Minimum Exposure Motion Planning

2026-03-17

AI Summary

This study addresses cooperative motion planning for unit-square robots in the plane, aiming to minimize their total exposure time in unobstructed regions. It introduces, for the first time, the Min-Exposure optimization criterion and develops an exact algorithm with running time $O(n^4 \log n)$ under an $L_1$-metric motion model, leveraging computational geometry and parameterized complexity analysis. Additionally, an XP algorithm is provided that applies to any number of robots. The work establishes fixed-parameter tractability for both Min-Makespan and Min-Sum when parameterized by the number of robots, generalizing prior results on grid graphs. Furthermore, it achieves efficient optimal scheduling for two robots and lays a parameterized tractability framework for multi-robot scenarios, thereby extending the theoretical foundations of cooperative motion planning.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We investigate multiple fundamental variants of the classic coordinated motion planning (CMP) problem for unit square robots in the plane under the $L_1$ metric. In coordinated motion planning, we are given two arrangements of $k$ robots and are tasked with finding a movement schedule that minimizes a certain objective function. The two most prominent objective functions are the sum of distances traveled (Min-Sum) and the latest time of arrival (Min-Makespan). Both objectives have previously been studied extensively. We introduce a new objective function for CMP in the plane. The proposed Min-Exposure objective function defines a set of polygonal regions in the plane that provide cover and asks for a schedule with minimal elapsed time during which at least one robot is partially or fully outside of these regions. We give an $\mathcal{O}(n^4\log n)$ time algorithm that computes exposure-minimal schedules for $k=2$ robots, and an XP algorithm for arbitrary $k$. As a result of independent interest, we leverage new insights to prove that both the Min-Makespan and Min-Sum objectives are fixed-parameter tractable (FPT) parameterized by the number of robots. Our parameterized complexity results generalize known FPT results for rectangular grid graphs [Eiben, Ganian, and Kanj, SoCG'23].

Problem

• coordinated motion planning
• minimum exposure
• robot scheduling
• polygonal cover
• L1 metric

Innovation

• Minimum Exposure
• Coordinated Motion Planning
• Fixed-Parameter Tractability
• L1 Metric
• Unit Square Robots