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The Interplay Between Symmetries and Impact Effects on Hybrid Mechanical Systems

William Clark, Leonardo Colombo, Anthony Bloch

Abstract

Hybrid systems are dynamical systems with continuous-time and discrete-time components in their dynamics. When hybrid systems are defined on a principal bundle we are able to define two classes of impacts for the discrete-time transition of the dynamics: interior impacts and exterior impacts. In this paper we define hybrid systems on principal bundles, study the underlying geometry on the switching surface where impacts occur and we find conditions for which both exterior and interior impacts are preserved by the mechanical connection induced in the principal bundle.

The Interplay Between Symmetries and Impact Effects on Hybrid Mechanical Systems

Abstract

Hybrid systems are dynamical systems with continuous-time and discrete-time components in their dynamics. When hybrid systems are defined on a principal bundle we are able to define two classes of impacts for the discrete-time transition of the dynamics: interior impacts and exterior impacts. In this paper we define hybrid systems on principal bundles, study the underlying geometry on the switching surface where impacts occur and we find conditions for which both exterior and interior impacts are preserved by the mechanical connection induced in the principal bundle.
Paper Structure (16 sections, 6 theorems, 33 equations, 1 figure)

This paper contains 16 sections, 6 theorems, 33 equations, 1 figure.

Table of Contents

  1. Intoduction
  2. Motivating example: pendulum on the cart
  3. Interior Impacts
  4. Exterior Impacts
  5. Structure of the paper:
  6. Preliminaries on the geometric formulation of mechanical systems
  7. Variational Corner Conditions
  8. Lie group actions
  9. Principal Bundles
  10. Momentum Maps and the Mechanical Connection
  11. Hybrid mechanical systems
  12. Hybrid Noether theorem
  13. Hybrid Mechanical Systems on Principal Bundles
  14. Interior vs Exterior Impacts
  15. Interpretation for the Pendulum on a Cart
  16. ...and 1 more sections

Key Result

Lemma 1

$\mathcal{S}$ is vertical if and only if $\mathcal{S}$ is invariant under the group action of left translations, i.e. $h.\mathcal{S} = \mathcal{S}$ for all $h\in G$.

Figures (1)

  • Figure 1: The pendulum on the cart.

Theorems & Definitions (23)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: colombo2021generalized
  • Definition 5: colombo2021generalized
  • Definition 6
  • Definition 7
  • Lemma 1
  • proof
  • ...and 13 more