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# fmt: off
import numpy as np
class Quaternion:
def __init__(self, qin=[1, 0, 0, 0]):
assert len(qin) == 4
self.q = np.array(qin)
def __str__(self):
return self.q.__str__()
def __mul__(self, other):
sw, sx, sy, sz = self.q
ow, ox, oy, oz = other.q
return Quaternion([sw * ow - sx * ox - sy * oy - sz * oz,
sw * ox + sx * ow + sy * oz - sz * oy,
sw * oy + sy * ow + sz * ox - sx * oz,
sw * oz + sz * ow + sx * oy - sy * ox])
def conjugate(self):
return Quaternion(-self.q * np.array([-1., 1., 1., 1.]))
def rotate(self, vector):
"""Apply the rotation matrix to a vector."""
qw, qx, qy, qz = self.q[0], self.q[1], self.q[2], self.q[3]
x, y, z = vector[0], vector[1], vector[2]
ww = qw * qw
xx = qx * qx
yy = qy * qy
zz = qz * qz
wx = qw * qx
wy = qw * qy
wz = qw * qz
xy = qx * qy
xz = qx * qz
yz = qy * qz
return np.array(
[(ww + xx - yy - zz) * x + 2 * ((xy - wz) * y + (xz + wy) * z),
(ww - xx + yy - zz) * y + 2 * ((xy + wz) * x + (yz - wx) * z),
(ww - xx - yy + zz) * z + 2 * ((xz - wy) * x + (yz + wx) * y)])
def rotation_matrix(self):
qw, qx, qy, qz = self.q[0], self.q[1], self.q[2], self.q[3]
ww = qw * qw
xx = qx * qx
yy = qy * qy
zz = qz * qz
wx = qw * qx
wy = qw * qy
wz = qw * qz
xy = qx * qy
xz = qx * qz
yz = qy * qz
return np.array([[ww + xx - yy - zz, 2 * (xy - wz), 2 * (xz + wy)],
[2 * (xy + wz), ww - xx + yy - zz, 2 * (yz - wx)],
[2 * (xz - wy), 2 * (yz + wx), ww - xx - yy + zz]])
def axis_angle(self):
"""Returns axis and angle (in radians) for the rotation described
by this Quaternion"""
sinth_2 = np.linalg.norm(self.q[1:])
if sinth_2 == 0:
# The angle is zero
theta = 0.0
n = np.array([0, 0, 1])
else:
theta = np.arctan2(sinth_2, self.q[0]) * 2
n = self.q[1:] / sinth_2
return n, theta
def euler_angles(self, mode='zyz'):
"""Return three Euler angles describing the rotation, in radians.
Mode can be zyz or zxz. Default is zyz."""
# https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0276302
if mode == 'zyz':
a, b, c, d = self.q[0], self.q[3], self.q[2], -self.q[1]
elif mode == 'zxz':
a, b, c, d = self.q[0], self.q[3], self.q[1], self.q[2]
else:
raise ValueError(f'Invalid Euler angles mode {mode}')
beta = 2 * np.arccos(
np.sqrt((a**2 + b**2) / (a**2 + b**2 + c**2 + d**2))
)
gap = np.arctan2(b, a) # (gamma + alpha) / 2
gam = np.arctan2(d, c) # (gamma - alpha) / 2
if np.isclose(beta, 0):
# gam is meaningless here
alpha = 0
gamma = 2 * gap - alpha
elif np.isclose(beta, np.pi):
# gap is meaningless here
alpha = 0
gamma = 2 * gam + alpha
else:
alpha = gap - gam
gamma = gap + gam
return np.array([alpha, beta, gamma])
def arc_distance(self, other):
"""Gives a metric of the distance between two quaternions,
expressed as 1-|q1.q2|"""
return 1.0 - np.abs(np.dot(self.q, other.q))
@staticmethod
def rotate_byq(q, vector):
"""Apply the rotation matrix to a vector."""
qw, qx, qy, qz = q[0], q[1], q[2], q[3]
x, y, z = vector[0], vector[1], vector[2]
ww = qw * qw
xx = qx * qx
yy = qy * qy
zz = qz * qz
wx = qw * qx
wy = qw * qy
wz = qw * qz
xy = qx * qy
xz = qx * qz
yz = qy * qz
return np.array(
[(ww + xx - yy - zz) * x + 2 * ((xy - wz) * y + (xz + wy) * z),
(ww - xx + yy - zz) * y + 2 * ((xy + wz) * x + (yz - wx) * z),
(ww - xx - yy + zz) * z + 2 * ((xz - wy) * x + (yz + wx) * y)])
@staticmethod
def from_matrix(matrix):
"""Build quaternion from rotation matrix."""
m = np.array(matrix)
assert m.shape == (3, 3)
# Now we need to find out the whole quaternion
# This method takes into account the possibility of qw being nearly
# zero, so it picks the stablest solution
if m[2, 2] < 0:
if (m[0, 0] > m[1, 1]):
# Use x-form
qx = np.sqrt(1 + m[0, 0] - m[1, 1] - m[2, 2]) / 2.0
fac = 1.0 / (4 * qx)
qw = (m[2, 1] - m[1, 2]) * fac
qy = (m[0, 1] + m[1, 0]) * fac
qz = (m[0, 2] + m[2, 0]) * fac
else:
# Use y-form
qy = np.sqrt(1 - m[0, 0] + m[1, 1] - m[2, 2]) / 2.0
fac = 1.0 / (4 * qy)
qw = (m[0, 2] - m[2, 0]) * fac
qx = (m[0, 1] + m[1, 0]) * fac
qz = (m[1, 2] + m[2, 1]) * fac
else:
if (m[0, 0] < -m[1, 1]):
# Use z-form
qz = np.sqrt(1 - m[0, 0] - m[1, 1] + m[2, 2]) / 2.0
fac = 1.0 / (4 * qz)
qw = (m[1, 0] - m[0, 1]) * fac
qx = (m[2, 0] + m[0, 2]) * fac
qy = (m[1, 2] + m[2, 1]) * fac
else:
# Use w-form
qw = np.sqrt(1 + m[0, 0] + m[1, 1] + m[2, 2]) / 2.0
fac = 1.0 / (4 * qw)
qx = (m[2, 1] - m[1, 2]) * fac
qy = (m[0, 2] - m[2, 0]) * fac
qz = (m[1, 0] - m[0, 1]) * fac
return Quaternion(np.array([qw, qx, qy, qz]))
@staticmethod
def from_axis_angle(n, theta):
"""Build quaternion from axis (n, vector of 3 components) and angle
(theta, in radianses)."""
n = np.array(n, float) / np.linalg.norm(n)
return Quaternion(np.concatenate([[np.cos(theta / 2.0)],
np.sin(theta / 2.0) * n]))
@staticmethod
def from_euler_angles(a, b, c, mode='zyz'):
"""Build quaternion from Euler angles, given in radians. Default
mode is ZYZ, but it can be set to ZXZ as well."""
q_a = Quaternion.from_axis_angle([0, 0, 1], a)
q_c = Quaternion.from_axis_angle([0, 0, 1], c)
if mode == 'zyz':
q_b = Quaternion.from_axis_angle([0, 1, 0], b)
elif mode == 'zxz':
q_b = Quaternion.from_axis_angle([1, 0, 0], b)
else:
raise ValueError(f'Invalid Euler angles mode {mode}')
return q_c * q_b * q_a
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