On Time-optimal Trajectories for a Car-like Robot with One Trailer

TL;DR

This paper analytically characterizes time-optimal trajectories for a car-like robot with a trailer using Pontryagin's Maximum Principle, revealing new extremal curves including elastica and merging curves for efficient motion planning.

Contribution

It provides the first detailed analytical characterization of extremal trajectories for a car-like robot with a trailer, including new classes of motion such as elastica and merging curves.

Findings

Identified planar elastica as a key extremal motion

Derived merging curves connecting maximum curvature turns to straight segments

Provided analytical integration of the system and adjoint equations

Abstract

In addition to the theoretical value of challenging optimal control problmes, recent progress in autonomous vehicles mandates further research in optimal motion planning for wheeled vehicles. Since current numerical optimal control techniques suffer from either the curse of dimens ionality, e.g. the Hamilton-Jacobi-Bellman equation, or the curse of complexity, e.g. pseudospectral optimal control and max-plus methods, analytical characterization of geodesics for wheeled vehicles becomes important not only from a theoretical point of view but also from a prac tical one. Such an analytical characterization provides a fast motion planning algorithm that can be used in robust feedback loops. In this work, we use the Pontryagin Maximum Principle to characterize extremal trajectories, i.e. candidate geodesics, for a car-like robot with one trailer. We use time as the distance function. In spite of partial progress, this problem has remained open in the past two decades. Besides straight motion and turn with maximum allowed curvature, we identify planar elastica as the third piece of motion that occurs along our extr emals. We give a detailed characterization of such curves, a special case of which, called \emph{merging curve}, connects maximum curvature turns to straight line segments. The structure of extremals in our case is revealed through analytical integration of the system and adjoint equations.