Decentralized Formation Control Part I: Geometric Aspects

In this paper, we develop new methods for the analysis of decentralized control systems and we apply them to formation control problems. The basic set-up consists of a system with multiple agents represented by the nodes of a directed graph whose edges represent an available communication channel for the agents. We address the question of whether the information flow defined by the graph is sufficient for the agents to accomplish a given task. Formation control is concerned with problems in which agents are required to stabilize at a given distance of other agents. In this context, the graph of a formation encodes both the information flow, as described above, and the distance constraints, by fixing the lengths of the edges. A formation is said to be rigid if it cannot be continuously deformed with the distance constraints satisfied; a formation is minimally rigid if no distance constraint can be omitted without the formation losing its rigidity. Hence, the graph underlying minimally rigid formation provides just enough constraints to yield a rigid formation. An open question we will settle is whether the information flow afforded by a minimally rigid graph is sufficient to define global stabilizing control laws. We show that the answer is negative in the case of directed information flow. In this first part, we establish basic properties of formation control in the plane. Formations and the associated control problems are defined modulo rigid transformations. This fact has strong implications on the geometry of the space of formations and on the feedback control laws, since they need to respect this invariance. We study both aspects here. In detail, we show that the space of formations of n agents is CP(n − 2) × (0,∞) where CP(n) is the complex-projective space of complex dimension n. We subsequently illustrate how the non-trivial topology of this space relates to the parametrization of the formation by inter-agent distances. We then establish conditions feedback control laws need to satisfy in order to yield a closed-loop system that respects both the invariance under the action of the Euclidean group and the constraints on the information flow. ∗M.-A. Belabbas is with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 [email protected]