A Complete and Bounded-Suboptimal Algorithm for a Moving Target Traveling Salesman Problem with Obstacles in 3D

TL;DR

This work introduces FMC*-TSP, a complete and bounded-suboptimal algorithm for the moving target traveling salesman problem with obstacles in 3D. The method interleaves a high-level GTSP-TW search over target-windows with a low-level planning stage on a graph of convex sets (GCS) to generate obstacle-free trajectories that intercept moving targets within their time windows. Completeness and a bounded-suboptimality guarantee are achieved using two layers of lower bounds and a focal-search framework, along with a dictionary-based pruning strategy and occasional reuse of search state across FMC* calls. Numerical results on 280 problem instances with up to 40 targets show FMC*-TSP often achieves faster medians than a sampling-based baseline while maintaining the suboptimality bound, highlighting the practicality of GCS-based planning for complex 3D MT-TSP-O scenarios.

Abstract

The moving target traveling salesman problem with obstacles (MT-TSP-O) seeks an obstacle-free trajectory for an agent that intercepts a given set of moving targets, each within specified time windows, and returns to the agent's starting position. Each target moves with a constant velocity within its time windows, and the agent has a speed limit no smaller than any target's speed. We present FMC*-TSP, the first complete and bounded-suboptimal algorithm for the MT-TSP-O, and results for an agent whose configuration space is $\mathbb{R}^3$. Our algorithm interleaves a high-level search and a low-level search, where the high-level search solves a generalized traveling salesman problem with time windows (GTSP-TW) to find a sequence of targets and corresponding time windows for the agent to visit. Given such a sequence, the low-level search then finds an associated agent trajectory. To solve the low-level planning problem, we develop a new algorithm called FMC*, which finds a shortest path on a graph of convex sets (GCS) via implicit graph search and pruning techniques specialized for problems with moving targets. We test FMC*-TSP on 280 problem instances with up to 40 targets and demonstrate its smaller median runtime than a baseline based on prior work.

Figures (3)

• Figure 1: Targets depicted as stars move along piecewise-linear trajectories. Portions of a target's trajectory within time windows are highlighted in bold colors. Agent trajectory shown in blue begins and ends at depot, intercepting each target within one of its time windows while avoiding obstacles.
• Figure 2: Computing $f(P')$ for $P' = (\mathcal{X}_{\Omega[1]}, \mathcal{X}_{\mathcal{A}^5}, \mathcal{X}_{A^4}, \mathcal{X}_{\Omega[2]}, \mathcal{X}_{\mathcal{A}^4}. \mathcal{X}_{\mathcal{A}^3})$. $P_{aux} = (\mathcal{X}_{\Omega[2]}, \mathcal{X}_{\mathcal{A}^4}, \mathcal{X}_{\mathcal{A}^3}, \mathcal{Q} \times \mathbb{R})$. Regions decomposing $\mathcal{Q}_{free}$ are shown as pink boxes. Trajectory segment endpoints are shown as yellow diamonds. $\tau^1$ is a zero-length segment between $\LTraj(P)$ and $\tau^2$, and $\tau^3$ is a zero-length segment between $\tau^2$ and $\tau^4$. Concatenating $\LTraj(P)$ with $\tau^1, \tau^2$, and $\tau^3$ gives an obstacle-free trajectory $\tau'$ with cost $g(P')$. $\tau^4$ and its extension ignore obstacles. Extension of $\tau^4$ terminates on purple target at time $f(P')$.
• Figure 3: All vertical axes are on a log-scale. (a) Varying the number of targets and sum of time window lengths per target. (b) Varying the agent's speed limit. (c) Varying FMC*-TSP's suboptimality factor.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2