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

William Clark, Leonardo Colombo, Anthony Bloch

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.

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

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 -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.

Paper Structure

This paper contains 14 sections, 2 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hybrid Mechanical Systems with Symmetries
  3. Hybrid Hamiltonian systems
  4. Impact geometry on principal bundles
  5. Controlled impacts and problem statement
  6. Example: pendulum on a cart
  7. Interior and Exterior Impacts
  8. Interior impacts and geometric obstruction
  9. Exterior impacts and controlled resets
  10. Periodic Orbit Generation and Stability
  11. Floquet analysis of hybrid orbits
  12. Dissipation-assisted stabilization
  13. A geometric criterion for impact-induced stabilization
  14. Conclusions and Future Work

Key Result

Proposition 1

Let $\pi:Q\to Q/G$ be a principal $G$-bundle with mechanical connection $\mathcal{A}$, and let $\mathcal{S}_{\mathrm{ext}}\subset Q$ be a non-vertical impact surface. Assume that the controlled exterior impact map $\Delta_u:TQ|_{\mathcal{S}_{\mathrm{ext}}}\to TQ$ depends smoothly on a scalar control

Figures (4)

  • Figure 3: Pendulum on a cart. The cart position $x$ defines the symmetry direction, while $\theta$ is the pendulum angle and $\ell$ the rod length.
  • Figure 4: A periodic orbit generated by controlled exterior impacts. The distance between the two walls is $\varepsilon$.
  • Figure 5: Largest Floquet multiplier for the reset law under consideration. In the computation shown here, $\kappa_{p_x}=0$, so the control input depends only on $p_\theta$ and $\theta$.
  • Figure 6: Heat map indicating the number of laps completed before destabilization.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof