Wahba's problem
Encyclopedia
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 magnetometer
s and multi-antenna GPS receivers
. The cost function that Wahba's problem seeks to minimise is as follows:
where
is a set of 
vectors in the reference frame, 
is the corresponding set of vectors in the body frame and 
is the rotation matrix between coordinate frames. 
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.
as reported by Markley
1. Obtain a matrix
as follows:
2. Find the SVD of
3. The rotation matrix is simply:
where
Grace Wahba
Grace Wahba is the I. J. Schoenberg Professor of Statistics at the University of Wisconsin–Madison. She is a pioneer in methods for smoothing noisy data...
in 1965, seeks to find a rotation matrix (special orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
attitude determination utilising sensors such as magnetometer
Magnetometer
A magnetometer is a measuring instrument used to measure the strength or direction of a magnetic field either produced in the laboratory or existing in nature...
s and multi-antenna GPS receivers
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
. The cost function that Wahba's problem seeks to minimise is as follows:
where
is a set of vectors in the reference frame, is the corresponding set of vectors in the body frame and is the rotation matrix between coordinate frames. 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
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
-based methods.
Solution by Singular Value Decomposition
One solution can be found using a Singular value decompositionSingular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
as reported by Markley
1. Obtain a matrix
as follows:2. Find the SVD of
3. The rotation matrix is simply:
where