Attitude estimation based on solution of system of polynomials via homotopy continuation

The optimal attitude estimate is often defined as the solution to a minimization problem. When the objective function of the minimization problem is quadratic in the attitude matrix or equivalently quartic in the attitude quaternion, for example, in GPS attitude determination, gradient-based iterative algorithms are usually used, which can only find the local minimizer. A novel numerical method is presented that is guaranteed to find the global minimizer of the minimization problem. It first finds all the stationary points of the minimization problem by solving the polynomial equations satisfied by the stationary points using the homotopy continuation based solver and then chooses the global minimizer from them. Two numerical examples are used to show the viability of the method for attitude estimation.