Leaderless Collective Motion in Affine Formation Control over the Complex Plane

TL;DR

This paper introduces a leaderless method for controlling affine formations of robot swarms in the plane, enabling complex collective motions through Laplacian weight modifications and complex number representations.

Contribution

It presents a novel leaderless approach for affine formation control, explicitly designing the Laplacian eigenstructure for desired collective motions.

Findings

Analytic solutions for collective motion based on Laplacian eigenstructure.

Use of complex numbers simplifies analysis and design of 2D affine formations.

Simulations with up to 20 agents validate the theoretical framework.

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.