Remarks on a Simple Control Law for Point Robot Formations with Exponential Complexity

This paper considers a simple decentralized control law which stabilizes rigid formations of point robots in a way that is consistent with recent work B.D.O. Anderson and a number of others. The feedback law is simple in that it is shown to make parsimonious use of feedback information in a sense that will be made precise, and it is also simple in that it is the obvious one would try to control the relative positions of a group of robots. The law is exponentially complex, however, in the sense that the number of stable equilibria grows exponentially as a function of the number of point robots involved in the formation. Assuming only one formation of the exponentially many possibilities is of interest, we discuss the problem of achieving the desired formation using the feedback law. The paper will also discuss the combinatorial problem of determining all possible formation topologies as well as the critical point theory of the proposed control law