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.

翻译：我们提出了一种在平面上实现仿射编队群集机动的方法，该方法通过修改用于实现机器人群体静态编队的拉普拉斯矩阵的原始权重实现。具体而言，由此产生的群集运动被表征为参考构型（或形状）的时变仿射变换。与传统的领航-跟随策略不同，我们的无领导方案允许智能体保持不同且可能时变的速度，从而支持更广泛的群集运动，包括参考形状的所有线性组合（平移、旋转、缩放和剪切）。我们的分析给出了控制最终群集运动的解析解，通过显式设计定义该运动的特征向量和特征值，并将其表示为新拉普拉斯矩阵中修改权重的函数。为便于二维仿射编队的分析与设计，我们提出使用复数表示所有相关信息。最多20个智能体的仿真验证了理论结果。