无需领导者的机器人编队可实现任意平面仿射变换运动。
Leaderless Collective Motion in Affine Formation Control over the Complex Plane
- 通过调整拉普拉斯矩阵权重，实现无领袖编队的动态运动控制。
- 理论证明可精确生成平移、旋转、缩放和剪切等复合运动形态。
- 采用复数建模简化二维编队设计，适合多智能体系统研究者。
我们提出一种方法，通过修改用于实现机器人集群静态编队的拉普拉斯矩阵原始权重，来控制平面内仿射编队的集体运动。所得集体运动被表征为参考构型（或形状）的时间变仿射变换。与传统领导者-跟随者策略不同，本无领袖方案允许各智能体保持不同且可能随时间变化的速度，从而实现更广泛的集体运动，包括参考形状的所有平移、旋转、缩放和剪切的线性组合。我们的分析提供了描述该集体运动的解析解，明确设计了决定此运动的特征向量和特征值，这些参数作为新拉普拉斯矩阵中修改后权重的函数。为便于在二维空间中对仿射编队进行更易处理的分析与设计，我们提出使用复数表示所有相关信息。模拟实验最多包含20个智能体，验证了理论结果。
原文摘要 · Abstract (English)
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.
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