Limitations of employing undirected information flow graphs for the maintenance of rigid formations for heterogeneous vehicles

In this paper, we consider the problem of decentralized control of a collection of heterogeneous vehicles trying to maintain a rigid formation. In a rigid formation of vehicles, the separation distance between any pair of vehicles does not change throughout their motion and equals a pre-specified value that defines the formation. In each formation, there is a set of reference vehicles and a set of following vehicles. A reference vehicle may not be influenced by the motion of the following vehicle, but the motion of the following vehicle is influenced by the motion of the reference vehicle. Since we will be primarily dealing with translational maneuvers of the formation, we will consider formations where there is only one reference vehicle whose motion specifies the desired motion for all the vehicles in the formation. Each following vehicle attempts to maintain a specified constant safe distance from its adjacent vehicles in the collection. We call a vehicle B adjacent to a vehicle A if the relative position of vehicle B is known to vehicle A either by communication or by sensing. We only consider information flow graphs that are undirected when restricted to the set of following vehicles, i.e., graphs where a following vehicle A is adjacent to a following vehicle B if and only if the following vehicle B is adjacent to the following vehicle A. We model each vehicle as a point mass of one unit with two types of forces act on each vehicle – a controlled force, which is the output of an actuation system and a disturbing force over which there is no control. The actuation system is assumed to be linear and time invariant, and may be representable by a rational and strictly proper transfer function. The input to the control system of a vehicle in the formation is based on the error in maintaining the desired separation from its adjacent vehicles and the output of the control system is a command to the actuator. An earlier result showed that spacing errors due to disturbances amplify in a collection of identical vehicles with identically structured linear controllers if the reference vehicle information is not available to Ω(n) (A function f(n) is Ω(g(n)) if there is a constant c > 0 and a N > 0 such that ∣f(n)∣ > c∣g(n)∣ for all n > N.) vehicles, n being the size of the collection [1]. From the viewpoint of tolerance to communication failures, it is therefore necessary that there be at least two vehicles in the formation that are adjacent to Ω(n) vehicles in the formation. In this paper, we consider a broad class of heterogeneously structured controllers defined by a pre-specified bound, say Bc > 0 as follows: the allowable controller in conjunction with the actuation systems on-board any vehicle will produce a force no greater than Bc in response to a constant input to the control system. If this class of controllers were to be employed with the aforesaid undirected information flow graphs, we show that there is a critical size of the formation beyond which the motion of the vehicles in the formation will be unstable. This result shows the inability to scale such controllers for maintenance of rigid formations in conjunction with undirected information flow graphs.