Traveing Salesperson Problems for a double integrator

TL;DR

This paper introduces novel path planning algorithms for a double integrator in TSP scenarios, providing asymptotic bounds, probabilistic performance guarantees, and stable policies for dynamic target environments.

Contribution

It presents new algorithms for double integrator TSP, including asymptotic bounds, probabilistic guarantees, and stable policies for dynamic target tracking.

Findings

Algorithms perform within a constant factor of the optimal.

Asymptotic bounds on worst-case tour times are established.

Stable policies ensure bounded unvisited targets over time.

Abstract

In this paper we propose some novel path planning strategies for a double integrator with bounded velocity and bounded control inputs. First, we study the following version of the Traveling Salesperson Problem (TSP): given a set of points in $\real^d$, find the fastest tour over the point set for a double integrator. We first give asymptotic bounds on the time taken to complete such a tour in the worst-case. Then, we study a stochastic version of the TSP for double integrator where the points are randomly sampled from a uniform distribution in a compact environment in $\real^2$ and $\real^3$. We propose novel algorithms that perform within a constant factor of the optimal strategy with high probability. Lastly, we study a dynamic TSP: given a stochastic process that generates targets, is there a policy which guarantees that the number of unvisited targets does not diverge over time? If such stable policies exist, what is the minimum wait for a target? We propose novel stabilizing receding-horizon algorithms whose performances are within a constant factor from the optimum with high probability, in $\real^2$ as well as $\real^3$. We also argue that these algorithms give identical performances for a particular nonholonomic vehicle, Dubins vehicle.