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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 {} and {}".format(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 {} and {}".format(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((1.0 - 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) | |