Reduction by Symmetry in Obstacle Avoidance Problems on Riemannian Manifolds

TL;DR

This paper explores symmetry reduction techniques for obstacle avoidance problems on Riemannian manifolds, deriving conditions and applying them to rigid body motion on SO(3) and the sphere S^2.

Contribution

It introduces a novel reduction framework for obstacle avoidance on Riemannian manifolds using symmetry and connections, with explicit solutions in special cases.

Findings

Derived reduced necessary conditions for obstacle avoidance problems.

Applied reduction methods to rigid body motion on SO(3).

Analyzed obstacle avoidance on the sphere S^2.

Abstract

This paper studies the reduction by symmetry of a variational obstacle avoidance problem. We derive the reduced necessary conditions in the case of Lie groups endowed with a left-invariant metric, and for its corresponding Riemannian homogeneous spaces by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the Lie group. A number of special cases where the obstacle avoidance potential can be computed explicitly are studied in detail, and these ideas are applied to the obstacle avoidance task for a rigid body evolving on SO$(3)$ and for the unit sphere $S^2$.