#### Abstract

This paper aims to investigate the formation control of leader-follower multiagent systems, where the problem of collision avoidance is considered. Based on the graph-theoretic concepts and locally distributed information, a neural fuzzy formation controller is designed with the capability of online learning. The learning rules of controller parameters can be derived from the gradient descent method. To avoid collisions between neighboring agents, a fuzzy separation controller is proposed such that the local minimum problem can be solved. In order to highlight the advantages of this fuzzy logic based collision-free formation control, both of the static and dynamic leaders are discussed for performance comparisons. Simulation results indicate that the proposed fuzzy formation and separation control can provide better formation responses compared to conventional consensus formation and potential-based collision-avoidance algorithms.

#### 1. Introduction

Recently, distributed multiagent coordination has attracted much attention in many fields, where only the information available locally from its neighbors is required for each agent [1–6]. There are many applications of multiagent systems, such as autonomous unmanned aerial vehicles [7], autonomous formation flight [8], congestion-controlled communication network [9], wireless sensor network [10], and autonomous multivehicle formations [11]. Graph theory has been used to characterize network topologies for consensus studies. A general consensus problem solving is to find a distributed control strategy such that the states of agents converge to a common value. Average-consensus problem was investigated for distributed networks with fixed and switching topologies [12]. Relying on graph theory, matrix theory, and control theory, the analysis of consensus protocols was thoroughly discussed. In [13], an impulsive control protocol was presented for multiagent linear dynamic systems with fixed topology based on the local information of agents. A fuzzy sliding-mode controller was proposed to investigate the formation control problem in directed graphs [14]. Cai et al. [15] addressed the controllability improvement problem for two types of linear time-invariant dynamic multiagent systems by adjusting the configuration of graphs. A general case of leader-following consensus problems under fixed and switching topologies was discussed in [16]. The second-order agents under switching topology were concerned in [17], where the condition of communication delays was determined for achieving consensus. The consensus problem for a group of high-order dynamic agents with switching topology and time-varying communication delays was discussed in [18]. In the work of [19], linear consensus protocol and saturated consensus protocol were presented for the consensus problem of heterogeneous multiagent system. The heterogeneous multiagent system consists of first-order and second-order integrator agents.

Lately, the fuzzy logic control which consists of linguistic control rules is a technique to design controllers based on human expert knowledge and experience. This technique is a good alternative to overcome the difficulties in the requirement of exact mathematics models for plants with unexpected complex dynamics and external disturbances. Although the method has been practically successful, it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems. Recently, based on Takagi-Sugeno (T-S) fuzzy techniques, there have appeared in the literature a great number of results concerning stability analysis and design. T-S fuzzy techniques have been applied to many applications of interest [20–25]. However, the membership functions of the above method are through manual adjustment. Lately, neural fuzzy control, combining with the capability of fuzzy reasoning to handle uncertain information and the capability of artificial neural networks to learn from processes, has been popularly addressed. In [26], a robust fuzzy neural network control (FNN) scheme including a parameter tuning algorithm was designed for a linear Maglev rail system to achieve the objective of model-free control. A neural network-based self-learning control strategy including an FNN controller and a recurrent neural network identifier was proposed for electronic throttle valves [27]. Lin and Shen [28] proposed an adaptive fuzzy neural network control scheme for a field-oriented control permanent magnet linear synchronous motor servo-drive system to track periodic reference trajectories. In [29], an adaptive network based fuzzy inference system was presented for speed and position estimations of a permanent-magnet synchronous generator. In [30], an adaptive neurofuzzy controller was presented for the tracking control of dynamic systems with unknown nonlinearities. A recurrent fuzzy neural controller for the robust tracking of a robot manipulator with adaptive observers was addressed in [31]. A robust self-organizing neural fuzzy control scheme was proposed for a class of uncertain nonlinear MIMO systems [32]. In the study of [33], an adaptive neurofuzzy inference system was employed to identify hand motion commands based on surface electromyogram signals. In [34], a new approach was proposed for machine health condition prognosis with the integration of neurofuzzy system and Bayesian algorithms. Moreover, a T-S fuzzy-neural model was adopted for identification and robust adaptive control of an antilock braking system [35]. In addition, a hybrid evolutionary algorithm using fuzzy rules to adjust optimization parameters was proposed in [36].

In multiagent networks, the problem of collision avoidance is also an important and interesting topic that is worthy of being discussed. A cooperative control law was proposed for general nonlinear dynamic models to guarantee collision-free conflict resolution [37]. In [38], a fuzzy logic was designed for potential functions to achieve the separation control with input constrains. In the work of [39], a flocking algorithm was presented for separation forces generated to avoid collisions with external obstacles. A modified avoidance function was proposed for nonlinear Lagrange systems to achieve collision avoidance with bounded disturbances [40]. In [41], a potential field method was discussed for mobile robots to solve the local minimum problem. In addition, Wang and Gu [42] presented a fuzzy potential force for the separating potential function in flocking control. However, only few of existing results have been presented to solve the problem of local minima in multiagent systems.

This paper aims to investigate the formation control of leader-follower multiagent systems, where the problem of collision avoidance is also considered. The graph theory is used to model the communication topology between agents. To improve the control performance, a novel formation algorithm, neural-fuzzy formation controller, is proposed for multiagent systems in directed graphs. The neural-fuzzy control parameter consists of input Gaussian membership functions and output fuzzy singletons, where the parameters of input and output membership functions can be adaptively adjusted. The proposed neural-fuzzy formation controller has the capability of on-line learning, and the adaptive rules can be derived using the gradient descent method. Moreover, a fuzzy based separation control is presented for collision avoidance, and the local minimum problem of traditional potential-based separation control can be solved. The fuzzy based separation control consists of triangular input membership functions and singleton output membership functions, where the control output provides an alternative moving direction for agents to achieve collision-free tasks. Numerical simulations are provided to validate the collision-free formation responses.

This paper is organized as follows. In Section 2, some essential graph-theoretic concepts and a network of single-integrator agents are introduced. In Section 3, the framework of a neural-fuzzy formation controller is investigated, where the updating rules for controller parameters are derived. In Section 4, the conventional potential-based collision avoidance is introduced. Moreover, a novel fuzzy-oriented separation control is presented. In Section 5, simulation results are provided for performance validations. Some concluding remarks are given in Section 6.

#### 2. Preliminaries

##### 2.1. Graph Theory

Considering a multiagent system of agents, let be a directed graph (digraph), consisting of a vertex set and an edge set . The vertexes and represent the th and th agents, respectively. In digraphs, an edge of is an ordered pair of distinct nodes , in which and are the head and tail of the edge, respectively [43]. The weighted adjacency matrix of a digraph is denoted as where is the link weight; , if , and , if .

In this paper, a leader-follower problem will be dealt with, where the multiagent system consists of agents, one leader, and followers. In notations, the agents indexed by are followers and the item is the leader. Assume that the leader agent has only transmitting capability, that is, the leader acquires no information from followers, . In this case, let the topology relationship of follower agents be denoted as , a subgraph of . Then, the associated adjacency matrix of is represented as Consequently, the connection relationship between the leader and followers can be described as , where , .

##### 2.2. Single-Integrator Multiagent System

In this paper, a single-integrator network is considered as where is the position vector and is the control input vector of the th agent, . It is assumed that all agents have the same environment sensing capability. In addition to formation keeping, each agent is not allowed to collide with other agents during the whole moving process.

The geometric relationship between agents is shown in Figure 1, where is the sensing radius. The node is a neighboring agent of the th node if the Euclidean distance between two agents is less or equal to the sensing radius, . Let stand for the neighbor set of the th agent. Once the th agent moves inside the sensing radius of the th agent, the collision-avoidance mechanism starts to work. In Figure 1, the notation denotes the avoidance radius of which the minimum distance allowed between two agents is . In this case, it is reasonable that for preventing collisions.

**(a)**

**(b)**

#### 3. Neural Fuzzy Formation Control

##### 3.1. Structure of Neural Fuzzy Controller

In this section, a neural fuzzy control (NFC) is proposed to deal with the leader-following formation problem, where the single-integrator model of (3) is considered. First, let the and -axis error functions be, respectively, defined as where , and are coordinate positions regarding a desired formation pattern in and -axes. It is noticed that means that the th agent can send position information to the th agent, and means that the th agent can receive position information from the leader. In this study, the leader is maneuvered along a prespecified trajectory, and the design of formation controller is focused on followers. Let the controller inputs of the th follower agent be designated as follows where and are positive constants, .

The network structure of NFC is shown in Figure 2. The fuzzy rules are given in Table 1, where the input and output spaces are fuzzily partitioned into six fuzzy sets, Negative Big (NB), Negative Medium (NM), Negative Small (NS), Positive Small (PS), Positive Medium (PM), and Positive Big (PB). The input and output membership functions are depicted in Figure 3. The corresponding if-then fuzzy rules for the th agent are expressed as where and are the fuzzy sets of the antecedent and consequence parts, respectively, . In Figure 3, the th nodes of the membership layer are Gaussian functions, where . In (8) and (9), and are means and standard deviations, respectively, . By using the centroid defuzzification technique, the NFC output can be calculated as follows: where are the values corresponding to singleton outputs, in which and .

**(a)**

**(b)**

##### 3.2. Parameter-Learning Algorithms

In this section, the idea of gradient descent will be adopted to derive on-line learning algorithms to update NFC parameters. First, energy functions and are defined as follows: where . Then, the update laws of layer parameters are described in the following.

*(1) Output Layer.* According to the gradient decent method [44], and are updated by the following rules:
where is a learning rate, . From (11), it can be obtained that
In (14), the error term to be propagated can be reformulated as
However, the terms and in (16) cannot be analytically determined. To overcome this problem, the following adaptive laws are adopted [45]:
where and are positive constants.

In summary, from (15) and (17), the parameters of the output layer can be adaptively updated.

*(2) Membership Layer.* The parameters and are updated by the following rules:
where is a learning rate, , . From (8), (9), and (11), it can be obtained that
Similarly, and the are updated by the following amounts:
where is a learning rate, , . From (8), (9), and (11), it can be obtained that
In summary, from (18)–(21), the parameters of the input layer can be adaptively updated.

#### 4. Separation Control for Collision Avoidance

##### 4.1. Potential-Based Separation Control

Let be the separation force between the th and th agents. Then, the integrated separation force from all its neighboring agents can be formulated as To derive a proper separation force between two connected agents, a smooth potential function is considered, such that In a two-dimensional case, , the gradient computations of can be obtained as follows: where is the maximum allowable separation force. Integrating the formation and separation forces, the net control action to an agent can be obtained as where is the neural fuzzy formation action.

##### 4.2. Fuzzy Separation Control

In a multiagent system, an agent could be standstill or move back and forth if the net force acting on this agent is balanced. This phenomenon is known as the local minimum problem. In Figure 4, the case of agent with two neighboring agents is considered, where and are neighboring agents, and the related target is located at . The notation is denoted as the attractive force to the target, and and are separation forces corresponding to neighboring agents. Then, the integrated separation force to the th agent is the vector sum of and , where is the related direction of to the -axis. In addition, is the net force of the th agent, and is angle between the agent and its target. In case , the direction of is opposite to the attractive action . Moreover, if the magnitude of is less or equal to the magnitude of , the th agent will be stuck in the local minimum. To solve this problem, a fuzzy separation control method will be presented, and the key idea is depicted in Figure 5, where an extra angle is added to the original separation force, . In Figure 5, is the modified separation force, of which the magnitude keeps unchanged but the direction is changed because of . Consequently, the integrated net force of attractive force and separation force can be represented as Basically, can provide a new route to bypass the neighboring agents when a local minimum situation happens. It is noticed that the target of a follower is its temporary destination for next movement during the process of avoiding collision. For those follower agents, communication-connected to the leader, their respective targets can be obtained according to the leader position and designated formation pattern; however, a substitute solution is required for other followers. Alternatively, from (4) and (5), targets can be equivalently viewed as the propagated errors from other followers, where follower is not communicated to the leader, , .

The design of fuzzy separation controller will be illustrated in the following. First, let the fuzzy inputs of the th agent be where is the position of the center gravity of neighboring agents, where is the cardinality of a set, that is, is the number of neighboring agents of the th agent.

The fuzzy rules are given in Table 2, where the input and output spaces are fuzzily partitioned into ten fuzzy sets, Negative Big (NB), Negative Small (NS), Zero (ZO), Positive Small (PS), Positive Big (PB), Very Close (VC), Close (C), Medium (M), Far (F), and Very Far (VF). The input and output membership functions are depicted in Figures 6 and 7, respectively. The corresponding if-then fuzzy rules for the th agent are expressed as where and are the fuzzy sets of the antecedent part, and is the fuzzy set of the consequence part, , and . By using the centroid defuzzification technique, the defuzzified fuzzy output is calculated as where a min-max operation is performed over all rules mapping to the same output fuzzy set:

**(a)**

**(b)**

#### 5. Simulation Results

In the following, all the agents are assumed to be homogeneous with the same specifications, (m) and (m). To verify the feasibility of proposed neurofuzzy formation controller and fuzzy separation controller, both the collision avoidance and formation preservation are considered.

##### 5.1. Collision Avoidance with Static Target

One dynamical agent is initially placed at the point (m), and four fixed agents are located at , , , and (m), respectively. The target position is given as (m). It is desired that the dynamic agent can reach the designated target without colliding with fixed agents. Figures 8 and 9 illustrate the collision-avoidance responses corresponding to potential-based separation control (PSC) and the proposed fuzzy separation control (FSC), where the formation control of each case is the conventional consensus algorithm [46]. In Figure 8, it can be seen that the agent lingers in front of neighboring agents with PSC. On the other hand, the agent can successfully bypass neighboring agents with the proposed FSC. The responses of position errors, shown in the bottom two subplots of Figure 8, indicate that the desired target can be asymptotically achieved with the proposed FSC. However, a local-minimum behavior exists by using the PSC, and the related separation force keeps oscillation. The oscillation behavior of the separation force is coincided with the response of , shown in Figure 9(a). In Figure 9(b), the angle after 8.26 (sec) means that the dynamic agent successfully bypasses the fixed agents, and thus, there is no separation force.

**(a)**

**(b)**

**(a)**

**(b)**

##### 5.2. Formation Control and Collision Avoidance with Static Leader

In leader-follower formation control, the case of five agents, one static leader and four followers, is considered. The communication topology is shown in Figure 10, where the circles labelled 1 to 4 denote the follower agents and the circle represents the leader agent. From (2), the information exchanges between leader and followers can be modelled as The followers are initially placed at the points , , , and (m), respectively, and the position of the leader is (m). The formation pattern is designated as , , , , and (m). The parameters of the NFC are originally chosen as , , , , , , , and . Three unique distributed strategies, consensus algorithm with potential-based separation control (CA + PSC), consensus algorithm with fuzzy separation control (CA + FSC), and neurofuzzy formation control with fuzzy separation control (NFC + FSC), are considered. Accordingly, simulation results are shown in Figures 11, 12, and 13, including the error trajectories of - and -axis, and the relative distances between a pair of agents. Since the relative distance between any two agents is greater than , shown in Figures 11(b), 12(b), and 13(b), the desired collision avoidance can be accomplished using these three control strategies. In addition, the formation responses are shown in Figures 11(a), 12(a), and 13(a). It can be seen that there exist significant steady-state errors using CA + PSC. Conversely, the formation pattern can be asymptotically achieved with the use of NFC + FSC. The position errors, , are summarized in Table 3, where IAE is the integral absolute error, ISE is the integral square error, ITAE is the integral time absolute error, and ITAE stands for the integral time square error. In summary, the proposed NFC + FSC can provide better performance than the counterparts of the other two methods.

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

##### 5.3. Formation Control and Collision Avoidance with Dynamic Leader

In the following, previous leader-follower system is addressed, where all initial settings are the same except that one fixed agent is added at the position (m). This fixed agent can be viewed as a standstill obstacle. Besides, a time-varying leader is considered, where the velocity vector is (m/s). The formation responses are shown in Figures 14, 15, and 16, where the methods of CA + PSC, CA + FSC, and NFC + FSC are considered. Similar to the previously mentioned illustrations, the moving trajectories and the relative distances between two follower agents are depicted in Figures 14, 15, and 16. It can be seen that even the collision avoidance can be achieved with CA + PSC and CA + FSC; however, these two strategies eventually fail to preserve the desired formation pattern. On the other hand, from Figure 16, collision-free formation can be obtained by using the proposed NFC + FSC control scheme. In particular, it can be seen that the third follower can successfully bypass the fixed agent and keep its way to form a designated pattern. The formation errors of different control strategies are summarized in Table 4. It can be concluded that the proposed neurofuzzy formation and fuzzy separation combined controller can provide better responses than the counterparts of conventional consensus algorithm with potential-based separation control.

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

*Remark 1. *From simulation results, it can be seen that the collision-free formation task can be achieved for single-integrator agents. In particular, the local minimum problem can be overcome using the proposed fuzzy separation control. It is promising that the proposed works can be applied to some practical applications, such as multirobot systems and unmanned vehicle systems.

#### 6. Conclusion

This paper presents a neural fuzzy formation controller for multiagent systems. The learning rules for controller parameters can be derived from the use of gradient decent method. In addition, a fuzzy separation control is proposed to achieve collision avoidance such that the problem of local minima can be solved. Simulation results are provided for the cases of static leader and dynamic leader. The collision-free leader-follower formation can be accomplished by the proposed fuzzy formation and separation control strategy. Performance comparisons indicate that the proposed fuzzy-based control scheme has better formation responses compared to the counterparts of conventional consensus algorithm and potential-based separation control. The current results are mainly limited in single-integrator multiagent systems. It should be challenging and interesting to investigate the collision avoidance problem for high-order agents with nonlinear dynamics. For example, a fuzzy based separation control for kinematic agents will be undertaken in our future work.

#### Acknowledgments

This work is partially supported by the Taiwan National Science Council under the Grants NSC 101-2221-E-182-041-MY2 and NSC 100-2918-I-182-002. The authors also want to appreciate the High Speed Intelligent Communication Research Center, Chang Gung University, Taiwan, for continuing support in the study of multiagent systems.