Variational Collision Avoidance on Riemannian Manifolds

TL;DR

This paper develops a variational framework for multi-agent collision avoidance on Riemannian manifolds, establishing existence of solutions and conditions for safe trajectories with bounded derivatives.

Contribution

It introduces a novel variational approach for collision avoidance on Riemannian manifolds, including existence results and safety conditions under derivative constraints.

Findings

Proves the existence of minimizers for the collision avoidance functional.

Provides conditions for collision avoidance within specified tolerances.

Derives safety conditions based on bounds on derivatives of trajectories.

Abstract

This paper studies variational collision avoidance problems for multi-agents systems on complete Riemannian manifolds. That is, we minimize an energy functional, among a set of admissible curves, which depends on an artificial potential function used to avoid collision between the agents. We show the global existence of minimizers to the variational problem and we provide conditions under which it is possible to ensure that agents will avoid collision within some desired tolerance. We also study the problem where trajectories are constrained to have uniform bounds on the derivatives, and derive alternate safety conditions for collision avoidance in terms of these bounds - even in the case where the artificial potential is not sufficiently regular to ensure existence of global minimizers.