Limitations of Employing Undirected Information Flow Graphs for the Maintenance of Rigid Vehicular Formations

In this paper, we consider the problem of decentralized control of a collection of homogeneous vehicles trying to maintain a rigid formation. Each vehicle attempts to maintain a specified constant safe distance from its adjacent vehicles in the collection. We consider an identical structure for each decentralized controller so that it is simpler from an implementation viewpoint as it does not depend on collection size or vehicle indices. We call a vehicle B adjacent to vehicle A if the relative position of vehicle B is known to vehicle A either by communication or by sensing. In this paper, we only consider undirected information flow graphs, i.e., graphs where vehicle A is adjacent to vehicle B if and only if vehicle B is adjacent to vehicle A. We consider a point mass model for each vehicle and assume the actuation transfer function, which relates the control input to the force supplied to the vehicle, to be a strictly proper rational transfer function. It is known that spacing errors due to disturbances amplify if the reference vehicle information is not available to Ω(n) vehicles, n being the size of the collection [1]. In this paper, we generalize this result to show the following: If there are two or more vehicles in the collection that are adjacent to Ω(n) vehicles, then there is a critical size N* so that the motion of the collection will be unstable if the size of the collection exceeds N*. Practical issues of fault tolerance indicate that there be at least two vehicles that are adjacent to Ω(n) vehicles in the collection. We also further show that the use of a kinematic vehicle model for analysis of disturbance propagation yields results which may not agree with what is observed in practice and hence are inappropriate.