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This paper investigates the path following control problem for an underactuated unmanned surface vehicle (USV) in the presence of dynamical uncertainties and time-varying external disturbances. Based on fuzzy optimization algorithm, an improved adaptive line-of-sight (ALOS) guidance law is proposed, which is suitable for straight-line and curve paths. On the basis of guidance information provided by LOS, a three-degree-of-freedom (DOF) dynamic model of an underactuated USV has been used to design a practical path following controller. The controller is designed by combining backstepping method, neural shunting model, neural network minimum parameter learning method, and Nussbaum function. Neural shunting model is used to solve the problem of “explosion of complexity,” which is an inherent illness of backstepping algorithm. Meanwhile, a simpler neural network minimum parameter learning method than multilayer neural network is employed to identify the uncertainties and time-varying external disturbances. In particular, Nussbaum function is introduced into the controller design to solve the problem of unknown control gain coefficient. And much effort is made to obtain the stability for the closed-loop control system, using the Lyapunov stability theory. Simulation experiments demonstrate the effectiveness and reliability of the improved LOS guidance algorithm and the path following controller.

USV is attracting more and more attention of researchers from all over the world because of its extensive applications in the military and civilian areas [

Generally speaking, the path following control for USV can be divided into three different system modules: navigation, guidance, and motion control [

Visualization of the proposed path following strategy.

The LOS guidance law is the most widely used one in guidance module due to its simplicity and intuitiveness, and it had been applied to surface ships by McGookin et al. in [

From the point of view of the whole path following system, guidance module and motion control module are independent, so we can design the controller individually. In the field of motion control, there are many excellent algorithms used to design controllers [

Motivated by the above-mentioned observations, the goal of this article is that, based on ALOS algorithm with a time-varying

Fuzzy logic is hired to optimize the

Neural shunting model and neural network minimum parameter learning method are used to deal with “explosion of complexity” and model uncertainties problems, respectively. The qualitative analysis of these two methods can reduce the computational burden of the controller to some extent.

Nussbaum function is first applied to solve the problem of unknown gain of path following controller for USV.

The rest of the paper is organized as follows. The problem formulation and preliminaries are presented in Section

According to past experience, the complex model increases the difficulty of controller design, and the simple model can not fully describe the physical characteristics of the object. So in this paper three-DOF dynamic model of underactuated USV is employed to describe its planar motion characteristics. The model of USV can be represented as [

In actual engineering,

RBF neural network is one of the most commonly used tools, which is used to approximate the unknown dynamics and uncertain parameters in the system. However, the computational complexity of multilayer neural network increases the difficulty in engineering implementation. In order to reduce the computational burden of the controller, on the basis of neural network, [

For any continuous function

The essence of neural network minimum parameter learning method is that defining

Neural shunting model belongs to the field of biology. In the early days, it was used to describe the responses of neurons to external stimuli. With the continuous progress of science and technology, it has been applied to path planning, mechanical arm control, and other fields [

The neural shunting model can be represented as

In order to deal with the unknown sign of the control gain, Nussbaum function [

If a continuous function

then

Define

The LOS algorithm transforms the desired path into physical quantities that can be controlled, where the adaptive sideslip angle and the fuzzy optimization

LOS guidance geometry for curved paths.

For the USV, the errors (along tracking error

According to (

If

The ALOS guidance algorithm can be chosen as

The origin

This is because

Design the Lyapunov function

Using (

We can choose that

In this subchapter, fuzzy logic is used to optimize the value of

The fuzzy subsets of

The control rules are shown in Figure

Fuzzy reasoning adopts Zadeh and max-min, and defuzzification uses the method of centroid area center of gravity.

Fuzzy control rules.

Yaw rate controller and surge speed controller are designed to complete USV path following when the guidance module has been designed. Controller design is the most difficult part of the whole path following system. Many practical conditions need to be taken into account.

In the control system, the first- and second-order derivatives of all the error variables and reference signals are bounded.

All variables needed (such as position, speed, and direction) of the USV are available for feedback.

The goal of designing yaw rate controller is to make the actual heading follow the target value well. In other words, the heading error

Define

By differentiating

For

Using (

The corresponding control law is chosen as

The adaptive law of neural network minimum parameter learning method is

In this article, one assumes that

When time

Using (

The corresponding control law is chosen as

The adaptive law of neural network minimum parameter learning method is

In previous literature, there are several ways to deal with control gain:

This paper mainly reduces computation from three aspects:

The stability of proposed control strategy and the closed-loop path following system are demonstrated in this chapter.

Define error variable

The time derivative of

Consider the kinematic and dynamic models of underactuated USV which are given by (

Define the second Lyapunov function candidate.

The time derivative of (

Substituting (

Because

Define

From Young’s inequality, we have

Then

Define

From (

Define

Solving inequality (

Then the above inequality means that

To analyze the stability of closed-loop system, construct the following Lyapunov function.

Its time derivative is computed as

Therefore, all signals in the closed-loop network are UUB.

Nussbaum function is widely used in dealing with the problem of unknown control gain [

In this section, numerical simulations on CyberShip II are carried out to prove the correctness and effectiveness of the whole path following system. CyberShip II is a 1 : 70 scale replica of a supply ship, and for its main parameters please refer to [

First, a straight-line path following simulation is carried out and its desired geometrical path is expressed as

Straight-line path simulation results are plotted in Figures

Path following performance (straight-line).

Heading control performance (straight-line).

Speed control performance (straight-line).

The value of

The estimated value of sideslip angle (straight-line).

Control input

Control input

Figure

In order to verify the versatility of the whole system, the curve path following is executed when the control parameters and initial state are not changed. The desired geometrical path is a curve expressed as

Path following performance (curve).

Heading control performance (curve).

Speed control performance (curve).

The value of

The estimated value of sideslip angle (curve).

Control input

Control input

The result of curve path following is shown in Figure

Through the above numerical simulations and theoretical analysis, one can see that in the presence of external disturbances USV has a good performance for the same set of guidance and control parameters regardless of whether the desired path is straight-line or curve, which shows the effectiveness and correctness of the whole path following strategy.

A complete path following strategy is proposed for underactuated USV in this paper. First of all, an optimized LOS algorithm is proposed. Then, in the process of designing the controller, the unknown dynamics and the external disturbances are compensated online by neural network minimum parameter learning method. Meanwhile, the “explosion of complexity” problem caused by the derivation of the virtual control law is addressed by introducing the neural shunting model. Aiming at the problem of unknown control gain, Nussbaum function is introduced to guarantee that the control gain is positive. Stability analysis using a Lyapunov function showed that all error signals in the closed-loop system are guaranteed to be uniformly ultimately bounded. Finally, the correctness of the proposed path following strategy for underactuated USV is proved by numerical simulations.

The authors declare that they have no conflicts of interest.

This work was partially supported by the Natural Science Foundation of Liaoning Province of China (Grand no. 2015020022), the National Natural Science Foundation of China (Grand no. 51609033), and the Fundamental Research Funds for the Central Universities (Grand nos. 3132014321, 3132016312, and 3132017133).