Discrete Optimal Control of Interconnected Mechanical Systems

Abstract

This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. Exploiting the left-trivialization of the cotangent bundle, and assuming the time-step of discrete evolution is small enough to exploit the diffeomorphism feature of the exponential map in a neighbourhood of the identity of the Lie group, that enables a mapping of the group variables to the Lie algebra, a variational approach is adopted to obtain the first order necessary conditions that characterise optimal trajectories. The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments.