Rigid graph control architectures for autonomous formations

This article sets out the rudiments of a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles. The theory rests on ideas of rigid graph theory, some but not all of which are old. The theory, however, has some gaps in it, and their elimination would help in applications. Some of the gaps in the relevant graph theory are as follows. First, there is as yet no analogue for three-dimensional graphs of Laman's theorem, which provides a combinatorial criterion for rigidity in two-dimensional graphs. Second, for three-dimensional graphs there is no analogue of the two-dimensional Henneberg construction for growing or deconstructing minimally rigid graphs although there are conjectures. Third, global rigidity can easily be characterized for two-dimensional graphs, but not for three-dimensional graphs.