Geometric descent algorithms for attitude determination using the global positioning system

This paper describes a set of numerical gradient-based optimization algorit hms for solving the Global Positioning System (GPS)-based attitude determin ation problem. We pose the problem as one of minimizing the function tr(The ta N Theta(T) Q - 2 Theta W) with respect to the rotation matrix Theta, whe re N, Q, and W are given 3 x 3 matrices, and tr(.) denotes the matrix trace . Both the method of steepest descent and Newton's method are generalized t o the rotation group by taking advantage of its underlying Lie group struct ure, Analytic solutions to the line search procedure are also derived, Resu lts of numerical experiments for the class of geometric descent algorithms proposed here are presented and compared with those of traditional vector s pace-based constrained optimization algorithms.