpcdcbf = ParamNMPCDCBF(8, 8, gammalist(gammaindex), 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\n simulator_nmpcdcbf.setOpt('nmpcdcbf', param_nmpcdcbf);\n % iterate over states\n x1list = linspace(-2,0,11);\n x2list = linspace(0,2,11);\n x3list = linspace(0,2,11);\n % data collection\n feas_points_mpcdcbf = [];\n feas_points_nmpcdcbf = [];\n infeas_points_all = [];\n \n %% Test feasibility with sampling states among controllers\n \n for k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with MPC-DCBF\n simulator_mpcdcbf.xcurr = xfeas;\n [feas_mpcdcbf, ~, ~, ~] = simulator_mpcdcbf.solve;\n % solve the problem with NMPC-DCBF\n simulator_nmpcdcbf.xcurr = xfeas;\n [feas_nmpcdcbf, ~, ~, ~] = simulator_nmpcdcbf.solve;\n if feas_mpcdcbf == 1\n feas_points_mpcdcbf = [feas_points_mpcdcbf, xfeas];\n end\n if feas_nmpcdcbf == 1\n feas_points_nmpcdcbf = [feas_points_nmpcdcbf, xfeas];\n end\n if feas_mpcdcbf == 0 && feas_nmpcdcbf == 0\n infeas_points_all = [infeas_points_all, xfeas];\n end\n end\n end\n end\n \n %% Plotting\n close all\n outer_color = [0.3010, 0.7450, 0.9330];\n inner_color = [1, 0, 0];\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n scatter3(feas_points_mpcdcbf(1,:),feas_points_mpcdcbf(2,:),feas_points_mpcdcbf(3,:), 5, 'o','LineWidth',1.0, 'MarkerEdgeColor',inner_color,'MarkerFaceColor',inner_color);\n scatter3(feas_points_nmpcdcbf(1,:),feas_points_nmpcdcbf(2,:),feas_points_nmpcdcbf(3,:), 20, 'o','LineWidth',1.2, 'MarkerEdgeColor',outer_color);\n axis equal\n if gammaindex == 1\n h=get(gca,'Children');\n h_legend = legend(h([end, end-1]),...\n {'MPC-DCBF', 'NMPC-DCBF'}, 'Location', 'NorthEast');\n set(h_legend, 'Interpreter','latex');\n end\n set(gca,'LineWidth', 1.0, 'FontSize', 15);\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$v (m/s)$','interpreter','latex','FontSize',20);\n zlabel('$a (m/s^2)$','interpreter','latex','FontSize',20);\n xlim([-2, 0]);\n ylim([0, 2]);\n zlim([0, 2]);\n view(70, 4);\n grid on;\n figurename_eps = \"feasibility-mpcdcbf\" + gammaindex + \".eps\";\n figurename_png = \"feasibility-mpcdcbf\" + gammaindex + \".png\";\n dataname = \"feasibility-mpcdcbf\" + gammaindex + \".mat\";\n % save data and generate figures\n print(gcf,strcat('figures/',figurename_eps), '-depsc');\n print(gcf,strcat('figures/',figurename_png), '-dpng', '-r800');\n save(strcat('data/',dataname));\nend" }, { "path": "matlab/cdc2021/testFeasibilityMPCGCBF.m", "content": "clc\nclose all\nclear\n% This file tests feasibility between MPC-GCBF and NMPC-DCBF.\n\n%% System setup\n\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [0;0;0]; % this value will be overrided\nt0 = 0.0;\n\ngammalist = [0.05, 0.1, 0.15, 0.2];\nfor gammaindex = 1:length(gammalist)\n %% Problem setup with different hyperparameters\n \n % MPC-GCBF simulator\n simulator_mpcgcbf = CBFDT(system_param, x0, t0);\n param_mpcgcbf = ParamMPCGCBF(8, gammalist(gammaindex), 10.0*eye(3), 10.0*eye(3), 1.0);\n simulator_mpcgcbf.setOpt('mpcgcbf', param_mpcgcbf);\n % NMPC-DCBF simulator\n simulator_nmpccbf = CBFDT(system_param, x0, t0);\n param_nmpcdcbf = ParamNMPCDCBF(8, 3, gammalist(gammaindex), 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\n simulator_nmpccbf.setOpt('nmpcdcbf', param_nmpcdcbf);\n % iterate over states\n x1list = linspace(-2,0,11);\n x2list = linspace(0,2,11);\n x3list = linspace(0,2,11);\n % data collection\n feas_points_mpcgcbf = [];\n feas_points_nmpcdcbf = [];\n infeas_points_all = [];\n \n %% Test feasibility with sampling states among controllers\n \n for k1 = 1:length(x1list)\n for k2 = 1:length(x2list)\n for k3 = 1:length(x3list)\n xfeas = [x1list(k1); x2list(k2); x3list(k3)];\n % solve the problem with MPC-GCBF\n simulator_mpcgcbf.xcurr = xfeas;\n [feas_mpcgcbf, ~, ~, ~] = simulator_mpcgcbf.solve;\n % solve the problem with NMPC-DCBF\n simulator_nmpccbf.xcurr = xfeas;\n [feas_nmpcdcbf, ~, ~, ~] = simulator_nmpccbf.solve;\n if feas_mpcgcbf == 1\n feas_points_mpcgcbf = [feas_points_mpcgcbf, xfeas];\n end\n if feas_nmpcdcbf == 1\n feas_points_nmpcdcbf = [feas_points_nmpcdcbf, xfeas];\n end\n if feas_mpcgcbf == 0 && feas_nmpcdcbf == 0\n infeas_points_all = [infeas_points_all, xfeas];\n end\n end\n end\n end\n \n %% Plotting\n close all\n outer_color = [0.3010, 0.7450, 0.9330];\n inner_color = [1, 0, 0];\n figure('Renderer', 'painters', 'Position', [0 0 400 400]);\n set(gca,'LooseInset',get(gca,'TightInset'));\n hold on;\n scatter3(feas_points_mpcgcbf(1,:),feas_points_mpcgcbf(2,:),feas_points_mpcgcbf(3,:),...\n 5, 'o', 'LineWidth', 1.0, 'MarkerEdgeColor', inner_color, 'MarkerFaceColor', inner_color);\n scatter3(feas_points_nmpcdcbf(1,:),feas_points_nmpcdcbf(2,:),feas_points_nmpcdcbf(3,:),...\n 20, 'o', 'LineWidth', 1.2, 'MarkerEdgeColor', outer_color);\n axis equal\n if gammaindex == 1\n h=get(gca,'Children');\n h_legend = legend(h([end, end-1]),...\n {'MPC-GCBF', 'NMPC-DCBF'}, 'Location', 'NorthEast');\n set(h_legend, 'Interpreter','latex');\n end\n set(gca,'LineWidth', 1.0, 'FontSize', 15);\n xlabel('$x (m)$','interpreter','latex','FontSize',20);\n ylabel('$v (m/s)$','interpreter','latex','FontSize',20);\n zlabel('$a (m/s^2)$','interpreter','latex','FontSize',20);\n xlim([-2, 0]);\n ylim([0, 2]);\n zlim([0, 2]);\n view(70, 4);\n grid on;\n figurename_eps = \"feasibility-mpcgcbf\" + gammaindex + \".eps\";\n figurename_png = \"feasibility-mpcgcbf\" + gammaindex + \".png\";\n dataname = \"feasibility-mpcgcbf\" + gammaindex + \".mat\";\n % save data and generate figures\n print(gcf,strcat('figures/',figurename_eps), '-depsc');\n print(gcf,strcat('figures/',figurename_png), '-dpng', '-r800');\n save(strcat('data/',dataname));\nend" }, { "path": "matlab/cdc2021/testSafety.m", "content": "clc\nclose all\nclear\n% This file tests safety performance for various approaches\n\n%% Setup and simulations\ntimestep = 0.1;\nsystem_param.A = [[1 ,timestep, 0]; [0, 1, timestep]; [0, 0, 1]];\nsystem_param.B = [0; 0; timestep];\nsystem_param.ul = -1;\nsystem_param.uu = 1;\nsystem_param.timestep = timestep;\nx0 = [-2.0;0.0;1.0];\nt0 = 0.0;\n\ngammalist = [0.05, 0.10, 0.15, 0.2];\n\n%% MPC-DCBF simulator\nmpcdcbf_simulators = {};\nfor index = 1:length(gammalist)\n mpcdcbf_simulators{index} = CBFDT(system_param, x0, t0);\n param_mpcdcbf = ParamMPCDCBF(8, gammalist(index), 10.0*eye(3), 10.0*eye(3), 1.0);\n mpcdcbf_simulators{index}.setOpt('mpcdcbf', param_mpcdcbf);\n mpcdcbf_simulators{index}.sim(10.0);\nend\n\n%% MPC-GCBF simulator\nmpcgcbf_simulators = {};\nfor index = 1:length(gammalist)\n mpcgcbf_simulators{index} = CBFDT(system_param, x0, t0);\n param_mpcgcbf = ParamMPCGCBF(8, gammalist(index), 10.0*eye(3), 10.0*eye(3), 1.0);\n mpcgcbf_simulators{index}.setOpt('mpcgcbf', param_mpcgcbf);\n mpcgcbf_simulators{index}.sim(10.0);\nend\n\n%% NMPC-DCBF simulator\nnmpcdcbf_simulators = {};\nfor index = 1:length(gammalist)\n nmpcdcbf_simulators{index} = CBFDT(system_param, x0, t0);\n param_nmpcdcbf = ParamNMPCDCBF(8, 8, gammalist(index), 10.0*eye(3), 10.0*eye(3), 1.0, 10.0);\n nmpcdcbf_simulators{index}.setOpt('nmpcdcbf', param_nmpcdcbf);\n nmpcdcbf_simulators{index}.sim(10.0);\nend\n\n%% Plotting\nfigure('Renderer', 'painters', 'Position', [0 0 500 400]);\ncolor1 = '[0, 0.4470, 0.7410]';\ncolor2 = '[0.8500, 0.3250, 0.0980]';\ncolor3 = '[0.9290, 0.6940, 0.1250]';\ncolor4 = '[0.4940, 0.1840, 0.5560]';\nset(gca,'LineWidth', 0.2, 'FontSize', 20);\nhold on;\ngrid on;\nset(gca, 'YScale', 'log');\nplot(mpcdcbf_simulators{2}.tlog, -mpcdcbf_simulators{2}.xlog(1, :), 'Color', color2, 'LineWidth', 1.0);\nplot(mpcdcbf_simulators{3}.tlog, -mpcdcbf_simulators{3}.xlog(1, :), 'Color', color3, 'LineWidth', 1.0);\nplot(mpcdcbf_simulators{4}.tlog, -mpcdcbf_simulators{4}.xlog(1, :), 'Color', color4, 'LineWidth', 1.0);\nh_legend = legend('MPC-DCBF ($\\gamma=0.10$)', 'MPC-DCBF ($\\gamma=0.15$)', 'MPC-DCBF ($\\gamma=0.20$)');\nset(h_legend, 'Interpreter','latex', 'Location', 'SouthWest');\nxlim([0, 10]);\n% save data and generate figures\nprint(gcf, 'figures/safety-mpcdcbf.eps', '-depsc');\nprint(gcf, 'figures/safety-mpcdcbf.png', '-dpng', '-r800');\nsave('data/safety-mpcdcbf.mat');\n\nfigure('Renderer', 'painters', 'Position', [0 0 500 400]);\nset(gca,'LineWidth', 0.2, 'FontSize', 20);\nhold on;\ngrid on;\nset(gca, 'YScale', 'log');\nplot(mpcgcbf_simulators{2}.tlog, -mpcgcbf_simulators{2}.xlog(1, :), 'Color', color2, 'LineWidth', 1.0);\nplot(mpcgcbf_simulators{3}.tlog, -mpcgcbf_simulators{3}.xlog(1, :), 'Color', color3, 'LineWidth', 1.0);\nplot(mpcgcbf_simulators{4}.tlog, -mpcgcbf_simulators{4}.xlog(1, :), 'Color', color4, 'LineWidth', 1.0);\nh_legend = legend('MPC-GCBF ($\\gamma=0.10$)', 'MPC-GCBF ($\\gamma=0.15$)', 'MPC-GCBF ($\\gamma=0.20$)');\nset(h_legend, 'Interpreter','latex', 'Location', 'SouthWest');\nxlim([0, 10]);\nylim([0.01, 2]);\n% save data and generate figures\nprint(gcf, 'figures/safety-mpcgcbf.eps', '-depsc');\nprint(gcf, 'figures/safety-mpcgcbf.png', '-dpng', '-r800');\nsave('data/safety-mpcgcbf.mat');\n\nfigure('Renderer', 'painters', 'Position', [0 0 500 400]);\nset(gca,'LineWidth', 0.2, 'FontSize', 20);\nhold on;\ngrid on;\nset(gca, 'YScale', 'log');\nplot(nmpcdcbf_simulators{1}.tlog, -nmpcdcbf_simulators{1}.xlog(1, :), 'Color', color1, 'LineWidth', 1.0);\nplot(nmpcdcbf_simulators{2}.tlog, -nmpcdcbf_simulators{2}.xlog(1, :), 'Color', color2, 'LineWidth', 1.0);\nplot(nmpcdcbf_simulators{3}.tlog, -nmpcdcbf_simulators{3}.xlog(1, :), 'Color', color3, 'LineWidth', 1.0);\nplot(nmpcdcbf_simulators{4}.tlog, -nmpcdcbf_simulators{4}.xlog(1, :), 'Color', color4, 'LineWidth', 1.0);\nh_legend = legend('NMPC-DCBF ($\\gamma=0.05$)', 'NMPC-DCBF ($\\gamma=0.10$)', 'NMPC-DCBF ($\\gamma=0.15$)', 'NMPC-DCBF ($\\gamma=0.20$)');\nset(h_legend, 'Interpreter','latex', 'Location', 'SouthWest');\nxlim([0, 10]);\n% save data and generate figures\nprint(gcf, 'figures/safety-nmpcdcbf.eps', '-depsc');\nprint(gcf, 'figures/safety-nmpcdcbf.png', '-dpng', '-r800');\nsave('data/safety-nmpccbf.mat');\n" } ]