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{{Expert-subject|Spaceflight|date=June 2011}}
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In applied mathematics, '''Wahba's problem''', first posed by [[Grace Wahba]] in 1965, seeks to find a [[rotation matrix]] ([[Orthogonal 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]] [[Flight dynamics (aircraft)|attitude determination]] utilising sensors such as [[magnetometer]]s and multi-antenna [[Global Positioning System|GPS receivers]]. The cost function that Wahba's problem seeks to minimise is as follows:
 
: <math>J(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k|| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k ||^2 </math>
 
where <math>\mathbf{w}_k</math> is a set of <math>k</math> vectors in the reference frame, <math>\mathbf{v}_k</math> is the corresponding set of vectors in the body frame and <math>\mathbf{R}</math> is the rotation matrix between coordinate frames. <math>a_k</math> 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 [http://www.control.auc.dk/~tb/best/aug23-Bak-svdalg.pdf Markley]
 
1. Obtain a matrix <math>\mathbf{B}</math> as follows:
 
<math>\mathbf{B} = \sum_{i=1}^{n} a_i \mathbf{w}_i {\mathbf{v}_i}^T</math>
 
2. Find the [[singular value decomposition]] of <math>\mathbf{B}</math>
 
<math>\mathbf{B} = \mathbf{U} \mathbf{S} \mathbf{V}^T</math>
 
3. The rotation matrix is simply:
 
<math>\mathbf{R} = \mathbf{U} \mathbf{M} \mathbf{V}^T</math>
 
where <math>\mathbf{M} = diag(\begin{bmatrix} 1 & 1 & det(\mathbf{U}) det(\mathbf{V})\end{bmatrix})</math>
 
== 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
 
== See also ==
* [[Kabsch algorithm]]
 
[[Category:Applied mathematics]]
 
 
{{Mathapplied-stub}}

Latest revision as of 19:23, 12 January 2015

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