Graph Theoretic Methods in the Stability of Vehicle Formations G. Lafferriere, J. Caughman, A. Williams gerardoL@pdx.edu, caughman@pdx.edu, ancaw@pdx.edu Abstract— This paper investigates the stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified (undirected) communication graph, G. The feedback control is based only on relative information about vehicle states shared via the communication links. We prove that a linear stabilizing feedback always exists provided that G is connected. Moreover, we show how the rate of convergence to formation is governed by the size of the smallest positive eigenvalue of the Laplacian of G. Several numerical simulations are used to illustrate the results. I. I NTRODUCTION From minisatellites to drone planes, the need to control the coordinated motion of multiple autonomous vehicles has received increasing attention recently [2], [3], [4], [6], [7], [8], [13], [14], [16]. One of the main goals is to distribute the control activity as much as possible while still achieving a coordinated objective. The objective investigated in this paper is that of attaining a moving formation. That is, the goal of the vehicles is to achieve and maintain pre-specified relative positions and orientations with respect to each other. Each vehicle is provided information only from a subset of the group. The specific subset is given through the set of “neighbors” in the communication graph. This graph need not be related to the actual formation geometry. The feedback scheme investigated is inspired by the motion of aggregates of individuals in nature. Flocks of birds and schools of fish achieve coordinated motions of large numbers of individuals without the use of a central controlling mechanism [10]. A computer graphics model to simulate flock behavior is presented in [11]. In a different context, a simple model is proposed in [15] that captures the observed motions of self-driven particles. These models employ feedback laws in which the motions of nearest neighbors are averaged. The notion of a communication graph is introduced in [2], and an averaging feedback law is proposed based on the flow of information. Keeping with the information flow approach, a proba- bilistic model for communication losses is introduced in [4] where it is shown that if the probability of losing a link is not too low, the formation is still achieved. A discrete averaging law is used in [6] to achieve a common heading. There, the communication graphs are allowed to change, All three authors are with the Department of Mathematics and Statistics, Portland State University, Portland, OR 97207. G. Lafferriere and A. Williams were supported in part by a grant from Honeywell, Inc. provided that they remain connected. In [7], [8], [9], [12], [14] the authors use artificial potentials to generate feedback laws. The resulting nonlinear feedback laws can be shown to stabilize the formation under various geometric consistency criteria. In this paper we study the communication graph ap- proach. We consider a general vehicle model and use state- space techniques to prove that stabilizability of a formation can be achieved, provided that the communication graph is connected. We show that the rate of convergence to formation is governed by the smallest positive eigenvalue of the Laplacian matrix of the communication graph. We also demonstrate how, for a fixed feedback gain matrix, convergence can be improved by choosing alternative com- munication graphs. The paper is organized as follows. In section II we set up the basic model. The relevant graph theoretic definitions and results are collected in section III. The main results are proved in section IV. Numerical simulations are presented in section V. II. MODEL We assume given N vehicles with the same dynamics ˙ x i = A veh x i + B veh u i i = 1 ... N x i ∈ R 2n where the entries of x i represent n configuration variables for vehicle i and their derivatives. We are also given a graph G which captures the com- munication links between vehicles (see next section for precise definitions of graph theoretic concepts). Each vertex represents a vehicle and two vertices are connected by an edge if the corresponding vehicles communicate directly with each other. We refer to such vehicles as “neighbors”. For each vehicle i, J i denotes the set of its neighbors. In this model, each vehicle only knows its state relative to its neighbors. That is, u i is a function of x j − x i for each j ∈ J i . The study will focus on the simplest such rule: use as input an average based on the neighbors’ states. To make this more precise we make the following definitions. Definition 2.1: A formation is a vector h = h p ⊗  1 0  ∈ R 2nN (where ⊗ denotes the Kronecker product). The N vehicles are in formation h if there are vectors q, w ∈ R n such that (x p ) i − (h p ) i = q, (x v ) i = w, i = 1 ... N, where the