Robot swarms stop guessing whether holes will close: the initial state alone yields a lower bound on cumulative hole area
Lead
A distributed feedback built on harmonic edge signals turns hole information into robot forces in planar swarms whose sensing disks cover a region; under stated assumptions, model and initial state alone yield offline rational lower bounds on cumulative hole area, and in a 15-second numerical demo both initial holes close by 9.9s.
Story
Hole repair shifts from running the swarm to see what happens toward computing a number first. Earlier coverage-repair methods do not provide a criterion for hole closure based solely on the model and initial state. Under the controlled dynamics and assumptions, offline finite calculations produce a sequence of rational lower bounds on cumulative hole area without querying trajectory data.
The controller converts harmonic signals on edges into two-dimensional forces at nodes. The ideal law needs the exact harmonic projector of the binary Čech complex, and computing it directly requires centralized kernel eigenvectors. The distributed realization replaces it with a fixed-degree polynomial of the regularized Hodge Laplacian, evaluated through local operator applications, with probe covariance from outer products of random probe columns.
The closure criterion is stated in a time-averaged sense. The lower-bound sequence converges to the cumulative hole area whether that area is finite or infinite. When the limit is finite, the average normalized hole area over every fixed-length time window tends to zero as time tends to infinity.
What to watch
The report covers planar multi-agent systems in which each agent senses an open disk and the control objective is to remove the bounded uncovered regions left by the union of those disks. The intended readers are engineers and researchers working on distributed control, algebraic topology, and coverage control. The authors' stated next steps include weakening assumptions A2, A3, and A4 and testing an actual distributed implementation rather than the current synchronous emulation.
The conclusions rest on assumptions: the normalized hole area is measurable and bounded, holes evolve continuously without splitting or merging, and only finitely many temporary holes appear after time zero. Closure in the time-averaged sense implies neither pointwise convergence to zero nor finite-time closure. The numerical demo evaluates the same finite-degree local operators with centralized single-process sparse operations, emulating the algebra of the distributed recurrence without measuring message delays or executing a networked message-passing protocol.