Dissipation-assisted stabilization of periodic orbits via actuated exterior impacts in hybrid mechanical systems with symmetry
This work addresses stabilization challenges in hybrid mechanical systems, such as robotics, but is incremental as it builds on known distinctions between impact types.
The paper tackled the problem of stabilizing periodic orbits in hybrid mechanical systems with symmetry by using actuated exterior impacts, finding that reset action alone was insufficient but adding dissipation in the continuous flow led to exponentially stable periodic behavior for appropriate feedback gains.
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.