Smooth Sliding Mode Control for Path Following of Underactuated Surface Vehicles Based on LOS Guidance

Abstract

In this paper, a solution to the problem of following a curved path for underactuated unmanned surface vehicles (USVs) with unknown sideslip angle and model uncertainties is studied. A novel smooth sliding mode control (SSMC) based on a finite-time extended state observer (FTESO) for heading control is proposed. Firstly, the model of a USV with rudderless double thrusters is established. Secondly, the path-following error dynamics of a USV is established in a path-tangential reference frame. Thirdly, a finite-time observer is introduced to estimate the unidentified sideslip angle, and the line-of-sight (LOS) guidance law is applied to produce the desired heading angle. Finally, an SSMC controller is proposed to force USV tracking at the desired heading angle and surge speed, in which FTESO is used to estimate and compensate the unknown disturbance in sliding mode dynamics. The theoretical analysis for FTESO-SSMC verifies that the controller can provide finite-time convergence to and stability on the sliding surface. Simulation studies and contrast test are conducted to demonstrate the robustness and rapidity of the proposed FTESO-SSMC controller.

1. Introduction

Unmanned surface vehicles (USVs) provide unique capabilities for military and security applications, including harbor patrol, maritime interdiction, and riverine operations [1]. Therefore, motion control of such autonomous vehicles is considered an important area within the marine control research community. The problems related to motion control of USVs can be classified into three basic groups, such as objective tracking, path following, and trajectory tracking. In the literature, among most of the scenarios including stabilization, trajectory tracking, and path following, it is of practical importance to follow a predefined path [2]. For the path-following problem, the vehicle needs to converge to and follow a predefined path without any temporal constraints. Compared with the trajectory tracking and target tracking, the path following controller can provide a smoother trajectory for USVs to move to and follow the desired path so that it is generally not possible to saturate the actuators of USVs [3]. Therefore, it is extremely important for a ship to perform tasks at sea, such as maritime search, resource exploration, and nautical charting.

Conventional USVs are commonly equipped with one central propeller for surge speed control and one rudder for heading control [4]. However, the mechanical structure of a combination of thrusters and rudders is sophisticated, and the rudders can be damaged easily due to frequent steering processes in curved path-following tasks [5]. To overcome this problem, the structure of double propellers without a rudder is used. In this paper, the curved path-following problem for an underactuated USV with rudderless double thrusters is investigated. The model uncertainties and unknown exterior disturbance are also considered in this paper. The demand for underactuated USVs to move on a desired path under severe unknown disturbances and model uncertainties creates heavy challenges for robust controller design. For the path following, there are two main research directions: advanced guidance strategies and advanced control methods. They are introduced as follows.

For the guidance system, a widely used method for path following is the line-of-sight (LOS) approach. The main advantages of a LOS guidance law are simplicity and a small computational footprint [4]. However, the conventional LOS guidance has limitations when the vehicle is under the unknown drift forces that are generated by ocean waves, ocean wind, ocean currents, or other exterior disturbances. In [6], the sideslip angle was measured by accelerometers. However, the sensor is expensive and the measured data may be noisy and distorted. Therefore, plenty of research work has been carried out to improve LOS. The methods to improve LOS guidance can be divided into two types. The first type is based on adaptive law to compensate unknown sideslip angle [7,8,9,10,11,12]. The second type is based on an estimator to estimate and compensate the sideslip angle. A multifarious sideslip observer was proposed to strengthen the robustness of the path-following controller in [13,14,15,16,17,18,19]. In [13], a predictor-based LOS (PLOS) was developed for the estimation of vehicle sideslip under the assumption of small sideslip angle. In [14], the proposed observer could estimate large sideslip angle, and its estimation error was asymptotically convergent. However, it will be singular when the difference between heading angle and path tangential angle is $\pi /2$. In [16], a finite-time predictor-based LOS (FPLOS) guidance law was presented for the problem of path following. Compared to PLOS, the FPLOS can make sideslip estimation error convergent to zeros in a finite time and speed up the convergence process. In [19], a fixed-time predictor-based LOS (FTPLOS) was designed to ensure effective convergence of tracking sideslip angle. In this paper, the FPLOS is introduced to estimate the unknown sideslip angle and produce desired heading angle for the control system.

For the control system, lots of methods, such as PID with extended Kalman filtering techniques [20,21], adaptive disturbance rejection control (ADRC) [22], model predictive control (MPC) [23,24], deep reinforcement learning control [25,26], and sliding mode control (SMC) [27,28,29], are used to force USVs to follow the desired heading angle produced by the LOS guidance law. Compared to other control methods, SMC has attracted a significant interest due to its simplicity, high robustness to external disturbances, and low sensitivity to the system parameter variations [30]. In [26,27], a policy based on adaptive sliding-mode control was proposed for the path-following control of USVs. In [28], sliding-mode dynamic surface and adaptive techniques were employed to compensate for the uncertainty from parameters varying and constant bias caused by the exterior disturbances. In [29], a disturbance-observer-based sliding mode control was designed to reach good tracking performance, where the observer was designed to estimate and compensate for the modeling uncertainties and external disturbance. In the above SMC techniques, a high frequent switch function is often used to obtain a high robustness. In [31,32], a finite-time extended state observer (FTESO) is proposed to estimate system state and disturbances, and FTESO is verified to be effective in disturbances estimation. Different from the above methods, in this paper, the FTESO and SMC are combined to the design controller to realize the heading and surge control, which is called the smooth sliding mode controller (SSMC). The unknown term in the sliding manifold differential equation is taken as total disturbance, and extended state observer (ESO) is applied to estimate and compensate for the disturbance, then, a second-order control law is designed to satisfy the existence condition of the sliding mode.

The main contributions of this paper are given as follows.

(1)

In the kinematic task, FPLOS is introduced to calculate the desired heading angle with sideslip angle compensation. Compared to the traditional LOS guidance law, the FPLOS can improve the performance of path-following control of USVs under the unknown ocean currents. The finite-time sideslip angle observer can make estimation error of sideslip angle convergent to zeros in finite time and speed up the convergence process.

(2)

In the kinetic task, a novel smooth second-order control law is proposed to satisfy the existence condition of the sliding mode. For the disturbance in sliding mode, FTESO is applied to estimate and compensate for it instead of using a strong discontinuous control signal, which will cause a strong chattering phenomenon.

The rest of this paper is organized as follows. Section 2 formulates preliminaries and the path-following problem of an USV under unknown sideslip angle. Section 3 derives the LOS guidance law based on sideslip angle observer. Then, the SSMC algorithm is designed for surge and heading control in kinetics in Section 4. The theoretical analysis verifies that the controller is semi-globally asymptotically stable in Section 5. Simulation studies are conducted in Section 6. Section 7 concludes this paper.

2. Problem Formulation and Preliminaries

2.1. USV Model

2.1.1. USV Model

To describe the motion of the USV, the North-East-Down coordinate system {$XI$-$OI$-$YI$} and body coordinate system {$YB$-$OB$-$YB$} are used in this paper, as shown in Figure 1.

In Figure 1, $u$ and $v$ are surge and yaw velocity, $U$ is the total speed, $\beta$ is the sideslip angle, which is unknown, $\psi$ is the heading angle of the USV, $u_c$ and $v_c$ are velocities of ocean currents, and $V_c$ is the total speed.

The vehicle’s model introduced in this subsection is similar to that given in [33]. This model can be used to describe an autonomous surface vehicle or an autonomous underwater vehicle moving in the plane. Under the coordinated system coordinate system {$XI$-$OI$-$YI$} and {$XB$-$OB$-$YB$}, the dynamics of the USV can be described as follows:

$$\begin{gathered} \dot{\mathit{\eta }}=\mathit{R}\left(\psi \right)\mathit{v} \\ \mathit{M}\dot{\mathit{v}}=-\mathit{C}\left(\mathit{v}\right){\mathit{v}}_{\mathit{r}}-\mathit{D}{\mathit{v}}_{\mathit{r}}+\mathit{H}\mathit{\tau }+{\mathit{\tau }}_{\mathit{w}} \end{gathered}$$

(1)

where $\mathit{\eta }={\left[x,y,\psi \right]}^{\mathrm{T}}$ describes the position and the orientation of the vehicle with respect to the inertial frame {I}, ${\mathit{\nu }}_{\mathit{r}}={\left[u-u_c,v-v_c,r\right]}^{\mathrm{T}}$ contains the surge, the sway, and the yaw velocities under current disturbance $u_c,v_c$, respectively (see Figure 1); M > 0 is the inertia matrix, $\mathit{C}\left(\mathit{v}\right)$ is the total Coriolis and centripetal acceleration matrix, D is the linear hydrodynamic damping matrix, $\mathit{\tau }=[{\tau }_u,{\tau }_n]$ is the control input produced by two propellers, ${\mathit{\tau }}_{\mathit{w}}=[{\tau }_{w1},{\tau }_{w2},{\tau }_{w3}]$ is environmental disturbance, and the matrix $\mathit{R}\left(\psi \right)$, M, $\mathit{C}\left(\mathit{v}\right)$, D, H is given by:

$$\begin{gathered} \mathit{R}\left(\psi \right)=\left[\begin{array}{ccc}\cos \left(\psi \right)& -\sin \left(\psi \right)& 0\\ \sin \left(\psi \right)& \cos \left(\psi \right)& 0\\ 0& 0& 1\end{array}\right], \mathit{M}=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{32}& m_{33}\end{array}\right], \\ \mathit{C}\left(\mathit{v}\right)=\left[\begin{array}{ccc}0& 0& c_{13}\\ 0& 0& c_{23}\\ -c_{13}& -c_{23}& 0\end{array}\right], \mathit{D}=\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& 0\\ 0& 0& d_{33}\end{array}\right], \mathit{H}=\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 1\end{array}\right] \end{gathered}$$

(2)

where $m_{11}=m-X_{\dot{u}}$, $m_{22}=m-Y_{\dot{v}}$, $m_{23}=m{x}_g-Y_{\dot{r}}$, $m_{32}=m{x}_g-N_{\dot{v}}$, $m_{33}=m-N_{\dot{r}}$, $c_{13}=-m_{22}v-m_{23}r$, $c_{23}=m_{11}u$. Here, $m$ is total mass of the USV, $X_{\dot{u}}$, $Y_{\dot{v}}$, $Y_{\dot{r}}$, $N_{\dot{v}}$, $N_{\dot{r}}$ are the added masses due to hydrodynamics, and $x_g$ is the $XB$-coordinate of the vehicle center of gravity. $d_{11}$, $d_{22}$, $d_{33}$ denote the damping terms. It is worth noting that the linear damping matrix in (2) is reasonable for low-speed motion. Furthermore, since $\mathit{\tau }\in {\mathbb{R}}^2$, namely, the vehicle is underactuated in the workspace ${\mathbb{R}}^3$.

The model uncertainties are expressed by $\mathit{\zeta }$, i.e., $\mathit{\zeta }=-{\mathit{M}}^{-1}[\mathit{C}\left(\mathit{v}\right){\mathit{v}}_{\mathit{r}}-\mathit{D}{\mathit{v}}_{\mathit{r}}+{\mathit{\tau }}_{\mathit{w}}]={\left[{\zeta }_u,{\zeta }_v,{\zeta }_r\right]}^T$, then, (1) can be rewritten as:

$$\dot{\mathit{v}}=\mathit{\zeta }+{\mathit{M}}^{-1}\mathit{H}\mathit{\tau }$$

(3)

Substituting (2) into (1), the dynamic model (1) can be also expressed by component style as follows:

$$\begin{gathered} \dot{x}=ucos\left(\psi \right)-vsin\left(\psi \right) \\ \dot{y}=usin\left(\psi \right)+vcos\left(\psi \right) \\ \dot{\psi }=r \\ \dot{u}={\zeta }_u+\frac{{\tau }_u}{m_{11}} \\ \dot{v}={\zeta }_v \\ \dot{r}={\zeta }_r+\frac{{\tau }_n}{I_z} \end{gathered}$$

(4)

where $I_z=\frac{m_{22}}{m_{22}{m}_{33}-m_{23}{m}_{32}}$.

To facilitate the design and analysis of the control system, the following assumptions are taken.

Assumption 1.

The ocean current in the inertial frame ${\mathit{V}}_{\mathit{c}}={\left[{\mathit{u}}_{\mathit{c}},{\mathit{v}}_{\mathit{c}}\right]}^{\mathit{T}}$ is constant, irrotational, and bounded, and there exists a constant ${\mathit{V}}_{\mathit{m}\mathit{a}\mathit{x}}>\mathbf{0}$ such that $\sqrt{{\mathit{u}}_{\mathit{C}}^2+{\mathit{v}}_{\mathit{c}}^2}\le {\mathit{V}}_{\mathit{m}\mathit{a}\mathit{x}}$ [34]. The environmental disturbance ${\mathit{\tau }}_{\mathit{w}}$ and its derivative ${\dot{\mathit{\tau }}}_{\mathit{w}}$ are bounded.

Remark 1.

Assumption 1 assures that the vehicle has enough thrust to resist the negative effect from ocean currents and environmental disturbances.

Assumption 2.

For the unknown term  ${\zeta }_i$, $i=u,v,r$ are bounded and differentiable, i.e., there exists a positive constant ${\zeta }_i^{*}$ satisfying ${|\zeta }_i|\le {\zeta }_i^{*}$, $\left|{\dot{\zeta }}_i\right|\le {{\dot{\zeta }}^{*}}_i$.

Remark 2.

Note that  $\mathit{\zeta }=-{\mathit{M}}^{-1}\left[\mathit{C}\left(\mathit{v}\right){\mathit{v}}_{\mathit{r}}-\mathit{D}{\mathit{v}}_{\mathit{r}}+{\mathit{\tau }}_{\mathit{w}}\right]$, the velocity $\mathit{v}$, the derivative of $\mathit{v}$, and control input $\mathit{\tau }$ are all bounded due to the physical limits, and, according to Assumption 1, ${\mathit{v}}_{\mathit{r}}$ and ${\mathit{\tau }}_{\mathit{w}}$ are also bounded, so $\mathit{\zeta }$ and $\dot{\mathit{\zeta }}$ are also bounded.

2.1.2. Propeller Model

A catamaran USV is driven by two propellers with shaft speed $n_1$ and $n_2$ in rad/s. The thrusters generate surge force ${\tau }_u$ and yaw moment ${\tau }_n$ given as follows [30].

$$\left[\begin{array}{c}{\tau }_u\\ {\tau }_n\end{array}\right]=\left[\begin{array}{cc}1& 1\\ l_1& -l_1\end{array}\right]\left[\begin{array}{c}{T}_1\\ T_2\end{array}\right]$$

(5)

where $l_1$ is the transverse distance from the center line of the USV to the center line of each propeller and T₁ and T₂ are the thrust force produced by left and right propeller, respectively. The relation between $T_i$ and $n_i$ (i = 1, 2) can be written as follows

$$\begin{gathered} T_i=k_{pos}{n}_i\left|n_i\right|,\hspace{1em}\mathrm{if}{n}_i\ge 0 \\ {T}_i=k_{neg}{n}_i\left|n_i\right|,\hspace{1em}\mathrm{if}{n}_i<0 \end{gathered}$$

(6)

where $k_{pos}$ and $k_{neg}$ represent the scale coefficient of thrust; $n_i$ ($n_{min}\le n_i\le n_{max},n_{min}<0,n_{max}>0$) is the thrust shaft propeller revolving speed (rad/s).

2.2. Path-Following Problem

The curved path-following problem for an underactuated USV can be decomposed into two parts as follows [35].

(1)

Geometric Task: Force the vehicle position to converge to a desired path by appropriate LOS guidance and path parameter update law. The LOS guidance law produces a desired heading angle.

(2)

Dynamic Task: Force the vehicle to track the desired heading angle and desired forward speed by appropriate dynamic controller.

The above problems will be solved in Section 3 and Section 4 separately. The following part establishes the error dynamics of path following. The schematic diagram of curved path following is shown in Figure 2.

As illustrated in Figure 2, the path P is parameterized with a path variable $\theta$. Moreover, for each virtual point on the given path, $(x_p\left(\theta \right),y_p\left(\theta \right))\in P$, a path tangential frame {XP-OP-YP} is constructed to describe the position of the vessel, as illustrated in Figure 2. Hence, the path-following errors can be expressed by the coordinates of the USV in the frame {XP-OP-YP} denoted by ${\mathit{P}}_{b|p}={[x_{b|p},y_{b|p}]}^T$, which are calculated by:

$$\left[\begin{array}{c}{x}_{b|p}\\ y_{b|p}\end{array}\right]=\left[\begin{array}{cc}\cos \left({\psi }_p\right)& \sin \left({\psi }_p\right)\\ -\sin \left({\psi }_p\right)& \cos \left({\psi }_p\right)\end{array}\right]\left[\begin{array}{c}{x-x}_p\left(\theta \right)\\ y-y_p\left(\theta \right)\end{array}\right]$$

(7)

where ${\psi }_p$ is the angle of the path at the point $(x_p\left(\theta \right),y_p\left(\theta \right))\in P$ with respect to the inertial XI-axis, ${\psi }_p=\arctan (y_p^{\prime }\left(\theta \right)/x_p^{\prime }\left(\theta \right))$. The error dynamic can be calculated by substituting (4) in the derivative of (7), which is given by:

$$\begin{gathered} {\dot{x}}_{b|p}=u\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)-v\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)+\left(k_c{y}_{b|p}-1\right)u_p \\ {\dot{y}}_{b|p}=u\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_{\mathrm{e}}\right)+v\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)-k_c{u}_p{\mathrm{x}}_{b|p} \end{gathered}$$

(8)

To facilitate the guidance law and controller design, Equation (8) can be rewritten as follows.

$$\begin{gathered} {\dot{x}}_{b|p}=U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)\cos \left(\beta \right)-U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\sin \left(\beta \right)+(k_c{y}_{b|p}-1)u_p \\ {\dot{y}}_{b|p}=U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\cos \left(\beta \right)+U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)\sin \left(\beta \right)-k_c{u}_p{x}_{b|p} \end{gathered}$$

(9)

where $\beta =\arctan (v/u)$ is sideslip angle; $U=\sqrt{u^2+v^2}$ is course speed; $k_c$ is curvature of the path at point $(x_p\left(\theta \right),y_p\left(\theta \right))$; ${\psi }_e=\psi -{\psi }_p$; $u_p$ is the speed of the virtual point on the desired path, which is calculated as:

$$u_p=\dot{\theta }\sqrt{{\left(x_p^{\prime }\left(\theta \right)\right)}^2+{\left(y_p^{\prime }\left(\theta \right)\right)}^2}$$

(10)

2.3. Preliminaries

Lemma 1 ([36]).

For$\left(x,y\right)\in R^2$, the following Young’s inequality holds:

$$xy\le \frac{{\epsilon }^p}{p}{\left|x\right|}^p+\frac{1}{p{\epsilon }^q}{\left|y\right|}^q$$

(11)

where $\epsilon$  is a positive constant. The constants p and q should satisfy the conditions as p > 1, q > 1, and (p − 1)(q − 1) = 1.

Lemma 2 ([37]).

For  $\forall x_i\in \mathbb{R}$, $i=\mathrm{1,2},\dots ,n$ and $0<q\le 1$, then

$${\left(\sum _{i=1}^n\left|x_i\right|\right)}^q\le \sum _{i=1}^n{\left|x_i\right|}^q\le n^{1-q}{\left(\sum _{i=1}^n\left|x_i\right|\right)}^q$$

(12)

Lemma 3 ([38]).

Consider the system of differential equations $\dot{x}=f\left(x\right)$,  $f\left(0\right)=0$,  $x\in {\mathbb{R}}^n$, where  $f\left(·\right):{\mathbb{R}}^n\to {\mathbb{R}}^n$  is a continuous function. Suppose that there exists a continuous function  $V\left(x\right):U\to \mathbb{R}$  such that the following conditions hold:

(i) V is positive definite.

(ii) There exist real numbers  $c>0$  and  $\alpha \in \left(\mathrm{0,1}\right)$, and an open neighborhood  $U_0$  of the origin such that:

$$\dot{V}\left(x\right)\le -{c\left(V\left(x\right)\right)}^{\alpha },xϵU_0\backslash \left\{0\right\}$$

(13)

Then, the origin is a finite-time-stable equilibrium, and T is the settling-time function, then

$$T\left(x\right)\le \frac{1}{c\left(1-\alpha \right)}{V\left(x\right)}^{\alpha }$$

(14)

Lemma 4 ([39]).

Suppose that there is a positive definite continuous Lyapunov function  $V(x,t)$ defined on  $U_1\subset {\mathbb{R}}^n$  of the origin, and

$$V\left(x,t\right)\le -c_1{V}^{\alpha }\left(x,t\right)+c_{2}V(x,t),\forall x\in U_1\left\{0\right\}$$

(15)

where  $c_1,c_2>0$  and  $0<\alpha <1$  . Thus, the origin of system  $\dot{x}=f\left(x\right)$  is locally finite-time stable. The set  $U_2=\left\{x\right|V^{1-\alpha }\left(x,t\right)\le c_1/c_2\}$  is contained in the domain of attraction of the origin. The settling time satisfies

$$T\left(x\right)\le \mathrm{l}\mathrm{n}(1-(\frac{c_2}{c_1})V^{1-\alpha }\left(x_0,t_0\right))/(c_2\alpha -c_2)$$

(16)

for the given initial condition $x\left(t_0\right)=\{U_1\cap U_2\}$.

3. Guidance System Design

The path-following control system can be divided into two parts: guidance system and control system. In this section, the guidance system will be designed to produce the desired heading angle to lead the USV to follow the desired path. To show the relation of the guidance system and control system, the block diagram of the whole control system is given as Figure 3.

In Figure 3, the blue dashed box shows the guidance system and the red dashed box shows the control system. For the guidance system, the LOS guidance law with sideslip angle observer is designed to generate the desired heading angle for the USV and update the law for the path parameter. In addition, the unknown sideslip angle is estimated by the designed observer. For the control system, the heading controller and surge controller are designed for the USV to track the desired heading angle and desired surge speed based on SSMC and FTESO. The guidance system is designed in Section 3, and the control system will be designed in Section 4. The guidance system will be designed in two steps. For the first step, the sideslip angle observer is presented to estimate sideslip angle, and the finite-time stability of the observer is analyzed, which is given in Section 3.1. For the second step, the LOS guidance law is designed to produce the desired heading angle and the desired update law based on the estimated sideslip angle, which is given in Section 3.2.

3.1. Sideslip Angle Observer

In practice, the sideslip angle of the USV is not more than $20°$ for a USV with double thrusters. If the sideslip angle is small enough, then there are $\cos \left(\beta \right)\approx 1$ and $\sin \left(\beta \right)\approx \beta$. In addition, according to [16], the derivative of sideslip angle equals to zero, i.e., $\dot{\beta }=0$. Although the sideslip angle is relatively small (typically less than $20°$), it largely affects the path-following properties of the vehicle, and if it is not properly compensated, this results in significant deviations from the desired path. Therefore, a finite-time observer is given in this subsection to estimate sideslip angle.

Due to the small sideslip angle, the tracking error dynamics (9) can be rewritten as follows:

$$\begin{gathered} {\dot{x}}_{b|p}=U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)-U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\beta +\left(k_c{y}_{b|p}-1\right)u_p \\ {\dot{y}}_{b|p}=U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)+U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)\beta -k_c{u}_p{x}_{b|p} \end{gathered}$$

(17)

Then, the sideslip angle observer can be designed as:

$$\begin{gathered} {\dot{\widehat{x}}}_{b|p}=U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)-U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\widehat{\beta }+\left(k_c{\widehat{y}}_{b|p}-1\right)u_p-k_{\widehat{x}}si{g}^{\rho }\left({\tilde{x}}_{b|p}\right) \\ {\dot{\widehat{y}}}_{b|p}=U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)+U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)\widehat{\beta }-k_c{u}_p{\widehat{x}}_{b|p}{-k}_{\widehat{y}}si{g}^{\rho }\left({\tilde{y}}_{b|p}\right) \\ \dot{\widehat{\beta }}=k_{\beta }(U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right){\tilde{x}}_{b|p}-U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right){\tilde{y}}_{b|p}) \end{gathered}$$

(18)

where ${\tilde{x}}_{b|p}={\widehat{x}}_{b|p}-x_{b|p}$, ${\tilde{y}}_{b|p}={\widehat{y}}_{b|p}-y_{b|p}$, $\mathrm{s}\mathrm{i}{\mathrm{g}}^{\rho }\left(\ast \right)={\left|\ast \right|}^{\rho }\mathrm{s}\mathrm{g}\mathrm{n}(\ast )$, and $\mathrm{s}\mathrm{g}\mathrm{n}\left(\mathrm{\ast }\right)$ is the sign function. Define the sideslip angle estimate error as $\tilde{\beta }=\widehat{\beta }-\beta$, as mentioned above, there is $\dot{\tilde{\beta }}=\dot{\widehat{\beta }}$, then the estimation error dynamics of (19) can be rewritten as:

$$\begin{gathered} {\dot{\tilde{x}}}_{b|p}=-U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\tilde{\beta }+\left(k_c{y}_{b|p}-1\right)u_p-k_{\widehat{x}}{\mathrm{s}\mathrm{i}\mathrm{g}}^{\rho }\left({\tilde{x}}_{b|p}\right) \\ {\dot{\tilde{y}}}_{b|p}=U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)\tilde{\beta }-k_c{u}_p{\tilde{x}}_{b|p}{-k}_{\widehat{y}}\mathrm{s}\mathrm{i}{\mathrm{g}}^{\rho }\left({\tilde{y}}_{b|p}\right) \\ \dot{\tilde{\beta }}=k_{\widehat{\beta }}\left(U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right){\tilde{x}}_{b|p}-U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right){\tilde{y}}_{b|p}\right) \end{gathered}$$

(19)

where the variables have the constraints as $U_{min}\le U\le U_{max}$, $\left|{\psi }_e\right|\le \pi$. $U_{min}$ and $U_{max}$ are the speed limits of the ship. $u_p$ is a designed virtual point of the path, which is also bounded.

Theorem 1.

Consider the sideslip angle observer (18), the unknown term  $\beta$ can be identified very well, and the estimate error  $\tilde{\beta }$,  ${\tilde{x}}_{b|p}$, and  ${\tilde{y}}_{b|p}$  in (19) asymptotically converge to zeros within finite time.

Proof. 

Consider the positive definite and radially unbounded Lyapunov function candidate

$$V_1=\frac{1}{2}{{\tilde{x}}^2}_{b|p}+\frac{1}{2}{{\tilde{y}}^2}_{b|p}+\frac{1}{2k_{\widehat{\beta }}}{\tilde{\beta }}^2$$

(20)

According to error dynamics (19), the time differentiation of $V_1$ can be obtained as follows:

$${\dot{V}}_1=-k_{\widehat{x}}{\left|{\tilde{x}}_{b|p}\right|}^{\rho +1}{-k}_{\widehat{y}}{\left|{\tilde{y}}_{b|p}\right|}^{\rho +1}$$

(21)

The system described in Equation (19) is an autonomous system, so by applying the LaSalle theory, the set $\{{\tilde{x}}_{b|p},{\tilde{y}}_{b|p},\tilde{\beta }|V_1=0\}$ consists of the axis ${\tilde{x}}_{b|p}=0,{\tilde{y}}_{b|p}=0$, and only the invariant set inside ${\tilde{x}}_{b|p}=0,{\tilde{y}}_{b|p}=0$ is the origin ${\tilde{x}}_{b|p}={\tilde{y}}_{b|p}=\tilde{\beta }=0$. Thus, the asymptotic convergence of ${\tilde{x}}_{b|p},{\tilde{y}}_{b|p},\tilde{\beta }$ to zero is assured, i.e., there is $\left|\tilde{\beta }\right|\le {\mathsf{ϵ}}_{\mathsf{\beta }},{ϵ}_{\beta }>0$. Then, (21) can be rewritten as follows:

$$\begin{array}{cc}{\dot{V}}_1& =-k_{\widehat{x}}{\left|{\tilde{x}}_{b|p}\right|}^{\rho 1+1}{-k}_{\widehat{y}}{\left|{\tilde{y}}_{b|p}\right|}^{\rho 1+1}{-\left(\frac{1}{k_{\widehat{\beta }}}\right)}^{\frac{\rho +1}{2}}{\left|\tilde{\beta }\right|}^{\rho +1}+{\left(\frac{1}{k_{\widehat{\beta }}}\right)}^{\frac{\rho +1}{2}}{\left|\tilde{\beta }\right|}^{\rho +1}\\ & \le -k_1\left(\begin{array}{c}{\left|{\tilde{x}}_{b|p}\right|}^{\rho +1}+{\left|{\tilde{y}}_{b|p}\right|}^{\rho +1}\\ +{\left(\frac{1}{k_{\widehat{\beta }}}\right)}^{\frac{\rho +1}{2}}{\left|\tilde{\beta }\right|}^{\rho +1}\end{array}\right)+\gamma (\tilde{\beta })\end{array}$$

(22)

where $k_1=\min (k_{\widehat{x}},k_{\widehat{y}},1)$, $\gamma \left(\tilde{\beta }\right)={\left(\frac{1}{k_{\widehat{\beta }}}\right)}^{\frac{\rho +1}{2}}{\left|\tilde{\beta }\right|}^{\rho +1}$.

Using the inequality in Lemma 2, ${\dot{V}}_1$ can be calculated as:

$$\begin{array}{cc}{\dot{V}}_1& \le -{2^{\frac{\rho +1}{2}}{k}_1\left({\frac{1}{2}\left|{\tilde{x}}_{b|p}\right|}^2+{\frac{1}{2}\left|{\tilde{y}}_{b|p}\right|}^2+\frac{1}{{2k}_{\widehat{\beta }}}{\left|\tilde{\beta }\right|}^2\right)}^{\frac{\rho +1}{2}}+\gamma \left(\tilde{\beta }\right)\\ & \le -k_1{V}_1^{\frac{\rho +1}{2}}+\gamma \left(\tilde{\beta }\right)\end{array}$$

(23)

According to Lemma 2, $\gamma \left(\tilde{\beta }\right)$ has the relations as follows:

$$\begin{array}{cc}\gamma \left(\tilde{\beta }\right)& \le {\left(\frac{1}{k_{\widehat{\beta }}}\right)}^{\frac{\rho +1}{2}}{\left|\tilde{\beta }\right|}^{\rho 1+1}{+\left|{\tilde{x}}_{b|p}\right|}^{\rho +1}+{\left|{\tilde{y}}_{b|p}\right|}^{\rho +1}\\ & \le 3^{\frac{1-\rho }{2}}{\left({\left|{\tilde{x}}_{b|p}\right|}^2+{\left|{\tilde{y}}_{b|p}\right|}^2+\frac{1}{k_{\widehat{\beta }}}{\left|\tilde{\beta }\right|}^2\right)}^{\frac{\rho +1}{2}}\\ & =2^{\frac{\rho +1}{2}}{3}^{\frac{1-\rho }{2}}{\left(\begin{array}{c}\frac{1}{2}{\left|{\tilde{x}}_{b|p}\right|}^2+{\frac{1}{2}\left|{\tilde{y}}_{b|p}\right|}^2\\ +\frac{1}{{2k}_{\widehat{\beta }}}{\left|\tilde{\beta }\right|}^2\end{array}\right)}^{\frac{\rho +1}{2}}\end{array}$$

(24)

Combining (23) and (24), we can obtain

$$\begin{array}{cc}{\dot{V}}_1& \le {-2}^{\frac{\rho +1}{2}}\left(k_1-3^{\frac{1-\rho }{2}}\right)V_1^{\frac{\rho +1}{2}}\\ & =-c_1{V}_1^{{\alpha }_1}\end{array}$$

(25)

The term $\gamma \left(\tilde{\beta }\right)$ will approach to zero asymptotically. According to Lemma 3, we know the origin of observer error dynamics is locally finite-time stable. The settling time satisfies $T_1\le \frac{V_0^{1-{\alpha }_1}}{c\left(1-{\alpha }_1\right)}$, $V_0=V({\tilde{x}}_{b|p}\left(t_0\right),{\tilde{y}}_{b|p}\left(t_0\right),\tilde{\mathsf{\beta }}\left(t_0\right),t_0)$.

The proof is concluded. □

3.2. LOS Guidance Law

To stabilize the tracking error $x_{b|p}$, an update law is designed of the curved path parameter as an extra degree of freedom in the controller design. The update law is designed as follows:

$$u_p=U\mathrm{c}\mathrm{o}\mathrm{s}\left({\psi }_e\right)-U\mathrm{s}\mathrm{i}\mathrm{n}\left({\psi }_e\right)\widehat{\beta }-k_{x_{b|p}}{x}_{b|p}$$

(26)

where $k_{x_{b|p}}>0$ is a positive constant.

To stabilize the tracking error $y_{b|p}$, the LOS law is chosen as the heading guidance law, and the form is as follows:

$${\psi }_d={\psi }_p-\arctan (\frac{y_{b|p}+\Delta \widehat{\beta }}{\Delta })$$

(27)

where $\Delta >0$.

Assumption 3.

The heading autopilot tracks the desired heading angle perfectly such that  $\psi ={\psi }_d$.

Theorem 2.

The tracking error dynamics (18) can be stabilized by adaptive LOS law (28) and the update law (27). Under Assumption 3, the origin $\left(x_e,y_e\right)=\left(\mathrm{0,0}\right)$ is semi-globally asymptotically stable.

Proof. 

Substitute guidance law (26) and (27) into (17), respectively, we can obtain

$$\begin{gathered} {\dot{x}}_{b|p}=\Phi \left(t,y_{b|p},U\right)\left(y_{b|p}+\Delta \widehat{\beta }\right)\tilde{\beta }+\dot{{\psi }_p}-k_{x_{b|p}}{x}_{b|p} \\ {\dot{y}}_{b|p}=-\Phi \left(t,y_{b|p},U\right)y_{b|p}-\Phi \left(t,y_{b|p},U\right)\Delta \tilde{\beta }-\dot{{\psi }_p}{y}_{b|p} \end{gathered}$$

(28)

where $\Phi \left(t,y_{b|p},U\right)=\frac{u}{\sqrt{{\Delta }^2+{\left(y_{b|p}+\Delta \widehat{\beta }\right)}^2}}>0$.

Consider the positive definite and radially unbounded Lyapunov function candidate

$$V_2=\frac{1}{2}{x}_{b|p}^2+\frac{1}{2}{y}_{b|p}^2$$

(29)

The derivative of $V_2$ is

$$\begin{array}{cc}{\dot{V}}_2& =-k_{xe}{x}_{b|p}^2-\Phi \left(t,y_{b|p},U\right)y_{b|p}^2-\Phi \left(t,y_{b|p},U\right)\Delta \tilde{\beta }{y}_{b|p}-\Phi \left(t,y_{b|p},U\right)\left(y_{b|p}+\Delta \widehat{\beta }\right)\tilde{\beta }{x}_{b|p}\\ & \le -k_{xe}{x}_{b|p}^2-\Phi \left(t,y_{b|p},U\right)y_{b|p}^2+\Phi \left(t,y_{b|p},U\right)\Delta \left|\tilde{\beta }\right|\left|y_{b|p}\right|+\Phi \left(t,y_{b|p},U\right)\left(\left|y_{b|p}\right|+\Delta \widehat{\beta }\right)\left|\tilde{\beta }\right|\left|x_{b|p}\right|\end{array}$$

(30)

Using the inequality in Lemma 1, the following equation can be obtained:

$$\left|x_{b|p}\right|\left|y_{b|p}\right|\le \frac{1}{2}{\left|x_{b|p}\right|}^2+\frac{1}{2}{\left|y_{b|p}\right|}^2$$

(31)

Substitute (31) into (30), it can be calculated as:

$$\begin{array}{cc}{\dot{V}}_2& \le -(k_{x_{b|p}}-\frac{1}{2}\Phi \left(t,y_{b|p},U\right)|\tilde{\beta }\left|\right))x_{b|p}^2-\Phi \left(t,y_{b|p},U\right)(1-\frac{1}{2}\left|\tilde{\beta }\right|)y_{b|p}^2+\gamma (x_{b|p},y_{b|p},\left|\tilde{\beta }\right|)\\ & \le -k_2{V}_2+\gamma \left(x_{b|p},y_{b|p},\left|\tilde{\beta }\right|\right)\end{array}$$

(32)

where $\gamma \left(x_{b|p},y_{b|p},\left|\tilde{\beta }\right|\right)=\Phi \left(t,y_{b|p},U\right)\Delta \left|\tilde{\beta }\right|(\widehat{\beta }\left|x_{b|p}\right|+\left|y_{b|p}\right|)$, and $\gamma \left(x_{b|p},y_{b|p},0\right)=0$, ${\mathrm{k}}_2$ satisfies that $k_2=\frac{1}{2}\min \left\{k_{x_{b|p}}-\frac{1}{2}\Phi \left(t,y_{b|p},U\right)\left|\tilde{\beta }\right|\right),\Phi \left(t,y_{b|p},U\right)(1-\frac{1}{2}|\tilde{\beta }\left|\right)y_{b|p}^2\}$.

It can be noted that $\left|\tilde{\beta }\right|$ will reach to zero in a finite time, therefore, $\gamma \left(x_{b|p},y_{b|p},\left|\tilde{\beta }\right|\right)$ will reach to zero in a finite time. According to the Lyapunov theorem, it is obvious that the equivalent point $\left(x_{b|p},y_{b|p}\right)=\left(\mathrm{0,0}\right)$ is semi-globally exponential stable.

The proof is concluded. □

4. Control System Design

Thanks to the LOS guidance system, the whole path-following control system for the USV can be decoupled to two independent part to design: heading controller design and surge controller design. The heading controller is designed to generate the desired yaw moment to force the USV to track the desired heading angle given in Section 3.2. The surge controller is designed to generate the desired longitudinal thrust to force the USV to follow the desired surge speed, which is set manually. In the following, the heading controller and surge controller will be designed in Section 4.1 and Section 4.2 based on the SSMC and FTESO techniques.

4.1. Heading Controller Design

The reference heading signal ${\psi }_d$ is provided by the guidance system. The next step is to design the heading controller with the SSMC technique. Define $e_{\psi }=\psi -{\psi }_d$, then, the task of the SSMC controller includes the selection of sliding-mode surface and control strategy to stabilize the tracking error $e_{\psi }$.

Step 1: Select the sliding manifold in system state space according to the relative degree r.

Considering the USV dynamical model (4), let the second-order derivative of the plant output be proportional to the heading control input ${\tau }_r$, $\ddot{\psi }~{\tau }_r$, so the relative degree is r = 2, then, the sliding manifold can be chosen as

$$\sigma ={\dot{e}}_{\psi }+c_1{e}_{\psi }$$

(33)

where $c_1$ is a positive constant.

Step 2: Design a disturbance compensation observer. The derivative of $\sigma$ can be calculated as:

$$\begin{array}{cc}\dot{\sigma }& ={\zeta }_r+\frac{{\tau }_n}{Iz}-{\ddot{\psi }}_d+c_1\left(r-{\dot{\psi }}_d\right)\\ & ={\zeta }_{\sigma }+\frac{{\tau }_n}{Iz}\end{array}$$

(34)

where ${\zeta }_{\sigma }={\zeta }_r-{\ddot{\psi }}_d+c_1(r-{\dot{\psi }}_d)$ is the total disturbance.

Define the estimation error $\tilde{\sigma }=\widehat{\sigma }-\sigma$, $\tilde{\zeta }=\widehat{\zeta }-\zeta$. Then, the disturbance observer can be designed as follows:

$$\begin{gathered} \dot{\widehat{\sigma }}=\widehat{\zeta }+\frac{{\tau }_n}{Iz}-k_{o1}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o1}}\left(\tilde{\sigma }\right)-L_{o1}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right) \\ \dot{\widehat{\zeta }}=-k_{o2}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o2}}\left(\tilde{\sigma }\right)-L_{o2}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right) \end{gathered}$$

(35)

Step 3: Design a heading controller to make the sliding surface hold that $\sigma =0$.

According to Lemma 5, the heading control law can be chosen as follows:

$$\begin{gathered} {\tau }_n=I_z\left(-{\widehat{\zeta }}_{\sigma }-k_{\sigma 1}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{{\sigma }_1}}+w_1-L_{\sigma }\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)\right) \\ \dot{w_1}=-k_{\sigma 2}{\left|\sigma \right|}^{{\rho }_{\sigma 2}}\mathrm{s}\mathrm{g}\mathrm{n}\left(\sigma \right) \end{gathered}$$

(36)

where $0.5<{\rho }_{\sigma 1}<1$; ${\rho }_{\sigma 2}={2\rho }_{\sigma 1}-1$; $k_{\sigma 1}>0$ and ${\mathrm{k}}_{\mathsf{\sigma }2}$ > 0 are constants.

Theorem 3.

For the sliding mode dynamics (34) with the total unknown dynamic  ${\zeta }_{\sigma }$, the FTESO is established in (35), which can estimate the unknown disturbance simultaneously, and the estimation error can converge to a bounded domain of zero.

Proof. 

By combining with (34) and (35), the estimate error dynamics can be obtained as follows:

$$\begin{gathered} \dot{\tilde{\sigma }}={\tilde{\zeta }}_{\sigma }-k_{o1}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o1}}\left(\tilde{\sigma }\right)-L_{o1}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right) \\ {\dot{\tilde{\zeta }}}_{\sigma }=-k_{o2}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o2}}\left(\tilde{\sigma }\right)-L_{o2}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)-{\dot{\zeta }}_{\sigma } \end{gathered}$$

(37)

Note that ${\zeta }_{\sigma }={\zeta }_r-{\ddot{\psi }}_d+c_1(r-{\dot{\psi }}_d)$ and ${\psi }_d$ is the expected heading angle, which is third-order derivative bounded, and according to Assumption 2, the ${\dot{\zeta }}_r$ is also bounded. Then, the total disturbance ${\dot{\zeta }}_{\sigma }$ is bounded, so there exists a positive constant $l_1$ satisfying $\left|{\dot{\zeta }}_{\sigma }\right|\le l_1$. First, the terms $–L_{\mathrm{o}1}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)$ and $–L_{o2}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)-{\dot{\zeta }}_{\sigma }$ are removed in Equation (38) under the condition of $L_{o2}>l_1$, where $l_1$ is a positive constant satisfying $\left|{\dot{\zeta }}_{\sigma }\right|\le l_1$. Then it can be obtained:

$$\begin{gathered} \dot{\tilde{\sigma }}=\tilde{\zeta }-k_{o1}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o1}}\left(\tilde{\sigma }\right) \\ \dot{\tilde{\zeta }}=-k_{o2}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o2}}\left(\tilde{\sigma }\right) \end{gathered}$$

(38)

Define $f_a$ as the vector filed of system (38), and $f_a$ is homogeneous of degree ${\rho }_{o1}-1$ with respect to the dilation, ${\Delta }_{\mathrm{k}}\left(\tilde{\sigma },\tilde{\zeta }\right)=(k\tilde{\sigma },k^{{\rho }_{o1}}\tilde{\zeta })$, where $k>0$. The Lyapunov function candidate is designed as follows:

$$V_3=\frac{1}{2}{Z}^{T}Z$$

(39)

where $Z={\left[si{g}^{\frac{1}{r}}\left(\tilde{\sigma }\right),{\mathrm{s}\mathrm{i}\mathrm{g}}^{\frac{1}{r{\rho }_{o1}}}\left(\tilde{\zeta }\right)\right]}^{\mathrm{T}}$, $r={\rho }_{o1}{\rho }_{o2}$, and we define $L_{fo1}{V}_{3a}$ as the Lie derivative of $V_{3a}$ along the $f_a$. Then, $V_{3a}$ is homogeneous of degree $\frac{2}{r}$, with respect to ${\Delta }_{\mathrm{k}}\left(\tilde{\sigma },\tilde{\zeta }\right)=(k\tilde{\sigma },k^{{\rho }_{o1}}\tilde{\zeta })$, and $L_{fa}{V}_3\le -c_2{V}_3^{{\alpha }_2}$, where ${\mathrm{c}}_2=-{\mathrm{m}\mathrm{a}\mathrm{x}}_{\left\{\left(\tilde{\sigma },\tilde{\zeta }\right):V_3=1\right\}}{L}_{fa}{V}_3$, ${\alpha }_2=1+\frac{r{\rho }_{o1}}{2}-\frac{r}{2}$.

Then, take the time of derivative of (39) along (37), ${\dot{V}}_3$ can be computed as

$$\begin{array}{cc}{\dot{V}}_3& =L_{f_a}{V}_3+Z^T\left[\begin{array}{c}-\frac{1}{2}\mathrm{s}\mathrm{i}{\mathrm{g}}^{\frac{1}{r}-1}\left(\tilde{\sigma }\right)L_{o1}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)\\ \frac{1}{r{\rho }_{o1}}\mathrm{s}\mathrm{i}{\mathrm{g}}^{\frac{1}{r{\rho }_{o1}}-1}\left(\tilde{\zeta }\right)\left(-\dot{\zeta }-L_{o2}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{\sigma }\right)\right)\end{array}\right]\\ & \le {-c}_2{V}_3^{{\alpha }_2}+\frac{L_{o1}}{r}{\left|\tilde{\sigma }\right|}^{\frac{2}{r}-1}+\frac{L_{o2}+l_1}{r{\rho }_{o1}}{\left|\tilde{\zeta }\right|}^{\frac{2}{{r\rho }_{o1}}-1}\end{array}$$

(40)

According to Lemma 2, the following relationship can be obtained as:

$$\begin{gathered} {\left|\tilde{\sigma }\right|}^{\frac{2}{r}-1}\le {\left|\tilde{\sigma }\right|}^{\frac{2}{r}\left(1-\frac{r}{2}\right)}+{\left|\tilde{\zeta }\right|}^{\frac{2}{{r\rho }_{o1}}\left(1-\frac{r}{2}\right)}\le 2^{\frac{r}{2}}{2}^{1-\frac{r}{2}}{V}_3^{1-\frac{r}{2}} \\ {\left|\tilde{\zeta }\right|}^{\frac{2}{{r\rho }_{o1}}-1}\le {\left|\tilde{\sigma }\right|}^{\frac{2}{r}\left(1-\frac{r{\rho }_{o1}}{2}\right)}+{\left|\tilde{\zeta }\right|}^{\frac{2}{{r\rho }_{o1}}\left(1-\frac{r{\rho }_{o1}}{2}\right)}\le {2^{1-\frac{r{\rho }_{o1}}{2}}V}_3^{1-\frac{r{\rho }_{o1}}{2}} \end{gathered}$$

(41)

Combining (40) and (41), the following inequality is given as:

$${\dot{V}}_3{\le -c}_2{V}_3^{{\alpha }_2}+c_3{V}_3^{{\alpha }_3}+c_4{V}_3^{{\alpha }_4}$$

(42)

where $c_3=2\frac{L_{o1}}{r},c_4=2\frac{L_{o2}+l_1}{r{\rho }_{o1}}$, ${\alpha }_3=1-\frac{r}{2}$, ${\alpha }_4=1-\frac{r{\rho }_{o1}}{2}$, and there is $0<{\alpha }_3<{\alpha }_4<{\alpha }_2<1$. The stability of $V_3$ can be divided into two parts [31,32]:

(1) If $V_3\ge 1$, ${\dot{V}}_3{\le -c}_2{V}_3^{{\alpha }_2}+c_5{V}_3$, where $c_5=c_3+c_4$, according to Lemma 5, and $V_3$ will converges to 1 within finite time $t_1\le \mathrm{l}\mathrm{n}[1-\left(\frac{c_5}{c_2}\right)V_3(t_0\left)\right]/(c_4{\alpha }_2-c_4)$.

(2) If $V_3<1$, ${{\dot{V}}_3\le -c}_2{V}_3^{{\alpha }_2}+c_5{V}_3^{{\alpha }_3}$. Select $c_0$, which satisfies $0<c_0<1-c_5/c_2$, then, ${\dot{\mathrm{V}}}_3\le -c_2{c}_0{V}_3^{{\alpha }_2}-\left[{\mathrm{c}}_2\left(1-{\mathrm{c}}_0\right){\mathrm{V}}_3^{{\mathsf{\alpha }}_2-{\alpha }_3}-{\mathrm{c}}_5\right]V_3^{{\alpha }_3}$. If $V_3^{{\alpha }_2-{\alpha }_3}$ satisfies $V_3^{{\alpha }_2-{\alpha }_3}>\frac{c_5}{c_2(1-c_0)}$, then there is ${{\dot{V}}_3\le -c}_2{c}_0{V}_3^{{\alpha }_2}\le 0$. According to Lemma 4, $V_3$ will converge into $V_3^{{\alpha }_2-{\alpha }_3}<\frac{c_5}{c_2(1-c_0)}$ within finite time $t_2\le \frac{V_3^{1-{\alpha }_2}\left(t_1\right)}{c_2{c}_0(1-{\alpha }_2)}$.

Finally, $V_3$ will converge into $V_3\le {\left(\frac{c_5}{c_2\left(1-c_0\right)}\right)}^{\frac{1}{{\alpha }_2-{\alpha }_3}}$ within finite time $T_2\le t_1+t_2$. Then, the convergence domain of observer error can be obtained as follows:

$$‖(\tilde{\sigma },\tilde{\zeta })‖\le \sqrt{2}{\left(\frac{c_5}{c_2\left(1-c_0\right)}\right)}^{\frac{1}{2\left({\alpha }_2-{\alpha }_3\right)}}$$

(43)

Finally, the estimation error can converge into a compact set $\mathsf{\Omega }=\left\{\left(\tilde{\sigma },\tilde{\zeta }\right)\right|||\left(\tilde{\sigma },\tilde{\zeta }\right)||\le \sqrt{2}{\left(\frac{c_5}{c_2\left(1-c_0\right)}\right)}^{\frac{1}{2({\alpha }_2-{\alpha }_3)}}\}$.

The proof is concluded. □

Theorem 4.

The control law (32) can make the sliding manifold $\sigma$ in (38) approach zero within finite time.

Proof. 

Substituting the control law (33) into (31), it can be calculated as:

$$\begin{gathered} \dot{\sigma }=-\tilde{\zeta }-k_{\sigma 1}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{{\sigma }_1}}\left(\sigma \right)+w_1-L_{\sigma }\mathrm{s}\mathrm{g}\mathrm{n}\left(\sigma \right) \\ {\dot{w}}_1=-k_{\sigma 2}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{{\sigma }_2}}\left(\sigma \right) \end{gathered}$$

(44)

Consider the Lyapunov function candidate as follows:

$$V_4=\frac{1}{2}{w}_1^2+k_{\sigma 2}{\int }_0^{\sigma }si{g}^{{{\rho }_{\sigma }}_2}\left(z\right)dz$$

(45)

Combining Equation (45), the derivative of $V_4$ in (46) can be written as follows:

$$\begin{array}{cc}{\dot{V}}_4& =w_1{\dot{w}}_1+k_{\sigma 2}si{g}^{{\rho }_{{\sigma }_2}}\left(\sigma \right)\left(\begin{array}{c}-\tilde{\zeta }-k_{\sigma 1}si{g}^{{\rho }_{{\sigma }_1}}\left(\sigma \right)\\ +w_1-L_{\sigma }sgn\left(\sigma \right)\end{array}\right)\\ & \le -k_{\sigma 1}{k}_{\sigma 2}{\left|\sigma \right|}^{{\rho }_{\sigma 1}+{\rho }_{\sigma 2}}-\left(L_{\sigma }-l_2\right)k_{\sigma 2}{\left|\sigma \right|}^{{\rho }_{\sigma 2}}\\ & \le -k_{\sigma 1}{k}_{\sigma 2}{\left|\sigma \right|}^{{\rho }_{\sigma 1}+{\rho }_{\sigma 2}}\end{array}$$

(46)

Apply the Lasalle theory. The set $\{\sigma ,w_1|{\dot{V}}_4=0\}$ consists of the axis $\mathsf{\sigma }=0$, and only the invariant set inside $\sigma =0$ is the origin $\sigma =w_1=0$. Thus, the asymptotic convergence of $\sigma$ and $w_1$ to zero is assured.

It can be seen from the above analysis that the term $-\tilde{\zeta }-L_{\sigma }sgn\left(\sigma \right)$ in (44) can be omitted reasonably. The $\dot{\sigma }$ and ${\dot{w}}_1$ can be obtained as:

$$\begin{gathered} \dot{\sigma }=-k_{\sigma 1}{\left|\sigma \right|}^{{\rho }_{\sigma 1}}sgn\left(\sigma \right)+w_1 \\ {\dot{w}}_1=-k_{\sigma 2}{\left|\sigma \right|}^{{\rho }_{\sigma 2}}sgn\left(\sigma \right) \end{gathered}$$

(47)

According to the analysis process of the stability of the system in (38), the dynamics described by (47) are finite-time stable.

The proof is concluded. □

4.2. Surge Controller Design

In this subsection, the surge controller is designed for the USV to track the desired surge speed, which is set manually in advance.

To estimate the unknown term ${\zeta }_u$, FTESO is introduced as follows:

$$\begin{gathered} \dot{\widehat{u}}={\widehat{\zeta }}_u+\frac{{\tau }_u}{m_{11}}-k_{o3}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o3}}\left(\tilde{u}\right)-L_{o3}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{u}\right) \\ {\dot{\widehat{\zeta }}}_u=-k_{o4}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o4}}\left(\tilde{u}\right)-L_{o4}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{u}\right) \end{gathered}$$

(48)

where $k_{o3},k_{o4}$ are positive constants.

Define estimation errors as $\tilde{u}=\widehat{u}-u$, ${\tilde{\zeta }}_u={\widehat{\zeta }}_u-{\zeta }_u$. The derivatives of $\tilde{\mathrm{u}}$ and ${\tilde{\mathsf{\zeta }}}_u$ can be obtained as:

$$\begin{gathered} \dot{\tilde{u}}={\tilde{\zeta }}_u-k_{o3}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o3}}\left(\tilde{u}\right)-L_{o3}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{u}\right) \\ {\dot{\tilde{\zeta }}}_u=-k_{o4}{\mathrm{s}\mathrm{i}\mathrm{g}}^{{\rho }_{o4}}\left(\tilde{u}\right)-L_{o4}\mathrm{s}\mathrm{g}\mathrm{n}\left(\tilde{u}\right) \end{gathered}$$

(49)

Define speed tracking error as $e_u=u-u_d$, where $u_d$ is the desired surge speed, which is given manually in advance, then the surge control law can be given by:

$$\begin{gathered} {\tau }_u=m_{11}(-{\widehat{\zeta }}_u-k_{u1}{\left|e_u\right|}^{{\rho }_{u1}}\mathrm{s}\mathrm{g}\mathrm{n}\left(e_u\right)+w_2-L_u\mathrm{s}\mathrm{g}\mathrm{n}(e_u\left)\right) \\ {\dot{w}}_2=-k_{u2}{\left|e_u\right|}^{{\rho }_{u2}}\mathrm{s}\mathrm{g}\mathrm{n}\left(e_u\right) \end{gathered}$$

(50)

Substituting (50) into (4), ${\dot{e}}_u$ can be obtained as:

$$\begin{gathered} {\dot{e}}_u=-k_{u1}{\left|e_u\right|}^{{\rho }_{u1}}\mathrm{s}\mathrm{g}\mathrm{n}\left(e_u\right)+w_2-L_u\mathrm{s}\mathrm{g}\mathrm{n}\left(e_u\right) \\ {\dot{w}}_2=-k_{u2}{\left|e_u\right|}^{{\rho }_{u2}}\mathrm{s}\mathrm{g}\mathrm{n}\left(e_u\right) \end{gathered}$$

(51)

Theorem 5

For the surge motion dynamics (4) with the total unknown dynamic ${\zeta }_u$, the FTESO established in (48) can be employed to obtain the value of the unknown disturbance simultaneously, and the estimation error in (50) is able to converge to a bounded domain of zero.

Theorem 6.

The control law (50) can make the tracking error  $e_u$ converge to zero within finite time.

Remark 3.

The proof processes of Theorems 5 and 6 are omitted because they are similar to the proof of Theorems 3 and 4.

5. Stability Analysis

The stability of the guidance system and control system are given in Section 3 and Section 4. In the following, the stability of the closed-loop system of the path-following control for the USV based on the proposed control method are explained.

Theorem 7.

The path-following errors  $x_{b|p}$ and $y_{b|p}$  are uniformly ultimately bounded.

Proof. 

According to where the variables have the constraints as ${\mathrm{U}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{U}\le {\mathrm{U}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, $\left|{\mathsf{\psi }}_{\mathrm{e}}\right|\le \mathsf{\pi }$, ${\mathrm{U}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{U}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the speed limits of the ship. ${\mathrm{u}}_{\mathrm{p}}$ is a designed virtual point of the path, which is also bounded.

Theorems 1 and 3–6, the errors $\tilde{\beta }$, $e_u$, and sliding surface $\sigma$ can converge to zero within finite time. Then, under the conditions of $\tilde{\beta }=0$, $e_u=0,$ and $\sigma =0$, the Lyapunov function candidate can be selected as $V=\frac{1}{2}{x}_{b|p}^2+\frac{1}{2}{y}_{b|p}^2+\frac{1}{2}{e}_{\psi }^2$, where $e_{\psi }=\psi -{\psi }_d$. Taking the time of derivative of $V$ along (17), with the guidance law (26) and (27), it can be obtained:

$$\begin{array}{cc}\dot{V}& =-c_1{e}_{\psi }^2-k_{x_{b|p}}{x}_{b|p}^2+U{y}_{b|p}\mathrm{s}\mathrm{i}\mathrm{n}\left(e_{\psi }+{\psi }_d-{\psi }_p\right)+U{y}_{b|p}\mathrm{c}\mathrm{o}\mathrm{s}\left(e_{\psi }+{\psi }_d-{\psi }_p\right)\beta \\ & =-c_1{e}_{\psi }^2-k_{x_{b|p}}{x}_{b|p}^2+\Phi \left(t,y_{b|p},U\right)\gamma \left(e_{\psi },y_{b|p}\right)-\cos \left(e_{\psi }\right)\Phi \left(t,y_{b|p},U\right)y_{b|p}^2\\ & \le {-k}_{3}V+\Phi \left(t,y_{b|p},U\right)\gamma (e_{\psi },y_{b|p})\end{array}$$

(52)

where $k_3=2\min \left(c_1,k_{x_{b|p}},\cos \left(e_{\psi }\right)\Phi \left(t,y_{b|p},U\right)\right)$, which satisfies $\gamma \left(\mathrm{0,0}\right)=0$. With a proper $k_3$, $\dot{V}\le 0$. According to the Lyapunov stability theorem, the whole system is uniformly ultimately bounded.

The proof is concluded. □

6. Numerical Simulations

For the simulation, the mathematical model of the Otter USV is applied [40], with length L = 2.0 m, width 1.07 m, and weight 65 kg. The model is given in [41], and the parameters are $m_{11}=70.5\mathrm{k}\mathrm{g}$, $m_{22}=147.5\mathrm{k}\mathrm{g}$, $m_{23}=m_{32}=11\mathrm{k}\mathrm{g}$, $m_{33}=43.2394\mathrm{k}\mathrm{g}$, $l_1=0.395\mathrm{m}$, $k_{pos}=0.0111\mathrm{k}\mathrm{g}\mathrm{m}$, $k_{neg}=0.0064\mathrm{k}\mathrm{g}\mathrm{m}$. The desired path is a sinusoidal path described as $x_p\left(\theta \right)=5\cos \left(0.8\theta \right)$, $y_p\left(\theta \right)=5\theta$. The surge speed of the Otter USV is set as $u_d=1.2\mathrm{m}/\mathrm{s}$, and the constant ocean current speed and direction are set as $v_c=0.2\mathrm{m}/\mathrm{s}$ and ${\beta }_c=30°$, respectively. In addition, the time-varying exterior disturbances are selected as:

$${\tau }_w=\left[\begin{array}{c}10+5\sin \left(0.8t\right)\cos \left(0.2t\right)\\ 1.5\sin \left(0.8t\right)\cos \left(0.2t\right)\\ 5+15\sin \left(0.8t\right)cos\left(0.2t\right)\end{array}\right]$$

(53)

In order to verify the performance of SSMC for heading control, two other controllers are applied to the USV. The first one is the PID controller with reference feedback proposed as [20]:

$${\tau }_n={\tau }_{ff}-K_p\left(e_{\psi }+\frac{1}{T_i}{\int }_0^t{e}_{\psi }d\tau +T_d{\dot{e}}_{\psi }\right)$$

(54)

The reference feedback signal is defined as:

$${\tau }_{ff}=\frac{T}{K}{\dot{\psi }}_d+\frac{1}{T}{\psi }_d$$

(55)

where $K_p,T_i,T_d$ are proportional gain, integral time constant, and derivative constant, respectively. Nomoto gain and time constants K and T can be chosen according to the multiple maneuvering experiment.

The second one is the conventional SMC controller, which is defined as:

$${\tau }_n=-({\beta }_1+d)\mathrm{s}\mathrm{a}\mathrm{t}(a,{\sigma }_1)$$

(56)

where ${\beta }_1$ and $d$ are designed positive constants; sliding mode is ${\sigma }_1$, which satisfies ${\sigma }_1={\dot{e}}_{\psi }+c_0{e}_{\psi }$; $\mathrm{s}\mathrm{a}\mathrm{t}(a,\ast )$ is the saturation function.

The control system and guidance system parameters are designed as

Table 1. Heading controller parameters.

Control Strategies	Parameters
SSMC	$k_{o1}=10$, $k_{o2}=25$, ${\rho }_{o1}=0.9$, ${\rho }_{o2}=0.8$, $L_{o1}=0.01$, $L_{o2}=0.1$,
	$k_{\sigma 1}=2$, $k_{\sigma 2}=0.05$, ${\rho }_{\sigma 1}=0.8$, ${\rho }_{\sigma 2}=0.6$, $L_{\sigma }=0.001$
PID	$K_p=93.15$, $T_i=0.89$, $T_d=6.67$,
	$K=0.0242$, $T=1$
SMC	${\beta }_1=1.2$, $d=0.3$, $a=0.05$

,

Table 2. Surge controller parameters.

Parameters
$k_{o1}=10$, $k_{o2}=25$, ${\rho }_{o1}=0.9$, ${\rho }_{o2}=0.8$, $L_{o1}=0.01$, $L_{o2}=0.1$, $k_{\sigma 1}=2$, $k_{\sigma 2}=0.05$, ${\rho }_{\sigma 1}=0.8$, ${\rho }_{\sigma 2}=0.6$, $L_{\sigma }=0.001$, $K_p=93.15$, $T_i=0.89$, $T_d=6.67$, $K=0.0242$, $T=1$, ${\beta }_1=1.2$, $d=0.3$, $a=0.05$

and

Table 3. Guidance system parameters.

Parameters
$\Delta =3\mathrm{m}$, $k_{x_{b|p}}=2$, $k_{\widehat{x}}=5$, $k_{\widehat{y}}=5$. $\rho =0.8$, $k_{\beta }=10$

. It should be noted that the same surge controller is used to track the desired surge speed, and the same guidance system is chosen to produce expected heading angle, where the only difference is the heading controller. The control performances of PID, conventional SMC, and SSMC proposed by this paper are compared for USV curved path following.

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. The desired path and the actual trajectories are shown in Figure 4. It is apparent that SSMC and SMC heading controllers have better performance than the PID controller under severe external disturbances. Around 20 s, the SSMC controller has higher tracking accuracy than the SMC controller. The sliding-mode dynamics of SMC and SSMC are shown in Figure 5. Both systems can reach the region of 0 bounded by $\pm 0.02$ within 4 s. However, the convergence of SSMC is smoother than SMC. The path-following errors are plotted in Figure 6, and SSMC and SMC controllers have almost the same tracking performance. The heading tracking errors of SSMC and SMC converge to a small neighborhood of 0 from $20°$ within 8 s. The tracking errors of the PID controller are relatively large and choppy because of the influence of ocean current. It proves that SSMC and SMC controllers are more robust than the PID controller. The velocities of USVs are shown in Figure 7. As shown in the third figure in Figure 7, the yaw velocities with SMC exhibit more severe oscillations than SSMC.

The propeller shaft speeds of the three controllers are shown in Figure 8. It is obvious that the very high frequency oscillation phenomenon occurs in the SMC controller. The changes of propeller shaft speeds of SSMC and PID controllers are comparatively gentle. It is shown that the high frequency auto-oscillation can be effectively avoided by the SSMC controller. The effect of disturbances estimation is shown in Figure 9. FTESO-based errors of ${\tilde{\zeta }}_{\sigma }$ reach the region of 0 bounded by ±0.1 within 2.5 s, and the errors of ${\tilde{\zeta }}_u$ reach the region of 0 bounded by ±0.01 within 2.1 s. It is illustrated that the estimation errors of FTESO are able to reach to a small bounded domain of zero within a finite time.

In summary, with the comparison analysis, the SSMC based on FTESO is more robust than the PID controller and smoother than the conventional SMC, thus the SSMC has the best path-following performance compared to the PID and SMC.

7. Conclusions

This paper develops a novel SSMC method based on FTESO for path following of the USV with unknown dynamics and exterior disturbances. The guidance and control scheme have simple and clear structures, and both the smoothness and robustness of the system are improved to visible levels, with the least possible modeling information and environmental disturbances. For the guidance system, the finite-time sideslip angle observer is incorporated into the LOS guidance law, which can make sideslip estimation error convergent to zero in finite time and speed up the convergence process. For the control system, FTESO is designed to estimate and compensate for the unknown disturbances on the sliding-mode surface instead of using a strong discontinuous control signal, which will cause a strong chattering phenomenon. At the same time, all errors of the dynamic system are proved to be bounded. The simulation results prove that the designed strategy has a better effectiveness than PID and conventional SMC, and the USV has satisfactory performance for path following under the unknown disturbances. It should be noted that the obstacle avoidance is not considered in the proposed control method. In the future, the artificial potential fields will be incorporated in the guidance system to realize collision avoidance.