Tracking control on homogeneous spaces: the Equivariant Regulator (EqR)

TL;DR

This paper introduces the Equivariant Regulator (EqR), a control method leveraging symmetry properties of systems on homogeneous spaces to improve trajectory tracking performance in robotics.

Contribution

The paper develops a novel control design framework for systems on homogeneous spaces using symmetry lifting and LQR, enhancing tracking accuracy.

Findings

EqR achieves superior tracking performance in simulations.

The method effectively handles perturbations and system symmetries.

Comparison shows EqR outperforms standard approaches.

Abstract

Accurate tracking of planned trajectories in the presence of perturbations is an important problem in control and robotics. Symmetry is a fundamental mathematical feature of many dynamical systems and exploiting this property offers the potential of improved tracking performance. In this paper, we investigate the tracking problem for systems on homogeneous spaces, manifolds which admit symmetries with transitive group actions. We show that there is natural manner to lift any desired trajectory of such a system to a lifted trajectory on the symmetry group. This construction allows us to define a global tracking error and apply LQR design to obtain an approximately optimal control in a single coordinate chart. The resulting control is then applied to the original plant and shown to yield excellent tracking performance. We term the resulting design methodology the Equivariant Regulator (EqR). We provide an example system posed on a homogeneous space, derive the trajectory linearisation in error coordinates and demonstrate the effectiveness of EqR compared to standard approaches in simulation.