2024/01/30 by Matthew Hampsey, Hampsey, Matthew, Pieter van Goor +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Electrical engineering #Homotopy and Cohomology in Algebraic Topology #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2401.16725
openalex publication_date 2024/01/30 · openalex created_date 2024/02/01 · openalex updated_date 2026/07/28
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.