Optimal control on Lie groups : implementations details of the projection operator approach

This paper discusses key implementation details required for computing the solution of a continuous-time optimal control problem on a Lie group using the projection operator approach. In particular, we provide the explicit formulas to compute the time-varying linear quadratic problem which defines the search direction step of the algorithm. We also show that the projection operator approach on Lie groups generates a sequence of adjoint state trajectories that converges, as a local minimum is approached, to the adjoint state trajectory of the first order necessary conditions of the Pontryagin's Maximum Principle, placing it between direct and indirect optimization methods. As illustrative example, an optimization problem on SO(3) is introduced and numerical results of the projection operator approach are presented, highlighting second order converge rate of the method.