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transform.py
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transform.py
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import numpy as np
import math
import robosuite.utils.transform_utils as T
def skew_sym(x):
""" convert 3D vector to skew-symmetric matrix form """
x1, x2, x3 = x.ravel()
return np.array([
[0, -x3, x2],
[x3, 0, -x1],
[-x2, x1, 0]
])
def rand_quat():
""" uniform sampling of unit quaternion """
u1 = np.random.uniform(0, 1)
u2 = np.random.uniform(0, 1)
u3 = np.random.uniform(0, 1)
q1 = np.sqrt(1-u1) * np.sin(2*np.pi*u2)
q2 = np.sqrt(1-u1) * np.cos(2*np.pi*u2)
q3 = np.sqrt(u1) * np.sin(2*np.pi*u3)
q4 = np.sqrt(u1) * np.cos(2*np.pi*u3)
return np.array((q1, q2, q3, q4))
q = rand_quat()
R = T.quat2mat(q)
p = np.array([1, 2, 3])
def pose2mat(R, p):
""" convert pose to transformation matrix """
p0 = p.ravel()
H = np.block([
[R, p0[:, np.newaxis]],
[np.zeros(3), 1]
])
return H
def mat2pose(T):
""" convert transformation matrix T to pose """
R = T[:3,:3]
p = T[:3,3]
return (R, p)
def adjoint(T):
""" adjoint representation of transformation """
R, p = mat2pose(T)
pR = np.matmul(skew_sym(p), R)
return np.block([
[R, np.zeros((3, 3))],
[pR, R],
])
def exp2rot(w, theta):
"""Matrix exponential of rotations (Rodrigues' Formula)
Convert exponential coordinates to rotation matrix
"""
ss_w = skew_sym(w)
R = np.eye(3) + np.sin(theta) * ss_w + (1-np.cos(theta)) * np.matmul(ss_w, ss_w)
return R
def rot2exp(R):
"""Matrix logarithm of rotations
Convert rotation matrix to exponential coordinates
"""
if np.allclose(R, np.eye(3)):
return (np.zeros(3), 0) # w is undefined
if np.isclose(np.trace(R), -1):
if not np.isclose(R[2][2], -1):
w = np.array([R[0][2], R[1][2], R[2][2] + 1])
w /= np.sqrt(2 * (1 + R[2][2]))
return (w, np.pi)
elif not np.isclose(R[1][1], -1):
w = np.array([R[0][1], R[1][1] + 1, R[2][1]])
w /= np.sqrt(2 * (1 + R[1][1]))
return (w, np.pi)
else:
w = np.array([R[0][0] + 1, R[1][0], R[2][0]])
w /= np.sqrt(2 * (1 + R[0][0]))
return (w, np.pi)
theta = np.arccos(0.5 * (np.trace(R) - 1))
ss_w = (R - R.T) / (2 * np.sin(theta))
w = np.array([ss_w[2][1], ss_w[0][2], ss_w[1][0]])
return (w, theta)
def exp2mat(w, v, theta):
"""Matrix exponential of rigid-body motions
Convert exponential coordinates to transformation matrix
"""
w_norm = np.linalg.norm(w)
v_norm = np.linalg.norm(v)
if np.isclose(w_norm, 0):
assert np.isclose(v_norm, 1), 'norm(v) must be 1'
new_v = v.ravel() * theta
return np.vstack([
np.hstack([
exp2rot(w, theta), new_v[:,np.newaxis]
]),
np.array([[0, 0, 0, 1]]),
])
assert np.isclose(w_norm, 1), 'norm(w) must be 1'
ss_w = skew_sym(w)
new_v = (np.eye(3)*theta + (1-np.cos(theta))*ss_w + (theta-np.sin(theta))*np.matmul(ss_w, ss_w)).dot(v)
return np.vstack([
np.hstack([
exp2rot(w, theta), new_v[:,np.newaxis]
]),
np.array([[0, 0, 0, 1]]),
])
def mat2exp(T):
"""Matrix logarithm of rigid-body motions
Convert transformation matrix to exponential coordinates
"""
R, p = mat2pose(T)
if np.allclose(R, np.eye(3)):
p_norm = np.linalg.norm(p)
w = np.zeros(3)
return (w, p/p_norm, p_norm)
w, theta = rot2exp(R)
ss_w = skew_sym(w)
G_inv = 1/theta*np.eye(3) - 0.5*ss_w + (1/theta-0.5/np.tan(theta/2))*np.matmul(ss_w, ss_w)
v = G_inv.dot(p)
return (w, v, theta)