Control of a Class of Underactuated Systems by Successive Submanifold Stabilization

Abstract

Many control design methods for underactuated systems require solving a partial differential equation, which can be complex for systems with many degrees of freedom. In order to reduce the complexity, it is proposed to decompose the dynamics into several subsystems. The problem then reduces to the successive stabilization of the individual subsystems, i.e., each step is a submanifold stabilization problem of reduced dimension. In this way, control methods which are only practicable for lower dimensional systems can be applied to the overall complex dynamical system. To ensure that the subsystems can be stabilized independently, the dynamics are transformed by a change of coordinates to a form with block-diagonal inertia metric. For the unactuated part kinetic symmetries can be utilized, whereas for the actuated part null space projectors are employed to decouple the dynamics with respect to the inertia metric. The subsystems are then stabilized by optimal control or PD-like feedback. In the stability analysis semidefinite Lyapunov functions are employed. The procedure is demonstrated for a manipulator on an elastic base and validated in simulation.