Finite-Time Control for Automatic Berthing of Pod-Driven Unmanned Surface Vessel with an Event-Triggering Mechanism

Abstract

Coastal water accidents have occurred frequently in recent years, and human factors are still the main cause of these accidents. The purpose of this study is to provide a better and safer solution for the symmetry problem of unmanned surface vessels during automatic berthing in coastal waters. In automatic berthing, the symmetry problem refers to whether the USVs can maintain a stable state during motion and positioning, including dynamic symmetry and environmental symmetry. A finite-time controller based on a global nonsingular terminal sliding mode is designed to improve response speed and state consistency. Dynamic uncertainty and disturbance are considered in the design process, which can optimize the control law and effectively ensure the symmetry of the vessel in different states. Then, an event-triggering mechanism based on a dynamic threshold is adopted. In practice, there is excessive operation of the actuator. This mechanism ensures that the actuator is triggered only when the threshold is reached. USVs can adaptively adjust control strategies based on the real-time status, thereby improving symmetry during berthing. In simulation analysis, a pod-driven unmanned surface vessel with good maneuverability is taken as the research object. The results indicate that this control strategy can ensure rapid and consistent response of the vessel when subjected to external disturbances, which helps to maintain the symmetry of the automatic berthing motion under different conditions.

1. Introduction

According to the European Maritime Safety Agency’s Annual Overview of Marine Casualties and Incidents for 2023, from 2014 to 2022, 80.7% of accident factors were related to human factors, and 45.3% of accidents occurred in coastal waters. In this context, people have turned their attention to the development of unmanned surface vessels (USVs), as their maneuverability issues in coastal waters urgently need to be addressed. USVs refer to vessels that navigate on the water surface and are fully automated without crew members. They utilize advanced intelligent perception algorithms, data fusion, communication technology, the Internet of Things, control technology, etc., to achieve automatic operation, navigation, and berthing. USVs play an increasingly important role in various complex maritime tasks, including marine resource exploration, water pollution removal, disaster rescue, maritime patrols, and exploration [1,2,3,4,5], due to their low cost, high efficiency, and flexibility.

As an important task of USVs in the coastal waters, automatic berthing is a great challenge to the symmetry and stability of ship movement. Symmetry refers to the property or structure of a system that remains unchanged under certain transformations. In automatic berthing control, symmetry usually refers to the fact that the control characteristics of the system should remain consistent under different motion conditions (such as different heading or speed). Ensuring symmetry can enable the control system to exhibit better consistency and reliability in different situations. If a control system has good symmetry, the consistency of control strategies in different states may help improve the overall stability of the system [6].

The most commonly studied method for automatic berthing is artificial neural networks (ANN). In [7], the authors used neural networks for automatic ship berthing, using two parallel controllers instead of a centralized controller, and achieved good control results. Later, Im [8] proposed the concept of designing ANN controllers separately for multiple areas in ports, enabling ships to automatically berth at any position. Ahmed [9] introduced a virtual window to create consistent data for the neural network training process and validated the ANN controller for ship berthing under strong wind conditions. Nguyen [10] suggested that existing ANN controllers use parameters composed of ship position and ship heading as inputs and highlighted that they cannot be applied to control ships to berth at different ports. To solve this problem, parameters such as the bearing and the distance from the ship to the berth calculated by ARPA (Automatic Radar Plotting Aid) can be used as inputs to the controller. This study developed an automatic berthing controller that does not require retraining of neural networks using parameters measured by ranging systems. Li [11] applied an ANN to an automatic berthing model with established predetermined routes. With the help of training data from a certain port, this model can be applied to ship berthing with different berth layouts. However, in the actual berthing process of ships, the allowed control convergence time is relatively short, and the real-time performance of this method is difficult to guarantee. Additionally, due to the lack of universality of the target ship training samples that need to be obtained via neural networks, it is difficult to apply ANN control methods to actual ship berthing.

Due to the highly nonlinear model of ships during low-speed berthing, it is difficult to calculate analytical solutions. Therefore, the nonlinear optimal control theory is widely used for numerical methods when automatically berthing ships. Optimal control is a comprehensive strategy that optimizes the performance indicators of a controlled system. The optimal control strategy can be identified from a class of allowed control schemes for a controlled dynamic system or motion process so that the performance indicator value of the system is optimal while transitioning from an initial state to a specified target state. At present, the main research methods for optimal control include model predictive control [12,13], data-driven methods [14], and covariance matrix adaptive evolutionary strategies [15]. Nonlinear optimal control methods can solve nonlinear control problems in real time, usually requiring the establishment of accurate mathematical models. However, establishing mathematical models for ships is difficult [16]. Based on the actual berthing environment, researchers often focus on establishing mathematical models in certain aspects, but lack unified and modular research on models. The automatic berthing mathematical model is mainly based on the MMG modeling idea, and also involves environmental (wind, waves, currents) interference models. The motion of a ship is influenced by various factors, including its center of gravity, buoyancy center, rudder angle, velocity, load state, etc. The interaction between these factors makes the modeling process more complex. The impact of different marine environments (such as wave height, wavelength, tides, etc.) on ship motion varies, so modeling is required for different situations, which increases the complexity of the model.

Based on the existing relevant research and research being conducted in other related studies [17,18,19], this study mainly focuses on the symmetry issue of the automatic berthing process of USVs. A finite time controller is designed to enable USVs to autonomously move to designated berths while maintaining dynamic positioning. During this process, an event triggering mechanism based on dynamic thresholds is added, reducing the number of actuator actions and effectively extending its service life without affecting control effectiveness. The design method can comprehensively improve the maneuverability of ships in different environments and maintain symmetry.

2. Principle of Automatic Berthing and Mathematical Model of Pod-Driven USV

2.1. Principle of Automatic Berthing

The berthing of USVs is a relatively complex and difficult control movement in many ship maneuvering situations. In the actual process of ship berthing, under the influence of several factors, such as the shore wall effect, shallow water, and weak rudder effect, ships usually need a tugboat to provide lateral force and moment to reach the designated berth. With the significant improvement of ship automation, automatic berthing of USVs can be realized under certain conditions [20,21,22,23]. To realize the whole process of autonomous intelligent unmanned transportation, research on the automatic berthing of USVs has important practical significance.

Specifically, automatic berthing refers to the process of berthing and positioning in a harbor. At present, due to the large inertia of ships, even in the presence of side thrusters, maneuvering in harbors is completed mainly with the assistance of tugboats. The process of tugboat action is as follows: when a ship applies for berthing, to prevent the ship from colliding with the wharf, the tugboat is used to pull the bow first, and then another tugboat is used to pull the stern after reaching a suitable position to gradually make the ship enter the berth. Many scholars have performed extensive research on ship motion control. Automatic berthing control involves path planning, track tracking, dynamic positioning and so on. The specific control methods include the sliding mode method, neural network method, backstepping method, and adaptive method. Scholars worldwide have made different emphases on the task and target of automatic berthing, so the control methods are given in different ways.

In this study, the automatic berthing problem is transformed into a stabilization control problem. A schematic diagram of automatic berthing is shown in Figure 1.

A diagram of the automatic berthing process is shown in Figure 2.

The automatic berthing system consists of a measurement system, a control system and a thrust distribution system. In navigation, the control system generates control signals to resist the influence of various disturbances. $x^{\ast }$, $y^{\ast }$, and ${\psi }^{\ast }$ are the expected position and course; $x$, $y$, and $\psi$ are the actual position and course; $X_T$, $Y_T$, and $N_T$ are the longitudinal force, transverse force, and moment produced by the controller; and $n_d$ and $n$ are the expected and actual revolution speed of propeller (revolutions per minute). Due to the ship being equipped with side thrusters and other devices, control forces can be generated at each degree of freedom to achieve the above control strategy. Previous research on ship trajectory tracking and dynamic positioning has been an important reference method for solving automatic berthing. The ship speed control is also gradually reduced to zero when the ship arrives at the berth. In essence, automatic berthing can be regarded as a stabilization control problem involving speed in all directions.

2.2. Symmetry Problem in Automatic Berthing of USVs

In automatic berthing, the symmetry problem is manifested as dynamic symmetry and environmental symmetry.

Dynamic symmetry: When a vessel is berthed, it needs to adjust its status based on its own dynamic characteristics (such as speed, turning radius, etc.) to approach the target berth in a balanced manner.

Environmental symmetry: During the berthing process, it is necessary to consider the symmetry of environmental factors such as water flow, wind, and waves. These factors may have asymmetric effects on the trajectory of USVs, leading to instability during berthing.

The symmetry problem is important for the berthing of unmanned surface vessels in the following aspects:

Safety: Asymmetric berthing processes may lead to collisions between ships and other vessels or port facilities, resulting in safety hazards, especially in busy coastal waters.

Accuracy: Symmetry issues affect the ability of a vessel to adjust its position and angle when approaching a berth. If precise control is not possible, it may result in the vessel being unable to berth safely or causing accidents.

Control strategy: Whether it is the processing of sensor data or the design of control algorithms, symmetry issues will affect the response speed and accuracy of the automatic control system, ensuring real-time correction of heading deviation.

Coastal waters are often accompanied by highly dynamic environmental changes, such as tides, waves, and other surface traffic, which can lead to asymmetric behavior in both the external environment and the USV itself. By studying and improving symmetry issues, the robustness of the system in the face of uncertainty and disturbance can be enhanced, thereby improving the adaptability of automatic berthing. During the automatic berthing process of USVs, symmetry is an important key to ensuring safe, accurate, and efficient berthing. In-depth research and resolution of this issue could help improve the operational capabilities of USVs in complex environments, reduce potential safety risks, and promote their application and development in coastal waters.

2.3. Mathematical Model of Pod-Driven USV

The object of this study is a ship with a pair of pod propellers installed at the stern. The dynamic equation was calculated using (1).

$$\left\{\begin{array}{l}(m+m_x)\dot{u}-(m+m_y)vr=X_H+X_P+X_D\\ (m+m_y)\dot{v}+(m+m_x)ur=Y_H+Y_P+Y_D\\ (I_{zz}+J_{zz})\dot{r}=N_H+N_P+N_D\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\end{array}\right.$$

(1)

where $m$, $m_x$, and $m_y$ are the mass and additional mass; $I_{zz}$ and $J_{zz}$ are the moment of inertia and additional moment of inertia; $X$, $Y$, and $N$ are forces and moments acting on the hull; $P$ represents the pod; $H$ represents the hull; $D$ represents the marine environmental disturbance; u is the longitudinal velocity; v is the lateral velocity; and r is the bow turning angular velocity.

After the Taylor expansion of (1), the equation that retains the first-order quantity is (2).

$$\left\{\begin{array}{l}(m-X_{\dot{u}})\dot{u}=X_u(u-u_0)+X_P+X_D\\ (m-Y_{\dot{v}})\dot{v}+Y_{\dot{r}}\dot{r}=Y_{v}v+(Y_r-m{u}_0)r+Y_P+Y_D\\ N_{\dot{v}}\dot{v}+(I_{zz}-N_{\dot{r}})\dot{r}=N_{v}v+N_{r}r+N_P+N_D\hspace{1em}\hspace{1em}\hspace{1em},\end{array}\right.$$

(2)

When the USV arrives at the berth, its speed is 0, so $u_0=0$ is the equilibrium state; then, (2) can be transformed into (3).

$$\left\{\begin{array}{l}(m-X_{\dot{u}})\dot{u}=X_{u}u+X_P+X_D\\ (m-Y_{\dot{v}})\dot{v}+Y_{\dot{r}}\dot{r}=Y_{v}v+Y_{r}r+Y_P+Y_D\\ N_{\dot{v}}\dot{v}+(I_{zz}-N_{\dot{r}})\dot{r}=N_{v}v+N_{r}r+N_P+N_D ,\end{array}\right.$$

(3)

Equation (3) is abbreviated as (4):

$$\mathit{M}\dot{\mathit{V}}+\mathit{D}\mathit{V}=\mathit{\tau }.$$

(4)

In (4), $\tau ={[X_P+X_D,Y_P+Y_D,N_P+N_D]}^T$, $V={[u,v,r]}^T$, $\mathit{M}$ = $\left[\begin{array}{ccc}m-X_{\dot{u}}& 0& 0\\ 0& m-Y_{\dot{v}}& Y_{\dot{r}}\\ 0& -N_{\dot{v}}& I_{zz}-N_{\dot{r}}\end{array}\right]$, and $\mathit{D}$ = $\left[\begin{array}{ccc}-X_u& 0& 0\\ 0& -Y_v& -Y_r\\ 0& -N_v& -N_r\end{array}\right]$. The parameters $\mathit{M}$ and $\mathit{D}$ are hydrodynamic derivatives. The formulas are shown below.

$$\left\{\begin{array}{ll}\hfill X_u^{\prime }=& S^{\prime }{C}_t\hfill \\ \hfill X_{\dot{u}}^{\prime }=& -0.01[0.398+11.97C_b(1+3.73T/B)\hfill \\ & -2.89C_{b}L/B(1+1.13T/B)\hfill \\ & +0.175C_b{(L/B)}^2(1+0.541d_m/B)-1.107L{d}_m/B^2]m^{\prime }\hfill \\ \hfill Y_{\dot{v}}\prime =& -[1+0.16C_{b}B/T-5.1{(B/L)}^2]\cdot \pi {(T/L)}^2\hfill \\ \hfill Y_{\dot{r}}=& -[0.67B/L-0.0033{(B/T)}^2]\cdot \pi {(T/L)}^2\hfill \\ \hfill N_{\dot{v}}^{\prime }=& -[1.1B/L-0.041B/T]\cdot \pi {(T/L)}^2\hfill \\ \hfill N_{\dot{r}}^{\prime }=& -[1/12+0.017C_{b}B/T-0.33B/L]\cdot \pi {(T/L)}^2\hfill \\ \hfill Y_v^{\prime }=& -[1+0.40C_{b}B/T]\cdot \pi {(T/L)}^2\hfill \\ \hfill Y_r^{\prime }=& -[-1/2+2.2B/L-0.080B/T]\cdot \pi {(T/L)}^2\hfill \\ \hfill N_v^{\prime }=& -[1/2+2.4T/L]\cdot \pi {(T/L)}^2\hfill \\ \hfill N_r^{\prime }=& -[1/4+0.039B/T-0.56B/L]\cdot \pi {(T/L)}^2\hfill \end{array}\right.,$$

(5)

In (5), $L$ is the length of the vessel; $T$ is the draft; $B$ is the width of the ship; $C_b$ is the square coefficient; and $C_t$ is the damping coefficient. $S$ can be calculated according to the following formula:

$$S=(1.54T+0.45B+0.904B{C}_b+0.026C_b{B}^2/T)L,$$

(6)

With the continuous development of pod propulsion, the advantages of pod propulsion in terms of economy and environmental protection have led to rapid development. This kind of propulsion device is suspended at the stern. The thrust direction of the pod thruster can be changed, which can improve maneuverability but also increase control difficulty, especially in terms of signal delay and response time. Under different navigation conditions, the pod thruster may apply lateral forces, which can affect the stability of the heading. The pod propulsor adopts electric propulsion, and the cable is used to connect to the marine generator, which saves space in the transmission shafting. This approach has important practical significance for the construction of ships, as container ships can carry more goods, and warships can carry more weapons and ammunition [24,25,26]. The concept ship “Revolt”, developed by Det Norske Veritas, adopts a pod propulsion device, which is a real unmanned transport ship. The ship has no superstructure, and all the containers are in the cabin. The whole ship uses electrical energy, which reduces the loss of energy from 85% to 40%. The purpose of autonomous navigation, collision avoidance, and maritime navigation can be achieved by using the automatic control system of the whole ship and the good maneuverability of the pod-type propeller. It can be seen that pod propulsion is feasible for USVs.

Compared with that of the traditional rudder and propeller propulsion mode, the complexity of the mathematical analysis of the pod is increased. It is a rotatable propulsion device. When the direction of the force generated by the rotation of its propeller is towards the stern, it pushes the ship forward; If the direction of the pod thruster is changed to direct the force generated by the propeller towards the bow of the ship, it can pull the ship backwards and play an important role in ship maneuvering. Thus, it is necessary to establish a thrust vector model. The distribution of ship pods is shown in Figure 3.

The thrust vector model equation is shown in (7).

$$\left\{\begin{array}{l}{X}_P=(1-t_p)(T_P\cos {\theta }_P+T_s\cos {\theta }_s)\\ Y_P=(1-t_p)(T_P\sin {\theta }_P+T_s\sin {\theta }_s)\\ N_P=(1-t_p)(T_P\cos {\theta }_P-T_s\cos {\theta }_s)0.5L_{PS}+Y_P{L}_{OP}\end{array}\right.,$$

(7)

where ${\theta }_P$ and ${\theta }_s$ are the steering angles of the pods; $L_{PS}$ is the distance between the two pods; $T_P$ and $T_s$ are the thrusts generated by the pods; $L_{OP}$ is the distance from stern to the center of the ship; and $t_p$ is the thrust derating factor.

The thrust calculation formula of a single POD propeller is shown in (8).

$$T_p=\rho n_p^2{D}^4{K}_T(J_p),$$

(8)

where $\rho$ is the density of seawater; $n_p$ is the speed of the propeller; $D$ is the propeller diameter; and $J_P$ is the advance ratio. $K_T$ can be calculated according to the open water performance map [27] shown in Figure 4.

The fitting equation is shown in (9):

$$K_T=0.7+0.3589J_P+0.1875J_P^2.$$

(9)

2.4. Model of Marine Environmental Disturbances

Wind force and moment can be calculated using the formula (10):

$$\left\{\begin{array}{l}{X}_{wind}=0.5{\rho }_a{U}_R^2{A}_T{C}_{wx}\\ Y_{wind}=0.5{\rho }_a{U}_R^2{A}_L{C}_{wy}\\ N_{wind}=0.5{\rho }_a{U}_R^2{A}_{L}L{C}_{wn}\end{array}\right..$$

(10)

In (10), $C_{wx}$, $C_{wy}$ and $C_{wz}$ are wind pressure coefficients; ${\rho }_a$ is the air density; $U_R$ is the relative wind speed; $A_T$ is the orthogonal projection area of the hull surface; and $A_L$ is the side of the projected area of the waterline, which is shown in Figure 5.

Regarding waves, a second-order wave mathematical model is used. The calculation formula is seen in (11):

$$\left\{\begin{array}{l}{X}_{wave}^D={\displaystyle \frac{1}{2}}\rho L{a}^2\cos \chi C_{Xw}^D(\lambda )\\ Y_{wave}^D={\displaystyle \frac{1}{2}}\rho L{a}^2\sin \chi C_{Yw}^D(\lambda )\\ N_{wave}^D={\displaystyle \frac{1}{2}}\rho L^2{a}^2\sin \chi C_{Nw}^D(\lambda )\end{array}\right.,$$

(11)

In (11), $a$ is the average wave amplitude; $\chi$ is the wave encounter angle; and $\lambda$ is the wavelength. $C_{Xw}^D$, $C_{Yw}^D$, and $C_{Nw}^D$ are the experimental coefficients obtained by summarizing. The calculation formula is (12):

$$\left\{\begin{array}{l}{C}_{Xw}^D(\lambda )=0.05-0.2({\displaystyle \frac{\lambda }{L}})+0.75({\displaystyle \frac{\lambda }{L}}{)}^2-0.51({\displaystyle \frac{\lambda }{L}}{)}^3\\ C_{Yw}^D(\lambda )=0.46+6.83({\displaystyle \frac{\lambda }{L}})-15.65({\displaystyle \frac{\lambda }{L}}{)}^2+8.44({\displaystyle \frac{\lambda }{L}}{)}^3\\ C_{Nw}^D(\lambda )=-0.11+0.68({\displaystyle \frac{\lambda }{L}})-0.79({\displaystyle \frac{\lambda }{L}}{)}^2+0.21({\displaystyle \frac{\lambda }{L}}{)}^3\end{array}\right.,$$

(12)

Regarding ocean current, a modeling approach is adopted for uniform current. The effect of ocean current can cause USVs to change their longitudinal and transverse velocities, causing them to deviate from their course. The formula is (13):

$$\left\{\begin{array}{l}{u}_c=U_c\cos \left({\psi }_c-\psi \right)\\ v_c=U_c\sin \left({\psi }_c-\psi \right)\end{array}\right..$$

(13)

In (13), $u_c$ and $v_c$ are the component of the velocity of the ocean current in the longitudinal and transverse directions; $U_c$ is the velocity of the ocean current; and ${\psi }_c$ is the direction angle of the ocean current.

The lateral and longitudinal velocities of a USV in an ocean current can be calculated using the following formula:

$$\left\{\begin{array}{l}{\dot{u}}_r=\dot{u}-r{v}_c\\ {\dot{v}}_r=\dot{v}-r{u}_c\end{array}\right.,$$

(14)

In (14), the transverse and longitudinal components of the USV’s speed relative to the earth are $u$ and $v$, and the relative speed under the influence of ocean currents becomes $u_r$ and $v_r$.

3. Finite-Time Controller Design Based on the Global Fast Nonsingular Terminal Sliding Mode

3.1. Finite-Time Stability Theory

In general, control engineering systems require stability in a limited time, which entails relatively stringent requirements concerning the rapidity of the control system. It is theoretically assumed that the system state converges to a range small enough for the balance position when time tends to infinity. Based on this idea, advanced intelligent control algorithms, which are being studied and widely applied at present, have improved the control performance in different aspects. In theory, it is not easy to determine the specific duration of control stability. With the advent of homogeneous theory, the development of finite-time control theory has accelerated [28,29,30,31]. The finite time control method is widely believed to have better rapidity and anti-disturbance performance, and it can be used to solve the symmetry problem in ship motion control [32].

The methods for designing controllers using finite time control theory are roughly as follows: the open-loop finite time control method is an early proposed method with relatively simple control laws. Like classical control theory, open-loop control has poor stability due to the lack of feedback in the system, making it unsuitable for practical applications. Non-continuous feedback control methods, such as bang-bang control, are not easy to implement in practical applications and suffer from severe chattering. The continuous feedback control method is a feasible way to achieve real-time control. This paper proposes a globally fast non-singular terminal sliding mode control law based on the idea of continuous feedback finite time control.

Terminal sliding mode control can achieve finite time stability of the system, making the stability analysis of ordinary sliding mode control transition from asymptotic stability with time approaching infinity to local stability analysis within finite time. In the process of designing a terminal sliding mode controller, it must be considered that there is no switching term function in ordinary sliding mode control, which reduces the chattering of the control signal. Therefore, considering many advantages, terminal sliding mode control is a control method with strong practical application value.

The system expressions can be described as in (15):

$$\left\{\begin{array}{l}\dot{x}=A(x)+Bu\\ y=C(x)\end{array}\right..$$

(15)

Equation (15) is the standard state equation of the control system. $x$ is the state variable, and its balance state is set to $x=0$.

Definition 1.

If the system reaches the equilibrium point at a finite moment, that is, for time $t$, when $t\to G(x_0)$, $\underset{t\to G(x_0)}{\lim }x(t,x_0)=0$, when $t>G(x_0)$, $x(t,x_0)=0$, $G(x)$ is a stable time function, then the system is finite-time stable.

Theorem 1.

Suppose that there is a continuously differentiable function $V(x)$ in the system, which is a positive definite function. If there exists any positive number l and a float number m between 0 and 1, make equation (16) hold:

$$\dot{V}(x)+l{V}^m(x)\le 0,x\in U_0\backslash \left\{0\right\}.$$

(16)

Then, system (15) is finite-time stable [33,34].

3.2. The Design of the Finite-Time Controller

The mathematical model of the automatic berthing system for a USV is as follows:

$$\left\{\begin{array}{l}\dot{\mathit{\eta }}=\mathit{R}\mathit{V}\\ \dot{\mathit{V}}=-{\mathit{M}}^{-1}\mathit{D}\mathit{V}+{\mathit{M}}^{-1}\mathit{\tau },\end{array}\right.$$

(17)

In (17), $\mathit{\eta }={[x,y,\psi ]}^T$ is the position vector of the USV. $\mathit{V}={[u,v,r]}^T$ is the velocity vector. $\mathit{R}=\left[\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]$ is the transition matrix. Let ${\mathit{e}}_1=\mathit{\eta }-{\mathit{\eta }}_d$, where ${\mathit{\eta }}_d={[x_d,y_d,{\psi }_d]}^T$ is the expected position. When ${\mathit{e}}_2={\dot{\mathit{e}}}_1$, then (17) can be derived:

$$\left\{\begin{array}{l}{\dot{\mathit{e}}}_1={\mathit{e}}_2\\ {\dot{\mathit{e}}}_2=-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2+\mathit{R}{\mathit{M}}^{-1}\mathit{\tau }\end{array}\right..$$

(18)

Therefore, the automatic berthing system of a USV is transformed into a stabilizing system, as shown above.

A global fast non-singular terminal sliding mode dynamic surface is proposed, whose form is shown in (19).

$$\mathit{s}={\mathit{e}}_2+U{\mathit{e}}_1+I{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}},$$

(19)

In (19), $I>0$, $U>0$; $1<{\displaystyle \frac{\alpha }{\beta }}<2(\alpha ,\beta >0)$; ${\mathit{e}}_1=\mathit{\eta }-{\mathit{\eta }}_{\mathit{d}}$ is the position error vector, and ${\mathit{e}}_2={\dot{\mathit{e}}}_1$. The nonlinear term ensures that the system state can quickly approach the equilibrium state when it is far away from the equilibrium state, while the linear term enables the system state to converge quickly when it approaches the equilibrium state. Moreover, due to the nonlinear term $I{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}}$, the obtained control law can ensure that no singular point problems occur. If the distance between the state variable and the sliding surface $s=0$ is relatively large, the linear sliding mode ${\mathit{e}}_2+U{\mathit{e}}_1$ plays an important role in the convergence speed of the state variable. When the state variable is relatively close to the sliding surface $s=0$, $U{e}_1$ can be ignored, and $e_2+I{{\mathit{e}}_1}^{{\displaystyle \frac{\alpha }{\beta }}}$, as the key to sliding mode motion, can stabilize the system at the equilibrium position.

The law of convergence is

$$\dot{\mathit{s}}=-\epsilon sgn(\mathit{s})-k\mathit{s},\epsilon >0,k>0.$$

(20)

The control law can be designed as (21):

$$\mathit{\tau }=\mathit{M}{\mathit{R}}^{-1}(-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s}-U{\mathit{e}}_2-{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2+\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2-\mathit{s}{\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_{\mathit{b}}^{\mathit{T}}{\mathit{k}}_{\mathit{b}}-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}).$$

(21)

In (21), ${\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_{\mathit{b}}^{\mathit{T}}{\mathit{k}}_{\mathit{b}}-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}$ is the Lyapunov obstacle function. The function tends to infinity when the restricted quantity is close to the constraint value. Therefore, if the control law is designed properly, the Lyapunov function of the system will not diverge to infinity, and the restricted quantity of the system will not reach the constraint value.

The characteristics of the global fast nonsingular terminal sliding mode control are summarized as follows:

(1)

The singularity problem in the control law is solved.

(2)

This approach can ensure the stability of the system in finite time.

(3)

This ensures that the system position variable is limited to a fixed area.

The proof process is as follows. Let the Lyapunov function be

$${\mathit{V}}_{\mathit{L}}={\displaystyle \frac{1}{2}}{\mathit{s}}^{\mathit{T}}\mathit{s}+{\displaystyle \frac{1}{2}}({\mathit{s}}^{\mathit{T}}\mathit{s})\ln {\displaystyle \frac{{\mathit{k}}_{\mathit{b}}^{\mathit{T}}{\mathit{k}}_{\mathit{b}}}{{\mathit{k}}_{\mathit{b}}^{\mathit{T}}{\mathit{k}}_{\mathit{b}}-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}},$$

(22)

$$\begin{array}{l}{\dot{V}}_L={\mathit{s}}^{\mathit{T}}\dot{\mathit{s}}+({\displaystyle \frac{1}{2}}({\mathit{s}}^{\mathit{T}}\mathit{s})\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}{)}^{\prime }\\ ={\mathit{s}}^{\mathit{T}}({\dot{\mathit{e}}}_2+U{\dot{\mathit{e}}}_1+{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\dot{\mathit{e}}}_1)+{\mathit{s}}^{\mathit{T}}\dot{\mathit{s}}\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ ={\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2+\mathit{R}{\mathit{M}}^{-1}\mathit{\tau }+U{\mathit{e}}_2+{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2)\\ +{\mathit{s}}^{\mathit{T}}\dot{\mathit{s}}\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ \le {\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2+\mathit{R}{\mathit{M}}^{-1}\mathit{M}{\mathit{R}}^{-1}(-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s}-U{\mathit{e}}_2-{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2\\ +\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2)-\mathit{s}{\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}})+U{\mathit{e}}_2+{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2)+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ \le {\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s}-U{\mathit{e}}_2-{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2\\ +\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2-\mathit{s}{\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}+U{\mathit{e}}_2+{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2)+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ ={\mathit{s}}^{\mathit{T}}(-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s})=-\epsilon {\mathit{s}}^{\mathit{T}}\mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k{\mathit{s}}^{\mathit{T}}\mathit{s}\le 0\end{array},$$

(23)

Because

$${\dot{V}}_L\le -\epsilon {\mathit{s}}^{\mathit{T}}\mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k{\mathit{s}}^{\mathit{T}}\mathit{s}\le -\epsilon {\mathit{s}}^{\mathit{T}}\mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s}),$$

(24)

$${\mathrm{V}}_L={\displaystyle \frac{1}{2}}{\mathit{s}}^{\mathit{T}}\mathit{s}+{\displaystyle \frac{1}{2}}({\mathit{s}}^{\mathit{T}}\mathit{s})\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}=({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}){\mathit{s}}^{\mathit{T}}\mathit{s},$$

(25)

$${{\mathrm{V}}_L}^{{\displaystyle \frac{1}{2}}}={({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}})}^{{\displaystyle \frac{1}{2}}}‖\mathit{s}‖,$$

(26)

Because of the existence of the inequality relation $\left|a\right|+\left|b\right|+\left|c\right|\ge$$\sqrt{a^2+b^2+c^2}$, when $\epsilon \ge {({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{{\mathit{k}}_b^T{\mathit{k}}_b}{{\mathit{k}}_b^T{\mathit{k}}_b-{\mathit{e}}_1^T{\mathit{e}}_1}})}^{{\displaystyle \frac{1}{2}}}$ is satisfied, the control law can satisfy Theorem 1. Therefore, it is proven that the system can realize finite-time control.

The next step is to analyze the stability time. Take any one of the degrees of freedom as an example for calculation. $e_1$ is the error of any one of the degrees of freedom. The following derivation can be made from (19):

$${\dot{e}}_1+U{e}_1+I{e}_1^{{\displaystyle \frac{\alpha }{\beta }}}=0,$$

(27)

$$e_1^{-{\displaystyle \frac{\alpha }{\beta }}}{\dot{e}}_1+U{e}_1^{1-{\displaystyle \frac{\alpha }{\beta }}}+I=0,$$

(28)

Let $y=e_1^{1-{\displaystyle \frac{\beta }{\alpha }}}$, so $\dot{y}={\displaystyle \frac{\alpha -\beta }{\alpha }}{e}_1^{-{\displaystyle \frac{\beta }{\alpha }}}{\dot{e}}_1$. When $U=A$ and $I=B$, (21) can be abbreviated as (29):

$$\dot{y}+{\displaystyle \frac{\alpha -\beta }{\alpha }}Ay={\displaystyle \frac{\beta -\alpha }{\alpha }}B.$$

(29)

Because the general solution of $\dot{y}+P(x)y=Q(x)$ is

$$y=e^{-{\displaystyle \int P(x)d(x)}}({{\displaystyle \int Q(x)e}}^{{\displaystyle \int P(x)d(x)}}d(x)+C),$$

(30)

Then, the solution of (29) is

$$\begin{array}{l}y=e^{{\displaystyle {\int }_0^t(\frac{\beta -\alpha }{\alpha }A)dt}}({{\displaystyle {\int }_0^t(-\frac{\alpha -\beta }{\alpha }Be}}^{{\displaystyle {\int }_0^t\frac{\alpha -\beta }{\alpha }Adt}})dt+C)\\ =e^{{\displaystyle \frac{\beta -\alpha }{\alpha }}At}({{\displaystyle {\int }_0^t(\frac{\beta -\alpha }{\alpha }Be}}^{{\displaystyle \frac{\alpha -\beta }{\alpha }}At})dt+C)\end{array}.$$

(31)

When $t=0$, $C=y(0)$ can be obtained. Therefore, (31) can be calculated as (32):

$$\begin{array}{cc}\hfill y& =e^{{\displaystyle \frac{\beta -\alpha }{\alpha }}At}({\displaystyle \frac{\beta -\alpha }{\alpha }}B{\displaystyle \frac{\alpha }{(\alpha -\beta )A}}{e}^{{\displaystyle \frac{\alpha -\beta }{\alpha }}At}{{\displaystyle |}}_0^t+y(0))\hfill \\ & =-{\displaystyle \frac{B}{A}}+{\displaystyle \frac{B}{A}}{e}^{{\displaystyle \frac{\beta -\alpha }{\alpha }}At}+y(0)e^{{\displaystyle \frac{\beta -\alpha }{\alpha }}At}\hfill \end{array}.$$

(32)

When $e_1=0$, $y=0$ and $t=t_s$ can be obtained. Therefore, (32) can be calculated as (33):

$${\displaystyle \frac{B}{A}}{e}^{{\displaystyle \frac{\beta -\alpha }{\alpha }}A{t}_s}+y(0)e^{{\displaystyle \frac{\beta -\alpha }{\alpha }}A{t}_s}={\displaystyle \frac{B}{A}}.$$

(33)

That is,

$${\displaystyle \frac{B+Ay(0)}{B}}=e^{{\displaystyle \frac{\beta -\alpha }{\alpha }}A{t}_s}.$$

(34)

The finite time of the steady state is

$$t_s={\displaystyle \frac{\alpha }{A(\alpha -\beta )}}\ln {\displaystyle \frac{A{e}_1{(0)}^{\frac{\beta -\alpha }{\alpha }}+B}{B}}.$$

(35)

Firstly, the design of the terminal sliding surface enables the system state to reach the equilibrium point within a finite time. It is different from traditional linear sliding mode surfaces. In traditional linear sliding mode control, the system state asymptotically converges, while the terminal sliding mode can ensure convergence in finite time. Secondly, the non-singular characteristics ensure the rationality of the control law. In the process of terminal sliding mode control, if singular problems occur, it will lead to infinite control variables, which is unacceptable in practical systems. The global non-singular terminal sliding mode controller avoids the occurrence of singular points through a special control law design, enabling effective and stable control of the system in finite time from any initial state throughout the entire control process.

4. Design of an Automatic Berthing System with an Event-Triggering Mechanism

An event-triggered control system is a kind of nonperiodic control system driven by events. The traditional control system design is continuously triggered. The signal is sampled and calculated continuously and periodically, which ensures the reliability of the control; however, in some cases, network resources are wasted, increasing the computational burden of the controller. The event-triggering mechanism only drives the actuator when the system meets the preset conditions, which reduces the calculation of the controller to a certain extent and improves the utilization of the network. There are many kinds of event settings. The threshold can also be a constant or a function. In brief, the design of an event-triggered control system is highly flexible, and the control signal is time-varying and aperiodic [35,36,37,38].

The application of event-triggered control in automatic berthing control systems also has practical engineering significance. In the process of berthing, the low-speed and weak rudder effect usually requires frequent maneuvering, and the application of an event-triggering mechanism can improve the berthing efficiency. In the traditional continuous trigger control system, the actuator is periodically driven at equal intervals, and the control signal remains unchanged between two adjacent time points through the function of the holder. When the disturbance at sea is low, continuous triggering will increase the consumption of the actuator. Previous studies have shown that the application of event triggering mechanisms in control systems can effectively optimize the number of executions of actuators to reduce wear and tear. Compared to traditional time-triggered control, event-triggered control significantly reduces the execution frequency of the controller, thereby reducing the wear and tear of the actuator. The relevant experimental results show that the system using event-triggered control can reduce the number of executions of the actuator by 30–50% under the same operating conditions. There are also studies indicating the impact of reducing execution frequency on overall system maintenance costs, with results showing that by implementing event-triggering mechanisms, maintenance costs of the system were reduced by 15%. Thus, in the design of control systems, it is possible to consider how to improve the reliability and economy of the system through event-triggered methods [39,40,41]. The basic block diagram is as follows in Figure 6:

Suppose the event trigger time is ${\left\{t_k\right\}}_{k=0}^{\infty }$ and $t_k<t_{k+1}$. Let the first trigger time be $t_0$ and $t_0=0$.

To ensure system stability, the event-triggering condition for designing dynamic thresholds is

$$t_{k+1}=\inf \left\{t>t_k\left|‖{\mathit{e}}_{\tau }(t)‖\ge T\right.\right\}.$$

(36)

In (36), T is the dynamic threshold:

$$0\le T\le ‖{\displaystyle \frac{M{R}^{-1}(\epsilon \mathrm{sgn}(s)+ks)}{-\lambda (t)}}‖,$$

(37)

where $\lambda (t)$ is an adjustable parameter, and $‖\lambda (t)‖\le 1$. The main factor affecting the dynamic threshold is $s$, which is the designed sliding surface composed of system errors. The threshold is adjusted in real-time based on the calculation results of the sliding surface. Compared to a fixed threshold, this approach can effectively reduce the number of triggers while ensuring system stability.

$e_{\tau }(t)={\tau }_s(t)-\tau (t)$, $t\in [t_k,t_{k+1})$ is the defined controller comparison error, $\tau (t)$ represents the force exerted by the controller at the current moment, and ${\tau }_s(t)$ is the force exerted by the controller at the triggering moment.

According to the event-triggering conditions, after a certain triggering moment, when the controller comparison error $e_{\tau }$ exceeds the dynamic threshold $T$, the event-triggering moment will be updated from $t_k$ to $t_{k+1}$. Moreover, the controller is updated from ${\tau }_s(t_k)$ to ${\tau }_s(t_{k+1})$, and the updated control signal is transmitted to the actuator to control the motion of the USV. By reducing the update frequency of the controller, the service life of the actuator is extended. The controller remains unchanged in $[t_k,t_{k+1})$, and the value of the controller is ${\tau }_s(t_k)$.

With $[t_k,t_{k+1})$ and $‖{\mathit{e}}_{\tau }(t)‖<T$, the following equation can be derived:

$$‖{\tau }_s(t)-\tau (t)‖<T,$$

(38)

$$\tau (t)={\tau }_s(t)-T\lambda (t),$$

(39)

According to (23), the derivative of the Lyapunov function after adding the event-triggering mechanism can be calculated as follows:

$$\begin{array}{l}{\dot{V}}_L={\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2+\mathit{R}{\mathit{M}}^{-1}({\tau }_{\mathit{s}}(\mathit{t})-\mathit{T}\lambda (\mathit{t}))+U{\mathit{e}}_2\\ +{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2)+{\mathit{s}}^{\mathit{T}}\dot{\mathit{s}}\ln {\displaystyle \frac{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ \le {\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s}-U{\mathit{e}}_2-{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2\\ +\mathit{R}{\mathit{M}}^{-1}\mathit{D}{\mathit{R}}^{-1}{\mathit{e}}_2-\mathit{s}{\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}+U{\mathit{e}}_2+{\displaystyle \frac{I\alpha }{\beta }}{\mathit{e}}_1^{{\displaystyle \frac{\alpha }{\beta }}-1}\ast {\mathit{e}}_2\\ -\mathit{R}{\mathit{M}}^{-1}\mathit{T}\lambda (\mathit{t}))+({\mathit{s}}^{\mathit{T}}\mathit{s}){\displaystyle \frac{{\mathit{e}}_1^{\mathit{T}}{\dot{\mathit{e}}}_1}{{\mathit{k}}_b^{\mathit{T}}{\mathit{k}}_b-{\mathit{e}}_1^{\mathit{T}}{\mathit{e}}_1}}\\ ={\mathit{s}}^{\mathit{T}}(-\mathit{R}{\mathit{M}}^{-1}\mathit{T}\lambda (\mathit{t})-\epsilon \mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k\mathit{s})\\ =-{\mathit{s}}^{\mathit{T}}\mathit{R}{\mathit{M}}^{-1}\mathit{T}\lambda (\mathit{t})-\epsilon {\mathit{s}}^{\mathit{T}}\mathrm{s}\mathrm{g}\mathrm{n}(\mathit{s})-k{\mathit{s}}^{\mathit{T}}\mathit{s}\end{array},$$

(40)

According to Lyapunov’s second law, to ensure that ${\dot{V}}_L\le 0$, when s > 0, it is necessary to ensure that $T\ge 0$. When s < 0, it is necessary to ensure that $T\le {\displaystyle \frac{M{R}^{-1}(\epsilon \mathrm{sgn}(s)+ks)}{-\lambda (t)}}$. The minimum trigger time is discussed below; that is, $t^{\ast }$ satisfies $\forall k\in z^{+}$, $t_{k+1}-t_k\ge t^{\ast }$. Because ${\mathit{e}}_1(t)=\mathit{\eta }(t)-{\mathit{\eta }}_d$,

$${\displaystyle \frac{d‖{\mathit{e}}_1(t)‖}{dt}}={\displaystyle \frac{d‖\eta (t)-{\eta }_d‖}{dt}}=‖\dot{\eta }(t)‖.$$

(41)

Since the stability has been proven, it can be seen that $\dot{\mathit{\eta }}(t)$ is a bounded function; that is, there must be a constant $P>0$ satisfying $‖\dot{\eta }(t)‖\le P$. The above formula can be written as follows:

$$\underset{\Delta t\to 0}{{\displaystyle \lim }}{\displaystyle \frac{‖{\mathit{e}}_1(t_{k+1})‖-‖{\mathit{e}}_1(t_k)‖}{t_{k+1}-t_k}}\le P.$$

(42)

5. Numerical Experiment

The “Taian Kou” ship is used as an example for this simulation. Her main parameters are shown in

Table 1. Ship size parameters.

Parameters	Numerical Value
Length	156 m
$L_{OP}$	78 m
$L_{PS}$	21 m
Beam	32.2 m
Draft	7.5 (m)
Displacement	26,426 t
Square coefficient	0.755
Prismatic coefficient	2.17

.

The calculation results of the ship model parameters M and D are as follows:

$$M=\left[\begin{array}{ccc}0.3418& 0& 0\\ 0& 0.3336& 0.0007\\ 0& -0.0005& 0.0228\end{array}\right], D=\left[\begin{array}{ccc}0.0044& 0& 0\\ 0& 0.0186& -0.0029\\ 0& 0.0051& 0.0024\end{array}\right].$$

(43)

The initial position and heading angle of the USV are ${\left[x,y,\psi \right]}^T={\left[0 {\mathrm{m} 0 \mathrm{m} 0}^{\circ }\right]}^T$, and the position and heading angle of the berth are ${\left[x_d,y_d,{\psi }_d\right]}^T={\left[50 {\mathrm{m} 50 \mathrm{m} 45}^{\circ }\right]}^T$. The controller parameters are $U=5$, $I=3$, $\beta ={\left[1,1,75\right]}^T$, $\epsilon ={\left[100,100,10\right]}^T$, $k={\left[50,50,50\right]}^T$, and $\alpha ={\left[1,1,5\right]}^T$. The parameters of the wave model are $\lambda /L=0.5$, ${\varsigma }_D=0.05 \mathrm{m}$, and $\chi ={90}^{\circ }$. The current relative velocity of the earth is 1 kn, and the direction is ${30}^{\circ }$. The relative velocity of the wind is 10 kn, and the wind direction is $60^{\circ }$.

The simulation experiment is divided into two cases. The first case is a comparative analysis of finite time control (FTC) and conventional sliding mode control (CSMC) techniques in the automatic berthing process of the USV. The second case is a comparative analysis after adding an event-triggering mechanism. Figure 7 and Figure 8 show comparison curves of velocity and trajectory. Finite time control technology improves the convergence speed of the system by using control laws with exponential forms.

From Figure 9, it can be seen that both finite time control and sliding mode control can generate reasonable control force and moment. From the results, it can be seen that the control force and moment calculated by the finite time controller can better ensure the speed of the system.

In

Table 2. Comparison of control performance between FTC and CSMC in Case 1.

Indicators	Finite Time Controller	Conventional Sliding Mode Controller
adjust time	24.6 s	37.8 s
${\overline{e}}_x$	0.75 m	1.21 m
${\overline{e}}_y$	0.52 m	1.07 m
${\overline{e}}_{\phi }$	0.37°	1.96°

, ${\overline{e}}_x={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|e_x\right|}dt$, ${\overline{e}}_y={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|e_y\right|}dt$, and ${\overline{e}}_{\psi }={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|e_{\psi }\right|}dt$. The finite-time controllers have demonstrated significant advantages over traditional control methods. One of the most notable benefits is their superior convergence speed, which allows the system to reach the desired state more rapidly. This characteristic is particularly crucial in applications where time is of the essence, as it enables real-time adjustments and enhances overall performance. Moreover, finite-time controllers are designed to minimize adjustment time, resulting in swift stabilization of the system. This reduces the impact of transient behaviors and ensures that the system behaves as intended without prolonged oscillations or delays. Additionally, the errors across various degrees of freedom are consistently lower when employing finite-time control strategies. This precision in error reduction contributes to enhanced system reliability and accuracy, making finite-time controllers an attractive option in fields demanding high performance and efficiency.

The following are the simulation results of the second case (comparative analysis after adding an event-triggering mechanism), adjusting the wind speed to 17 kn. Figure 10 and Figure 11 are the curves of velocity and trajectory under the disturbances.

As shown in Figure 10 and Figure 11, in the whole process of automatic berthing, the control effect of event-triggering control is basically the same as that of continuous drive control. Due to the particularity of the event-triggering mechanism, the control effect occurs only when it reaches threshold. Incorporating an event-driven mechanism significantly enhances the stability of system state transitions by minimizing the frequency of unnecessary operations. This approach allows for a more efficient response to relevant inputs, resulting in smoother transitions between different states. By effectively filtering out extraneous triggers, the system can focus on pertinent events, which not only conserves resources but also elevates overall performance. Consequently, the addition of this mechanism leads to a reduction in latency and disruption during state changes, fostering a more reliable operational framework. This improvement is particularly beneficial in dynamic environments where rapid and fluid adjustments are crucial for maintaining system integrity and user satisfaction. Thus, the integration of an event-triggering system serves as a pivotal strategy in optimizing the responsiveness and stability of the overall architecture.

Figure 12 shows that the implementation of an event-triggering mechanism significantly enhances the stability of control force and moment in our system. This selective activation leads to smoother transitions and minimizes wear on components, ultimately resulting in a more efficient and reliable operation.

Figure 13 shows the trigger time, that is, the time when the actuator works.

In Figure 13, the amplitude of “1” indicates that the controller has been triggered. From Figure 14, it can be seen, during the entire process of automatic berthing, the finite time controller with the event-triggering mechanism triggered 2713 times, while the control method without this mechanism triggered 5000 times. During the automatic berthing process, due to the low speed and slow response of the ship, the response of the previous control signal is insufficient, and the next control information is obtained. To avoid such meaningless losses, event-triggering mechanisms have practical significance.

For this case, some quantitative indicators have been summarized and presented in

Table 3. Comparison of control performance between FTC and EFTC in Case 2.

Indicators	Finite Time Controller	Finite Time Controller with Event-Triggering Mechanism
${\overline{e}}_x$	0.86 m	0.47 m
${\overline{e}}_y$	0.83 m	0.59 m
${\overline{e}}_{\phi }$	1.17°	0.96°
${\overline{T}}_1$	13.73 KN	14.37 KN
${\overline{T}}_2$	12.81 KN	13.21 KN
${\overline{T}}_3$	102.94 KNm	153.57 KNm
Trigger time	2713	5000

.

In Table 3, ${\overline{T}}_1={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|T_1\right|}dt$, ${\overline{T}}_2={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|T_2\right|}dt$, and ${\overline{T}}_3={\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t\left|T_3\right|}dt$. The event-triggering mechanism has significantly reduced system oscillations, effectively leading to a decrease in the errors across various degrees of freedom. This reduction in triggering rates not only lessens the overall control force but also diminishes the required moment, thereby improving the system’s efficiency.

Based on the above theoretical and simulation analysis, the designed automatic berthing control method can effectively ensure symmetry, with the following three main factors.

(1) Event-triggered control can adaptively adjust control strategies based on the real-time status of the system, thereby achieving more precise control. This is very important for USVs to maintain symmetry during berthing, as it can correct heading and position deviations in real time.

(2) By setting event-triggering conditions reasonably, frequent changes in control signals can be reduced, thereby reducing shaking, maintaining system stability, and improving symmetry during berthing.

(3) The core of finite time control lies in the ability to achieve the expected state of the system within a specified time. This feature is very suitable for berthing, as time constraints can be designed to ensure reaching the ideal symmetrical state in a short period of time.

6. Conclusions

This article focuses on the automatic berthing problem of pod-driven USVs and proposes a finite-time control algorithm based on event triggering. First, a global fast nonsingular terminal sliding-mode finite-time control law is designed. The singularity of terminal sliding mode control is avoided, and the rapidity and anti-disturbance performance of the system increase. This algorithm has better tracking performance, strong robustness, and a smooth control input curve, which can compensate for the uncertainty impact of USVs. Then, to address the issue of actuator loss caused by frequent controller updates, a dynamic threshold-based event-triggering mechanism is designed to reduce the frequency of controller updates and actuator loss. Finally, theoretical analysis and simulation experiments verify that the errors of the USV converge exponentially and effectively ensuring the symmetry of the system. In practical applications, there will be various challenges, especially sensor limitations, real-time computing, and actuator accuracy. Here are some strategies and solutions to address these issues: 1; by combining data from multiple sensors, the overall measurement reliability can be improved through weighted averaging or data fusion algorithms; 2. allocating some computing tasks to edge devices can reduce the computational burden on the main control system and improve response speed; and 3. designing fault-tolerant control strategies to ensure continuous operation through redundant or alternative control systems in the event of actuator abnormalities. In practical applications, attention should be paid to collaborative work in multiple aspects to achieve optimal control effects.