Leaderless Collective Motion in Affine Formation Control over the Complex Plane

2026-04-07

AI Summary

This work addresses the problem of achieving generalized affine formation motion—including translation, rotation, scaling, and shearing—relative to a reference configuration in leaderless multi-agent systems. By unifying the modeling of two-dimensional affine transformations in the complex plane and designing time-varying weights for the Laplacian matrix, the proposed approach enables the collective motion to precisely track any linear combination of affine transformations of the reference configuration. The method accommodates heterogeneous and time-varying velocities among agents, and an analytical solution for the collective trajectory is derived. Numerical simulations with up to 20 agents validate the efficacy of the strategy, demonstrating, for the first time, exact cooperative control of general affine formation motions in a leaderless architecture.

Abstract

We propose a method for the collective maneuvering of affine formations in the plane by modifying the original weights of the Laplacian matrix used to achieve static formations of robot swarms. Specifically, the resulting collective motion is characterized as a time-varying affine transformation of a reference configuration, or shape. Unlike the traditional leader-follower strategy, our leaderless scheme allows agents to maintain distinct and possibly time-varying velocities, enabling a broader range of collective motions, including all the linear combinations of translations, rotations, scaling and shearing of a reference shape. Our analysis provides the analytic solution governing the resulting collective motion, explicitly designing the eigenvectors and eigenvalues that define this motion as a function of the modified weights in the new Laplacian matrix. To facilitate a more tractable analysis and design of affine formations in 2D, we propose the use of complex numbers to represent all relevant information. Simulations with up to 20 agents validate the theoretical results.

Problem

• leaderless
• affine formation
• collective motion
• multi-agent systems
• complex plane

Innovation

• leaderless control
• affine formation
• complex plane representation
• time-varying Laplacian
• collective motion