Wahba's problem

Applied mathematics 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 ( R ) = 1 2 ∑ k = 1 N a k ‖ w k − R v k ‖ 2 {\displaystyle J(\mathbf {R} )={\frac {1}{2}}\sum _{k=1}^{N}a_{k}\|\mathbf {w} _{k}-\mathbf {R} \mathbf {v} _{k}\|^{2}} for N ≥ 2 {\displaystyle N\geq 2} where w k {\displaystyle \mathbf {w} _{k}} is the k-th 3-vector measurement in the reference frame, v k {\displaystyle \mathbf {v} _{k}} is the corresponding k-th 3-vector measurement in the body frame and R {\displaystyle \mathbf {R} } is a 3 by 3 rotation matrix between the coordinate frames. a k {\displaystyle 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 methods based on the singular value decompo