Trajectory Tracking and Docking Control Strategy for Unmanned Surface Vehicles in Water-Based Search and Rescue Missions

Abstract

This paper investigates a global fixed-time control strategy for a search and rescue unmanned surface vehicle (SRUSV) targeting water rescue missions. Firstly, an improved time allocation trajectory generation (ITATG) method is proposed to generate a smooth and continuous desired trajectory, incorporating position, velocity, and acceleration information. Secondly, a fixed-time sideslip angle observer-based adaptive line-of-sight (FTSOALOS) guidance law is designed. This law integrates time-varying look-ahead distances with a fixed-time sideslip angle observer (FTSO) to ensure rapid convergence of positional errors within a fixed timeframe. Additionally, this paper employs a first-order fixed-time disturbance observer (FOFTDO) to accurately estimate external disturbances. To alleviate network pressure and reduce actuator failure rates, a fixed-time event-triggered sliding mode control (FTETSMC) mechanism is developed, ensuring the convergence of tracking errors within a fixed timeframe. Finally, using Lyapunov theory, this paper proves that the entire control system designed possesses consistent global fixed-time stability. Comparative simulation experiments validate the effectiveness and superiority of the FTSOALOS guidance law and the FTETSMC controller.

1. Introduction

In recent years, the continuous improvement in living standards has led to an increase in water-based industrial and recreational activities. This has led to a corresponding annual rise in the incidence of water-related accidents. According to data from the China Maritime Search and Rescue Center, the number of coordinated search and rescue operations from 2015 to 2023 was 1887, 1996, 2053, 1925, 1918, 1745, 1924, and 2112, respectively. During these operations, the number of distressed vessels rescued was 1589, 1766, 1773, 1642, 1578, 1375, 1404, and 1449, respectively. The number of people rescued was 14,636, 15,112, 14,999, 13,242, 14,387, 11,269, 13,198, and 14,298, respectively, thereby significantly escalating the demand for waterborne rescue operations. Currently, water rescue missions predominantly employ water rescue equipment, such as lifeboats and salvage ships, which still require human intervention, resulting in low rescue efficiency and posing significant risks to the safety of rescuers. Consequently, there is an urgent need to develop an unmanned intelligent comprehensive water rescue platform to reduce the operational burden on rescue personnel and improve the survival rate of rescue missions. Equipping unmanned surface vehicles with advanced autonomous operating systems, navigation and communication systems, and search and rescue equipment can create a search and rescue unmanned surface vehicle. The SRUSV can perform extensive and prolonged search and rescue missions in complex and hazardous water conditions, offering high efficiency and maneuverability. When there is no emergency, the SRUSV remains in a cruising state. In the event of an emergency, the SRUSV will navigate to the location of the person in distress along a desired trajectory, pass through preset waypoints, and ultimately arrive at the target rescue point with zero speed to complete the rescue mission. However, current research on the application of USVs in the field of water rescue is relatively scarce. Most studies on USV motion control focus primarily on trajectory tracking, automatic berthing, and dynamic positioning control algorithms.

The trajectory tracking control of the USV is subject to time constraints, requiring the USV to reach the corresponding positions on the desired trajectory within the specified time. However, during the trajectory tracking process, the USVs are highly susceptible to the effects of wind, waves, and ocean currents, which can lead to the generation of sideslip angles and significantly reduce tracking accuracy. In early studies, ref. [1] discussions revolved around its application to USV, but the influence of sideslip was ignored. In [2,3], PLOS and ILOS were subsequently proposed, considering the effect of the sideslip on LOS guidance performance but failing to consider convergence time performance metrics of the guidance system. Based on the IoT framework, ref. [4] proposed a method that utilizes low-cost sensors to accurately estimate the sideslip angle while reducing in-vehicle network usage through an event-triggering mechanism. In [5], Huang et al. designed a novel trajectory tracking guidance law for underactuated USVs based on the LOS (line-of-sight) path-following guidance law. This guidance law took into account the effects of external disturbances such as wind, waves, and currents on the system, effectively enhancing the tracking performance of underactuated USVs. To achieve high-performance trajectory tracking control over a large range of look-ahead distances and vehicle speeds, ref. [6] proposed a time-varying look-ahead distance profile. References [7,8] introduced a finite-time sliding mode controller that achieves trajectory tracking within a finite time frame. However, in recent years, the limitations of finite-time theory—specifically, the dependence of convergence time on the system’s initial state—have led scholars to adopt fixed-time theory, which is independent of initial system conditions. Reference [9] proposed a fixed-time sliding mode control (FTSMC) law by combining a fixed-time extended state observer (FESO) and a fixed-time differentiator. Reference [10] presented a fixed-time prescribed performance control technique to address the challenge of precise trajectory tracking control for unmanned surface vessels in the presence of external time-varying disturbances and input saturation. However, current trajectory tracking control algorithms only enable the USV to continuously navigate at the desired trajectory’s tangential speed. For water rescue missions, it is necessary for the SRUSV to reduce its speed to zero when approaching the location of the person in the water. Clearly, traditional trajectory tracking control algorithms cannot achieve this objective.

Dynamic positioning (DP) refers to the technology by which a vessel uses its own propulsion system to counteract the effects of external disturbances such as wind, waves, and currents, in order to maintain a fixed position on the water surface or to navigate along a desired trajectory with a certain attitude [11]. Early DP systems employed control strategies based on PID linear controllers. In [12], Fang et al. applied a self-tuning PID controller based on neural network theory to adjust the optimal angle, aiming to reduce the random motion of the USV in waves. In recent years, with the rapid development of nonlinear control theory, many nonlinear control methods have gradually been applied to DP systems. To address modeling uncertainties and marine environmental disturbances, ref. [13] presented a composite anti-disturbance control scheme by designing a disturbance observer to estimate disturbances with partially known information and using $H_{\infty }$ robust control to attenuate unknown bounded disturbances. Reference [14] proposed a robust nonlinear model predictive control method to solve time-varying environmental disturbances and input saturation. In [15], a novel robust adaptive finite-time fault-tolerant control (FFC) scheme was presented for the DP of vessels under thruster faults with unknown model parameters and environmental disturbances. Reference [16] focused on ship adaptive dynamic positioning output feedback control, considering the dynamics of the thruster system; this approach addresses the issues of measurement noise, unmeasurable ship speed, unknown time-varying disturbances, model parameter ingestion, and input saturation simultaneously. In order to simultaneously address the issues of ship operating area limitations, unknown time-varying disturbances, immeasurable ship speed, unknown dynamics, and input saturation, ref. [17] investigated position-constrained ship dynamic positioning output feedback control, taking thruster system dynamics into account. Aiming at the difficulties of the dynamic positioning process and trajectory control under the influence of complex disturbances, ref. [18] studied ship dynamic positioning control based on a nonlinear fuzzy algorithm. Reference [19] designed an adaptive backstepping fast terminal sliding mode control (ABFTSMC) to address the issues of low positioning accuracy, slow convergence speed, and poor anti-disturbance performance in vessel positioning caused by dynamic model parameter uncertainties and unknown time-varying environmental disturbances, providing a robust solution for precise vessel positioning. All the aforementioned research achievements in DP control aim to achieve low-speed convergence to the desired position within a relatively small water area. However, they fail to meet the requirement of high-speed movement to the target point and executing berthing rescue operations in vast water areas. Therefore, for surface rescue missions, it is necessary to consider other technologies or methods to compensate for the limitations of DP control.

Current research on automatic berthing control can be broadly classified into three categories: berth-out stabilization, direct berthing, and a combined approach involving berth-out stabilization followed by direct berthing. The process of automatic berthing for USV involves two stages: trajectory planning and tracking control [20]. To ensure the consistency of training data, AHMED et al. in [21] created the concept of a virtual window. They utilized two independent feedforward neural networks to train the rudder angle output and the propeller speed output separately. The ANN was trained for both windless and windy conditions, achieving berth-out stabilization control under gust disturbances. Reference [22] designed a fuzzy controller, employing bow thrusters and tug assistance to achieve automatic parallel berthing, but it did not consider wind and current disturbances. In [23], Maki et al. employed a covariance matrix adaptation evolution strategy to obtain the optimal solution for the automatic berthing problem of underactuated ships. Reference [24] utilized Bezier curves to plan multi-segment berthing paths and designed an automatic berthing system that applies a path-following control algorithm to address the automatic berthing problem of underactuated unmanned ships under wind disturbances. Reference [25] proposed an automatic berthing control scheme for the fully actuated USV, utilizing an A* algorithm with the Bezier curve (AB) approach, and verified the effectiveness and superiority of the proposed scheme by performing comparative simulation experiments in two scenarios. In [26], a trajectory generation scheme that combines Bezier curve planning, least squares fitting, and a speed allocation mechanism was designed to provide a continuous and smooth reference trajectory for a backstepping-based automatic berthing control system. However, automatic berthing technology faces issues, including speed discontinuity during position transitions and the inability to pass through specified waypoints. Moreover, in rescue missions, distressed individuals are typically located in offshore areas rather than nearshore areas. Therefore, a communication mechanism capable of enabling offshore long-distance transmission needs to be proposed. References [4,6,27] introduced an event-triggered method, which can effectively reduce data transmission and network usage. Additionally, it helps to minimize failures caused by the frequent actions of the actuators. Previous researchers on automatic berthing had primarily focused on berthing vessels at nearshore berths, and these two scenarios present significant differences in control algorithm design.

Based on the summary of the three main USV motion control methods mentioned above and inspired by them, this paper proposes a fixed-time event-triggered trajectory tracking control strategy combined with a trajectory generation method [26,28,29,30] to address the docking control problem of SRUSV in the context of water search and rescue missions. The main contributions of the proposed control scheme are summarized as follows:

(1)

This paper proposes an ITATG method based on [28], which outputs a high-order continuously differentiable trajectory considering endpoint velocity and acceleration constraints, thus addressing the issue of discontinuity at waypoints. Compared to the trajectory in [28], the proposed trajectory generation method significantly alleviate the convex hull situation and make the generated trajectory significantly close to multi-segment polylines.

(2)

This paper presents an FTSOALOS guidance law that contains heading guidance and longitudinal velocity guidance. The FTSOALOS integrates FTSO for guaranteed positional error convergence within a fixed time around the origin. Moreover, considering the limitations of a constant forward line-of-sight distance during the actual operation of SRUSV, a variable forward line-of-sight distance strategy that changes with the positional error is designed to expedite the system’s convergence speed further.

(3)

A new FTETSMC strategy is proposed for SRUSV, which integrates a fixed-time sliding mode controller with a relative threshold event-triggering strategy. This strategy aims to alleviate network pressure and reduce actuator failure rates by reducing the number of actuator actions. Additionally, it considers the impact of external disturbances on control accuracy. A FOFTDO is proposed to observe external disturbances. The first-order form of this observer helps alleviate computational pressure on terminal devices.

The structure of the section is as follows: Section 2 covers preliminaries, lemma development, problem formulation, the SRUSV model, and the control objective. Section 3 presents a method for the trajectory generation process using the ITATG. Section 4 addresses the design and stability proof of the fixed-time LOS guidance law. Section 5 presents the design and stability proof of the proposed fixed-time event-triggered control mechanism. Section 6 presents comparative simulations to verify the effectiveness and performance of the above strategy.

2. Preliminaries, Lemma, and Problem Formulation

2.1. Preliminaries and Lemma

Preliminary 1.

Notation $si{g}^{\alpha }\left({\mu }_i\right)$ is defined as $si{g}^{\alpha }\left({\mu }_i\right)=|{\mu }_i{|}^{\alpha }sgn\left({\mu }_i\right),i=1,2,\dots ,n$.

Preliminary 2.

The following nonlinear system is considered:

$$\dot{x}\left(t\right)=f\left(x\left(t\right)\right),x\left(0\right)=0,f\left(0\right)=0$$

(1)

where $x\left(t\right)\in R^n$ represents the state variable of the system, and $f(·)$ is a continuous nonlinear function.

Definition 1

([31]). If system (1) is Lyapunov-stable, and there exists a convergence time $T_s$, which is independent of the system states, when $t>T_s$, $x\left(t\right)=0$ holds, then system (1) is considered fixed-time stability.

Lemma 1.

For system (1), consider the Lyapunov candidate function (LCF), $V\left(x\right(t\left)\right)$ satisfying the following:

$$\dot{V}\left(x\left(t\right)\right)\le -{\delta }_1{V}^{\alpha }\left(x\left(t\right)\right)-{\delta }_2{V}^{\beta }\left(x\left(t\right)\right)$$

(2)

There exist variables ${\delta }_1,{\delta }_2,\alpha \in (0,1),\beta \in (1,+\infty )$, then system (1) is fixed-time stable, where the setting stable time is expressed as follows:

$$T\le T_{max}:={\displaystyle \frac{1}{{\delta }_1(1-\alpha )}}+{\displaystyle \frac{1}{{\delta }_2(\beta -1)}}$$

(3)

Lemma 2

([32]). If there exists ${\delta }_1,{\delta }_2,\alpha ,\theta \in (0,1),\beta \in (1,+\infty ),\vartheta \in (0,+\infty )$, and consider LCF, $V\left(x\right(t\left)\right)$, satisfying the following:

$$\dot{V}\left(x\left(t\right)\right)\le -{\delta }_1{V}^{\alpha }\left(x\left(t\right)\right)-{\delta }_2{V}^{\beta }\left(x\left(t\right)\right)+\vartheta$$

(4)

Then the system is practically fixed-time stable and the setting time is expressed as follows:

$$T\le T_{max}:={\displaystyle \frac{1}{{\delta }_1\theta (1-\alpha )}}+{\displaystyle \frac{1}{{\delta }_2\theta (\beta -1)}}$$

(5)

Lemma 3

([33]). Assume $x_1,x_2>0,0<p<1,q>1$, then we have the following:

$$\left\{\begin{array}{c}{x}_1^p+x_2^p\ge {\left(x_1+x_2\right)}^p\hfill \\ x_1^q+x_2^q\ge 2^{1-q}{\left(x_1+x_2\right)}^q\hfill \end{array}\right.$$

(6)

Lemma 4.

If $\delta =e^{-(\delta +1)}\left(\mathrm{i}.e.,\delta =0.2785\right)$, the inequality $0\le \left|\mu \right|-\mu tanh\left({\displaystyle \frac{\mu }{ϵ}}\right)\le \delta ϵ$ is valid for any $ϵ>0$ and for any $\mu \in \mathrm{R}$.

Lemma 5

([33]). The following system is as follows:

$$\left\{\begin{array}{c}{\dot{z}}_1=z_2-{\xi }_{1}si{g}^{{\alpha }_1}\left(z_1\right)-{\zeta }_{1}si{g}^{{\beta }_1}\left(z_1\right)\hfill \\ \dots \hfill \\ {\dot{z}}_{\iota }=z_{\iota +1}-{\xi }_{\iota }si{g}^{{\alpha }_{\iota }}\left(z_1\right)-{\zeta }_{\iota }si{g}^{{\beta }_{\iota }}\left(z_1\right)\hfill \\ \iota =1,\dots ,n-1\hfill \\ \dots \hfill \\ {\dot{z}}_n=-{\xi }_{n}si{g}^{{\alpha }_n}\left(z_1\right)-{\zeta }_{n}si{g}^{{\beta }_n}\left(z_1\right)\hfill \end{array}\right.$$

(7)

where the exponents ${\alpha }_{\iota }\in (0,1)$, ${\beta }_{\iota }\in (1,+\infty )$, $\iota =1,\dots ,n$, ${\alpha }_{\iota }=\iota \mu -(\iota -1)$, ${\beta }_{\iota }=\iota \nu -(\iota -1)$, $\iota =1,\dots ,n$, $\mu \in (1-\epsilon ,1)$, $\nu \in (1+o,+\infty )$ with extremely small constants $\epsilon >0$, $o>0$. If the parameters $\xi \iota ,\zeta \iota ,\iota =1,\dots ,n$ render the following two matrices, $A_1$, $A_2$, as Hurwitz matrices, then the system will be stable within a fixed time.

$A_1=\left[\begin{array}{ccccc}-{\xi }_1& 1& 0& \cdots & 0\\ -{\xi }_2& 0& 1& \cdots & 0\\ \cdots & & & & \\ -{\xi }_{n-1}& 0& 0& \cdots & 1\\ -{\xi }_n& 0& 0& \cdots & 0\end{array}\right]$  $A_2=\left[\begin{array}{ccccc}-{\zeta }_1& 1& 0& \cdots & 0\\ -{\zeta }_2& 0& 1& \cdots & 0\\ \cdots \\ -{\zeta }_{n-1}& 0& 0& \cdots & 1\\ -{\zeta }_n& 0& 0& \cdots & 0\end{array}\right]$

The system stabilizes within time $T\le {\displaystyle \frac{{\lambda }_{max}^{\rho }\left(P\right)}{r\rho }}+{\displaystyle \frac{1}{r_1\sigma {\gamma }^{\sigma }}}$, where $\rho =1-\alpha$, $\sigma =\beta -1$, $r={\lambda }_{min}\left(Q\right)/{\lambda }_{max}\left(P\right)$, $r_1={\lambda }_{min}\left(Q_1\right)/{\lambda }_{max}\left(P_1\right)$, $\gamma \le {\lambda }_{min}\left(P_1\right)$ is a positive number, and $Q_1\in {\mathbb{R}}^{nxn}$ are symmetric positive definite matrices. The symmetric positive matrix $P_1$ satisfies a Lyapunov equation $P_1{A}_1+A_1^T{P}_1=-Q_1$.

Lemma 6

([34]). If there exists a high-order nonlinear system, then we have the following:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=x_3\hfill \\ \cdots \hfill \\ {\dot{x}}_{n-1}=x_n\hfill \\ {\dot{x}}_n=g\left(x\right)+d+u\hfill \end{array}\right.$$

(8)

System (8) is of the order $n\ge 2$; $g\left(x\right)$ is a smooth and known nonlinear function; $u\in R$ is the control input; the external disturbance d is presented and assumed to have upper bounds of $\left|d\right|\le \kappa$ and $|\dot{d}|\le \overline{\kappa }$, where $\kappa ,\overline{\kappa }>0$ are known constants. If designing the disturbance observer in this way, then we have the following:

$$\left\{\begin{array}{c}\Xi =x_n-\Pi \hfill \\ \Pi =g\left(x\right)+u+\widehat{d}\hfill \\ \widehat{d}=r_{1}si{g}^{(m_1/n_1)}(\Xi )+r_{2}si{g}^{(p_1/q_1)}(\Xi )+r_{3}tanh(\Xi /{\rm Y})\hfill \end{array}\right.$$

(9)

where $r_1,r_2,r_3,{\rm Y}>0$, $m_1/n_1<1$, $p_1/q_1>1$, and the disturbance estimation error $\tilde{d}=d-\widehat{d}$ will converge to a small region surrounding the balance point within a fixed time.

2.2. SRUSV Model

$X_n{O}_n{Y}_n$ represents the north–east–down (NED) coordinate, while $X_b{O}_b{Y}_b$ represents the body-fixed (BF) coordinate. NED and BF will be used in this paper to describe the motion of the SRUSV. The NED frame refers to a tangent plane on the surface of the earth that moves with the SRUSV. This frame is adequate for local operations and is defined by an origin point denoted by $o_n$. The x-axis of the NED frame points toward the true north, while the y-axis points toward the east. The body-fixed frame moves with the SRUSV and is defined by an origin point $o_b$, which coincides with the center of gravity of the SRUSV.

The 3-DOF kinematics and dynamics mathematical model of SRUSV is given by the following:

$$\left\{\begin{array}{c}\dot{\eta }=R\left(\psi \right)\nu \hfill \\ M\dot{\nu }+C\left(\nu \right)\nu +D\left(\nu \right)\nu =\tau +{\tau }_d\hfill \end{array}\right.$$

(10)

where $\nu ={\left[\mathrm{u}\mathrm{v}\mathrm{r}\right]}^T$, represents surge, sway velocity, and yaw rate, $\eta ={\left[xy\psi \right]}^T$ represents the USV position and heading angle under the NED coordinates, $\tau =\left[{\tau }_{u}0{\tau }_r\right]$ in which ${\tau }_u$ is the surge force and ${\tau }_r$ denotes yaw torque. And ${\tau }_d={\left[{\tau }_{du}{\tau }_{d\mathrm{v}}{\tau }_{dr}\right]}^T$ represents the environmental forces. $R\left(\psi \right)$ represents the rotation matrix, M represents the inertia coefficient matrix with added mass. $C\left(\nu \right)$ represents the Coriolis and centripetal matrix, and $D\left(\nu \right)$ represents the damping matrix. Their specific forms are as follows:

$$\begin{array}{ccc}\hfill & R\left(\psi \right)=\left[\begin{array}{ccc}cos\psi & -sin\psi & 0\\ sin\psi & cos\psi & 0\\ 0& 0& 1\end{array}\right]\hfill & \hfill M=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& 0\\ 0& 0& m_{33}\end{array}\right]\\ \hfill & C\left(\nu \right)=\left[\begin{array}{ccc}0& 0& -m_{22}\mathrm{v}\\ 0& 0& m_{11}u\\ m_{22}\mathrm{v}& -m_{11}u& 0\end{array}\right]\hfill & \hfill D\left(\nu \right)=\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& 0\\ 0& 0& d_{33}\end{array}\right]\end{array}$$

(11)

Substituting (11) into (10) and (10) can be expanded as follows:

$$\left\{\begin{array}{c}\dot{x}=ucos\psi -\mathrm{v}sin\psi \hfill \\ \dot{y}=usin\psi +\mathrm{v}cos\psi \hfill \\ \dot{\psi }=r\hfill \end{array}\right.$$

(12)

$$\left\{\begin{array}{c}\dot{u}={\displaystyle \frac{m_{22}}{m_{11}}}\mathrm{v}r-{\displaystyle \frac{d_{11}}{m_{11}}}u+{\displaystyle \frac{{\tau }_u+{\tau }_{du}}{m_{11}}}\hfill \\ \dot{\mathrm{v}}=-{\displaystyle \frac{m_{11}}{m_{22}}}ur-{\displaystyle \frac{d_{22}}{m_{22}}}\mathrm{v}+{\displaystyle \frac{{\tau }_{d\mathrm{v}}}{m_{22}}}\hfill \\ \dot{r}={\displaystyle \frac{m_{11}-m_{22}}{m_{33}}}u\mathrm{v}-{\displaystyle \frac{d_{33}}{m_{33}}}r+{\displaystyle \frac{{\tau }_r+{\tau }_{d\dot{r}}}{m_{33}}}\hfill \end{array}\right.$$

(13)

2.3. Control Objectives

The control objective of this paper is to enable the SRUSV to follow preset waypoints and trajectories, ultimately arriving at the target rescue point with zero speed, while avoiding collisions with surrounding obstacles. Under conditions of external disturbances and unknown sideslip angles, a corresponding trajectory tracking controller for the underactuated SRUSV is designed. This ensures that during the process of guiding the SRUSV along the trajectory to the target point, all errors converge to the smallest neighborhood of the origin within a fixed time. Figure 1 describes the system control diagram.

3. ITATG Trajectory Generation Design

The waypoints are represented as $x=\left(\begin{array}{c}{x}_0{x}_1\cdots x_k\end{array}\right)$, $y=\left(\begin{array}{c}{y}_0{y}_1\cdots y_k\end{array}\right)$, and the trajectory passing through waypoints includes k segments. The n-order polynomial describing k segments of trajectories $p\left(t\right)$ can be expressed as (14), where $i,j\in \{0,1,2,\dots ,n\}$, $g\in \{0,1,2,\dots ,k\}$.

$$p\left(t\right)=\left\{\begin{array}{c}{p}_1\left(t\right)={\displaystyle \sum _{j=0}^n}{B}_1{t}^j{\mathrm{t}}_0\le t<t_1\hfill \\ ⋮\hfill \\ p_2\left(t\right)={\displaystyle \sum _{j=0}^n}{B}_2{t}^j{\mathrm{t}}_{g-1}\le t<t_g\hfill \\ ⋮\hfill \\ p_k\left(t\right)={\displaystyle \sum _{j=0}^n}{B}_k{t}^j{\mathrm{t}}_{k-1}\le t<t_k\hfill \end{array}\right.$$

(14)

where $B_k={\left[b_{k0}{b}_{k1}\dots b_{kn}\right]}^T$ represents the trajectory parameters of the k-th segment. The trajectory function of an n-order polynomial with time as the variable is represented as (15), The first-, second-, third-, and fourth-order derivatives of the trajectory are represented in (16), as follows:

$$y\left(t\right)=b_{k0}+b_{k1}t+b_{k2}{t}^2+\cdots +b_{kn}{t}^n=\sum _{i=0}^n{b}_{ki}{t}^i$$

(15)

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)=\left[\begin{array}{ccccccc}0& 1& 2t& 3t^2& 4t^3& \dots & n{t}^{n-1}\end{array}\right]\times B_k\hfill \\ \hfill \\ y^{″}\left(t\right)=\left[\begin{array}{ccccccc}0& 0& 2& 6t& 12t^2& \dots & n(n-1)t^{n-2}\end{array}\right]\times B_k\hfill \\ \hfill \\ y^{\left(3\right)}\left(t\right)=\left[\begin{array}{ccccccc}0& 0& 0& 6& 24t& \dots & {\displaystyle \frac{n!}{(n-3)!}}{t}^{n-3}\end{array}\right]\times B_k\hfill \\ \hfill \\ y^{\left(4\right)}\left(t\right)=\left[\begin{array}{ccccccc}0& 0& 0& 0& 24& \dots & {\displaystyle \frac{n!}{(n-4)!}}{t}^{n-4}\end{array}\right]\times B_k\hfill \end{array}\right.$$

(16)

where $t_{k-1}\le t<t_k$ represents the time execution interval of the kth segment of the trajectory, and the allocation of time $t_i$ for reaching each desired turning point is assigned according to the following (17):

$$\left\{\begin{array}{c}{T}_i=T{\displaystyle \frac{{\displaystyle \sum _{j=0}^i}\sqrt{{(x_j-x_{j-1})}^2+(y_j-y_{j-1})}}{{\displaystyle \sum _{i=0}^k}\sqrt{{(x_i-x_{i-1})}^2+(y_i-y_{i-1})}}};i=0,1,\dots ,k\hfill \\ \left[t_0{t}_1{t}_2\dots t_k\right]=diag\left(T_0{T}_1{T}_2\dots T_k\right)\times {\lambda }_T\hfill \end{array}\right.$$

(17)

where ${\lambda }_T=[{\lambda }_{T0},{\lambda }_{T1},{\lambda }_{T2},\dots ,{\lambda }_{Tk}]$ denotes the time allocation parameters, T denotes the total expected time from $(x_0,y_0)$ to $(x_k,y_k)$, and $T_i$ denotes the time allocated to reach each waypoint according to the distance.

To ensure continuity of position, velocity, and acceleration at waypoints, the objective cost function based on (16) is expressed as follows:

$$\begin{array}{cc}\hfill & \underset{B_k}{min}{\int }_0^T{∥p^{\left(4\right)}\left(t\right)∥}^{2}dt\hfill \\ \hfill & =min\sum _{i=1}^k{\int }_{t_{i+1}}^{t_i}{([000024\dots {\displaystyle \frac{\mathrm{n}!}{(n-4!)}}{\mathbf{t}}^{n-4}]·B_k)}^T([000024\dots {\displaystyle \frac{\mathrm{n}!}{(n-4!)}}{\mathbf{t}}^{n-4}]·B_k)dt\hfill \\ \hfill & =min\sum _{i=1}^k{B}_k^T{Q}_g{B}_k\hfill \end{array}$$

(18)

where the output of the optimization objective function is $B_k$, and the matrix $Q_g$ denotes the coefficient matrix for the d-th segment of the trajectory; $Q_g$ is expressed as follows:

$$\begin{array}{c}{Q}_g={\int }_{t_{i-1}}^{t_i}{\left[000024\dots {\displaystyle \frac{\mathrm{n}!}{\left(n-4!\right)}}{\mathbf{t}}^{n-4}\right]}^T\left[000024\dots {\displaystyle \frac{\mathrm{n}!}{\left(n-4!\right)}}{\mathbf{t}}^{n-4}\right]dt\end{array}$$

(19)

$\begin{array}{c}\hfill Q=diag(Q_1,Q_2,\cdots ,Q_k)\end{array}$ represents the coefficient matrix. Therefore, the minimum cost function can be equivalently represented as follows: $min{\int }_0^T{\parallel p^{\left(4\right)}\left(t\right)\parallel }^{2}dt=min{B}_k^{T}Q{B}_k$.

Trajectory generation needs to address the issue of trajectory smoothness, which means the position, velocity, and acceleration at waypoints should exhibit continuity. That is, the endpoint of the g-th segment of the trajectory and the starting point of the (g+1)-th segment have the same position, velocity, and acceleration at time $t_d$, which can be described as $p_i^{\left(m\right)}\left(t_i\right)=p_{i+1}^{\left(m\right)}\left(t_i\right)$, $m=1,2,3$. Therefore, the constraint matrices $A_0$ and $A_k$ for the initial and final positions, velocities, and accelerations are shown in (20) and (21). The constraint matrices $A_m$ for the intermediate point positions are illustrated in (22), while the continuity constraint matrices $A_{mc}$ for the intermediate point positions, velocities, and accelerations are depicted in (23), where $A_{mcp}=[1t_i{t}_i^2\dots n{t}_i^n-1-t_i-t_i^2$ … $-n{t}_i^n]$, $A_{mc\mathrm{v}}=[012t_i\dots n{t}_i^{n-1}0-1-2t_i\dots -n{t}_i^{n-1}]$, $A_{mca}=[002\dots {\displaystyle \frac{n!}{\left(n-2\right)!}}{t}_i^{n-2}00-2\dots -{\displaystyle \frac{n!}{\left(n-2\right)!}}{t}_i^{n-2}]$.

$$A_0=\left[\begin{array}{c}1{\mathit{t}}_0{\mathit{t}}_0^2\dots {\mathit{t}}_0^n\underset{(k-1)(n+1)}{\underbrace{\mathbf{0}\dots \mathbf{0}}}\\ 012{\mathit{t}}_0\dots \mathit{n}{\mathit{t}}_0^{n-1}\underset{(k-1)(n+1)}{\underbrace{\mathbf{0}\dots \mathbf{0}}}\\ 002\dots \mathit{n}(n-1){\mathit{t}}_0^{n-2}\underset{(k-1)(n+1)}{\underbrace{\mathbf{0}\dots \mathbf{0}}}\end{array}\right]$$

(20)

$$A_k=\left[\begin{array}{c}\underset{(k-1)(n+1)}{\underbrace{0\dots 0}}1t_k{t}_k^2\dots t_k^n\\ \underset{(k-1)(n+1)}{\underbrace{0\dots 0}}012t_k\dots n{t}_k^{n-1}\\ \underset{(k-1)(n+1)}{\underbrace{0\dots 0}}002\dots n(n-1)t_k^{n-2}\end{array}\right]$$

(21)

$$A_m=\left[\begin{array}{c}\underset{3}{\underbrace{0\dots 0}}1t_1{t}_1^2\dots t_1^n\underset{6}{\underbrace{0\dots 0}}\\ \dots \\ \underset{3}{\underbrace{0\dots 0}}1t_{k-1}{t}_{k-1}^2\dots t_{k-1}^n\underset{6}{\underbrace{0\dots 0}}\end{array}\right]$$

(22)

$$A_{mc}=\left[\begin{array}{c}\underset{(i-1)(n+1)}{\underbrace{0\dots 0}},A_{mcp},\underset{(k-i-1)(n+1)}{\underbrace{0\dots 0}}\\ \\ \underset{(i-1)(n+1)}{\underbrace{0\dots 0}},A_{mc\mathrm{v}},\underset{(k-i-1)(n+1)}{\underbrace{0\dots 0}}\\ \\ \underset{(i-1)(n+1)}{\underbrace{0\dots 0}},A_{mca},\underset{(k-i-1)(n+1)}{\underbrace{0\dots 0}}\end{array}\right].$$

(23)

The equality constraint is as follows: $A_{eq}p=d_{eq}$, when calculating the polynomial equation in the X-direction, $d_{eq}={[x_0{U}_0{a}_0\dots x_i\dots x_k{U}_k{a}_{k}0\dots 0]}^T$, when calculating the polynomial equation in the Y-direction, ${\left.d_{eq}=\left[\begin{array}{c}{y}_0{U}_0{a}_0\dots y_i\dots y_k{U}_k{a}_{k}0\dots ,0\end{array}\right.\right]}^T$; where $U_k$ represents the specified initial and final velocities, $a_0,a_k$ represents the initial and final set accelerations, $A_{eq}={\left[A_0{A}_m{A}_k{A}_{mc}\right]}^{\mathrm{T}}$. Finally, the trajectory optimization problem can be described as follows:

$$\begin{array}{cc}\hfill & \underset{B_k}{min}{p}^{T}Qp\hfill \\ \hfill \mathrm{s}.& \mathrm{t}.A_{eq}{B}_k=d_{eq}\hfill \end{array}$$

(24)

4. Fixed-Time LOS Guidance Law Design

4.1. Positional Error Dynamics

Consider a USV at the position $(x,y)$ moving with the following speed:

$$U=\sqrt{u^2+{\mathrm{v}}^2}$$

(25)

${\alpha }_k$ represents the trajectory’s tangent angle as follows:

$${\alpha }_k=atan2(y_d^{\prime }\left(t\right),x_d^{\prime }\left(t\right))\in \left[\begin{array}{cc}-\pi & \pi \end{array}\right]$$

(26)

$U_p$ in (27) represents the virtual input velocity, indicating the tangential velocity of the desired trajectory at a specific point.

$$U_p=\sqrt{{\dot{x}}_d^2+{\dot{y}}_d^2}$$

(27)

$\beta$ in (28) represents the sideslip angle, which is usually caused by external disturbances and changes slowly over time. Therefore, it can be assumed that $\dot{\beta }\approx 0$. Additionally, since the sideslip angle is very small, it can be approximated as $cos\left(\beta \right)\approx 0$, $sin\left(\beta \right)\approx \beta$.

$$\beta =atan2(\mathrm{v},\mathrm{u})$$

(28)

We define the trajectory errors as follows:

$$\left[\begin{array}{c}{x}_e\\ y_e\end{array}\right]={\left[\begin{array}{cc}cos{\alpha }_k& -sin{\alpha }_k\\ sin{\alpha }_k& cos{\alpha }_k\end{array}\right]}^T\left[\begin{array}{c}x-x_d\left(t\right)\\ y-y_d\left(t\right)\end{array}\right]$$

(29)

4.2. FTSO Design

Considering expensive sensors, unavoidable measurement noise, and sway velocity, obtaining the value of $\beta$ requires a high cost. In order to estimate $\beta$, in summary, the FTSO is designed to estimate the sideslip angle as follows:

$$\left\{\begin{array}{c}{\dot{\widehat{y}}}_e=Usin(\phi -{\alpha }_k)-{\dot{\alpha }}_k+\widehat{h}-{\theta }_{1}si{g}^{{\nu }_1}\left({\tilde{y}}_e\right)-{\theta }_{2}si{g}^{{\nu }_2}\left({\tilde{y}}_e\right)\hfill \\ \dot{\widehat{h}}=-{\theta }_{3}si{g}^{{\nu }_3}\left({\tilde{y}}_e\right)-{\theta }_{4}si{g}^{{\nu }_4}\left({\tilde{y}}_e\right)-\Lambda sgn\left({\tilde{y}}_e\right)\hfill \end{array}\right.$$

(30)

where ${\tilde{y}}_e:={\widehat{y}}_e-y_e$, ${\tilde{h}}_e:={\widehat{y}}_e-y_e$, so the estimated sideslip angle denotes $\widehat{\beta }=\widehat{h}/Ucos\left(e_{\psi }\right)$. By substituting the derivatives of $x_e$ and $y_e$ into (30), then, the following formula can be obtained:

$$\left\{\begin{array}{c}{\dot{\tilde{y}}}_e=\tilde{h}-{\theta }_{1}si{g}^{{\nu }_1}\left({\tilde{y}}_e\right)-{\theta }_{2}si{g}^{{\nu }_2}\left({\tilde{y}}_e\right)\hfill \\ \dot{\tilde{h}}=-{\theta }_{3}si{g}^{{\nu }_3}\left({\tilde{y}}_e\right)-{\theta }_{4}si{g}^{{\nu }_4}\left({\tilde{y}}_e\right)\hfill \end{array}\right.$$

(31)

$\Lambda sgn\left({\tilde{y}}_e\right)$ is a robust term added to FTSO to enhance robustness. According to Lemma 5, the setting time $T_1$ satisfies $T_1\le {\displaystyle \frac{{\lambda }_{max}^{\rho }\left(P\right)}{r\rho }}+{\displaystyle \frac{1}{r_1\sigma {\gamma }^{\sigma }}}$.

4.3. FTSOALOS Guidance Law Design

Figure 2 shows a diagram of SRUSV following a desired curved path, where the time-varying looking-ahead distance $f(\Delta )$ is defined as follows:

$$\mathrm{f}(\Delta )=\Delta +{\mathrm{k}}_{\Delta }tanh\left(\mathrm{s}\mathrm{i}{g}^m\left({\mathrm{x}}_e\right)+\mathrm{s}\mathrm{i}{g}^n\left({\mathrm{y}}_e\right)\right)$$

(32)

The desired heading angle and longitudinal velocity are designed as follows:

$$\left\{\begin{array}{c}{\psi }_d={\alpha }_k+arctan\left({\displaystyle \frac{f\left(y_e\right)}{f(\Delta )}}\right)-\widehat{\beta }\hfill \\ \\ u_d=\sqrt{U_d^2-{\mathrm{v}}^2}\hfill \end{array}\right.$$

(33)

where $U_d={\displaystyle \frac{\left(U_p-k_{1}si{g}^m\left(x_e\right)-k_{2}si{g}^n\left(x_e\right)\right)\sqrt{f^2\left({\gamma }_e\right)+f^2\left(\Delta \right)}}{f(\Delta )}}$, $f\left(y_e\right)=-k_{3}si{g}^m\left(y_e\right)-k_{4}si{g}^n\left(y_e\right)$, $k_1, k_2, k_3, k_4>0$ and satisfies $U_p>k_{1}si{g}^m\left(x_e\right)+k_{2}si{g}^n\left(x_e\right)$, $0<m<1$ and $n>1$. According to [9], the trajectory errors reach zero within a fixed time.

5. FTETSMC Design

5.1. FOFTDO Design

Depending on (11), it can be rewritten as follows:

$$\left\{\begin{array}{c}\dot{\eta }=\xi \hfill \\ \dot{\xi }=RS\nu -R{M}^{-1}D\left(\nu \right)\nu -R{M}^{-1}C\left(\nu \right)v+R{M}^{-1}\tau +d\hfill \end{array}\right.$$

(34)

where $\xi =R\nu$, $d=R{M}^{-1}{\tau }_d$.

An auxiliary error variable is designed as follows: $z=\xi -\widehat{\xi }$, where the designed variable $\widehat{\xi }$ is developed by the following:

$$\begin{array}{cc}\hfill \dot{\widehat{\xi }}& =RS\nu -R{M}^{-1}D\left(\nu \right)\nu -R{M}^{-1}C\left(\nu \right)\nu +R{M}^{-1}\tau +r_1{\mathrm{sig}}^{k_{\mathrm{o}}1}\left(z\right)+r_2{\mathrm{sig}}^{k_{\mathrm{o}}2}\left(z\right)+r_{3}tanh\left({\displaystyle \frac{z}{{\mu }_o}}\right)\hfill \end{array}$$

(35)

where $0<k_{o_1}<1$, $k_{o_2}>1$, $r_1,r_2,r_3>0$, $r_3\ge \kappa$, according to Lemma 6, $\left|d\right|\le \kappa$, $\kappa$ is an unknown constant. FOFTDO is designed as follows:

$$\widehat{d}=r_{1}si{g}^{k_{o_1}}\left(z\right)+r_{2}si{g}^{k_{o_2}}\left(z\right)+r_{3}tanh\left({\displaystyle \frac{z}{{\mu }_o}}\right)$$

(36)

Taking the derivative of z and substituting it into (34) and (35) yields the following:

$$\begin{array}{cc}\hfill \dot{z}& =\dot{\xi }-\dot{\widehat{\xi }}\hfill \\ \hfill & =d-r_{1}si{g}^{k_{o_1}}\left(z\right)+r_{2}si{g}^{k_{o_2}}\left(z\right)+r_{3}tanh\left({\displaystyle \frac{z}{{\mu }_o}}\right)\hfill \\ \hfill & =d-\widehat{d}\hfill \end{array}$$

(37)

According to [33], the auxiliary error variable z reaches zero within a fixed time.

According to Lemma 2, there exists ${\theta }_0\in (0,1)$, such that $V_2$ will decrease to a residual set for all $t>T_3$, $T_3={\displaystyle \frac{1}{r_1{\theta }_0{2}^{\frac{k_{o_1}+1}{2}}(1-\frac{k_{o_1}+1}{2})}}+{\displaystyle \frac{1}{r_2{\theta }_0{2}^{\frac{k_{o_2}+1}{2}}(\frac{k_{o_2}+1}{2}-1)}}$.

The expression $\tilde{d}=\dot{\xi }-\dot{\widehat{\xi }}=\dot{Z}$ denotes the measurement error of the disturbance observer. According to Lemma 6, assuming that the measurement error has an upper bound $\overline{\kappa }$, then $\tilde{d}$ will converge to a small neighborhood around the origin when $t>T_3$. $\overline{\kappa }$ can be determined by $\tilde{d}=\dot{Z}\le |\kappa -r_{1}si{g}^{k_{o1}}\left(0.2785r_3{\mu }_o\right)+r_{2}si{g}^{k_{o2}}\left(0.2785r_3{\mu }_o\right)+r_{3}tanh\left(0.2785r_3\right)|$.

5.2. Fixed-Time Intermediate Sliding Mode Controller Design

The surge velocity error variable and heading error variable are defined as follows:

$$\left\{\begin{array}{cc}\hfill u_e& =u-u_d\hfill \\ \hfill {\psi }_e& =\psi -{\psi }_d\hfill \end{array}\right.$$

(38)

The sliding mode surfaces are designed as follows:

$$\left\{\begin{array}{cc}\hfill S_u=& u_e\left(t\right)+{\lambda }_1{\int }_0^t{u}_e\left(\tau \right)d\tau \hfill \\ \hfill S_{\psi }=& \dot{{\psi }_e}\left(t\right)+{\lambda }_2{\psi }_e\left(t\right)+{\lambda }_3{\int }_0^t{\psi }_e\left(\tau \right)d\tau \hfill \end{array}\right.$$

(39)

where ${\lambda }_1,{\lambda }_2,{\lambda }_3>0$ is a design constant. The equivalent control law can be obtained as follows:

$$\left\{\begin{array}{cc}\hfill {\tau }_{ueq}& =m_{11}\left[-f_u\left(u,\mathrm{v},\mathrm{r}\right)-{\widehat{d}}_u+{\dot{u}}_d-{\lambda }_1{u}_e\right]\hfill \\ \hfill {\tau }_{req}& =m_{33}\left[-f_r\left(u,\mathrm{v},\mathrm{r}\right)-{\widehat{d}}_r+{\ddot{\psi }}_d-{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)-{\lambda }_3{\psi }_e\right]\hfill \end{array}\right.$$

(40)

The convergence laws of the derivative of (39) is as follows:

$$\left\{\begin{array}{cc}\hfill {\tau }_{\iota \nu ss}& =-{\alpha }_{1}si{g}^a\left(S_u\right)-{\alpha }_{2}si{g}^b\left(S_u\right)\hfill \\ \hfill {\tau }_{r\nu ss}& =-{\alpha }_{3}si{g}^p\left(S_{\psi }\right)-{\alpha }_{4}si{g}^q\left(S_{\psi }\right)\hfill \end{array}\right.$$

(41)

where ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4>0$, $0<\mathrm{a}<1$, $\mathrm{b}>1$, $0<p<1$, $\mathrm{q}>1$ are design constants.

And the intermediate sliding mode controller can be obtained as follows:

$$\begin{array}{cc}\hfill {\omega }_u\left(t\right)& =m_{11}\left[-{\alpha }_{1}si{g}^a\left(S_u\right)-{\alpha }_{2}si{g}^b\left(S_u\right)\right.-f_u\left(u,\mathrm{v},\mathrm{r}\right)-{\widehat{d}}_u+{\dot{u}}_d-{\lambda }_1{u}_e]\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\omega }_r\left(t\right)=m_{33}& \left[-{\alpha }_{3}si{g}^p\left(S_{\phi }\right)-{\alpha }_{4}si{g}^q\left(S_{\phi }\right)\right.-f_r\left(u,\mathrm{v},\mathrm{r}\right)\left.-{\widehat{d}}_r+{\ddot{\psi }}_d-{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)-{\lambda }_s{\psi }_e\right]\hfill \end{array}$$

(43)

5.3. Relative Threshold Event-Triggered Controller Design

We define the controller execution measurement error as follows:

$$\left\{\begin{array}{cc}\hfill e_u\left(t\right)& =u_u\left(t\right)-{\tau }_u\left(t\right)\hfill \\ \hfill e_r\left(t\right)& =u_r\left(t\right)-{\tau }_r\left(t\right)\hfill \end{array}\right.$$

(44)

where $t\in [t_k,t_{k+1})$, and ${\tau }_u\left(t_{\mathrm{k}}\right)$, ${\tau }_r\left(t_{\mathrm{k}}\right)$ is the value of the controller at the previous triggering instant. ${\tau }_u\left(t_{\mathrm{k}}\right)$, ${\tau }_r\left(t_{\mathrm{k}}\right)$ start from the previous triggering instant $t_k$ and will be held constant until the next triggering instant $t_{\mathrm{k}+1}$ to update.

We design relative threshold event-triggered control laws based on (44):

$$\left\{\begin{array}{cc}\hfill u_u\left(t\right)& =-\left(1+{\rho }_u\right)\left[{\omega }_{u}tanh\left({\displaystyle \frac{S_u{\omega }_u}{{\mu }_u}}\right)+{\overline{l}}_{u}tanh\left({\displaystyle \frac{S_u{\overline{l}}_u}{{\mu }_u}}\right)\right]\hfill \\ \hfill u_r\left(t\right)& =-\left(1+{\rho }_r\right)\left[{\omega }_{r}tanh\left({\displaystyle \frac{S_{\psi }{\omega }_r}{{\mu }_r}}\right)+{\overline{l}}_{r}tanh\left({\displaystyle \frac{S_{\psi }{\overline{l}}_r}{{\mu }_r}}\right)\right]\hfill \end{array}\right.$$

(45)

We define the event-triggering condition as follows:

$$\left\{\begin{array}{c}{\tau }_u\left(t\right)=u_u\left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right)\hfill \\ t_{k+1}=inf\left\{t\in R\left.∥e_u\left(t\right)\right|\ge {\rho }_u\left|{\tau }_u\left(t\right)\right|+l_u\right\}\hfill \end{array}\right.$$

(46)

$$\left\{\begin{array}{c}{\tau }_r\left(t\right)=u_r\left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right)\hfill \\ t_{k+1}=inf\left\{t\in R\left|\left|e_r\left(t\right)\right|\ge {\rho }_r\left|{\tau }_r\left(t\right)\right|+l_r\right.\right\}\hfill \end{array}\right.$$

(47)

where $t_{\mathrm{k}}$ is the update time of the controller. $l_{u,}{l}_r>0$, $0<{\rho }_r<1$, $0<{\rho }_u<1$, ${\overline{l}}_u>{\displaystyle \frac{l_u}{1-{\rho }_u}}$, ${\overline{l}}_r>{\displaystyle \frac{l_r}{1-{\rho }_r}}$, ${\mu }_u,{\mu }_r>0$ are design parameters.

When the above event trigger signal is triggered, the controller update time is denoted as $t_{\mathrm{k}+1}$, ${\tau }_u\left(t_{\mathrm{k}+1}\right)$, and ${\tau }_r\left(t_{\mathrm{k}+1}\right)$ will be used as control inputs. When the system’s operating time is $t\in [t_k,t_{k+1})$, the control signal will be held constant at a value of ${\tau }_u\left(t_{\mathrm{k}}\right)$, ${\tau }_r\left(t_{\mathrm{k}}\right)$.

For any $t\in [t_k,t_{k+1})$, according to (46) and (47), the following formula can be obtained: $\left|u_u\left(t\right)-{\tau }_u\left(t\right)\right|\le {\rho }_u\left|{\tau }_u\left(t\right)\right|+l_u$, $\left|u_r\left(t\right)-{\tau }_r\left(t\right)\right|\le {\rho }_r\left|{\tau }_r\left(t\right)\right|+l_r$. There exists a continuous time-varying parameter ${\theta }_{u1}$, ${\theta }_{u2}$, ${\theta }_{r1}$, ${\theta }_{r2}$, which satisfies the following: $|{\theta }_{u1}|,|{\theta }_{u2}|,|{\theta }_{r1}|,|{\theta }_{r2}|\le 1$. Based on $u_u\left(t\right)=(1+{\theta }_{u1}\left(t\right){\rho }_u){\tau }_u\left(t\right)+{\theta }_{u2}\left(t\right)l_u$ and $u_r\left(t\right)=(1+{\theta }_{r1}\left(t\right){\rho }_r){\tau }_r\left(t\right)+{\theta }_{r2}\left(t\right)l_r$, it can be obtained that:

$${\tau }_u\left(t\right)={\displaystyle \frac{u_u\left(t\right)}{1+{\theta }_{u1}\left(t\right){\rho }_u}}-{\displaystyle \frac{{\theta }_{u2}\left(t\right)l_u}{1+{\theta }_{u1}\left(t\right){\rho }_u}}$$

(48)

$${\tau }_r\left(t\right)={\displaystyle \frac{u_r\left(t\right)}{1+{\theta }_{r1}\left(t\right){\rho }_r}}-{\displaystyle \frac{{\theta }_{r2}\left(t\right)l_r}{1+{\theta }_{r1}\left(t\right){\rho }_r}}$$

(49)

Proof. 

Considering the following LFC:

$$V_3={\displaystyle \frac{1}{2}}{S}_u^2+{\displaystyle \frac{1}{2}}{S}_{\psi }^2$$

(50)

Taking the derivative of $V_3$, and substituting the derivatives of (39), (48), and (49), we have the following:

$$\begin{array}{cc}\hfill {\dot{V}}_3& =S_{\psi }{\dot{S}}_{\psi }+S_u{\dot{S}}_u\hfill \\ \hfill & =S_u\left[f_u\left(u,\mathrm{v},\mathrm{r}\right)+{\displaystyle \frac{1}{m_{11}}}\left({\displaystyle \frac{u_u\left(t\right)}{1+{\theta }_{u1}\left(t\right){\rho }_u}}-{\displaystyle \frac{{\theta }_{u2}\left(t\right)l_u}{1+{\theta }_{u1}\left(t\right){\rho }_u}}\right)\right.+{\widehat{d}}_u-{\dot{u}}_d+{\lambda }_1{u}_e]\hfill \\ \hfill & +S_{\psi }\left[f_r\left(u,\mathrm{v},\mathrm{r}\right)+{\displaystyle \frac{1}{m_{33}}}\left({\displaystyle \frac{u_r\left(t\right)}{1+{\theta }_{r1}\left(t\right){\rho }_r}}-{\displaystyle \frac{{\theta }_{r2}\left(t\right)l_r}{1+{\theta }_{r1}\left(t\right){\rho }_r}}\right)\right.+{\widehat{d}}_r-{\ddot{\psi }}_d+{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)+{\lambda }_3{\psi }_e]\hfill \end{array}$$

(51)

Since (53) holds, by substituting (42), (43), and (53) into (51), and according to Lemma 4, we can obtain (52).

$$\begin{array}{cc}\hfill {\dot{V}}_3\le & S_u{f}_u\left(u,\mathrm{v},\mathrm{r}\right)+S_u{\widehat{d}}_u-S_u{\dot{u}}_d+S_u{\lambda }_1{u}_e-{\displaystyle \frac{1}{m_{11}}}[{\displaystyle \frac{S_u}{1+{\rho }_u}}(1+{\rho }_u)({\omega }_{u}tanh\left({\displaystyle \frac{S_u{\omega }_u}{{\mu }_u}}\right)+{\overline{l}}_{u}tanh\left({\displaystyle \frac{S_u{\overline{l}}_u}{{\mu }_u}}\right))\hfill \\ \hfill & -\left|{\displaystyle \frac{S_u{l}_u}{1+{\rho }_u}}\right|]+S_{\psi }{f}_r\left(u,\mathrm{v},\mathrm{r}\right)+S_{\psi }{\widehat{d}}_r-S_{\psi }{\ddot{\psi }}_d+S_{\psi }{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)+S_{\psi }{\lambda }_3{\psi }_e-{\displaystyle \frac{1}{m_{33}}}[{\displaystyle \frac{S_{\psi }}{1+{\rho }_r}}\left(1+{\rho }_r\right)({\omega }_{r}tanh\left({\displaystyle \frac{S_{\psi }{\omega }_r}{{\mu }_r}}\right)\hfill \\ \hfill & +{\overline{l}}_{r}tanh\left({\displaystyle \frac{S_{\psi }{\overline{l}}_r}{{\mu }_r}}\right))-\left|{\displaystyle \frac{S_{\psi }{l}_r}{1+{\rho }_r}}\right|]\hfill \\ \hfill \le & S_u{f}_u\left(u,\mathrm{v},\mathrm{r}\right)+S_u{\widehat{d}}_u-S_u{\dot{u}}_d+S_u{\lambda }_1{u}_e-{\displaystyle \frac{1}{m_{11}}}0.557{\mu }_u-S_u(-{\alpha }_{1}si{g}^a\left(S_u\right)-{\alpha }_{2}si{g}^b\left(S_u\right)-f_u\left(u,\mathrm{v},\mathrm{r}\right)\hfill \\ \hfill & -{\widehat{d}}_u+{\dot{u}}_d-{\lambda }_1{u}_e)+S_{\psi }{f}_r\left(u,\mathrm{v},\mathrm{r}\right)+S_{\psi }{\widehat{d}}_r-S_{\psi }{\ddot{\psi }}_d+S_{\psi }{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)+S_{\psi }{\lambda }_3{\psi }_e-{\displaystyle \frac{1}{m_{33}}}0.557{\mu }_2\hfill \\ \hfill & -S_{\psi }(-{\alpha }_{3}si{g}^p\left(S_{\psi }\right)-{\alpha }_{4}si{g}^q\left(S_{\psi }\right)-f_r\left(u,\mathrm{v},\mathrm{r}\right)-{\widehat{d}}_r+{\ddot{\psi }}_d-{\lambda }_2\left(\dot{\psi }-{\dot{\psi }}_d\right)-{\lambda }_3{\psi }_e)\hfill \\ \hfill =& -S_u{\alpha }_{1}si{g}^p\left(S_u\right)-S_u{\alpha }_{2}si{g}^q\left(S_u\right)-S_{\psi }{\alpha }_{3}si{g}^p\left(S_{\psi }\right)-S_{\psi }{\alpha }_{4}si{g}^q\left(S_{\psi }\right)-{\displaystyle \frac{1}{m_{11}}}0.557{\mu }_u-{\displaystyle \frac{1}{m_{33}}}0.557{\mu }_r\hfill \\ \hfill =& -{\alpha }_1{\left|S_u\right|}^{p+1}-{\alpha }_2{\left|S_u\right|}^{q+1}-{\alpha }_3{\left|S_{\psi }\right|}^{p+1}-{\alpha }_4{\left|S_{\psi }\right|}^{q+1}+\Xi \hfill \\ \hfill \le & -{\eta }_1{({\displaystyle \frac{1}{2}}{\left|S_u\right|}^2+{\displaystyle \frac{1}{2}}{\left|S_{\psi }\right|}^2)}^{{\displaystyle \frac{p+1}{2}}}-2^{{\displaystyle \frac{1-q}{2}}}{\eta }_2({\displaystyle \frac{1}{2}}{\left|S_u\right|}^{q+1}+{\displaystyle \frac{1}{2}}{\left|S_{\psi }\right|}^{q+1}{)}^{{\displaystyle \frac{q+1}{2}}}+\Xi \hfill \\ \hfill =& -{\eta }_1{V}_3^{{\displaystyle \frac{p+1}{2}}}-2^{{\displaystyle \frac{1-q}{2}}}{\eta }_2{V}_3^{{\displaystyle \frac{q+1}{2}}}+\Xi \hfill \end{array}$$

(52)

$$\left\{\begin{array}{c}{\displaystyle \frac{S_u{u}_u\left(t\right)}{1+{\theta }_{u1}\left(t\right){\rho }_u}}\le {\displaystyle \frac{S_u{u}_u\left(t\right)}{1+{\rho }_u}}\hfill \\ \left|-{\displaystyle \frac{S_u{\theta }_{u2}\left(t\right)l_u}{1+{\theta }_{u1}\left(t\right){\rho }_u}}\right|\le \left|{\displaystyle \frac{S_u{l}_u}{1+{\rho }_u}}\right|\hfill \\ {\displaystyle \frac{S_{\psi }{u}_r\left(t\right)}{1+{\theta }_{r1}\left(t\right){\rho }_r}}\le {\displaystyle \frac{S_{\psi }{u}_r\left(t\right)}{1+{\rho }_r}}\hfill \\ \left|-{\displaystyle \frac{S_{\psi }{\theta }_{r2}\left(t\right)l_r}{1+{\theta }_{r1}\left(t\right){\rho }_r}}\right|\le \left|{\displaystyle \frac{S_{\psi }{l}_r}{1+{\rho }_r}}\right|\hfill \end{array}\right.$$

(53)

where $\Xi ={\displaystyle \frac{1}{m_{11}}}0.557{\mu }_u+{\displaystyle \frac{1}{m_{33}}}0.557{\mu }_r$, ${\eta }_1=min({\alpha }_1,{\alpha }_3)$, ${\eta }_2=min({\alpha }_2,{\alpha }_3)$. According to Lemma 2, there exists a constant ${\theta }_1\in (0,1)$, $V_3$ will decrease to a residual set for all $t>T_4$, where $T_4:={\displaystyle \frac{1}{{\eta }_1{\theta }_1(1-\frac{1-p}{2})}}+{\displaystyle \frac{1}{{\eta }_2{\theta }_1(1-\frac{1-q}{2})}}$. □

6. Simulation Studies

redThe algorithm in this paper was implemented and executed using MATLAB-Simulink 2023a. The simulation experiments were conducted on a system with an Intel(R) Core(TM) i7-6820HQ processor running at 2.71 GHz and 32.0 GB of RAM. Both the sampling frequency of the observer in the simulation software and the control frequency of the controller were set to 100 Hz.

In this paper, the “Lan Xin” USV depicted in Figure 1 is selected as the SRUSV model for the simulation study. Based on the maritime environment, the waypoints set for SRUSV are $x=\left[2550130180200200220330\right]$, $y=\left[10100150200280320330330\right]$, the initial point velocity ${\mathrm{v}}_0=0$, the initial point acceleration $a_0=0$, the endpoint velocity ${\mathrm{v}}_7=0$, endpoint acceleration $a_7=0$, and total time $T=400$, ${\lambda }_{T_0}=1$, ${\lambda }_{T1}=1$, ${\lambda }_{T2}=0.95$, ${\lambda }_{T3}={\lambda }_{T4}={\lambda }_{T5}={\lambda }_{T6}=0.7$, ${\lambda }_{T7}=0.8$. The inertia parameters of the ‘Lan Xin’ USV are ${\mathrm{m}}_{11}=2652.52$ kg, $m_{22}=2825.57$ kg, $m_{33}=4201.68$ kg, $d_{11}=848.13$ kg/s, $d_{22}$ = 10,162.44 kg/s, $d_{33}$ = 22,721.63 kg/s. Initial states are $x\left(0\right)=70$, $y\left(0\right)=55$, $\psi \left(0\right)=0$, $u\left(0\right)=0$, $\mathrm{v}\left(0\right)=0$, $r\left(0\right)=0$.

The external disturbances are as follows:

$$\left[\begin{array}{c}{\tau }_{du}\\ {\tau }_{d\mathrm{v}}\\ {\tau }_{dr}\end{array}\right]=\left[\begin{array}{c}1000+500sin\left(0.02t\right)+100sin\left(0.1t\right)\\ 1000+500sin(0.02t-\pi /6)+100sin\left(0.3t\right)\\ 1000+500sin\left(0.05t\right)+100sin\left(0.1t\right)\end{array}\right]$$

(54)

To verify the superiority of the control mechanism proposed in this paper, IAE and ITAE indicators are employed to quantify the tracking error, $e_i\left(i=x,y\right)$, as expressed in (55), where t represents the simulation time:

$$\left\{\begin{array}{cc}\hfill IAE& ={\int }_0^t|e_i\mid d\tau \hfill \\ \hfill ITAE& ={\int }_0^{t}t\mid e_i\mid d\tau \hfill \end{array}\right.$$

(55)

To further demonstrate, the designed fixed-time event-triggered trajectory tracking control method is compared with the FTSMC + FTLOS [16] and SMC + LOS methods. Moreover, the controllers follow the desired trajectory provided by ITATG.

Table 1. Initial states and control parameters.

Name	Parameters
FTSO	${\theta }_1={\theta }_2=20$, ${\theta }_3={\theta }_4=0.001$,
	${\upsilon }_1={\upsilon }_3=0.8$, ${\upsilon }_2={\upsilon }_4=1.2$, $\Lambda =-0.01$
FTSOALOS	$\Delta =14$, $k_{\Delta }=8$, $k_1=k_2=0.13$,
	$k_3=1.7$, $k_4=1.2$, $m=0.8$, $n=1.2$
FOFTDO	$r_1=r_2=r_3=10$, $k_{o1}=0.9$, $k_{o2}=20$, ${\mu }_o=5$
FTETSMC	${\lambda }_1=1$, ${\lambda }_2={\lambda }_3=3$, ${\alpha }_1={\alpha }_2=0.5$,
	${\alpha }_3={\alpha }_4=50$, $a=p=0.8$, $b=q=1.2$
	${\beta }_1=0.001$, ${\beta }_2=0.01$, ${\mu }_r$ = 20,000,
	${\theta }_{r1}={\theta }_{u1}=0$, ${\theta }_{r2}={\theta }_{u2}=1$,
	$l_r=l_u=12$, ${\overline{l}}_r={\overline{l}}_u=13$

presents the designed control parameters. From

Table 2. Performance index of LOS and control mechanisms.

Method	IAE	ITAE
FTETSMC + FTSOALOS	IAE($x_e$) = 26.8	ITAE($x_e$) = 428
	IAE($y_e$) = 35.21	ITAE($y_e$) = 976.7
FTSMC + FTLOS	IAE($x_e$) = 71.8	ITAE($x_e$) = 2920
	IAE($y_e$) = 141.9	ITAE($y_e$) = 1306
SMC + LOS	IAE($x_e$) = 90.2	ITAE($x_e$) = 3259
	IAE($y_e$) = 160	ITAE($y_e$) = 2994

, the IAE and ITAE of FTETSMC are smaller than those of FTSMC and SMC, demonstrating that the proposed control method exhibits superior tracking performance and robustness. Figure 3 shows the effectiveness of three different control methods for USV-tracking trajectories generated by ITATG. Although all methods can track the desired trajectory, the FTETSMC + FTSOALOS method has higher tracking accuracy. During the initial stage of the trajectory, FTSOALOS guidance law exhibits a faster convergence response speed, which means the proposed control method has a better transient response. Additionally, at the rescue location at the end of the trajectory, it is evident that the SRUSV, under the FTETSMC control algorithm, can approach the rescue point with zero velocity more accurately. Figure 4 illustrates the positional error $x_e$ and $y_e$ under three different control methods. We can observe that FTETSMC can converge both track errors ($x_e$,$y_e$) into arbitrary small neighborhoods around zero in fixed time, faster than FTSMC and SMC, with smaller overshoot and oscillation. Figure 5 represents the number of updates of FTETSMC. It can be observed that the FTETSMC effectively reduces the number of controller updates.

Figure 6 illustrates the trend of time-varying forward line-of-sight $\mathrm{f}\left(\Delta \right)$ as designed in (33), which changes reasonably with varying $x_e$ and $y_e$, promoting rapid convergence of positional errors. In Figure 7, both actual external disturbances and estimated external disturbances are depicted. It can be observed that the FOFTDO is capable of efficiently approximating external disturbances, and the estimation error converges within the minimum neighborhood of zero with a sufficiently small response time without obvious oscillation. Figure 8 shows the control input. It can be seen that the FTETSMC algorithm will maintain the control input of the previous moment until the triggering condition is met, under the action of the event-triggering mechanism. From Figure 9, it can be observed that the designed FTSO can timely and accurately observe the sideslip angle with a minimal initial state oscillation. Figure 10 shows $\psi$ and u tracking effects under the action of the proposed control method. It can be observed that the proposed control method enables the rapid convergence of surge velocity and heading errors in a fixed time.

7. Conclusions

In the context of water search and rescue, this paper proposes an ITATG trajectory generation strategy for the control system of SRUSV. This strategy not only resolves this problem but also effectively alleviates the convex hull issue inherent to the original trajectory generation strategy. Consequently, it prevents overlaps between trajectories and obstacles. In order to converge the position of the SRUSV to the desired path generated by the ITATG method, a global fixed-time trajectory tracking control strategy is proposed. This strategy ensures that all errors within the closed-loop control system converge to a small neighborhood around the origin within a fixed-time interval. Firstly, a combined FTSOALOS guidance law incorporating FTSO is proposed. The designed FTSO ensures that the estimated sideslip angle error converges within a fixed-time interval, providing real-time desired heading angle and forward velocity for the SRUSV in the ship’s heading direction. The variable forward line-of-sight facilitates faster and smoother convergence of positional errors, building upon the principles outlined in previous research. Next, FTETSMC is introduced, which—by reducing the number of controller updates—decreases the actuator failure rate and network pressure, with errors converging to a smaller neighborhood near the balance point in a fixed time. Additionally, it facilitates the convergence of heading and forward velocity errors within a fixed-time interval. The FOFTDO ensures that estimated disturbance errors converge within a fixed-time interval while minimizing controller resource utilization. Simulation results indicate that when tracking trajectories generated by ITATG, the FTETSMC + FTSOALOS strategy outperforms the FTSMC + FTLOS and SMC + LOS control strategies. Specifically, the integral of absolute error (IAE) is reduced by 70.1%, and the integral of the time-weighted absolute error (ITAE) is reduced by 66.7%, indicating a higher level of control accuracy.