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import numpy as np 

from scipy.linalg import solve_banded 

from .rotation import Rotation 

 

 

def _create_skew_matrix(x): 

"""Create skew-symmetric matrices corresponding to vectors. 

 

Parameters 

---------- 

x : ndarray, shape (n, 3) 

Set of vectors. 

 

Returns 

------- 

ndarray, shape (n, 3, 3) 

""" 

result = np.zeros((len(x), 3, 3)) 

result[:, 0, 1] = -x[:, 2] 

result[:, 0, 2] = x[:, 1] 

result[:, 1, 0] = x[:, 2] 

result[:, 1, 2] = -x[:, 0] 

result[:, 2, 0] = -x[:, 1] 

result[:, 2, 1] = x[:, 0] 

return result 

 

 

def _matrix_vector_product_of_stacks(A, b): 

"""Compute the product of stack of matrices and vectors.""" 

return np.einsum("ijk,ik->ij", A, b) 

 

 

def _angular_rate_to_rotvec_dot_matrix(rotvecs): 

"""Compute matrices to transform angular rates to rot. vector derivatives. 

 

The matrices depend on the current attitude represented as a rotation 

vector. 

 

Parameters 

---------- 

rotvecs : ndarray, shape (n, 3) 

Set of rotation vectors. 

 

Returns 

------- 

ndarray, shape (n, 3, 3) 

""" 

norm = np.linalg.norm(rotvecs, axis=1) 

k = np.empty_like(norm) 

 

mask = norm > 1e-4 

nm = norm[mask] 

k[mask] = (1 - 0.5 * nm / np.tan(0.5 * nm)) / nm**2 

mask = ~mask 

nm = norm[mask] 

k[mask] = 1/12 + 1/720 * nm**2 

 

skew = _create_skew_matrix(rotvecs) 

 

result = np.empty((len(rotvecs), 3, 3)) 

result[:] = np.identity(3) 

result[:] += 0.5 * skew 

result[:] += k[:, None, None] * np.matmul(skew, skew) 

 

return result 

 

 

def _rotvec_dot_to_angular_rate_matrix(rotvecs): 

"""Compute matrices to transform rot. vector derivatives to angular rates. 

 

The matrices depend on the current attitude represented as a rotation 

vector. 

 

Parameters 

---------- 

rotvecs : ndarray, shape (n, 3) 

Set of rotation vectors. 

 

Returns 

------- 

ndarray, shape (n, 3, 3) 

""" 

norm = np.linalg.norm(rotvecs, axis=1) 

k1 = np.empty_like(norm) 

k2 = np.empty_like(norm) 

 

mask = norm > 1e-4 

nm = norm[mask] 

k1[mask] = (1 - np.cos(nm)) / nm ** 2 

k2[mask] = (nm - np.sin(nm)) / nm ** 3 

 

mask = ~mask 

nm = norm[mask] 

k1[mask] = 0.5 - nm ** 2 / 24 

k2[mask] = 1 / 6 - nm ** 2 / 120 

 

skew = _create_skew_matrix(rotvecs) 

 

result = np.empty((len(rotvecs), 3, 3)) 

result[:] = np.identity(3) 

result[:] -= k1[:, None, None] * skew 

result[:] += k2[:, None, None] * np.matmul(skew, skew) 

 

return result 

 

 

def _angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot): 

"""Compute the non-linear term in angular acceleration. 

 

The angular acceleration contains a quadratic term with respect to 

the derivative of the rotation vector. This function computes that. 

 

Parameters 

---------- 

rotvecs : ndarray, shape (n, 3) 

Set of rotation vectors. 

rotvecs_dot: ndarray, shape (n, 3) 

Set of rotation vector derivatives. 

 

Returns 

------- 

ndarray, shape (n, 3) 

""" 

norm = np.linalg.norm(rotvecs, axis=1) 

dp = np.sum(rotvecs * rotvecs_dot, axis=1) 

cp = np.cross(rotvecs, rotvecs_dot) 

ccp = np.cross(rotvecs, cp) 

dccp = np.cross(rotvecs_dot, cp) 

 

k1 = np.empty_like(norm) 

k2 = np.empty_like(norm) 

k3 = np.empty_like(norm) 

 

mask = norm > 1e-4 

nm = norm[mask] 

k1[mask] = (-nm * np.sin(nm) - 2 * (np.cos(nm) - 1)) / nm ** 4 

k2[mask] = (-2 * nm + 3 * np.sin(nm) - nm * np.cos(nm)) / nm ** 5 

k3[mask] = (nm - np.sin(nm)) / nm ** 3 

 

mask = ~mask 

nm = norm[mask] 

k1[mask] = 1/12 - nm ** 2 / 180 

k2[mask] = -1/60 + nm ** 2 / 12604 

k3[mask] = 1/6 - nm ** 2 / 120 

 

dp = dp[:, None] 

k1 = k1[:, None] 

k2 = k2[:, None] 

k3 = k3[:, None] 

 

return dp * (k1 * cp + k2 * ccp) + k3 * dccp 

 

 

def _compute_angular_rate(rotvecs, rotvecs_dot): 

"""Compute angular rates given rotation vectors and its derivatives. 

 

Parameters 

---------- 

rotvecs : ndarray, shape (n, 3) 

Set of rotation vectors. 

rotvecs_dot : ndarray, shape (n, 3) 

Set of rotation vector derivatives. 

 

Returns 

------- 

ndarray, shape (n, 3) 

""" 

return _matrix_vector_product_of_stacks( 

_rotvec_dot_to_angular_rate_matrix(rotvecs), rotvecs_dot) 

 

 

def _compute_angular_acceleration(rotvecs, rotvecs_dot, rotvecs_dot_dot): 

"""Compute angular acceleration given rotation vector and its derivatives. 

 

Parameters 

---------- 

rotvecs : ndarray, shape (n, 3) 

Set of rotation vectors. 

rotvecs_dot : ndarray, shape (n, 3) 

Set of rotation vector derivatives. 

rotvecs_dot_dot : ndarray, shape (n, 3) 

Set of rotation vector second derivatives. 

 

Returns 

------- 

ndarray, shape (n, 3) 

""" 

return (_compute_angular_rate(rotvecs, rotvecs_dot_dot) + 

_angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot)) 

 

 

def _create_block_3_diagonal_matrix(A, B, d): 

"""Create a 3-diagonal block matrix as banded. 

 

The matrix has the following structure: 

 

DB... 

ADB.. 

.ADB. 

..ADB 

...AD 

 

The blocks A, B and D are 3-by-3 matrices. The D matrices has the form 

d * I. 

 

Parameters 

---------- 

A : ndarray, shape (n, 3, 3) 

Stack of A blocks. 

B : ndarray, shape (n, 3, 3) 

Stack of B blocks. 

d : ndarray, shape (n + 1,) 

Values for diagonal blocks. 

 

Returns 

------- 

ndarray, shape (11, 3 * (n + 1)) 

Matrix in the banded form as used by `scipy.linalg.solve_banded`. 

""" 

ind = np.arange(3) 

ind_blocks = np.arange(len(A)) 

 

A_i = np.empty_like(A, dtype=int) 

A_i[:] = ind[:, None] 

A_i += 3 * (1 + ind_blocks[:, None, None]) 

 

A_j = np.empty_like(A, dtype=int) 

A_j[:] = ind 

A_j += 3 * ind_blocks[:, None, None] 

 

B_i = np.empty_like(B, dtype=int) 

B_i[:] = ind[:, None] 

B_i += 3 * ind_blocks[:, None, None] 

 

B_j = np.empty_like(B, dtype=int) 

B_j[:] = ind 

B_j += 3 * (1 + ind_blocks[:, None, None]) 

 

diag_i = diag_j = np.arange(3 * len(d)) 

i = np.hstack((A_i.ravel(), B_i.ravel(), diag_i)) 

j = np.hstack((A_j.ravel(), B_j.ravel(), diag_j)) 

values = np.hstack((A.ravel(), B.ravel(), np.repeat(d, 3))) 

 

u = 5 

l = 5 

result = np.zeros((u + l + 1, 3 * len(d))) 

result[u + i - j, j] = values 

return result 

 

 

class RotationSpline(object): 

"""Interpolate rotations with continuous angular rate and acceleration. 

 

The rotation vectors between each consecutive orientation are cubic 

functions of time and it is guaranteed that angular rate and acceleration 

are continuous. Such interpolation are analogous to cubic spline 

interpolation. 

 

Refer to [1]_ for math and implementation details. 

 

Parameters 

---------- 

times : array_like, shape (N,) 

Times of the known rotations. At least 2 times must be specified. 

rotations : `Rotation` instance 

Rotations to perform the interpolation between. Must contain N 

rotations. 

 

Methods 

------- 

__call__ 

 

References 

---------- 

.. [1] `Smooth Attitude Interpolation 

<https://github.com/scipy/scipy/files/2932755/attitude_interpolation.pdf>`_ 

 

Examples 

-------- 

>>> from scipy.spatial.transform import Rotation, RotationSpline 

 

Define the sequence of times and rotations from the Euler angles: 

 

>>> times = [0, 10, 20, 40] 

>>> angles = [[-10, 20, 30], [0, 15, 40], [-30, 45, 30], [20, 45, 90]] 

>>> rotations = Rotation.from_euler('XYZ', angles, degrees=True) 

 

Create the interpolator object: 

 

>>> spline = RotationSpline(times, rotations) 

 

Interpolate the Euler angles, angular rate and acceleration: 

 

>>> angular_rate = np.rad2deg(spline(times, 1)) 

>>> angular_acceleration = np.rad2deg(spline(times, 2)) 

>>> times_plot = np.linspace(times[0], times[-1], 100) 

>>> angles_plot = spline(times_plot).as_euler('XYZ', degrees=True) 

>>> angular_rate_plot = np.rad2deg(spline(times_plot, 1)) 

>>> angular_acceleration_plot = np.rad2deg(spline(times_plot, 2)) 

 

On this plot you see that Euler angles are continuous and smooth: 

 

>>> import matplotlib.pyplot as plt 

>>> plt.plot(times_plot, angles_plot) 

>>> plt.plot(times, angles, 'x') 

>>> plt.title("Euler angles") 

>>> plt.show() 

 

The angular rate is also smooth: 

 

>>> plt.plot(times_plot, angular_rate_plot) 

>>> plt.plot(times, angular_rate, 'x') 

>>> plt.title("Angular rate") 

>>> plt.show() 

 

The angular acceleration is continuous, but not smooth. Also note that 

the angular acceleration is not a piecewise-linear function, because 

it is different from the second derivative of the rotation vector (which 

is a piecewise-linear function as in the cubic spline). 

 

>>> plt.plot(times_plot, angular_acceleration_plot) 

>>> plt.plot(times, angular_acceleration, 'x') 

>>> plt.title("Angular acceleration") 

>>> plt.show() 

""" 

# Parameters for the solver for angular rate. 

MAX_ITER = 10 

TOL = 1e-9 

 

def _solve_for_angular_rates(self, dt, angular_rates, rotvecs): 

angular_rate_first = angular_rates[0].copy() 

 

A = _angular_rate_to_rotvec_dot_matrix(rotvecs) 

A_inv = _rotvec_dot_to_angular_rate_matrix(rotvecs) 

M = _create_block_3_diagonal_matrix( 

2 * A_inv[1:-1] / dt[1:-1, None, None], 

2 * A[1:-1] / dt[1:-1, None, None], 

4 * (1 / dt[:-1] + 1 / dt[1:])) 

 

b0 = 6 * (rotvecs[:-1] * dt[:-1, None] ** -2 + 

rotvecs[1:] * dt[1:, None] ** -2) 

b0[0] -= 2 / dt[0] * A_inv[0].dot(angular_rate_first) 

b0[-1] -= 2 / dt[-1] * A[-1].dot(angular_rates[-1]) 

 

for iteration in range(self.MAX_ITER): 

rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates) 

delta_beta = _angular_acceleration_nonlinear_term( 

rotvecs[:-1], rotvecs_dot[:-1]) 

b = b0 - delta_beta 

angular_rates_new = solve_banded((5, 5), M, b.ravel()) 

angular_rates_new = angular_rates_new.reshape((-1, 3)) 

 

delta = np.abs(angular_rates_new - angular_rates[:-1]) 

angular_rates[:-1] = angular_rates_new 

if np.all(delta < self.TOL * (1 + np.abs(angular_rates_new))): 

break 

 

rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates) 

angular_rates = np.vstack((angular_rate_first, angular_rates[:-1])) 

 

return angular_rates, rotvecs_dot 

 

def __init__(self, times, rotations): 

from scipy.interpolate import PPoly 

 

if len(rotations) == 1: 

raise ValueError("`rotations` must contain at least 2 rotations.") 

 

times = np.asarray(times, dtype=float) 

if times.ndim != 1: 

raise ValueError("`times` must be 1-dimensional.") 

 

if len(times) != len(rotations): 

raise ValueError("Expected number of rotations to be equal to " 

"number of timestamps given, got {} rotations " 

"and {} timestamps." 

.format(len(rotations), len(times))) 

 

dt = np.diff(times) 

if np.any(dt <= 0): 

raise ValueError("Values in `times` must be in a strictly " 

"increasing order.") 

 

rotvecs = (rotations[:-1].inv() * rotations[1:]).as_rotvec() 

angular_rates = rotvecs / dt[:, None] 

 

if len(rotations) == 2: 

rotvecs_dot = angular_rates 

else: 

angular_rates, rotvecs_dot = self._solve_for_angular_rates( 

dt, angular_rates, rotvecs) 

 

dt = dt[:, None] 

coeff = np.empty((4, len(times) - 1, 3)) 

coeff[0] = (-2 * rotvecs + dt * angular_rates 

+ dt * rotvecs_dot) / dt ** 3 

coeff[1] = (3 * rotvecs - 2 * dt * angular_rates 

- dt * rotvecs_dot) / dt ** 2 

coeff[2] = angular_rates 

coeff[3] = 0 

 

self.times = times 

self.rotations = rotations 

self.interpolator = PPoly(coeff, times) 

 

def __call__(self, times, order=0): 

"""Compute interpolated values. 

 

Parameters 

---------- 

times : float or array_like 

Times of interest. 

order : {0, 1, 2}, optional 

Order of differentiation: 

 

* 0 (default) : return Rotation 

* 1 : return the angular rate in rad/sec 

* 2 : return the angular acceleration in rad/sec/sec 

 

Returns 

------- 

Interpolated Rotation, angular rate or acceleration. 

""" 

if order not in [0, 1, 2]: 

raise ValueError("`order` must be 0, 1 or 2.") 

 

times = np.asarray(times, dtype=float) 

if times.ndim > 1: 

raise ValueError("`times` must be at most 1-dimensional.") 

 

singe_time = times.ndim == 0 

times = np.atleast_1d(times) 

 

rotvecs = self.interpolator(times) 

if order == 0: 

index = np.searchsorted(self.times, times, side='right') 

index -= 1 

index[index < 0] = 0 

n_segments = len(self.times) - 1 

index[index > n_segments - 1] = n_segments - 1 

result = self.rotations[index] * Rotation.from_rotvec(rotvecs) 

elif order == 1: 

rotvecs_dot = self.interpolator(times, 1) 

result = _compute_angular_rate(rotvecs, rotvecs_dot) 

elif order == 2: 

rotvecs_dot = self.interpolator(times, 1) 

rotvecs_dot_dot = self.interpolator(times, 2) 

result = _compute_angular_acceleration(rotvecs, rotvecs_dot, 

rotvecs_dot_dot) 

else: 

assert False 

 

if singe_time: 

result = result[0] 

 

return result