Puneet Varma (Editor)

Wahba's problem

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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 ) = 1 2 ∑ k = 1 N a k | | w k − R v k | | 2

where w k is the k-th 3-vector measurement in the reference frame, v k is the corresponding k-th 3-vector measurement in the body frame and R is a 3 by 3 rotation matrix between the 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 = ∑ i = 1 n a i w i v i T

2. Find the singular value decomposition of B

B = U S V T

3. The rotation matrix is simply:

R = U M V T

where M = diag ⁡ ( [ 1 1 det ( U ) det ( V ) ] )

References

Wahba's problem Wikipedia
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