Reduction by Symmetry in Obstacle Avoidance Problems on Riemannian Manifolds

2022/07/27 by Jacob R. Goodman, Goodman, Jacob R., Leonardo Colombo +1
Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Electrical engineering #FOS: Mathematics #Metric Geometry (math.MG) #Morphological variations and asymmetry #Optimization and Control (math.OC) #Robotic Mechanisms and Dynamics #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · pdf · doi:10.48550/arxiv.2207.13574

openalex publication_date 2022/07/27 · openalex created_date 2022/07/30 · openalex updated_date 2026/08/01

Abstract

This paper studies the reduction by symmetry of a variational obstacle avoidance problem. We derive the reduced necessary conditions in the case of Lie groups endowed with a left-invariant metric, and for its corresponding Riemannian homogeneous spaces by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the Lie group. A number of special cases where the obstacle avoidance potential can be computed explicitly are studied in detail, and these ideas are applied to the obstacle avoidance task for a rigid body evolving on SO(3) and for the unit sphere S².

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