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Equivariant Tracking Control for Fully Actuated Mechanical Systems on Matrix Lie Groups

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

TL;DR

This work tackles trajectory tracking for mechanical systems with state spaces on Lie groups by formulating extended Lie-Poisson dynamics on a semidirect product $\mathbf{G} \ltimes \mathfrak{g}^*$. By introducing a right-invariant error on this extended space, the authors show the error dynamics inherit a Lie-Poisson structure with time-varying inertia, enabling an energy-shaping tracking design that combines a feedforward term, damping, and a navigation-function-based potential. The theory is instantiated for $\mathbf{SO}(3)$, yielding an explicit EqT control law and a comparative analysis with a classical GT controller, including discussions of feedforward differences and energy implications. The framework provides a general, geometry-preserving approach to tracking on Lie groups with time-varying inertia, with potential broad applicability to aerial, marine, and space robotics, and future extensions to more complex semidirect-product configurations.

Abstract

Mechanical control systems such as aerial, marine, space, and terrestrial robots often naturally admit a state-space that has the structure of a Lie group. The kinetic energy of such systems is commonly invariant to the induced action by the Lie group, and the system dynamics can be written as a coupled ordinary differential equation on the group and the dual space of its Lie algebra, termed a Lie-Poisson system. In this paper, we show that Lie-Poisson systems can also be written as a left-invariant system on a semi-direct Lie group structure placed on the trivialised cotangent bundle of the symmetry group. The authors do not know of a prior reference for this observation and we are confident the insight has never been exploited in the context of tracking control. We use this representation to build a right-invariant tracking error for the full state of a Lie-Poisson mechanical system and show that the error dynamics for this error are themselves of Lie-Poisson structure, albeit with time-varying inertia. This allows us to tackle the general trajectory tracking problem using an energy shaping design metholodology. To demonstrate the approach, we apply the proposed design methodology to a simple attitude tracking control.

Equivariant Tracking Control for Fully Actuated Mechanical Systems on Matrix Lie Groups

TL;DR

This work tackles trajectory tracking for mechanical systems with state spaces on Lie groups by formulating extended Lie-Poisson dynamics on a semidirect product . By introducing a right-invariant error on this extended space, the authors show the error dynamics inherit a Lie-Poisson structure with time-varying inertia, enabling an energy-shaping tracking design that combines a feedforward term, damping, and a navigation-function-based potential. The theory is instantiated for , yielding an explicit EqT control law and a comparative analysis with a classical GT controller, including discussions of feedforward differences and energy implications. The framework provides a general, geometry-preserving approach to tracking on Lie groups with time-varying inertia, with potential broad applicability to aerial, marine, and space robotics, and future extensions to more complex semidirect-product configurations.

Abstract

Mechanical control systems such as aerial, marine, space, and terrestrial robots often naturally admit a state-space that has the structure of a Lie group. The kinetic energy of such systems is commonly invariant to the induced action by the Lie group, and the system dynamics can be written as a coupled ordinary differential equation on the group and the dual space of its Lie algebra, termed a Lie-Poisson system. In this paper, we show that Lie-Poisson systems can also be written as a left-invariant system on a semi-direct Lie group structure placed on the trivialised cotangent bundle of the symmetry group. The authors do not know of a prior reference for this observation and we are confident the insight has never been exploited in the context of tracking control. We use this representation to build a right-invariant tracking error for the full state of a Lie-Poisson mechanical system and show that the error dynamics for this error are themselves of Lie-Poisson structure, albeit with time-varying inertia. This allows us to tackle the general trajectory tracking problem using an energy shaping design metholodology. To demonstrate the approach, we apply the proposed design methodology to a simple attitude tracking control.

Paper Structure

This paper contains 10 sections, 5 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie-Poisson Systems
  4. Energy shaping control of systems with invariant kinetic energy Hamiltonians
  5. Extended Lie-Poisson Systems
  6. Error Systems
  7. Tracking Control for Lie-Poisson Systems
  8. Attitude control on $\mathbf{SO}(3)$
  9. Conclusion
  10. Semidirect structure

Key Result

Theorem 3.2

Consider the Lie-Poisson dynamics eq:system corresponding to a Hamiltonian $h$ of the form $h = \frac{1}{2}\langle P , \mathbb{I}_t^{{-1}} [P]\rangle = \frac{1}{2}\langle P , S^{-1}_t \mathbb{I}_0^{{-1}} S^{-\ast} [P]\rangle$, where $\mathbb{I}_t : \mathfrak{g} \times \mathfrak{g} \to \mathbb{R}_+$ Then, the closed-loop system eq:system with control $\tau = \tau_{\mathrm{ff}} + \tau_{\mathrm{pd}}

Figures (3)

  • Figure 1: Equivariance as seen as a transformation of the input space commuting with the transformation of vector fields $\phi_\ast(Y,f_u) := \mathrm{D} \phi_Y \circ f_u \circ \phi_{Y^{-1}}$ induced by the group action $\phi$.
  • Figure 2: Equivariance of extended Lie-Poisson systems.
  • Figure 3: $\mathbf{SO}(3)$ tracking error averaged over 200 iterations.

Theorems & Definitions (15)

  • Definition 2.1
  • Example 3.1: Dynamics on $\mathbf{SO}(3)$
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Example 3.4: Stabilising control on $\mathbf{SO}(3)$
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 5 more