Almost-globally stable tracking for on compact Riemannian manifolds

TL;DR

This paper develops a control law for almost-global asymptotic tracking of trajectories on Riemannian manifolds, extending existing results beyond Lie groups and demonstrating effectiveness on spherical pendulum and particle motion.

Contribution

It introduces a novel intrinsic configuration error and error dynamics framework for Riemannian manifolds, enabling AGAT analysis beyond Lie group cases.

Findings

Successfully achieves AGAT on a spherical pendulum on S^2

Demonstrates tracking on a particle on a Lissajous curve in R^3

Extends tracking results to general Riemannian manifolds

Abstract

In this article, we propose a control law for almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving on a Riemannian manifold which can be embedded in a Euclidean space. The existing results on tracking for an SMS are either local, or almost-global, only in the case the manifold is a Lie group. In the latter case, the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis. However, such a notion is not intrinsic to a Riemannian manifold. In this paper, we define a configuration error followed by error dynamics on a Riemannian manifold, and then prove AGAT. The results are demonstrated for a spherical pendulum which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb{R}^3$.