# Variational obstacle avoidance problem on Riemannian manifolds

## TL;DR

This paper formulates and analyzes variational obstacle avoidance problems on Riemannian manifolds, deriving conditions for optimal curves that avoid obstacles, with applications to robotic motion planning on Lie groups and specific vehicle models.

## Contribution

It introduces a novel variational framework for obstacle avoidance on Riemannian manifolds, including sub-Riemannian and Lie group cases, with derived necessary optimality conditions.

## Key findings

- Derived necessary conditions for obstacle avoidance extremals.
- Applied the theory to planar rigid body and unicycle models.
- Extended the framework to Lie groups with invariant metrics.

## Abstract

We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and covariant acceleration, among a set of admissible curves, and also depending on a navigation function used to avoid an obstacle on the workspace, a Riemannian manifold. We study two different scenarios, a general one on a Riemannian manifold and, a sub-Riemannian problem. By introducing a left-invariant metric on a Lie group, we also study the variational obstacle avoidance problem on a Lie group. We apply the results to the obstacle avoidance problem of a planar rigid body and an unicycle.