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Lie Group Variational Collision Integrators for a Class of Hybrid Systems

Khoa Tran, Melvin Leok

TL;DR

This work develops Lie Group Variational Collision Integrators (LGVCI) for a class of hybrid systems where a convex rigid body elastically bounces on a plane. By formulating the dynamics on the configuration space $SE(3)$ and using nonsmooth Lagrangian mechanics, it derives both continuous-time jump conditions and discrete-time LGVCI that are symplectic and momentum-preserving, delivering near-energy conservation over long times. A central innovation is the collision-detection function $\Phi$ built from a signed distance formulation, enabling robust handling of corner impacts via $\epsilon$-rounding and avoiding nonsmooth convex analysis. The framework is extended to tilted planes and unions/intersections of convex bodies, with detailed numerical experiments validating energy transfer, Zeno considerations, and sensitivity to initial conditions, highlighting the practical impact for geometric mechanics and physics-based simulation on SE(3).

Abstract

The problem of 3-dimensional, convex rigid-body collision over a plane is fully investigated; this includes bodies with sharp corners that is resolved without the need for nonsmooth convex analysis of tangent and normal cones. In particular, using nonsmooth Lagrangian mechanics, the equations of motion and jump equations are derived, which are largely dependent on the collision detection function. Following the variational approach, a Lie group variational collision integrator (LGVCI) is systematically derived that is symplectic, momentum-preserving, and has excellent long-time, near energy conservation. Furthermore, systems with corner impacts are resolved adeptly using $ε$-rounding on the sign distance function (SDF) of the body. Extensive numerical experiments are conducted to demonstrate the conservation properties of the LGVCI.

Lie Group Variational Collision Integrators for a Class of Hybrid Systems

TL;DR

This work develops Lie Group Variational Collision Integrators (LGVCI) for a class of hybrid systems where a convex rigid body elastically bounces on a plane. By formulating the dynamics on the configuration space and using nonsmooth Lagrangian mechanics, it derives both continuous-time jump conditions and discrete-time LGVCI that are symplectic and momentum-preserving, delivering near-energy conservation over long times. A central innovation is the collision-detection function built from a signed distance formulation, enabling robust handling of corner impacts via -rounding and avoiding nonsmooth convex analysis. The framework is extended to tilted planes and unions/intersections of convex bodies, with detailed numerical experiments validating energy transfer, Zeno considerations, and sensitivity to initial conditions, highlighting the practical impact for geometric mechanics and physics-based simulation on SE(3).

Abstract

The problem of 3-dimensional, convex rigid-body collision over a plane is fully investigated; this includes bodies with sharp corners that is resolved without the need for nonsmooth convex analysis of tangent and normal cones. In particular, using nonsmooth Lagrangian mechanics, the equations of motion and jump equations are derived, which are largely dependent on the collision detection function. Following the variational approach, a Lie group variational collision integrator (LGVCI) is systematically derived that is symplectic, momentum-preserving, and has excellent long-time, near energy conservation. Furthermore, systems with corner impacts are resolved adeptly using -rounding on the sign distance function (SDF) of the body. Extensive numerical experiments are conducted to demonstrate the conservation properties of the LGVCI.
Paper Structure (47 sections, 8 theorems, 96 equations, 8 figures, 1 table)

This paper contains 47 sections, 8 theorems, 96 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. The Problem: Dynamics of a Bouncing Ellipsoid
  5. Notation
  6. Skew Map, Trace, and Inner products
  7. Special Euclidean Group SE3
  8. Distance between an arbitrary ellipsoid and the plane
  9. Background
  10. Continuous-Time Model.
  11. Legendre Transform
  12. Discrete-Time Model
  13. Discrete Legendre Transforms
  14. Dynamics of the Bouncing Ellipsoid
  15. Equations of Motion: Euler--Lagrange Equations
  16. ...and 32 more sections

Key Result

Proposition 2.1

\newlabelprop:traceProp0 For all skew-symmetric matrices $\Omega \in \mathbb{R}^{3 \times 3}$, $\mathop{\mathrm{tr }}\limits [\Omega^T P] = 0$ if and only if $P \in \mathbb{R}^{3 \times 3}$ is symmetric, i.e., $\mathop{\mathrm{Asym}}\limits(P) = 0$.

Figures (8)

  • Figure 1: The discrete variational principle for collisions
  • Figure 1: Examples of neighborhood with non-unique closest points on convex bodies with $C^\infty$ boundary.
  • Figure 1: Snapshots of configurations, including some collisions, of the four hybrid systems.
  • Figure 2: $\epsilon$-rounding figures
  • Figure 2: Plots of total energy (T.E.), translational $+$ potential energy (T.P.E), and rotational energy (R.E) of different hybrid systems demonstrating the exchange of energies between T.P.E and R.E. after each collision.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Proof 1
  • Remark 4.2
  • Theorem 4.3
  • Proof 2
  • Corollary 4.4
  • ...and 2 more