Cut time in sub-Riemannian problem on Engel group

TL;DR

This paper analyzes the sub-Riemannian problem on the Engel group, establishing the structure of optimal trajectories and proving that the cut time equals the first Maxwell time, which is linked to symmetries of the exponential map.

Contribution

It provides a detailed geometric analysis of the Engel group sub-Riemannian problem and proves the cut time equals the first Maxwell time due to symmetries.

Findings

The exponential mapping is globally diffeomorphic.

Cut time equals the first Maxwell time.

Symmetries determine optimality boundaries.

Abstract

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.