jaxtransform3d.rotations.norm_matrix#

Orthonormalize rotation matrix.

A rotation matrix is defined as

\[\begin{split}\boldsymbol R = \left( \begin{array}{ccc} r_{11} & r_{12} & r_{13}\\ r_{21} & r_{22} & r_{23}\\ r_{31} & r_{32} & r_{33}\\ \end{array} \right) \in SO(3)\end{split}\]

and must be orthonormal, which results in 6 constraints:

column vectors must have unit norm (3 constraints)
and must be orthogonal to each other (3 constraints)

A more compact representation of these constraints is \(\boldsymbol R^T \boldsymbol R = \boldsymbol I\). In addition, to ensure right-handedness of the basis \(det(R) = 1\).

Because of numerical problems, a rotation matrix might not satisfy the constraints anymore. This function will enforce them with Gram-Schmidt orthonormalization optimized for 3 dimensions.

Parameters:

Rarray-like, shape (3, 3): Rotation matrix with small numerical errors.

Returns:

Rarray, shape (3, 3): Orthonormalized rotation matrix.

See also

robust_polar_decomposition: A more expensive orthonormalization method that spreads the error more evenly between the basis vectors.
norm_quaternion: Normalizes a quaternion.

Examples

>>> import jax.numpy as jnp
>>> from jaxtransform3d.rotations import norm_matrix
>>> norm_matrix(jnp.array([[0.5, 0., 0.], [0., 0.5, 0.], [0., 0., 0.5]]))
Array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]], dtype=float32)