Stability and Control of Chaplygin Beanies Coupled to a Platform Through Nonholonomic Constraints

Abstract

Many multi-agent systems in nature are comprised of agents that interact with, and respond to, the dynamics of their environment. In this paper, we approach the study of such agent-environment interactions through the study of passively compliant vehicles coupled to their environment via simple nonholonomic constraints. We first consider a single passively compliant Chaplygin beanie atop a platform having translational compliance, introduce the reduced equations for the system using the notion of nonholonomic momentum, and provide proof for its stability under arbitrary deformations of the elastic element modeling its compliance. We then direct our focus to results concerning the frequency response and control of passive Chaplygin beanies under actuation of the platform, discuss rich dynamical features arising from periodic actuation, and develop rules by which control can be exerted to collect and disperse multiple passive vehicles. We then discuss how the latter of these results clarifies the extent to which stable behavior can be excited in the system through exogenous control.