Distributed Dynamic Event-Triggered Formation Control for Multiple Unmanned Surface Vehicles

This paper investigates the problem of distributed dynamic event-triggered formation control for multiple unmanned surface vehicles (USVs) subject to external disturbances. Firstly, a novel event-based distributed formation control scheme is proposed, under an undirected communication graph. Wherein, a dynamic event-triggered data transmission mechanism is designed for communication resource reduction. The time-varying dynamic parameter ensures fewer triggering instants and a larger triggering time interval compared with the traditional static one. The adaptive laws are incorporated into the formation control scheme to address the effects of external disturbances and uncertainties. Subsequent to the theoretical analysis, it can be concluded that the relative tracking errors can converge to the origin, and all closed-loop signals are bounded. Moreover, it is proved that the triggering time sequences do not exhibit Zeno behavior. Finally, a numerical simulation is performed to verify the effectiveness of the proposed distributed formation controller.


I. INTRODUCTION
Over the last few decades, USVs have attracted growing interest from the academic community due to their great system reliability and high mission completion efficiency [1], [2].In inspection, monitoring, and oceanographic research tasks, especially given the harsh marine environmental disturbance, USV formation control technology is of greater importance than a single USV due to its better anti-jamming properties, greater adaptability, and higher reliability [3], [4].Nevertheless, the ever-increasing complexity and automation pose a variety of challenges for controller design.In this context, researchers have put forward a great diversity of research methods for USV formation, involving leader-follower strategy [5], [6], [7], behavior-based [8], [9], and virtual structure [10], [11], [12].Particularly, due to its high control accuracy The associate editor coordinating the review of this manuscript and approving it for publication was Zhiguang Feng .and decent system stability, the virtual structure control strategy has become an overwhelmingly popular research topic today.
In the virtual structure control method, a key aspect is the sharing of information among members, because the formation is considered as a cohesive unit to maintain the desired shape.Thus, different communication schemes have been proposed to schedule information transmission among members.Traditional communication strategies usually assume that an ideal communication network with unlimited bandwidth is required, which is not easy to satisfy in practice, especially when there are many members [13], [14], [15], [16].In addition, traditional communication schemes usually rely on time delays, i.e., so-called timetriggered communication techniques [13], [14].For example, in [13], all the formation members are transmitting periodic information in real-time.Similar to the information transmission method of [13], data sampling, processing, and execution are also performed periodically in [14].Such solutions may cause unnecessary waste of communication resources in actual operation.To make up for the above problem, the event-triggered mechanism has received wide attraction recently.In this approach, a triggering strategy is utilized to determine when data transmission occurs.Therefore, the event-triggered approach, including static-based and dynamic-based methods, provides significant advantages in terms of decreasing communication occupation and saving energy usage.In engineering practice, it is necessary to consider whether there is a Zeno phenomenon, which means that the control behavior is triggered indefinitely within a limited time.The occurrence of the Zeno phenomenon will make the corresponding control behavior impossible to execute and even cause system instability.In view of this, the control algorithm theoretically needs to eliminate the Zeno phenomenon.
In preliminary work, static event-triggered communication schemes [17], [18], [19], [20] have been continuously investigated with the aim of reducing the frequency of communication while ensuring consensus.For instance, in [18], a compensator-based command-filtered formation control algorithm is proposed, and an event-triggering communication strategy is developed to govern the communications between the leader AUV and follower AUVs.Using the same approach to dealing with wasted communication resources as presented in the literature [18], a distributed event-triggered adaptive coordinated trajectory tracking controller for multiple unmanned surface vehicles is proposed in [17], based on the undirected communication topology.As a technical extension of this result, the fixed-time control method is introduced to design the event-based distributed formation controller for time-oriented missions [19].In conclusion, the majority of the event-triggered control algorithm is to update the control protocol when the norm of the measurement error is no less than an admissible bound or the norm of the system states.In the above studies, the Zeno phenomenon is avoided.However, in the context of USV systems, the development of a triggering scheme that can effectively reduce the frequency of communication becomes a valuable pursuit under ensuring satisfactory system performance.
Compared to static event-triggered control, a class of dynamic event-triggered control strategies containing dynamic parameters can increase the event trigger time interval.In addition, it has the advantage of ensuring that the minimum event interval is at least one sampling period, as clearly stated in the works of [21], [22], [23], and [24].Specifically, distributed dynamic triggering laws involving internal dynamic variables are designed in [21] and [22] to reduce the internal communication pressure.Based on the above distributed control methods, in [23], a fully distributed event-triggered dynamic output feedback control law is proposed based on the feedforward design approach so that each agent can determine when to broadcast its information to its neighbors.Unlike the distributed dynamic eventtriggered approach, a centralized dynamic event-triggered mechanism is introduced in [24] specifically for addressing the leader-following consensus problem.The results show that the above literature uses a dynamic event-triggering mechanism that extends the minimum event interval between any two consecutive triggering moments and that none of the members exhibit the Zeno phenomenon.Nevertheless, it is noteworthy that the above method is only for multi-agent systems.There are certain difficulties in the realization from the agent to the practical vehicle due to the highly nonlinear dynamics of the vehicle and the complex environment of the ocean.Therefore, there is still no literature on event-triggered strategies in the field of USV.Especially, developing intermittent communication among multiple USVs using a dynamic triggering scheme presents even greater challenges.So, it is worthwhile to research how the dynamic event-triggered mechanism can be realized in multiple USV systems.
Inspired by the above discussion, this paper is devoted to addressing the distributed formation control problem of USVs with intermittent communication and subject to external disturbances.To eliminate frequent data transmission among inter-USVs, a novel dynamic event-triggered communication mechanism based on a hyperbolic tangent function is constructed.Furthermore, by combining a sliding mode control strategy with an adaptive control algorithm, an adaptive distributed formation controller for USVs is designed to achieve trajectory tracking.Compared with existing results, the main advantages of the developed control scheme are summarized as follows.
i) Compared to the works in [25] and [26] where continuous communication is required between the neighbors of USVs, the event-triggered method is introduced in this paper to manage the communication among the USVs, achieving intermittent communication.In the communication network, information transmission only occurs when the designed event-triggered mechanism is satisfied.Thus, the communication frequency can be significantly reduced, alleviating the burden on communication energy.
ii) A new dynamic event-triggered communication strategy is proposed by introducing a dynamic auxiliary variable.The event-triggered mechanism, with the introduced auxiliary variable, can ensure that the inter-event times are greater than 0. This can be explicitly calculated and helps prevent Zeno behavior.Because the event-triggered scheme can be implemented successfully, each USV only needs to obtain the states of neighbors at a specific trigger moment.Compared with the static event-triggered control [17], [18], [19], [20], the trigger time interval of this triggering scheme has significantly increased.
iii) The adaptive laws are employed to estimate the lumped external disturbances and uncertainties in this paper.In previous research [26], the RBFNNs and MLP algorithms were adopted to approximate uncertain system dynamics.However, the lumped uncertainties are approximated using a single parameter in this paper.Then an adaptive law estimates the single parameter, simplifying the controller design.The robustness of the USVs system is effectively ensured with the help of estimating the lumped disturbances using an adaptive law.
The rest of the paper is described below.In Section II, the corresponding preparatory work for this paper is given.Section III explains the design of the dynamic event-triggered formation controller.In Section V, numerical simulations verify the validity of the presented approach.Finally, the conclusions of this paper can be drawn in Section 5.

II. PRELIMINARIES
In this section, we presented the dynamic model of the USVs, the description of the interaction graph, as well as mathematical preliminary knowledge and related lemmas.Specifically, the interaction diagram will be used to represent the information transmission between multiple USVs, while the preliminaries and lemmas will be used to verify the stability of the proposed controller.
Notations: R n denotes the n dimensional real space.For a vector a ∈ R n , ∥a∥ denotes its Euclidean norm.For a matrix A, ∥A∥ is the induced norm of this matrix.The nation T denotes the transpose operation.If A is a positive matrix, λ max (A) stand for the maximum eigenvalue of the matrix A. Given the vector a = [a 1 , a 2 , . . ., a n ] T , we defined tanh (a) = [tanh (a 1 ) , tanh (a 2 ) , . . ., tanh (a n )] T .

A. DYNAMICS OF USVs
In this brief, we consider that the tracking trajectory of the formation system is defined on the horizontal plane.Therefore, two coordinates are first given in the earth-fixed frame O, X E , Y E and body-fixed frame O, X B , Y B respectively to describe the motion of ith USV.Before proceeding, the graphical illustration of a multi-USVs system with two frames is presented in Fig. 1.
Then, for a network containing n USVs, the kinematic and dynamic models of the ith vehicle in 3 degrees of freedom under these two coordinates are described as [25]: where η i = [x i , y i , ψ i ] T is a combination of the position vector (x i , y i ) and the heading angle ψ i of the surface vehicle in the earth-fixed frame (EFF); v i = [u i , v i , r i ] T is composed of the corresponding linear velocity (u i , v i ) and angular velocity r i in the body-fixed frame (BFF); R (ψ i ) represents the rotation matrix from BFF to EFF, its description is given as: What's more, T denotes the lumped time-varying disturbances caused by winds, waves, and ocean currents in Eq. ( 1); τ i = [τ ui , τ vi , τ ri ] T represents the actual control input of the closed-loop system; g i = [g ui , g vi , g ri ] T represents the unmodeled dynamics and is unknown during the controller design process.M i is the inertia matrix of the ith surface vehicle; C i (v i ) and D i (v i ) signify the Coriolis matrix and Hydrodynamic damping matrix, respectively.The detailed expressions of the parameters for given as: Remark 1: From a practical viewpoint, there are three main factors that affect the development of formation control for USVs.First, the nonlinear unmodeled dynamics g i = [g ui , g vi , g ri ] T cannot be accurately obtained due to unknown parameter variations.Second, the marine environment is both complex and non-predictable.The time-varying disturbances d(t) are challenging to observe accurately in real time.Third, the Coriolis matrix C (v) and Hydrodynamic damping matrix D (v) exhibit strong nonlinearity owing to the complexity of hydrodynamics.Consequently, it is necessary to consider external disturbances and model nonlinearity when designing a formation controller.
Suppose that there is a virtual leader with a constant desired relative position defined by η d (t) ∈ R 3 , and all of the follower members are required to keep track of the virtual leader while maintaining the prescribed geometry.Therefore, tracking errors e 1i and e 2i are derived by introducing the desired relative formation shapes vector i ∈ R 3 .Then, the tracking errors of the leader-follower distributed formation control are written as follows: By position tracking error e 1i and velocity tracking error e 2i , the objective of the distributed formation control in this paper is described as: Control objective: Considering a network comprising n USVs as described in Eq. ( 1) along with a virtual leader, the primary aim is to design a distributed control scheme denoted by τ i for each USV.This scheme should enable the USVs to track their desired relative positions accurately, ensuring that the tracking errors e 1i and e 2i ultimately achieve asymptotic stability, i.e., lim t→∞ e 1i = 0, lim t→∞ e 2i = 0. Understanding this control target in a physical sense means that the follower USVs in the formation will track the desired trajectory and maintain the preset formation shape.

B. BASIC GRAPH THEORY
All the USVs in a formation system are connected to each other through the network.In this paper, an undirected connected graph is used to describe the communication topology of the formation members.The graph is denoted as G = (N , E, A), where the node set N = {n 1 , n 2 , . . ., n n } represents n USVs in the formation; the edgeset E ⊆ N × N signifies the communication paths among the USVs; A = a ij ∈ R n×n is the weighted adjacency matrixwith the non-negative element a ij .In an undirected connected graph, for any (n i , n j ) ∈ E, the (n j , n i ) ∈ E exists, correspondingly.The information exchange occurs between the ith vehicle and the jth vehicle which are neighbors of each other.The element a ij denotes the degree of the interaction communication, it satisfies: It is worth noting that each node does not exchange information with itself, i.e., a ij = 0, (i = j).Generally, for any undirected graph, A is a symmetric matrix with a ij = a ji .
Lemma 2 ( [27]): For any vector x ∈ R n , the inequations involving the hyperbolic tangent function are held as follows: Assumption 1: The communication topology of the formation system is an undirected connected graph.Therefore, it is assumed that the reference trajectory η d (t) of the virtual leader can be completely obtained for all vehicles through information network exchange, where the time delays and packet drops are negligible.
Assumption 2: The reference trajectory η d (t)of the virtual leader is second-order differentiable, and η d , ηd , ηd are all considered to be bounded.
Assumption 3: The marine environmental disturbances d i possess an unknown upper bound, i.e., there is a positive constant D i that ensures the existence of inequality Remark 2: As explained in the literature [29], in order to ensure bidirectional connectivity of the communication network, Assumption 1 must exist and be reasonable.In practical engineering, the value of reference trajectory η d (t) is determined by the designers.Thus, the second-order differentiability of the reference trajectory η d (t) exists and the items η d , ηd , ηd are all considered to be bounded in Assumption 2. Due to the fact that marine environmental disturbances are caused by winds, waves, and ocean currents, it is necessary to assume the unknown and time-varying external disturbances are bounded.It can be found that Assumption 3 has been recognized as an appropriate option by the majority of designers in the formation control problem under external disturbances.
Assumption 4: The event-triggered error γ i has an upper bound so that ∥γ i ∥ ⩽ γ , where γ is a positive constant.Define the state vector during the event-sampling instants as S i t i ki = S i (t).

III. DISTRIBUTED EVENT-TRIGGERED FORMATION CONTROLLER DESIGN
In this section, the distributed formation control problem of multiple USVs is solved by using a sliding mode control method.Firstly, the idea of dynamic event-triggered is proposed to overcome the limited problem of information exchange and communication between vehicles.Then, appropriate Lyapunov functions are designed to prove the stability of the proposed controller theoretically.Finally, it is proved that the closed-loop system does not exhibit Zeno behavior under the proposed event-triggered formation controller.To demonstrate the control algorithm more visually, the block diagram of the closed-loop system with a dynamic event-triggered communication framework is illustrated in Fig. 2.

A. CONTROL LAW DERIVATION
To facilitate the subsequent stability analysis of the USVs formation system, the nonlinear functions in the dynamics of the ith vehicle described by Eq. ( 1) are approximated.The dynamics of the ith USV in formation are further described: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The nonlinear function Y i (v i ) is defined as the unknown dynamics, which is expressed as: where Here, the nonlinear function Y i is approximated by using a single parameter Y i .In this way, the dynamic of the ith USV, as expressed in Eq. ( 8), will be used as the foundation for both the controller design and the corresponding stability analysis.Then, we can reduce the computational complexity of the controller implementation even in a formation system.In relation to the disturbance denoted as the item M i d i and following the condition ∥M i d i ∥ ⩽ D i in Assumption 3, the estimation of the item D i will be carried out through the utilization of an adaptive law.So, this adaptive law is employed to address the effect of the item M i d i .
The aim of this paper is to formulate a distributed event-triggered control scheme that enables the USV formation system to maintain a consistent formation configuration and maneuver to the reference position, all while reducing inter-vehicle communication pressure.So, based on the tracking errors defined in Eq. ( 4), a sliding mode surface (SMS) auxiliary state for the ith vehicle is first defined as: where k 1 is the positive parameter.
Taking the time derivative of S i and substituting Eqs. ( 4), (8), and (9) yields: Remark 3: In the subsequent stability analysis, the SMS expressed in Eq. ( 10) will converge to a steady state.As can be found later, these two tracking errors e 1i and e 2i are forced into a residual set once the SMS is stabilized.
On the basis of the sliding mode control method and hyperbolic tangent function, the formation controller Eq. ( 12) and adaptive laws Eqs. ( 13)-( 14) for the ith vehicle is designed as: where k 2 , k 3 , k 4 , k 5 , k 6 are all positive constants; µ exp (−σ t) > 0 with µ, σ are positive constants and chosen by designer; Di and Ŷi stand for the estimation of D i and Y i , respectively.
Then, the estimation errors of D i and Y i are defined as: Remark 4: The formation controller defined by Eq. ( 12) is composed of three items.1) The item−k 2 S i − M i ¨ i − ηd + k 1 ė1i will ensure stability for the tracking error systems.2) The item− n j=1 a ij tanh S i t i ki − S j t j kj is proposed by resorting to the hyperbolic tangent function.
Compared with [30], the proposed control structure possesses the virtue of non-singularity.The advantage of the item S i t i ki − S j t j kj will be given in the following Remark 7.
3) The item can not only compensate for nonlinear dynamics and external disturbances but also suppress chattering.
Remark 5: The major superiorities of these adaptive laws can be summarized as three aspects: 1) The term ) is here introduced to perform the function of disturbance rejection.The parameters µ and σ in Eqs. ( 13) and ( 14) can be changed by designers to adjust control accuracy and rate.2) It must be noted that the chattering phenomenon will cause much damage to actuators.To tackle this issue, the term ∥S i ∥ 2 (∥S i ∥ + µ exp (−σ t)) is developed to alleviate the chattering and singularity problem.
3) From term µ exp (−σ t), it can be seen that µ exp (−σ t) = 0 when t → ∞, so no additional steady-state error is introduced to affect the control accuracy.
To proceed, the event-triggered error for the ith vehicle is presented as follows: where γ i (t) is used to gauge the mismatch between the states in real-time and the latest event instant, which should be restricted by a set threshold.t i ki is the ith vehicle triggering instant.
Here, we presented a dynamic event-triggered strategy in which an auxiliary variable ξ i is introduced.The trigger time sequence t i k is established by the event generator with the following dynamic event-triggering conditions: where β i is a positive constant with 0 < β i < 1, α i is a positive constant with 0 < α i < β i , and ξ i is an auxiliary variable satisfying the following equation: a ij tanh S i t i ki − S j t j kj (19) with ξ i (0) > 0, ρ i > 0, which are correlated in the triggering law.
From dynamic event triggering condition Eq. ( 18) and auxiliary variable equation ( 19), we have where ρ i + 1 > 0. Thus, it follows from the comparison lemma that According to Eq. ( 21), the triggering condition in the dynamic event-triggered scheme Eq. ( 18) is more stringent, since ξ i (t) is positive.Therefore, it can be expected that for the previous triggering instant t i k , the time interval of the next triggering instant t i k+1 generated by dynamic the event-triggered strategy will become longer.
Remark 6: The event-triggered information transmission strategy can be described as follows.When , an event is triggered for the ith USV.Then, the state information of the ith vehicle is allowed to broadcast to its neighbors, meaning that there is no communication for information between the ith vehicle and its neighbors at t ∈ t i k , t i k+1 .Remark 7: Under the developed event-triggered control framework, each USV only needs to communicate with its neighbors intermittently.In fact, only when the corresponding trigger condition is met, the current state information of USV will be passed to its neighbors.At the same time, the measurement error of each USV only depends on its own sampling state and actual state.Therefore, it provides a more robust and efficient utilization of the communication network.
Remark 8: The proposed mechanism Eq. ( 18) is called the dynamic event-triggered mechanism since it involves an internal variable ξ i (t).Besides, due to the fact that the internal variable ξ i (t) is always nonnegative, the measurement error γ i (t) of the static mechanism will reach its threshold earlier than that of the dynamic mechanism when the same parameters are selected.This implies that the dynamic event-triggered scheme presented here offers larger triggering time intervals.What's more, different from the static eventtriggered scheme, based on Eqs. ( 20) and ( 21), the dynamic scheme involving an internal dynamic variable ξ i (t) plays an essential role in excluding Zeno behavior.

B. STABILITY ANALYSIS
Theorem 1: For the closed-loop dynamics Eq. ( 8) with Assumptions 1-4, suppose that the communication topology is undirected and connected.If the SMS is designed as Eq. ( 10), adaptive control laws Eqs. ( 12)-( 14) and event-triggered error Eq. ( 17) are employed, and the closed-loop dynamic system can achieve the desired trajectory tracking under the constraints of limited communication network resources and unknown external disturbances.Meanwhile, 1) S i and adaptive estimation error Di and Ỹi will converge to a tiny region containing the origin.2) The stability of e 1i can also be guaranteed when k 1 > 1 held.
Proof: The Lyapunov candidate function (LCF) is selected as: Then, in light of Eq. ( 11), the time derivative of V 1 can be written as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. − Substituting formation controller Eq. ( 12) and adaptive laws Eqs. ( 13)-( 14) into Eq.( 23), it can be obtained that Furthermore, note that the terms Thus, the Eq. ( 24) can be further rewritten as: Since µ exp (−σ t) > 0 when the parameter µ > 0, thus we have: Then Eq. ( 26) can be changed to: Furthermore, according to Young's inequality, note that Di Di satisfy the following relations: Thus, the Eq. ( 28) can be further rewritten as: Then, recalling the definition of γ i (t), and using the symmetry of the bidirectional communication graph and accompanying Lemma 1, Eq. ( 31) can be presented as Taking Lemma 2 into account, Eq. ( 32) further becomes: Substituting Eq. ( 19) into Eq.( 33), one has: From event-triggered function Eq. ( 18), it follows that: Due to the design parameters 0 < α i < β i < 1, it follows that α i −β i 2β i < 0 holds.Then, one has: where . According to Eq. (34), it follows the conclusion that V 1 will converge to a tiny region containing zero.Then, in view of the definition of V 1 , the stability of the SMS S i could be obtained.
Thus far, the validity of the first point of Theorem 1 is illustrated.
Next, prove the second point of Theorem 1.To facilitate the subsequent analysis, the convergence region of S i is defined as .Based on the above result, the convergence region of tracking errors will be demonstrated subsequently.
The relevant LCF is selected as: Differentiating V 2 with respect to time and using Eq. ( 10) yield: According to Eq. ( 38), it shows that V 2 will move into a residual set when k 1 − 1 > 0 is satisfied.From the definition of V 2 , it concludes that e 1i will also converge into a residual set.At this point, the proof of Theorem 1 is completed.Remark 9: It follows from Theorem 1 that the S i , and tracking errors e 1i and e 2i can converge to zero and become satisfactorily small, respectively.This implies that the goal of distributed dynamic event-triggered formation control can be achieved.Additionally, numerous design parameters in the controller significantly determine the control performance.By trial and error, the selection of these parameters mainly affects the control accuracy, system response speed, and chattering issue.k 1 and k 2 can determine the control accuracy and convergence rate.Tuning the parameters µ, σ and using 106404 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the hyperbolic tangent function can obtain desired control accuracy and reduce chattering.

C. ZENO-FREE ANALYSIS
As the information among USVs cannot be transferred unlimitedly in a finite time period, the Zeno behavior is inadmissible and should be ruled out.For the ith USV, we define T = t i ki+1 − t i ki as the time interval between any two adjacent triggering events.Then, we will prove that the event-triggered closed-loop system does not exhibit the Zeno behavior.
Theorem 2: Under the distributed event-based formation control scheme with the event-triggered strategy Eq. ( 18), the USVs formation system Eq.( 8) is Zeno-free, and the inter-event time interval T has a lower bound that is a positive constant.
Proof.From Eq. ( 36), This demonstrates that V1 < 0 is applicable when the sliding mode vector S i is beyond the set: which guarantees that the sliding mode vector eventually converges to the set C.
In the light of event-triggered error Eq. ( 17), as t ∈ t i k , t i k=1 , it is obtained that Since S i (t) converges to a small set containing the origin, it follows that where B 1 is the upper bound of Ṡi (t) .
According to event triggering condition Eq. ( 18), it follows from Eqs. ( 40)-(41) that where δ > 0 and 0 < α i < β i < 1.Thus, the inter-event time intervals are strictly greater than zero.So, the closed-loop system does not exhibit the Zeno behavior under the proposed event-triggered formation control law.

IV. SIMULATION RESULTS
In this section, simulation examples for the formation system Eq.( 8) under external disturbances are provided to demonstrate the efficiency and availability of the controller Eq. (12).Given the presence of three followers and one virtual leader within the formation system, the internal communication topology is depicted using an undirected connected graph, as illustrated in Fig. 3: Furthermore, the corresponding weighted matrix A is given as: For the followers, the mass of the model USV is 23.8 kg, and its length and breadth are 1.255 m and 0.29 m, respectively.The model parameters of USVs are given in Table 1, which can be found in the reference [25], [26].The desired trajectory of the virtual leader, initial state vectors, and desired   shape configuration of multiple USV systems are presented in Table 2.It implies that the formation finally forms a standard shape of an isosceles right triangle.To simplify the matter, the reference trajectory of the virtual leader is predefined as a straight line parallel to the X-axis.The performances of controller Eq. ( 12) will be tested through a numerical simulation.According to Theorem 1, all the followers in formation will follow the virtual leader by internal communication along the prescribed shape.In order to achieve satisfactory control performance and reduce system communication occupancy, the design parameters of the controller and event-triggered function are chosen in Table 3.What's more, to better illustrate the robustness of the proposed method, external environmental disturbances are 106406 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.imposed on the formation system of USVs to simulate the influence of windy, waves and ocean currents.The simulated where lumped external disturbances are introduced into the simulation after 50 s, to verify the robustness and stability of the proposed scheme.
The detailed results of the numerical simulation are shown in Figs.5-15.Fig. 5 shows the tracking trajectory of the USV formation in the two-dimensional plane, in which three followers  curves of tracking errors for each USV in the formation are given in Figs.6-7.In the figures, the beginning of the introduction of lumped disturbances is indicated by the red dashed line.As shown in these two figures, the position tracking errors e 1i , (i = 1, 2, 3) and velocity tracking errors e 2i , (i = 1, 2, 3) converge to the near origin within the 50 s.Only the velocity tracking mistakes will show modest changes even if the external time-varying interferences are applied at the 50s, while the position tracking errors will be unchanged.These findings demonstrate that the planned adaptive control strategy successfully achieves the anti-disturbance goal.Fig. 8 depicts the curve of sliding mode surface auxiliary states.Obviously, this control strategy will ensure that the sliding mode surface S i will reach satisfactory stability within the 50 s, despite the presence of unknown system dynamics.Even if external disturbances are added at the 50s, the sliding mode surface S i is still stable in a bounded region.The control input signals are illustrated in Fig. 9.The bounded control torques will move into a small stable region even if disturbances are added at the 50s thanks to the proposed controller's use of the hyperbolic tangent function rather than the sign function, effectively removing undesirable chattering effects and enhancing the formation system's overall performance.The simulation result in Figs.8-9 provides more evidence of the anti-disturbance capability.Fig. 10 depicts the time evolutions of dynamic threshold parameters ξ i (t).It is worth noting that the parameter ξ i (t) is always positive and converges to zero eventually at around 0.5s.The adaptive estimations for D and Y are described in Figs.11-12, in which all the estimated parameters can finally converge to a compact set under the proposed controller.The simulation findings thus confirm the robustness and efficiency of the developed control technique.
To illustrate the advantage of a longer interval between dynamic event-triggering and static event-triggering.Under the premise that the controller Eq. ( 12), adaptive law Eqs.( 13)-( 14) and design parameters remain unchanged.Then, rewrite the dynamic event-triggering conditions Eq. ( 18) to static event-trigger condition Eq. (43): where χ i is a constant.The LCF Eq. ( 22) is reselected in the absence of the auxiliary variables ξ i : Substituting controller Eq. ( 12) and adaptive learning laws Eqs. ( 13)-( 14) into Eq.(44), the derivative of V 3 can be obtained as follows: After statistics, Fig. 13 shows the static event-triggered average time interval that is 0.265 s.In Fig. 14, the dynamic event-triggered average time interval is 1.25 s.Obviously, it can be seen from Figs. 13-14 that the trigger time interval is longer and the triggered numbers are less between USVs under dynamic event triggering conditions, which saves more communication resources.Fig. 15 shows the data size of the communication transmission among three follower USVs under three different control schemes.Specifically, three different control schemes include the traditional sample-continuous control approach in [25], the static event-triggered control method, and the dynamic event-triggered control strategy proposed in this paper.
In numerical simulation, the simulation time is chosen as 150 s, and the sample time is chosen as 0.125 s.Assume that the storage size of each data is 4 bytes.In the traditional formation control scheme [25] without the event-triggered method, the data size sent through the communication channel is up to 38400 bytes for each USV during the whole simulation time.In the static event-triggered control scheme, the data size sent through the communication channel among USVs is 17500, 18100, 18200 bytes.However, applying the dynamic event-triggered control method proposed in this paper, the data size sent through the communication channel among USVs is only 4380, 4580, 2780 bytes.Compared with the scheme in [25], this means that the communication data is greatly reduced by 88.6% , 88.1% and 92.8% , respectively, which is a clear advantage over continuous sampling controllers.Compared with the static event-triggered scheme, it can also reduce by 75% , 74.7% and 84.7% , respectively.
According to Figs. 5-15, these pictures illustrate that the purposes of reducing communication resources, stable tracking, and formation maintenance are achieved simultaneously.

V. CONCLUSION
In this paper, the problem of distributed formation control for USVs in an undirected connected graph is addressed.To reduce the communication frequency, a novel dynamic event-triggered transmission strategy is proposed.With the trigger policy, each formation USV only needs the intermittent status of its neighbors, thus greatly avoiding frequent message transfers between USVs and reducing communication consumption.Then, a distributed formation sliding mode control scheme is derived in conjunction with the adaptive method, which effectively suppresses external disturbances.Rigorous stability analysis is carried out using the Lyapunov stability theory.Moreover, a discussion ensues concerning the matter that the triggering time sequence does not exhibit Zeno behavior.Simulation results show that the proposed control scheme can greatly reduce communication frequency, save energy consumption, and improve performance.In the future, we will focus on the edge-based event-triggered control problem for USVs, while considering more constraints, such as input saturation and actuator faults.

FIGURE 2 .
FIGURE 2. Dynamic event-triggered communication framework for distributed formation control of USVs.

FIGURE 13 .
FIGURE 13.Trigger numbers and Trigger interval of three USVs (a), (b), (c) under the static event-triggering condition.

FIGURE 14 .
FIGURE 14. Trigger numbers and Trigger interval of three USVs (a), (b), (c) under the dynamic event-triggering condition.

FIGURE 15 .
FIGURE 15.Date size of communication transmission among USV.

TABLE 1 .
The model parameters of USVs.

TABLE 2 .
The initial and desired formation configuration.

TABLE 3 .
The parameters in the controller and event-triggered function.