Adaptive Neural Path Following Control of Underactuated Surface Vessels With Input Saturation

In this paper, considering the input saturation, off-diagonal mass matrix, model uncertainties and time-varying environment disturbances, an adaptive neural path following control strategy, which is based on the surge-heading line-of-sight guidance law, is presented for underactuated surface vessels. In view of the practical situation, we consider that the mass and damping matrices are off-diagonal. For the sake of better path following performance, a surge-heading line-of-sight guidance law is established, where the surge-heading line-of-sight guidance law not only generates the desired heading angle, but also designs the desired surge speed for the control system. Then, adaptive neural path following controllers are designed to track the referenced signals, where the input saturation nonlinearity is handled by a hyperbolic tangent function, and the lumped disturbances including external environment disturbances, approximation errors and model uncertainties are approximated by adaptive radial basis function neural network. On the basis of the proposed control scheme, all error signals of the whole system are proven to be uniformly ultimately bounded, so that the target of path following problem is realized. At last, simulation results are applied to indicate that the presented approach is effective.


I. INTRODUCTION
In recent years, with the requirement of developing marine resources, motion control of underactuated surface vessels (USVs) has gained great attention [1]- [6]. The purpose of path following control problem, which is an important part of motion control, is to design a control strategy for an USV to arrive and sail on a predefined path [7]. In order to obtain better path following performance, numerous control schemes, which are made up of guidance system and control system, have been applied widely.
Based on the path information, the guidance system is developed to generate the desired signals for control system to accomplish the path following task. In the past years, the line-of-sight (LOS) guidance law, which is a simple and convenient method, has been researched extensively. In [8], The associate editor coordinating the review of this manuscript and approving it for publication was Jianyong Yao . a proportional LOS (PLOS) guidance law was designed and the uniform semiglobal exponential stability was guaranteed. However, the PLOS guidance law did not consider the environment disturbances, which can create the sideslip angle and damage the path following performance. In order to deal with the sideslip angle, some improved LOS guidance laws were developed, such as integral LOS (ILOS) [9], [10], adaptive LOS (ALOS) [11], extended state observer based-LOS (ELOS) [12], and sideslip-tangent LOS (SLOS) [13], [14]. In addition, the above guidance laws only concluded the desired heading angle signals, and the referenced surge speeds were predefined as constants, so that the maneuverabilities of USVs were decreased. For improving the path following performance, the surge-varying LOS guidance law was developed to generate the desired surge speed and heading angle simultaneously in [15].
It should be noted that the aforementioned methods did not consider the input saturation. In practice, actuator saturation VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ nonlinearity exists. The control performance will be degraded and instablity may occur, while the input saturation is not handled. In this situation, the input saturation should be considered in the control design to obtain better performance [16].
In [17], an auxiliary design system in the form of anti-windup compensator was applied to deal with actuator saturation. In [18], an auxiliary system was employed to compensate for the effect of actuator saturation. Then, the auxiliary system method was applied in the fullactuated vessels [19]. Moreover, this method was combined with path following control law to deal with the input saturation for underactuated surface vessels [20]. In [21], a hyperbolic tangent function was applied to approximate the effect of input saturation nonlinearity, so that the backstepping technique was used in the control design. Besides input saturation, another challenge of path following problem is the model uncertainties, which is a common problem in engineering application [22], [23]. In order to deal with the model uncertainties, the neural network (NN) was used to approximate a nonlinear function based on the universal approximation property in [24]. In [25], a multilayer neural-networks (MLPNN) was employed to compensate the model uncertainties. However, the structure of MLPNN is complicated, so that the design parameters are difficult to choose and the learning speed is slow. In [26], the radial basis function neural network (RBFNN) was applied to estimate the unknown model dynamics of fullactuated vessels for dynamic position. For path following problem of USVs, the RBFNN was used to compensate the model uncertainties and environment disturbances in [27] and [28]. In addition, most path following studies assumed that the mass matrices of USVs were diagonal, which was a simplification of the practical condition. In practice, the assumption is not reasonable and causes a problem that the sway and yaw dynamics are coupled, which means that the yaw moment can also control the sway dynamic. In this case, the path following control designs become difficult. In order to solve the problem, the coordinate transformation method was applied in [29], [30].
Motivated by the aforementioned considerations, we consider the path following problem of USVs with off-diagonal mass matrix in the presence of model uncertainties, input saturation and time-varying environment disturbances. An adaptive neural path following control (ANPFC) scheme, which is based on the surge-heading line-of-sight (SHLOS) guidance law, is proposed in this paper. The main contributions of this paper are summarized as follows.
1) Different from the conventional LOS guidance law, a SHLOS guidance law is presented for the path following problem of USVs, while the SHLOS guidance law can generate desired surge speed and heading angle simultaneously. Compared with the designed guidance law in [15], the condition of off-diagonal mass matrix is considered in the SHLOS guidance law.
2) The input saturation is handled by a hyperbolic tangent function, so that the dynamic surface control (DSC) technique can be used in the control design. Then the adaptive neural path following control laws are designed with the lumped disturbances approximated by a RBFNN.
3) It is proven that all states of closed-loop control system are uniform ultimately bounded by using the proposed control scheme.
The paper is organized as follows. The preliminaries and problem formation are stated in Section II. The designs of guidance system and control system are given in Section III and Section IV, respectively. Then, Section V formulates the stability analysis. In addition, Section VI shows the simulation results. At last, Section VII concludes the paper.

A. NOTATION
The following notations will be used throughout this paper.

B. RBFNN
On the basis of the universal approximation property, unknown continuous function can be approximated by the RBFNN, which is a popular method. For any nonlinear continuous function f (x) : R n → R over a compact set x ⊆ R n , it can be approximated by the RBFNN, which is described as [31]: where x ∈ x is the input vector, χ is the approximation error, which is a bound parameter, i.e. there exist an unknown positive constant χ * , such that |χ | ≤ χ * . In addition, W * = [w 1 , w 2 , · · · , w m ] ∈ R m denotes the optimal weight vector, and m is the node number of the hidden neurons. The optimal weight vector is unknown in practice, and can be given as: whereŴ is the estimation value of W * , and can be applied to estimate unknown function. Then, we can get the estimation of f (x) with the estimation valueŴ : where (x) = [φ 1 (x), φ 1 (x), · · · , φ m (x)] is the basis function, and can be chosen as Gaussian function with the form as: where c i is the center vector and b i is the width of the Gaussian function.

C. MODEL OF UNDERACTUATED SURFACE VESSEL
Based on [32], only consider the motion on the horizontal plane, and the mathematic model of the USV is expressed by:η where η = [x, y, ψ] T stands for the position and heading angle of the USV in the north-east-down (NED) reference frame; υ = [u, v, r] T stands for velocities in the body-fixed frame; R (ψ) is a state dependent rotation matrix and given as: In addition, δ (t) = [δ u , δ v , δ r ] T denotes the unknown time-varying environment disturbances produced by wind, waves and ocean current. M represents the ship inertia matrix including added mass; C (υ) represents the Coriolis and centripetal matrix; and D is the hydrodynamic damping matrix. The M, C (υ) and D are given as: The vector τ = [τ 1 , 0, τ 2 ] T represents the surge force and yaw moment, which are the inputs of the USV and outputs of actuators, so that the vessel is underactuated. Due to the saturation nonlinearities of actuators, the inputs of the USV can be described as follows: where σ i , i = 1, 2 are the control commands; τ Mi , i = 1, 2 are the bounds of the τ i . Remark 1: In general, the conformation of the bow of vessels is different from that of the stern. The mass matrix M is off-diagonal in this context, which is different from the most studies. The sway and yaw dynamics are coupled, so that the yaw moment can control the sway dynamics, which makes the control design become difficult.
Remark 2: Sharp corners exist when |σ i | = τ Mi , so that backstepping technique cannot be applied directly.
In order to deal with the problem that saturation control law cannot be applied directly in the DSC design, a smooth function is employed to approximate the saturation characteristic, which is described as follows [16]: Then τ i can be expressed as: which are the approximate errors between τ i and h i (σ i ), are bounded.
In order to facilitate the following written, we omit σ i without confusion, such as For the sake of handling the situation that sway dynamics and yaw dynamics are controlled by the yaw moment concurrently, a coordinate transformation is applied. The coordinate transformation is described as:x = x + cos ψ,  [30]. The mathematic model of the USV is rewritten as: where

D. PATH FOLLOWING ERROR DYNAMIC
As shown in Figure 1, the path following problem can be described as the USV follows a predefined path, which is parameterized by a path variable θ. Denote a Serret-Frenet (SF) frame at the point x p (θ) , y p (θ) along the desired path. To arrive at the SF frame, the NED frame should be rotated with an angle as: where x p (θ) = ∂x p /∂θ and y p (θ) = ∂y p /∂θ .
When the position of the USV is denoted as (x, y), and define the path following along-track error and cross-track error built in the SF frame as x e and y e , then the error vector is represented by: Differentiating (19), the path following error dynamics model in the SF frame is written as follows: where u x is the speed of virtual target along the desired path and is expressed by: Considering the sideslip angle β = atan 2(v, u), the path following error dynamics are rewritten as: where U = √ u 2 +v 2 is the speed of the USV. In the presence of off-diagonal mass matrix, model uncertainties, unknown external disturbances and input saturation, the path following objective of this study is to ensure that the USV arrives and follows the predefined path. In this situation, a SHLOS guidance law, which is used to generate the desired surge speed and heading angle, and a control system, which is applied to produce the suitable surge force and yaw moment are designed in this paper.
In order to accomplish the design of guidance and control laws for the USV, the following assumptions are made.
Assumption 1: All states are available for measuring. Assumption 2: The damping matrix is unknown. Assumption 3:The environment disturbances δ i are bounded, i.e. there exist unknown positive constantsδ i , such that |δ i | ≤δ i .
Remark 3: Assumption 1 is a common precondition in the field of control design of vessels [4], [5], [19], [28]. In Assumption 2, the damping terms are difficult to obtain [29]. The energy of environment disturbances is limited, so that Assumption 3 is reasonable. In practice, the velocities of USV are bounded. In addition, the approximation errors ρ 1 and ρ 2 are bounded, so that the unknown nonlinear terms D u and D r are bounded, and can be approximated by RBFNN.

III. GUIDANCE SYSTEM
From the previous studies, it is known that most guidance laws only generate desired heading angles for control systems and referenced surge speed signals are predefined. In this situation, the cross-track error is only influenced by the yaw dynamics, which will lead to more wear and tear on the rudder. For obtaining more prefect path following performance, not only the desired heading angle but also referenced surge speed are produced by the SHLOS guidance law in this section.
In order to deal with the singularity of sideslip angle when surge and sway speeds are zero simultaneously, the desired sideslip angle is defined as: where u d is the desired surge speed, which is generated by the SHLOS guidance law and always greater than zero. Define the surge speed tracking error and the heading angle tracking error as:ũ where ψ d is the desired heading angle. Substitute (23), (24) and (25) into (22), and we obtain: where Since cos ψ e −1 ψ e < 0.73, sin ψ e ψ e ≤ 1, ω is bounded, and the maximum value is determined as d 1 .
In order to stabilize path following errors, the desired surge speed, heading angle and the virtual input with the SHLOS guidance law are designed as: where k 1 > 0, k 2 > 0 are the design parameters, and is the look-ahead distance.
And we can see that, u d ≥ k 2 , so that the singularity of desired sideslip angle is avoid, which means that β d ∈ (− π 2 , π 2 ). According to (29), it is obtained that: Substituting (27), (28), (29) into (26), the path following error dynamics can be rewritten as: x e = −k 1 x e +γ p y e +ũ cos ψ − γ p , Choose the Lyapunov function candidate as follows: Taking the derivative of V 1 with respect to time, we havė The result in (33) will be used for the stability analysis of the whole control system in Section V.

IV. CONTROL SYSTEM
In this section, the tracking control laws, which are made up of the surge control force and yaw control moment, are designed under the lumped disturbances and input saturation. The unknown nonlinear terms D u and D r contain the environment disturbances, model uncertainties and approximation errors, which are compensated by RBFNN.

A. SURGE SPEED CONTROL
On the basis of the SHLOS guidance law, the surge speed control force is designed for tracking the desired surge speed.
Define the first error surface as: where k 3 > 0 is the design constant.

Remark 4:
As (17) shown, we can not get the control input σ 1 directly, so that we introduce an auxiliary signal to handle the problem.
The time derivative of u e is given as: Then the desired surge control σ * 1 is designed as: where k u > 0, and D u is the unknown lumped disturbance in surge, which can be approximated by RBFNN: where W * 1 is the optimal weight value and χ 1 is the approximation error. Therefore, the desired surge control law σ * 1 is designed as: Since the W * 1 and χ 1 are unknown, the control law σ 1 can be expressed as: whereŴ 1 is the estimation of W * 1 . Consider the Lyapunov function as: whereW 1 =Ŵ 1 − W * 1 is the estimation error of the weight value, and γ u is the design parameter.
Taking the time derivative of V 2 , we obtaiṅ Design the update law ofŴ 1 as: where k 4 > 0 is the design constant. Then, (42) is rewritten as: The result in (44) will be used for the stability analysis of the whole control system in Section V.

B. HEADING CONTROL
In this subsection, the yaw control moment is designed to track the desired heading angle, which is generated by SHLOS guidance law.
Step 1: Consider the heading tracking error ψ e and take the time derivative, then we havė To stabilize the dynamic surface ψ e , a virtual input r d is chosen as: where k ψ > 0 is a design parameter. In order to avoid the complexity explosion problem induced by the repeated differentiations of the virtual control law, the DSC technology is applied. Let the virtual control input r d pass through a first-order filter where ι 1 is the design time constant, and α r is the output of the first-order filter. Define the output error of the filter as z 1 = α r − r d , and take the derivative respect with timė where the maximum value denotes d 2 .
Step 2: Define the second error surface r e r e = r − α r − ξ 2 (49) where k 5 > 0 is the design constant. Then the derivative of (49) is given as follows: Then the desired yaw control σ * 2 is designed as: where k r > 0, and D r is the unknown lumped disturbance in yaw, which can be approximated by RBFNN: where W * 2 is the optimal weight value and χ 2 is the approximation error. Then the desired yaw control law σ * 2 is designed as: Since the W * 2 and χ 2 are unknown, the control law σ 2 can be expressed as: whereŴ 2 is the estimation of W * 2 . Consider the Lyapunov function as: whereW 2 =Ŵ 2 − W * 2 is the estimation error of the weight value, and γ r is the design parameter.
The time derivative of V 3 is given by: Design the update law ofŴ 2 as: where k 6 > 0 is the design constant. Then, (57) is rewritten as: The result in (59) will be used for the stability analysis of the whole control system in Section V.

V. STABILITY ANALYSIS
In the presence of the off-diagonal mass matrix, environment disturbances, model uncertainties and input saturation, the SHLOS guidance law and tracking control laws are applied to obtain prefect path following performance for an USV. As the result, the stability analysis is presented as following.
proof: Choose the following Lyapunov function: Based on (33), (44) and (59), the time derivative of V is derived as: In virtue of Young's inequality, we obtain x eũ cos ψ − γ p ≤ x e (u e + ξ 1 ) In addition, consider the fact as follows: Substituting (62)-(73) into (61), we geṫ . Then, choose the suitable design parameters satisfy K > 0, such that We can obtain the solution of (74) It is concluded that V is bounded, which means that the error signals are uniformly ultimately bounded.
As a result, the proof is concluded. Remark 5: Considering the sway dynamics, choose the Lyapunov function The time derivative can obtain as: Since D v is bounded, so thatv is uniformly ultimately bounded [28]. Consider the coordinate transformationv = v + r, and r is bounded, then we can get the sway velocity v is bounded. Remark 6: A parameter selection guide is provided as follows. First, the parameter determines the convergence speed of cross-path error. The convergence speed can be fast by increasing . Second, the parameter k 1 influences the convergence speed of along-path error. Third, the parameters γ u , γ r determine the learning rate of RBFNNs. Forth, the parameters k u , k ψ and k r influence the output response, and a tradeoff should be made between response speed and stability margin. Finally, optimal parameters can be obtained with running simulation.
In order to show the advantages of the proposed strategy, a comparison between the ANPFC with the ANPFC/SA, which is defined as the ANPFC method without considering the input saturation is applied. In addition, we also conduct a comparison between ANPFC based on the SHLOS guidance with the ELOS guidance law [12], while the desired surge speed of the ELOS law is defined as 0.25m/s, and the observer gain K o = 10.

A. COMPARISON WITH ANPFC/SA
In this subsection, the importance of the method with considering input saturation is revealed by the comparison analysis between ANPFC with ANPFC/SA. The simulation results of the different methods are shown in Figure 2-Figure 6. As we can see that, the ANPFC/SA method with the unconstrained   controller has a better path following performance. Due to the input saturation, the surge speed based ANPFC is smaller than ANPFC/SA, then the convergence time of ANPFC is more than ANPFC/SA. However, the control inputs break through the limitations of actuators, so that this situation is not practical. In addition, the ANPFC can force the USV to arrive   and follow the desired path, and the desired heading angle and surge speed can be tracked accurately, while the control inputs via ANPFC method stay in the limitations, as shown in Figure 2-Figure 6. Based on the simulation results, we can conclude that the presented ANPFC method is more efficient and operable.

B. COMPARISON WITH ELOS
In this subsection, a comparison between the proposed SHLOS guidance law with ELOS is applied. The results of simulation are shown in Figure 7- Figure 12. From Figure 7- Figure 8, we can see that the SHLOS performs better than ELOS, while the USV based on SHLOS guidance law arrive the desired path faster than ELOS. In addition, the steady-state path tracking errors of ELOS are larger than SHLOS. The heading angle and surge speed tracking performance are shown in Figure 9- Figure 10, and illustrate that the two methods can track the desired values. In addition, the surge speed based on SHLOS guidance law is bigger than ELOS, so that the convergence time of SHLOS is less than ELOS. In addition, from Figure 11- Figure 12, we can conclude that the lumped disturbances can be estimated with RBFNN accurately, and the control inputs are smooth and keep in the limits. The simulation results demonstrate the efficiency of the designed control strategy.

VII. CONCLUSION
In this paper, an adaptive neural path following control scheme based on the SHLOS guidance law has been proposed for path following problem of an USV in the presence of off-diagonal mass matrix, unknown time-varying environment disturbances, model uncertainties and input saturation. The SHLOS guidance law can generate the reference surge speed and heading angle simultaneously. The effect of input saturation is handled by the hyperbolic tangent function, and the lumped disturbances are compensated by the RBFNN. Then, the proposed ANPFC scheme has been designed via DSC technique, so that the USV can arrive and follow the desired path. It has been proven that all error signals of the whole system are uniformly ultimately bounded. In addition, the simulation results confirm the effectiveness of the proposed control method. In the future, the path following tracking errors constraint problem will be considered in path following control design.
GUOQING XIA received the Ph.D. degree in control theory and control engineering from Harbin Engineering University (HEU), in 2001. He is currently working as a Professor with HEU. His research interests are ship dynamic positioning control technique, intelligent control theory, and system simulation technique.
XINWEI WANG received the bachelor's degree in electrical engineering and automation from Harbin Engineering University (HEU), in 2015, where he is currently pursuing the Ph.D. degree in control science and engineering. His research interests are ship motion control, nonlinear control theory, and intelligent control theory.
BO ZHAO received the bachelor's degree in automation and the master's degree in navigation, guidance and control from the Beijing University of Aeronautics and Astronautics (BUAA), in 2006 and 2009, respectively, and the Ph.D. degree in marine cybernetics from the Norwegian University of Science and Technology, in 2015. From 2013 to 2018, he served at Global Maritime AS as a Senior Marine System Advisor, and developed hardware-in-the-loop testing for dynamic positioning systems. He currently works as an Associate Professor at Harbin Engineering University. His research interests are applying advanced control and artificial intelligence in the control of vessel, underwater robotics, and other marine systems. VOLUME 8, 2020 ZHIWEI HAN received the bachelor's degree in electrical engineering and automation from Northeast Agricultural University (NEAU), in 2015. He is currently pursuing the Ph.D. degree in control science and engineering from Harbin Engineering University. His research interests are intelligent optimization algorithm, path planning and collision avoidance of unmanned surface vehicle, and thrust allocation of DP vessels.
LINHE ZHENG was born in Suihua, China, in 1993. He is currently pursuing the Ph.D. degree with the College of Automation, Harbin Engineering University, China. His research interest includes ship motion stabilization control.