Variational collision avoidance problems on Riemannian manifolds

TL;DR

This paper develops a variational method for planning collision-free trajectories of multiple agents on Riemannian manifolds, deriving necessary conditions and validating results through numerical experiments.

Contribution

It introduces a novel variational framework for collision avoidance on Riemannian manifolds, including necessary conditions for optimal trajectories.

Findings

Successful numerical validation on Euclidean and spherical manifolds

Effective collision avoidance using energy functional minimization

Derivation of necessary conditions for extremal trajectories

Abstract

In this article we introduce a variational approach to collision avoidance of multiple agents evolving on a Riemannian manifold and derive necessary conditions for extremals. The problem consists of finding non-intersecting trajectories of a given number of agents, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision among the agents. The results are validated through numerical experiments on the manifolds $\mathbb{R}^{2}$ and $S^2$.