Discrete optimal control on lie groups and applications to robotic vehicles

This paper is concerned with optimal trajectory generation for robotic multi-body systems. The focus is on discrete optimal control methods which operate intrinsically in the state space system manifold and do not require coordinate charts or projections. This is accomplished by defining both the dynamics and the optimal control solution as sequences of vector fields mapping to curves on the Lie group through retraction maps, and defining variations and differentiation with respect to such vector fields. As a result, standard trajectory optimization methods can be easily extended to the Lie group setting without loss of efficiency. The methods are illustrated with three numerical examples: a quadrotor, an aerial vehicle with manipulators, and a simple nonholonomic system.