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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 a set of vectors in the reference frame, is the corresponding set of vectors in the body frame and is the rotation matrix between 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. 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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