Min-Sum Uniform Coverage Problem by Autonomous Mobile Robots

TL;DR

The paper tackles min-sum uniform coverage for anonymous, oblivious robots constrained to a line segment or circle, operating in an asynchronous Look-Compute-Move model without memory or explicit communication. It proves tight solvability conditions: a deterministic, collision-free, cost-optimal solution exists for line segments; on circles, solvability depends on symmetry, with a complete impossibility result for certain symmetric cases and constructive, cost-optimal algorithms for all solvable configurations, including line-of-symmetry and rotationally symmetric cases. The work introduces extremal-robot concepts and a rigorous assignment framework to preserve optimality under non-rigid, asynchronous motion, delivering finite-time convergence proofs and demonstrating first distributed min-sum optimality under such strong constraints. These results advance energy-efficient, scalable formation and coverage in 1D geometric environments and lay groundwork for extensions to more complex topologies and higher dimensions.

Abstract

We study the \textit{min-sum uniform coverage} problem for a swarm of $n$ mobile robots on a given finite line segment and on a circle having finite positive radius, where the circle is given as an input. The robots must coordinate their movements to reach a uniformly spaced configuration that minimizes the total distance traveled by all robots. The robots are autonomous, anonymous, identical, and homogeneous, and operate under the \textit{Look-Compute-Move} (LCM) model with \textit{non-rigid} motion controlled by a fair asynchronous scheduler. They are oblivious and silent, possessing neither persistent memory nor a means of explicit communication. In the \textbf{line-segment setting}, the \textit{min-sum uniform coverage} problem requires placing the robots at uniformly spaced points along the segment so as to minimize the total distance traveled by all robots. In the \textbf{circle setting} for this problem, the robots have to arrange themselves uniformly around the given circle to form a regular $n$-gon. There is no fixed orientation or designated starting vertex, and the goal is to minimize the total distance traveled by all the robots. We present a deterministic distributed algorithm that achieves uniform coverage in the line-segment setting with minimum total movement cost. For the circle setting, we characterize all initial configurations for which the \textit{min-sum uniform coverage} problem is deterministically unsolvable under the considered robot model. For all the other remaining configurations, we provide a deterministic distributed algorithm that achieves uniform coverage while minimizing the total distance traveled. These results characterize the deterministic solvability of min-sum coverage for oblivious robots and achieve optimal cost whenever solvable.

Figures (14)

• Figure 3: An illustration of configuration view of a robot. Let $\mathcal{R}=\{r_1,r_2,\ldots,r_7\}$. The clockwise view of $r_4$ is $\mathscr{V}^+(r_4) = (0,\alpha_1,\alpha_2,\ldots,\alpha_6)$, while the counterclockwise view is $\mathscr{V}^-(r_4) =(0,\beta_1,\beta_2,\ldots,\beta_6)$. Using lexicographic ordering, the view of robot $r_4$ is $\mathscr{V}(r_4)=\min\bigl(\mathscr{V}^+(r_4),\mathscr{V}^-(r_4)\bigr)= \mathscr{V}^+(r_4)$.
• Figure 4: Illustration of candidate robots (red dots) on the line segment $[a,b]$ with unobstructed paths to their target positions $x_i^{*}$ (red crosses), and hence allowed to move. The remaining robots (blue dots), whose paths are obstructed, remain stationary until a free path becomes available.
• Figure 5: Illustrates the assignment tree used in this paper. Starting from the initial robot configuration, configurations are first classified according to their symmetry properties. Each class is further refined according to the cardinality of optimal assignments and the presence of an extremal robot on a line of symmetry. Each leaf node corresponds to a distinct algorithmic strategy (green) or an impossibility result (red).
• Figure 6: An illustration of the initial configuration of five robots $\mathcal{R}(t_0)={r_1(t_0), r_2(t_0), \ldots, r_5(t_0)}$, where $r_1(t_0)$ is the extremal robot (red dot). Fixing $r_1(t_0)$ determines the destination set ${p_1^1(t_0), p_2^1(t_0), \ldots, p_5^1(t_0)}$ (red crosses). (A) The destination $p_2^1(t_0)$ for robot $r_2(t_0)$ is assigned in the clockwise direction. (B) The destination $p_2^1(t_0)$ for robot $r_2(t_0)$ is assigned in the counterclockwise direction.
• Figure 7: An illustration of the initial configuration of five robots $\mathcal{R}(t_0)=\{r_1(t_0), r_2(t_0), \ldots, r_5(t_0)\}$, where the extremal robot $r_1(t_0)$ is shown as a red dot and the candidate robot $r_2(t_0)$ as a black dot. The destination point set of the regular $n$-gon $\mathcal{N}_1(t_0)$, determined by fixing the extremal robot $r_1(t_0)$, is $\mathcal{P}^1(t_0)=\{p_1^1(t_0), p_2^1(t_0), \ldots, p_5^1(t_0)\}$ (red crosses). The destination point sets of the regular $n$-gons $\mathcal{N}_2(t_0)$ and $\mathcal{N}_2(t_1)$, determined by fixing the candidate robot $r_2(t_0)$ at times $t_0$ and $t_1$, are given by $\mathcal{P}^2(t_0)=\{p_1^2(t_0), p_2^2(t_0), \ldots, p_5^2(t_0)\}$ (black crosses) and $\mathcal{P}^2(t_1)=\{p_1^2(t_1), p_2^2(t_1), \ldots, p_5^2(t_1)\}$ (pink crosses), respectively. Considering the clockwise assignment of $r_2(t_0)$ to $p_2^1(t_0)$: (A) none of the vertices of the regular polygon traced by $r_2(t_1)$ at time $t_1$ passes through or coincides with the position of any other robot; (B) some vertices of the polygon traced by $r_2(t_1)$ at time $t_1$ pass through the initial positions of robots $r_3(t_0)$ and $r_5(t_0)$, respectively.

Theorems & Definitions (16)

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