Geometric descent algorithms for attitude determination using the global positioning system

This paper describes a set of numerical gradient-based optimization algorithms for solving the Global Positioning System (GPS)-based attitude determination problem. We pose the problem as one of minimizing the function tr(Theta N Theta(T) Q - 2 Theta W) with respect to the rotation matrix Theta, where 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 to the rotation group by taking advantage of its underlying Lie group structure, Analytic solutions to the line search procedure are also derived, Results of numerical experiments for the class of geometric descent algorithms proposed here are presented and compared with those of traditional vector space-based constrained optimization algorithms.