Controllability of a class of underactuated mechanical systems with symmetry

Abstract

In this paper we develop results based on geometric mechanics to study the controllability of a class of controlled under-actuated left invariant mechanical systems on Lie groups. We exploit the invariance of the controlled nonlinear dynamics to the group action (symmetry) to derive a set of reduced dynamics for the system. We first present sufficient conditions for the controllability of the reduced dynamics. We prove conditions (boundedness of coadjoint orbits and existence of a radially unbounded Lyapunov function) under which the drift vector field (of the reduced system) is weakly positively Poisson stable (WPPS). The WPPS nature of the drift vector field along with the Lie algebra rank condition is used to show controllability of the reduced system. We then extend these results to the unreduced dynamics, considering separately the cases when the symmetry group is compact and noncompact. In the noncompact case, under further assumptions of equilibrium controllability, sufficient conditions for controllability of the unreduced dynamics are derived. Jetpuck (hovercraft) and underwater vehicles are used as mechanical systems to motivate our work and illustrate our theoretical results.