def jac_left(xi_vec):
"""Compute the left derivative of Exp(xi_vec) with respect to xi_vec.
:param xi_vec: The tangent space 6D column vector xi_vec = [rho_vec, theta_vec]^T.
:return: The Jacobian (6x6 matrix)
"""
theta_vec = xi_vec[3:]
J_l_theta = SO3.jac_left(theta_vec)
Q_l = SE3._Q_left(xi_vec)
return np.block([[J_l_theta, Q_l], [np.zeros((3, 3)), J_l_theta]])
def test_jacobian_right_inverse():
theta_vec = 3 * np.pi / 4 * np.array([[1, -1, 0]]).T / np.sqrt(2)
X = SO3.Exp(theta_vec)
J_r_inv = SO3.jac_right_inverse(theta_vec)
# Should have J_l * J_r_inv = Exp(theta_vec).adjoint().
J_l = SO3.jac_left(theta_vec)
np.testing.assert_almost_equal(J_l @ J_r_inv,
SO3.Exp(theta_vec).adjoint(), 14)
# Test the Jacobian numerically.
delta = 1e-3 * np.ones((3, 1))
taylor_diff = X.oplus(delta).Log() - (X.Log() + J_r_inv @ delta)
np.testing.assert_almost_equal(taylor_diff, np.zeros((3, 1)), 5)
def test_jacobian_left():
theta_vec = np.pi / 4 * np.array([[-1, -1, 1]]).T / np.sqrt(3)
# Should have J_l(theta_vec) == J_r(-theta_vec).
np.testing.assert_almost_equal(SO3.jac_left(theta_vec),
SO3.jac_right(-theta_vec), 14)