Minimum Exposure Motion Planning

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].

Figures (7)

• Figure 1: An example instance and schedule, including a timeline, for Min-Exposure CMP. While both robots are in a covered region (light blue), they can move to any position within this region for free. The robots are exposed (red) when at least one of the robots is not fully covered.
• Figure 2: Configurations are (a) feasible or (b) infeasible. (c) A trajectory $m$ from $a$ to $b$ with four turns.
• Figure 3: (a) A polygonal domain $S$, its inner Minkowski sum $\textsf{inner}(S)$, and a feasible configuration of two robots in $S$. (b) The horizontal decomposition of $\textsf{inner}(S)$.
• Figure 4: We divide the configuration space into ordering groups based on the order of robots wrt. axis-aligned bisectors (dashed) that separate them. Shown here is $\mathcal{F}_2$, split into four orderings.
• Figure 5: Two states. In (b), the robots cannot change their relative order along the $y$-axis.

Theorems & Definitions (23)

• Definition 1
• Lemma 2
• Corollary 3
• Theorem 4
• Corollary 5
• Theorem 6
• Definition 7
• Theorem 8
• Corollary 9
• Theorem 10