Local Minimizers for Variational Obstacle Avoidance on Riemannian manifolds

TL;DR

This paper investigates obstacle avoidance on Riemannian manifolds by minimizing an action functional, generalizing bi-Jacobi fields and biconjugate points, and classifies local minimizers with conditions for optimality and uniqueness.

Contribution

It introduces a generalized framework for obstacle avoidance on Riemannian manifolds, extending bi-Jacobi theory and providing classification and optimality conditions for local minimizers.

Findings

Generalized bi-Jacobi fields and biconjugate points on Riemannian manifolds.

Derived necessary and sufficient conditions for optimality of local minimizers.

Established local uniqueness results for different categories of minimizers.

Abstract

This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases.