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

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

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

Wahba's problem

Solution by Singular Value Decomposition

References

See also