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| import numpy as np | |
| class Quaternions: | |
| """ | |
| Quaternions is a wrapper around a numpy ndarray | |
| that allows it to act as if it were an narray of | |
| a quaternion data type. | |
| Therefore addition, subtraction, multiplication, | |
| division, negation, absolute, are all defined | |
| in terms of quaternion operations such as quaternion | |
| multiplication. | |
| This allows for much neater code and many routines | |
| which conceptually do the same thing to be written | |
| in the same way for point data and for rotation data. | |
| The Quaternions class has been desgined such that it | |
| should support broadcasting and slicing in all of the | |
| usual ways. | |
| """ | |
| def __init__(self, qs): | |
| if isinstance(qs, np.ndarray): | |
| if len(qs.shape) == 1: qs = np.array([qs]) | |
| self.qs = qs | |
| return | |
| if isinstance(qs, Quaternions): | |
| self.qs = qs.qs | |
| return | |
| raise TypeError('Quaternions must be constructed from iterable, numpy array, or Quaternions, not %s' % type(qs)) | |
| def __str__(self): return "Quaternions("+ str(self.qs) + ")" | |
| def __repr__(self): return "Quaternions("+ repr(self.qs) + ")" | |
| """ Helper Methods for Broadcasting and Data extraction """ | |
| def _broadcast(cls, sqs, oqs, scalar=False): | |
| if isinstance(oqs, float): return sqs, oqs * np.ones(sqs.shape[:-1]) | |
| ss = np.array(sqs.shape) if not scalar else np.array(sqs.shape[:-1]) | |
| os = np.array(oqs.shape) | |
| if len(ss) != len(os): | |
| raise TypeError('Quaternions cannot broadcast together shapes %s and %s' % (sqs.shape, oqs.shape)) | |
| if np.all(ss == os): return sqs, oqs | |
| if not np.all((ss == os) | (os == np.ones(len(os))) | (ss == np.ones(len(ss)))): | |
| raise TypeError('Quaternions cannot broadcast together shapes %s and %s' % (sqs.shape, oqs.shape)) | |
| sqsn, oqsn = sqs.copy(), oqs.copy() | |
| for a in np.where(ss == 1)[0]: sqsn = sqsn.repeat(os[a], axis=a) | |
| for a in np.where(os == 1)[0]: oqsn = oqsn.repeat(ss[a], axis=a) | |
| return sqsn, oqsn | |
| """ Adding Quaterions is just Defined as Multiplication """ | |
| def __add__(self, other): return self * other | |
| def __sub__(self, other): return self / other | |
| """ Quaterion Multiplication """ | |
| def __mul__(self, other): | |
| """ | |
| Quaternion multiplication has three main methods. | |
| When multiplying a Quaternions array by Quaternions | |
| normal quaternion multiplication is performed. | |
| When multiplying a Quaternions array by a vector | |
| array of the same shape, where the last axis is 3, | |
| it is assumed to be a Quaternion by 3D-Vector | |
| multiplication and the 3D-Vectors are rotated | |
| in space by the Quaternions. | |
| When multipplying a Quaternions array by a scalar | |
| or vector of different shape it is assumed to be | |
| a Quaternions by Scalars multiplication and the | |
| Quaternions are scaled using Slerp and the identity | |
| quaternions. | |
| """ | |
| """ If Quaternions type do Quaternions * Quaternions """ | |
| if isinstance(other, Quaternions): | |
| sqs, oqs = Quaternions._broadcast(self.qs, other.qs) | |
| q0 = sqs[...,0]; q1 = sqs[...,1]; | |
| q2 = sqs[...,2]; q3 = sqs[...,3]; | |
| r0 = oqs[...,0]; r1 = oqs[...,1]; | |
| r2 = oqs[...,2]; r3 = oqs[...,3]; | |
| qs = np.empty(sqs.shape) | |
| qs[...,0] = r0 * q0 - r1 * q1 - r2 * q2 - r3 * q3 | |
| qs[...,1] = r0 * q1 + r1 * q0 - r2 * q3 + r3 * q2 | |
| qs[...,2] = r0 * q2 + r1 * q3 + r2 * q0 - r3 * q1 | |
| qs[...,3] = r0 * q3 - r1 * q2 + r2 * q1 + r3 * q0 | |
| return Quaternions(qs) | |
| """ If array type do Quaternions * Vectors """ | |
| if isinstance(other, np.ndarray) and other.shape[-1] == 3: | |
| vs = Quaternions(np.concatenate([np.zeros(other.shape[:-1] + (1,)), other], axis=-1)) | |
| return (self * (vs * -self)).imaginaries | |
| """ If float do Quaternions * Scalars """ | |
| if isinstance(other, np.ndarray) or isinstance(other, float): | |
| return Quaternions.slerp(Quaternions.id_like(self), self, other) | |
| raise TypeError('Cannot multiply/add Quaternions with type %s' % str(type(other))) | |
| def __div__(self, other): | |
| """ | |
| When a Quaternion type is supplied, division is defined | |
| as multiplication by the inverse of that Quaternion. | |
| When a scalar or vector is supplied it is defined | |
| as multiplicaion of one over the supplied value. | |
| Essentially a scaling. | |
| """ | |
| if isinstance(other, Quaternions): return self * (-other) | |
| if isinstance(other, np.ndarray): return self * (1.0 / other) | |
| if isinstance(other, float): return self * (1.0 / other) | |
| raise TypeError('Cannot divide/subtract Quaternions with type %s' + str(type(other))) | |
| def __eq__(self, other): return self.qs == other.qs | |
| def __ne__(self, other): return self.qs != other.qs | |
| def __neg__(self): | |
| """ Invert Quaternions """ | |
| return Quaternions(self.qs * np.array([[1, -1, -1, -1]])) | |
| def __abs__(self): | |
| """ Unify Quaternions To Single Pole """ | |
| qabs = self.normalized().copy() | |
| top = np.sum(( qabs.qs) * np.array([1,0,0,0]), axis=-1) | |
| bot = np.sum((-qabs.qs) * np.array([1,0,0,0]), axis=-1) | |
| qabs.qs[top < bot] = -qabs.qs[top < bot] | |
| return qabs | |
| def __iter__(self): return iter(self.qs) | |
| def __len__(self): return len(self.qs) | |
| def __getitem__(self, k): return Quaternions(self.qs[k]) | |
| def __setitem__(self, k, v): self.qs[k] = v.qs | |
| def lengths(self): | |
| return np.sum(self.qs**2.0, axis=-1)**0.5 | |
| def reals(self): | |
| return self.qs[...,0] | |
| def imaginaries(self): | |
| return self.qs[...,1:4] | |
| def shape(self): return self.qs.shape[:-1] | |
| def repeat(self, n, **kwargs): | |
| return Quaternions(self.qs.repeat(n, **kwargs)) | |
| def normalized(self): | |
| return Quaternions(self.qs / self.lengths[...,np.newaxis]) | |
| def log(self): | |
| norm = abs(self.normalized()) | |
| imgs = norm.imaginaries | |
| lens = np.sqrt(np.sum(imgs**2, axis=-1)) | |
| lens = np.arctan2(lens, norm.reals) / (lens + 1e-10) | |
| return imgs * lens[...,np.newaxis] | |
| def constrained(self, axis): | |
| rl = self.reals | |
| im = np.sum(axis * self.imaginaries, axis=-1) | |
| t1 = -2 * np.arctan2(rl, im) + np.pi | |
| t2 = -2 * np.arctan2(rl, im) - np.pi | |
| top = Quaternions.exp(axis[np.newaxis] * (t1[:,np.newaxis] / 2.0)) | |
| bot = Quaternions.exp(axis[np.newaxis] * (t2[:,np.newaxis] / 2.0)) | |
| img = self.dot(top) > self.dot(bot) | |
| ret = top.copy() | |
| ret[ img] = top[ img] | |
| ret[~img] = bot[~img] | |
| return ret | |
| def constrained_x(self): return self.constrained(np.array([1,0,0])) | |
| def constrained_y(self): return self.constrained(np.array([0,1,0])) | |
| def constrained_z(self): return self.constrained(np.array([0,0,1])) | |
| def dot(self, q): return np.sum(self.qs * q.qs, axis=-1) | |
| def copy(self): return Quaternions(np.copy(self.qs)) | |
| def reshape(self, s): | |
| self.qs.reshape(s) | |
| return self | |
| def interpolate(self, ws): | |
| return Quaternions.exp(np.average(abs(self).log, axis=0, weights=ws)) | |
| def euler(self, order='xyz'): | |
| q = self.normalized().qs | |
| q0 = q[...,0] | |
| q1 = q[...,1] | |
| q2 = q[...,2] | |
| q3 = q[...,3] | |
| es = np.zeros(self.shape + (3,)) | |
| if order == 'xyz': | |
| es[...,0] = np.arctan2(2 * (q0 * q1 + q2 * q3), 1 - 2 * (q1 * q1 + q2 * q2)) | |
| es[...,1] = np.arcsin((2 * (q0 * q2 - q3 * q1)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q3 + q1 * q2), 1 - 2 * (q2 * q2 + q3 * q3)) | |
| elif order == 'yzx': | |
| es[...,0] = np.arctan2(2 * (q1 * q0 - q2 * q3), -q1 * q1 + q2 * q2 - q3 * q3 + q0 * q0) | |
| es[...,1] = np.arctan2(2 * (q2 * q0 - q1 * q3), q1 * q1 - q2 * q2 - q3 * q3 + q0 * q0) | |
| es[...,2] = np.arcsin((2 * (q1 * q2 + q3 * q0)).clip(-1,1)) | |
| else: | |
| raise NotImplementedError('Cannot convert from ordering %s' % order) | |
| """ | |
| # These conversion don't appear to work correctly for Maya. | |
| # http://bediyap.com/programming/convert-quaternion-to-euler-rotations/ | |
| if order == 'xyz': | |
| es[...,0] = np.arctan2(2 * (q0 * q3 - q1 * q2), q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q1 * q3 + q0 * q2)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q1 - q2 * q3), q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) | |
| elif order == 'yzx': | |
| es[...,0] = np.arctan2(2 * (q0 * q1 - q2 * q3), q0 * q0 - q1 * q1 + q2 * q2 - q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q1 * q2 + q0 * q3)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q2 - q1 * q3), q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3) | |
| elif order == 'zxy': | |
| es[...,0] = np.arctan2(2 * (q0 * q2 - q1 * q3), q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q0 * q1 + q2 * q3)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q3 - q1 * q2), q0 * q0 - q1 * q1 + q2 * q2 - q3 * q3) | |
| elif order == 'xzy': | |
| es[...,0] = np.arctan2(2 * (q0 * q2 + q1 * q3), q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q0 * q3 - q1 * q2)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q1 + q2 * q3), q0 * q0 - q1 * q1 + q2 * q2 - q3 * q3) | |
| elif order == 'yxz': | |
| es[...,0] = np.arctan2(2 * (q1 * q2 + q0 * q3), q0 * q0 - q1 * q1 + q2 * q2 - q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q0 * q1 - q2 * q3)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q1 * q3 + q0 * q2), q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) | |
| elif order == 'zyx': | |
| es[...,0] = np.arctan2(2 * (q0 * q1 + q2 * q3), q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) | |
| es[...,1] = np.arcsin((2 * (q0 * q2 - q1 * q3)).clip(-1,1)) | |
| es[...,2] = np.arctan2(2 * (q0 * q3 + q1 * q2), q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3) | |
| else: | |
| raise KeyError('Unknown ordering %s' % order) | |
| """ | |
| # https://github.com/ehsan/ogre/blob/master/OgreMain/src/OgreMatrix3.cpp | |
| # Use this class and convert from matrix | |
| return es | |
| def average(self): | |
| if len(self.shape) == 1: | |
| import numpy.core.umath_tests as ut | |
| system = ut.matrix_multiply(self.qs[:,:,np.newaxis], self.qs[:,np.newaxis,:]).sum(axis=0) | |
| w, v = np.linalg.eigh(system) | |
| qiT_dot_qref = (self.qs[:,:,np.newaxis] * v[np.newaxis,:,:]).sum(axis=1) | |
| return Quaternions(v[:,np.argmin((1.-qiT_dot_qref**2).sum(axis=0))]) | |
| else: | |
| raise NotImplementedError('Cannot average multi-dimensionsal Quaternions') | |
| def angle_axis(self): | |
| norm = self.normalized() | |
| s = np.sqrt(1 - (norm.reals**2.0)) | |
| s[s == 0] = 0.001 | |
| angles = 2.0 * np.arccos(norm.reals) | |
| axis = norm.imaginaries / s[...,np.newaxis] | |
| return angles, axis | |
| def transforms(self): | |
| qw = self.qs[...,0] | |
| qx = self.qs[...,1] | |
| qy = self.qs[...,2] | |
| qz = self.qs[...,3] | |
| x2 = qx + qx; y2 = qy + qy; z2 = qz + qz; | |
| xx = qx * x2; yy = qy * y2; wx = qw * x2; | |
| xy = qx * y2; yz = qy * z2; wy = qw * y2; | |
| xz = qx * z2; zz = qz * z2; wz = qw * z2; | |
| m = np.empty(self.shape + (3,3)) | |
| m[...,0,0] = 1.0 - (yy + zz) | |
| m[...,0,1] = xy - wz | |
| m[...,0,2] = xz + wy | |
| m[...,1,0] = xy + wz | |
| m[...,1,1] = 1.0 - (xx + zz) | |
| m[...,1,2] = yz - wx | |
| m[...,2,0] = xz - wy | |
| m[...,2,1] = yz + wx | |
| m[...,2,2] = 1.0 - (xx + yy) | |
| return m | |
| def ravel(self): | |
| return self.qs.ravel() | |
| def id(cls, n): | |
| if isinstance(n, tuple): | |
| qs = np.zeros(n + (4,)) | |
| qs[...,0] = 1.0 | |
| return Quaternions(qs) | |
| if isinstance(n, int) or isinstance(n, long): | |
| qs = np.zeros((n,4)) | |
| qs[:,0] = 1.0 | |
| return Quaternions(qs) | |
| raise TypeError('Cannot Construct Quaternion from %s type' % str(type(n))) | |
| def id_like(cls, a): | |
| qs = np.zeros(a.shape + (4,)) | |
| qs[...,0] = 1.0 | |
| return Quaternions(qs) | |
| def exp(cls, ws): | |
| ts = np.sum(ws**2.0, axis=-1)**0.5 | |
| ts[ts == 0] = 0.001 | |
| ls = np.sin(ts) / ts | |
| qs = np.empty(ws.shape[:-1] + (4,)) | |
| qs[...,0] = np.cos(ts) | |
| qs[...,1] = ws[...,0] * ls | |
| qs[...,2] = ws[...,1] * ls | |
| qs[...,3] = ws[...,2] * ls | |
| return Quaternions(qs).normalized() | |
| def slerp(cls, q0s, q1s, a): | |
| fst, snd = cls._broadcast(q0s.qs, q1s.qs) | |
| fst, a = cls._broadcast(fst, a, scalar=True) | |
| snd, a = cls._broadcast(snd, a, scalar=True) | |
| len = np.sum(fst * snd, axis=-1) | |
| neg = len < 0.0 | |
| len[neg] = -len[neg] | |
| snd[neg] = -snd[neg] | |
| amount0 = np.zeros(a.shape) | |
| amount1 = np.zeros(a.shape) | |
| linear = (1.0 - len) < 0.01 | |
| omegas = np.arccos(len[~linear]) | |
| sinoms = np.sin(omegas) | |
| amount0[ linear] = 1.0 - a[linear] | |
| amount1[ linear] = a[linear] | |
| amount0[~linear] = np.sin((1.0 - a[~linear]) * omegas) / sinoms | |
| amount1[~linear] = np.sin( a[~linear] * omegas) / sinoms | |
| return Quaternions( | |
| amount0[...,np.newaxis] * fst + | |
| amount1[...,np.newaxis] * snd) | |
| def between(cls, v0s, v1s): | |
| a = np.cross(v0s, v1s) | |
| w = np.sqrt((v0s**2).sum(axis=-1) * (v1s**2).sum(axis=-1)) + (v0s * v1s).sum(axis=-1) | |
| return Quaternions(np.concatenate([w[...,np.newaxis], a], axis=-1)).normalized() | |
| def from_angle_axis(cls, angles, axis): | |
| axis = axis / (np.sqrt(np.sum(axis**2, axis=-1)) + 1e-10)[...,np.newaxis] | |
| sines = np.sin(angles / 2.0)[...,np.newaxis] | |
| cosines = np.cos(angles / 2.0)[...,np.newaxis] | |
| return Quaternions(np.concatenate([cosines, axis * sines], axis=-1)) | |
| def from_euler(cls, es, order='xyz', world=False): | |
| axis = { | |
| 'x' : np.array([1,0,0]), | |
| 'y' : np.array([0,1,0]), | |
| 'z' : np.array([0,0,1]), | |
| } | |
| q0s = Quaternions.from_angle_axis(es[...,0], axis[order[0]]) | |
| q1s = Quaternions.from_angle_axis(es[...,1], axis[order[1]]) | |
| q2s = Quaternions.from_angle_axis(es[...,2], axis[order[2]]) | |
| return (q2s * (q1s * q0s)) if world else (q0s * (q1s * q2s)) | |
| def from_transforms(cls, ts): | |
| d0, d1, d2 = ts[...,0,0], ts[...,1,1], ts[...,2,2] | |
| q0 = ( d0 + d1 + d2 + 1.0) / 4.0 | |
| q1 = ( d0 - d1 - d2 + 1.0) / 4.0 | |
| q2 = (-d0 + d1 - d2 + 1.0) / 4.0 | |
| q3 = (-d0 - d1 + d2 + 1.0) / 4.0 | |
| q0 = np.sqrt(q0.clip(0,None)) | |
| q1 = np.sqrt(q1.clip(0,None)) | |
| q2 = np.sqrt(q2.clip(0,None)) | |
| q3 = np.sqrt(q3.clip(0,None)) | |
| c0 = (q0 >= q1) & (q0 >= q2) & (q0 >= q3) | |
| c1 = (q1 >= q0) & (q1 >= q2) & (q1 >= q3) | |
| c2 = (q2 >= q0) & (q2 >= q1) & (q2 >= q3) | |
| c3 = (q3 >= q0) & (q3 >= q1) & (q3 >= q2) | |
| q1[c0] *= np.sign(ts[c0,2,1] - ts[c0,1,2]) | |
| q2[c0] *= np.sign(ts[c0,0,2] - ts[c0,2,0]) | |
| q3[c0] *= np.sign(ts[c0,1,0] - ts[c0,0,1]) | |
| q0[c1] *= np.sign(ts[c1,2,1] - ts[c1,1,2]) | |
| q2[c1] *= np.sign(ts[c1,1,0] + ts[c1,0,1]) | |
| q3[c1] *= np.sign(ts[c1,0,2] + ts[c1,2,0]) | |
| q0[c2] *= np.sign(ts[c2,0,2] - ts[c2,2,0]) | |
| q1[c2] *= np.sign(ts[c2,1,0] + ts[c2,0,1]) | |
| q3[c2] *= np.sign(ts[c2,2,1] + ts[c2,1,2]) | |
| q0[c3] *= np.sign(ts[c3,1,0] - ts[c3,0,1]) | |
| q1[c3] *= np.sign(ts[c3,2,0] + ts[c3,0,2]) | |
| q2[c3] *= np.sign(ts[c3,2,1] + ts[c3,1,2]) | |
| qs = np.empty(ts.shape[:-2] + (4,)) | |
| qs[...,0] = q0 | |
| qs[...,1] = q1 | |
| qs[...,2] = q2 | |
| qs[...,3] = q3 | |
| return cls(qs) | |