Adaptive Platoon Control for Marine Surface Vehicles in the Presence of Constraints

+is paper presents the platoon formation control design for unmanned surface vehicles with input and output constraints. First, Barrier Lyapunov Function (BLF) is employed to ensure the desired line-of-sight (LOS) range. +e LOS range should be maintained within the predefined regions between each marine vehicle and its predecessor while the connected platoons track the trajectory. Next, to handle the model uncertainties and unknown external disturbance, we propose an adaptive neural network controller that approximates the unknown nonlinearities. Furthermore, the Nussbaum function is applied to compensate for input saturations. In addition, formation errors can be guaranteed by stability analysis. +e distance between two consecutive agents does not violate collision avoidance and connectivity in the presence of input saturation. Finally, the feasibility of the theoretical results is illustrated through the simulation results.


Introduction
Multiagent systems have been an active area due to higher efficiency, stronger robustness, and less communication requirement [1][2][3]. Recently, many effective schemes have been developed to solve the formation control problem, such as sliding mode control [4][5][6] and backstepping control [7][8][9][10][11]. Platoon control strategy has played an increasingly important role in decentralized formation fields [12][13][14], which brings many benefits such as less energy consumption and more channel capacity. e robust H ∞ control method was proposed to design platoon velocity profile which promotes smoothness of traffic flow [15]. Traffic efficiency was studied with the help of vehicular networking to optimize traffic flow [16]. Existing works mostly focus platoon formation on the development of the automobile industry. However, extensive applications on marine control are seldom considered. Inland navigation and channel transshipment are important components of waterborne transport. Marine vessel train consists of a lead vessel followed by a series of unmanned follower vessels. Motivated by the previous discussion, this paper investigates the issue of adaptive platoon formation control of marine surface vehicles.
Dynamic disturbances and unmodeled dynamics widely exist in many physical systems, such as power industries [17,18] and aerospace systems [19,20]. Inherently, nonlinear and uncertainty characteristics of system dynamics are difficult to model. In addition, ignoring the unknown disturbances in controller design, the precision and robustness will have an effect on the performance of the closed-loop system. Neural network has the ability to adjust the parameters adaptively [21]. Comparing with the existing work, modelbased control [22] and the linearity-in-parameter assumption [23,24] are not required. Due to extreme or varying environments, the external forces such as wind, wave, and current are involved into vessel motion. Many academicians and engineers have dedicated to adaptive control of marine surface vehicles. To overcome the nonlinear uncertainty and external disturbance, adaptive neural control schemes are proposed for multiple marine surface vehicles with model uncertainties in platoon formation cases.
Driven by practical needs and theoretical challenges, the constrained control design of practical systems has become an interesting research topic [25][26][27]. In practical systems, input saturation and output constraint cause performance degradation or even instability [28]. How to achieve and maintain a desired formation geometry in these situations is one of the fundamental issues in platoon. If the relative motion states are constrained within the predefined regions, two consecutive vehicles can avoid undesired collision in the procedure of platoon phase. is problem is generally described by line-of-sight (LOS) range and angle as output constraint. To ensure output constraint, barrier Lyapunov function (BLF) is a remarkable concept to deal with output constraints. Various schemes based on BLF have been reported for marine vessels [29][30][31]. On the basis of a previous successful work, BLF in the current study is further applied to MSVs by solving constraints of LOS range and angle. Furthermore, the requirements of limited input amplitude should be incorporated into the platoon formation design. Actuator saturation is another important factor to a practical system. Many research results have been carried out on input saturation. In [32], adaptive fuzzy tracking was developed for uncertain multi-input multi-output (MIMO) nonlinear systems in the presence of input saturation. In [33], the auxiliary system based on backstepping design is employed to cope with input constraints. No assumption on the uncertain parameters is needed within the compact set. Decentralized neural network based on adaptive backstepping scheme was presented in [34] to deal with time delay and input nonsmooth constraint. However, in all the papers mentioned above, there is little information on how to handle the two challenging constraints for the platoon formation. erefore, it is meaningful to consider the platoon formation subject to output constraint and input saturation.
Marine vessels have characteristics of coupled dynamics, nonlinearities, and parametric uncertainties. Actually, the requirements of safety, reliability, robustness, and limited input amplitude should be improved in multiple marine vessels. Motivated by the aforementioned discussion, a platoon formation control approach of multiple marine vessels has been developed to simultaneously deal with output constraint and input saturation. BLF in adaptive backstepping design is proposed to ensure the platoon in the safe ranges. e unknown external disturbance and unmodeled dynamics are approximated by using neural network. e problem of multiple marine vehicles with actuator saturations is also considered in this paper. e main contributions of the proposed schemes are highlighted as follows: (i) Output constraints on marine vessels positioning within line-of-sight cone is proceeded under the framework of adaptive backstepping design. LOS range and angle errors in formation operations of unmanned surface vehicles are handled by barrier Lyapunov function. (ii) Adaptive backstepping is applied to parametric-uncertain systems. Unknown disturbances and parametric uncertainties are compensated by adaptive neural network control. Backstepping control provides the systematic and recursive design. (iii) An auxiliary system is employed to cope with physical constraints of the actuators. In order to ensure the constraints on thrust magnitude, Nussbaum function is introduced to facilitate the closed-loop system design. Compared with other algorithms dealing with input constraints, this approach avoids the discontinuous switching mechanism [32] and singularity in [35]. e rest of the paper is divided into four sections. Section 2 describes the problem formation for platoon formation and preliminaries. Section 3 presents the controller design process, and the stability of the multiple marine vessels is rigorously analyzed. In Section 4, numerical simulations are conducted to support the theoretical developments. Finally, conclusions are drawn in Section 5.

Problem Description.
We consider n + 1 marine surface vehicles labeled as 0 to n. e formation pattern of this class multiagent system is shown in Figure 1. e kinematics and dynamics of the i-th MSV can be modeled as follows: where η i � [x i , y i, ψ i ] T denotes the MSV position and yaw angle with respect to earth-fixed frame, υ i � [u i , ] i , r i ] T represents the orientation known as surge, sway, and yaw velocities in the body-fixed frame, d i is external disturbances induced by wind, wave, and ocean currents, Δ i represents the unmodeled dynamics, τ i denotes the actual control inputs of the i-th MSV, J i (ψ i ) is a nonsingular transformation matrix, M i is the inertia matrix, C i is the skew-symmetric matrix of Coriolis and centripetal terms, and D i is the damping matrix. From a practical point of view, the control forces and moments of the marine vessels are limited by the physical properties of thrusters. e saturation nonlinearities can be described as where τ imax stands for the maximum control forces or moments of vessels' propulsion systems: en, we obtain e relationship between the saturation and approximation function is shown in Figure 2. en, the augmented system considering the saturation approximation function is 2 Complexity where τ ci describes the control command signal that we should design, T i ∈ R 3×3 is a diagonal matrix of thruster time constant, and the 'disturbance-like' term is denoted as Assumption 1. e desired reference trajectory η 0 � [x 0 , y 0 , ψ 0 ] T and its first time derivative _ η 0 are bounded functions.

Assumption 2.
e external disturbances d i are bounded with unknown constants. en, according to equation (7), there exist positive constants D i max such that d i ≤ D i max .
In this section, the platoon formation problem is considered for MSVs modeled by (1) and (2) with uncertainties and constraints. We will design a control strategy for the control objectives as follows: (1) A desired line-of-sight range between two consecutive MSVs can be achieved under collision and connectivity constraints. (2) Modelling uncertainties and external disturbances should be compensated, which can improve the system performance. (3) A string of MSVs proceed along a given trajectory in a platoon configuration. All MSVs can maintain the desired position in a decentralized formation.

Preliminaries
Lemma 1 (see [31]). For any constant x ∈ R n , if |x| < k and k is a constant, then the following inequality holds Lemma 2 (see [36]). Let N(χ) be an even Nussbaum function and V(·) and χ i (·) be smooth functions defined on where c > 0, r χ > 0, ε i > 0, and M is a positive constant, then V(·) and χ i (·) and

Formation Control of MSVs.
In this section, we consider the communications among n + 1 MSVs involving a LOS guidance system. First, a LOS guidance procedure is derived in this section. e desired output is reduced from e LOS projection algorithm will guarantee that the task of tracking trajectory is satisfied. LOS geometric task is to force the vessel position (x i , y i ) to converge to a desired path by following its leader. Line-of-sight (LOS) range, d i , and angle, φ i , between two MSVs are defined as e formation errors of the MSVs are designed as where d i,des is a desired distance between two MSVs. According to equation (11), differentiating e di and e ψi with respect to time, we can obtain   Complexity In order to avoid collision and connectivity maintenance among vehicles, the desired distance during the whole moving process must satisfy the following equations: where d i, min col and d i, max com represent the minimum safety distance and maximum effective communication distance respectively. For convenience, we define the maximum and minimum error distance as e bounds of yaw angle errors are defined as e ψi and e ψi . Kinematic constraints are generally considered from the motion planning. Based on coordinate transformation, kinematic constraints have been transferred to collision avoidance and connectivity maintenance constraints based on LOS range and angle. e constraints of the errors are given as follows: Define the tracking errors as where α i � [α 1i , α 2i , α 3i ] T is a stabilizing function designed as follows: Differentiating z 2i with respect to time, we can get e continuous nonlinear function related to the speed D i (v i )v i and the unmodeled dynamics of the MSV are all unknown. To solve these problems, RBFNN is used to estimate the unknown dynamics and hydrodynamic damping terms. We give Here, Z i � v i is the inputs of the NN, W * i is the true constant weight value, H i (Z i ) is the radial basis function, ε i (Z i ) ≤ ε i is the approximate error, ε i > 0 is an unknown arbitrary small constant, and W T i H i (Z i ) is used to approximate W * T i H i (Z i ). e adaptive update law is designed as follows: where Γ i � Γ T i > 0 are adaptive gain matrices and σ i is a positive constant. e adaptive neural network control law is proposed as follows: where ( * ) + is the Moore-Penrose pseudoinverse of ( * ) and the control gain matrix K 2i ∈ R 3×3 > 0. Denote the 3rd tracking error as e saturation of the thrusters has been considered in the abovementioned equation. en, we have where ] T , and an auxiliary dynamic system is defined as follows: where j � 1, 2, 3 and design parameter c χi > 0. e control law is designed as where control gain matrix K 3i � diag(k 31i , k 32i , k 33i ) > 0. (1) and (2) with Assumptions 1 and 2, if the initial condition satisfies V i (t) ≤ B 0 with B 0 being any positive constant, by adjusting the design parameters k di , k ψi , K 2i , K 3i , Γ i , σ i , and c χi ; then, under virtual control law (17) and control law (25) with adaptive updating law (20), the following properties hold: e output constraint is never violated; i.e.,

Theorem 1. Consider a string of N + 1 USVs modeled as
e constraints on collision avoidance and connectivity maintenance are guaranteed between two consecutive marine vehicles. e formation errors e di and e ψi , the tracking error z 2i , and weigh error W i converge to a small neighborhood of zero.

Complexity
All signals of the MSVs platoon formation closed-loop are bounded.
Remark 1. Vessel train configurations of inland waterways and port areas are an innovative shipping scheme to meet the transportation demand. Although similar works have been developed in robotics and vehicular technology domains, few researchers investigated the waterborne platooning. In the platoon-operating mode, the follower vessels can track the leader in LOS guidance. By employing BLF, collision avoidance is handled to keep a safe distance. Effective communication in predetermined configuration is also taken into account in the design. Besides, input saturation has been dealt with due to the requirement on thrusters.
We present here an analysis based on the obtained platoon controller structure and derive a set of constraints to satisfy platoon objectives. To guarantee the output constraints, it is necessary to ensure that the errors defined in (11) are not violated and all signals are bounded. erefore, BLF can be utilized in constraints of LOS rang and angle for MSVs to prevent constraint violation. Consider the symmetric barrier Lyapunov function candidate as where k ai and k bi are positive constants used to constrain e di and e ψi , i.e., |e di | < k ai and |e ψi | < k bi , respectively. Differentiating V 1i with respect to time, we have Substituting equations (12) and (17) into equation (27) where Consider the Lyapunov function candidate V 2i as follows: where where For adaptive law (20), there exists a compact set where ‖H i (Z i )‖ ≤ s i with s i > 0, such that Ω wi , ∀t ≥ 0, provided that W i (0) ∈ Ω wi . From Lemma 2, we can know that Substituting control law (25), then according to the Young inequality and Lemma 1, we have Complexity 5 Consider the following Lyapunov function candidate as follows: Taking the time derivative of V 3i yields e overall Lyapunov function is So we can obtain where where the minimum and maximum eigenvalues of matrix ⋆ are denoted by λ min (⋆) and λ max (⋆), respectively.
From (42) and Lemma 2, it is obvious that V(t) and χ i are uniformly ultimately bounded for all V(0) ≤ B 0 on [0, t f ). erefore, we can deduce that the closed-loop errors e di , e ψi , z 2i , and W i are also uniformly ultimately bounded.

Simulation Examples
In this section, a string of 4 MSVs is used for the numerical simulations to demonstrate the effectiveness of the proposed method. e communication relationship of the 4 MSVs is shown in Figure 3. e 3-DOF rotation matrix J i is defined as M i ∈ R 3×3 is a symmetric positive definite inertia matrix specified as where ri , m 33i � I zi − N _ ri , and the mass of the i-th marine vehicle is m i , and the i − th marine vehicle's inertia matrix in the body-fixed frame is I zi . D i (υ i ) is a hydrodynamic damping matrix specified as  Complexity e matrix C i (υ i ) is specified as  Figure 4. Figure 5 shows the control inputs, which satisfy the design e distances between the follower and the leader are shown in Figure 6. e outputs are constrained between the boundaries effectively. e desired distance between the successive vehicles is 5 m. e minimum collision distance is set as 4 m, and the maximum connectivity distance is set as 6 m. Figure 7 shows the velocities information among MSVs. Simulation results demonstrate the efficacy of the proposed method.

Conclusions
In this paper, decentralized formation control with input and output constraints have been proposed for a string MSVs. In order to achieve collision avoidance and connectivity maintenance, LOS range and angle errors were presented in the controller design. A BLF has been employed to prevent constraint violation. Next, to deal with the model uncertainties, the adaptive neural networks are applied in the backstepping control scheme due to their superior approximation capability. Furthermore, the nonlinear term arising from the input saturation is compensated by using a Nussbaum function. e analysis of uniformly ultimate boundedness for the closed-loop platoon system was provided by choosing the design parameters. Finally, the proposed algorithm was applied to MSVs to illustrate the feasibility of the theoretical results.

Data Availability
e simulation data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.