Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 948086, 17 pages

http://dx.doi.org/10.1155/2015/948086

## Finite-Time Formation Control without Collisions for Multiagent Systems with Communication Graphs Composed of Cyclic Paths

CINVESTAV, Electrical Engineering Department, Mechatronics Section, AP 14-740, 07000 México, DF, Mexico

Received 26 November 2014; Accepted 20 January 2015

Academic Editor: Luis Rodolfo Garcia Carrillo

Copyright © 2015 J. F. Flores-Resendiz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper addresses the formation control problem without collisions for multiagent systems. A general solution is proposed for the case of any number of agents moving on a plane subject to communication graph composed of cyclic paths. The control law is designed attending separately the convergence to the desired formation and the noncollision problems. First, a normalized version of the directed cyclic pursuit algorithm is proposed. After this, the algorithm is generalized to a more general class of topologies, including all the balanced formation graphs. Once the finite-time convergence problem is solved we focus on the noncollision complementary requirement adding a repulsive vector field to the previous control law. The repulsive vector fields display an unstable focus structure suitably scaled and centered at the position of the rest of agents in a certain radius. The proposed control law ensures that the agents reach the desired geometric pattern in finite time and that they stay at a distance greater than or equal to some prescribed lower bound for all times. Moreover, the closed-loop system does not exhibit undesired equilibria. Numerical simulations and real-time experiments illustrate the good performance of the proposed solution.

#### 1. Introduction

During the last years, formation control in multiagent systems has received much attention due to the wide range of applications in which they can be used as exploration, rescue tasks, toxic residues cleaning, and so forth, [1, 2]. A very important issue in formation control is the collision avoidance problem, with either other agents or obstacles [3]. If the formation control algorithms are designed in a totally centralized way, that is, with information exchange among all the agents, the computational load can increase seriously. On the other hand, an additional constraint arises if the communication among agents is restricted. Then, in most cases it is assumed that every agent in the group knows for all time the state or simply the position of a specific subset of robots and, eventually, can sense the position of any robot within a certain radius, [4]. Taking into account this difficulty several types of communications, as cyclic or balanced formations graphs, have been studied, [5–8].

Initially, the proposed solutions to the noncollision problem were designed as the sum of attractive and repulsive vector fields, in most cases obtained as the negative gradient of potential functions. Attractive potential functions are centred, for each agent, at its desired position, while the repulsive potential functions are centred at the positions of the rest of agents or even at obstacles’ positions, [9, 10]. Under this approach, one main drawback is the fact that the combination of gradients of attractive and repulsive potential functions could result in the appearance of undesired equilibrium points, leading the agents to get stuck at an undesired formation. Attending this problem, a solution has been proposed with the requirement of having totally centralized schemes [11]. Moreover, the repulsive vector fields are designed in such a way that they appear smoothly as the distance between any pair of agents becomes smaller and tends to infinity when this distance tends to zero. Then, although the collision avoidance is ensured, the position among agents can be arbitrarily small, which could imply a collision in real applications where physical dimensions cannot be ignored.

Related works can be found in [12] where authors consider formation control problems under limited and intermittent sensing. Based on a navigation function framework, a decentralized hybrid controller is developed to ensure network connectivity and collision avoidance while controlling the formation. In [13] the differential game approach is used for a group of agents to reach desired target positions while avoiding collisions among them. A methodical approach to the problem of collision avoidance of mobile robots taking advantages of multiagent systems has been presented in [14]. In order to achieve the trajectory, a control strategy based on a pure pursuit algorithm was implemented in the robots. The collision avoidance in the leader-follower multiagent systems was studied in [15]. The graph theory is used to model the communication topology between agents. To avoid collisions between neighboring agents, a fuzzy separation controller is proposed.

In a recent work, a new strategy for designing the repulsive vector fields has been proposed [16]. This approach differs from the classical one on the use of scaled unstable focus structures centred at the position of others agents. These functions cannot be obtained as the gradient of a scalar function of the distance between agents. Although this technique can also lead to undesired equilibria, these can be removed. The key point is to use an unstable focus scaled by a function depending on the distance among agents. This scaling function vanishes when the agents are far enough and tend to infinity as distance tends to zero. The analysis in [16] has been presented for the case of two agents only, while in [5] an extension to an arbitrary number of agents has been presented for the case of a directed cyclic pursuit communication graph.

In this paper we study the noncollision problem in formation control using discontinuous vector fields for an arbitrary number of agents. In one hand we undertake the design of attractive vector fields based on the well known cyclic pursuit algorithm but, unlike the results reported in the literature [7], we focus our analysis on normalized vector fields. That is, regarding only the attractive part, the agents move at constant known velocity and they reach the desired formation in finite time. Moreover, the case of more general communication graphs is analysed as the combination of single cyclic pursuit schemes. On the other hand, the repulsive vector fields have the unstable focus structure scaled by a suitable constant. As mentioned before, the general problem of an arbitrary number of robots is treated and the designed controllers are proven to be effective from the case where no collision risk exists to the one when a robot is rounded by a set of robots and there could be collision with any of them. It will be shown in this paper that this is the most complex situation that can occur.

This paper is organized as follows. We start in Section 2 stating some useful definitions and technical preliminaries. After this, in Section 3 we present the problem statement along with a couple of standing assumptions. The main contribution is given in Section 4, initially regarding the finite time convergence problem. Then, we take into account all the possible scenarios of collisions, starting with the simplest case of two robots in danger of collision. Based on this simple case we extend the study to risk of multiple collisions. Numerical simulations and real-time experiments are presented in Sections 5 and 6, respectively. Finally, in Section 7, we list the conclusions and outlooks of this research.

#### 2. Preliminaries

As we mentioned before, in this section we state some useful definitions [17, 18] and a technical lemma that we will use in the rest of the paper.

*Definition 1 (formation graph). *A formation graph that describes the communication among the agents consists of a set of vertices corresponding to each of the agents in the group and a set of edges , which denotes the agent receives information about . Finally, a set of constant vectors that represent the relative desired position of agent with respect to .

For a directed communication graph, implies that . For an undirected formation graph implies that .

*Definition 2 (paths and cycles). *There exists a path between the vertices and in the formation graph if there is a sequence of edges for some . We call a “cycle” to some path that starts and finishes in the same vertex.

If there is a path between any two vertices of the formation graph, then the graph is called connected. If a formation graph is connected and the vector satisfies the so-called closed-formation condition [19], that is, , then the formation control problem is solvable and the formation graph is said to be well-defined.

*Definition 3 (Laplacian matrix). *The Laplacian matrix associated with a formation graph is given by where is the degree matrix defined as , where is the number of edges directed to , , and is the adjacency matrix of defined as follows:

Proposition 4. *Consider the dynamical system , where and the matrix is Hurwitz. Then the normalized system with is stable with finite time convergence.*

*Proof. *Since is Hurwitz, then, for every matrix there exists a matrix such that the Lyapunov equation holds and is a Lyapunov function for the system . Taking as a Lyapunov function candidate and evaluating the time derivative along the trajectories of the normalized system we haveNote that, since is diagonal, it follows that and , where ; thereforeMoreover, since the matrix is Hurwitz, the Lyapunov equation always admits a positive definite solution for every positive definite matrix . Taking the time derivative is bounded from above by Now, since is diagonal, we haveOn the other hand, knowing that it is true that which implies directly thatThen, is bounded as follows: If we regard the quadratic form as a norm for vector , we can write where is a proportionality constant. This leads to finally write the last expression as that, according to [20], ensures convergence in finite time.

#### 3. Problem Statement

Consider a group of mobile agents denoted by moving on a plane. The cartesian coordinates of agent are given by , . Every robot is described by the kinematic modelwhere are the velocities along the - and -axes. In this paper we consider a decentralized general scheme. We assume that robot can detect the position of a subset of robots , where , . Therefore, the desired position of robot , say , with respect to is defined bywhere is the cardinality of and , . Throughout the paper, the following assumptions are supposed to hold.

*Assumption 5. *Agent measures the position of agents for all time and, eventually, can detect the presence of any other agent within a circle of radius . More precisely, can detect the agents in the set .

*Assumption 6. *The initial conditions of all agents satisfy , .

##### 3.1. Control Goal

The goal is to design control laws , , such that(i)the agents reach a desired formation; that is, ;(ii)there are no collisions among agents; moreover, at all times robots remain at some distance greater than or equal to a predefined distance from each other; that is, , , .

#### 4. Control Design

The control design is presented in two parts, one of each attending a different objective according to the control goal. We start proposing a control law based on normalized attractive vector fields to ensure finite time convergence of the agents to the desired formation.

##### 4.1. Attractive Vector Fields

The control law to reach the desired formation pattern is designed based on attractive vector fields proportional to the position error; that is,where corresponds to the position errors and is a design parameter. In this paper, we consider a normalized version of (15) to treat a suitable system where all the agents move at the same known velocity; namely,where is the constant velocity of all agents. In real time experiments, the control law (16) can induce chattering effects which can be avoided as it is shown in Section 6. Now, we can state our first main result regarding a cyclic pursuit directed communication graph, as shown in Figure 1(a), which is the basement for further cases.