[Paper Review] Coverage Optimization using Generalized Voronoi Partition

K. R. Guruprasad, Debasish Ghose|arXiv (Cornell University)|Aug 25, 2009

Distributed Control Multi-Agent Systems31 references4 citations

TL;DR

This paper proposes a generalized Voronoi partition framework for optimal deployment of heterogeneous autonomous agents with varying sensor capabilities, using node functions instead of Euclidean distance to define Voronoi cells. The method ensures local optimality through a centroidal configuration and guarantees continuous partition evolution under specific initial conditions, validated via simulations for improved sensor coverage in multi-agent systems.

ABSTRACT

In this paper a generalization of the Voronoi partition is used for optimal deployment of autonomous agents carrying sensors with heterogeneous capabilities, to maximize the sensor coverage. The generalized centroidal Voronoi configuration, in which the agents are located at the centroids of the corresponding generalized Voronoi cells, is shown to be a local optimal configuration. Simulation results are presented to illustrate the presented deployment strategy.

Motivation & Objective

• To address the limitation of standard Voronoi partitions in handling sensors with heterogeneous capabilities in multi-agent coverage problems.
• To develop a generalized Voronoi partition using strictly decreasing analytic node functions in place of distance metrics to model sensor heterogeneity.
• To establish conditions under which the generalized Voronoi partition remains continuous during agent motion.
• To derive a control law that ensures local optimality via centroidal configuration of agents in generalized Voronoi cells.
• To validate the proposed method through simulations demonstrating improved sensor coverage in heterogeneous environments.

Proposed method

• The generalized Voronoi partition is defined using strictly decreasing analytic node functions $ f_i: \mathbb{R}^+ \to \mathbb{R} $, replacing the standard Euclidean distance measure.
• Each agent's Voronoi cell $ V^g_i $ is defined as the set of points $ q \in Q $ where $ f_i(\|p_i - q\|) \geq f_j(\|p_j - q\|) $ for all $ j \neq i $.
• A control law is proposed to drive agents toward the centroids of their generalized Voronoi cells, ensuring convergence to a local optimal configuration.
• The continuity of the generalized Voronoi partition is analyzed, with a condition (Condition A) identified to prevent discontinuities when agents with identical node functions approach each other.
• Lemmas and theorems are established to prove that the partition depends continuously on agent configuration $ \mathcal{P} $ if Condition A is satisfied.
• Theoretical analysis confirms that if agents start with distinct positions $ p_i(0) \neq p_j(0) $ for agents with identical $ f_i = f_j $, the control law prevents collision and maintains partition continuity.

Experimental results

Research questions

• RQ1 How can Voronoi-based coverage optimization be generalized to handle sensors with heterogeneous capabilities?
• RQ2 What conditions ensure the continuity of the generalized Voronoi partition during agent motion in multi-agent systems?
• RQ3 Can a control law be designed to drive agents to the centroids of their generalized Voronoi cells, achieving local optimality?
• RQ4 How does the use of node functions instead of distance metrics affect the topological structure of Voronoi cells?
• RQ5 Under what initial configurations does the generalized Voronoi partition remain stable and continuous during deployment?

Key findings

• The generalized centroidal Voronoi configuration, where agents are positioned at the centroids of their generalized Voronoi cells, is proven to be a local optimal solution for sensor coverage in heterogeneous systems.
• The generalized Voronoi partition depends continuously on agent configuration $ \mathcal{P} $ if Condition A is satisfied, which prevents agents with identical node functions from merging.
• The proposed control law ensures that agents with identical node functions do not collide, preserving partition continuity, provided they start at distinct positions.
• Theoretical analysis confirms that discontinuities in the generalized Voronoi partition only occur when agents with identical node functions converge, which is prevented under the given initial conditions.
• Simulations demonstrate the effectiveness of the proposed deployment strategy in achieving improved sensor coverage for heterogeneous agents.
• The method generalizes prior work on standard and weighted Voronoi partitions, extending applicability to a broader class of heterogeneous locational optimization problems.