The combinatorial graph theory of structured formations

Graphical methods for the design and analysis of distributed control laws for multiagent robot formations have been proposed by many researchers. This paper extends recent results on the rigidity theory of directed graphs that have been developed to characterize feedback patterns used in a class of formation control problems. While the theory arises from an attempt to understand rigid formation motions, it is argued that the main ideas can be applied to a wider class of problems in which it is desired to make maximally parsimonious use of sensor information in controlling the relative positions of each robot throughout the motion. The paper is based on the recently established characterization of a family of digraphs which describe parsimonious relative distance sensing patterns that are used in controlling rigid robot formations of arbitrary geometry. We discuss the complexity of the corresponding distributed control algorithms in terms of an enumeration of the formation digraphs. The enumeration problem is discussed in detail, and explicit results are given for low-order formations. An algorithmic procedure for carrying out the enumeration is given, but a general closed-form solution remains to be found.