Distributed Circumferential Coverage Control in Non-Convex Annulus Environments

It has long been a prominent challenge in multi-agent systems to achieve distributed coverage of non-convex annulus environments while ensuring workload equalization among agents. To address this challenge, a distributed circumferential coverage control formulation is developed in this note by constructing a Riemannian metric for the navigation in the non-convex subregion while avoiding collisions with the region boundary. In addition, a distributed partition law is designed to balance the workload on the entire coverage region by endowing each agent with a virtual partition bar that slides along the inner boundary of coverage region. Theoretical analysis is conducted to ensure the exponential convergence of workload partition and asymptotic convergence of each agent towards the local optimum in its subregion. Finally, a case study is presented to demonstrate the effectiveness of the proposed coverage control approach.

💡 Research Summary

This paper tackles the longstanding problem of achieving distributed coverage in non‑convex annular environments while guaranteeing equal workload distribution among multiple agents. The authors propose a novel circumferential coverage control framework that does not rely on a common reference point, making it applicable to non‑star‑shaped, non‑convex regions. Each agent is equipped with a virtual partition bar that slides along the inner boundary of the annulus. The bar’s dynamics are defined by (\dot s_i = -\kappa_s (m_i - m_{i-1}) \nu_i), where (m_i) denotes the workload of the (i)-th subregion and (\nu_i) is the unit tangent of the inner boundary at the bar’s anchor point. By constructing a Lyapunov function (V = \frac12\sum_i (m_i-\bar m)^2) and showing (\dot V \le -c V), the authors prove exponential convergence of the workload partition, i.e., all subregions eventually carry the same amount of work.

For navigation, a Riemannian metric (g_{ij}= \delta_{ij}/h^2(q)) is introduced, where (h(q)) measures the distance from a point (q) to the region boundary. This metric inflates distances near the boundary, naturally repelling agents from collisions. An energy‑like function (E(t)=\frac12 d_g^2(p_i,q_i^*)) is defined, where (q_i^*) is the optimal point (centroid) of subregion (\Omega_i) that minimizes the local coverage cost (f(p_i,q)). The control law (u_i = -\kappa_p \nabla_{p_i}E) drives each agent toward its local optimum while respecting the metric‑induced safety constraints.

A series of lemmas establish: (1) the partial derivative of workload with respect to the bar position equals the integral of the density over the bar’s intersection with the boundary; (2) the partition dynamics guarantee workload equalization; (3) exponential decay of workload differences; (4) convergence of the bar positions (s_i) to steady values; (5) convergence of the optimal points (q_i^*) and absolute integrability of their velocities; (6) convergence of the energy function (E(t)) to a finite limit; and (7) uniform continuity of (\dot E). Using Barbalat’s lemma, the authors finally show that (\dot E\to0) implies (| \nabla_{p_i}E|_g\to0), i.e., each agent’s position converges to its subregion’s optimal point. Collision avoidance follows from the unbounded nature of the metric near the boundary, making boundary contact mathematically impossible.

The theoretical results are validated through a simulation of a non‑convex annulus (inner boundary convex, outer boundary star‑shaped) with multiple agents starting from random positions. The virtual partition bars quickly balance the workload (differences drop below 1 % within seconds), and each agent follows a metric‑shortest path to its centroid without ever touching the boundary or obstacles. Visualizations illustrate simultaneous convergence of both the partition bars and the agents.

In summary, the paper’s contributions are threefold: (1) a fully distributed circumferential coverage formulation that works without a common reference point; (2) a sliding virtual partition bar mechanism that achieves exponential workload equalization; (3) a Riemannian‑metric‑based navigation scheme that guarantees collision‑free convergence to local optima. The approach is theoretically sound and practically demonstrated, opening avenues for multi‑robot coverage in complex, non‑convex settings such as disaster response, structural health monitoring, and border security. Future work is suggested on extending the method to three‑dimensional environments, handling non‑holonomic dynamics, and incorporating sensor uncertainties.