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Multi-Agent Coverage Control on Surfaces Using Conformal Mapping

Chao Zhai, Yuming Wu

TL;DR

This work addresses real time multi-agent coverage on general surfaces by transforming a surface $Q$ into a disk $D$ through a conformal mapping, enabling planar distributed control and avoiding direct 3D path planning. It introduces a distortion correction framework based on the Beltrami coefficient $mu$ and uses the Linear Beltrami Solver with a Cayley transform to produce a bijective conformal map that minimizes distortion. A disk based coverage optimization is developed where agents partition the disk via Voronoi cells, with the control law $u_i = -k_i (f(p_i) - C_{V_i})$ and the mapping back to $Q$ preserving continuity and preventing partition intersections. Simulations on surface deformation scenarios demonstrate effective coverage and deformation detection, suggesting practical applications in geohazard monitoring such as landslide precursor feature extraction.

Abstract

Real-time environmental monitoring using a multi-agent system (MAS) has long been a focal point of cooperative control. It is still a challenging task to provide cost-effective services for potential emergencies in surface environments. This paper explores the transformation of a general surface into a two-dimensional (2D) disk through the construction of a conformal mapping. Multiple agents are strategically deployed within the mapped convex disk, followed by mapping back to the original surface environment. This approach circumvents the complexities associated with handling the difficulties and intricacies of path planning. Technical analysis encompasses the design of distributed control laws and the method to eliminate distortions introduced by the mapping. Moreover, the developed coverage algorithm is applied to a scenario of monitoring surface deformation. Finally, the effectiveness of the proposed algorithm is validated through numerical simulations.

Multi-Agent Coverage Control on Surfaces Using Conformal Mapping

TL;DR

This work addresses real time multi-agent coverage on general surfaces by transforming a surface into a disk through a conformal mapping, enabling planar distributed control and avoiding direct 3D path planning. It introduces a distortion correction framework based on the Beltrami coefficient and uses the Linear Beltrami Solver with a Cayley transform to produce a bijective conformal map that minimizes distortion. A disk based coverage optimization is developed where agents partition the disk via Voronoi cells, with the control law and the mapping back to preserving continuity and preventing partition intersections. Simulations on surface deformation scenarios demonstrate effective coverage and deformation detection, suggesting practical applications in geohazard monitoring such as landslide precursor feature extraction.

Abstract

Real-time environmental monitoring using a multi-agent system (MAS) has long been a focal point of cooperative control. It is still a challenging task to provide cost-effective services for potential emergencies in surface environments. This paper explores the transformation of a general surface into a two-dimensional (2D) disk through the construction of a conformal mapping. Multiple agents are strategically deployed within the mapped convex disk, followed by mapping back to the original surface environment. This approach circumvents the complexities associated with handling the difficulties and intricacies of path planning. Technical analysis encompasses the design of distributed control laws and the method to eliminate distortions introduced by the mapping. Moreover, the developed coverage algorithm is applied to a scenario of monitoring surface deformation. Finally, the effectiveness of the proposed algorithm is validated through numerical simulations.
Paper Structure (10 sections, 6 theorems, 22 equations, 8 figures, 1 algorithm)

This paper contains 10 sections, 6 theorems, 22 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coverage Performance Function
  4. Mapping construction
  5. Main Results
  6. Distortion Correction
  7. Algorithm for Disk Conformal Mapping
  8. Coverage Optimization
  9. Simulations
  10. Conclusions

Key Result

Lemma 3.1

(Jacobian choi2015fast) For the conformal mapping $f$, the Jacobi matrix of $f$ is $J_{f}=\left | \frac{\partial f}{\partial z} \right | ^{2} (1-\left | \mu _{f} \right | ^{2} )$, due to $\left \| \mu _{f} \right \| _{\infty } < 1$, one obtains $\frac{\partial f}{\partial z} ^{2} \ne 0$ and $(1-\

Figures (8)

  • Figure 1: Illustration on surface environments.
  • Figure 2: Schematic diagram reflecting the degree of contraction.
  • Figure 3: Schematic representation of weighting factors $k_{uv}$.
  • Figure 4: Comparison between original and deformed surfaces.
  • Figure 5: Multi-agent coverage on the disk.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 2 more