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Node-wise monotone barrier coupling law for formation control

Jin Gyu Lee, Cyrus Mostajeran, Graham Van Goffrier

TL;DR

This work introduces a node-wise monotone barrier coupling law for agents on the circle, enabling formation control with multiple, switchable central patterns. By embedding a barrier effect in the coupling, the state space is partitioned into finitely many regions, each admitting a unique phase-locked formation, with the common frequency $\\bar{\\omega}$ determined by $F(\\bar{\\omega}) = 0$. The analysis establishes invariant sets, a finite count of central patterns, and a framework for viability and design—showing how to realize or switch patterns via kicks and to minimize interconnections while preserving desired formations. The approach broadens prior edge-based methods, offers robustness to heterogeneity, and provides practical guidance for applications such as drone clusters, where rapid, reliable pattern transitions are valuable.

Abstract

We study a node-wise monotone barrier coupling law, motivated by the synaptic coupling of neural central pattern generators. It is illustrated that this coupling imitates the desirable properties of neural central pattern generators. In particular, the coupling law 1) allows us to assign multiple central patterns on the circle and 2) allows for rapid switching between different patterns via simple `kicks'. In the end, we achieve full control by partitioning the state space by utilizing a barrier effect and assigning a unique steady-state behavior to each element of the resulting partition. We analyze the global behavior and study the viability of the design.

Node-wise monotone barrier coupling law for formation control

TL;DR

This work introduces a node-wise monotone barrier coupling law for agents on the circle, enabling formation control with multiple, switchable central patterns. By embedding a barrier effect in the coupling, the state space is partitioned into finitely many regions, each admitting a unique phase-locked formation, with the common frequency determined by . The analysis establishes invariant sets, a finite count of central patterns, and a framework for viability and design—showing how to realize or switch patterns via kicks and to minimize interconnections while preserving desired formations. The approach broadens prior edge-based methods, offers robustness to heterogeneity, and provides practical guidance for applications such as drone clusters, where rapid, reliable pattern transitions are valuable.

Abstract

We study a node-wise monotone barrier coupling law, motivated by the synaptic coupling of neural central pattern generators. It is illustrated that this coupling imitates the desirable properties of neural central pattern generators. In particular, the coupling law 1) allows us to assign multiple central patterns on the circle and 2) allows for rapid switching between different patterns via simple `kicks'. In the end, we achieve full control by partitioning the state space by utilizing a barrier effect and assigning a unique steady-state behavior to each element of the resulting partition. We analyze the global behavior and study the viability of the design.
Paper Structure (5 sections, 2 theorems, 9 equations)

This paper contains 5 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Motivation of the Barrier Effect
  3. Graph Theoretical Preliminaries
  4. Node-Wise Monotone Barrier Coupling Law
  5. Analysis of Central Patterns

Key Result

Theorem 1

Under Assumptions assum:graph and assum:coupling, for any given $\{n_i\}_\mathcal{N}$, each solution trajectory of eq:circ_con starting from the extended space $\Theta_{\{n_i\}_\mathcal{N}}^\mathbb{R}$uniquely exists and resides inside $\Theta_{\{n_i\}_\mathcal{N}}^\mathbb{R}$(hence, forward complet In particular, there exists a steady-state solution of eq:circ_con corresponding to $(\bar{\omega},

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Theorem 2