Aspects of globally-stable tracking for certain classes of simple mechanical systems

Abstract

Proportional derivative plus feed forward (PD+FF) feedback control laws exist to locally asymptotically track a smooth reference trajectory having bounded velocity for a fully actuated simple mechanical system (SMS) on a Riemannian manifold. Almost-global asymptotically stable (AGAS) tracking for an SMS on a compact Lie group is also addressed in the literature by observing that the closed loop error dynamics is obtained by lifting the gradient vector field of a navigation function and stabilizing the error dynamics about the Lie group identity. Almost-global setpoint stabilization for an SMS on a Riemannian manifold is often tackled by introducing a gradient field of a Morse function as a feedback control force for the dissipative SMS. In this paper we address the almost-global tracking problem for an SMS on a compact Riemannian manifold embedded in Euclidean space by explicitly introducing the error dynamics. A control law, including both feedback and feedforward terms, is then synthesized for the purpose of almost-global stabilization of the error dynamics.