Variational collision avoidance problems on Riemannian manifolds
It provides a theoretical foundation for multi-agent collision avoidance on curved spaces, but the results are preliminary and validated only on simple manifolds.
This paper introduces a variational framework for collision avoidance of multiple agents on Riemannian manifolds, deriving necessary conditions for extremals and validating the approach with numerical experiments on ℝ² and S².
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 $S^2$.