classmethod Rotation.match_vectors(a, b, weights=None, normalized=False)[source]

Estimate a rotation to match two sets of vectors.

Find a rotation between frames A and B which best matches a set of unit vectors a and b observed in these frames. The following loss function is minimized to solve for the direction cosine matrix \(C\):

\[L(C) = \frac{1}{2} \sum_{i = 1}^{n} w_i \lVert \mathbf{a}_i - C \mathbf{b}_i \rVert^2 ,\]

where \(w_i\)’s are the weights corresponding to each vector.

The rotation is estimated using Markley’s SVD method [1].

aarray_like, shape (N, 3)

Vector components observed in initial frame A. Each row of a denotes a vector.

barray_like, shape (N, 3)

Vector components observed in another frame B. Each row of b denotes a vector.

weightsarray_like shape (N,), optional

Weights describing the relative importance of the vectors in a. If None (default), then all values in weights are assumed to be equal.

normalizedbool, optional

If True, assume input vectors a and b to have unit norm. If False, normalize a and b before estimating rotation. Default is False.

estimated_rotationRotation instance

Best estimate of the rotation that transforms b to a.

sensitivity_matrixndarray, shape (3, 3)

Scaled covariance of the attitude errors expressed as the small rotation vector of frame A. Multiply with harmonic mean [3] of variance in each observation to get true covariance matrix. The error model is detailed in [2].



F. Landis Markley, “Attitude determination using vector observations: a fast optimal matrix algorithm”, Journal of Astronautical Sciences, Vol. 41, No.2, 1993, pp. 261-280.


F. Landis Markley, “Attitude determination using vector observations and the Singular Value Decomposition”, Journal of Astronautical Sciences, Vol. 38, No.3, 1988, pp. 245-258.