Dissipation-assisted stabilization of periodic orbits via actuated exterior impacts in hybrid mechanical systems with symmetry

TL;DR

This paper explores how exterior impacts combined with dissipation can stabilize periodic orbits in hybrid mechanical systems, using a pendulum-on-a-cart as a case study.

Contribution

It introduces a novel approach of using exterior impacts with dissipation to achieve exponential stability in hybrid mechanical systems.

Findings

Exterior impacts can be exploited for control in hybrid systems.

Dissipation in continuous flow enhances stability of periodic orbits.

Reset actions alone are insufficient for stabilization.

Abstract

Impulsive mechanical systems exhibit discontinuous jumps in their state, and when such jumps are triggered by spatial events, the geometry of the impact surface carries information about the controllability of the hybrid dynamics. For mechanical systems defined on principal $G$-bundles, two qualitatively distinct types of impacts arise: interior impacts, associated with events on the shape space, and exterior impacts, associated with events on the fibers. A key distinction is that interior impacts preserve the mechanical connection, whereas exterior impacts generally do not. In this paper, we exploit this distinction by allowing actuation through exterior impacts. We study the pendulum-on-a-cart system, derive controlled reset laws induced by moving-wall impacts, and analyze the resulting periodic motions. Our results show that reset action alone does not provide a convincing stabilizing regime, whereas the addition of dissipation in the continuous flow yields exponentially stable periodic behavior for suitable feedback gains.