Distributed Harmonic-Covariance Feedback for Hole Closure in Planar Robot Swarms

Summary

Abstract Summary

We consider hole closure in a planar multi-agent system whose sensing disks form a time-varying \v{C}ech complex. We propose a harmonic-covariance feedback that uses harmonic edge signals to locate holes and converts those signals into agent forces. For distributed evaluation, we use a fixed-degree polynomial of the regularized Hodge Laplacian. This polynomial is evaluated through local operator applications. The closed loop has a unique global trajectory from every safe initial state. The model data and initial state are supplied with procedures for approximation to any requested accuracy with explicit error bounds. We assume that hole area is measurable and bounded, holes evolve continuously without splitting or merging, and only finitely many temporary new holes appear. Under these conditions, offline finite calculations produce rational lower bounds for the cumulative hole area on the time horizon without querying trajectory data. These bounds converge to that cumulative area, whether finite or infinite. If it is finite, holes close in the time-averaged sense: the average normalized hole area over every fixed-length time window tends to zero as time tends to infinity. In a 15-second numerical demo, all holes close by 9.9s and remain closed through the end.

Key Points

We consider hole closure in a planar multi-agent system whose sensing disks form a time-varying \v{C}ech complex.
We propose a harmonic-covariance feedback that uses harmonic edge signals to locate holes and converts those signals into agent forces.
For distributed evaluation, we use a fixed-degree polynomial of the regularized Hodge Laplacian
This polynomial is evaluated through local operator applications
The closed loop has a unique global trajectory from every safe initial state

Problem Formulation and Sensing Model

We consider $N$ mobile sensing agents. Agent $i$ has position $q_{i}(t)\in\mathbb{R}^{2}$ and velocity $v_{i}(t)\in\mathbb{R}^{2}$; we stack these variables as $q,v\in\mathbb{R}^{2N}$ and write the state as $z=(q,v)$. Each agent possesses a sensing disk centered at itself, and the control objective is to remove the bounded regions left uncovered by the union of those disks.

Section IV-A defines these regions, which we call holes, and their cumulative cost. Each agent senses the open disk of radius $r_{s}>0$ centered at its position.

Problem Formulation and Sensing Model — Distributed Harmonic-Covariance Feedback for Hole Closure in Planar Robot Swarms — Fig. 1: Left: the union of sensing disks of the 30-agent annulus; right: its strict Čech complex.

Controller Design

Harmonic information is stored as scalar values on edges, while each agent needs a two-dimensional force at a node. We first convert edge values into node vectors and use these vectors to define force laws on each agent. The ideal law uses the exact harmonic projector of the binary Čech complex; Section III-B replaces it with a fixed-degree polynomial of $L_{w}$ from Section II-A .

Here “exact” and “ideal” refer only to the projector. Let $u(q)=F^{H}(q)$ be the ideal harmonic-covariance force.

Controller Design — Distributed Harmonic-Covariance Feedback for Hole Closure in Planar Robot Swarms — Fig. 2: Two orthonormal harmonic forms of the weighted active-edge restriction of $L_{w}$ for the same connected two-hole complex. Triangles are shown lightly; colored edges give signed coefficients on the common scale, while gray denotes the zero level. Orientations are fixed by $i

Well-Posedness and Bounds on Cumulative Hole Areas

After the regularized controller is defined, it is natural to raise the following questions: Given a safe initial state, does the controller yield one unique trajectory? Given an initial state, without simulation, can we determine whether the holes will eventually close under the controlled dynamics?

Section IV-C gives the first question a positive answer, and Section IV-D answers the second question with an offline construction of a rational sequence $\{c_{T}\}_{T}$ converging to the all-time cumulative hole area, with each term obtained by fi...

Well-Posedness and Bounds on Cumulative Hole Areas — Distributed Harmonic-Covariance Feedback for Hole Closure in Planar Robot Swarms — Fig. 3: Edge forces of the harmonic-covariance law.

Numerical Demo

This section shows how the controller behaves on a 60-agent configuration with two initial holes. The snapshots show the agent motion and the numerical Čech complex, while the area plot follows the two holes that are present at the initial time. $t=0.0$ s $t=5.0$ s $t=10.0$ s $t=15.0$ s The initial positions form the two-hole peanut shown on the left of Fig. 4 .

Table I lists the parameters and integrator settings, where the Chebyshev polynomial degree $d=343$ is determined by the stated polynomial-tail tolerance.

Numerical Demo — Distributed Harmonic-Covariance Feedback for Hole Closure in Planar Robot Swarms — Fig. 4: The two-hole peanut with state-dependent damping. Each column shows the configuration and numerical Čech complex from the same accepted trajectory; the terminal saved frame has $\beta_{1}=0$.

Sources and Demonstrations

Figures are reproduced from Liu et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.