Stabilizing a Multi-Agent System to an Equilateral Polygon Formation

The problem of stabilizing a group of agents in the plane to a stationary formation is analyzed. A local control scheme is proposed to stabilize the agents to the vertices of an equilateral polygon. The centroid of the agents is stationary during the evolution. For three agents a full stability analysis is performed: If three agents start in distinct and non-collinear locations, they converge to the vertices of a stationary equilateral triangle, while maintaining a stationary centroid. I. I NTRODUCTION There has been a considerable amount of work on formation stabilization in the multi-agent systems litera ture. The two main types of formation stabilization that are studied are stabilization to a moving formation, and stabilization to a stationary formation. Much of the work has been on stabilizing a group of agents (most commonly unicycles) to a moving formation. For example, Justh and Krishnaprasad [1] develop a control law to stabilize two unicycles moving at constant speed to a common heading. In [8] by Paley et al., the same problem is approached by using potential functions to maintain spacing between unicycles. Marshall et al. [5] create a control strategy bas ed on cyclic pursuit in which the unicycles can converge to a circle formation. In formation the unicycles are moving around the circle, equally spaced. In [11], [12] by Tanner et al., moving formations are studied in the context of flocking. The agents are modeled as double integrators (i.e., the control input is the acceleration) and the stabil ity of a flocking control law is studied for both fixed and dynamic communication topologies. In the area of stabilization to a stationary formation there are some interesting results. Sugihara and Suzuki [10] propose a heuristic distributed algorithm to stabiliz e a group of agents (modeled as point masses) to stationary positions, equally spaced around a circle. Each agent adjusts its position based on the position of the nearest agent and the farthest agent. Through simulation, it is shown that the agents form a rough approximation of a circle. The formation stabilization problem has also been studied using graph theory, as in, for example, [6], [7]. In this work a formation is viewed as a rigid graph, where the links on the graph represent the distance constraints S.L. Smith is with the Department of Mechanical Engineering, U niversity of California at Santa Barbara, Santa Barbara, CA 93 106 USA (e-mail: stephen@engineering.ucsb.edu). M.E. Broucke and B.A. Francis are with the Department of Elect rical and Computer Engineering, University of Toronto, ON, Canada , M5S 3G4 (e-mail:{broucke, francis }@control.utoronto.ca). between agents. Problems such as determining the best way to split a large rigid formation into smaller rigid formations are studied in this framework. Another scheme for formation stabilization of point masses is given by Lin et al. [3]. This scheme requires that each agent be equipped with a compass, so that they share a common direction. If the agents have this property then a local control strategy can be designed to stabilize to any stationary formation. In [4] this idea is extended to unicycles. In this paper we look at the problem of stabilizing a group of agents to a stationary formation. We model the agents as point masses, and we uniquely identify each of then agents with a number between 1 and n. The position of the i agent is given by the vector zi = (xi, yi) in R. The input to each agent is a velocity vector ui: żi = ui. The agents’ positions can also be represented as points in the complex planezi = xi + jyi, i = 1, . , n. The agents are not equipped with a compass, and thus they do not share a common heading (i.e., the agents are disoriented). The fact that the agents are disoriented makes formation stabilization significantly more complicated then when the agents are oriented. The problem we address is to find a local control strategy such that for each i, agent i is stabilized to a distance b > 0 from agentsi+ 1 and i− 1. The organization of the paper is as follows. In Section II we briefly examine the strategy in [3] in order to contrast with the present scheme. In Section III we introduce the control strategy, which is based on the linear polygon shortening scheme of [9], and analyze the system for n agents. Finally, in Section IV we study the special case of three agents and show that they stabilize to an equilateral triangle. II. FORMATION STABILIZATION WITH A COMPASS Consider a strategy in which each agent pursues a displacement of the next żi = (zi+1 + di) − zi, i = 1, . , n, where the indexi is evaluated modulon and ∑n i=1 di = 0. In vector form, this can be written aṡ z = A1z+ d, where A1 = circ(−1, 1, 0, . . . , 0). A result from [3] is that the centroid ofz1(t), . , zn(t) is stationary, and there exists a unique vectorh orthogonal tokerA1 such thatA1h + d = 0. Every zi(t) converges to the stationary centroid displaced byhi. By appropriate choice of d, a group of agents can be stabilized to a formation about their centroid. For example , let di = e . Notice that ∑n i=1 di = 0 and therefore the centroid of the n points is stationary. In equilibriumzi+1 − zi = e. Therefore, this stabilizes a group of agents to the vertices of a regular polygon centered at the centroid. However, notice that in order to implement this scheme, each agent must be able to calculate the vector e. This vector resides in the global coordinate system, which in this case is a global complex plane. Therefore, in order to implement this scheme, each agent must agree on a real and imaginary axis. Hence, each agent must be equipped with a compass. In this paper, the agents are not equipped with compasses. This makes the problem considerably more difficult. III. L OCAL CONTROL SCHEME AND STABILITY ANALYSIS Consider a group of n agents, numbered from1, . , n, lying in the plane. Thei agents’ position is given by (xi, yi), which we can represent in the complex plane as zi = xi + jyi. We can view the group of agents as the vertices of ann-gon by joining consecutive pairs of points z1, z2, . , zn to create the sides z1z2, z2z3, . , znz1. In this section we will introduce a control scheme for stabilizing the agents to an equilateral n-gon and study the stability of its equilibria. A. The z dynamics In order to stabilize a group of agents to an equilateral polygon, consider the following control strategy: żi = ui = 1 2 (zi+1 − zi) ( 1 − b 2 |zi+1 − zi|2 ) + 1 2 (zi−1 − zi) ( 1 − b 2 |zi−1 − zi|2 )