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quaternion.py
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quaternion.py
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import numpy as np
from scipy.spatial.transform import Rotation
def quaternion(w,x,y,z):
return np.array([w,x,y,z])
def vector(x,y,z):
return np.array([x,y,z])
def normalize(q):
return q/np.sqrt(q.dot(q)) # q/|q|
# https://stackoverflow.com/questions/39000758/how-to-multiply-two-quaternions-by-python-or-numpy
def quaternion_multiply(Q1, Q2):
w0, x0, y0, z0 = Q2
w1, x1, y1, z1 = Q1
return np.array([-x1 * x0 - y1 * y0 - z1 * z0 + w1 * w0,
x1 * w0 + y1 * z0 - z1 * y0 + w1 * x0,
-x1 * z0 + y1 * w0 + z1 * x0 + w1 * y0,
x1 * y0 - y1 * x0 + z1 * w0 + w1 * z0])
def quat_mult_nnp(Q1,Q2):
w0, x0, y0, z0 = Q2
w1, x1, y1, z1 = Q1
return [-x1 * x0 - y1 * y0 - z1 * z0 + w1 * w0,
x1 * w0 + y1 * z0 - z1 * y0 + w1 * x0,
-x1 * z0 + y1 * w0 + z1 * x0 + w1 * y0,
x1 * y0 - y1 * x0 + z1 * w0 + w1 * z0]
# https://www.mathworks.com/help/aeroblks/quaternioninverse.html
def inverse(q):
# negate imaginary components to get inverse of unit quat
return quaternion(q[0],-q[1],-q[2],-q[3])
def conjugate(q):
# negate imaginary components to get inverse of unit quat
return quaternion(q[0],-q[1],-q[2],-q[3])
def rotate_vector(q, v):
q2 = [0, v[0],v[1],v[2]]
return quaternion_multiply(quaternion_multiply(q,q2), conjugate(q))[1:]
def rotate_vector_fast(q,v):
q2 = [0, v[0],v[1],v[2]]
return np.array(quat_mult_nnp(quat_mult_nnp(q,q2), [q[0],-q[1],-q[2],-q[3]])[1:])
def rotate_vector_standalone(q,v):
wxyz = q ** 2
wx,wy,wz = q[0] * q[1:]
xy = q[1] * q[2]
xz = q[1] * q[3]
yz = q[2] * q[3]
# Formula from http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/transforms/index.htm
# p2.x = w*w*p1.x + 2*y*w*p1.z - 2*z*w*p1.y + x*x*p1.x + 2*y*x*p1.y + 2*z*x*p1.z - z*z*p1.x - y*y*p1.x;
# p2.y = 2*x*y*p1.x + y*y*p1.y + 2*z*y*p1.z + 2*w*z*p1.x - z*z*p1.y + w*w*p1.y - 2*x*w*p1.z - x*x*p1.y;
# p2.z = 2*x*z*p1.x + 2*y*z*p1.y + z*z*p1.z - 2*w*y*p1.x - y*y*p1.z + 2*w*x*p1.y - x*x*p1.z + w*w*p1.z;
r1 = wxyz[0]*v[0] + 2*wy*v[2] - 2*wz*v[1] + \
wxyz[1]*v[0] + 2*xy*v[1] + 2*xz*v[2] - \
wxyz[3]*v[0] - wxyz[2]*v[0]
r2 = 2*xy*v[0] + wxyz[2]*v[1] + 2*yz*v[2] + \
2*wz*v[0] - wxyz[3]*v[1] + wxyz[0]*v[1] - \
2*wx*v[2] - wxyz[1]*v[1]
r3 = 2*xz*v[0] + 2*yz*v[1] + wxyz[3]*v[2] - \
2*wy*v[0] - wxyz[2]*v[2] + 2*wx*v[1] - \
wxyz[1]*v[2] + wxyz[0]*v[2]
return np.array([r1,r2,r3])
def rot_between(q1, q2):
"""Compute rotation quaternion from q1 to q2"""
# https://www.gamedev.net/forums/topic/423462-rotation-difference-between-two-quaternions/
return quaternion_multiply(inverse(q1), q2)
# https://en.wikipedia.org/wiki/Slerp
def slerp(v0, v1, t_array):
"""Spherical linear interpolation."""
# >>> slerp([1,0,0,0], [0,0,0,1], np.arange(0, 1, 0.001))
t_array = np.array(t_array)
v0 = np.array(v0)
v1 = np.array(v1)
dot = np.sum(v0 * v1)
if dot < 0.0:
v1 = -v1
dot = -dot
DOT_THRESHOLD = 0.9995
if dot > DOT_THRESHOLD:
result = v0[np.newaxis,:] + t_array[:,np.newaxis] * (v1 - v0)[np.newaxis,:]
return (result.T / np.linalg.norm(result, axis=1)).T
theta_0 = np.arccos(dot)
sin_theta_0 = np.sin(theta_0)
theta = theta_0 * t_array
sin_theta = np.sin(theta)
s0 = np.cos(theta) - dot * sin_theta / sin_theta_0
s1 = sin_theta / sin_theta_0
return (s0[:,np.newaxis] * v0[np.newaxis,:]) + (s1[:,np.newaxis] * v1[np.newaxis,:])
def single_slerp(v0, v1, t):
v0 = np.array(v0)
v1 = np.array(v1)
dot = np.sum(v0 * v1)
if dot < 0.0:
v1 = -v1
dot = -dot
DOT_THRESHOLD = 0.9995
if dot > DOT_THRESHOLD:
result = v0 + t * (v1 - v0)
return result / np.linalg.norm(result)
theta_0 = np.arccos(dot)
sin_theta_0 = np.sin(theta_0)
theta = theta_0 * t
sin_theta = np.sin(theta)
s0 = np.cos(theta) - dot * sin_theta / sin_theta_0
s1 = sin_theta / sin_theta_0
return (s0 * v0) + (s1 * v1)
def angle_between(q1, q2):
z = quaternion_multiply(inverse(q1), q2)
angle = 2 * np.arccos(min(z[0], 1))
return angle
if __name__ == "__main__":
import time
q = np.array([0.5,0.7,0.5,0.5])
v = np.array([1,2,3])
start = time.time()
for i in range(100000):
rotate_vector(q,v)
stop = time.time()
print((stop-start) * 1000)
start = time.time()
for i in range(100000):
rotate_vector_fast(q,v)
stop = time.time()
print((stop-start) * 1000)
print(rotate_vector_fast(q,v))
exit()
a = pyquaternion.Quaternion([1,0,0,0])
b = pyquaternion.Quaternion([0,1,0,0])
start = time.time()
for i in range(10000):
c = pyquaternion.Quaternion.slerp(a,b,0.5)
stop = time.time()
print((stop - start) * 1000)
a = quaternion(1, 0, 0, 0)
b = quaternion(0, 1, 0, 0)
start = time.time()
for i in range(10000):
c = slerp(a,b,[0.5])[0]
stop = time.time()
print((stop - start) * 1000)
a = quaternion(1, 0, 0, 0)
b = quaternion(0, 1, 0, 0)
start = time.time()
for i in range(10000):
c = single_slerp(a,b,0.5)
stop = time.time()
print((stop - start) * 1000)
for i in range(1):
a = np.random.random(4)
b = np.random.random(4)
c = single_slerp(a,b,0.5)
d = slerp(a,b,[0.5])[0]
print(d-c)