Rigidity and persistence of three and higher dimensional formations

Hendrickx, Julien [UCL] Fidan, B. [Australian National University] Yu, C. [Australian National University] Anderson, B.D.O. [Australian National University] Blondel, Vincent [UCL]

In this paper,we generalize the notion of persistence, which has been originally introduced for two-dimensional formations, to Rd for d ≥ 3, seeking to provide a theoretical framework for real world applications, which often are in three-dimensional space as opposed to the plane. We verify that many of the properties of rigid and/or persistent formations established in R2 are also valid for higher dimensions. Analysing the closed subgraphs and directed paths in persistent graphs, we derive some further properties of persistent formations. We also provide an easily checkable necessary condition for persistence.