Wahba's problem

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 the k-th 3-vector measurement in the reference frame, ${\mathbf {v}}_{k}$ is the corresponding k-th 3-vector measurement in the body frame and $\mathbf {R}$ 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 $\mathbf {B}$ 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 ${\mathbf {M}}=\operatorname {diag}({\begin{bmatrix}1&1&\det({\mathbf {U}})\det({\mathbf {V}})\end{bmatrix}})$

References

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

Wahba's problem

Solution by Singular Value Decomposition

References

See also