Multi-Agent Coverage Control in Non-Convex Annulus Region with Conformal Mapping

TL;DR

This work addresses multi-agent coverage in non-convex, non-star-shaped regions by mapping the irregular workspace to a topologically equivalent annulus Xi via a diffeomorphic conformal pipeline. It introduces a decentralized sectorial partition controller operating in Xi, coupled with a length metric d_l that guides agents toward geodesic-centroid centroids and guarantees equitable workload with exponential convergence. An iterative, non-convex search framework identifies near-optimal deployment under the mapping, with theoretical stability results and numerical demonstrations validating load balancing and path efficiency. The approach outperforms Euclidean-metric baselines and standard Voronoi schemes in complex environments, offering a scalable, distributed solution for practical MAS coverage tasks.

Abstract

Efficiently fulfilling coverage tasks in non-convex regions has long been a significant challenge for multi-agent systems (MASs). By leveraging conformal mapping, this paper introduces a novel sectorial coverage formulation to transform a non-convex annulus region into a topologically equivalent one. This approach enables the deployment of MASs in a non-star-shaped region while optimizing coverage performance and achieving load balance among sub-regions. It provides a unique perspective on the partitioned sub-regions to highlight the geodesic convex property of the non-star-shaped region. By utilizing the sectorial partition mechanism and the diffeomorphism property of conformal mapping, a decentralized control law is designed to drive MASs towards a desired configuration, which not only optimizes the global coverage cost but also ensures exponential convergence of equitable workload. Moreover, an iterative search algorithm is developed to identify the optimal approximation of multi-agent deployment in the non-star-shaped region. Theoretical analysis is conducted to confirm the asymptotic stability and global convergence with arbitrary small tolerance of the closed-loop system. Finally, numerical simulations demonstrate the practicality of the proposed coverage formulation with conformal mapping.

Figures (9)

• Figure 1: Illustration of non-convex coverage region $\Omega$, whose obstacle has non-smooth boundary. It is not easy to directly deploy the multi-agent system in this region because of the absence of a common reference point for the inner and outer star-shaped sets, which implies that it is infeasible to realize the sectorial partition.
• Figure 2: An illustration of the conformal mapping process. At first, we use the harmonic mapping $H_i$ for the sub-region $V^i$, establishing a harmonic diffeomorphism between $V^i$ and unit disk $\mathbbm{D}$. Using $\sigma_i$ to transform the unit disk $\mathbbm{D}$ to the rectangular region $[0, L_i] \times [0,1]$. Through Algorithm \ref{['Algorithm1']}, we obtain the optimum length $L_*$. Then, the rectangular conformal inverse mapping $(\vartheta _i^*)^{-1}$ maps the region $[0, L_i] \times [0,1]$ to point cloud $V^i_*$. According to the corresponding mechanism among agents and the ICP point cloud registration, we obtain all the coordinates in ($\Omega$-$O$)$\cup$$\partial O$. Slicing the mesh ($\Omega$-$O$)$\cup$$\partial O$ along a path $v_1v_2$ yields a simply connected open region $\tilde{\Omega}$. Secondly, we construct a rectangular conformal mapping $\vartheta_i^*$ between $\tilde{\Omega}$ and the rectangular region $[0, L_*] \times [0,1]$ with an optimal length $L_*$ and width 1. The rectangle is subsequently mapped to an annulus using an exponential map $\iota ={e^{2\pi \left( {z -L_*} \right)}}$. Finally, we identify the cut vertices $v_1v_2$ and compose the quasi-conformal map $\varpi$ to reduce angle distortion, which forms a conformal parameterization. The last region is $\Xi$.
• Figure 3: The density distribution in the original space ($\Omega$-$O$)$\cup$$\partial O$ and the mapped region $\Xi$. $\tau$ is the conformal mapping. Due to the harmonic mapping would rotate the region, the density correspondence relationship would present the phenomenon.
• Figure 4: Simulation result on the mapped region $\Xi$. Blue pointers represent the partition bar between adjacent sub-regions. Blue circles denote the mobile agents, and green stars refer to the sub-region centroids. The result illustrates the sectorial partition can finish the coverage mission and achieve workload balance in the hollow environment with a rapid convergence rate.
• Figure 5: Simulation result on the original region ($\Omega$-$O$)$\cup$$\partial O$. Due to the bijective of the conformal mapping $\tau$, we can find the pre-image of the partition bar, the agent's position, and the optimal centroids. The red pointers represent the pre-image of the blue pointers in $\Xi$. The purple, orange, cyan, and yellow lines show the moving paths of the multi-agent. The phenomenon illustrates the effectiveness of conformal mapping in the coverage problem of non-star-shaped region.

Theorems & Definitions (39)

• Definition 2.1: Conformal Map computerconformalGeometry07
• Definition 2.2: Quasi-Conformal Map computerconformalGeometry07
• Definition 2.3: Coordinate Chart Topology99
• Definition 2.4: Riemannian Metric Riemannian geometry
• Definition 2.5: Cut Locus Cut loci85
• Definition 2.6: Geodesically Convex Region Riemannian geometry
• Lemma 2.1: Poincare-Koebe Uniformization computerconformalGeometry07
• Remark 2.2
• Remark 2.3
• Remark 2.4