Conjugate points in nilpotent sub-Riemannian problem on the Engel group

TL;DR

This paper investigates conjugate points and local optimality of extremal trajectories in a sub-Riemannian problem on the Engel group, relevant for modeling control systems like mobile trailer robots.

Contribution

It provides new estimates of conjugate times and analyzes local optimality in the nilpotent sub-Riemannian problem on the Engel group.

Findings

Identified conjugate points along extremal trajectories.

Estimated conjugate times for optimality loss.

Applied results to control systems like mobile trailer robots.

Abstract

The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes movement of mobile trailer robot. We study the local optimality of extremal trajectories and estimate conjugate time in this article.