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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:

$J(𝐑)=\frac{1}{2}\sum _{k=1}^N{a}_k||{𝐰}_k-𝐑{𝐯}_k|{|}^2$

where ${𝐰}_k$ is a set of $k$ vectors in the reference frame, ${𝐯}_k$ is the corresponding set of vectors in the body frame and $𝐑$ is the rotation matrix between coordinate frames. $a_k$ 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 $𝐁$ as follows:

$𝐁=\sum _{i=1}^n{a}_i{𝐰}_i{{𝐯}_i}^T$

2. Find the singular value decomposition of $𝐁$

$𝐁=𝐔𝐒{𝐕}^T$

3. The rotation matrix is simply:

$𝐑=𝐔𝐌{𝐕}^T$

where $𝐌=diag(\left[\begin{array}{ccc}1& 1& det(𝐔)det(𝐕)\end{array}\right])$

References

• Markley, F. L. Attitude Determination using Vector Observations and the Singular Value Decomposition Journal of the Astronautical Sciences, 1988, 38, 245-258

• Wahba, G. Problem 65–1: A Least Squares Estimate of Spacecraft Attitude, SIAM Review, 1965, 7(3), 409

See also

• Kabsch algorithm

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