Collision Avoidance and Connectivity Preservation for Time-Varying Formation of Second-Order Multi-Agent Systems With a Dynamic Leader

In this paper, the problem of time-varying formation tracking control, with collision avoidance (CA) and connectivity preservation (CP), is studied for second-order multi-agent systems. Consider the unknown control input of virtual leader, an estimator is introduced to design a distributed time-varying formation tracking control protocol, while relaxing the condition the leader’s velocity is constant. Different from some existing results, relative velocity-based action function is designed such that both CA and CP with small negative effects on formation are achieved, and a simpler method is proposed to eliminate the large jump and discontinuity of potential field force. Furthermore, the Lyapunov-like approach is used for the stability analysis and the theoretical results are derived. Accordingly, it is demonstrated that velocity tracking errors can asymptotically converge to 0, and steady-state position tracking errors are bounded and related to initial state. Finally, two simulation examples verify the effectiveness of theoretical results.


I. INTRODUCTION
With the superiority in terms of distribution, coordination and robustness, multi-agent systems (MASs) have numerous applications in the fields of intelligent logistics, military defense and disaster rescue, and have become the hottest research topic of distributed control systems in recent years. Generally, in the investigation of MASs, our attention is focused on the design of model-based or model-free control protocols such that cooperative tasks (e.g. consensus, formation, containment and flocking) are completed, while satisfying the system performance requirements [1]- [4]. Notice that, an interesting problem of MASs is how to realize CA and CP, because safety assurance is a prerequisite for other tasks.
When mobile agent performs task, it cannot obtain state information and surrounding environmental information in presence of unknown interference factors, which may cause The associate editor coordinating the review of this manuscript and approving it for publication was Zhan Bu . path conflict and collision with other objects. Compared with obstacle avoidance, CA between agents is more difficult and challenging as a result of the unknown movement trends of moving agents. At present, for the collision problem of MASs, the popular solutions are behavior control method based on null space, model predictive control method and artificial potential field-based method. In behavior control approach, the missions to be accomplished have different priorities, and the low-priority task is projected onto the null space of CA task, so as to achieve the purpose of CA. At first, a series of results [5]- [7] were reported to detail the method design and theoretical proof. And then, behavior control approachs were applied to complete formation and collision tasks for MASs, UAVs and multiple spacecrafts [8]- [13]. Model predictive control approach considers distance constraints between agents for solving the optimal controller. Therefore, it can be realized that distances between agents are greater than predetermined safety distance [14]- [17]. In [18] and [19], the problem of formation control with CA for MASs was addressed by using distributed model predictive control scheme, and the latter made a trade-off between the conflict objectives through introducing relaxation parameters into control Lyapunov function. Under model predictive control architecture, a bi-directional CA system was designed to minimize collision risk, and a dynamic weighted tuning strategy was also proposed in [20].
Compared with aforementioned two methods, the principle and implementation of artificial potential field-based method are simpler. In short, repulsive forces are applied to agents such that the agent separates from other agent's potential field [21]- [23], when their relative distances are less than the predefined value. In [24], a novel adaptive potential function was proposed to tune the critical variable. For networked Lagrangian systems, cooperative surrounding control problem with CA was handled in [25], by combining the adaptive technique and the potential functions. In the presence of unknown velocity and uncertainty, an observer-based sliding mode control law was proposed in [26], while ensuring that the relative distance of agents met the specified constraints. Different from the other works, the energy function including the relative position variable in [27] was presented to certify the effectiveness of the proposed CA method. In [28], based on distance and velocity, five cases of possible collision are firstly summarized, and a time-varying formation control law is proposed. However, the above results did not consider the controller jump problem, as a result of the discontinuities of potential field forces at potential field boundary. As is well known, the high-frequency change of control input will accelerate the wear of actuator, and has adverse effect on control performance. Therefore, it is of practical significance to propose an artificial potential field-based CA method, without large jump phenomenon.
In practical application scenarios, each agent is equipped with communication equipment, of which the communication ability is usually limited. For this reason, communication will be interrupted, if the distance between agents exceeds the threshold of communication range [29]- [31]. Though communication interruption can be restored automatically with appropriate relative distance, it has declined coordination performance of MASs. With second-order double-integral dynamics, a formation control law was proposed in [32], and the range of potential field function parameters was given, while achieving both CA and CP. Yu et al. [33] proposed a practical time-varying formation control protocol for highorder MASs, and the smooth of potential field forces was ensured. In [34], the trajectory tracking control law for unicycle robots was developed such that interagent collisions were avoided, and communications among agents were ensured. For autonomous surface vehicles [35], the problem of CA and CP was investigated based on output feedback. It is worth noting that the results of this paragraph only studied fixed formation with limitations in practical application. Furthermore, the dynamic of virtual leader was considered without control input, that is, its velocity was constant, which shows that the assumption condition is too strong.
In light of the above discussion and analysis, this paper proposes a time-varying formation protocol with CA and CP for second-order MASs, and a dynamic virtual leader is considered. The main innovations and contributions are summarized as follows: 1) Both CA and CP for time-varying formation are addressed, rather than only the CA problem was investigated in [24]- [27]. Furthermore, compared with [32] and [33], relative velocity is introduced to determine the risk of collision and communication interruption, as well as reduce the adverse effect of potential field forces on time-varying formation. 2) A novel and effective method is proposed to eliminate the large jump and discontinuity of potential field forces, and it is simpler compared with [33]. 3) A dynamic leader is considered by using an estimator, which relaxes the condition in [28] that the leader's velocity is costant. Consequently, the application scenarios of time-varying foramtion are expanded and the corresponding theoretical results are derived. The rest of this paper is arranged as follows. The preliminaries and problem formulation are given in Section II to more clearly illustrate the main results in Section III. Section IV is the important stability analysis results. Simulation results and conclusion are shown in Section V and Section VI, respectively.
Notation: In this paper, the symbols are all standard definitions. sign(· ) is the sign function. · p represents the p norm, and p is positive integer. I n stands for the n-dimension identity matrix. x denotes the gradient along vector x.
⊗ is kronecker product. The index (t) for all the variables is omitted for simplicity in the appropriate case throughout the paper.

II. PRELIMINARIES AND PROBLEM FORMULATION
In this section, we first introduce graph theory, which will be used in subsection. And then, system model description and the objectives of this paper are provided.

A. GRAPH THEORY
In this paper, the communication between followers are twoway, so an undirected graph G = (W, ξ, A) is used to describe the interaction. W and ξ represent the edge set and vertex set, respectively. When there is an edge between agents i and j, one has a ij = a ji = 1, namely, they can exchange information with each other, otherwise a ij = a ji = 0. Notably, a ii = 0. The adjacency matrix A ∈ R N ×N is composed of element a ij . For the verification of resulting closed-loop system stability, the Laplace matrix L = D−A is necessary to be introduced, where D = diag{d 1 , d 2 , · · · , d N } is the in-degree matrix with d i = N j=1 a ij , and the number of followers is N .
Assumption 1: At least one follower can obtain information from virtual leader, and the communication topology G is connected. VOLUME 10, 2022 B. SYSTEM MODEL Consider the MAS modeled by the following linear secondorder double-integral system where p i ∈ R m , v i ∈ R m and u i ∈ R m are the position, velocity and control input of agent i, i = 0, 1, · · · , N , respectively. The label 0 represents the virtual leader, and others are the followers. Remark 1: It is noted that the velocity of virtual leader is time-varying, and so the motion forms of formation are varied. As a result, with the external reference signal (p 0 and v 0 ), the method design of formation tracking control is more complex, and application scenarios are wider compared with the existing works [24], [28], [29], [32], where the velocity of virtual leader is constant.
Assumption 2: [36] The control input of virtual leader is unknown to its followers, and it is bounded, namely, u 0 2 ≤ ξ .
In order to describe the geometry of time-varying formation, τ pi ∈ R m is introduced, and it is the desired position offset between the follower i and virtual leader. Define the formation tracking error as where τ vi ∈ R m is the desired velocity offset between the follower i and virtual leader. Obviously, τ vi can't always be 0 with the property of time-varying formation. The goal of this paper is to provide a formation tracking control protocol for MAS (2) to achieve the desired formation, and two cases are presented. Furthermore, there are no collisions between followers, and the distance between followers does not exceed the maximum communication distance. One case is that, if u 0 = 0, lim t→∞ e pi (t) 2 = 0 and lim t→∞ e vi (t) 2 = 0 for all followers should be assured. Nonzero control input of virtual leader is considered in the other case, and it can be obtained lim t→∞ e pi (t) 2 is bounded, as well as lim t→∞ e vi (t) 2 = 0.

III. MAIN RESULTS
In this section, we first give the time-varying formation control protocol. Next, based on the principle of artificial potential field, the CA and CP scheme without large jump phenomenon of potential field forces, are presented, respectively.

A. TIME-VARYING FORMATION CONTROL PROTOCOL DESIGN
Consider that only a portion of followers connect directly with virtual leader, the following estimator is implemented to obtain the state estimation of virtual leader as wherep 0,i andv 0,i are the ith follower's estimates of the position and velocity of virtual leader. 1 and 2 are parameters to be designed [36]. b i = 1 denotes that follower i can obtain information from the virtual leader, otherwise b i = 0. For analysis and description, the following auxiliary variables are introduced Then the timevarying formation control protocol is designed as follows where k 1 , k 2 and k 3 are the parameters to be designed.

Remark 2:
The third term of (6) ensures that the velocity tracking error asymptotically converges to 0 in presence of the non-zero control input of virtual leader. So it can be removed with in the autonomous virtual leader case. As is known to all, lim t→∞ê vi (t) ≡ 0 is not established in practical application as a result of measure noise and estimated accuracy. Hence, the presence of sign(ê vi ) has an adverse effect on control input whenê vi is small enough. To deal with this problem, we can remove the third term of (6) if the velocity tracking error e vi satisfies our predefined precision.

B. COLLISION AVOIDANCE SCHEME DESIGN
As shown in Fig. 1, there is a virtual potential field around agent j with a radius r out , and r out is the safe distance between agents. When agent i is in the potential field of agent j, and the condition v i 2 cosθ ij + v j 2 cosθ ji > 0 holds, it will be subject to a repulsive force along − → p ij . Likewise, agent j will be subject to a repulsive force along − → p ji . θ ij is the intersection angle between − → v i and − → p ji . The chattering-free potential field function is designed as follows: r out φ ca (s)ds, p ij 2 ∈ (2r, r out ] and ρ > 0 ca p ij 2 r φ ca (s)ds, p ij 2 ∈ (r out ,r] and ρ > 0 0, otherwise where ρ = v i 2 cosθ ij + v j 2 cosθ ji , andr > r out . Besides, ca is the chattering-free factor to be given, and p ij 2 represents the distance between agents i and j. The action function φ ca is designed as where δ ca is a positive constant, and µ is a small enough positive constant to prevent that the denominator of (8) is 0. The chattering-free factor ca is given by Remark 3: Different from the previous works [24], [27], [28], [32], the potential field function (7) is divided into three sections to achieve chattering-free effect. We can have that from the action function (8), when the distance between agents is certain, the greater the relative velocity, the greater the potential field force. Therefore, the magnitude of the potential field force is determined jointly by the distance and the relative velocity. From Fig. 1 and Eqs. (7)-(9), it can be seen that it is a transient process, when p ij 2 varies from r out tor. To be specific, ca approaches 1 as p ij 2 → r out , and it approaches 0 as p ij 2 →r, which ensures that Ψ ca is continuous at p ij 2 =r and p ij 2 = r out , respectively. Therefore, the chattering phenomenon can be eliminated, and Eq. (9) is simpler than the results in [33].
Remark 4: ρ is the relative velocity between agents i and j, and it is introduced such that collision risk is accurately judged. The truth is, distance cannot be the only condition to determine the collision risk. Though the distance between two agents is very small, there is no collision risk in the case that two agents are stationary, as well as they move in the direction of increasing relative distance. Thus, the relative velocity and distance jointly determine the collision risk.
Consider other agents may collide with agent i at the same time, we get the CA control law of agent i as where N ca i is the CA neighbors, i.e., N ca i = {j| p ij 2 ≤r}.

C. CONNECTIVITY MAINTENANCE SCHEME DESIGN
With the limited communication capacity, CP is necessary to complete the collaborative task. Consider that CP is the dual problem of CA, the scheme design procedure is similar to CA case. As shown in Fig. 2, withR ≤ p ij 2 ≤ R out and ρ < 0, a potential field attractive force arises, which will make the distance between agents decrease. It is worth reminding that R out is the radius of the maximum communication range, andR is the communication interrupt warning radius. Based on the above description, the potential field function  for CP is where the adaptive action function φ conn is designed as where δ conn is a positive constant. The chattering-free factor is given as follows The CP control input for the agent i is where N conn i = {j :R ≤ p ij 2 < R out and j ∈ N i }. Combine the formation term (6), the CA term (10) and CP (14), we have (15) VOLUME 10, 2022
Proof: Consider the following Lyapunov-like function V 1 as where E p = [e T p1 e T p2 · · · e T pN ] T , and then the time derivative of V 1 iṡ With the undirected topology, the potential field force has no effect on the overall system (see [27], [28], [32]). Therefore, we have Note thatp 0,i = p 0 ,v 0,i = v 0 with the introduced estimator (4) when t ≥ T 1 [36], and thus Eq. (17) can be rewritten aṡ where e vi 1 > e vi 2 and the property of L are used. From (19), one has that V 1 is bounded and satisfies V 1 ≤ V 1 (0). Also,V 1 → 0 as t → ∞ is implied. Then, e vi → 0 as t → ∞ are obtained with Eqs. (16) and (19).
If p ij 2 → 2r or p ij 2 → R out , Ψ ca or Ψ conn will be infinite. Nevertheless, it has been proved V 1 is bounded. Consequently, CA and CP are ensured.
Remark 5: It is noted that the states of system (2) will not escape during [0, T 1 ], and the detailed reason has been given in [37]. Hence, the proof process is omitted for simplicity.
Remark 6: To the best knowledge of authors, one of two following conditions should be satisfied such that the leaderfollower formation tracking control can achieve zero steadystate tracking error. One is that all the followers can know the control input of the leader or estimate the states of the leader (see [39]- [41]), and the other is that the leader is autonomous (i.e. u 0 = 0) (see [42]- [44]).
Remark 7: For the method proposed in this paper, we can place agents in the positions in advance to make the initial energy (i.e. V 1 (0)) as small as possible, so we can obtain the purpose of small formation steady-state position tracking error.

V. SIMULATION RESULTS
In this section, two simulation examples are employed to verify the effectiveness of the proposed scheme, of which the undirected communication topology is shown in Fig. 3. First, the fixed formation with an autonomous leader is considered in 2-D environment, and CA as well as CP is verified. Next, to be more general, the time-varying formation case is presented, with a dynamic leader in 3-D environment.

A. FIXED FORMATION WITH AN AUTONOMOUS LEADER IN 2-D ENVIRONMENT
In this example, the distances between agents are timeinvariant if the formation is achieved. The desired formation is described by The initial velocity of followers and the state of estimator (4), are all 0. The initial velocity of virtual leader is [1.5 1.5] T , and the initial position is According to Theorem 1, relative parameters are selected as The simulation results are exhibited in Figs. 4-6. Fig. 4 shows the fixed formation trajectory of the MAS in 2-D environment. We can see that the five followers form a fixed formation like a boat, and the trajectory is a little abnormal in the initial stage due to the potential field forces. The evolution of distances between followers with/without CA and CP is revealed in Fig. 5, respectively. It can be seen from Fig. 5(a), agent 2 collides with agent 3 (e.g. p 23 2 < 2r) and the distances p 12 2 and p 13 2 are all greater than the radius of the maximum communication range (e.g. p 12 2 > R out and p 13 2 > R out ), when the algorithms of CA and CP are not introduced. For comparison, both CA and CP are achieved in Fig. 5(b), which reveals the validity of our proposed method. Furthermore, it is confirmed that in Fig. 6 zero steady-state formation tracking error can be achieved, when the control input u 0 of virtual leader is 0.

B. TIME-VARYING FORMATION WITH A DYNAMIC LEADER IN 3-D ENVIRONMENT
In this example, the initial position as well as the initial velocity of virtual leader, and the initial state of estimator, are all set to 0. The initial positions and velocities of followers are    The time-varying formation are described as 3cos(0.5t), −0.5cos(0.5t), 0] Ṫ τ v5 = [0, 0, 0] T .
The control input of virtual leader is u 0 = [0, 0.2sin(t) + 0.02, 0.5sin(t) + 0.1] T . To handle it, we set k 3 = 5, and other parameters are the same as the fixed formation case.   The simulation results are presented in Figs. 7-9. As shown in Fig. 7, the formation shape is time-varying in 3-D environment. In Fig. 8, the distances between followers accord with our prespecified formation (see τ p1 (0) andτ v1 ). Note that, the algorithms of CA and CP do not work, because the distances between followers are always within a reasonable range in this simulation. Obviously, Fig. 9 is different from Fig. 6 in 2-D environment, because the virtual leader is dynamic. The velocity tracking errors converge to 0, and the steady-state position tracking errors satisfy the condition lim t→∞ N i=1 e pi 2 ≤ 2V 1 (0)/ξ in Theorem 1. Therefore, we can reduce the steady-state position tracking errors by selecting appropriate initial state.

VI. CONCLUSION
This paper has investigated the time-varying formation problems of second-order MASs with a dynamic leader. Compared with existing works, the proposed formation control scheme can handle the unknown control input of virtual leader, in the case that not all followers can receive information from virtual leader. As a result, zero steady-state velocity tracking errors can be guaranteed, and the steadystate position tracking errors are within allowable domain with appropriate initial state. Also, CA as well as CP was ensured, and the large jump phenomenon of potential field force was eliminated. To validate the above results, two simulation examples including fixed formation and time-varying formation, were implemented. In the future, we will focus on the time-varying formation control problem with CA and CP for higher-order nonlinear MASs, while using the multiobjective optimization method [45]- [48].