多智能体网络中广义速度刚性的宏观运动模式
Macroscopic Motion Patterns from Generalized Velocity Rigidity in Multi-Agent Networks
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中文总结 AI 辅助
本文提出广义速度刚性概念，将刚性约束从位置空间移至速度空间，证明通过速度刚性矩阵零空间的代数约束可精确生成平移、旋转、缩放等宏观运动模式，实现多智能体群体的灵活形状控制。
中文摘要 AI 辅助
传统图刚性理论在位置空间中施加结构约束，有效地将多智能体编队限制为静态几何形状。在本文中，我们引入了一个根本性的范式转变，将刚性约束直接应用于智能体的连续时间速度空间。我们提出了广义速度刚性的概念，证明多智能体网络的宏观物理运动模式完全由底层静态图拓扑决定。通过将网络的加速度剖面严格限制在速度刚性矩阵的零空间中，我们将该零空间的平凡运动数学映射为精确的物理轨迹。具体来说，我们证明平移、旋转和缩放平凡运动分别无缝集成为内聚的弯曲群集飞行、同步螺旋和圆形SO(d)轨道运动以及指数空间位似变换。此外，我们分析了速度空间奇异性，表明速度共识引发有效的拓扑收缩而非结构失效。最后，我们提供了全面的数值模拟来验证这些理论映射，证明复杂的宏观机动可以仅通过速度空间中的代数约束来编排，为多智能体群体解锁了前所未有的空间形状灵活性。
英文摘要
Traditional graph rigidity theory enforces structural constraints in the position space, effectively restricting a multi-agent formation to a static geometric shape. In this paper, we introduce a fundamental paradigm shift by applying rigidity constraints directly to the agents' continuous-time velocity space. We propose the concept of generalized velocity rigidity, demonstrating that the macroscopic physical motion patterns of a multi-agent network are entirely dictated by the underlying static graph topology. By strictly confining the network's acceleration profile to the null space of the velocity rigidity matrix, we mathematically map the trivial motions of this null space into exact physical trajectories. Specifically, we prove that translational, rotational, and scaling trivial motions seamlessly integrate into cohesive curved flocking, synchronized helical and circular $\mathrm{SO}(d)$ orbiting, and exponential spatial homothety, respectively. Furthermore, we analyze velocity-space singularities, showing that velocity consensus induces a valid topological contraction rather than a structural failure. Finally, we provide comprehensive numerical simulations to validate these theoretical mappings, demonstrating that complex macroscopic maneuvers can be orchestrated purely through algebraic constraints in the velocity space, unlocking unprecedented spatial shape flexibility for multi-agent swarms.
发表机构
- City University of Hong Kong（香港城市大学）
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