Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Density-Driven Optimal Control for Non-Uniform Area Coverage in Decentralized Multi-Agent Systems Using Optimal Transport

Sungjun Seo, Kooktae Lee

TL;DR

The paper tackles non-uniform area coverage in multi-agent systems by formulating Density-Driven Optimal Control (D$^2$OC), which fuses Optimal Transport with decentralized control to track a mission-specific density. It constructs a three-stage framework (Stage A: optimal control, Stage B: weight update, Stage C: weight sharing) and provides analytic solutions for linear dynamics while accommodating nonlinear systems via a horizon-based, Lagrangian approach; a novel decentralized weight-sharing rule reduces work redundancy. Simulations against SMC and D$^2$C show that D$^2$OC achieves lower Wasserstein distances to the reference density and requires less computation, while handling energy constraints and heterogeneous agents. The work demonstrates practical, scalable, and decentralized density-oriented coverage with potential for applications in surveillance, environmental monitoring, and search-and-rescue, supported by public MATLAB code.

Abstract

This paper addresses the fundamental problem of non-uniform area coverage in multi-agent systems, where different regions require varying levels of attention due to mission-dependent priorities. Existing uniform coverage strategies are insufficient for realistic applications, and many non-uniform approaches either lack optimality guarantees or fail to incorporate crucial real-world constraints such as agent dynamics, limited operation time, the number of agents, and decentralized execution. To resolve these limitations, we propose a novel framework called Density-Driven Optimal Control (D2OC). The central idea of D2OC is the integration of optimal transport theory with multi-agent coverage control, enabling each agent to continuously adjust its trajectory to match a mission-specific reference density map. The proposed formulation establishes optimality by solving a constrained optimization problem that explicitly incorporates physical and operational constraints. The resulting control input is analytically derived from the Lagrangian of the objective function, yielding closed-form optimal solutions for linear systems and a generalizable structure for nonlinear systems. Furthermore, a decentralized data-sharing mechanism is developed to coordinate agents without reliance on global information. Comprehensive simulation studies demonstrate that D2OC achieves significantly improved non-uniform area coverage performance compared to existing methods, while maintaining scalability and decentralized implementability.

Density-Driven Optimal Control for Non-Uniform Area Coverage in Decentralized Multi-Agent Systems Using Optimal Transport

TL;DR

The paper tackles non-uniform area coverage in multi-agent systems by formulating Density-Driven Optimal Control (DOC), which fuses Optimal Transport with decentralized control to track a mission-specific density. It constructs a three-stage framework (Stage A: optimal control, Stage B: weight update, Stage C: weight sharing) and provides analytic solutions for linear dynamics while accommodating nonlinear systems via a horizon-based, Lagrangian approach; a novel decentralized weight-sharing rule reduces work redundancy. Simulations against SMC and DC show that DOC achieves lower Wasserstein distances to the reference density and requires less computation, while handling energy constraints and heterogeneous agents. The work demonstrates practical, scalable, and decentralized density-oriented coverage with potential for applications in surveillance, environmental monitoring, and search-and-rescue, supported by public MATLAB code.

Abstract

This paper addresses the fundamental problem of non-uniform area coverage in multi-agent systems, where different regions require varying levels of attention due to mission-dependent priorities. Existing uniform coverage strategies are insufficient for realistic applications, and many non-uniform approaches either lack optimality guarantees or fail to incorporate crucial real-world constraints such as agent dynamics, limited operation time, the number of agents, and decentralized execution. To resolve these limitations, we propose a novel framework called Density-Driven Optimal Control (D2OC). The central idea of D2OC is the integration of optimal transport theory with multi-agent coverage control, enabling each agent to continuously adjust its trajectory to match a mission-specific reference density map. The proposed formulation establishes optimality by solving a constrained optimization problem that explicitly incorporates physical and operational constraints. The resulting control input is analytically derived from the Lagrangian of the objective function, yielding closed-form optimal solutions for linear systems and a generalizable structure for nonlinear systems. Furthermore, a decentralized data-sharing mechanism is developed to coordinate agents without reliance on global information. Comprehensive simulation studies demonstrate that D2OC achieves significantly improved non-uniform area coverage performance compared to existing methods, while maintaining scalability and decentralized implementability.

Paper Structure

This paper contains 16 sections, 6 theorems, 42 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary and Problem description
  3. Illustrative Example for Non-Uniform Coverage
  4. Optimal Transport Theory and Wasserstein Distance
  5. Multi-Agent Coverage Strategy Using Wasserstein Distance
  6. Density-Driven Optimal Control (D$^2$OC)
  7. Optimal Control Stage
  8. Linear Time-Invariant system case
  9. Nonlinear system case
  10. Weight Update Stage
  11. Weight-Sharing Stage for Decentralized Control
  12. Simulation
  13. Performance Comparison between SMC and D$^2$OC
  14. Performance Comparison between D$^2$C and D$^2$OC
  15. Performance Comparison in Weight Sharing
  16. ...and 1 more sections

Key Result

Proposition 1

Let the symmetric matrix E be defined by where $E_{ij}$ is a real submatrix, $E_{11}\in \mathbb{R}^{n\times n}$ and $E_{33}\in \mathbb{R}^{m\times m}$ are square matrices, and $E_{12} \in \mathbb{R}^{n\times n}$ is invertible. Then, the matrix $E$ is invertible if $E_{11}$ is symmetric positive semidefinite and $E_{33}$ is positive definit where

Figures (13)

  • Figure 1: Uniform vs. Non-Uniform Coverage: Satellite image of Petabo, Indonesia Nishan2018. (a) before the tsunami; (b) after the tsunami; (c) uniform coverage path planning for a search and rescue mission; (d) a probability map of the victims' location based on pre-acquired information; (e) discretized point clouds generated from the probability map; (f) non-uniform coverage path utilizing the probability (or density)-based data.
  • Figure 2: Schematic drawing of each stage in the D$^2$OC scheme: (a) Stage A - Optimal control stage: Local sample-points (red circles) are chosen from sample-points (hollow black circles). Optimal control input ${}^{r}u^{*|k}$ is applied and the agent moves to the new location ${}^{r}y^{k+1}$; (b) Stage B - Weight update stage: The optimal transportation ${}^{r}\gamma^{*|k+1}_j$ taken out of the remaining weight ${}^{r}\beta^k_j$ is transported to the newly created agent-point (hollow green circles).
  • Figure 3: Schematic illustrating the exchange of the coverage progress in the proposed weight-sharing method.
  • Figure 4: Schematic illustrating both weight-sharing methods for the scenario of $k=k_2$ in Fig. \ref{['fig: schematic_proposed_sharing']}.
  • Figure 5: Trajectory comparison between the SMC method and the D$^2$OC scheme. The initial and final locations of agents are indicated by blue-cross and yellow-circle marks, respectively: (a) D$^2$OC with a terminal time of 20 seconds; (b) D$^2$OC with a terminal time of 200 seconds; (c) SMC with a terminal time of 20 seconds; (d) SMC with a terminal time of 200 seconds.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Remark 4
  • ...and 6 more