def test_creation(self):
# from dual quaternion array: careful, need to supply a normalized DQ
dql = np.array([
0.7071067811, 0.7071067811, 0, 0, -3.535533905, 3.535533905,
1.767766952, -1.767766952
])
dq1 = DualQuaternion.from_dq_array(dql)
dq2 = DualQuaternion.from_dq_array(dql)
self.assertEqual(dq1, dq2)
# from quaternion + translation array
dq3 = DualQuaternion.from_quat_pose_array(
np.array([1, 2, 3, 4, 5, 6, 7]))
dq4 = DualQuaternion.from_quat_pose_array([1, 2, 3, 4, 5, 6, 7])
self.assertEqual(dq3, dq4)
# from homogeneous transformation matrix
T = np.array([[1, 0, 0, 2], [0, 1, 0, 3], [0, 0, 1, 1], [0, 0, 0, 1]])
dq7 = DualQuaternion.from_homogeneous_matrix(T)
self.assertEqual(dq7.q_r, quaternion.one)
self.assertEqual(dq7.translation(), [2, 3, 1])
try:
np.testing.assert_array_almost_equal(dq7.homogeneous_matrix(), T)
except AssertionError as e:
self.fail(e)
# from a point
dq8 = DualQuaternion.from_translation_vector([4, 6, 8])
self.assertEqual(dq8.translation(), [4, 6, 8])
def test_screw(self):
# test unit
l, m, theta, d = self.identity_dq.screw()
self.assertEqual(d, 0)
self.assertEqual(theta, 0)
# test pure translation
trans = [10, 5, 0]
dq_trans = DualQuaternion.from_translation_vector(trans)
l, m, theta, d = dq_trans.screw()
self.assertAlmostEqual(d, np.linalg.norm(trans), 2)
self.assertAlmostEqual(theta, 0)
# test pure rotation
theta1 = np.pi / 2
dq_rot = DualQuaternion.from_dq_array(
[np.cos(theta1 / 2),
np.sin(theta1 / 2), 0, 0, 0, 0, 0, 0])
l, m, theta, d = dq_rot.screw()
self.assertAlmostEqual(theta, theta1)
# test simple rotation and translation: rotate in the plane of a coordinate system with the screw axis offset
# along +y. Rotate around z axis so that the coordinate system stays in the plane. Translate along z-axis
theta2 = np.pi / 2
dq_rot2 = DualQuaternion.from_dq_array(
[np.cos(theta2 / 2), 0, 0,
np.sin(theta2 / 2), 0, 0, 0, 0])
dist_axis = 5.
displacement_z = 3.
dq_trans = DualQuaternion.from_translation_vector([
dist_axis * np.sin(theta2), dist_axis * (1. - np.cos(theta2)),
displacement_z
])
dq_comb = dq_trans * dq_rot2
l, m, theta, d = dq_comb.screw()
try:
# the direction of the axis should align with the z axis of the origin
np.testing.assert_array_almost_equal(l,
np.array([0, 0, 1]),
decimal=3)
# m = p x l with p any point on the line
np.testing.assert_array_almost_equal(
np.cross(np.array([[0, dist_axis, 0]]), l).flatten(), m)
except AssertionError as e:
self.fail(e)
self.assertAlmostEqual(
d, displacement_z) # only displacement along z should exist here
self.assertAlmostEqual(theta, theta2) # the angles should be the same
def test_homogeneous_conversion(self):
# 1. starting from a homogeneous matrix
theta1 = np.pi / 2 # 90 deg
trans = [10., 5., 0.]
H1 = np.array([[1., 0., 0., trans[0]],
[0., np.cos(theta1), -np.sin(theta1), trans[1]],
[0., np.sin(theta1),
np.cos(theta1), trans[2]], [0., 0., 0., 1.]])
# check that if we convert to DQ and back to homogeneous matrix, we get the same result
double_conv1 = DualQuaternion.from_homogeneous_matrix(
H1).homogeneous_matrix()
try:
np.testing.assert_array_almost_equal(H1, double_conv1)
except AssertionError as e:
self.fail(e)
# check that dual quaternions are also equal
dq1 = DualQuaternion.from_homogeneous_matrix(H1)
dq_double1 = DualQuaternion.from_homogeneous_matrix(double_conv1)
self.assertEqual(dq1, dq_double1)
# 2. starting from a DQ
dq_trans = DualQuaternion.from_translation_vector([10, 5, 0])
dq_rot = DualQuaternion.from_dq_array(
[np.cos(theta1 / 2),
np.sin(theta1 / 2), 0, 0, 0, 0, 0, 0])
dq2 = dq_trans * dq_rot
# check that this is the same as the previous DQ
self.assertEqual(dq2, dq1)
# check that if we convert to homogeneous matrix and back, we get the same result
double_conv2 = DualQuaternion.from_homogeneous_matrix(
dq2.homogeneous_matrix())
self.assertEqual(dq2, double_conv2)
def blend_poses(poses: Sequence[np.ndarray]) -> np.ndarray:
"""Blend a list of 4x4 transformation matrices together using dual
quaternion blending.
See: https://www.cs.utah.edu/~ladislav/kavan06dual/kavan06dual.pdf
:param poses: A list of poses to blend.
:return: The result of DQB applied on the input poses.
"""
dq = DualQuaternion.from_dq_array([0, 0, 0, 0, 0, 0, 0])
# We use a constant weight for all poses.
weight = 1.0 / len(poses)
for p in poses:
dq += weight * DualQuaternion.from_homogeneous_matrix(p)
dq.normalize()
return dq.homogeneous_matrix()