Repository: aipiano/ESEKF_IMU Branch: master Commit: 38320a8617cd Files: 11 Total size: 11.9 MB Directory structure: gitextract_w7e7o7_e/ ├── .gitignore ├── README.md ├── data/ │ ├── imu_noise.txt │ ├── imu_noise_free.txt │ ├── params.yaml │ ├── traj_esekf_out.txt │ ├── traj_gt.txt │ └── traj_gt_out.txt ├── esekf.py ├── main.py └── transformations.py ================================================ FILE CONTENTS ================================================ ================================================ FILE: .gitignore ================================================ # Byte-compiled / optimized / DLL files __pycache__/ *.py[cod] *$py.class # C extensions *.so # Distribution / packaging .Python build/ develop-eggs/ dist/ downloads/ eggs/ .eggs/ lib/ lib64/ parts/ sdist/ var/ wheels/ *.egg-info/ .installed.cfg *.egg MANIFEST # PyInstaller # Usually these files are written by a python script from a template # before PyInstaller builds the exe, so as to inject date/other infos into it. *.manifest *.spec # Installer logs pip-log.txt pip-delete-this-directory.txt # Unit test / coverage reports htmlcov/ .tox/ .coverage .coverage.* .cache nosetests.xml coverage.xml *.cover .hypothesis/ .pytest_cache/ # Translations *.mo *.pot # Django stuff: *.log local_settings.py db.sqlite3 # Flask stuff: instance/ .webassets-cache # Scrapy stuff: .scrapy # Sphinx documentation docs/_build/ # PyBuilder target/ # Jupyter Notebook .ipynb_checkpoints # pyenv .python-version # celery beat schedule file celerybeat-schedule # SageMath parsed files *.sage.py # Environments .env .venv env/ venv/ ENV/ env.bak/ venv.bak/ # Spyder project settings .spyderproject .spyproject # Rope project settings .ropeproject # mkdocs documentation /site # mypy .mypy_cache/ ================================================ FILE: README.md ================================================ # ESEKF_IMU A python implemented error-state extended Kalman Filter. Suit for learning EKF and IMU integration. Use simulated imu data (./data/imu_noise.txt) as input. Output an trajectory estimated by esekf (./data/traj_esekf_out.txt) and a ground truth trajectory (./data/traj_gt_out.txt). You can use [evo](https://github.com/MichaelGrupp/evo) to show both trajectories above. > evo_traj tum ./data/traj_esekf_out.txt --ref ./data/traj_gt_out.txt -p ![trajectory](https://github.com/aipiano/ESEKF_IMU/blob/master/images/trajectory.png?raw=true) ![xyz](https://github.com/aipiano/ESEKF_IMU/blob/master/images/xyz.png?raw=true) ![rpy](https://github.com/aipiano/ESEKF_IMU/blob/master/images/rpy.png?raw=true) # Reference 1. [Quaternion kinematics for the error-state Kalman filter](https://arxiv.org/abs/1711.02508) 2. State Estimation for Robotics. 3. 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8.253340000000000121e-01 -1.010129999999999926e-03 -5.646419999999999773e-01 -1.476510000000000105e-03 5.999360000000000070e+01 9.999980000000000757e+00 -2.006830000000000072e-02 1.996790000000000020e+01 8.253350000000000408e-01 -5.665719999999999958e-04 -5.646419999999999773e-01 -8.281550000000000153e-04 5.999860000000000326e+01 1.000000000000000000e+01 -4.357009999999999703e-03 1.999299999999999855e+01 8.253359999999999586e-01 -1.230079999999999986e-04 -5.646419999999999773e-01 -1.798000000000000065e-04 ================================================ FILE: esekf.py ================================================ import numpy as np import numpy.linalg as la import transformations as tr import math class ImuParameters: def __init__(self): self.frequency = 200 self.sigma_a_n = 0.0 # acc noise. m/(s*sqrt(s)), continuous noise sigma self.sigma_w_n = 0.0 # gyro noise. rad/sqrt(s), continuous noise sigma self.sigma_a_b = 0.0 # acc bias m/sqrt(s^5), continuous bias sigma self.sigma_w_b = 0.0 # gyro bias rad/sqrt(s^3), continuous bias sigma class ESEKF(object): def __init__(self, init_nominal_state: np.array, imu_parameters: ImuParameters): """ :param init_nominal_state: [ p, q, v, a_b, w_b, g ], a 19x1 or 1x19 vector :param imu_parameters: imu parameters """ self.nominal_state = init_nominal_state if self.nominal_state[3] < 0: self.nominal_state[3:7] *= 1 # force the quaternion has a positive real part. self.imu_parameters = imu_parameters # initialize noise covariance matrix noise_covar = np.zeros((12, 12)) # assume the noises (especially sigma_a_n) are isotropic so that we can precompute self.noise_covar and save it. noise_covar[0:3, 0:3] = (imu_parameters.sigma_a_n**2) * np.eye(3) noise_covar[3:6, 3:6] = (imu_parameters.sigma_w_n**2) * np.eye(3) noise_covar[6:9, 6:9] = (imu_parameters.sigma_a_b**2) * np.eye(3) noise_covar[9:12, 9:12] = (imu_parameters.sigma_w_b**2) * np.eye(3) G = np.zeros((18, 12)) G[3:6, 3:6] = -np.eye(3) G[6:9, 0:3] = -np.eye(3) G[9:12, 6:9] = np.eye(3) G[12:15, 9:12] = np.eye(3) self.noise_covar = G @ noise_covar @ G.T # initialize error covariance matrix self.error_covar = 0.01 * self.noise_covar self.last_predict_time = 0.0 def predict(self, imu_measurement: np.array): """ :param imu_measurement: [t, w_m, a_m] :return: """ if self.last_predict_time == imu_measurement[0]: return # we predict error_covar first, because __predict_nominal_state will change the nominal state. self.__predict_error_covar(imu_measurement) self.__predict_nominal_state(imu_measurement) self.last_predict_time = imu_measurement[0] # update timestamp def __update_legacy(self, gt_measurement: np.array, measurement_covar: np.array): """ An old implementation of the updating procedure. :param gt_measurement: [p, q], a 7x1 or 1x7 vector :param measurement_covar: a 7x7 symmetrical matrix :return: """ """ Hx = dh/dx = [[I, 0, 0, 0, 0, 0] [0, I, 0, 0, 0, 0]] """ Hx = np.zeros((7, 19)) Hx[0:3, 0:3] = np.eye(3) Hx[3:7, 3:7] = np.eye(4) """ X = dx/d(delta_x) = [[I_3, 0, 0], [0, Q_d_theta, 0], [0, 0, I_12] """ X = np.zeros((19, 18)) q = self.nominal_state[3:7] X[0:3, 0:3] = np.eye(3) X[3:7, 3:6] = 0.5 * np.array([[-q[1], -q[2], -q[3]], [q[0], -q[3], q[2]], [q[3], q[0], -q[1]], [-q[2], q[1], q[0]]]) X[7:19, 6:18] = np.eye(12) H = Hx @ X # 7x18 PHt = self.error_covar @ H.T # 18x7 # compute Kalman gain. HPH^T, project the error covariance to the measurement space. K = PHt @ la.inv(H @ PHt + measurement_covar) # update error covariance matrix self.error_covar = (np.eye(18) - K @ H) @ self.error_covar # force the error_covar to be a symmetrical matrix self.error_covar = 0.5 * (self.error_covar + self.error_covar.T) """ compute errors. restrict quaternions in measurement and state have positive real parts. this is necessary for errors computation since we subtract quaternions directly. """ if gt_measurement[3] < 0: gt_measurement[3:7] *= -1 # NOTE: subtracting quaternion directly is tricky. that's why we abandon this implementation. errors = K @ (gt_measurement.reshape(-1, 1) - Hx @ self.nominal_state.reshape(-1, 1)) # inject errors to the nominal state self.nominal_state[0:3] += errors[0:3, 0] # update position dq = tr.quaternion_about_axis(la.norm(errors[3:6, 0]), errors[3:6, 0]) # print(dq) self.nominal_state[3:7] = tr.quaternion_multiply(q, dq) # update rotation self.nominal_state[3:7] /= la.norm(self.nominal_state[3:7]) if self.nominal_state[3] < 0: self.nominal_state[3:7] *= 1 self.nominal_state[7:] += errors[6:, 0] # update the rest. """ reset errors to zero and modify the error covariance matrix. we do nothing to the errors since we do not save them. but we need to modify the error_covar according to P = GPG^T """ G = np.eye(18) G[3:6, 3:6] = np.eye(3) - tr.skew_matrix(0.5 * errors[3:6, 0]) self.error_covar = G @ self.error_covar @ G.T def update(self, gt_measurement: np.array, measurement_covar: np.array): """ :param gt_measurement: [p, q], a 7x1 or 1x7 vector :param measurement_covar: a 6x6 symmetrical matrix = diag{sigma_p^2, sigma_theta^2} :return: """ """ we simulate a system that measure the errors between the nominal state and ground-truth state directly, so that we can avoid the direct subtracting of quaternions. we define q1 - q2 = conjugate(q2) x q1, so that q2 x (q1 - q2) = q1. ground_truth - nominal_state = delta = H @ error_state + noise """ H = np.zeros((6, 18)) H[0:3, 0:3] = np.eye(3) H[3:6, 3:6] = np.eye(3) PHt = self.error_covar @ H.T # 18x6 # compute Kalman gain. HPH^T, project the error covariance to the measurement space. K = PHt @ la.inv(H @ PHt + measurement_covar) # 18x6 # update error covariance matrix self.error_covar = (np.eye(18) - K @ H) @ self.error_covar # force the error_covar to be a symmetrical matrix self.error_covar = 0.5 * (self.error_covar + self.error_covar.T) # compute the measurements according to the nominal state and ground-truth state. if gt_measurement[3] < 0: gt_measurement[3:7] *= -1 gt_p = gt_measurement[0:3] gt_q = gt_measurement[3:7] q = self.nominal_state[3:7] delta = np.zeros((6, 1)) delta[0:3, 0] = gt_p - self.nominal_state[0:3] delta_q = tr.quaternion_multiply(tr.quaternion_conjugate(q), gt_q) if delta_q[0] < 0: delta_q *= -1 angle = math.asin(la.norm(delta_q[1:4])) if math.isclose(angle, 0): axis = np.zeros(3,) else: axis = delta_q[1:4] / la.norm(delta_q[1:4]) delta[3:6, 0] = angle * axis # compute state errors. errors = K @ delta # inject errors to the nominal state self.nominal_state[0:3] += errors[0:3, 0] # update position dq = tr.quaternion_about_axis(la.norm(errors[3:6, 0]), errors[3:6, 0]) # print(dq) self.nominal_state[3:7] = tr.quaternion_multiply(q, dq) # update rotation self.nominal_state[3:7] /= la.norm(self.nominal_state[3:7]) if self.nominal_state[3] < 0: self.nominal_state[3:7] *= 1 self.nominal_state[7:] += errors[6:, 0] # update the rest. """ reset errors to zero and modify the error covariance matrix. we do nothing to the errors since we do not save them. but we need to modify the error_covar according to P = GPG^T """ G = np.eye(18) G[3:6, 3:6] = np.eye(3) - tr.skew_matrix(0.5 * errors[3:6, 0]) self.error_covar = G @ self.error_covar @ G.T def __predict_nominal_state(self, imu_measurement: np.array): p = self.nominal_state[:3].reshape(-1, 1) q = self.nominal_state[3:7] v = self.nominal_state[7:10].reshape(-1, 1) a_b = self.nominal_state[10:13].reshape(-1, 1) w_b = self.nominal_state[13:16] g = self.nominal_state[16:19].reshape(-1, 1) w_m = imu_measurement[1:4].copy() a_m = imu_measurement[4:7].reshape(-1, 1).copy() dt = imu_measurement[0] - self.last_predict_time """ dp/dt = v dv/dt = R(a_m - a_b) + g dq/dt = 0.5 * q x(quaternion product) (w_m - w_b) a_m and w_m are the measurements of IMU. a_b and w_b are biases of acc and gyro, respectively. R = R{q}, which bring the point from local coordinate to global coordinate. """ w_m -= w_b a_m -= a_b # use the zero-order integration to integrate the quaternion. # q_{n+1} = q_n x q{(w_m - w_b) * dt} angle = la.norm(w_m) axis = w_m / angle R_w = tr.rotation_matrix(0.5 * dt * angle, axis) q_w = tr.quaternion_from_matrix(R_w, True) q_half_next = tr.quaternion_multiply(q, q_w) R_w = tr.rotation_matrix(dt * angle, axis) q_w = tr.quaternion_from_matrix(R_w, True) q_next = tr.quaternion_multiply(q, q_w) if q_next[0] < 0: # force the quaternion has a positive real part. q_next *= -1 # use RK4 method to integration velocity and position. # integrate velocity first. R = tr.quaternion_matrix(q)[:3, :3] R_half_next = tr.quaternion_matrix(q_half_next)[:3, :3] R_next = tr.quaternion_matrix(q_next)[:3, :3] v_k1 = R @ a_m + g v_k2 = R_half_next @ a_m + g # v_k3 = R_half_next @ a_m + g # yes. v_k2 = v_k3. v_k3 = v_k2 v_k4 = R_next @ a_m + g v_next = v + dt * (v_k1 + 2 * v_k2 + 2 * v_k3 + v_k4) / 6 # integrate position p_k1 = v p_k2 = v + 0.5 * dt * v_k1 # k2 = v_next_half = v + 0.5 * dt * v' = v + 0.5 * dt * v_k1(evaluate at t0) p_k3 = v + 0.5 * dt * v_k2 # v_k2 is evaluated at t0 + 0.5*delta p_k4 = v + dt * v_k3 p_next = p + dt * (p_k1 + 2 * p_k2 + 2 * p_k3 + p_k4) / 6 self.nominal_state[:3] = p_next.reshape(3,) self.nominal_state[3:7] = q_next self.nominal_state[7:10] = v_next.reshape(3,) # print(q_next) def __predict_error_covar(self, imu_measurement: np.array): w_m = imu_measurement[1:4] a_m = imu_measurement[4:7] a_b = self.nominal_state[9:12] w_b = self.nominal_state[12:15] q = self.nominal_state[3:7] R = tr.quaternion_matrix(q)[:3, :3] F = np.zeros((18, 18)) F[0:3, 6:9] = np.eye(3) F[3:6, 3:6] = -tr.skew_matrix(w_m - w_b) F[3:6, 12:15] = -np.eye(3) F[6:9, 3:6] = -R @ tr.skew_matrix(a_m - a_b) F[6:9, 9:12] = -R # use 3rd-order truncated integration to compute transition matrix Phi. dt = imu_measurement[0] - self.last_predict_time Fdt = F * dt Fdt2 = Fdt @ Fdt Fdt3 = Fdt2 @ Fdt Phi = np.eye(18) + Fdt + 0.5 * Fdt2 + (1. / 6.) * Fdt3 """ use trapezoidal integration to integrate noise covariance: Qd = 0.5 * dt * (Phi @ self.noise_covar @ Phi.T + self.noise_covar) self.error_covar = Phi @ self.error_covar @ Phi.T + Qd operations above can be merged to the below for efficiency. """ Qc_dt = 0.5*dt*self.noise_covar self.error_covar = Phi @ (self.error_covar + Qc_dt) @ Phi.T + Qc_dt ================================================ FILE: main.py ================================================ import numpy as np import matplotlib.pyplot as plt import transformations as tr import yaml import math from esekf import * def load_imu_parameters(): f = open('./data/params.yaml', 'r') yml = yaml.load(f.read()) params = ImuParameters() params.frequency = yml['IMU.frequency'] params.sigma_a_n = yml['IMU.acc_noise_sigma'] # m/sqrt(s^3) params.sigma_w_n = yml['IMU.gyro_noise_sigma'] # rad/sqrt(s) params.sigma_a_b = yml['IMU.acc_bias_sigma'] # m/sqrt(s^5) params.sigma_w_b = yml['IMU.gyro_bias_sigma'] # rad/sqrt(s^3) f.close() return params def main(): imu_data = np.loadtxt('./data/imu_noise.txt') gt_data = np.loadtxt('./data/traj_gt.txt') imu_parameters = load_imu_parameters() init_nominal_state = np.zeros((19,)) init_nominal_state[:10] = gt_data[0, 1:] # init p, q, v init_nominal_state[10:13] = 0 # init ba init_nominal_state[13:16] = 0 # init bg init_nominal_state[16:19] = np.array([0, 0, -9.81]) # init g estimator = ESEKF(init_nominal_state, imu_parameters) test_duration_s = [0., 61.] start_time = imu_data[0, 0] mask_imu = np.logical_and(imu_data[:, 0] <= start_time + test_duration_s[1], imu_data[:, 0] >= start_time + test_duration_s[0]) mask_gt = np.logical_and(gt_data[:, 0] <= start_time + test_duration_s[1], gt_data[:, 0] >= start_time + test_duration_s[0]) imu_data = imu_data[mask_imu, :] gt_data = gt_data[mask_gt, :] traj_est = [gt_data[0, :8]] update_ratio = 10 # control the frequency of ekf updating. sigma_measurement_p = 0.02 # in meters sigma_measurement_q = 0.015 # in rad sigma_measurement = np.eye(6) sigma_measurement[0:3, 0:3] *= sigma_measurement_p**2 sigma_measurement[3:6, 3:6] *= sigma_measurement_q**2 for i in range(1, imu_data.shape[0]): timestamp = imu_data[i, 0] estimator.predict(imu_data[i, :]) if i % update_ratio == 0: # we assume the timestamps are aligned. assert math.isclose(gt_data[i, 0], timestamp) gt_pose = gt_data[i, 1:8].copy() # gt_pose = [p, q] # add position noise gt_pose[:3] += np.random.randn(3,) * sigma_measurement_p # add rotation noise, u = [1, 0.5 * noise_angle_axis] # u = 0.5 * np.random.randn(4,) * sigma_measurement_q # u[0] = 1 u = np.random.randn(3, ) * sigma_measurement_q qn = tr.quaternion_about_axis(la.norm(u), u / la.norm(u)) gt_pose[3:] = tr.quaternion_multiply(gt_pose[3:], qn) # update filter by measurement. estimator.update(gt_pose, sigma_measurement) print('[%f]:' % timestamp, estimator.nominal_state) frame_pose = np.zeros(8,) frame_pose[0] = timestamp frame_pose[1:] = estimator.nominal_state[:7] traj_est.append(frame_pose) # save trajectory to TUM format traj_est = np.array(traj_est) np.savetxt('./data/traj_gt_out.txt', gt_data[:, :8]) np.savetxt('./data/traj_esekf_out.txt', traj_est) if __name__ == '__main__': main() ================================================ FILE: transformations.py ================================================ # -*- coding: utf-8 -*- # transformations.py # Copyright (c) 2006-2015, Christoph Gohlke # Copyright (c) 2006-2015, The Regents of the University of California # Produced at the Laboratory for Fluorescence Dynamics # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions are met: # # * Redistributions of source code must retain the above copyright # notice, this list of conditions and the following disclaimer. # * Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer in the # documentation and/or other materials provided with the distribution. # * Neither the name of the copyright holders nor the names of any # contributors may be used to endorse or promote products derived # from this software without specific prior written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" # AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE # IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE # ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE # LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR # CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF # SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS # INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN # CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) # ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE # POSSIBILITY OF SUCH DAMAGE. """Homogeneous Transformation Matrices and Quaternions. A library for calculating 4x4 matrices for translating, rotating, reflecting, scaling, shearing, projecting, orthogonalizing, and superimposing arrays of 3D homogeneous coordinates as well as for converting between rotation matrices, Euler angles, and quaternions. Also includes an Arcball control object and functions to decompose transformation matrices. :Author: `Christoph Gohlke `_ :Organization: Laboratory for Fluorescence Dynamics, University of California, Irvine :Version: 2015.07.18 Requirements ------------ * `CPython 2.7 or 3.4 `_ * `Numpy 1.9 `_ * `Transformations.c 2015.07.18 `_ (recommended for speedup of some functions) Notes ----- The API is not stable yet and is expected to change between revisions. This Python code is not optimized for speed. Refer to the transformations.c module for a faster implementation of some functions. Documentation in HTML format can be generated with epydoc. Matrices (M) can be inverted using numpy.linalg.inv(M), be concatenated using numpy.dot(M0, M1), or transform homogeneous coordinate arrays (v) using numpy.dot(M, v) for shape (4, \*) column vectors, respectively numpy.dot(v, M.T) for shape (\*, 4) row vectors ("array of points"). This module follows the "column vectors on the right" and "row major storage" (C contiguous) conventions. The translation components are in the right column of the transformation matrix, i.e. M[:3, 3]. The transpose of the transformation matrices may have to be used to interface with other graphics systems, e.g. with OpenGL's glMultMatrixd(). See also [16]. Calculations are carried out with numpy.float64 precision. Vector, point, quaternion, and matrix function arguments are expected to be "array like", i.e. tuple, list, or numpy arrays. Return types are numpy arrays unless specified otherwise. Angles are in radians unless specified otherwise. Quaternions w+ix+jy+kz are represented as [w, x, y, z]. A triple of Euler angles can be applied/interpreted in 24 ways, which can be specified using a 4 character string or encoded 4-tuple: *Axes 4-string*: e.g. 'sxyz' or 'ryxy' - first character : rotations are applied to 's'tatic or 'r'otating frame - remaining characters : successive rotation axis 'x', 'y', or 'z' *Axes 4-tuple*: e.g. (0, 0, 0, 0) or (1, 1, 1, 1) - inner axis: code of axis ('x':0, 'y':1, 'z':2) of rightmost matrix. - parity : even (0) if inner axis 'x' is followed by 'y', 'y' is followed by 'z', or 'z' is followed by 'x'. Otherwise odd (1). - repetition : first and last axis are same (1) or different (0). - frame : rotations are applied to static (0) or rotating (1) frame. Other Python packages and modules for 3D transformations and quaternions: * `Transforms3d `_ includes most code of this module. * `Blender.mathutils `_ * `numpy-dtypes `_ References ---------- (1) Matrices and transformations. Ronald Goldman. In "Graphics Gems I", pp 472-475. Morgan Kaufmann, 1990. (2) More matrices and transformations: shear and pseudo-perspective. Ronald Goldman. In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991. (3) Decomposing a matrix into simple transformations. Spencer Thomas. In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991. (4) Recovering the data from the transformation matrix. Ronald Goldman. In "Graphics Gems II", pp 324-331. Morgan Kaufmann, 1991. (5) Euler angle conversion. Ken Shoemake. In "Graphics Gems IV", pp 222-229. Morgan Kaufmann, 1994. (6) Arcball rotation control. Ken Shoemake. In "Graphics Gems IV", pp 175-192. Morgan Kaufmann, 1994. (7) Representing attitude: Euler angles, unit quaternions, and rotation vectors. James Diebel. 2006. (8) A discussion of the solution for the best rotation to relate two sets of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828. (9) Closed-form solution of absolute orientation using unit quaternions. BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642. (10) Quaternions. Ken Shoemake. http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf (11) From quaternion to matrix and back. JMP van Waveren. 2005. http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm (12) Uniform random rotations. Ken Shoemake. In "Graphics Gems III", pp 124-132. Morgan Kaufmann, 1992. (13) Quaternion in molecular modeling. CFF Karney. J Mol Graph Mod, 25(5):595-604 (14) New method for extracting the quaternion from a rotation matrix. Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087. (15) Multiple View Geometry in Computer Vision. Hartley and Zissermann. Cambridge University Press; 2nd Ed. 2004. Chapter 4, Algorithm 4.7, p 130. (16) Column Vectors vs. Row Vectors. http://steve.hollasch.net/cgindex/math/matrix/column-vec.html Examples -------- >>> alpha, beta, gamma = 0.123, -1.234, 2.345 >>> origin, xaxis, yaxis, zaxis = [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1] >>> I = identity_matrix() >>> Rx = rotation_matrix(alpha, xaxis) >>> Ry = rotation_matrix(beta, yaxis) >>> Rz = rotation_matrix(gamma, zaxis) >>> R = concatenate_matrices(Rx, Ry, Rz) >>> euler = euler_from_matrix(R, 'rxyz') >>> numpy.allclose([alpha, beta, gamma], euler) True >>> Re = euler_matrix(alpha, beta, gamma, 'rxyz') >>> is_same_transform(R, Re) True >>> al, be, ga = euler_from_matrix(Re, 'rxyz') >>> is_same_transform(Re, euler_matrix(al, be, ga, 'rxyz')) True >>> qx = quaternion_about_axis(alpha, xaxis) >>> qy = quaternion_about_axis(beta, yaxis) >>> qz = quaternion_about_axis(gamma, zaxis) >>> q = quaternion_multiply(qx, qy) >>> q = quaternion_multiply(q, qz) >>> Rq = quaternion_matrix(q) >>> is_same_transform(R, Rq) True >>> S = scale_matrix(1.23, origin) >>> T = translation_matrix([1, 2, 3]) >>> Z = shear_matrix(beta, xaxis, origin, zaxis) >>> R = random_rotation_matrix(numpy.random.rand(3)) >>> M = concatenate_matrices(T, R, Z, S) >>> scale, shear, angles, trans, persp = decompose_matrix(M) >>> numpy.allclose(scale, 1.23) True >>> numpy.allclose(trans, [1, 2, 3]) True >>> numpy.allclose(shear, [0, math.tan(beta), 0]) True >>> is_same_transform(R, euler_matrix(axes='sxyz', *angles)) True >>> M1 = compose_matrix(scale, shear, angles, trans, persp) >>> is_same_transform(M, M1) True >>> v0, v1 = random_vector(3), random_vector(3) >>> M = rotation_matrix(angle_between_vectors(v0, v1), vector_product(v0, v1)) >>> v2 = numpy.dot(v0, M[:3,:3].T) >>> numpy.allclose(unit_vector(v1), unit_vector(v2)) True """ from __future__ import division, print_function import math import numpy __version__ = '2015.07.18' __docformat__ = 'restructuredtext en' __all__ = () # TODO evo: added to suppress annoying warnings, see README.md import warnings warnings.filterwarnings("ignore", message="failed to import module _transformations") def identity_matrix(): """Return 4x4 identity/unit matrix. >>> I = identity_matrix() >>> numpy.allclose(I, numpy.dot(I, I)) True >>> numpy.sum(I), numpy.trace(I) (4.0, 4.0) >>> numpy.allclose(I, numpy.identity(4)) True """ return numpy.identity(4) def skew_matrix(a): return numpy.array([ [0, -a[2], a[1]], [a[2], 0, -a[0]], [-a[1], a[0], 0] ]) def translation_matrix(direction): """Return matrix to translate by direction vector. >>> v = numpy.random.random(3) - 0.5 >>> numpy.allclose(v, translation_matrix(v)[:3, 3]) True """ M = numpy.identity(4) M[:3, 3] = direction[:3] return M def translation_from_matrix(matrix): """Return translation vector from translation matrix. >>> v0 = numpy.random.random(3) - 0.5 >>> v1 = translation_from_matrix(translation_matrix(v0)) >>> numpy.allclose(v0, v1) True """ return numpy.array(matrix, copy=False)[:3, 3].copy() def reflection_matrix(point, normal): """Return matrix to mirror at plane defined by point and normal vector. >>> v0 = numpy.random.random(4) - 0.5 >>> v0[3] = 1. >>> v1 = numpy.random.random(3) - 0.5 >>> R = reflection_matrix(v0, v1) >>> numpy.allclose(2, numpy.trace(R)) True >>> numpy.allclose(v0, numpy.dot(R, v0)) True >>> v2 = v0.copy() >>> v2[:3] += v1 >>> v3 = v0.copy() >>> v2[:3] -= v1 >>> numpy.allclose(v2, numpy.dot(R, v3)) True """ normal = unit_vector(normal[:3]) M = numpy.identity(4) M[:3, :3] -= 2.0 * numpy.outer(normal, normal) M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal return M def reflection_from_matrix(matrix): """Return mirror plane point and normal vector from reflection matrix. >>> v0 = numpy.random.random(3) - 0.5 >>> v1 = numpy.random.random(3) - 0.5 >>> M0 = reflection_matrix(v0, v1) >>> point, normal = reflection_from_matrix(M0) >>> M1 = reflection_matrix(point, normal) >>> is_same_transform(M0, M1) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=False) # normal: unit eigenvector corresponding to eigenvalue -1 w, V = numpy.linalg.eig(M[:3, :3]) i = numpy.where(abs(numpy.real(w) + 1.0) < 1e-8)[0] if not len(i): raise ValueError("no unit eigenvector corresponding to eigenvalue -1") normal = numpy.real(V[:, i[0]]).squeeze() # point: any unit eigenvector corresponding to eigenvalue 1 w, V = numpy.linalg.eig(M) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not len(i): raise ValueError("no unit eigenvector corresponding to eigenvalue 1") point = numpy.real(V[:, i[-1]]).squeeze() point /= point[3] return point, normal def rotation_matrix(angle, direction, point=None): """Return matrix to rotate about axis defined by point and direction. >>> R = rotation_matrix(math.pi/2, [0, 0, 1], [1, 0, 0]) >>> numpy.allclose(numpy.dot(R, [0, 0, 0, 1]), [1, -1, 0, 1]) True >>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(angle-2*math.pi, direc, point) >>> is_same_transform(R0, R1) True >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(-angle, -direc, point) >>> is_same_transform(R0, R1) True >>> I = numpy.identity(4, numpy.float64) >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc)) True >>> numpy.allclose(2, numpy.trace(rotation_matrix(math.pi/2, ... direc, point))) True """ sina = math.sin(angle) cosa = math.cos(angle) direction = unit_vector(direction[:3]) # rotation matrix around unit vector R = numpy.diag([cosa, cosa, cosa]) R += numpy.outer(direction, direction) * (1.0 - cosa) direction *= sina R += numpy.array([[ 0.0, -direction[2], direction[1]], [ direction[2], 0.0, -direction[0]], [-direction[1], direction[0], 0.0]]) M = numpy.identity(4) M[:3, :3] = R if point is not None: # rotation not around origin point = numpy.array(point[:3], dtype=numpy.float64, copy=False) M[:3, 3] = point - numpy.dot(R, point) return M def rotation_from_matrix(matrix): """Return rotation angle and axis from rotation matrix. >>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> angle, direc, point = rotation_from_matrix(R0) >>> R1 = rotation_matrix(angle, direc, point) >>> is_same_transform(R0, R1) True """ R = numpy.array(matrix, dtype=numpy.float64, copy=False) R33 = R[:3, :3] # direction: unit eigenvector of R33 corresponding to eigenvalue of 1 w, W = numpy.linalg.eig(R33.T) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not len(i): raise ValueError("no unit eigenvector corresponding to eigenvalue 1") direction = numpy.real(W[:, i[-1]]).squeeze() # point: unit eigenvector of R33 corresponding to eigenvalue of 1 w, Q = numpy.linalg.eig(R) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not len(i): raise ValueError("no unit eigenvector corresponding to eigenvalue 1") point = numpy.real(Q[:, i[-1]]).squeeze() point /= point[3] # rotation angle depending on direction cosa = (numpy.trace(R33) - 1.0) / 2.0 if abs(direction[2]) > 1e-8: sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2] elif abs(direction[1]) > 1e-8: sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1] else: sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0] angle = math.atan2(sina, cosa) return angle, direction, point def scale_matrix(factor, origin=None, direction=None): """Return matrix to scale by factor around origin in direction. Use factor -1 for point symmetry. >>> v = (numpy.random.rand(4, 5) - 0.5) * 20 >>> v[3] = 1 >>> S = scale_matrix(-1.234) >>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3]) True >>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S = scale_matrix(factor, origin) >>> S = scale_matrix(factor, origin, direct) """ if direction is None: # uniform scaling M = numpy.diag([factor, factor, factor, 1.0]) if origin is not None: M[:3, 3] = origin[:3] M[:3, 3] *= 1.0 - factor else: # nonuniform scaling direction = unit_vector(direction[:3]) factor = 1.0 - factor M = numpy.identity(4) M[:3, :3] -= factor * numpy.outer(direction, direction) if origin is not None: M[:3, 3] = (factor * numpy.dot(origin[:3], direction)) * direction return M def scale_from_matrix(matrix): """Return scaling factor, origin and direction from scaling matrix. >>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S0 = scale_matrix(factor, origin) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True >>> S0 = scale_matrix(factor, origin, direct) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=False) M33 = M[:3, :3] factor = numpy.trace(M33) - 2.0 try: # direction: unit eigenvector corresponding to eigenvalue factor w, V = numpy.linalg.eig(M33) i = numpy.where(abs(numpy.real(w) - factor) < 1e-8)[0][0] direction = numpy.real(V[:, i]).squeeze() direction /= vector_norm(direction) except IndexError: # uniform scaling factor = (factor + 2.0) / 3.0 direction = None # origin: any eigenvector corresponding to eigenvalue 1 w, V = numpy.linalg.eig(M) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not len(i): raise ValueError("no eigenvector corresponding to eigenvalue 1") origin = numpy.real(V[:, i[-1]]).squeeze() origin /= origin[3] return factor, origin, direction def projection_matrix(point, normal, direction=None, perspective=None, pseudo=False): """Return matrix to project onto plane defined by point and normal. Using either perspective point, projection direction, or none of both. If pseudo is True, perspective projections will preserve relative depth such that Perspective = dot(Orthogonal, PseudoPerspective). >>> P = projection_matrix([0, 0, 0], [1, 0, 0]) >>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:]) True >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> P1 = projection_matrix(point, normal, direction=direct) >>> P2 = projection_matrix(point, normal, perspective=persp) >>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> is_same_transform(P2, numpy.dot(P0, P3)) True >>> P = projection_matrix([3, 0, 0], [1, 1, 0], [1, 0, 0]) >>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20 >>> v0[3] = 1 >>> v1 = numpy.dot(P, v0) >>> numpy.allclose(v1[1], v0[1]) True >>> numpy.allclose(v1[0], 3-v1[1]) True """ M = numpy.identity(4) point = numpy.array(point[:3], dtype=numpy.float64, copy=False) normal = unit_vector(normal[:3]) if perspective is not None: # perspective projection perspective = numpy.array(perspective[:3], dtype=numpy.float64, copy=False) M[0, 0] = M[1, 1] = M[2, 2] = numpy.dot(perspective-point, normal) M[:3, :3] -= numpy.outer(perspective, normal) if pseudo: # preserve relative depth M[:3, :3] -= numpy.outer(normal, normal) M[:3, 3] = numpy.dot(point, normal) * (perspective+normal) else: M[:3, 3] = numpy.dot(point, normal) * perspective M[3, :3] = -normal M[3, 3] = numpy.dot(perspective, normal) elif direction is not None: # parallel projection direction = numpy.array(direction[:3], dtype=numpy.float64, copy=False) scale = numpy.dot(direction, normal) M[:3, :3] -= numpy.outer(direction, normal) / scale M[:3, 3] = direction * (numpy.dot(point, normal) / scale) else: # orthogonal projection M[:3, :3] -= numpy.outer(normal, normal) M[:3, 3] = numpy.dot(point, normal) * normal return M def projection_from_matrix(matrix, pseudo=False): """Return projection plane and perspective point from projection matrix. Return values are same as arguments for projection_matrix function: point, normal, direction, perspective, and pseudo. >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, direct) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False) >>> result = projection_from_matrix(P0, pseudo=False) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> result = projection_from_matrix(P0, pseudo=True) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=False) M33 = M[:3, :3] w, V = numpy.linalg.eig(M) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not pseudo and len(i): # point: any eigenvector corresponding to eigenvalue 1 point = numpy.real(V[:, i[-1]]).squeeze() point /= point[3] # direction: unit eigenvector corresponding to eigenvalue 0 w, V = numpy.linalg.eig(M33) i = numpy.where(abs(numpy.real(w)) < 1e-8)[0] if not len(i): raise ValueError("no eigenvector corresponding to eigenvalue 0") direction = numpy.real(V[:, i[0]]).squeeze() direction /= vector_norm(direction) # normal: unit eigenvector of M33.T corresponding to eigenvalue 0 w, V = numpy.linalg.eig(M33.T) i = numpy.where(abs(numpy.real(w)) < 1e-8)[0] if len(i): # parallel projection normal = numpy.real(V[:, i[0]]).squeeze() normal /= vector_norm(normal) return point, normal, direction, None, False else: # orthogonal projection, where normal equals direction vector return point, direction, None, None, False else: # perspective projection i = numpy.where(abs(numpy.real(w)) > 1e-8)[0] if not len(i): raise ValueError( "no eigenvector not corresponding to eigenvalue 0") point = numpy.real(V[:, i[-1]]).squeeze() point /= point[3] normal = - M[3, :3] perspective = M[:3, 3] / numpy.dot(point[:3], normal) if pseudo: perspective -= normal return point, normal, None, perspective, pseudo def clip_matrix(left, right, bottom, top, near, far, perspective=False): """Return matrix to obtain normalized device coordinates from frustum. The frustum bounds are axis-aligned along x (left, right), y (bottom, top) and z (near, far). Normalized device coordinates are in range [-1, 1] if coordinates are inside the frustum. If perspective is True the frustum is a truncated pyramid with the perspective point at origin and direction along z axis, otherwise an orthographic canonical view volume (a box). Homogeneous coordinates transformed by the perspective clip matrix need to be dehomogenized (divided by w coordinate). >>> frustum = numpy.random.rand(6) >>> frustum[1] += frustum[0] >>> frustum[3] += frustum[2] >>> frustum[5] += frustum[4] >>> M = clip_matrix(perspective=False, *frustum) >>> numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1]) array([-1., -1., -1., 1.]) >>> numpy.dot(M, [frustum[1], frustum[3], frustum[5], 1]) array([ 1., 1., 1., 1.]) >>> M = clip_matrix(perspective=True, *frustum) >>> v = numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1]) >>> v / v[3] array([-1., -1., -1., 1.]) >>> v = numpy.dot(M, [frustum[1], frustum[3], frustum[4], 1]) >>> v / v[3] array([ 1., 1., -1., 1.]) """ if left >= right or bottom >= top or near >= far: raise ValueError("invalid frustum") if perspective: if near <= _EPS: raise ValueError("invalid frustum: near <= 0") t = 2.0 * near M = [[t/(left-right), 0.0, (right+left)/(right-left), 0.0], [0.0, t/(bottom-top), (top+bottom)/(top-bottom), 0.0], [0.0, 0.0, (far+near)/(near-far), t*far/(far-near)], [0.0, 0.0, -1.0, 0.0]] else: M = [[2.0/(right-left), 0.0, 0.0, (right+left)/(left-right)], [0.0, 2.0/(top-bottom), 0.0, (top+bottom)/(bottom-top)], [0.0, 0.0, 2.0/(far-near), (far+near)/(near-far)], [0.0, 0.0, 0.0, 1.0]] return numpy.array(M) def shear_matrix(angle, direction, point, normal): """Return matrix to shear by angle along direction vector on shear plane. The shear plane is defined by a point and normal vector. The direction vector must be orthogonal to the plane's normal vector. A point P is transformed by the shear matrix into P" such that the vector P-P" is parallel to the direction vector and its extent is given by the angle of P-P'-P", where P' is the orthogonal projection of P onto the shear plane. >>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S = shear_matrix(angle, direct, point, normal) >>> numpy.allclose(1, numpy.linalg.det(S)) True """ normal = unit_vector(normal[:3]) direction = unit_vector(direction[:3]) if abs(numpy.dot(normal, direction)) > 1e-6: raise ValueError("direction and normal vectors are not orthogonal") angle = math.tan(angle) M = numpy.identity(4) M[:3, :3] += angle * numpy.outer(direction, normal) M[:3, 3] = -angle * numpy.dot(point[:3], normal) * direction return M def shear_from_matrix(matrix): """Return shear angle, direction and plane from shear matrix. >>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S0 = shear_matrix(angle, direct, point, normal) >>> angle, direct, point, normal = shear_from_matrix(S0) >>> S1 = shear_matrix(angle, direct, point, normal) >>> is_same_transform(S0, S1) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=False) M33 = M[:3, :3] # normal: cross independent eigenvectors corresponding to the eigenvalue 1 w, V = numpy.linalg.eig(M33) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-4)[0] if len(i) < 2: raise ValueError("no two linear independent eigenvectors found %s" % w) V = numpy.real(V[:, i]).squeeze().T lenorm = -1.0 for i0, i1 in ((0, 1), (0, 2), (1, 2)): n = numpy.cross(V[i0], V[i1]) w = vector_norm(n) if w > lenorm: lenorm = w normal = n normal /= lenorm # direction and angle direction = numpy.dot(M33 - numpy.identity(3), normal) angle = vector_norm(direction) direction /= angle angle = math.atan(angle) # point: eigenvector corresponding to eigenvalue 1 w, V = numpy.linalg.eig(M) i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0] if not len(i): raise ValueError("no eigenvector corresponding to eigenvalue 1") point = numpy.real(V[:, i[-1]]).squeeze() point /= point[3] return angle, direction, point, normal def decompose_matrix(matrix): """Return sequence of transformations from transformation matrix. matrix : array_like Non-degenerative homogeneous transformation matrix Return tuple of: scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix Raise ValueError if matrix is of wrong type or degenerative. >>> T0 = translation_matrix([1, 2, 3]) >>> scale, shear, angles, trans, persp = decompose_matrix(T0) >>> T1 = translation_matrix(trans) >>> numpy.allclose(T0, T1) True >>> S = scale_matrix(0.123) >>> scale, shear, angles, trans, persp = decompose_matrix(S) >>> scale[0] 0.123 >>> R0 = euler_matrix(1, 2, 3) >>> scale, shear, angles, trans, persp = decompose_matrix(R0) >>> R1 = euler_matrix(*angles) >>> numpy.allclose(R0, R1) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=True).T if abs(M[3, 3]) < _EPS: raise ValueError("M[3, 3] is zero") M /= M[3, 3] P = M.copy() P[:, 3] = 0.0, 0.0, 0.0, 1.0 if not numpy.linalg.det(P): raise ValueError("matrix is singular") scale = numpy.zeros((3, )) shear = [0.0, 0.0, 0.0] angles = [0.0, 0.0, 0.0] if any(abs(M[:3, 3]) > _EPS): perspective = numpy.dot(M[:, 3], numpy.linalg.inv(P.T)) M[:, 3] = 0.0, 0.0, 0.0, 1.0 else: perspective = numpy.array([0.0, 0.0, 0.0, 1.0]) translate = M[3, :3].copy() M[3, :3] = 0.0 row = M[:3, :3].copy() scale[0] = vector_norm(row[0]) row[0] /= scale[0] shear[0] = numpy.dot(row[0], row[1]) row[1] -= row[0] * shear[0] scale[1] = vector_norm(row[1]) row[1] /= scale[1] shear[0] /= scale[1] shear[1] = numpy.dot(row[0], row[2]) row[2] -= row[0] * shear[1] shear[2] = numpy.dot(row[1], row[2]) row[2] -= row[1] * shear[2] scale[2] = vector_norm(row[2]) row[2] /= scale[2] shear[1:] /= scale[2] if numpy.dot(row[0], numpy.cross(row[1], row[2])) < 0: numpy.negative(scale, scale) numpy.negative(row, row) angles[1] = math.asin(-row[0, 2]) if math.cos(angles[1]): angles[0] = math.atan2(row[1, 2], row[2, 2]) angles[2] = math.atan2(row[0, 1], row[0, 0]) else: #angles[0] = math.atan2(row[1, 0], row[1, 1]) angles[0] = math.atan2(-row[2, 1], row[1, 1]) angles[2] = 0.0 return scale, shear, angles, translate, perspective def compose_matrix(scale=None, shear=None, angles=None, translate=None, perspective=None): """Return transformation matrix from sequence of transformations. This is the inverse of the decompose_matrix function. Sequence of transformations: scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix >>> scale = numpy.random.random(3) - 0.5 >>> shear = numpy.random.random(3) - 0.5 >>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi) >>> trans = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(4) - 0.5 >>> M0 = compose_matrix(scale, shear, angles, trans, persp) >>> result = decompose_matrix(M0) >>> M1 = compose_matrix(*result) >>> is_same_transform(M0, M1) True """ M = numpy.identity(4) if perspective is not None: P = numpy.identity(4) P[3, :] = perspective[:4] M = numpy.dot(M, P) if translate is not None: T = numpy.identity(4) T[:3, 3] = translate[:3] M = numpy.dot(M, T) if angles is not None: R = euler_matrix(angles[0], angles[1], angles[2], 'sxyz') M = numpy.dot(M, R) if shear is not None: Z = numpy.identity(4) Z[1, 2] = shear[2] Z[0, 2] = shear[1] Z[0, 1] = shear[0] M = numpy.dot(M, Z) if scale is not None: S = numpy.identity(4) S[0, 0] = scale[0] S[1, 1] = scale[1] S[2, 2] = scale[2] M = numpy.dot(M, S) M /= M[3, 3] return M def orthogonalization_matrix(lengths, angles): """Return orthogonalization matrix for crystallographic cell coordinates. Angles are expected in degrees. The de-orthogonalization matrix is the inverse. >>> O = orthogonalization_matrix([10, 10, 10], [90, 90, 90]) >>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10) True >>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7]) >>> numpy.allclose(numpy.sum(O), 43.063229) True """ a, b, c = lengths angles = numpy.radians(angles) sina, sinb, _ = numpy.sin(angles) cosa, cosb, cosg = numpy.cos(angles) co = (cosa * cosb - cosg) / (sina * sinb) return numpy.array([ [ a*sinb*math.sqrt(1.0-co*co), 0.0, 0.0, 0.0], [-a*sinb*co, b*sina, 0.0, 0.0], [ a*cosb, b*cosa, c, 0.0], [ 0.0, 0.0, 0.0, 1.0]]) def affine_matrix_from_points(v0, v1, shear=True, scale=True, usesvd=True): """Return affine transform matrix to register two point sets. v0 and v1 are shape (ndims, \*) arrays of at least ndims non-homogeneous coordinates, where ndims is the dimensionality of the coordinate space. If shear is False, a similarity transformation matrix is returned. If also scale is False, a rigid/Euclidean transformation matrix is returned. By default the algorithm by Hartley and Zissermann [15] is used. If usesvd is True, similarity and Euclidean transformation matrices are calculated by minimizing the weighted sum of squared deviations (RMSD) according to the algorithm by Kabsch [8]. Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9] is used, which is slower when using this Python implementation. The returned matrix performs rotation, translation and uniform scaling (if specified). >>> v0 = [[0, 1031, 1031, 0], [0, 0, 1600, 1600]] >>> v1 = [[675, 826, 826, 677], [55, 52, 281, 277]] >>> affine_matrix_from_points(v0, v1) array([[ 0.14549, 0.00062, 675.50008], [ 0.00048, 0.14094, 53.24971], [ 0. , 0. , 1. ]]) >>> T = translation_matrix(numpy.random.random(3)-0.5) >>> R = random_rotation_matrix(numpy.random.random(3)) >>> S = scale_matrix(random.random()) >>> M = concatenate_matrices(T, R, S) >>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20 >>> v0[3] = 1 >>> v1 = numpy.dot(M, v0) >>> v0[:3] += numpy.random.normal(0, 1e-8, 300).reshape(3, -1) >>> M = affine_matrix_from_points(v0[:3], v1[:3]) >>> numpy.allclose(v1, numpy.dot(M, v0)) True More examples in superimposition_matrix() """ v0 = numpy.array(v0, dtype=numpy.float64, copy=True) v1 = numpy.array(v1, dtype=numpy.float64, copy=True) ndims = v0.shape[0] if ndims < 2 or v0.shape[1] < ndims or v0.shape != v1.shape: raise ValueError("input arrays are of wrong shape or type") # move centroids to origin t0 = -numpy.mean(v0, axis=1) M0 = numpy.identity(ndims+1) M0[:ndims, ndims] = t0 v0 += t0.reshape(ndims, 1) t1 = -numpy.mean(v1, axis=1) M1 = numpy.identity(ndims+1) M1[:ndims, ndims] = t1 v1 += t1.reshape(ndims, 1) if shear: # Affine transformation A = numpy.concatenate((v0, v1), axis=0) u, s, vh = numpy.linalg.svd(A.T) vh = vh[:ndims].T B = vh[:ndims] C = vh[ndims:2*ndims] t = numpy.dot(C, numpy.linalg.pinv(B)) t = numpy.concatenate((t, numpy.zeros((ndims, 1))), axis=1) M = numpy.vstack((t, ((0.0,)*ndims) + (1.0,))) elif usesvd or ndims != 3: # Rigid transformation via SVD of covariance matrix u, s, vh = numpy.linalg.svd(numpy.dot(v1, v0.T)) # rotation matrix from SVD orthonormal bases R = numpy.dot(u, vh) if numpy.linalg.det(R) < 0.0: # R does not constitute right handed system R -= numpy.outer(u[:, ndims-1], vh[ndims-1, :]*2.0) s[-1] *= -1.0 # homogeneous transformation matrix M = numpy.identity(ndims+1) M[:ndims, :ndims] = R else: # Rigid transformation matrix via quaternion # compute symmetric matrix N xx, yy, zz = numpy.sum(v0 * v1, axis=1) xy, yz, zx = numpy.sum(v0 * numpy.roll(v1, -1, axis=0), axis=1) xz, yx, zy = numpy.sum(v0 * numpy.roll(v1, -2, axis=0), axis=1) N = [[xx+yy+zz, 0.0, 0.0, 0.0], [yz-zy, xx-yy-zz, 0.0, 0.0], [zx-xz, xy+yx, yy-xx-zz, 0.0], [xy-yx, zx+xz, yz+zy, zz-xx-yy]] # quaternion: eigenvector corresponding to most positive eigenvalue w, V = numpy.linalg.eigh(N) q = V[:, numpy.argmax(w)] q /= vector_norm(q) # unit quaternion # homogeneous transformation matrix M = quaternion_matrix(q) if scale and not shear: # Affine transformation; scale is ratio of RMS deviations from centroid v0 *= v0 v1 *= v1 M[:ndims, :ndims] *= math.sqrt(numpy.sum(v1) / numpy.sum(v0)) # move centroids back M = numpy.dot(numpy.linalg.inv(M1), numpy.dot(M, M0)) M /= M[ndims, ndims] return M def superimposition_matrix(v0, v1, scale=False, usesvd=True): """Return matrix to transform given 3D point set into second point set. v0 and v1 are shape (3, \*) or (4, \*) arrays of at least 3 points. The parameters scale and usesvd are explained in the more general affine_matrix_from_points function. The returned matrix is a similarity or Euclidean transformation matrix. This function has a fast C implementation in transformations.c. >>> v0 = numpy.random.rand(3, 10) >>> M = superimposition_matrix(v0, v0) >>> numpy.allclose(M, numpy.identity(4)) True >>> R = random_rotation_matrix(numpy.random.random(3)) >>> v0 = [[1,0,0], [0,1,0], [0,0,1], [1,1,1]] >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20 >>> v0[3] = 1 >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> S = scale_matrix(random.random()) >>> T = translation_matrix(numpy.random.random(3)-0.5) >>> M = concatenate_matrices(T, R, S) >>> v1 = numpy.dot(M, v0) >>> v0[:3] += numpy.random.normal(0, 1e-9, 300).reshape(3, -1) >>> M = superimposition_matrix(v0, v1, scale=True) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v = numpy.empty((4, 100, 3)) >>> v[:, :, 0] = v0 >>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0])) True """ v0 = numpy.array(v0, dtype=numpy.float64, copy=False)[:3] v1 = numpy.array(v1, dtype=numpy.float64, copy=False)[:3] return affine_matrix_from_points(v0, v1, shear=False, scale=scale, usesvd=usesvd) def euler_matrix(ai, aj, ak, axes='sxyz'): """Return homogeneous rotation matrix from Euler angles and axis sequence. ai, aj, ak : Euler's roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple >>> R = euler_matrix(1, 2, 3, 'syxz') >>> numpy.allclose(numpy.sum(R[0]), -1.34786452) True >>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1)) >>> numpy.allclose(numpy.sum(R[0]), -0.383436184) True >>> ai, aj, ak = (4*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R = euler_matrix(ai, aj, ak, axes) >>> for axes in _TUPLE2AXES.keys(): ... R = euler_matrix(ai, aj, ak, axes) """ try: firstaxis, parity, repetition, frame = _AXES2TUPLE[axes] except (AttributeError, KeyError): _TUPLE2AXES[axes] # validation firstaxis, parity, repetition, frame = axes i = firstaxis j = _NEXT_AXIS[i+parity] k = _NEXT_AXIS[i-parity+1] if frame: ai, ak = ak, ai if parity: ai, aj, ak = -ai, -aj, -ak si, sj, sk = math.sin(ai), math.sin(aj), math.sin(ak) ci, cj, ck = math.cos(ai), math.cos(aj), math.cos(ak) cc, cs = ci*ck, ci*sk sc, ss = si*ck, si*sk M = numpy.identity(4) if repetition: M[i, i] = cj M[i, j] = sj*si M[i, k] = sj*ci M[j, i] = sj*sk M[j, j] = -cj*ss+cc M[j, k] = -cj*cs-sc M[k, i] = -sj*ck M[k, j] = cj*sc+cs M[k, k] = cj*cc-ss else: M[i, i] = cj*ck M[i, j] = sj*sc-cs M[i, k] = sj*cc+ss M[j, i] = cj*sk M[j, j] = sj*ss+cc M[j, k] = sj*cs-sc M[k, i] = -sj M[k, j] = cj*si M[k, k] = cj*ci return M def euler_from_matrix(matrix, axes='sxyz'): """Return Euler angles from rotation matrix for specified axis sequence. axes : One of 24 axis sequences as string or encoded tuple Note that many Euler angle triplets can describe one matrix. >>> R0 = euler_matrix(1, 2, 3, 'syxz') >>> al, be, ga = euler_from_matrix(R0, 'syxz') >>> R1 = euler_matrix(al, be, ga, 'syxz') >>> numpy.allclose(R0, R1) True >>> angles = (4*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R0 = euler_matrix(axes=axes, *angles) ... R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes)) ... if not numpy.allclose(R0, R1): print(axes, "failed") """ try: firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()] except (AttributeError, KeyError): _TUPLE2AXES[axes] # validation firstaxis, parity, repetition, frame = axes i = firstaxis j = _NEXT_AXIS[i+parity] k = _NEXT_AXIS[i-parity+1] M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:3, :3] if repetition: sy = math.sqrt(M[i, j]*M[i, j] + M[i, k]*M[i, k]) if sy > _EPS: ax = math.atan2( M[i, j], M[i, k]) ay = math.atan2( sy, M[i, i]) az = math.atan2( M[j, i], -M[k, i]) else: ax = math.atan2(-M[j, k], M[j, j]) ay = math.atan2( sy, M[i, i]) az = 0.0 else: cy = math.sqrt(M[i, i]*M[i, i] + M[j, i]*M[j, i]) if cy > _EPS: ax = math.atan2( M[k, j], M[k, k]) ay = math.atan2(-M[k, i], cy) az = math.atan2( M[j, i], M[i, i]) else: ax = math.atan2(-M[j, k], M[j, j]) ay = math.atan2(-M[k, i], cy) az = 0.0 if parity: ax, ay, az = -ax, -ay, -az if frame: ax, az = az, ax return ax, ay, az def euler_from_quaternion(quaternion, axes='sxyz'): """Return Euler angles from quaternion for specified axis sequence. >>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0]) >>> numpy.allclose(angles, [0.123, 0, 0]) True """ return euler_from_matrix(quaternion_matrix(quaternion), axes) def quaternion_from_euler(ai, aj, ak, axes='sxyz'): """Return quaternion from Euler angles and axis sequence. ai, aj, ak : Euler's roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple >>> q = quaternion_from_euler(1, 2, 3, 'ryxz') >>> numpy.allclose(q, [0.435953, 0.310622, -0.718287, 0.444435]) True """ try: firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()] except (AttributeError, KeyError): _TUPLE2AXES[axes] # validation firstaxis, parity, repetition, frame = axes i = firstaxis + 1 j = _NEXT_AXIS[i+parity-1] + 1 k = _NEXT_AXIS[i-parity] + 1 if frame: ai, ak = ak, ai if parity: aj = -aj ai /= 2.0 aj /= 2.0 ak /= 2.0 ci = math.cos(ai) si = math.sin(ai) cj = math.cos(aj) sj = math.sin(aj) ck = math.cos(ak) sk = math.sin(ak) cc = ci*ck cs = ci*sk sc = si*ck ss = si*sk q = numpy.empty((4, )) if repetition: q[0] = cj*(cc - ss) q[i] = cj*(cs + sc) q[j] = sj*(cc + ss) q[k] = sj*(cs - sc) else: q[0] = cj*cc + sj*ss q[i] = cj*sc - sj*cs q[j] = cj*ss + sj*cc q[k] = cj*cs - sj*sc if parity: q[j] *= -1.0 return q def quaternion_about_axis(angle, axis): """Return quaternion for rotation about axis. >>> q = quaternion_about_axis(0.123, [1, 0, 0]) >>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0]) True """ q = numpy.array([0.0, axis[0], axis[1], axis[2]]) qlen = vector_norm(q) if qlen > _EPS: q *= math.sin(angle/2.0) / qlen q[0] = math.cos(angle/2.0) return q def quaternion_matrix(quaternion): """Return homogeneous rotation matrix from quaternion. >>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0]) >>> numpy.allclose(M, rotation_matrix(0.123, [1, 0, 0])) True >>> M = quaternion_matrix([1, 0, 0, 0]) >>> numpy.allclose(M, numpy.identity(4)) True >>> M = quaternion_matrix([0, 1, 0, 0]) >>> numpy.allclose(M, numpy.diag([1, -1, -1, 1])) True """ q = numpy.array(quaternion, dtype=numpy.float64, copy=True) n = numpy.dot(q, q) if n < _EPS: return numpy.identity(4) q *= math.sqrt(2.0 / n) q = numpy.outer(q, q) return numpy.array([ [1.0-q[2, 2]-q[3, 3], q[1, 2]-q[3, 0], q[1, 3]+q[2, 0], 0.0], [ q[1, 2]+q[3, 0], 1.0-q[1, 1]-q[3, 3], q[2, 3]-q[1, 0], 0.0], [ q[1, 3]-q[2, 0], q[2, 3]+q[1, 0], 1.0-q[1, 1]-q[2, 2], 0.0], [ 0.0, 0.0, 0.0, 1.0]]) def quaternion_from_matrix(matrix, isprecise=False): """Return quaternion from rotation matrix. If isprecise is True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used. >>> q = quaternion_from_matrix(numpy.identity(4), True) >>> numpy.allclose(q, [1, 0, 0, 0]) True >>> q = quaternion_from_matrix(numpy.diag([1, -1, -1, 1])) >>> numpy.allclose(q, [0, 1, 0, 0]) or numpy.allclose(q, [0, -1, 0, 0]) True >>> R = rotation_matrix(0.123, (1, 2, 3)) >>> q = quaternion_from_matrix(R, True) >>> numpy.allclose(q, [0.9981095, 0.0164262, 0.0328524, 0.0492786]) True >>> R = [[-0.545, 0.797, 0.260, 0], [0.733, 0.603, -0.313, 0], ... [-0.407, 0.021, -0.913, 0], [0, 0, 0, 1]] >>> q = quaternion_from_matrix(R) >>> numpy.allclose(q, [0.19069, 0.43736, 0.87485, -0.083611]) True >>> R = [[0.395, 0.362, 0.843, 0], [-0.626, 0.796, -0.056, 0], ... [-0.677, -0.498, 0.529, 0], [0, 0, 0, 1]] >>> q = quaternion_from_matrix(R) >>> numpy.allclose(q, [0.82336615, -0.13610694, 0.46344705, -0.29792603]) True >>> R = random_rotation_matrix() >>> q = quaternion_from_matrix(R) >>> is_same_transform(R, quaternion_matrix(q)) True >>> R = euler_matrix(0.0, 0.0, numpy.pi/2.0) >>> numpy.allclose(quaternion_from_matrix(R, isprecise=False), ... quaternion_from_matrix(R, isprecise=True)) True """ M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:4, :4] if isprecise: q = numpy.empty((4, )) t = numpy.trace(M) if t > M[3, 3]: q[0] = t q[3] = M[1, 0] - M[0, 1] q[2] = M[0, 2] - M[2, 0] q[1] = M[2, 1] - M[1, 2] else: i, j, k = 1, 2, 3 if M[1, 1] > M[0, 0]: i, j, k = 2, 3, 1 if M[2, 2] > M[i, i]: i, j, k = 3, 1, 2 t = M[i, i] - (M[j, j] + M[k, k]) + M[3, 3] q[i] = t q[j] = M[i, j] + M[j, i] q[k] = M[k, i] + M[i, k] q[3] = M[k, j] - M[j, k] q *= 0.5 / math.sqrt(t * M[3, 3]) else: m00 = M[0, 0] m01 = M[0, 1] m02 = M[0, 2] m10 = M[1, 0] m11 = M[1, 1] m12 = M[1, 2] m20 = M[2, 0] m21 = M[2, 1] m22 = M[2, 2] # symmetric matrix K K = numpy.array([[m00-m11-m22, 0.0, 0.0, 0.0], [m01+m10, m11-m00-m22, 0.0, 0.0], [m02+m20, m12+m21, m22-m00-m11, 0.0], [m21-m12, m02-m20, m10-m01, m00+m11+m22]]) K /= 3.0 # quaternion is eigenvector of K that corresponds to largest eigenvalue w, V = numpy.linalg.eigh(K) q = V[[3, 0, 1, 2], numpy.argmax(w)] if q[0] < 0.0: numpy.negative(q, q) return q def quaternion_multiply(quaternion1, quaternion0): """Return multiplication of two quaternions. quaternion1 x quaternion0 >>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7]) >>> numpy.allclose(q, [28, -44, -14, 48]) True """ w0, x0, y0, z0 = quaternion0 w1, x1, y1, z1 = quaternion1 return numpy.array([-x1*x0 - y1*y0 - z1*z0 + w1*w0, x1*w0 + y1*z0 - z1*y0 + w1*x0, -x1*z0 + y1*w0 + z1*x0 + w1*y0, x1*y0 - y1*x0 + z1*w0 + w1*z0], dtype=numpy.float64) def quaternion_conjugate(quaternion): """Return conjugate of quaternion. >>> q0 = random_quaternion() >>> q1 = quaternion_conjugate(q0) >>> q1[0] == q0[0] and all(q1[1:] == -q0[1:]) True """ q = numpy.array(quaternion, dtype=numpy.float64, copy=True) numpy.negative(q[1:], q[1:]) return q def quaternion_inverse(quaternion): """Return inverse of quaternion. >>> q0 = random_quaternion() >>> q1 = quaternion_inverse(q0) >>> numpy.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0]) True """ q = numpy.array(quaternion, dtype=numpy.float64, copy=True) numpy.negative(q[1:], q[1:]) return q / numpy.dot(q, q) def quaternion_real(quaternion): """Return real part of quaternion. >>> quaternion_real([3, 0, 1, 2]) 3.0 """ return float(quaternion[0]) def quaternion_imag(quaternion): """Return imaginary part of quaternion. >>> quaternion_imag([3, 0, 1, 2]) array([ 0., 1., 2.]) """ return numpy.array(quaternion[1:4], dtype=numpy.float64, copy=True) def quaternion_slerp(quat0, quat1, fraction, spin=0, shortestpath=True): """Return spherical linear interpolation between two quaternions. >>> q0 = random_quaternion() >>> q1 = random_quaternion() >>> q = quaternion_slerp(q0, q1, 0) >>> numpy.allclose(q, q0) True >>> q = quaternion_slerp(q0, q1, 1, 1) >>> numpy.allclose(q, q1) True >>> q = quaternion_slerp(q0, q1, 0.5) >>> angle = math.acos(numpy.dot(q0, q)) >>> numpy.allclose(2, math.acos(numpy.dot(q0, q1)) / angle) or \ numpy.allclose(2, math.acos(-numpy.dot(q0, q1)) / angle) True """ q0 = unit_vector(quat0[:4]) q1 = unit_vector(quat1[:4]) if fraction == 0.0: return q0 elif fraction == 1.0: return q1 d = numpy.dot(q0, q1) if abs(abs(d) - 1.0) < _EPS: return q0 if shortestpath and d < 0.0: # invert rotation d = -d numpy.negative(q1, q1) angle = math.acos(d) + spin * math.pi if abs(angle) < _EPS: return q0 isin = 1.0 / math.sin(angle) q0 *= math.sin((1.0 - fraction) * angle) * isin q1 *= math.sin(fraction * angle) * isin q0 += q1 return q0 def random_quaternion(rand=None): """Return uniform random unit quaternion. rand: array like or None Three independent random variables that are uniformly distributed between 0 and 1. >>> q = random_quaternion() >>> numpy.allclose(1, vector_norm(q)) True >>> q = random_quaternion(numpy.random.random(3)) >>> len(q.shape), q.shape[0]==4 (1, True) """ if rand is None: rand = numpy.random.rand(3) else: assert len(rand) == 3 r1 = numpy.sqrt(1.0 - rand[0]) r2 = numpy.sqrt(rand[0]) pi2 = math.pi * 2.0 t1 = pi2 * rand[1] t2 = pi2 * rand[2] return numpy.array([numpy.cos(t2)*r2, numpy.sin(t1)*r1, numpy.cos(t1)*r1, numpy.sin(t2)*r2]) def random_rotation_matrix(rand=None): """Return uniform random rotation matrix. rand: array like Three independent random variables that are uniformly distributed between 0 and 1 for each returned quaternion. >>> R = random_rotation_matrix() >>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4)) True """ return quaternion_matrix(random_quaternion(rand)) class Arcball(object): """Virtual Trackball Control. >>> ball = Arcball() >>> ball = Arcball(initial=numpy.identity(4)) >>> ball.place([320, 320], 320) >>> ball.down([500, 250]) >>> ball.drag([475, 275]) >>> R = ball.matrix() >>> numpy.allclose(numpy.sum(R), 3.90583455) True >>> ball = Arcball(initial=[1, 0, 0, 0]) >>> ball.place([320, 320], 320) >>> ball.setaxes([1, 1, 0], [-1, 1, 0]) >>> ball.constrain = True >>> ball.down([400, 200]) >>> ball.drag([200, 400]) >>> R = ball.matrix() >>> numpy.allclose(numpy.sum(R), 0.2055924) True >>> ball.next() """ def __init__(self, initial=None): """Initialize virtual trackball control. initial : quaternion or rotation matrix """ self._axis = None self._axes = None self._radius = 1.0 self._center = [0.0, 0.0] self._vdown = numpy.array([0.0, 0.0, 1.0]) self._constrain = False if initial is None: self._qdown = numpy.array([1.0, 0.0, 0.0, 0.0]) else: initial = numpy.array(initial, dtype=numpy.float64) if initial.shape == (4, 4): self._qdown = quaternion_from_matrix(initial) elif initial.shape == (4, ): initial /= vector_norm(initial) self._qdown = initial else: raise ValueError("initial not a quaternion or matrix") self._qnow = self._qpre = self._qdown def place(self, center, radius): """Place Arcball, e.g. when window size changes. center : sequence[2] Window coordinates of trackball center. radius : float Radius of trackball in window coordinates. """ self._radius = float(radius) self._center[0] = center[0] self._center[1] = center[1] def setaxes(self, *axes): """Set axes to constrain rotations.""" if axes is None: self._axes = None else: self._axes = [unit_vector(axis) for axis in axes] @property def constrain(self): """Return state of constrain to axis mode.""" return self._constrain @constrain.setter def constrain(self, value): """Set state of constrain to axis mode.""" self._constrain = bool(value) def down(self, point): """Set initial cursor window coordinates and pick constrain-axis.""" self._vdown = arcball_map_to_sphere(point, self._center, self._radius) self._qdown = self._qpre = self._qnow if self._constrain and self._axes is not None: self._axis = arcball_nearest_axis(self._vdown, self._axes) self._vdown = arcball_constrain_to_axis(self._vdown, self._axis) else: self._axis = None def drag(self, point): """Update current cursor window coordinates.""" vnow = arcball_map_to_sphere(point, self._center, self._radius) if self._axis is not None: vnow = arcball_constrain_to_axis(vnow, self._axis) self._qpre = self._qnow t = numpy.cross(self._vdown, vnow) if numpy.dot(t, t) < _EPS: self._qnow = self._qdown else: q = [numpy.dot(self._vdown, vnow), t[0], t[1], t[2]] self._qnow = quaternion_multiply(q, self._qdown) def next(self, acceleration=0.0): """Continue rotation in direction of last drag.""" q = quaternion_slerp(self._qpre, self._qnow, 2.0+acceleration, False) self._qpre, self._qnow = self._qnow, q def matrix(self): """Return homogeneous rotation matrix.""" return quaternion_matrix(self._qnow) def arcball_map_to_sphere(point, center, radius): """Return unit sphere coordinates from window coordinates.""" v0 = (point[0] - center[0]) / radius v1 = (center[1] - point[1]) / radius n = v0*v0 + v1*v1 if n > 1.0: # position outside of sphere n = math.sqrt(n) return numpy.array([v0/n, v1/n, 0.0]) else: return numpy.array([v0, v1, math.sqrt(1.0 - n)]) def arcball_constrain_to_axis(point, axis): """Return sphere point perpendicular to axis.""" v = numpy.array(point, dtype=numpy.float64, copy=True) a = numpy.array(axis, dtype=numpy.float64, copy=True) v -= a * numpy.dot(a, v) # on plane n = vector_norm(v) if n > _EPS: if v[2] < 0.0: numpy.negative(v, v) v /= n return v if a[2] == 1.0: return numpy.array([1.0, 0.0, 0.0]) return unit_vector([-a[1], a[0], 0.0]) def arcball_nearest_axis(point, axes): """Return axis, which arc is nearest to point.""" point = numpy.array(point, dtype=numpy.float64, copy=False) nearest = None mx = -1.0 for axis in axes: t = numpy.dot(arcball_constrain_to_axis(point, axis), point) if t > mx: nearest = axis mx = t return nearest # epsilon for testing whether a number is close to zero _EPS = numpy.finfo(float).eps * 4.0 # axis sequences for Euler angles _NEXT_AXIS = [1, 2, 0, 1] # map axes strings to/from tuples of inner axis, parity, repetition, frame _AXES2TUPLE = { 'sxyz': (0, 0, 0, 0), 'sxyx': (0, 0, 1, 0), 'sxzy': (0, 1, 0, 0), 'sxzx': (0, 1, 1, 0), 'syzx': (1, 0, 0, 0), 'syzy': (1, 0, 1, 0), 'syxz': (1, 1, 0, 0), 'syxy': (1, 1, 1, 0), 'szxy': (2, 0, 0, 0), 'szxz': (2, 0, 1, 0), 'szyx': (2, 1, 0, 0), 'szyz': (2, 1, 1, 0), 'rzyx': (0, 0, 0, 1), 'rxyx': (0, 0, 1, 1), 'ryzx': (0, 1, 0, 1), 'rxzx': (0, 1, 1, 1), 'rxzy': (1, 0, 0, 1), 'ryzy': (1, 0, 1, 1), 'rzxy': (1, 1, 0, 1), 'ryxy': (1, 1, 1, 1), 'ryxz': (2, 0, 0, 1), 'rzxz': (2, 0, 1, 1), 'rxyz': (2, 1, 0, 1), 'rzyz': (2, 1, 1, 1)} _TUPLE2AXES = dict((v, k) for k, v in _AXES2TUPLE.items()) def vector_norm(data, axis=None, out=None): """Return length, i.e. Euclidean norm, of ndarray along axis. >>> v = numpy.random.random(3) >>> n = vector_norm(v) >>> numpy.allclose(n, numpy.linalg.norm(v)) True >>> v = numpy.random.rand(6, 5, 3) >>> n = vector_norm(v, axis=-1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2))) True >>> n = vector_norm(v, axis=1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> v = numpy.random.rand(5, 4, 3) >>> n = numpy.empty((5, 3)) >>> vector_norm(v, axis=1, out=n) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> vector_norm([]) 0.0 >>> vector_norm([1]) 1.0 """ data = numpy.array(data, dtype=numpy.float64, copy=True) if out is None: if data.ndim == 1: return math.sqrt(numpy.dot(data, data)) data *= data out = numpy.atleast_1d(numpy.sum(data, axis=axis)) numpy.sqrt(out, out) return out else: data *= data numpy.sum(data, axis=axis, out=out) numpy.sqrt(out, out) def unit_vector(data, axis=None, out=None): """Return ndarray normalized by length, i.e. Euclidean norm, along axis. >>> v0 = numpy.random.random(3) >>> v1 = unit_vector(v0) >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) True >>> v0 = numpy.random.rand(5, 4, 3) >>> v1 = unit_vector(v0, axis=-1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2) >>> numpy.allclose(v1, v2) True >>> v1 = unit_vector(v0, axis=1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1) >>> numpy.allclose(v1, v2) True >>> v1 = numpy.empty((5, 4, 3)) >>> unit_vector(v0, axis=1, out=v1) >>> numpy.allclose(v1, v2) True >>> list(unit_vector([])) [] >>> list(unit_vector([1])) [1.0] """ if out is None: data = numpy.array(data, dtype=numpy.float64, copy=True) if data.ndim == 1: data /= math.sqrt(numpy.dot(data, data)) return data else: if out is not data: out[:] = numpy.array(data, copy=False) data = out length = numpy.atleast_1d(numpy.sum(data*data, axis)) numpy.sqrt(length, length) if axis is not None: length = numpy.expand_dims(length, axis) data /= length if out is None: return data def random_vector(size): """Return array of random doubles in the half-open interval [0.0, 1.0). >>> v = random_vector(10000) >>> numpy.all(v >= 0) and numpy.all(v < 1) True >>> v0 = random_vector(10) >>> v1 = random_vector(10) >>> numpy.any(v0 == v1) False """ return numpy.random.random(size) def vector_product(v0, v1, axis=0): """Return vector perpendicular to vectors. >>> v = vector_product([2, 0, 0], [0, 3, 0]) >>> numpy.allclose(v, [0, 0, 6]) True >>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]] >>> v1 = [[3], [0], [0]] >>> v = vector_product(v0, v1) >>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]]) True >>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]] >>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]] >>> v = vector_product(v0, v1, axis=1) >>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]]) True """ return numpy.cross(v0, v1, axis=axis) def angle_between_vectors(v0, v1, directed=True, axis=0): """Return angle between vectors. If directed is False, the input vectors are interpreted as undirected axes, i.e. the maximum angle is pi/2. >>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3]) >>> numpy.allclose(a, math.pi) True >>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False) >>> numpy.allclose(a, 0) True >>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]] >>> v1 = [[3], [0], [0]] >>> a = angle_between_vectors(v0, v1) >>> numpy.allclose(a, [0, 1.5708, 1.5708, 0.95532]) True >>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]] >>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]] >>> a = angle_between_vectors(v0, v1, axis=1) >>> numpy.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532]) True """ v0 = numpy.array(v0, dtype=numpy.float64, copy=False) v1 = numpy.array(v1, dtype=numpy.float64, copy=False) dot = numpy.sum(v0 * v1, axis=axis) dot /= vector_norm(v0, axis=axis) * vector_norm(v1, axis=axis) return numpy.arccos(dot if directed else numpy.fabs(dot)) def inverse_matrix(matrix): """Return inverse of square transformation matrix. >>> M0 = random_rotation_matrix() >>> M1 = inverse_matrix(M0.T) >>> numpy.allclose(M1, numpy.linalg.inv(M0.T)) True >>> for size in range(1, 7): ... M0 = numpy.random.rand(size, size) ... M1 = inverse_matrix(M0) ... if not numpy.allclose(M1, numpy.linalg.inv(M0)): print(size) """ return numpy.linalg.inv(matrix) def concatenate_matrices(*matrices): """Return concatenation of series of transformation matrices. >>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5 >>> numpy.allclose(M, concatenate_matrices(M)) True >>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T)) True """ M = numpy.identity(4) for i in matrices: M = numpy.dot(M, i) return M def is_same_transform(matrix0, matrix1): """Return True if two matrices perform same transformation. >>> is_same_transform(numpy.identity(4), numpy.identity(4)) True >>> is_same_transform(numpy.identity(4), random_rotation_matrix()) False """ matrix0 = numpy.array(matrix0, dtype=numpy.float64, copy=True) matrix0 /= matrix0[3, 3] matrix1 = numpy.array(matrix1, dtype=numpy.float64, copy=True) matrix1 /= matrix1[3, 3] return numpy.allclose(matrix0, matrix1) def _import_module(name, package=None, warn=True, prefix='_py_', ignore='_'): """Try import all public attributes from module into global namespace. Existing attributes with name clashes are renamed with prefix. Attributes starting with underscore are ignored by default. Return True on successful import. """ import warnings from importlib import import_module try: if not package: module = import_module(name) else: module = import_module('.' + name, package=package) except ImportError: if warn: warnings.warn("failed to import module %s" % name) else: for attr in dir(module): if ignore and attr.startswith(ignore): continue if prefix: if attr in globals(): globals()[prefix + attr] = globals()[attr] elif warn: warnings.warn("no Python implementation of " + attr) globals()[attr] = getattr(module, attr) return True _import_module('_transformations') if __name__ == "__main__": import doctest import random # used in doctests numpy.set_printoptions(suppress=True, precision=5) doctest.testmod()