Wahba's problem

In applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix (special orthogonal matrix) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers and multi-antenna GPS receivers. The cost function that Wahba's problem seeks to minimise is as follows:

where is the k-th 3-vector measurement in the reference frame, is the corresponding k-th 3-vector measurement in the body frame and is a 3 by 3 rotation matrix between the coordinate frames. is an optional set of weights for each observation.

A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and singular value decomposition-based methods.

Solution by Singular Value Decomposition

One solution can be found using a singular value decomposition as reported by Markley

1. Obtain a matrix as follows:

2. Find the singular value decomposition of

3. The rotation matrix is simply:

where

References

See also


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