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This paper introduces a leaderless control strategy for robot swarms to achieve time-varying affine transformations of a reference shape in 2D. By modifying the weights of the Laplacian matrix, the method enables collective motion characterized by translations, rotations, scaling, and shearing without relying on a leader-follower architecture. The authors provide an analytic solution for the collective motion, explicitly designing eigenvectors and eigenvalues as functions of the modified Laplacian weights, and validate the approach through simulations using complex number representations for simplified analysis.
Unlock complex 2D robot swarm maneuvers – translations, rotations, scaling, and shearing – without any designated leader.
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.
