Consensus Formation Control and Obstacle Avoidance of Multiagent Systems with Directed Topology

,is study addresses the problems of formation control and obstacle avoidance for a class of second-ordermultiagent systems with directed topology. Formation and velocity control laws are designed to solve the formation tracking problem. A new obstacle avoidance control law is also proposed to avoid obstacles.,en, the consensus control protocol consists of the formation, velocity, and obstacle avoidance control laws. ,e convergence of the proposed control protocol is analyzed by a redesigned Lyapunov function. Finally, the effectiveness of theoretical results is illustrated by simulation examples. ,e simulation results show that the formation tracking problem of the given multiagent systems can be realized and obstacles can be avoided under the proposed control protocol.


Introduction
e cooperative control problem of multiagent systems has been attracted outstanding attention in the past few years due to its widespread applications in multisensor systems [1], mobile robot systems [2,3], unmanned aerial vehicle systems [4,5], power distribution networks [6], and so forth. e basic problem of cooperative control is consensus, in which the objective of consensus is to design an appropriate control protocol, such that the output of all agents can achieve synchronization or track a desired trajectory.
Currently, the consensus control problems of multiagent systems have been extensively addressed in existing papers. Many control protocols have been designed to achieve the consensus control of multiagent systems. In [7], the distributed linear control protocol for the linear multiagent systems with limited interaction ranges was designed. Iterative learning control protocols were proposed in [8][9][10] to solve the consensus tracking problem of nonlinear multiagent systems. In [11], the consensus problem of nonlinear multiagent systems with directed topology and communication constraints was investigated, in which each agent communicated only with its neighbors. Moreover, in [12], a consensus protocol with the local state information was proposed to solve the event-triggered control problem of general linear multiagent systems. e finite-time consensus tracking control problem of multiagent systems with uncertain nonlinear dynamics and error constraints was investigated in [13], in which the nonsingular fast sliding mode control technique was used.
It is not difficult to see from the abovementioned papers that the research on the consensus control problem of multiagent systems has achieved rich results. However, these papers do not further analyze the formation control problem of multiagent systems. As an important research direction, the formation control problem has played an important role in many fields, such as the formation control of spacecraft [14], multiple aerial vehicles [15], multiple quadrotors [16], and mobile robots [17][18][19]. e formation control of multiagent systems has been discussed in some studies. In [20], the optimal formation problem of first-order multiagent systems with fixed communication topology was considered.
In [21], the formation tracking problem with distributed observer was addressed, in which the distributed formation tracking control protocol was constructed. e control protocol with communication time-varying delay was presented in [22]. Furthermore, the formation control strategy with position estimation [23] and the distributed formation iterative learning control protocol [24] were also addressed. Meanwhile, the consensus control protocols for the multiagent systems were proposed to solve the problem of collision avoidance [25][26][27]. It should be pointed out that the results on the formation control or the control of collision avoidance are discussed separately. To the best of the authors' knowledge, however, it should be paid attention to prevent the collision with obstacles while solving the formation problem of multiagent systems. However, the problem has received minimal attention in the existing literature.
Inspired by the abovementioned facts, this study investigates the formation control and obstacle avoidance for a class of second-order multiagent systems with directed topology. e main contributions of this work are as follows: (i) the formation control law and velocity control law are designed to solve the formation tracking problem of given multiagent systems with directed topology. Furthermore, a new obstacle avoidance strategy is proposed to guarantee that all agents avoid obstacles. (ii) By comparing with the control protocols proposed in [24][25][26], the current consensus control protocol consists of the designed formation control law, velocity consensus control law, and obstacle avoidance control law. e purpose of this is to solve the formation control and obstacle avoidance problems of the given multiagent systems at the same time. (iii) To prove the convergence of the proposed control protocol, a new Lyapunov function is structured in this paper. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed control protocol. e remainder of this paper is organized as follows. Graph theory is introduced in Section 2, and the problem formulation is given in Section 3. In Section 4, the control protocol design and convergence analysis are discussed. Next, the simulation examples are provided to illustrate the effectiveness of theoretical analysis in Section 4. Finally, conclusions are drawn in Section 5.

Graph Theory
Let G � (V, Ε, A) denote a directed graph with a set of nodes V � υ 1 , . . . , υ n and a set of directed edges E � (i, j), i, j ∈ V, and i ≠ j . e weighted adjacency matrix is A � [a ij ] ∈ R n×n , where a ij > 0 if and only if (i, j) ∈ E; otherwise, a ij � 0. Agent j is called the neighbor of i if agent i receives the information from agent j. e set of neighbors of agent i is defined as e graph G is connected if there exists a path between any two vertices.
In this paper, the multiagent systems with n agents are considered. Hence, the exchange information among agents can be modeled as the directed graph n with n nodes.
According to the related knowledge of the graph theory G, we can theoretically analyze the control problem of multiagent systems. In addition, in order to achieve the desired formation shape, the distance between agents should be set. Hence, in this paper, let the matrix h is defined as the desired formation shape of given multiagent systems. Here, the matrix h � [h 1 , . . . , h n ] and h i � [h i1 , . . . , h in ] T with h ij being the desired distance between agent i and agent j.

Problem Formulation
In this paper, a class of second-order multiagent systems with n agents is studied, and the ith agent's dynamics are described as are the position, velocity, and control input of agent i, respectively.
To facilitate the following discussion, the time variable t will be ignored if there is no ambiguity. In addition, some definitions and lemmas are given as follows.
is a nonnegative function if the following properties are satisfied at the same time: Lemma 1. For a given multiagent system, let ∇ x i σ(x ij ) be the gradient function of a continuous differentiable function σ(x ij ); then, the following property is held: Proof. For a given multiagent system, let Furthermore, we can obtain For the given multiagent systems, it is easy to obtain that where x ji represents the Euclidean distance from agent j to i, and then, we have Hence, on the basis of equations (4) and (5), one obtains Considering equation (6), we obtain and it is obtained that e proof is completed. e control objective of this study is to design a suitable control protocol u i (t) (i � 1, . . . , n), such that the output of all agents can achieve desired formation shape without obstacles, that is, Meanwhile, it is also guaranteed that the desired formation shape can be maintained after avoiding obstacles.
where α > 0 is the formation control coefficient. e curve of function Θ(x ij ) is shown in Figure 1. Figure 1 presents that Θ(x ij ) ≥ 0 regardless of how to change the distance between agents i and j. at is to say, is condition implies that agents i and j can hold the desired distance. erefore, the formation control law of the given multiagent systems can be designed as To maintain the velocity consensus, a velocity control law is designed as where Ψ(v i − v j ) is called velocity adjust function and represented as where β > 0 is the velocity control coefficient. On the basis of equation (12), the function Ψ(v i − v j ) will be equal to zero as v i � v j for i ≠ j ∈ 1, . . . , n { }. When obstacles exist in the environment, to prevent the collision with obstacles, a new control strategy U O bi for agent i is regarded as where c > 0 is the avoidance control coefficient; x i is the position of agent i; O b , r, and R represent the center, radius, and maximum detection radius of an obstacle, respectively; and ‖x i − O b ‖ is the Euclidean distance between agent i and an obstacle.

Mathematical Problems in Engineering
By taking gradient of function U O bi , By checking equations (13) and (14), we have (14), which indicates that ∇ On the basis of the above analysis, the new obstacle avoidance control law can be designed as Hence, considering the formation control law u ia , velocity control law u ib , and obstacle avoidance control law u ic , the consensus control protocol for agent i can be selected as u i � u ia + u ib + u ic , that is,

Convergence Analysis.
e main results of this work are shown in eorem 1. (16), and assume that multiple obstacle surroundings being considered or not; then, all agents can achieve the desired formation shape and maintain the velocity unchanged, i.e., lim t⟶∞ (x i (t) − x j (t)) � h ij and lim t⟶∞ v i (t) � v j (t) with i ≠ j ∈ 1, . . . , n { } and h ij ∈ h. Meanwhile, the desired formation shape will still be held after avoiding obstacles.

Theorem 1. Consider the multiagent system (1) with the directed communication topology and the consensus control protocol
Proof.
e Lyapunov function candidate is considered: On the basis of equation (17), (1/2) n i�1 v 2 i and (1/2) n i�1 n j�1,j≠i Θ(x ij ) are continuously differentiable in x and v. As shown in equation (13), n i�1 U O bi is also a differentiable function. Equation (17) presents that (1/2) n i�1 v 2 i ≥ 0. Figure 1 indicates that the second item in equation (17) (1/2) n i�1 n j�1,j≠i Θ(x ij ) ≥ 0 is easily obtained. Furthermore, function U O bi ≥ 0 is obtained from the description of equation (13). en, the third item in equation (17) n i�1 U O bi ≥ 0 can be acquired. Equation (17) is hence an effective Lyapunov function, and V(x, v) ≥ 0. By taking the derivative of function V(x, v) along with x and v, we have where the control protocol (16) Substituting equation (19) into (18) Substituting equation (12) into equation (20) yields then we have Γ ≜ (x, v): v 1 � · · · � v n . e preceding analysis implies that _ V(x, v) � 0 if and only if v 1 � · · · � v n . Moreover, let v 1 � · · · � v n � d, and obtain _ us, the consensus problem of the multiagent system (1) can be achieved.
Considering equation (9), let where ‖x i − x j ‖ � h ij is the equilibrium point of the multiagent system (1), and equation (17) has the minimum value under the equilibrium point. e analysis of equation (23) shows that each agent can maintain the desired distance h ij , that is, ‖x i − x j ‖ � h ij , which indicates that the desired formation is achieved. Function V(x, v) is a bounded function due to _ V(x, v) ≤ 0. erefore, the agents can avoid obstacles under the consensus control law (16). e proof is completed.

Simulation Analysis
A class of second-order multiagent systems with four agents is considered. e desired formation shape is defined as a square. e length of the square is set as 4. e number of obstacles is set as 3. e dynamics of the four agents are described as equation (1), that is, and Figure 2 presents the directed topology among agents. On the basis of Figure 2, adjacency matrix A and desired distance h are as follows:  Figures 3 and 4, it can be easily found that the consensus formation tracking problem of the multiagent system (1) with the designed control protocol (16) can be achieved. Figure 5 shows the desired formation shape, which indicates that the desired formation tracking problem can also be solved under the proposed control protocol (16). Figure 6 depicts the desired distance among agents. is is completely consistent with the theoretical results, which further illustrate that the control protocol designed in this paper is effective. In addition, the control input curves of the four agents are given in Figure 7.
Overall, the results of Figures 3-7 show that the control protocol designed in this paper is effective. Although the existence of obstacles is not considered, the four agents can still achieve consensus, and at the same time, they can achieve and maintain the desired formation shape.
Example B. Consensus analysis with multiple obstacles.
In this example, three obstacles are considered in the process of achieving formation tracking. e initial position and velocity of each agent are the same as those in Example A. Figures 8-12 display the formation tracking results. Figures 8 and 9 show the tracking results of position and velocity with three obstacles, respectively. Although three obstacles are considered in the process of achieving formation tracking control, the consensus can still be solved under the consensus control protocol (16). e tracking results of position and velocity of the four agents have changed due to the existence of obstacles. Figure 10 presents the desired formation shape, which implies that the formation can be maintained after avoiding obstacles. As can be seen from Figure 10, when an obstacle appears during the operation of the agent, the agent will bypass the obstacle under the action of the designed control protocol. After circumventing obstacles, the agents will continue to maintain the desired formation shape under the control protocol.
is also illustrates the effectiveness of the control protocol designed in this paper from another angle. Figure 11 exhibits the desired distance of the four agents. e results in Figures 10 and 11 illustrate the effectiveness of the theoretical results. Figure 12 displays control input curves.
In general, the consensus control protocol (16) designed in this study not only can achieve the desired formation control but also can avoid obstacles and maintain the formation shape after avoiding the obstacles.

Conclusions
e consensus problems of formation control and obstacle avoidance for a class of second-order multiagent systems with directed topology were considered in this study. e designed control protocol consisted of the formation, velocity consensus, and obstacle avoidance control laws. A designed Lyapunov function was applied to analyze the convergence of the designed consensus control protocol. Under the given directed topology, the formation control problem of the multiagent systems without obstacles was solved by using the designed consensus control protocol. At the same time, the desired formation control can be achieved and maintained despite the existence of obstacles in the environment.

Data Availability
e figure data used to support the findings of this study are included within the article.   Mathematical Problems in Engineering 9