Local minimizers for variational obstacle avoidance on Riemannian manifolds

Jacob R. Goodman; Jacob R. Goodman

doi:10.3934/jgm.2023003

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 59-72. doi: 10.3934/jgm.2023003

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Research article

Local minimizers for variational obstacle avoidance on Riemannian manifolds

Jacob R. Goodman ^,

Instituto de Ciencias Matematicas(CSIC-UAM-UC3M-UCM), Calle Nicolas Cabrera 13-15, 28049, Madrid, Spain

Received: 18 January 2022 Revised: 29 June 2022 Accepted: 29 June 2022 Published: 31 October 2022
Primary: 49K15, 53B20; Secondary: 53Z30

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—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.
- bi-Jacobi fields,
- biconjugate points,
- local minimizers,
- Riemannian geometry,
- path planning,
- obstacle avoidance
Citation: Jacob R. Goodman. Local minimizers for variational obstacle avoidance on Riemannian manifolds[J]. Journal of Geometric Mechanics, 2023, 15(1): 59-72. doi: 10.3934/jgm.2023003

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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—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.

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