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:

: $J(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k\| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k \|^2$ for $N\geq2$

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.The rotation $\mathbf{R}$ in the problem's definition transforms the body frame to the reference frame. Most publications define rotation in the reverse direction, i.e. from the reference to the body frame which amounts to $\mathbf{R}^T$ .

$a_k$ 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,{{Cite web |title=Davenport's Q-method (Finding an orientation matching a set of point samples) |url=https://math.stackexchange.com/q/1634113 |access-date=2020-07-23 |website=Mathematics Stack Exchange}} QUEST and methods based on the singular value decomposition (SVD). Several methods for solving Wahba's problem are discussed by Markley and Mortari.

This is an alternative formulation of the orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the alternative formulation). A compact and elegant derivation can be found in Appel (2015).{{Cite journal|last=Appel|first=M.|title=Robust Spoofing Detection and Mitigation based on Direction of Arrival Estimation|url=https://elib.dlr.de/99721/1/appel_ion_2015_v05.pdf|journal=Ion GNSS+ 2015|volume=28}}

Solution via SVD

One solution can be found using a singular value decomposition (SVD).

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

Notes

References

Wahba, G. Problem 65–1: [https://epubs.siam.org/doi/abs/10.1137/1007077 A Least Squares Estimate of Satellite Attitude], SIAM Review, 1965, 7(3), 409
Shuster, M. D. and Oh, S. D. [https://web.archive.org/web/20181121070028/http://www.malcolmdshuster.com/Pub_1981a_J_TRIAD-QUEST_scan.pdf Three-Axis Attitude Determination from Vector Observations], Journal of Guidance and Control, 1981, 4(1):70–77
Markley, F. L. [https://www.researchgate.net/profile/Landis_Markley/publication/243753921_Attitude_Determination_Using_Vector_Observations_and_the_Singular_Value_Decomposition/links/00463533d8a184d09d000000.pdf Attitude Determination using Vector Observations and the Singular Value Decomposition], Journal of the Astronautical Sciences, 1988, 38:245–258
Markley, F. L. and Mortari, D. [https://web.archive.org/web/20171201030630/https://pdfs.semanticscholar.org/f151/97510b1e501876a6f6d8683ac69ab1ef8d39.pdf Quaternion Attitude Estimation Using Vector Observations], Journal of the Astronautical Sciences, 2000, 48(2):359–380
Markley, F. L. and Crassidis, J. L. [http://ancs.eng.buffalo.edu/index.php/Fundamentals_of_Spacecraft_Attitude_Determination_and_Control Fundamentals of Spacecraft Attitude Determination and Control], Springer 2014
Libbus, B. and Simons, G. and Yao, Y. [https://www.researchgate.net/publication/313483723_Rotating_Multiple_Sets_of_Labeled_Points_to_Bring_Them_Into_Close_Coincidence_A_Generalized_Wahba_Problem Rotating Multiple Sets of Labeled Points to Bring Them Into Close Coincidence: A Generalized Wahba Problem], The American Mathematical Monthly, 2017, 124(2):149–160
Lourakis, M. and Terzakis, G. [https://www.researchgate.net/publication/326683452_Efficient_Absolute_Orientation_Revisited Efficient Absolute Orientation Revisited], IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2018, pp. 5813-5818.

Wahba's problem

Solution via SVD

Notes

References

See also