上海精品网站建设,网站说建设中,网站推广一站式服务,北京高端网站设计公司前几日研究scipy的旋转#xff0c;不知道具体里面怎么实现的#xff0c;因此搜索一番。
发现Rotation在scipy的表达是用四元数的 https://github.com/jgagneastro/coffeegrindsize/edit/master/App/dist/coffeegrindsize.app/Contents/Resources/lib/python3.7/scipy/spatia…前几日研究scipy的旋转不知道具体里面怎么实现的因此搜索一番。
发现Rotation在scipy的表达是用四元数的 https://github.com/jgagneastro/coffeegrindsize/edit/master/App/dist/coffeegrindsize.app/Contents/Resources/lib/python3.7/scipy/spatial/transform/rotation.py from __future__ import division, print_function, absolute_import
import re import warnings import numpy as np import scipy.linalg from scipy._lib._util import check_random_state _AXIS_TO_IND {x: 0, y: 1, z: 2} def _elementary_basis_vector(axis): b np.zeros(3) b[_AXIS_TO_IND[axis]] 1 return b def _compute_euler_from_dcm(dcm, seq, extrinsicFalse): # The algorithm assumes intrinsic frame transformations. For representation # the paper uses transformation matrices, which are transpose of the # direction cosine matrices used by our Rotation class. # Adapt the algorithm for our case by # 1. Instead of transposing our representation, use the transpose of the # O matrix as defined in the paper, and be careful to swap indices # 2. Reversing both axis sequence and angles for extrinsic rotations if extrinsic: seq seq[::-1] if dcm.ndim 2: dcm dcm[None, :, :] num_rotations dcm.shape[0] # Step 0 # Algorithm assumes axes as column vectors, here we use 1D vectors n1 _elementary_basis_vector(seq[0]) n2 _elementary_basis_vector(seq[1]) n3 _elementary_basis_vector(seq[2]) # Step 2 sl np.dot(np.cross(n1, n2), n3) cl np.dot(n1, n3) # angle offset is lambda from the paper referenced in [2] from docstring of # as_euler function offset np.arctan2(sl, cl) c np.vstack((n2, np.cross(n1, n2), n1)) # Step 3 rot np.array([ [1, 0, 0], [0, cl, sl], [0, -sl, cl], ]) res np.einsum(...ij,...jk-...ik, c, dcm) dcm_transformed np.einsum(...ij,...jk-...ik, res, c.T.dot(rot)) # Step 4 angles np.empty((num_rotations, 3)) # Ensure less than unit norm positive_unity dcm_transformed[:, 2, 2] 1 negative_unity dcm_transformed[:, 2, 2] -1 dcm_transformed[positive_unity, 2, 2] 1 dcm_transformed[negative_unity, 2, 2] -1 angles[:, 1] np.arccos(dcm_transformed[:, 2, 2]) # Steps 5, 6 eps 1e-7 safe1 (np.abs(angles[:, 1]) eps) safe2 (np.abs(angles[:, 1] - np.pi) eps) # Step 4 (Completion) angles[:, 1] offset # 5b safe_mask np.logical_and(safe1, safe2) angles[safe_mask, 0] np.arctan2(dcm_transformed[safe_mask, 0, 2], -dcm_transformed[safe_mask, 1, 2]) angles[safe_mask, 2] np.arctan2(dcm_transformed[safe_mask, 2, 0], dcm_transformed[safe_mask, 2, 1]) if extrinsic: # For extrinsic, set first angle to zero so that after reversal we # ensure that third angle is zero # 6a angles[~safe_mask, 0] 0 # 6b angles[~safe1, 2] np.arctan2( dcm_transformed[~safe1, 1, 0] - dcm_transformed[~safe1, 0, 1], dcm_transformed[~safe1, 0, 0] dcm_transformed[~safe1, 1, 1] ) # 6c angles[~safe2, 2] -np.arctan2( dcm_transformed[~safe2, 1, 0] dcm_transformed[~safe2, 0, 1], dcm_transformed[~safe2, 0, 0] - dcm_transformed[~safe2, 1, 1] ) else: # For instrinsic, set third angle to zero # 6a angles[~safe_mask, 2] 0 # 6b angles[~safe1, 0] np.arctan2( dcm_transformed[~safe1, 1, 0] - dcm_transformed[~safe1, 0, 1], dcm_transformed[~safe1, 0, 0] dcm_transformed[~safe1, 1, 1] ) # 6c angles[~safe2, 0] np.arctan2( dcm_transformed[~safe2, 1, 0] dcm_transformed[~safe2, 0, 1], dcm_transformed[~safe2, 0, 0] - dcm_transformed[~safe2, 1, 1] ) # Step 7 if seq[0] seq[2]: # lambda 0, so we can only ensure angle2 - [0, pi] adjust_mask np.logical_or(angles[:, 1] 0, angles[:, 1] np.pi) else: # lambda or - pi/2, so we can ensure angle2 - [-pi/2, pi/2] adjust_mask np.logical_or(angles[:, 1] -np.pi / 2, angles[:, 1] np.pi / 2) # Dont adjust gimbal locked angle sequences adjust_mask np.logical_and(adjust_mask, safe_mask) angles[adjust_mask, 0] np.pi angles[adjust_mask, 1] 2 * offset - angles[adjust_mask, 1] angles[adjust_mask, 2] - np.pi angles[angles -np.pi] 2 * np.pi angles[angles np.pi] - 2 * np.pi # Step 8 if not np.all(safe_mask): warnings.warn(Gimbal lock detected. Setting third angle to zero since it is not possible to uniquely determine all angles.) # Reverse role of extrinsic and intrinsic rotations, but let third angle be # zero for gimbal locked cases if extrinsic: angles angles[:, ::-1] return angles def _make_elementary_quat(axis, angles): quat np.zeros((angles.shape[0], 4)) quat[:, 3] np.cos(angles / 2) quat[:, _AXIS_TO_IND[axis]] np.sin(angles / 2) return quat def _compose_quat(p, q): product np.empty((max(p.shape[0], q.shape[0]), 4)) product[:, 3] p[:, 3] * q[:, 3] - np.sum(p[:, :3] * q[:, :3], axis1) product[:, :3] (p[:, None, 3] * q[:, :3] q[:, None, 3] * p[:, :3] np.cross(p[:, :3], q[:, :3])) return product def _elementary_quat_compose(seq, angles, intrinsicFalse): result _make_elementary_quat(seq[0], angles[:, 0]) for idx, axis in enumerate(seq[1:], start1): if intrinsic: result _compose_quat( result, _make_elementary_quat(axis, angles[:, idx])) else: result _compose_quat( _make_elementary_quat(axis, angles[:, idx]), result) return result class Rotation(object): Rotation in 3 dimensions. This class provides an interface to initialize from and represent rotations with: - Quaternions - Direction Cosine Matrices - Rotation Vectors - Euler angles The following operations on rotations are supported: - Application on vectors - Rotation Composition - Rotation Inversion - Rotation Indexing Indexing within a rotation is supported since multiple rotation transforms can be stored within a single Rotation instance. To create Rotation objects use from_... methods (see examples below). Rotation(...) is not supposed to be instantiated directly. Methods ------- __len__ from_quat from_dcm from_rotvec from_euler as_quat as_dcm as_rotvec as_euler apply __mul__ inv __getitem__ random match_vectors See Also -------- Slerp Notes ----- .. versionadded: 1.2.0 Examples -------- from scipy.spatial.transform import Rotation as R A Rotation instance can be initialized in any of the above formats and converted to any of the others. The underlying object is independent of the representation used for initialization. Consider a counter-clockwise rotation of 90 degrees about the z-axis. This corresponds to the following quaternion (in scalar-last format): r R.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)]) The rotation can be expressed in any of the other formats: r.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) r.as_rotvec() array([0. , 0. , 1.57079633]) r.as_euler(zyx, degreesTrue) array([90., 0., 0.]) The same rotation can be initialized using a direction cosine matrix: r R.from_dcm(np.array([ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]])) Representation in other formats: r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) r.as_rotvec() array([0. , 0. , 1.57079633]) r.as_euler(zyx, degreesTrue) array([90., 0., 0.]) The rotation vector corresponding to this rotation is given by: r R.from_rotvec(np.pi/2 * np.array([0, 0, 1])) Representation in other formats: r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) r.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) r.as_euler(zyx, degreesTrue) array([90., 0., 0.]) The from_euler method is quite flexible in the range of input formats it supports. Here we initialize a single rotation about a single axis: r R.from_euler(z, 90, degreesTrue) Again, the object is representation independent and can be converted to any other format: r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) r.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) r.as_rotvec() array([0. , 0. , 1.57079633]) It is also possible to initialize multiple rotations in a single instance using any of the from_... functions. Here we initialize a stack of 3 rotations using the from_euler method: r R.from_euler(zyx, [ ... [90, 0, 0], ... [0, 45, 0], ... [45, 60, 30]], degreesTrue) The other representations also now return a stack of 3 rotations. For example: r.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) Applying the above rotations onto a vector: v [1, 2, 3] r.apply(v) array([[-2. , 1. , 3. ], [ 2.82842712, 2. , 1.41421356], [ 2.24452282, 0.78093109, 2.89002836]]) A Rotation instance can be indexed and sliced as if it were a single 1D array or list: r.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) p r[0] p.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) q r[1:3] q.as_quat() array([[0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) Multiple rotations can be composed using the * operator: r1 R.from_euler(z, 90, degreesTrue) r2 R.from_rotvec([np.pi/4, 0, 0]) v [1, 2, 3] r2.apply(r1.apply(v)) array([-2. , -1.41421356, 2.82842712]) r3 r2 * r1 # Note the order r3.apply(v) array([-2. , -1.41421356, 2.82842712]) Finally, it is also possible to invert rotations: r1 R.from_euler(z, [90, 45], degreesTrue) r2 r1.inv() r2.as_euler(zyx, degreesTrue) array([[-90., 0., 0.], [-45., 0., 0.]]) These examples serve as an overview into the Rotation class and highlight major functionalities. For more thorough examples of the range of input and output formats supported, consult the individual methods examples. def __init__(self, quat, normalizedFalse, copyTrue): self._single False quat np.asarray(quat, dtypefloat) if quat.ndim not in [1, 2] or quat.shape[-1] ! 4: raise ValueError(Expected quat to have shape (4,) or (N x 4), got {}..format(quat.shape)) # If a single quaternion is given, convert it to a 2D 1 x 4 matrix but # set self._single to True so that we can return appropriate objects # in the to_... methods if quat.shape (4,): quat quat[None, :] self._single True if normalized: self._quat quat.copy() if copy else quat else: self._quat quat.copy() norms scipy.linalg.norm(quat, axis1) zero_norms norms 0 if zero_norms.any(): raise ValueError(Found zero norm quaternions in quat.) # Ensure norm is broadcasted along each column. self._quat[~zero_norms] / norms[~zero_norms][:, None] def __len__(self): Number of rotations contained in this object. Multiple rotations can be stored in a single instance. Returns ------- length : int Number of rotations stored in object. return self._quat.shape[0] classmethod def from_quat(cls, quat, normalizedFalse): Initialize from quaternions. 3D rotations can be represented using unit-norm quaternions [1]_. Parameters ---------- quat : array_like, shape (N, 4) or (4,) Each row is a (possibly non-unit norm) quaternion in scalar-last (x, y, z, w) format. normalized : bool, optional If False, input quaternions are normalized to unit norm before being stored. If True, quaternions are assumed to already have unit norm and are stored as given. Default is False. Returns ------- rotation : Rotation instance Object containing the rotations represented by input quaternions. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation Examples -------- from scipy.spatial.transform import Rotation as R Initialize a single rotation: r R.from_quat([1, 0, 0, 0]) r.as_quat() array([1., 0., 0., 0.]) r.as_quat().shape (4,) Initialize multiple rotations in a single object: r R.from_quat([ ... [1, 0, 0, 0], ... [0, 0, 0, 1] ... ]) r.as_quat() array([[1., 0., 0., 0.], [0., 0., 0., 1.]]) r.as_quat().shape (2, 4) It is also possible to have a stack of a single rotation: r R.from_quat([[0, 0, 0, 1]]) r.as_quat() array([[0., 0., 0., 1.]]) r.as_quat().shape (1, 4) By default, quaternions are normalized before initialization. r R.from_quat([0, 0, 1, 1]) r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) If unit norms are ensured, skip the normalization step. r R.from_quat([0, 0, 1, 0], normalizedTrue) r.as_quat() array([0., 0., 1., 0.]) return cls(quat, normalized) classmethod def from_dcm(cls, dcm): Initialize from direction cosine matrices. Rotations in 3 dimensions can be represented using 3 x 3 proper orthogonal matrices [1]_. If the input is not proper orthogonal, an approximation is created using the method described in [2]_. Parameters ---------- dcm : array_like, shape (N, 3, 3) or (3, 3) A single matrix or a stack of matrices, where dcm[i] is the i-th matrix. Returns ------- rotation : Rotation instance Object containing the rotations represented by the input direction cosine matrices. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions .. [2] F. Landis Markley, Unit Quaternion from Rotation Matrix, Journal of guidance, control, and dynamics vol. 31.2, pp. 440-442, 2008. Examples -------- from scipy.spatial.transform import Rotation as R Initialize a single rotation: r R.from_dcm([ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]) r.as_dcm().shape (3, 3) Initialize multiple rotations in a single object: r R.from_dcm([ ... [ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1], ... ], ... [ ... [1, 0, 0], ... [0, 0, -1], ... [0, 1, 0], ... ]]) r.as_dcm().shape (2, 3, 3) If input matrices are not special orthogonal (orthogonal with determinant equal to 1), then a special orthogonal estimate is stored: a np.array([ ... [0, -0.5, 0], ... [0.5, 0, 0], ... [0, 0, 0.5]]) np.linalg.det(a) 0.12500000000000003 r R.from_dcm(a) dcm r.as_dcm() dcm array([[-0.38461538, -0.92307692, 0. ], [ 0.92307692, -0.38461538, 0. ], [ 0. , 0. , 1. ]]) np.linalg.det(dcm) 1.0000000000000002 It is also possible to have a stack containing a single rotation: r R.from_dcm([[ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]]) r.as_dcm() array([[[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]]) r.as_dcm().shape (1, 3, 3) is_single False dcm np.asarray(dcm, dtypefloat) if dcm.ndim not in [2, 3] or dcm.shape[-2:] ! (3, 3): raise ValueError(Expected dcm to have shape (3, 3) or (N, 3, 3), got {}.format(dcm.shape)) # If a single dcm is given, convert it to 3D 1 x 3 x 3 matrix but set # self._single to True so that we can return appropriate objects in # the to_... methods if dcm.shape (3, 3): dcm dcm.reshape((1, 3, 3)) is_single True num_rotations dcm.shape[0] decision_matrix np.empty((num_rotations, 4)) decision_matrix[:, :3] dcm.diagonal(axis11, axis22) decision_matrix[:, -1] decision_matrix[:, :3].sum(axis1) choices decision_matrix.argmax(axis1) quat np.empty((num_rotations, 4)) ind np.nonzero(choices ! 3)[0] i choices[ind] j (i 1) % 3 k (j 1) % 3 quat[ind, i] 1 - decision_matrix[ind, -1] 2 * dcm[ind, i, i] quat[ind, j] dcm[ind, j, i] dcm[ind, i, j] quat[ind, k] dcm[ind, k, i] dcm[ind, i, k] quat[ind, 3] dcm[ind, k, j] - dcm[ind, j, k] ind np.nonzero(choices 3)[0] quat[ind, 0] dcm[ind, 2, 1] - dcm[ind, 1, 2] quat[ind, 1] dcm[ind, 0, 2] - dcm[ind, 2, 0] quat[ind, 2] dcm[ind, 1, 0] - dcm[ind, 0, 1] quat[ind, 3] 1 decision_matrix[ind, -1] quat / np.linalg.norm(quat, axis1)[:, None] if is_single: return cls(quat[0], normalizedTrue, copyFalse) else: return cls(quat, normalizedTrue, copyFalse) classmethod def from_rotvec(cls, rotvec): Initialize from rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation (in radians) [1]_. Parameters ---------- rotvec : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors, where rot_vec[i] gives the ith rotation vector. Returns ------- rotation : Rotation instance Object containing the rotations represented by input rotation vectors. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- from scipy.spatial.transform import Rotation as R Initialize a single rotation: r R.from_rotvec(np.pi/2 * np.array([0, 0, 1])) r.as_rotvec() array([0. , 0. , 1.57079633]) r.as_rotvec().shape (3,) Initialize multiple rotations in one object: r R.from_rotvec([ ... [0, 0, np.pi/2], ... [np.pi/2, 0, 0]]) r.as_rotvec() array([[0. , 0. , 1.57079633], [1.57079633, 0. , 0. ]]) r.as_rotvec().shape (2, 3) It is also possible to have a stack of a single rotaton: r R.from_rotvec([[0, 0, np.pi/2]]) r.as_rotvec().shape (1, 3) is_single False rotvec np.asarray(rotvec, dtypefloat) if rotvec.ndim not in [1, 2] or rotvec.shape[-1] ! 3: raise ValueError(Expected rot_vec to have shape (3,) or (N, 3), got {}.format(rotvec.shape)) # If a single vector is given, convert it to a 2D 1 x 3 matrix but # set self._single to True so that we can return appropriate objects # in the as_... methods if rotvec.shape (3,): rotvec rotvec[None, :] is_single True num_rotations rotvec.shape[0] norms np.linalg.norm(rotvec, axis1) small_angle (norms 1e-3) large_angle ~small_angle scale np.empty(num_rotations) scale[small_angle] (0.5 - norms[small_angle] ** 2 / 48 norms[small_angle] ** 4 / 3840) scale[large_angle] (np.sin(norms[large_angle] / 2) / norms[large_angle]) quat np.empty((num_rotations, 4)) quat[:, :3] scale[:, None] * rotvec quat[:, 3] np.cos(norms / 2) if is_single: return cls(quat[0], normalizedTrue, copyFalse) else: return cls(quat, normalizedTrue, copyFalse) classmethod def from_euler(cls, seq, angles, degreesFalse): Initialize from Euler angles. Rotations in 3 dimensions can be represented by a sequece of 3 rotations around a sequence of axes. In theory, any three axes spanning the 3D Euclidean space are enough. In practice the axes of rotation are chosen to be the basis vectors. The three rotations can either be in a global frame of reference (extrinsic) or in a body centred frame of refernce (intrinsic), which is attached to, and moves with, the object under rotation [1]_. Parameters ---------- seq : string Specifies sequence of axes for rotations. Up to 3 characters belonging to the set {X, Y, Z} for intrinsic rotations, or {x, y, z} for extrinsic rotations. Extrinsic and intrinsic rotations cannot be mixed in one function call. angles : float or array_like, shape (N,) or (N, [1 or 2 or 3]) Euler angles specified in radians (degrees is False) or degrees (degrees is True). For a single character seq, angles can be: - a single value - array_like with shape (N,), where each angle[i] corresponds to a single rotation - array_like with shape (N, 1), where each angle[i, 0] corresponds to a single rotation For 2- and 3-character wide seq, angles can be: - array_like with shape (W,) where W is the width of seq, which corresponds to a single rotation with W axes - array_like with shape (N, W) where each angle[i] corresponds to a sequence of Euler angles describing a single rotation degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- rotation : Rotation instance Object containing the rotation represented by the sequence of rotations around given axes with given angles. References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations Examples -------- from scipy.spatial.transform import Rotation as R Initialize a single rotation along a single axis: r R.from_euler(x, 90, degreesTrue) r.as_quat().shape (4,) Initialize a single rotation with a given axis sequence: r R.from_euler(zyx, [90, 45, 30], degreesTrue) r.as_quat().shape (4,) Initialize a stack with a single rotation around a single axis: r R.from_euler(x, [90], degreesTrue) r.as_quat().shape (1, 4) Initialize a stack with a single rotation with an axis sequence: r R.from_euler(zyx, [[90, 45, 30]], degreesTrue) r.as_quat().shape (1, 4) Initialize multiple elementary rotations in one object: r R.from_euler(x, [90, 45, 30], degreesTrue) r.as_quat().shape (3, 4) Initialize multiple rotations in one object: r R.from_euler(zyx, [[90, 45, 30], [35, 45, 90]], degreesTrue) r.as_quat().shape (2, 4) num_axes len(seq) if num_axes 1 or num_axes 3: raise ValueError(Expected axis specification to be a non-empty string of upto 3 characters, got {}.format(seq)) intrinsic (re.match(r^[XYZ]{1,3}$, seq) is not None) extrinsic (re.match(r^[xyz]{1,3}$, seq) is not None) if not (intrinsic or extrinsic): raise ValueError(Expected axes from seq to be from [x, y, z] or [X, Y, Z], got {}.format(seq)) if any(seq[i] seq[i1] for i in range(num_axes - 1)): raise ValueError(Expected consecutive axes to be different, got {}.format(seq)) seq seq.lower() angles np.asarray(angles, dtypefloat) if degrees: angles np.deg2rad(angles) is_single False # Prepare angles to have shape (num_rot, num_axes) if num_axes 1: if angles.ndim 0: # (1, 1) angles angles.reshape((1, 1)) is_single True elif angles.ndim 1: # (N, 1) angles angles[:, None] elif angles.ndim 2 and angles.shape[-1] ! 1: raise ValueError(Expected angles parameter to have shape (N, 1), got {}..format(angles.shape)) elif angles.ndim 2: raise ValueError(Expected float, 1D array, or 2D array for parameter angles corresponding to seq, got shape {}..format(angles.shape)) else: # 2 or 3 axes if angles.ndim not in [1, 2] or angles.shape[-1] ! num_axes: raise ValueError(Expected angles to be at most 2-dimensional with width equal to number of axes specified, got {} for shape.format( angles.shape)) if angles.ndim 1: # (1, num_axes) angles angles[None, :] is_single True # By now angles should have shape (num_rot, num_axes) # sanity check if angles.ndim ! 2 or angles.shape[-1] ! num_axes: raise ValueError(Expected angles to have shape (num_rotations, num_axes), got {}..format(angles.shape)) quat _elementary_quat_compose(seq, angles, intrinsic) return cls(quat[0] if is_single else quat, normalizedTrue, copyFalse) def as_quat(self): Represent as quaternions. Rotations in 3 dimensions can be represented using unit norm quaternions [1]_. The mapping from quaternions to rotations is two-to-one, i.e. quaternions q and -q, where -q simply reverses the sign of each component, represent the same spatial rotation. Returns ------- quat : numpy.ndarray, shape (4,) or (N, 4) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation Examples -------- from scipy.spatial.transform import Rotation as R Represent a single rotation: r R.from_dcm([ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]) r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) r.as_quat().shape (4,) Represent a stack with a single rotation: r R.from_quat([[0, 0, 0, 1]]) r.as_quat().shape (1, 4) Represent multiple rotaions in a single object: r R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]]) r.as_quat().shape (2, 4) if self._single: return self._quat[0].copy() else: return self._quat.copy() def as_dcm(self): Represent as direction cosine matrices. 3D rotations can be represented using direction cosine matrices, which are 3 x 3 real orthogonal matrices with determinant equal to 1 [1]_. Returns ------- dcm : ndarray, shape (3, 3) or (N, 3, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions Examples -------- from scipy.spatial.transform import Rotation as R Represent a single rotation: r R.from_rotvec([0, 0, np.pi/2]) r.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) r.as_dcm().shape (3, 3) Represent a stack with a single rotation: r R.from_quat([[1, 1, 0, 0]]) r.as_dcm() array([[[ 0., 1., 0.], [ 1., 0., 0.], [ 0., 0., -1.]]]) r.as_dcm().shape (1, 3, 3) Represent multiple rotations: r R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]]) r.as_dcm() array([[[ 1.00000000e00, 0.00000000e00, 0.00000000e00], [ 0.00000000e00, 2.22044605e-16, -1.00000000e00], [ 0.00000000e00, 1.00000000e00, 2.22044605e-16]], [[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]]) r.as_dcm().shape (2, 3, 3) x self._quat[:, 0] y self._quat[:, 1] z self._quat[:, 2] w self._quat[:, 3] x2 x * x y2 y * y z2 z * z w2 w * w xy x * y zw z * w xz x * z yw y * w yz y * z xw x * w num_rotations len(self) dcm np.empty((num_rotations, 3, 3)) dcm[:, 0, 0] x2 - y2 - z2 w2 dcm[:, 1, 0] 2 * (xy zw) dcm[:, 2, 0] 2 * (xz - yw) dcm[:, 0, 1] 2 * (xy - zw) dcm[:, 1, 1] - x2 y2 - z2 w2 dcm[:, 2, 1] 2 * (yz xw) dcm[:, 0, 2] 2 * (xz yw) dcm[:, 1, 2] 2 * (yz - xw) dcm[:, 2, 2] - x2 - y2 z2 w2 if self._single: return dcm[0] else: return dcm def as_rotvec(self): Represent as rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation (in radians) [1]_. Returns ------- rotvec : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- from scipy.spatial.transform import Rotation as R Represent a single rotation: r R.from_euler(z, 90, degreesTrue) r.as_rotvec() array([0. , 0. , 1.57079633]) r.as_rotvec().shape (3,) Represent a stack with a single rotation: r R.from_quat([[0, 0, 1, 1]]) r.as_rotvec() array([[0. , 0. , 1.57079633]]) r.as_rotvec().shape (1, 3) Represent multiple rotations in a single object: r R.from_quat([[0, 0, 1, 1], [1, 1, 0, 1]]) r.as_rotvec() array([[0. , 0. , 1.57079633], [1.35102172, 1.35102172, 0. ]]) r.as_rotvec().shape (2, 3) quat self._quat.copy() # w 0 to ensure 0 angle pi quat[quat[:, 3] 0] * -1 angle 2 * np.arctan2(np.linalg.norm(quat[:, :3], axis1), quat[:, 3]) small_angle (angle 1e-3) large_angle ~small_angle num_rotations len(self) scale np.empty(num_rotations) scale[small_angle] (2 angle[small_angle] ** 2 / 12 7 * angle[small_angle] ** 4 / 2880) scale[large_angle] (angle[large_angle] / np.sin(angle[large_angle] / 2)) rotvec scale[:, None] * quat[:, :3] if self._single: return rotvec[0] else: return rotvec def as_euler(self, seq, degreesFalse): Represent as Euler angles. Any orientation can be expressed as a composition of 3 elementary rotations. Once the axis sequence has been chosen, Euler angles define the angle of rotation around each respective axis [1]_. The algorithm from [2]_ has been used to calculate Euler angles for the rotation about a given sequence of axes. Euler angles suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- seq : string, length 3 3 characters belonging to the set {X, Y, Z} for intrinsic rotations, or {x, y, z} for extrinsic rotations [1]_. Adjacent axes cannot be the same. Extrinsic and intrinsic rotations cannot be mixed in one function call. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to: - [-90, 90] degrees if all axes are different (like xyz) - [0, 180] degrees if first and third axes are the same (like zxz) References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Malcolm D. Shuster, F. Landis Markley, General formula for extraction the Euler angles, Journal of guidance, control, and dynamics, vol. 29.1, pp. 215-221. 2006 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics Examples -------- from scipy.spatial.transform import Rotation as R Represent a single rotation: r R.from_rotvec([0, 0, np.pi/2]) r.as_euler(zxy, degreesTrue) array([90., 0., 0.]) r.as_euler(zxy, degreesTrue).shape (3,) Represent a stack of single rotation: r R.from_rotvec([[0, 0, np.pi/2]]) r.as_euler(zxy, degreesTrue) array([[90., 0., 0.]]) r.as_euler(zxy, degreesTrue).shape (1, 3) Represent multiple rotations in a single object: r R.from_rotvec([ ... [0, 0, np.pi/2], ... [0, -np.pi/3, 0], ... [np.pi/4, 0, 0]]) r.as_euler(zxy, degreesTrue) array([[ 90., 0., 0.], [ 0., 0., -60.], [ 0., 45., 0.]]) r.as_euler(zxy, degreesTrue).shape (3, 3) if len(seq) ! 3: raise ValueError(Expected 3 axes, got {}..format(seq)) intrinsic (re.match(r^[XYZ]{1,3}$, seq) is not None) extrinsic (re.match(r^[xyz]{1,3}$, seq) is not None) if not (intrinsic or extrinsic): raise ValueError(Expected axes from seq to be from [x, y, z] or [X, Y, Z], got {}.format(seq)) if any(seq[i] seq[i1] for i in range(2)): raise ValueError(Expected consecutive axes to be different, got {}.format(seq)) seq seq.lower() angles _compute_euler_from_dcm(self.as_dcm(), seq, extrinsic) if degrees: angles np.rad2deg(angles) return angles[0] if self._single else angles def apply(self, vectors, inverseFalse): Apply this rotation to a set of vectors. If the original frame rotates to the final frame by this rotation, then its application to a vector can be seen in two ways: - As a projection of vector components expressed in the final frame to the original frame. - As the physical rotation of a vector being glued to the original frame as it rotates. In this case the vector components are expressed in the original frame before and after the rotation. In terms of DCMs, this application is the same as self.as_dcm().dot(vectors). Parameters ---------- vectors : array_like, shape (3,) or (N, 3) Each vectors[i] represents a vector in 3D space. A single vector can either be specified with shape (3, ) or (1, 3). The number of rotations and number of vectors given must follow standard numpy broadcasting rules: either one of them equals unity or they both equal each other. inverse : boolean, optional If True then the inverse of the rotation(s) is applied to the input vectors. Default is False. Returns ------- rotated_vectors : ndarray, shape (3,) or (N, 3) Result of applying rotation on input vectors. Shape depends on the following cases: - If object contains a single rotation (as opposed to a stack with a single rotation) and a single vector is specified with shape (3,), then rotated_vectors has shape (3,). - In all other cases, rotated_vectors has shape (N, 3), where N is either the number of rotations or vectors. Examples -------- from scipy.spatial.transform import Rotation as R Single rotation applied on a single vector: vector np.array([1, 0, 0]) r R.from_rotvec([0, 0, np.pi/2]) r.as_dcm() array([[ 2.22044605e-16, -1.00000000e00, 0.00000000e00], [ 1.00000000e00, 2.22044605e-16, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]]) r.apply(vector) array([2.22044605e-16, 1.00000000e00, 0.00000000e00]) r.apply(vector).shape (3,) Single rotation applied on multiple vectors: vectors np.array([ ... [1, 0, 0], ... [1, 2, 3]]) r R.from_rotvec([0, 0, np.pi/4]) r.as_dcm() array([[ 0.70710678, -0.70710678, 0. ], [ 0.70710678, 0.70710678, 0. ], [ 0. , 0. , 1. ]]) r.apply(vectors) array([[ 0.70710678, 0.70710678, 0. ], [-0.70710678, 2.12132034, 3. ]]) r.apply(vectors).shape (2, 3) Multiple rotations on a single vector: r R.from_rotvec([[0, 0, np.pi/4], [np.pi/2, 0, 0]]) vector np.array([1,2,3]) r.as_dcm() array([[[ 7.07106781e-01, -7.07106781e-01, 0.00000000e00], [ 7.07106781e-01, 7.07106781e-01, 0.00000000e00], [ 0.00000000e00, 0.00000000e00, 1.00000000e00]], [[ 1.00000000e00, 0.00000000e00, 0.00000000e00], [ 0.00000000e00, 2.22044605e-16, -1.00000000e00], [ 0.00000000e00, 1.00000000e00, 2.22044605e-16]]]) r.apply(vector) array([[-0.70710678, 2.12132034, 3. ], [ 1. , -3. , 2. ]]) r.apply(vector).shape (2, 3) Multiple rotations on multiple vectors. Each rotation is applied on the corresponding vector: r R.from_euler(zxy, [ ... [0, 0, 90], ... [45, 30, 60]], degreesTrue) vectors [ ... [1, 2, 3], ... [1, 0, -1]] r.apply(vectors) array([[ 3. , 2. , -1. ], [-0.09026039, 1.11237244, -0.86860844]]) r.apply(vectors).shape (2, 3) It is also possible to apply the inverse rotation: r R.from_euler(zxy, [ ... [0, 0, 90], ... [45, 30, 60]], degreesTrue) vectors [ ... [1, 2, 3], ... [1, 0, -1]] r.apply(vectors, inverseTrue) array([[-3. , 2. , 1. ], [ 1.09533535, -0.8365163 , 0.3169873 ]]) vectors np.asarray(vectors) if vectors.ndim 2 or vectors.shape[-1] ! 3: raise ValueError(Expected input of shape (3,) or (P, 3), got {}..format(vectors.shape)) single_vector False if vectors.shape (3,): single_vector True vectors vectors[None, :] dcm self.as_dcm() if self._single: dcm dcm[None, :, :] n_vectors vectors.shape[0] n_rotations len(self) if n_vectors ! 1 and n_rotations ! 1 and n_vectors ! n_rotations: raise ValueError(Expected equal numbers of rotations and vectors , or a single rotation, or a single vector, got {} rotations and {} vectors..format( n_rotations, n_vectors)) if inverse: result np.einsum(ikj,ik-ij, dcm, vectors) else: result np.einsum(ijk,ik-ij, dcm, vectors) if self._single and single_vector: return result[0] else: return result def __mul__(self, other): Compose this rotation with the other. If p and q are two rotations, then the composition of q followed by p is equivalent to p * q. In terms of DCMs, the composition can be expressed as p.as_dcm().dot(q.as_dcm()). Parameters ---------- other : Rotation instance Object containing the rotaions to be composed with this one. Note that rotation compositions are not commutative, so p * q is different from q * p. Returns ------- composition : Rotation instance This function supports composition of multiple rotations at a time. The following cases are possible: - Either p or q contains a single rotation. In this case composition contains the result of composing each rotation in the other object with the single rotation. - Both p and q contain N rotations. In this case each rotation p[i] is composed with the corresponding rotation q[i] and output contains N rotations. Examples -------- from scipy.spatial.transform import Rotation as R Composition of two single rotations: p R.from_quat([0, 0, 1, 1]) q R.from_quat([1, 0, 0, 1]) p.as_dcm() array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]) q.as_dcm() array([[ 1., 0., 0.], [ 0., 0., -1.], [ 0., 1., 0.]]) r p * q r.as_dcm() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) Composition of two objects containing equal number of rotations: p R.from_quat([[0, 0, 1, 1], [1, 0, 0, 1]]) q R.from_rotvec([[np.pi/4, 0, 0], [-np.pi/4, 0, np.pi/4]]) p.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0.70710678, 0. , 0. , 0.70710678]]) q.as_quat() array([[ 0.38268343, 0. , 0. , 0.92387953], [-0.37282173, 0. , 0.37282173, 0.84971049]]) r p * q r.as_quat() array([[ 0.27059805, 0.27059805, 0.65328148, 0.65328148], [ 0.33721128, -0.26362477, 0.26362477, 0.86446082]]) if not(len(self) 1 or len(other) 1 or len(self) len(other)): raise ValueError(Expected equal number of rotations in both or a single rotation in either object, got {} rotations in first and {} rotations in second object..format( len(self), len(other))) result _compose_quat(self._quat, other._quat) if self._single and other._single: result result[0] return self.__class__(result, normalizedTrue, copyFalse) def inv(self): Invert this rotation. Composition of a rotation with its inverse results in an identity transformation. Returns ------- inverse : Rotation instance Object containing inverse of the rotations in the current instance. Examples -------- from scipy.spatial.transform import Rotation as R Inverting a single rotation: p R.from_euler(z, 45, degreesTrue) q p.inv() q.as_euler(zyx, degreesTrue) array([-45., 0., 0.]) Inverting multiple rotations: p R.from_rotvec([[0, 0, np.pi/3], [-np.pi/4, 0, 0]]) q p.inv() q.as_rotvec() array([[-0. , -0. , -1.04719755], [ 0.78539816, -0. , -0. ]]) quat self._quat.copy() quat[:, -1] * -1 if self._single: quat quat[0] return self.__class__(quat, normalizedTrue, copyFalse) def __getitem__(self, indexer): Extract rotation(s) at given index(es) from object. Create a new Rotation instance containing a subset of rotations stored in this object. Parameters ---------- indexer : index, slice, or index array Specifies which rotation(s) to extract. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. Returns ------- rotation : Rotation instance Contains - a single rotation, if indexer is a single index - a stack of rotation(s), if indexer is a slice, or and index array. Examples -------- from scipy.spatial.transform import Rotation as R r R.from_quat([ ... [1, 1, 0, 0], ... [0, 1, 0, 1], ... [1, 1, -1, 0]]) r.as_quat() array([[ 0.70710678, 0.70710678, 0. , 0. ], [ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) Indexing using a single index: p r[0] p.as_quat() array([0.70710678, 0.70710678, 0. , 0. ]) Array slicing: q r[1:3] q.as_quat() array([[ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) return self.__class__(self._quat[indexer], normalizedTrue) classmethod def random(cls, numNone, random_stateNone): Generate uniformly distributed rotations. Parameters ---------- num : int or None, optional Number of random rotations to generate. If None (default), then a single rotation is generated. random_state : int, RandomState instance or None, optional Accepts an integer as a seed for the random generator or a RandomState object. If None (default), uses global numpy.random random state. Returns ------- random_rotation : Rotation instance Contains a single rotation if num is None. Otherwise contains a stack of num rotations. Examples -------- from scipy.spatial.transform import Rotation as R Sample a single rotation: R.random(random_state1234).as_euler(zxy, degreesTrue) array([-110.5976185 , 55.32758512, 76.3289269 ]) Sample a stack of rotations: R.random(5, random_state1234).as_euler(zxy, degreesTrue) array([[-110.5976185 , 55.32758512, 76.3289269 ], [ -91.59132005, -14.3629884 , -93.91933182], [ 25.23835501, 45.02035145, -121.67867086], [ -51.51414184, -15.29022692, -172.46870023], [ -81.63376847, -27.39521579, 2.60408416]]) random_state check_random_state(random_state) if num is None: sample random_state.normal(size4) else: sample random_state.normal(size(num, 4)) return Rotation.from_quat(sample) classmethod def match_vectors(cls, a, b, weightsNone, normalizedFalse): Estimate a rotation to match two sets of vectors. Find a rotation between frames A and B which best matches a set of unit vectors a and b observed in these frames. The following loss function is minimized to solve for the direction cosine matrix :math:C: .. math:: L(C) \\frac{1}{2} \\sum_{i 1}^{n} w_i \\lVert \\mathbf{a}_i - C \\mathbf{b}_i \\rVert^2 , where :math:w_is are the weights corresponding to each vector. The rotation is estimated using Markleys SVD method [1]_. Parameters ---------- a : array_like, shape (N, 3) Vector components observed in initial frame A. Each row of a denotes a vector. b : array_like, shape (N, 3) Vector components observed in another frame B. Each row of b denotes a vector. weights : array_like shape (N,), optional Weights describing the relative importance of the vectors in a. If None (default), then all values in weights are assumed to be equal. normalized : bool, optional If True, assume input vectors a and b to have unit norm. If False, normalize a and b before estimating rotation. Default is False. Returns ------- estimated_rotation : Rotation instance Best estimate of the rotation that transforms b to a. sensitivity_matrix : ndarray, shape (3, 3) Scaled covariance of the attitude errors expressed as the small rotation vector of frame A. Multiply with harmonic mean [3]_ of variance in each observation to get true covariance matrix. The error model is detailed in [2]_. References ---------- .. [1] F. Landis Markley, Attitude determination using vector observations: a fast optimal matrix algorithm, Journal of Astronautical Sciences, Vol. 41, No.2, 1993, pp. 261-280. .. [2] F. Landis Markley, Attitude determination using vector observations and the Singular Value Decomposition, Journal of Astronautical Sciences, Vol. 38, No.3, 1988, pp. 245-258. .. [3] https://en.wikipedia.org/wiki/Harmonic_mean a np.asarray(a) if a.ndim ! 2 or a.shape[-1] ! 3: raise ValueError(Expected input a to have shape (N, 3), got {}.format(a.shape)) b np.asarray(b) if b.ndim ! 2 or b.shape[-1] ! 3: raise ValueError(Expected input b to have shape (N, 3), got {}..format(b.shape)) if a.shape ! b.shape: raise ValueError(Expected inputs a and b to have same shapes , got {} and {} respectively..format( a.shape, b.shape)) if b.shape[0] 1: raise ValueError(Rotation cannot be estimated using a single vector.) if weights is None: weights np.ones(b.shape[0]) else: weights np.asarray(weights) if weights.ndim ! 1: raise ValueError(Expected weights to be 1 dimensional, got shape {}..format(weights.shape)) if weights.shape[0] ! b.shape[0]: raise ValueError(Expected weights to have number of values equal to number of input vectors, got {} values and {} vectors..format( weights.shape[0], b.shape[0])) weights weights / np.sum(weights) if not normalized: a a / scipy.linalg.norm(a, axis1)[:, None] b b / scipy.linalg.norm(b, axis1)[:, None] B np.einsum(ji,jk-ik, weights[:, None] * a, b) u, s, vh np.linalg.svd(B) C np.dot(u, vh) zeta (s[0]s[1]) * (s[1]s[2]) * (s[2]s[0]) if np.abs(zeta) 1e-16: raise ValueError(Three component error vector has infinite covariance. It is impossible to determine the rotation uniquely.) kappa s[0]*s[1] s[1]*s[2] s[2]*s[0] sensitivity ((kappa * np.eye(3) np.dot(B, B.T)) / (zeta * a.shape[0])) return cls.from_dcm(C), sensitivity class Slerp(object): Spherical Linear Interpolation of Rotations. The interpolation between consecutive rotations is performed as a rotation around a fixed axis with a constant angular velocity [1]_. This ensures that the interpolated rotations follow the shortest path between initial and final orientations. 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__ See Also -------- Rotation Notes ----- .. versionadded:: 1.2.0 References ---------- .. [1] https://en.wikipedia.org/wiki/Slerp#Quaternion_Slerp Examples -------- from scipy.spatial.transform import Rotation as R from scipy.spatial.transform import Slerp Setup the fixed keyframe rotations and times: key_rots R.random(5, random_state2342345) key_times [0, 1, 2, 3, 4] Create the interpolator object: slerp Slerp(key_times, key_rots) Interpolate the rotations at the given times: times [0, 0.5, 0.25, 1, 1.5, 2, 2.75, 3, 3.25, 3.60, 4] interp_rots slerp(times) The keyframe rotations expressed as Euler angles: key_rots.as_euler(xyz, degreesTrue) array([[ 14.31443779, -27.50095894, -3.7275787 ], [ -1.79924227, -24.69421529, 164.57701743], [146.15020772, 43.22849451, -31.34891088], [ 46.39959442, 11.62126073, -45.99719267], [-88.94647804, -49.64400082, -65.80546984]]) The interpolated rotations expressed as Euler angles. These agree with the keyframe rotations at both endpoints of the range of keyframe times. interp_rots.as_euler(xyz, degreesTrue) array([[ 14.31443779, -27.50095894, -3.7275787 ], [ 4.74588574, -32.44683966, 81.25139984], [ 10.71094749, -31.56690154, 38.06896408], [ -1.79924227, -24.69421529, 164.57701743], [ 11.72796022, 51.64207311, -171.7374683 ], [ 146.15020772, 43.22849451, -31.34891088], [ 68.10921869, 20.67625074, -48.74886034], [ 46.39959442, 11.62126073, -45.99719267], [ 12.35552615, 4.21525086, -64.89288124], [ -30.08117143, -19.90769513, -78.98121326], [ -88.94647804, -49.64400082, -65.80546984]]) def __init__(self, times, rotations): if len(rotations) 1: raise ValueError(rotations must contain at least 2 rotations.) times np.asarray(times) if times.ndim ! 1: raise ValueError(Expected times to be specified in a 1 dimensional array, got {} dimensions..format(times.ndim)) if times.shape[0] ! len(rotations): raise ValueError(Expected number of rotations to be equal to number of timestamps given, got {} rotations and {} timestamps..format( len(rotations), times.shape[0])) self.times times self.timedelta np.diff(times) if np.any(self.timedelta 0): raise ValueError(Times must be in strictly increasing order.) self.rotations rotations[:-1] self.rotvecs (self.rotations.inv() * rotations[1:]).as_rotvec() def __call__(self, times): Interpolate rotations. Compute the interpolated rotations at the given times. Parameters ---------- times : array_like, 1D Times to compute the interpolations at. Returns ------- interpolated_rotation : Rotation instance Object containing the rotations computed at given times. # Clearly differentiate from self.times property compute_times np.asarray(times) if compute_times.ndim ! 1: raise ValueError(Expected times to be specified in a 1 dimensional array, got {} dimensions..format(compute_times.ndim)) # side left (default) excludes t_min. ind np.searchsorted(self.times, compute_times) - 1 # Include t_min. Without this step, index for t_min equals -1 ind[compute_times self.times[0]] 0 if np.any(np.logical_or(ind 0, ind len(self.rotations) - 1)): raise ValueError(Interpolation times must be within the range [{}, {}], both inclusive..format( self.times[0], self.times[-1])) alpha (compute_times - self.times[ind]) / self.timedelta[ind] return (self.rotations[ind] * Rotation.from_rotvec(self.rotvecs[ind] * alpha[:, None]))
