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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:
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
1. Obtain a matrix as follows:
2. Find the singular value decomposition of
3. The rotation matrix is simply:
- Markley, F. L. and Crassidis, J. L. Fundamentals of Spacecraft Attitude Determination and Control, Springer 2014
- 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
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