  By an undirected rigid formation of mobile autonomous agents is meant a
formation based on graph rigidity in which each pair of "neighboring" agents is
responsible for maintaining a prescribed target distance between them. In a
recent paper a systematic method was proposed for devising gradient control
laws for asymptotically stabilizing a large class of rigid, undirected
formations in two dimensional space assuming all agents are described by
kinematic point models. The aim of this paper is to explain what happens to
such formations if neighboring agents have slightly different understandings of
what the desired distance between them is supposed to be or equivalently if
neighboring agents have differing estimates of what the actual distance between
them is. In either case, what one would expect would be a gradual distortion of
the formation from its target shape as discrepancies in desired or sensed
distances increase. While this is observed for the gradient laws in question,
something else quite unexpected happens at the same time. It is shown that for
any rigidity-based, undirected formation of this type which is comprised of
three or more agents, that if some neighboring agents have slightly different
understandings of what the desired distances between them are suppose to be,
then almost for certain, the trajectory of the resulting distorted but rigid
formation will converge exponentially fast to a closed circular orbit in
two-dimensional space which is traversed periodically at a constant angular
speed.
