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(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k|| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k ||^2

where \mathbf{w}_k is a set of k vectors in the reference frame, \mathbf{v}_k is the corresponding set of vectors in the body frame and \mathbf{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[edit]

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

1. Obtain a matrix \mathbf{B} as follows:

\mathbf{B} = \sum_{i=1}^{n} a_i \mathbf{w}_i {\mathbf{v}_i}^T

2. Find the singular value decomposition of \mathbf{B}

\mathbf{B} = \mathbf{U} \mathbf{S} \mathbf{V}^T

3. The rotation matrix is simply:

\mathbf{R} = \mathbf{U} \mathbf{M} \mathbf{V}^T

where \mathbf{M} = diag(\begin{bmatrix} 1 & 1 & det(\mathbf{U}) det(\mathbf{V})\end{bmatrix})

References[edit]

  • 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[edit]