Distributed Coverage Control on Poriferous Surface via Poly-Annulus Conformal Mapping

TL;DR

This work tackles coverage control for multi-agent systems on multiply-connected poriferous surfaces by crafting a distributed poly-annulus conformal mapping that diffeomorphically maps the surface S to an n-holed disk Xi. A buffer-based partitioning scheme and a pull-back Riemannian length metric enable a distributed gradient controller that guarantees collision avoidance and convergence to optimal Riemannian centroids, with proven ISS of partition dynamics and global convergence. The approach is scalable, supports distributed computation of partial weldings, and remains robust to agent failures, as demonstrated by numerical simulations that also contrast with Voronoi-based baselines which fail on non-convex perforated domains. Theoretical results establish topological conjugacy between S and Xi, geometric safety, and convergence on a maximal genus-zero class of manifolds, providing a principled, distributed solution for complex non-convex coverage tasks with practical impact in environments with holes and obstacles.

Abstract

The inherent non-convexity of poriferous surfaces typically entraps agents in local minima and complicates workload distribution. To resolve this, we propose a distributed diffeomorphic coverage control framework for the multi-agent system (MAS) in such surfaces. First, we establish a distributed poly-annulus conformal mapping that transforms arbitrary poriferous surfaces into a multi-hole disk. Leveraging this topological equivalence, a collision-free sectorial partition mechanism is designed in the multi-hole disk, which rigorously induces strictly connected subregions and workload balance on the poriferous surfaces. This mechanism utilizes a buffer-based sequence mechanism to ensure strict topological safety when bypassing obstacles. Furthermore, a pull-back Riemannian metric is constructed to define the length metric that encodes safety constraints. Based on this metric, a distributed gradient-based control law is synthesized to drive agents toward optimal configurations, ensuring simultaneous obstacle avoidance and coverage optimization. Theoretical analyses guarantee the Input-to-State Stability (ISS) of the partition dynamics and the asymptotic convergence of the closed-loop system. Numerical simulations confirm the reachability and robustness of the proposed coverage algorithm, offering a scalable solution for distributed coverage in poriferous surfaces.

Figures (9)

• Figure 1: Topological relationship between the surface $S$ and $n$-holed disk $\Xi$. The color gradient illustrates the preservation of density distribution $\rho$.
• Figure 2: The workflow of Algorithm \ref{['Algorithm3']}. The process decomposes the complex topology into local sub-problems, stitches them via partial and global welding, and applies geometric rectification to generate a global coordinate system for the $n$-holed disk $\Xi$.
• Figure 3: Temporal evolution of the MAS in the ball world $\Xi$. Black radial lines denote the safe sectorial partitions, circles represent agent positions, and green stars indicate the optimal Riemannian centroids. The partition bars strictly bypass the circular obstacles, ensuring topological safety.
• Figure 4: Corresponding coverage evolution in the poly-annulus $S$. Through the inverse conformal mapping $\tau^{-1}$, the linear sectors in $\Xi$ are transformed into curvilinear regions in $S$ that naturally conform to the irregular obstacle boundaries while maintaining simply connected topologies.
• Figure 5: Evolution of performance indices under nominal conditions: (Top) Convergence of individual agent workloads $m_i$ to the equilibrium; (Middle) Asymptotic decay of control inputs $u_i$, indicating arrival at optimal centroids; (Bottom) Stabilization of partition phases $\psi_i$.

Theorems & Definitions (23)

• Definition 3.1: Partition Operator Admissibility
• Remark 3.1
• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof