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

Authors: Xiaoling Liang; Duansong Wang
Journal: Complexity
 (2020)

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## Abstract

This 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. The 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. The 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.


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## Body

## 1. Introduction

Multiagent systems have been an active area due to higher efficiency, stronger robustness, and less communication requirement [1–3]. Recently, many effective schemes have been developed to solve the formation control problem, such as sliding mode control [4–6] and backstepping control [7–11]. Platoon control strategy has played an increasingly important role in decentralized formation fields [12–14], which brings many benefits such as less energy consumption and more channel capacity. The 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, model-based 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 [25–27]. 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. This 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 [29–31]. 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. Therefore, 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. The unknown external disturbance and unmodeled dynamics are approximated by using neural network. The problem of multiple marine vehicles with actuator saturations is also considered in this paper. The 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].The rest of the paper is divided into four sections. Section2 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.


## 2. Preliminaries and Problem Description

### 2.1. Problem Description

We considern+1 marine surface vehicles labeled as 0 to n. The formation pattern of this class multiagent system is shown in Figure 1. The kinematics and dynamics of the i-th MSV can be modeled as follows:(1)η˙i=Jiηiυi,(2)Miυ˙i=−Ciυiυi−Diυiυi+d¯i+Δi+satτi,where ηi=xi,yi,ψiT denotes the MSV position and yaw angle with respect to earth-fixed frame, υi=ui,νi,riT 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, Δi represents the unmodeled dynamics, τi denotes the actual control inputs of the i-th MSV, Jiψi is a nonsingular transformation matrix, Mi is the inertia matrix, Ci is the skew-symmetric matrix of Coriolis and centripetal terms, and Di is the damping matrix.Figure 1
MSVs formation configuration.From a practical point of view, the control forces and moments of the marine vessels are limited by the physical properties of thrusters. The saturation nonlinearities can be described as(3)satτi=signτiτimax,ifτi>τimax,τi,ifτi<τimax,where τimax stands for the maximum control forces or moments of vessels’ propulsion systems:(4)giτi=τimax×tanhτiτimax=τimaxeτi/τimax−e−τi/τimaxeτi/τimax+eτi/τimax.Then, we obtain(5)satτi=giτi+d1i.The relationship between the saturation and approximation function is shown in Figure2. Then, the augmented system considering the saturation approximation function is(6)Miυ˙i=−Ciυiυi−Diυiυi+di+Δi+giτi,Tiτ˙i+τi=τci,where τci describes the control command signal that we should design, Ti∈ℝ3×3 is a diagonal matrix of thruster time constant, and the ‘disturbance-like’ term is denoted as(7)di=d1i+d¯i.Figure 2
Saturation (dotted line: approximate function; solid line: sat(v)).Assumption 1.
The desired reference trajectoryη0=x0,y0,ψ0T and its first time derivative η˙0 are bounded functions.Assumption 2.
The external disturbancesd¯i are bounded with unknown constants. Then, according to equation (7), there exist positive constants Dimax such that di≤Dimax.
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.


### 2.2. Preliminaries

Lemma 1.
(see [31]). For any constant x∈ℝn, if x<k and k is a constant, then the following inequality holds(8)lnk2k2−x2≤x2k2−x2.Lemma 2.
(see [36]). Let Nχ be an even Nussbaum function and V⋅ and χi⋅ be smooth functions defined on 0,tf, with Vt≥0,∀t∈0,tf. If the following inequality holds,(9)V≤V0e−ct+Mc1−e−ct+e−ctrχ∫0t∑i=1nεiNχiχ˙i−χ˙iecτdτ,where c>0,rχ>0,εi>0, and M is a positive constant, then V⋅andχi⋅ and ∫0t∑i=1nεiNχiχ˙i−χ˙iecτdτ are bounded on 0,tf.


## 2.1. Problem Description

We considern+1 marine surface vehicles labeled as 0 to n. The formation pattern of this class multiagent system is shown in Figure 1. The kinematics and dynamics of the i-th MSV can be modeled as follows:(1)η˙i=Jiηiυi,(2)Miυ˙i=−Ciυiυi−Diυiυi+d¯i+Δi+satτi,where ηi=xi,yi,ψiT denotes the MSV position and yaw angle with respect to earth-fixed frame, υi=ui,νi,riT 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, Δi represents the unmodeled dynamics, τi denotes the actual control inputs of the i-th MSV, Jiψi is a nonsingular transformation matrix, Mi is the inertia matrix, Ci is the skew-symmetric matrix of Coriolis and centripetal terms, and Di is the damping matrix.Figure 1
MSVs formation configuration.From a practical point of view, the control forces and moments of the marine vessels are limited by the physical properties of thrusters. The saturation nonlinearities can be described as(3)satτi=signτiτimax,ifτi>τimax,τi,ifτi<τimax,where τimax stands for the maximum control forces or moments of vessels’ propulsion systems:(4)giτi=τimax×tanhτiτimax=τimaxeτi/τimax−e−τi/τimaxeτi/τimax+eτi/τimax.Then, we obtain(5)satτi=giτi+d1i.The relationship between the saturation and approximation function is shown in Figure2. Then, the augmented system considering the saturation approximation function is(6)Miυ˙i=−Ciυiυi−Diυiυi+di+Δi+giτi,Tiτ˙i+τi=τci,where τci describes the control command signal that we should design, Ti∈ℝ3×3 is a diagonal matrix of thruster time constant, and the ‘disturbance-like’ term is denoted as(7)di=d1i+d¯i.Figure 2
Saturation (dotted line: approximate function; solid line: sat(v)).Assumption 1.
The desired reference trajectoryη0=x0,y0,ψ0T and its first time derivative η˙0 are bounded functions.Assumption 2.
The external disturbancesd¯i are bounded with unknown constants. Then, according to equation (7), there exist positive constants Dimax such that di≤Dimax.
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.


## 2.2. Preliminaries

Lemma 1.
(see [31]). For any constant x∈ℝn, if x<k and k is a constant, then the following inequality holds(8)lnk2k2−x2≤x2k2−x2.Lemma 2.
(see [36]). Let Nχ be an even Nussbaum function and V⋅ and χi⋅ be smooth functions defined on 0,tf, with Vt≥0,∀t∈0,tf. If the following inequality holds,(9)V≤V0e−ct+Mc1−e−ct+e−ctrχ∫0t∑i=1nεiNχiχ˙i−χ˙iecτdτ,where c>0,rχ>0,εi>0, and M is a positive constant, then V⋅andχi⋅ and ∫0t∑i=1nεiNχiχ˙i−χ˙iecτdτ are bounded on 0,tf.


## 3. Control Design with Barrier Lyapunov Function

### 3.1. Formation Control of MSVs

In this section, we consider the communications amongn+1 MSVs involving a LOS guidance system. First, a LOS guidance procedure is derived in this section. The desired output is reduced from xi,yi,ψi to di,ψi based on LOS projection algorithm. The LOS projection algorithm will guarantee that the task of tracking trajectory is satisfied. LOS geometric task is to force the vessel position xi,yi to converge to a desired path by following its leader. Line-of-sight (LOS) range, di, and angle, φi, between two MSVs are defined as(10)di=xi−1−xi2+yi−1−yi2,φi=arctan2yi−1−yi,xi−1−xi.The formation errors of the MSVs are designed as(11)edi=di−di,des,eψi=ψi−1−ψi,where di,des is a desired distance between two MSVs. According to equation (11), differentiating edi and eψi with respect to time, we can obtain(12)e˙di=−z21i+α1icosψi−φi+y˙i−1sinφi+z22i+α2isinψi−φi+x˙i−1cosφi,e˙ψi=ψ˙i−1−z23i+α3i.In order to avoid collision and connectivity maintenance among vehicles, the desired distance during the whole moving process must satisfy the following equations:(13)0<di,mincol<di≤di,maxcom,where di,mincol and di,maxcom represent the minimum safety distance and maximum effective communication distance respectively. For convenience, we define the maximum and minimum error distance as(14)e¯dit=di,mincol−di,des,e¯dit=di,maxcom−di,des.The bounds of yaw angle errors are defined ase¯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. The constraints of the errors are given as follows:(15)e¯di<edi<e¯di,e¯di<eψi<e¯ψi.Define the tracking errors as(16)z1i=z11i,z12iT=edi,eψiT,z2i=z21i,z22i,z23iT=vi−αi,where αi=α1i,α2i,α3iT is a stabilizing function designed as follows:(17)α1i=cosψi−φikdiedikai2−edi2+x˙i−1cosφi+y˙i−1sinφi,α2i=−sinψi−φikdiedikai2−edi2+x˙i−1cosφi+y˙i−1sinφi,α3i=kψieψi+ψ˙i−1.Differentiatingz2i with respect to time, we can get(18)Miz˙2i=−Ciυiυi−Diυiυi+giτi+di+Δi−Miα˙ci,The continuous nonlinear function related to the speedDivivi 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(19)Wi∗THiZi+εiZi=−Divivi+Δi.Here, Zi=vi is the inputs of the NN, Wi∗ is the true constant weight value, HiZi is the radial basis function, εiZi≤ε¯i is the approximate error, ε¯i>0 is an unknown arbitrary small constant, and W^iTHiZi is used to approximate Wi∗THiZi. The adaptive update law is designed as follows:(20)W^˙i=ΓiHiZiz2i−σiz2iW^i,i=1,2,3,where Γi=ΓiT>0 are adaptive gain matrices and σi is a positive constant.The adaptive neural network control law is proposed as follows:(21)αci=z2iT+−kdiedi2kai2−edi2−kψieψi2kbi2−eψi2+Civivi−Θ2i+Miα˙i−K2iz2i−W^iTHiZi,where ∗+ is the Moore–Penrose pseudoinverse of ∗ and the control gain matrix K2i∈ℝ3×3>0. Denote the 3rd tracking error as(22)z3i=giτi−αci.The saturation of the thrusters has been considered in the abovementioned equation. Then, we have(23)z˙3i=ξi−1Tiτi+1Tiτci−α˙ci,where ξi=diagξi1,ξi2,ξi3 and ξij=∂gijτij/∂τij,j=1,2,3. We introduce Nussbaum function matrix Ni=diagNi1χi1,Ni2χi2,Ni3χi3T, and an auxiliary dynamic system is defined as follows:(24)Nijχij=χij2cosχij,χ˙ij=γχiτ¯cijz3ij,where j=1,2,3 and design parameter γχi>0.The control law is designed as(25)τci=TiNiτ¯ci,τ¯ci=−K3iz3i+ξiTiτi+α˙i−z2i,where control gain matrix K3i=diagk31i,k32i,k33i>0.Theorem 1.
Consider a string ofN + 1 USVs modeled as (1) and (2) with Assumptions 1 and 2, if the initial condition satisfies Vit≤B0 with B0 being any positive constant, by adjusting the design parameters kdi,kψi,K2i,K3i,Γi,σi,andγχi; then, under virtual control law (17) and control law (25) with adaptive updating law (20), the following properties hold:The output constraint is never violated; i.e.,0<di,mincol<di≤di,maxcom,∀t>0. The constraints on collision avoidance and connectivity maintenance are guaranteed between two consecutive marine vehicles.The formation errorsediandeψi, the tracking error z2i, and weigh error W˜i converge to a small neighborhood of zero.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. Therefore, 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(26)V1i=12lnkai2kai2−edi2+12lnkbi2kbi2−eψi2,where kai and kbi are positive constants used to constrain edi and eψi, i.e., edi<kaiandeψi<kbi, respectively. Differentiating V1i with respect to time, we have(27)V˙1i=edie˙dikai2−edi2+eψie˙ψikbi2−eψi2.
Substituting equations (12) and (17) into equation (27) yields(28)V˙1i=−kdiedi2−kψieψi2+Θ1i,where(29)Θ1i=edi−z21icosψi−φi+z22isinψi−φikai2−edi2+eψi−z23ikbi2−eψi2.
Consider the Lyapunov function candidateV2i as follows:(30)V2i=V1i+12z2iTMiz2i+12∑i=13W˜iTΓi−1W˜i,where W˜i=W^i−Wi∗. Then, the derivation of V2i is(31)V˙2i=−kdiedi2−kψieψi2+z2iTΘ2+z2iTz3i−Civivi−Miα˙i+di+Wi∗THiZi+εiZi+αci+∑j=13W˜ijTΓij−1W^˙ij,where(32)Θ2i=−edicosψi−φikai2−edi2edisinψi−φikai2−edi2−eψikbi2−eψi2T.
For adaptive law (20), there exists a compact set(33)Ωwi=W^iW^i≤siσi,where HiZi≤si with si>0, such that Ωwi,∀t≥0, provided that W^i0∈Ωwi.
From Lemma2, we can know that(34)W˜i=W^i−Wi∗≤W^i+Wi∗≤siσi+Wi∗=ϖi.
Substituting control law (25), then according to the Young inequality and Lemma 1, we have(35)V˙2i=−kdiedi2−kψieψi2−kai2edi2kai2−edi2−kbi2eψi2kbi2−eψi2−z2iTK2iz2i−z2iTεiZi+z2iTz3i+z2iTdi+∑i=13W˜iTHiZiz2i−σiz2iW^i≤−kdilnkai2kai2−edi2−kψilnkbi2kbi2−eψi2+z2iTdi+z2iTz3i−z2iTK2i−Iz2i+12εiZi2+∑i=13σi28Wi∗4+σi28ϖi4−2Wi∗2W˜i2.
Consider the following Lyapunov function candidate as follows:(36)V3i=12z3iTz3i.
Taking the time derivative ofV3i yields(37)V˙3i=z3iTξi−1Tiτi+1Tiτci−α˙i=−z3iTK3iz3i−z3iTz2i+z3iTξiTiτci−τ¯ci=−z3iTK3iz3i−z3iTz2i+1γχi∑j=13ξijNij−1χ˙ij.
The overall Lyapunov function is(38)Vi=V2i+V3i.
So we can obtain(39)V˙i≤−kdilnkai2kai2−edi2−kψilnkbi2kbi2−eψi2−z3iTK3iz3i−z2iTK2i−32Iz2i+12εiZi2+∑i=13σi28Wi∗4+σi28ϖi4−2Wi∗2W˜i2+1γχi∑j=13ξijNij−1χ˙ij+12Dimax≤−ρiVi+Ci+1γχi∑j=13ξijNij−1χ˙ij,where(40)ρi=min2kdi,2kψi,2λminK2i−3λmaxMi,2λminK3i,σi2Wi∗22λmaxΓi−1,(41)Ci=12ε¯i2+12Dimax+∑i=13σi28Wi∗4+σi28ϖi4,where the minimum and maximum eigenvalues of matrix ⋆ are denoted by λmin⋆ and λmax⋆, respectively.(42)0≤Vit≤Ciρi+Vi0−Ciρie−ρit+e−ρitγχi∫0t∑j=13ξijNij−1χ˙ije−ρisds.
From (42) and Lemma 2, it is obvious that Vt and χi are uniformly ultimately bounded for all V0≤B0 on 0,tf. Therefore, we can deduce that the closed-loop errors edi,eψi,z2i,andW˜i are also uniformly ultimately bounded.


## 3.1. Formation Control of MSVs

In this section, we consider the communications amongn+1 MSVs involving a LOS guidance system. First, a LOS guidance procedure is derived in this section. The desired output is reduced from xi,yi,ψi to di,ψi based on LOS projection algorithm. The LOS projection algorithm will guarantee that the task of tracking trajectory is satisfied. LOS geometric task is to force the vessel position xi,yi to converge to a desired path by following its leader. Line-of-sight (LOS) range, di, and angle, φi, between two MSVs are defined as(10)di=xi−1−xi2+yi−1−yi2,φi=arctan2yi−1−yi,xi−1−xi.The formation errors of the MSVs are designed as(11)edi=di−di,des,eψi=ψi−1−ψi,where di,des is a desired distance between two MSVs. According to equation (11), differentiating edi and eψi with respect to time, we can obtain(12)e˙di=−z21i+α1icosψi−φi+y˙i−1sinφi+z22i+α2isinψi−φi+x˙i−1cosφi,e˙ψi=ψ˙i−1−z23i+α3i.In order to avoid collision and connectivity maintenance among vehicles, the desired distance during the whole moving process must satisfy the following equations:(13)0<di,mincol<di≤di,maxcom,where di,mincol and di,maxcom represent the minimum safety distance and maximum effective communication distance respectively. For convenience, we define the maximum and minimum error distance as(14)e¯dit=di,mincol−di,des,e¯dit=di,maxcom−di,des.The bounds of yaw angle errors are defined ase¯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. The constraints of the errors are given as follows:(15)e¯di<edi<e¯di,e¯di<eψi<e¯ψi.Define the tracking errors as(16)z1i=z11i,z12iT=edi,eψiT,z2i=z21i,z22i,z23iT=vi−αi,where αi=α1i,α2i,α3iT is a stabilizing function designed as follows:(17)α1i=cosψi−φikdiedikai2−edi2+x˙i−1cosφi+y˙i−1sinφi,α2i=−sinψi−φikdiedikai2−edi2+x˙i−1cosφi+y˙i−1sinφi,α3i=kψieψi+ψ˙i−1.Differentiatingz2i with respect to time, we can get(18)Miz˙2i=−Ciυiυi−Diυiυi+giτi+di+Δi−Miα˙ci,The continuous nonlinear function related to the speedDivivi 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(19)Wi∗THiZi+εiZi=−Divivi+Δi.Here, Zi=vi is the inputs of the NN, Wi∗ is the true constant weight value, HiZi is the radial basis function, εiZi≤ε¯i is the approximate error, ε¯i>0 is an unknown arbitrary small constant, and W^iTHiZi is used to approximate Wi∗THiZi. The adaptive update law is designed as follows:(20)W^˙i=ΓiHiZiz2i−σiz2iW^i,i=1,2,3,where Γi=ΓiT>0 are adaptive gain matrices and σi is a positive constant.The adaptive neural network control law is proposed as follows:(21)αci=z2iT+−kdiedi2kai2−edi2−kψieψi2kbi2−eψi2+Civivi−Θ2i+Miα˙i−K2iz2i−W^iTHiZi,where ∗+ is the Moore–Penrose pseudoinverse of ∗ and the control gain matrix K2i∈ℝ3×3>0. Denote the 3rd tracking error as(22)z3i=giτi−αci.The saturation of the thrusters has been considered in the abovementioned equation. Then, we have(23)z˙3i=ξi−1Tiτi+1Tiτci−α˙ci,where ξi=diagξi1,ξi2,ξi3 and ξij=∂gijτij/∂τij,j=1,2,3. We introduce Nussbaum function matrix Ni=diagNi1χi1,Ni2χi2,Ni3χi3T, and an auxiliary dynamic system is defined as follows:(24)Nijχij=χij2cosχij,χ˙ij=γχiτ¯cijz3ij,where j=1,2,3 and design parameter γχi>0.The control law is designed as(25)τci=TiNiτ¯ci,τ¯ci=−K3iz3i+ξiTiτi+α˙i−z2i,where control gain matrix K3i=diagk31i,k32i,k33i>0.Theorem 1.
Consider a string ofN + 1 USVs modeled as (1) and (2) with Assumptions 1 and 2, if the initial condition satisfies Vit≤B0 with B0 being any positive constant, by adjusting the design parameters kdi,kψi,K2i,K3i,Γi,σi,andγχi; then, under virtual control law (17) and control law (25) with adaptive updating law (20), the following properties hold:The output constraint is never violated; i.e.,0<di,mincol<di≤di,maxcom,∀t>0. The constraints on collision avoidance and connectivity maintenance are guaranteed between two consecutive marine vehicles.The formation errorsediandeψi, the tracking error z2i, and weigh error W˜i converge to a small neighborhood of zero.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. Therefore, 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(26)V1i=12lnkai2kai2−edi2+12lnkbi2kbi2−eψi2,where kai and kbi are positive constants used to constrain edi and eψi, i.e., edi<kaiandeψi<kbi, respectively. Differentiating V1i with respect to time, we have(27)V˙1i=edie˙dikai2−edi2+eψie˙ψikbi2−eψi2.
Substituting equations (12) and (17) into equation (27) yields(28)V˙1i=−kdiedi2−kψieψi2+Θ1i,where(29)Θ1i=edi−z21icosψi−φi+z22isinψi−φikai2−edi2+eψi−z23ikbi2−eψi2.
Consider the Lyapunov function candidateV2i as follows:(30)V2i=V1i+12z2iTMiz2i+12∑i=13W˜iTΓi−1W˜i,where W˜i=W^i−Wi∗. Then, the derivation of V2i is(31)V˙2i=−kdiedi2−kψieψi2+z2iTΘ2+z2iTz3i−Civivi−Miα˙i+di+Wi∗THiZi+εiZi+αci+∑j=13W˜ijTΓij−1W^˙ij,where(32)Θ2i=−edicosψi−φikai2−edi2edisinψi−φikai2−edi2−eψikbi2−eψi2T.
For adaptive law (20), there exists a compact set(33)Ωwi=W^iW^i≤siσi,where HiZi≤si with si>0, such that Ωwi,∀t≥0, provided that W^i0∈Ωwi.
From Lemma2, we can know that(34)W˜i=W^i−Wi∗≤W^i+Wi∗≤siσi+Wi∗=ϖi.
Substituting control law (25), then according to the Young inequality and Lemma 1, we have(35)V˙2i=−kdiedi2−kψieψi2−kai2edi2kai2−edi2−kbi2eψi2kbi2−eψi2−z2iTK2iz2i−z2iTεiZi+z2iTz3i+z2iTdi+∑i=13W˜iTHiZiz2i−σiz2iW^i≤−kdilnkai2kai2−edi2−kψilnkbi2kbi2−eψi2+z2iTdi+z2iTz3i−z2iTK2i−Iz2i+12εiZi2+∑i=13σi28Wi∗4+σi28ϖi4−2Wi∗2W˜i2.
Consider the following Lyapunov function candidate as follows:(36)V3i=12z3iTz3i.
Taking the time derivative ofV3i yields(37)V˙3i=z3iTξi−1Tiτi+1Tiτci−α˙i=−z3iTK3iz3i−z3iTz2i+z3iTξiTiτci−τ¯ci=−z3iTK3iz3i−z3iTz2i+1γχi∑j=13ξijNij−1χ˙ij.
The overall Lyapunov function is(38)Vi=V2i+V3i.
So we can obtain(39)V˙i≤−kdilnkai2kai2−edi2−kψilnkbi2kbi2−eψi2−z3iTK3iz3i−z2iTK2i−32Iz2i+12εiZi2+∑i=13σi28Wi∗4+σi28ϖi4−2Wi∗2W˜i2+1γχi∑j=13ξijNij−1χ˙ij+12Dimax≤−ρiVi+Ci+1γχi∑j=13ξijNij−1χ˙ij,where(40)ρi=min2kdi,2kψi,2λminK2i−3λmaxMi,2λminK3i,σi2Wi∗22λmaxΓi−1,(41)Ci=12ε¯i2+12Dimax+∑i=13σi28Wi∗4+σi28ϖi4,where the minimum and maximum eigenvalues of matrix ⋆ are denoted by λmin⋆ and λmax⋆, respectively.(42)0≤Vit≤Ciρi+Vi0−Ciρie−ρit+e−ρitγχi∫0t∑j=13ξijNij−1χ˙ije−ρisds.
From (42) and Lemma 2, it is obvious that Vt and χi are uniformly ultimately bounded for all V0≤B0 on 0,tf. Therefore, we can deduce that the closed-loop errors edi,eψi,z2i,andW˜i are also uniformly ultimately bounded.


## 4. Simulation Examples

In this section, a string of 4 MSVs is used for the numerical simulations to demonstrate the effectiveness of the proposed method. The communication relationship of the 4 MSVs is shown in Figure3. The 3-DOF rotation matrix Ji is defined as(43)Jiψi=cosψi−sinψi0sinψicosψi0001.Mi∈ℝ3×3 is a symmetric positive definite inertia matrix specified as(44)Mi=m11i000m22im23i0m32im33i,where m11i=mi−Xu˙i,m22i=mi−Yv˙i,m23i=m32i=mixgi−Yr˙i,m33i=Izi−Nr˙i, and the mass of the i-th marine vehicle is mi, and the i−th marine vehicle’s inertia matrix in the body-fixed frame is Izi. Diυi is a hydrodynamic damping matrix specified as(45)Diυi=d11i000d22id23i0d32id33i,where(46)d11ivi=−Xui+Xuiuiui+Xuiuiuiui2,d22ivi,ri=−Yvi+Yvivivi+Yriviri,d23ivi,ri=−Yri+Yvirivi+Yririri,d32ivi,ri=−Nvi+Nvivivi+Nriviri,d33ivi,ri=−Nri+Nvirivi+Nririri.Figure 3
Trajectories of the MSVs.The matrixCiυi is specified as(47)Ciυi=00−m22iυi−m23iri00m11iuim22iυ+m23iri−m11iui0.The marine vehicle model parameters in this simulation are exploited from Cybership-II [37].The initial states of the MSVs are given asη0=0,0,0T, η1=0,5.6,0T,η2=0,10.6,0T,η3=0,15.6,0T,andν1=ν2=ν3=0. From Figure 3, the MSVs formate in platoon during the moving process. The desired positions and the orientations of MSVs are shown in Figure 4. Figure 5 shows the control inputs, which satisfy the design requirement. The distances between the follower and the leader are shown in Figure 6. The outputs are constrained between the boundaries effectively. The desired distance between the successive vehicles is 5 m. The 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.Figure 4
Positions and orientations of the MSVs.
(a)(b)(c)Figure 5
Control inputs.
(a)(b)(c)Figure 6
The distances between successive vehicles and constraints.Figure 7
Velocities of the MSVs.
(a)(b)(c)


## 5. 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. The 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.



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