Dissipation-Induced Self-Recovery in Systems on Principal Bundles

Abstract

The “self-recovery” phenomenon is a seemingly curious property of certain underactuated dissipative systems in which dissipative forces always push the system to a pre-determined equilibrium state dependent on the initial conditions. The systems for which this has been studied are Abelian, with all system velocity interactions due entirely to inertial effects. In this paper we also consider Abelian systems, but in the context of principal bundles, and introduce drag in addition to inertial interactions, allowing us to show that the same conservation that induces self-recovery now depends on the trajectories of the system inputs in addition to initial conditions. We conclude by demonstrating an example illustrating the conditions derived from our proof, along with an observation that the present analysis is insufficient for self-recovery in non-Abelian systems.