Kernel

Kernels are mostly internal data structures for dealing with the exponential map and logarithm of Lie groups such as $SO(3)$ and $SE(3)$ . The gtsam.so3 classes evaluate the scalar functions used by the $SO(3)$ exponential-map Jacobian and its inverse: Kernel represents a forward kernel, while InvJKernel couples the inverse Jacobian to its forward counterpart for stable evaluation.

Authors: Frank Dellaert, et al. (see THANKS for the full author list)

See LICENSE for the license information

try:
    import google.colab
    %pip install --quiet gtsam-develop
except ImportError:
    pass

import gtsam
import numpy as np

Obtaining the kernels¶

Users normally obtain these objects from so3.DexpFunctor, which precomputes coefficients for a rotation vector and handles the near-zero series expansions.

omega = np.array([0.1, -0.2, 0.3])
dexp = gtsam.so3.DexpFunctor(omega)
jacobian_kernel = dexp.Jacobian()
inverse_kernel = dexp.InvJacobian()

`so3.Kernel`¶

left() and right() return the left and right Jacobian matrices. applyLeft(v) and applyRight(v) apply those matrices to a vector without requiring callers to form the product themselves. frechet() and applyFrechet() expose derivatives of the kernel action.

J_left = jacobian_kernel.left()
vector = np.array([1.0, 2.0, -1.0])
np.testing.assert_allclose(
    jacobian_kernel.applyLeft(vector), J_left @ vector, atol=1e-12
)
print("left Jacobian:")
print(J_left)

`so3.InvJKernel`¶

left() and right() return inverse Jacobians, and applyLeft()/applyRight() apply them. Its public J property retains the associated forward kernel.

J_left_inverse = inverse_kernel.left()
np.testing.assert_allclose(J_left_inverse @ J_left, np.eye(3), atol=1e-12)
np.testing.assert_allclose(
    inverse_kernel.applyLeft(vector), J_left_inverse @ vector, atol=1e-12
)

Source¶

Kernel.h

AI assistance caveat¶

AI was used to help draft this documentation, and inaccuracies could be present.

Obtaining the kernels¶

so3.Kernel¶

so3.InvJKernel¶

Source¶

AI assistance caveat¶

`so3.Kernel`¶

`so3.InvJKernel`¶