Complex Formation Control of Large-Scale Intelligent Autonomous Vehicles

Abstract

A new formation framework of large-scale intelligent autonomous vehicles is developed, which can realize complex formations while reducing data exchange. Using the proposed hierarchy formation method and the automatic dividing algorithm, vehicles are automatically divided into leaders and followers by exchanging information via wireless network at initial time. Then, leaders form formation geometric shape by global formation information and followers track their own virtual leaders to form line formation by local information. The formation control laws of leaders and followers are designed based on consensus algorithms. Moreover, collision-avoiding problems are considered and solved using artificial potential functions. Finally, a simulation example that consists of 25 vehicles shows the effectiveness of theory.

1. Introduction

Distributed formation control of intelligent vehicles, including unmanned aerial vehicles, unmanned ground vehicles, and unmanned underwater submarines, has attracted a great attention, due to its broad applications, such as cooperative searching, surveillance, and combating using multiple unmanned air or ground vehicles. Roughly speaking, formation control strategy falls into six categories, namely, the behavior strategy [1–3], the potentials strategy [4–6], the virtual structure strategy [7–10], the distributed optimize strategy [11–14], and the consensus-based strategy [15–20].

The performances of the above strategies are almost excellent and effective when vehicles’ number is small. However, when the number is large and formation is complex, formation problems become complicated. In [21], a formation control framework is developed through dividing information into two independent parts. It can greatly reduce data exchange and realize various kinds of static formations. But, framework is developed based on a first-order model and the information dividing criterion is not given. Another approach in controlling a large group of vehicles is flocking control strategies [22–26]. The essence difference between formation control and flocking is that flocking is concentrated on achieving three heuristic rules introduced by Reynolds [22]. The rules are cohesion, separation, and alignment in which formation requirements are not considered. In [23], a distributed flocking controller is proposed which only utilizes the neighbor information in both free and obstacle space. However, the flocking shape problems are studied in [24], a new piecewise smooth neighbor-based local controller is designed to regulate the flocking shape, three kinds of flocking shapes are obtained, such as, in a single line, vee shape or corner shape. Furthermore, flocking with formation within a constrained environment is studied in [25], a potential field-based method is proposed to achieve the specific formation, but each agent must know the relative positions of information. Also, the flocking shape problems are studied in the consensus framework [26] and in the Lie group setting [27].

Compared with the centralized formation, the communication and control structure of distributed formation is more flexible and is excluded from the restrictions of individual number. Consensus-based distributed formation control only needs local information acquired by wireless network and has received significant attention in the control community [28, 29]. Usually, intelligent vehicle acquires information through wireless network conveniently. However, communication time delays induced by network are inevitable. It is very important to study stability problems of time delay system. By using a small gain stability theorem and structured singular value in the frequency domain, a criterion to guarantee convergence is derived [30]. A leader-following consensus problem of autonomous agents with coupling delays is studied and system stability conditions are obtained by using Lyapunov functions [31]. Considering a finite set of bounded internal point and distributed delays, output-feedback stabilization is considered by linear dynamic controllers and an alternative stabilization technique on time delays in the controller is proposed in [32]. A multiestimation-based adaptive controller is proposed in [33] to improve the transparency property of the teleoperation scheme. The closed-loop system’s stability is analyzed and stability capabilities of the overall scheme are carried out. With internal constant point delays, [34] researched the global asymptotic stability properties of switched systems under not very stringent conditions on the matrix functions of parameters. Several important and valued asymptotic stability conditions are obtained in this paper. In [35], by using algebraic graph theory, a method for decentralized stabilization of vehicle formations is proposed and a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback is obtained. In [36], the problems of packet dropouts, transmission delays, and quantization effects existing in the decentralized networked control systems are studied deeply. An efficient decentralized state feedback controller is designed and system stability conditions are acquired by using linear matrix inequalities.

When the number is large and formation is complex, it is impractical for each vehicle to obtain all the other vehicles’ state information quickly. Moreover, there are many uncertainties and disturbances in environment, which may lead to information transmission failure. In fact, it is worth paying more attention to some key members in a formation. One simple example is the line formation with loose space constraint between vehicles. It is necessary and very important to design appropriate formation frameworks.

In this paper, we propose a new hierarchy decentralized formation framework. Our purpose is to realize large-scale vehicles’ complex formations. In order to make the framework more close to reality, the second-order model and time delay are considered in formation control algorithm. The study is motivated by the work [19], the consensus-based formation control strategy [13–15], and the flocking shape problems [22–26]. The basic idea is given as follows.(1)Formation is described using vertexes and edges, and the number of leaders and followers is ascertained, respectively.(2)Automatically, based on initial position states and the formation information, vehicles are divided into two parts through wireless ad hoc network.(3)Leaders receive global information from control center by tel-network and form desired geometric shape based on consensus control strategy, meanwhile, broadcast their states through local wireless ad hoc network.(4)Followers receive leaders’ information and track their own virtual leaders using tracking consensus control strategy separately.(5)All vehicles adjust their states in a distributed manner.

The rest of the paper is organized as follows: in Section 2, some definitions are presented; in Section 3, formation control framework and control laws are discussed in detail; in Section 4, system stability conditions are obtained; in Section 5, numerical simulations are given to illustrate the effectiveness of the theoretical results; finally, concluding remarks are made in Section 6.

2. Problem Description and Preliminaries

Suppose the system under consideration consists of 𝑁 autonomous vehicles, labeled 1−𝑁. All these vehicles share a common state space and interact with each other via wireless network. Moreover, each vehicle updates its current states based upon the information received from its neighbors. The dynamics of the vehicles are all the same and described by double integrators, the 𝑖th vehicle has the dynamics as follows: 𝐱̇_𝑖(𝑡)=𝐯_𝑖(𝑡),𝐯̇_𝑖(𝑡)=𝐮_𝑖(𝑡),𝑖=1,2,…,𝑁, (2.1) where 𝐱_𝑖(𝑡)∈ℝ^𝑛,𝐯_𝑖(𝑡)∈ℝ^𝑛,𝐮_𝑖(𝑡)∈ℝ^𝑛 denote the position state, velocity state, and control vector separately.

We use graphs to describe complex formations. Thus, a formation can be constructed by a sequence of vertexes and lines. Vehicles in the vertex positions are called leaders and determine the basic frame of a formation. Vehicles in the line positions are called followers and associate with two leaders. Some examples of formations are given in Figure 1.

Figure 1 

Leaders and followers in a formation.

Definition 2.1. In 𝑛 dimension Euclidean space, each agent’s position vector is 𝐡_𝑖∈ℝ^𝑛, 𝑖=1,2,…,𝑁, and relative position vector is 𝐡_𝑖𝑗=𝐡_𝑖−𝐡_𝑗, for all 𝑖,𝑗=1,2,…,𝑁; thus, a formation vector 𝐡∈ℝ^{𝑛×𝑁×𝑁} is defined as 𝐡=[^ℎ1₁₁^,ℎ1₁₂^,…,ℎ1_1𝑁^,…,ℎ𝑛_𝑁1^,…,ℎ𝑛_𝑁𝑁]^𝑇, where ℎ^𝑘_𝑖𝑗, 𝑘=1,2,…,𝑛 is the projection of 𝐡_𝑖𝑗 in each coordinate axis.

Definition 2.2. When 𝑡→∞, if for all 𝑖,𝑗∈𝑁,𝐱_𝑖(𝑡)−𝐱_𝑗(𝑡)=𝐡_𝑖𝑗,𝑣_𝑖(𝑡)=𝐯_𝑗(𝑡) holds, then formation is formed and maintained.

Definition 2.3. Potential function 𝑉_𝑖𝑗 is a differentiable, nonnegative function of the distance ℎ_𝑖𝑗 between vehicles 𝑖 and 𝑗 if 𝑗∈𝑁_𝑖 such that 𝑉_𝑖𝑗→∞ when |ℎ_𝑖𝑗|→0 and 𝑉_𝑖𝑗 attains its unique minimum when vehicles 𝑖 and 𝑗 are located at a desired distance.

Definition 2.4. The Kronecker of two matrixes 𝐏=[𝑝_𝑖𝑗],𝐐=[𝑞_𝑖𝑗] is defined as 𝐏⊗𝐐=[𝑝_𝑖𝑗𝐐].

It is natural to model the interaction among vehicles using directed graph. Let 𝐺=(𝑉,𝐸,𝐴) be a graph of order 𝑁, with a set of nodes 𝑉={𝑉₁,𝑉₂,…,𝑉_𝑛}, a set of edges 𝜀⊆𝑉×𝑉, and an adjacency matrix 𝐴=[𝑎_𝑖𝑗], for all 𝑖,𝑗,𝑛∈𝐼={1,2,…,𝑛}. If there is information exchange between node 𝑉_𝑖 and 𝑉_𝑗, 𝑎_𝑖𝑗=1, else 𝑎_𝑖𝑗=0. In a directed graph, if (𝑉_𝑖,𝑉_𝑗)∈𝜀, node 𝑉_𝑗 is 𝑉_𝑗’s out-degree neighbor, if (𝑉_𝑗,𝑉_𝑖)∈𝜀, node 𝑉_𝑗 is 𝑉_𝑗’s in-degree neighbor. A set of node 𝑉_𝑖’s out-degree neighbors is 𝑁_𝑖(out). A set of node 𝑉_𝑖’s in-degree neighbors is 𝑁_𝑖(in); similarly, a set of node 𝑉_𝑖’s neighbors is 𝑁_𝑖=𝑁_𝑖(out)⋃𝑁_𝑖(in). The degree matrix of graph 𝐺 is a diagonal matrix 𝐷=[𝑑_𝑖𝑗], 𝑑_𝑖𝑗=0, for all 𝑖≠𝑗 and 𝐿=𝐷−𝐴 is a Laplacian matrix of graph 𝐺. Furthermore, we can obtain the normalized Laplacian matrix 𝐿=𝐷⁻¹𝐿.

3. Main Results

3.1. Overview of Formation Framework

All vehicles equip the same wireless communication devices. That is they can interact with each other through local ad hoc wireless network and interact with control center through tel-network on demanding. It is necessary to design appropriate ad hoc network architecture and communication protocol to improve system’s performances. We adopt a cluster architecture ad hoc network and TDMA-type MAC protocol, where each vehicle is assigned to a specific time slot. We propose a distributed formation framework; see Figure 2.

Figure 2 

A distributed formation framework.

Formation information is divided into global information and local information. Global information including number of leaders and followers, formation shape, velocity, and acceleration is transmitted by tel-network. Corresponding to global information, vehicle’s position and velocity are defined as local information transmitted via a local ad hoc network.

Once a formation is given, global formation information is ascertained by a control center. Follow on, the control center sends global formation information to a leader by a tel-network and obtained by the other leaders via local Ad hoc network. It is worth to note that global information can be set, changed, and injected to leaders at any time by the control center. However, there is one exception that initial information must be known by each vehicle. Then, using the dividing method, vehicles are automatically divided into two groups according to initial position information. Based on consensus algorithm and artificial potentials’ functions, leaders form a basic formation using global formation information set by the control center, and followers track their leaders to form the entire formation.

Assumption 3.1. Suppose that communication delays in the system are uniform and acquiring own state information has no delay, then, communication delay can be denoted by 𝜏(𝑡) and 𝜏(𝑡)=𝜏,𝜏>0.

3.2. Automatic Dividing Algorithm

When the number of vehicles is large and the formation is complex, it is very convenience to design an efficient dividing algorithm. Under the assumption that network is in the connectivity state at initial time, vehicles can elect leaders and followers automatically using the following algorithm.

Step 1. At the first time period, using flooding protocol, each vehicle acquires all the other’s initial position information and the global formation information.

Step 2. Each vehicle calculates the center of initial positions 𝑥=(1/𝑁)∑^𝑁_𝑖=1𝑥_𝑖 separately.

Step 3. Each vehicle calculates its distance to the center 𝑑_𝑖=‖𝑥_𝑖−𝑥‖ and composites the results in descending order (small node ID has higher prior) separately. The first vehicle in the descending sequence is defined as a leader, written as 𝐿1.

Step 4. Each vehicle calculates the angle 𝜃_𝑖𝐿1 made by the center, local agent, and 𝐿1: 𝜃_𝑖𝐿1𝑥=arccos^𝑖−𝑥²+𝑥_𝐿1−𝑥²−𝑥_𝐿1−𝑥_𝑖²2𝑥_𝑖−𝑥𝑥_𝐿1−𝑥. (3.1)

Step 5. Each vehicle calculates 𝑘_𝑖𝑙1=𝑁₀𝜃_𝑖𝐿1/2𝜋, and composites the results in descending order (small node id has higher prior) separately. Tick out the vehicle 𝑘_𝑖𝐿1>1 and form a new sequence, then, the first vehicle in the descending sequence is defined as a leader, written as 𝐿2.

Step 6. Repeat Step 3 to Step 5 until find leader 𝐿𝑁, then, other vehicles are followers.

Step 7. Ascertain a followers’ leader. Compute the angle between leaders and followers 𝜃_𝑖𝐿𝑗 in turn. According to the global formation information and the angle 𝜃_𝑖𝐿𝑗, followers associate with their two leaders which have max 𝜃_𝑖𝐿𝑗: 𝜃_𝑖𝐿𝑗𝑥=arccos^𝑖−𝑥_𝐿𝑗²+𝑥_𝑖−𝑥_𝐿(𝑗+1)²−𝑥_𝐿𝑗−𝑥_𝐿(𝑗+1)²2𝑥_𝑖−𝑥_𝐿𝑗𝑥_𝑖−𝑥_𝐿(𝑗+1). (3.2)

Step 8. Ascertain virtual leaders. The position and velocity of virtual leader is written as 𝑥_𝑖𝑎, 𝑣_𝑖𝑎 and defined as follows: 𝑥_𝑖𝑎(𝑡)=1−𝑝_𝑖𝑥_𝑖𝐿1(𝑡)+𝑝_𝑖𝑥_𝑖𝐿2(𝑡),𝑣_𝑖𝑎(𝑡)=0.5𝑣_𝑖𝐿1(𝑡)+𝑣_𝑖𝐿2(𝑡), (3.3) where 𝑝_𝑖∈[0,1] is a proportion factor correlative with expectation position.
A virtual leader can maintain that the virtual leader is at the expectation position of the line. The line is made by its leader each time.

3.3. Formation Control Laws

The design method of formation control laws are shown in Figure 3. Formation control law depends on the role of the vehicle if it is a leader, 𝛼_𝑖=1, otherwise, 𝛼_𝑖=0. Thus, the control law for an individual vehicle 𝑢_𝑖(𝑡) is designed as follows: 𝑢_𝑖(𝑡)=𝛼_𝑖𝐮^𝐿_𝑖(𝑡)+1−𝛼_𝑖𝐮^𝐹_𝑖(𝑡)+𝐟_𝑖^𝑔(𝑡), (3.4) where 𝐮^𝐿_𝑖(𝑡), 𝐮^𝐹_𝑖(𝑡), 𝐟_𝑖^𝑔(𝑡) are leaders’ formation control law, followers’ formation control law, and collision avoidance control law separately.

(a)

(b)

Figure 3 

(a) Control of leaders. (b) Control of followers.

The leaders’ formation control law is designed based on consensus algorithm. The control purpose is used to form desired formation shape and movement. Considering communication delay, control law is designed as follows: 𝐮^𝐿_𝑖(𝑡)=𝛼𝐯̇_𝑑−𝛽𝐯_𝑖(𝑡)−𝐯_𝑑−1𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝐯_𝑗()−1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡)−𝐱_𝑗(𝑡−𝜏)−𝐡_𝑖𝑗. (3.5) Because of that, 1𝑁_𝑖−^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡−𝜏)−^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡)−𝐱_𝑗(𝑡−𝜏)−𝐡_𝑖𝑗=1𝑁_𝑖−^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑗(𝑡−𝜏)−^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑗(𝑡−𝜏)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐡_𝑖𝑗=1𝑁_𝑖−^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡)−^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑗(𝑡−𝜏)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑗(𝑡−𝜏)+^𝑁_𝑗=1𝑎_𝑖𝑗𝐡_𝑖𝑗=−𝐯_𝑖(𝑡)−𝐱_𝑖(𝑡)+1𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑗()+1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑗()+1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐡_𝑖𝑗=−𝐯_𝑖(𝑡)−𝐱_𝑖(𝑡)+𝐯_𝑖(𝑡−𝜏)+𝐱_𝑖()−1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡−𝜏)−𝐯_𝑗()−1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡−𝜏)−𝐱_𝑗()+1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐡_𝑖𝑗. (3.6) Then, we can rewrite 𝐮^𝐿_𝑖(𝑡) as follows: 𝐮^𝐿_𝑖(𝑡)=𝛼𝐯̇_𝑑−𝛽𝑣_𝑖(𝑡)−𝐯_𝑑−𝐯_𝑖(𝑡)−𝐱_𝑖(𝑡)+𝑣_𝑖(𝑡−𝜏)+𝐱_𝑖()+1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐡_𝑖𝑗−1𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡−𝜏)−𝐯_𝑗()−1𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐱_𝑖(𝑡−𝜏)−𝐱_𝑗(𝑡−𝜏), (3.7) where 𝐮^𝐿_𝑖(𝑡)∈ℝ^𝑛 is control vector; 𝐯_𝑑,𝐯̇_𝑑∈ℝ^𝑛 are the expected velocity and acceleration; 𝑎_𝑖𝑗 is the communication weight, if information flows from 𝑗 to 𝑖, 𝑎_𝑖𝑗=1, else 𝑎_𝑖𝑗=0; 𝛼>0, 𝛽>0 are control parameters; ℎ_𝑖𝑗 is desired relative distance vector and 𝑁_𝑖 is number of current neighbors.

According to the individual control law (3.7), we can get the leader system’s control vector 𝐮_𝑑(𝑡)𝐮_𝑑(𝑡)=𝛼𝟏_𝑁⊗𝐯̇_𝑑−𝛽𝐯(𝑡)−𝟏_𝑁⊗𝐯_𝑑(𝑡)(𝑡)()()−−𝐯−𝐱+𝐯𝑡−𝜏+𝐱𝑡−𝜏𝐋⊗𝐈_𝑛𝑣(𝑡−𝜏)−𝐋⊗𝐼_𝑛𝐱()𝟏𝑡−𝜏+𝐄=𝛼_𝑁⊗𝐯̇_𝑑+𝛽𝟏_𝑁⊗𝐯_𝑑−𝐯(𝑡)(𝑡)+𝐈𝛽+1−𝐱+𝐄_𝑛𝑁−𝐋⊗𝐈_𝑛𝐯(𝑡−𝜏)+𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱(𝑡−𝜏), (3.8) where 𝐈𝐄=^𝑛⊗𝐋₁𝐈_𝑛⊗𝐋₂⋯𝐈_𝑛⊗𝐋_𝑁𝐡. (3.9) Considering that, −𝐱(𝑡)−𝛽+1𝐯(𝑡)+𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱(𝑡−𝜏)+𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐯()=𝑡−𝜏𝟎𝟎−𝐈_𝑛𝑁−𝛽+1𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝐈𝟎𝟎_𝑛𝑁−𝐋⊗𝐈_𝑛𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱()𝐯().𝑡−𝜏𝑡−𝜏 (3.10) We can rewrite 𝐮_𝑑(𝑡) as follow: 𝐮_𝑑(𝑡)=𝛼𝟏_𝑁⊗𝐯̇_𝑑+𝛽𝟏_𝑁⊗𝐯_𝑑(𝑡)−𝐯(𝑡)+𝐈+𝐄−𝐱𝛽+1_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱(𝑡−𝜏)+𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐯()𝟏𝑡−𝜏=𝛼_𝑁⊗𝐯̇_𝑑+𝛽𝟏_𝑁⊗𝐯_𝑑++𝐄−𝐈_𝑛𝑁−𝛽+1𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱()𝐯().𝑡−𝜏𝑡−𝜏 (3.11) Note that system model (2.1) can be rewritten as follows: 𝐱̇(𝑡)𝐯̇(𝑡)=𝟎𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝟎𝟎𝟎𝐮_𝑑(𝑡). (3.12) Consequently, substituting equation (3.11) into system model (3.12), we have 𝐱̇(𝑡)𝐯̇(𝑡)=𝟎𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝟎𝟎𝟎𝛼𝟏_𝑁⊗𝐯̇_𝑑+𝛽𝟏_𝑁⊗𝐯_𝑑+𝐈+𝐄𝟎𝟎_𝑛𝑁−𝛽+1𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝐈𝟎𝟎_𝑛𝑁−𝐋⊗𝐈_𝑛𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱()𝐯()=𝑡−𝜏𝑡−𝜏𝟎𝐈_𝑛𝑁−𝐈_𝑛𝑁−𝛽+1𝐈_𝑛𝑁𝐱(𝑡)𝐯(𝑡)+𝐈𝟎𝟎_𝑛𝑁−𝐋⊗𝐈_𝑛𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛𝐱()𝐯()+𝟎𝛼𝟏𝑡−𝜏𝑡−𝜏_𝑁⊗𝐯̇_𝑑+𝛽𝟏_𝑁⊗𝐯_𝑑.+𝐄 (3.13) Furthermore, we define 𝝃(𝑡)=𝐱(𝑡)𝐯(𝑡),𝝃̇(𝑡)=𝐱̇(𝑡)𝐯̇(𝑡)((𝑡=𝐱()𝐯(),𝝃𝑡−𝜏))𝑡−𝜏𝑡−𝜏. (3.14) Thus, the close loop of the leader system state equation is derived: 𝝃̇(𝑡)=𝐀𝝃(𝑡)+𝐁𝝃(𝑡−𝜏)+𝐂, (3.15) where 𝐀=𝟎𝐈_𝑛𝑁−𝐈_𝑛𝑁−𝛽+1𝐈_𝑛𝑁𝐈,𝐁=𝟎𝟎_𝑛𝑁−𝐋⊗𝐈_𝑛𝐈_𝑛𝑁−𝐋⊗𝐈_𝑛,𝟎𝟏𝐂=𝐄+𝛽_𝑁⊗𝐯_𝑑+𝛼𝟏_𝑁⊗𝐯̇_𝑑. (3.16)

The followers’ formation control law is designed based on the consensus algorithm also. The control purpose is tracking its virtual leader, respectively. Considering communication delay, control law is designed as follows: 𝐮^𝐹_𝑖(𝑡)=𝐟_𝑖^𝑣(𝑡)+𝐟_𝑖^𝑐(𝑡),𝐟^𝑣_𝑖(𝑡)𝑐=−𝑁_𝑖^𝑁𝑖_𝑗=1𝐯_𝑖−𝐯_𝑗(),𝐟𝑡−𝜏_𝑐𝑖(𝑡)=−𝑣_𝑖(𝑡)−𝐯_𝑖𝑎(𝑡)−𝛾𝐱_𝑖(𝑡)−𝐱_𝑖𝑎(𝑡). (3.17)

The velocity feedback damp control term 𝐟^𝑣_𝑖(𝑡) is used to regulate the vehicles’ velocity and 𝑐 is the damp control parameter. The consensus tracking control term 𝐟^𝑐_𝑖(𝑡) is used to track its own virtual leader and 𝛾 is the consensus control parameter.

Let 𝑣_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝐯_𝑖𝑎(𝑡), 𝐱_𝑖(𝑡)=𝐱_𝑖(𝑡)−𝐱_𝑖𝑎(𝑡), then we have 𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡−𝜏)=𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡−𝜏)+𝐯_𝑖𝑎(𝑡)−𝐯_𝑗𝑎(𝑡−𝜏). (3.18)

Since that, −𝑐𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡)−𝐯_𝑗()𝑐𝑡−𝜏=−𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑖(𝑡−𝜏)+𝐯_𝑖𝑎(𝑡)−𝐯_𝑗𝑎()𝑐𝑡−𝜏=−𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡)+𝑐𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑗()−𝑐𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖𝑎(𝑡)−𝐯_𝑗𝑎(𝑡−𝜏). (3.19)

The followers’ formation control law is rewritten as follows: 𝐮^𝐹_𝑖(𝑡)=−𝐯_𝑖(𝑡)−𝛾𝐱_𝑖(𝑡)−𝑐𝐯_𝑖(𝑡)+𝑐𝐯_𝑖(𝑡−𝜏)−𝑐𝐯_𝑖𝑎(𝑡)+𝑐𝐯_𝑖𝑎()−𝑐𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖(𝑡−𝜏)−𝐯_𝑗()−𝑐𝑡−𝜏𝑁_𝑖^𝑁_𝑗=1𝑎_𝑖𝑗𝐯_𝑖𝑎(𝑡−𝜏)−𝐯_𝑗𝑎(𝑡−𝜏). (3.20)

Furthermore, we can acquire the formation control law of the follower system: 𝐮^𝐹(𝑡)=−𝑣(𝑡)−𝛾𝐱(𝑡)−𝑐𝑣(𝑡)+𝑐𝑣(𝑡−𝜏)−𝑐𝐋⊗𝐈_𝑛𝑣(𝑡−𝜏)−𝑐𝐯_𝑎(𝑡)+𝑐𝐋⊗𝐈_𝑛𝐯_𝑎()𝐱(𝑡)−()𝑣(𝑡)𝐈𝑡−𝜏=−𝛾1+𝑐+𝑐_𝑛𝑁−𝐋⊗𝐈_𝑛𝑣(𝑡−𝜏)−𝑐𝐯_𝑎(𝑡)+𝑐𝐯_𝑎(𝑡−𝜏)+𝑐𝐋⊗𝐈_𝑛𝐯_𝑎(𝑡−𝜏). (3.21)

Consequently, the close loop of the follower system state equation is derived: 𝝃̇(𝑡)=𝐃𝝃(𝑡)+𝐄𝝃(𝑡−𝜏)+𝐅(𝑡), (3.22) where 𝐃=𝟎𝐈_𝑛𝑁−𝛾𝐈_𝑛𝑁−(1+𝑐)𝐈_𝑛𝑁𝐈,𝐄=𝟎𝟎𝟎𝑐_𝑛𝑁−𝐋⊗𝐈_𝑛,𝐅(𝑡)=𝟎−𝑐𝐯_𝑎(𝑡)+𝑐𝐯_𝑎(𝑡−𝜏)+𝑐𝐋⊗𝐈_𝑛𝐯_𝑎().𝑡−𝜏 (3.23)

The collision avoidance control law is designed based on artificial potential functions and the design methods are showed in Figure 4.

(a)

(b)

Figure 4 

(a) Safe zone, communication zone, and collision zone. (b) Artificial potential function.

Consequently, the collision avoidance control law is designed as follows: 𝐟^𝑔_𝑖(𝑡)=1𝑁_{𝑖𝑗∈𝑁𝑖(𝑡)}𝑥_𝑖𝑗(𝑡)𝑉_𝑖𝑗𝑥_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)⁻¹,𝑉_𝑖𝑗𝑥_𝑖𝑗(𝑡)=𝑘𝜑𝑥_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)⁻¹,𝜑𝑥_𝑖𝑗(𝑡)=log²𝑥_𝑖𝑗(𝑡)𝑑_𝑒⁻¹𝑥_𝑖𝑗(𝑡)<𝑑_𝑒,0𝑥_𝑖𝑗(𝑡)≥𝑑_𝑒. (3.24)

The control term 𝐟^𝑔_𝑖(𝑡) is designed on artificial potential functions which are used to avoid collision, 𝑉_𝑖𝑗(‖𝑥_𝑖𝑗(𝑡)‖) is artificial potential function, 𝑑_𝑒 is the safe range, 𝑑 is communication range, and 𝑘 is positive control parameter.

4. Stability Analysis

In the following, we get two steps to study the system’s stability problems.

Lemma 4.1 (Schur complement). For a given symmetric matrix 𝐒 with the form 𝐒=^𝐒11𝐒12_𝐒12𝐒22, where 𝐒₁₁∈ℝ^𝑟×𝑟, 𝐒₂₂∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)}, 𝐒₁₂∈ℝ^{𝑟×(𝑛−𝑟)}, 𝐒₂₁∈ℝ^{(𝑛−𝑟)×𝑟}, 𝐒<0 if and only if 𝐒₂₂<0 and 𝐒₁₁−𝐒₁₂𝐒⁻¹₂₂𝐒^𝑇₁₂<0.

Step 1. The stability analysis of a leader system.

Theorem 4.2. Given communication delay 0<𝜏≤𝑑,0<𝑑<∞, formation vector ℎ, velocity 𝐯_𝑑, acceleration 𝐯̇_𝑑, and control parameters 𝛼>0, 𝛽>0 are bounded constants. Under the condition that directed network topology is connectivity, if there exit two symmetric matrixes 𝐏>0, 𝐐>0 subject to Γ₁<0, then for any time delay 𝜏, the leader system is stable: Γ₁=(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)+2𝜏𝐀^𝑇𝐐𝐀+𝐁^𝑇𝐁𝐐𝐁𝐏𝐁_𝑇𝐐𝐏−𝜏. (4.1)

Proof. When the formation vector 𝐡 and desired velocity and acceleration 𝐯_𝑑, 𝐯̇_𝑑 are known accurately and control parameters 𝛼>0 and 𝛽>0 are bounded constant, the bias vector 𝐂 is a bounded constant and plays no role in the system’s convergence. So, the leader system’s convergence (3.17) is the same as the following system: 𝝃̇(𝑡)=𝐀𝝃(𝑡)+𝐁𝝃(𝑡−𝜏). (4.2)
By the Newton-Leibniz formula, we have 𝝃̇=(𝐀+𝐁)𝝃(𝑡)−𝐁^𝑡_𝑡−𝜏𝝃̇(𝑠)𝑑𝑠. (4.3)
Define a Lyapunov-Krasovskii function for system (4.3) as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡),𝑉₁(𝑡)=𝝃^𝑇(𝑡)𝐏𝝃(𝑡),𝑉₂(𝑡)=^𝜏₀^𝑡_𝑡−𝛽𝝃̇^𝑇(𝑠)𝐐𝝃̇(𝑠)𝑉𝑑𝑠𝑑𝛽,₃(𝑡)=2^𝑡_𝑡−𝜏𝜏𝝃^𝑇(𝑠)𝐁^𝑇𝐐𝐁𝝃(𝑠)𝑑𝑠. (4.4)
Differentiating 𝑉(𝑡) along the system (4.3), we can acquire 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡),𝑉̇₁(𝑡)=𝝃̇^𝑇(𝑡)𝐏𝝃(𝑡)+𝝃^𝑇(𝑡)𝐏𝝃̇(𝑡)=(𝐀+𝐁)𝝃(𝑡)−𝐁^𝑡_𝑡−𝜏𝝃̇(𝑠)𝑑𝑠^𝑇𝐏𝝃(𝑡)+𝝃^𝑇(𝑡)𝐏(𝐀+𝐁)𝝃(𝑡)−𝐁^𝑡_𝑡−𝜏𝝃̇(𝑠)𝑑𝑠=𝝃^𝑇(𝑡)(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)𝝃(𝑡)−^𝑡_𝑡−𝜏𝝃̇^𝑇(𝑠)𝐁^𝑇𝐏𝝃(𝑡)+𝝃^𝑇(𝑡)𝝃̇(𝑠)𝑉̇𝐏𝐁𝑑𝑠,₂(𝑡)=^𝜏₀^𝑡_𝑡−𝛽𝝃̇^𝑇(𝑠)𝐐𝝃̇(𝑠)𝑑𝑠𝑑𝛽=𝜏𝝃̇^𝑇(𝑡)𝐐𝝃̇(𝑡)−^𝑡_𝑡−𝜏𝝃̇^𝑇(𝑠)𝐐𝝃̇(𝑠)𝑉̇𝑑𝑠,₃(𝑡)=2𝜏𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)−2𝜏𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐐𝐁𝝃(𝑡−𝜏). (4.5)
Sum the integral terms in 𝑉̇₁(𝑡),𝑉̇₂(𝑡),𝑉̇₃(𝑡), then, we have ^𝑡_𝑡−𝜏−𝝃̇^𝑇(𝑠)𝐁^𝑇𝐏𝝃(𝑡)−𝝃^𝑇(𝑡)𝐏𝐁𝝃̇(𝑠)−𝝃̇^𝑇(𝑠)𝐐𝝃̇(𝑠)=𝑑𝑠_{𝑡𝑡−𝜏}−𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)+𝝃^𝑇(𝑡)𝐏𝐁𝐐⁻¹𝐁^𝑇(𝑡)=𝐏𝝃𝑑𝑠_{𝑡𝑡−𝜏}−𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝝃^𝑇(𝑡)𝐏𝐁𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)𝑑𝑠=−^𝑡_𝑡−𝜏𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)𝑑𝑠+𝜏𝝃^𝑇(𝑡)𝐏𝐁𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡). (4.6)
Since that, 𝜏𝝃̇^𝑇(𝑡)𝐐𝝃̇(𝑡)+2𝜏𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)−2𝜏𝝃^𝑇(𝑡−𝜏)𝐁^𝑇()𝝃𝐐𝐁𝝃𝑡−𝜏=𝜏^𝑇(𝑡)𝐀^𝑇𝐐𝐀𝝃(𝑡)+𝝃^𝑇(𝑡)𝐀^𝑇𝐐𝐁𝝃(𝑡−𝜏)+𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐀𝝃(𝑡)+𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐁𝝃(𝑡−𝜏)+2𝜏𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)−2𝜏𝝃^𝑇(𝑡−𝜏)𝐁^𝑇(),𝝃𝐐𝐁𝝃𝑡−𝜏_𝑇(𝑡)𝐀^𝑇𝐐𝐀𝝃(𝑡)+𝝃^𝑇(𝑡)𝐀^𝑇𝐐𝐁𝝃(𝑡−𝜏)+𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐀𝝃(𝑡)+𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐁𝝃(𝑡−𝜏)−2𝝃^𝑇(𝑡)𝐀^𝑇𝐐𝐀𝝃(𝑡)−2𝝃^𝑇(𝑡−𝜏)𝐁^𝑇𝐐𝐁𝝃(𝑡−𝜏)+2𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)+2𝝃^𝑇(𝑡)𝐀^𝑇(𝑡)((𝑡)(𝐐𝐀𝝃=−𝐴𝝃−𝐁𝝃𝑡−𝜏))^𝑇𝐐(𝐀𝝃(𝑡)−𝐁𝝃(𝑡−𝜏))+2𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)+2𝝃^𝑇(𝑡)𝐀^𝑇𝐐𝐀𝝃(𝑡). (4.7)
According to (4.4) to (4.7), we have 𝑉̇(𝑡)=𝝃^𝑇(𝑡)(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)𝝃(𝑡)+𝜏𝝃^𝑇(𝑡)𝐏𝐁𝐐⁻¹𝐁^𝑇(𝑡)((𝑡)(𝐏𝝃−𝜏𝐀𝝃−𝐁𝝃𝑡−𝜏))^𝑇𝐐(𝐀𝝃(𝑡)−𝐁𝝃(𝑡−𝜏))+2𝜏𝝃^𝑇(𝑡)𝐁^𝑇𝐐𝐁𝝃(𝑡)+2𝜏𝝃^𝑇(𝑡)𝐀^𝑇𝐐𝐀𝝃(𝑡)−^𝑡_𝑡−𝜏𝐁^𝑇𝐏𝝃(𝐭)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)𝑑𝑠=𝝃^𝑇(𝑡)(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)+𝜏𝐏𝐁𝐐⁻¹𝐁^𝑇𝐏+2𝜏𝐀^𝑇𝐐𝐀+2𝜏𝐁^𝑇𝝃(𝑡)((𝑡)(𝐐𝐁−𝜏𝐀𝝃−𝐁𝝃𝑡−𝜏))^𝑇𝐐((𝑡)(−𝐀𝝃−𝐁𝝃𝑡−𝜏))_{𝑡𝑡−𝜏}𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)𝑑𝑠. (4.8)
According to (4.9), we get −(𝐀𝝃(𝑡)−𝐁𝝃(𝑡−𝜏))^𝑇𝐐((𝑡)(−𝐁𝐀𝝃−𝐁𝝃𝑡−𝜏))<0,_𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)^𝑇𝐐⁻¹𝐁^𝑇𝐏𝝃(𝑡)+𝐐𝝃̇(𝑠)<0. (4.9)
Let Ω=(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)+𝜏𝐏𝐁𝐐⁻¹𝐁^𝑇𝐏+2𝜏𝐀^𝑇𝐐𝐀+2𝜏𝐁^𝑇𝐐𝐁<0. (4.10)
Then we have 𝑉̇(𝑡)<0. By Lemma 4.1, we get Ω<0 which is equivalent to Γ₁<0: Γ₁=(𝐀+𝐁)^𝑇𝐏+𝐏(𝐀+𝐁)+2𝜏𝐀^𝑇𝐐𝐀+𝐁^𝑇𝐁𝐐𝐁𝐏𝐁_𝑇𝐐𝐏−𝜏. (4.11)
If Γ₁<0, then we have 𝑉̇(𝑡)<0. Therefore, the leader system is stable and a basic formation is formed and maintained.

Step 2. The stability analysis of followers.

Theorem 4.3. Under the condition of Theorem 4.2, if a leader system is stable and there exit two symmetric matrixes 𝐑>0,𝐒>0 subject to Γ₂<0, then for any delay 𝜏, a follower system is stable: Γ₂=(𝐃+𝐄)^𝑇𝐑+𝐑(𝐃+𝐄)+2𝜏𝐃^𝑇𝐒𝐃+𝐄^𝑇𝐄𝐒𝐄𝐑𝐄_𝑇𝐐𝐑−𝜏. (4.12)

Proof. Similar to the proof of Theorem 4.2, we can derive the follower system’s stability condition. It is that if there exit two symmetric matrixes 𝐑>0,𝐒>0 subject to Γ₂<0, then for any delay 𝜏, the follower system is stable. Γ₂=(𝐃+𝐄)^𝑇𝐑+𝐑(𝐃+𝐄)+2𝜏𝐃^𝑇𝐒𝐃+𝐄^𝑇𝐄𝐒𝐄𝐑𝐄_𝑇𝐐𝐑−𝜏. (4.13)
According to the follower system’s close-loop state equation, the equilibrium point of the dynamic system 𝝃_𝑒=𝐅(𝑡) is stable. If a leader system is stable, it means the condition 𝐯_𝑖(𝑡)=𝐯_𝑗(𝑡), for all 𝑖,𝑗∈𝑁, is true. Consequently, we can easily acquire the equation 𝐯_𝑖𝑎(𝑡)=𝐯_𝑗𝑎(𝑡)=𝐯_𝑖𝑎(𝑡−𝜏)=𝐯_𝑗𝑎(𝑡−𝜏). Thus, equations 𝑐𝐯_𝑎(𝑡)=𝑐𝐯_𝑎(𝑡−𝜏) and (𝐋⊗𝐈_𝑛)𝐯_𝑎(𝑡−𝜏)=0 are true which lead to 𝐅(𝑡)=0. Furthermore, it is easy to see that 𝑣_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝐯_𝑖𝑎(𝑡)=0 and 𝐱_𝑖(𝑡)=𝐱_𝑖(𝑡)−𝐱_𝑖𝑎(𝑡)=0 when 𝑡→∞. According to the virtual leader’s definition, local formation can be formed and maintained.

5. Simulations

In this section we give a simulation example of multiple vehicles formation in the plane to illustrate the effectiveness of the proposed algorithm in the paper. In the simulations, 25 vehicles are randomly distributed in [0,100] and [0,100]; initial velocities are randomly distributed in [0,1] and [0,1]. Expected velocity is 𝐕_𝑑=[6,6]^𝑇 and expected formation is shown as Figure 5. According to the proposed method, it is easily known that the number of leaders is 5 and the number of followers is 20. Leaders’ formation vector is 𝐡^𝑇: [].𝐡=0,−30,−30,30,30,0,40,100,100,40,30,0,0,60,60,−40,0,60,60,0,30,0,0,60,60,−100,−60,0,0,−60,−30,−60,−60,0,0,−100,−60,0,0,−60,−30,−60,−60,0,0,−40,0,60,60,0 (5.1)

(a)

(b)

Figure 5 

(a) The desired formation of all vehicles. (b) The desired formation shape of leaders.

Control parameters are 𝛼=1, 𝛽=5, 𝑘=10, damp feedback coefficient 𝑐=1, and velocity navigation coefficient 𝛾=0.4. Communication range is 20 and safe range is 6. Time delay is 0.2 second. Simulation step is not fixed and total time is 60 seconds.

The initial positions of all vehicles are randomly distributed and the initial network is connectivity as shown in Figure 6(a). In the designed formation framework, vehicles can elect leaders and followers automatically using the dividing algorithm. The states of all vehicles after grouping are shown in Figure 6(b).

(a)

(b)

Figure 6 

(a) Initial states of all vehicles. (b) States of all vehicles after grouping.

Formation movement of 25 vehicles can be seen clearly in Figure 7. At time 5 seconds, followers are tracking their own virtual leader shown in Figure 7(a). At time 10 seconds, followers have tracked their own virtual leaders shown in Figure 7(b). At time 15 seconds, leaders are forming desired formation geometric shape using the global formation information shown in Figure 7(c). Final states of all vehicles are shown in Figure 7(d); the desired formation is formed and maintained.

(a)

(b)

(c)

(d)

Figure 7 

(a) States of all vehicles at time 5 seconds. (b) States of all vehicles at time 10 seconds. (c) States of all vehicles at time 15 seconds. (d) Final states of all vehicles.

To show the simulation results clearly, we choose position and velocity on 𝑥 and 𝑦 axes to make the results fill in Figure 8. Trajectories of leader vehicles are shown in Figure 8(a); it shows that the desired formation is formed and maintained. Velocities of leader vehicles are shown in Figure 8(b); it shows that desired velocities are achieved. Trajectories of follower vehicles are shown in Figure 8(c), it shows that local formation is formed and maintained. Velocities of follower vehicles are shown in Figure 8(d); it shows that desired formation velocities are achieved.

(a)

(b)

(c)

(d)

Figure 8 

(a) Trajectories of leader vehicles. (b) Velocities of leader vehicles. (c) Trajectories of follower vehicles. (d) Velocities of follower vehicles.

6. Conclusions

In this paper, we proposed a distributed formation framework of large-scale intelligent autonomous vehicles. Under the assumption that the initial network is connected, the vehicles are divided into leaders and followers automatically. Global formation information is only injected to leaders which can greatly reduce data exchange. Using the designed formation control law, the vehicles are able to realize various kinds of formation asymptotically while avoiding collision. It is illustrated that the proposed formation framework is effective by simulation results.

Nevertheless, there are still some other problems that need to be discussed in further detai, such as introducing more complex dynamic model and considering obstacle avoidance problems.

Acknowledgment

The research is supported by the National Science Foundation of China (NSFC) under Grant 6001203.