Chordal Distance

ahrs.utils.metrics.chordal(R1: ndarray, R2: ndarray) → float | ndarray

The chordal distance between two rotations \(\mathbf{R}_1\) and \(\mathbf{R}_2\) in SO(3) is the Euclidean distance between them in the embedding space \(\mathbb{R}^{3\times 3}=\mathbb{R}^9\) [HTDL13]:

\[d(\mathbf{R}_1, \mathbf{R}_2) = \|\mathbf{R}_1-\mathbf{R}_2\|_F\]

where \(\|\mathbf{X}\|_F\) represents the Frobenius norm of the matrix \(\mathbf{X}\).

The error lies within: [0, \(2\sqrt{3}\)]

Parameters:

• R1 (numpy.ndarray) – 3-by-3 rotation matrix.

• R2 (numpy.ndarray) – 3-by-3 rotation matrix.

Returns:

d – Chordal distance between matrices.

Return type:

float or numpy.ndarray

Raises:

• ValueError – If the rotation matrices are not of the same shape.

• TypeError – If the rotation matrices are not of type list or numpy.ndarray.

Examples

>>> import ahrs
>>> R1 = ahrs.DCM(rpy=[0.0, 0.0, 0.0])
>>> R2 = ahrs.DCM(rpy=[90.0, 90.0, 90.0])
>>> ahrs.utils.chordal(R1, R2)
2.0
>>> R1 = ahrs.DCM(rpy=[10.0, -20.0, 30.0])
>>> R2 = ahrs.DCM(rpy=[-10.0, 20.0, -30.0])
>>> ahrs.utils.chordal(R1, R2)
1.6916338074634352