Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: He, Ronghai, Wan, Changhuang
Institution: City University of Hong Kong
Subfield: Computer Networks and Communications
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213514210
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