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A Continuification-Based Control Solution for Large-Scale Shepherding

Beniamino Di Lorenzo, Gian Carlo Maffettone, Mario di Bernardo

TL;DR

This approach transforms the microscopic agent-based dynamics into a macroscopic continuum model via partial differential equations (PDEs), which enables efficient, scalable control design for the herders' behavior, with guarantees of global convergence.

Abstract

In this paper, we address the large-scale shepherding control problem using a continuification-based strategy. We consider a scenario in which a large group of follower agents (targets) must be confined within a designated goal region through indirect interactions with a controllable set of leader agents (herders). Our approach transforms the microscopic agent-based dynamics into a macroscopic continuum model via partial differential equations (PDEs). This formulation enables efficient, scalable control design for the herders' behavior, with guarantees of global convergence. Numerical and experimental validations in a mixed-reality swarm robotics framework demonstrate the method's effectiveness.

A Continuification-Based Control Solution for Large-Scale Shepherding

TL;DR

This approach transforms the microscopic agent-based dynamics into a macroscopic continuum model via partial differential equations (PDEs), which enables efficient, scalable control design for the herders' behavior, with guarantees of global convergence.

Abstract

In this paper, we address the large-scale shepherding control problem using a continuification-based strategy. We consider a scenario in which a large group of follower agents (targets) must be confined within a designated goal region through indirect interactions with a controllable set of leader agents (herders). Our approach transforms the microscopic agent-based dynamics into a macroscopic continuum model via partial differential equations (PDEs). This formulation enables efficient, scalable control design for the herders' behavior, with guarantees of global convergence. Numerical and experimental validations in a mixed-reality swarm robotics framework demonstrate the method's effectiveness.

Paper Structure

This paper contains 17 sections, 3 theorems, 31 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. MODEL AND PROBLEM STATEMENT
  3. The model
  4. Problem statement
  5. CONTROL DESIGN
  6. Continuification
  7. Macroscopic control design
  8. Choice of $N^H$
  9. Design of $\mathbf{u}$
  10. herders' control design
  11. Targets' stability analysis
  12. Discretization
  13. VALIDATION
  14. Numerical validation
  15. Experimental validation
  16. ...and 2 more sections

Key Result

Proposition 1

Given $\Bar{\rho}^H$ as defined in eq:rho_bar_L and setting $A = \min_\mathbf{x} H(\mathbf{x})$, the quantity represents the minimum herders' mass required to make the problem feasible.

Figures (6)

  • Figure 1: Continuification control pipeline, inspired by nikitin2021continuation.
  • Figure 2: Feasibility plot showing the minimum amount of herders' mass $\widehat{M}^H$ varying $\mathbf{k}$ and $D$ of the Von-Mises distribution \ref{['eq:von_mises']} with fixed kernel length $L=\pi$. The red line denotes the curve where $\widehat{M}^H$ becomes greater than $1$. $\widehat{M}^H$ has been saturated to 1 for visualization purposes.
  • Figure 3: (a) Macroscopic feed-forward control scheme. (b) Detail of the feasibility analysis step.
  • Figure 4: Implementation scheme: adaptation of the macroscopic controller described in Sec. \ref{['sec:macro_control_design']} to cope with a discrete set of herders and targets; the "Discretization" and "Density estimation" blocks work across the micro and macro scale.
  • Figure 5: (a) Desired targets' density $\bar{\rho}^T$. (b) Desired herders' density $\bar{\rho}^H$. (c) Final agents' configuration: the goal region is represented by the dashed circle, blue diamonds represent the herders, magenta dots are the targets. (d) Percentage of targets inside the goal region.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • Theorem 1: Herders' global exponential convergence
  • Theorem 2: Targets' global exponential stability