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Exploiting Equivariance in the Design of Tracking Controllers for Euler-Poincare Systems on Matrix Lie Groups

Matthew Hampsey, Pieter van Goor, Ravi Banavar, Robert Mahony

TL;DR

This work develops a geometric, symmetry-based framework for trajectory tracking of fully-actuated mechanical systems whose configuration space is a matrix Lie group. By endowing the cotangent bundle with a semidirect-product group structure and defining an equivariant tracking error, the authors derive error dynamics that inherit the extended Euler-Poincaré form and construct a simple PD-type controller that achieves almost-global convergence. The approach is instantiated on SO(3), where attitude dynamics are written in a velocity-momentum form with an explicit inertia-map, yielding a Lyapunov-based proof of local asymptotic stability of the error at the identity and a demonstration of that behavior in simulation. Overall, the method provides a coordinate-free, globally meaningful error construct and a tractable control law with clear geometric interpretation for Euler-Poincaré systems on Lie groups, with potential impact on robust tracking and observer design in robotics and aerial systems.

Abstract

The trajectory tracking problem is a fundamental control task in the study of mechanical systems. A key construction in tracking control is the error or difference between an actual and desired trajectory. This construction also lies at the heart of observer design and recent advances in the study of equivariant systems have provided a template for global error construction that exploits the symmetry structure of a group action if such a structure exists. Hamiltonian systems are posed on the cotangent bundle of configuration space of a mechanical system and symmetries for the full cotangent bundle are not commonly used in geometric control theory. In this paper, we propose a group structure on the cotangent bundle of a Lie group and leverage this to define momentum and configuration errors for trajectory tracking drawing on recent work on equivariant observer design. We show that this error definition leads to error dynamics that are themselves ``Euler-Poincare like'' and use these to derive simple, almost global trajectory tracking control for fully-actuated Euler-Poincare systems on a Lie group state space.

Exploiting Equivariance in the Design of Tracking Controllers for Euler-Poincare Systems on Matrix Lie Groups

TL;DR

This work develops a geometric, symmetry-based framework for trajectory tracking of fully-actuated mechanical systems whose configuration space is a matrix Lie group. By endowing the cotangent bundle with a semidirect-product group structure and defining an equivariant tracking error, the authors derive error dynamics that inherit the extended Euler-Poincaré form and construct a simple PD-type controller that achieves almost-global convergence. The approach is instantiated on SO(3), where attitude dynamics are written in a velocity-momentum form with an explicit inertia-map, yielding a Lyapunov-based proof of local asymptotic stability of the error at the identity and a demonstration of that behavior in simulation. Overall, the method provides a coordinate-free, globally meaningful error construct and a tractable control law with clear geometric interpretation for Euler-Poincaré systems on Lie groups, with potential impact on robust tracking and observer design in robotics and aerial systems.

Abstract

The trajectory tracking problem is a fundamental control task in the study of mechanical systems. A key construction in tracking control is the error or difference between an actual and desired trajectory. This construction also lies at the heart of observer design and recent advances in the study of equivariant systems have provided a template for global error construction that exploits the symmetry structure of a group action if such a structure exists. Hamiltonian systems are posed on the cotangent bundle of configuration space of a mechanical system and symmetries for the full cotangent bundle are not commonly used in geometric control theory. In this paper, we propose a group structure on the cotangent bundle of a Lie group and leverage this to define momentum and configuration errors for trajectory tracking drawing on recent work on equivariant observer design. We show that this error definition leads to error dynamics that are themselves ``Euler-Poincare like'' and use these to derive simple, almost global trajectory tracking control for fully-actuated Euler-Poincare systems on a Lie group state space.
Paper Structure (19 sections, 6 theorems, 57 equations, 1 figure)

This paper contains 19 sections, 6 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Useful identities
  5. The Lie Group $\mathbf{SO}(3)$
  6. Group actions and Equivariance for Extended Euler-Poincare Equations
  7. Extended Euler-Poincare equations
  8. Symmetry and group action
  9. Equivariance
  10. Control Error
  11. Energy and Lyapunov analysis
  12. Example - controller design on $\mathbf{SO}(3)$
  13. Attitude dynamics and Lie group structure
  14. Semidirect group structure and equivariance
  15. Error structure and error dynamics
  16. ...and 4 more sections

Key Result

Lemma 3.1

Given the semidirect product group $\mathbf{G}^\ast_{\ltimes}$, consider the system function eq:f_def that assigns the vector field. Then $f$ is equivariant with respect to the group action $\phi$eq:phi_def and the input action $\psi$eq:psi_def in the sense that for all $X = (X_Q, X_P) \in \mathbf{G}_\ltimes^\ast$.

Figures (1)

  • Figure 1: A: The lyapunov function $\mathcal{L}(t)$ vs time for the simulation described in § \ref{['sec:simulation']}. B: The magnitudes of the individial errors $\lVert R_E \rVert$ and $\lVert p_E \rVert$ vs time.

Theorems & Definitions (12)

  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Proposition 5.1
  • proof
  • ...and 2 more