Variational point-obstacle avoidance on Riemannian manifolds

TL;DR

This paper develops a variational approach for obstacle avoidance on Riemannian manifolds, deriving equations for optimal paths that consider velocity, acceleration, and obstacle repulsion, with applications demonstrated on Lie groups and symmetric spaces.

Contribution

It introduces a novel variational framework for obstacle avoidance on Riemannian manifolds, including derivation of dynamical equations and numerical illustrations.

Findings

Derived equations for obstacle-avoiding curves on manifolds.

Applied the method to Lie groups and symmetric spaces.

Numerical examples validate the approach.

Abstract

In this letter we study variational obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid a static obstacle on the manifold, among a set of admissible curves. We derive the dynamical equations for extrema of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.