## 2 Problem Formulation and Preliminaries  
### Problem Formulation  
In this section, two reference frames are considered to study the USV model. The horizontal plane motion of USV is described by the position and orientation, namely, neglecting roll, pitch and heave, as illustrated in Figure 1, in which \(X_{E}O_{E}Y_{E}\) is the earth-fixed inertial frame \(\{i\}\) and \(x_{b}o_{b}y_{b}\) is the body-fixed frame \(\{b\}\). The position \((x,y)\) in \(\{i\}\) represents the actual location of USV, and \(\psi\in(0,2\pi)\) is course angle. In addition, \(u\), \(v\) and \(r\) denote the surge velocity, the sway velocity and the yaw rate of USV, respectively.  
Based on the above analysis, the kinematics and dynamics mathematical model of the three degrees of freedom [29] can be described as  
\[\dot{\eta}=J\left(\psi\right)v\] \[M\dot{v}+C\left(v\right)v+Dv=\tau+d\left(t\right) \tag{1}\]  
where \(\eta=[x,y,\psi]^{T}\), \(v=[u,v,r]^{T}\), \(J\left(\psi\right)=\left[\begin{array}{ccc}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{array}\right]\) is the rotation matrix from \(\{b\}\) to \(\{i\}\); \(M=\left[\begin{array}{ccc}m_{11}&0&0\\ 0&m_{22}&m_{23}\\ 0&m_{32}&m_{33}\end{array}\right]\) is the inertial matrix, and \(C\left(v\right)\in R^{3\times 3}\) is the matrix of Coriolis and  
Figure 1: The earth-fixed inertial and the body-fixed frame.  
centripetal terms; \(D=\left[\begin{array}{ccc}d_{11}&0&0\\ 0&d_{22}&d_{23}\\ 0&d_{32}&d_{33}\end{array}\right]\) is the damping matrix; \(\tau=\left[\tau_{u},0,\tau_{r}\right]^{T}\), in which \(\tau_{u}\) and \(\tau_{r}\) denote the surge force and yaw moment, respectively. In practice, the amplitude and change rate of the equivalent control resultant force and moment are possibly constrained due to physical limitations of the propulsion system, which can be described as  
\[\tau_{j}=\left\{\begin{array}{ll}\tau_{j\max},&if\ \tau_{cj}>\tau_{j\max}\\ \tau_{cj},&if\ \tau_{j\min}<\tau_{cj}<\tau_{j\max}\\ \tau_{j\min},&if\ \tau_{cj}<\tau_{j\min}\end{array}\right. \tag{2}\]  
where \(\tau_{j\max}\) and \(\tau_{j\min}\)\(\left(j=u,r\right)\) are the maximum and the minimum control forces or moments, \(\tau_{cj}\) is the commanded control calculated by adaptive sliding mode controller. In addition, \(d\left(t\right)=\left[d_{u},d_{v},d_{r}\right]^{T}\) represent unknown time-varying ocean disturbances.  
Assume that USV is symmetrical, the origin of the attached coordinate system and the barycentre of USV are coincident [10; 30; 31]. (1) can be rewritten as the following state-space equations  
\[\left\{\begin{array}{l}\dot{x}=u\cos\psi-v\sin\psi\\ \dot{y}=u\sin\psi+v\cos\psi\\ \dot{\psi}=r\end{array}\right. \tag{3}\]  
\[\left\{\begin{array}{l}\dot{u}=f_{u}+\frac{\Delta f_{u}}{m_{1}}+\frac{\tau_ {u}}{m_{1}}+\frac{d_{v}}{m_{11}}\\ \dot{v}=f_{v}+\frac{\Delta f_{v}}{m_{2}}+\frac{d_{v}}{m_{2}}\\ \dot{r}=f_{r}+\frac{\Delta f_{r}}{m_{33}}+\frac{\tau_{r}}{m_{33}}+\frac{d_{r}} {m_{33}}\end{array}\right. \tag{4}\]  
where \(f_{u}=\frac{m_{22}}{m_{11}}v\tau-\frac{d_{11}}{m_{11}}u\), \(f_{v}=-\frac{m_{11}}{m_{22}}ur-\frac{d_{22}}{m_{22}}v\), \(f_{r}=\frac{\left(m_{11}-m_{22}\right)}{m_{33}}ur-\frac{d_{33}}{m_{33}}r\), in which \(\Delta f=\left[\Delta f_{u},\Delta f_{v},\Delta f_{r}\right]^{T}\) represents the uncertain system dynamics.  
**Assumption 1**.: _The reference trajectory \(\left(x_{d},y_{d}\right)\) is smooth and differentiable, \(x_{d}\), \(\dot{x}_{d}\), \(y_{d}\) and \(\dot{y}_{d}\) are all bounded._  
**Assumption 2**.: _[_32_]_ _The external disturbances \(d_{i}\left(i=u,v,r\right)\) are bounded by \(\left|d_{i}\right|\leq d_{i}^{*}\), where \(d_{i}^{*}\) are unknown positive constants._  
Control objective: Considering model uncertainty and input saturation, the USV accurately tracks the reference path \(\left(x_{d},y_{d}\right)\) according to the design of the surge force \(\tau_{u}\) and the yaw moment \(\tau_{r}\). All state variables are proved to be semi-globally uniformly ultimately bounded.