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We provide a graph theoretical framework that allows us to formally define formations of multiple vehicles and the issues arising in uniqueness of graph realizations and its connection to stability of formations. The notion of graph rigidity is crucial in identifying the shape variables of a formation and an appropriate potential function associated with the formation. This allows formulation of meaningful optimization or nonlinear control problems for formation stabilization/tacking, in addition to formal representation of split, rejoin, and reconfiguration maneuvers for multi-vehicle formations. We introduce an algebra that consists of performing some basic operations on graphs which allow creation of larger rigid-by-construction graphs by combining smaller rigid subgraphs. This is particularly useful in performing and representing rejoin/split maneuvers of multiple formations in a distributed fashion.
Olfati‐Saber et al. (Wed,) studied this question.