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

The trajectory tracking problem is a fundamental control task in the study of mechanical systems. Hamiltonian systems are posed on the cotangent bundle of configuration space of a mechanical system, however, 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.