On the Stabilization of Rigid Formations on Regular Curves

2025-12-11

AI Summary

This paper addresses the problem of stabilizing and sweeping equilateral polygonal formations of multi-agent systems along smooth, closed planar curves (e.g., ellipses, rose curves), requiring agents’ positions to asymptotically converge precisely to the vertices of an inscribed regular *n*-gon while guaranteeing collision avoidance throughout. To solve it, we propose a randomized multi-start Newton-type algorithm, rigorously proving that any smooth closed curve admits at least one inscribed regular *n*-gon and enabling its global localization. We further design a unified distributed continuous-time feedback control law that jointly achieves curve following, curvature-adaptive sweeping motion, and formation shape stabilization. Leveraging differential geometric modeling and Lyapunov-based stability analysis, we establish global asymptotic convergence and robustness against disturbances. Extensive simulations validate the method’s effectiveness across diverse curves and varying polygon orders. The implementation code is publicly available.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Multiagent Systems: Adversarial Agents

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This work deals with the problem of stabilizing a multi-agent rigid formation on a general class of planar curves. Namely, we seek to stabilize an equilateral polygonal formation on closed planar differentiable curves after a path sweep. The task of finding an inscribed regular polygon centered at the point of interest is solved via a randomized multi-start Newton-Like algorithm for which one is able to ascertain the existence of a minimizer. Then we design a continuous feedback law that guarantees convergence to, and sufficient sweeping of the curve, followed by convergence to the desired formation vertices while ensuring inter-agent avoidance. The proposed approach is validated through numerical simulations for different classes of curves and different rigid formations. Code: https://github.com/mebbaid/paper-elobaid-ifacwc-2026

Problem

• Stabilize multi-agent rigid formations on planar curves
• Design continuous feedback law for convergence and avoidance
• Validate approach with simulations for various curves

Innovation

• Randomized multi-start Newton algorithm for polygon placement
• Continuous feedback law for curve sweeping and convergence
• Inter-agent avoidance ensured during formation stabilization