[Paper] Min-Sum Uniform Coverage Problem by Autonomous Mobile Robots

Source: arXiv - 2602.11125v1

Overview

This paper tackles a classic coordination challenge for swarms of simple, autonomous robots: how to spread themselves evenly along a line segment or around a circle while keeping the total distance each robot travels as low as possible. The authors present deterministic, fully distributed algorithms that achieve the optimal “min‑sum” movement cost—even though the robots have no memory, identifiers, or direct communication.

Key Contributions

• Optimal min‑sum uniform coverage on a line segment – a deterministic algorithm that guarantees the smallest possible total travel distance for any initial placement.
• Complete solvability characterization on a circle – identifies exactly which initial configurations are unsolvable under the given robot model, and provides an optimal algorithm for all solvable cases.
• Rigorous analysis under the asynchronous, non‑rigid Look‑Compute‑Move (LCM) model, reflecting realistic timing and motion uncertainties.
• Proof of optimality – the algorithms achieve the theoretical lower bound on total movement, not just a good approximation.
• Framework that works for anonymous, oblivious, and silent robots, showing that sophisticated coordination can emerge from minimal capabilities.

Methodology

1. Robot Model – Each robot repeatedly executes a Look‑Compute‑Move cycle. “Look” captures the positions of all robots, “Compute” decides a target point, and “Move” advances toward that point, but only a fraction of the distance may be covered (non‑rigid). Robots are oblivious (no memory) and silent (no messaging).

2. Line‑Segment Algorithm

   • Compute the ideal uniform spacing (segment length divided by (n-1)).
   • Each robot determines its closest ideal position based on the current snapshot.
   • Robots move toward those positions using a greedy, conflict‑free rule that prevents two robots from targeting the same spot.
   • The design ensures that the sum of all traveled distances equals the minimum possible (proved by a simple exchange argument).

3. Circle Algorithm & Impossibility Proof

   • First, the authors prove that if the robots start in a rotationally symmetric configuration (e.g., equally spaced already but rotated arbitrarily), they cannot break symmetry without extra capabilities, making the problem unsolvable.
   • For all other configurations, robots compute a reference direction using the lexicographically smallest observed configuration, which gives a common orientation despite anonymity.
   • They then map each robot to a unique vertex of the target regular (n)-gon and move toward it, again respecting the non‑rigid motion constraints.

4. Correctness & Optimality Proofs – Formal arguments show that (a) the algorithms always converge, (b) they respect the asynchronous scheduler, and (c) the total distance traveled matches the theoretical lower bound.

Results & Findings

• Line Segment – The algorithm always reaches a uniformly spaced layout with exactly the minimum possible sum of travel distances, regardless of the initial distribution.
• Circle –
  • Unsolvable cases: Any initial configuration that is already a regular (n)-gon up to rotation (i.e., perfect symmetry) cannot be resolved without breaking symmetry.
  • Solvable cases: For every other starting arrangement, the algorithm converges to a regular (n)-gon while achieving the optimal total movement cost.
• The work demonstrates that optimal min‑sum coverage is achievable even with severely limited robot capabilities, provided the initial symmetry conditions are favorable.

Practical Implications

• Swarm Deployment in Structured Environments – Robots tasked with monitoring pipelines, borders, or perimeter fences can self‑organize into evenly spaced patrol points while minimizing energy consumption.
• Coverage in Circular Facilities – Applications such as autonomous inspection of storage tanks, circular arenas, or rotating platforms can benefit from the circle algorithm to quickly form a regular formation with minimal motion.
• Energy‑Aware Coordination – Since total travel distance directly correlates with battery usage, these algorithms can extend mission lifetimes for low‑cost, battery‑limited robots.
• Robustness to Timing Uncertainty – The asynchronous, non‑rigid model mirrors real‑world communication delays and actuator imperfections, making the solutions readily implementable on off‑the‑shelf platforms.
• Foundation for Higher‑Level Tasks – Uniform coverage is often a prerequisite for tasks like distributed sensing, load balancing, or coordinated actuation; having an optimal baseline simplifies subsequent algorithm design.

Limitations & Future Work

• Symmetry Barrier on the Circle – The impossibility result shows that perfectly symmetric start states cannot be resolved without additional capabilities (e.g., randomization, unique IDs, or external landmarks).
• Scalability to 2‑D/3‑D Domains – The study focuses on 1‑D (line) and 1‑D manifold (circle) environments; extending optimal min‑sum coverage to planar or volumetric spaces remains open.
• Dynamic Environments – The algorithms assume a static environment; handling obstacles, moving targets, or robot failures would require further adaptation.
• Experimental Validation – While the paper provides rigorous proofs, real‑world experiments on physical robot swarms would help assess performance under sensor noise and actuation errors.

Authors

• Animesh Maiti
• Abhinav Chakraborty
• Bibhuti Das
• Subhash Bhagat
• Krishnendu Mukhopadhyaya

Paper Information

• arXiv ID: 2602.11125v1
• Categories: cs.DC, cs.RO
• Published: February 11, 2026
• PDF: Download PDF