## 2 Preliminaries and Problem Formulation
### USV Model  
As illustrated in Figure 1, the position and orientation describe the horizontal plane motion of a USV neglecting roll, pitch, and heave, where \(\{i\}\) and \(\{b\}\) represent the inertial frame and body fixed frame, respectively. The desired continuous path \(P_{d}(\theta)=\left[x_{d}(\theta),y_{d}(\theta)\right]^{T}\) is parameterized by a scalar variable \(\theta\) and \(P=\left[x,y\right]^{T}\) is the position coordinate. The kinematic equations of a USV can be described by relative velocities as follows [23]  
\[\left\{\begin{aligned} \dot{x}&=u_{r}\cos\psi-v_{r} \sin\psi+V_{x}\\ \dot{y}&=u_{r}\sin\psi+v_{r}\cos\psi+V_{y}\\ \dot{\psi}&=r\end{aligned}\right. \tag{4}\]  
where \(u_{r}\) and \(v_{r}\) are the relative surge and sway velocities; \(x\), \(y\), and \(\psi\) express the position and orientation in \(\{i\}\); and \(V_{x}\) and \(V_{y}\) are the ocean currents represented in \(\{b\}\). The dynamics of a USV is expressed as follows [25]  
\[\left\{\begin{aligned} \dot{u}_{r}&=\underbrace{\frac{m_{22}}{m_{11 }}v_{r}-\frac{d_{11}}{m_{11}}u_{r}-\sum_{i=2}^{3}\frac{d_{ui}}{m_{11}}\left|u_ {r}\right|^{i-1}u_{r}}_{f_{u}}+\frac{1}{m_{11}}\tau_{u}+\frac{1}{m_{11}}\tau_ {wu}\\ \dot{v}_{r}&=\underbrace{-\frac{m_{11}}{m_{22}}u_{r} -\frac{d_{22}}{m_{22}}v_{r}-\sum_{i=2}^{3}\frac{d_{vi}}{m_{11}}\left|v_{r} \right|^{i-1}v_{r}}_{f_{r}}+\frac{1}{m_{22}}\tau_{uvv}\\ \dot{r}&=\underbrace{\frac{(m_{11}-m_{22})}{m_{33}}u_ {r}v_{r}-\frac{d_{33}}{m_{33}}r-\sum_{i=2}^{3}\frac{d_{ri}}{m_{11}}\left|r \right|^{i-1}r}_{f_{r}}+\frac{1}{m_{33}}\tau_{r}+\frac{1}{m_{33}}\tau_{ur}\end{aligned}\right. \tag{5}\]  
where the positive constant parameters \(m_{jj}(j=1,2,3)\) are the inertia including added mass; \(d_{ii},d_{ui},d_{vi}\), and \(d_{ri}(i=2,3)\) are the linear and quadratic hydrodynamic damping in surge, sway, and yaw; \(\tau_{uu},\tau_{uv}\), and \(\tau_{ur}\) are the unknown time-varying environmental disturbances; and \(\tau_{u}\) and \(\tau_{r}\) are the available control inputs of the surge force and the yaw moment thereby viewing it as the underactuated control problem. Since the model parameters are directly related to the operation conditions [26], the parameters of the model is uncertain. Actually, due to the physical constraint, the control inputs are subject to nonlinear saturations, which is shown as follows  
\[\tau_{i}=\left\{\begin{aligned} \tau_{i}\max,&\tau_{i0}> \tau_{i}\max\\ \tau_{i0},&\tau_{i}\min\leqslant\tau_{i0}\leqslant\tau_{i \max}\\ \tau_{i\min},&\tau_{i0}<\tau_{i\min}\end{aligned}\right. \tag{6}\]  
where \(\tau_{i\min}\) and \(\tau_{i\max}\)\((i=u,r)\) are the minimum and maximum control inputs produced by the actuators, referring to actual constraints of the motor's rotational speed and rudder deflection; and \(\tau_{i0}\) is the command control input of the path following controller.  
**Assumption 1**.: _Assume that all states of a USV are measurable._  
**Assumption 2**.: _The time-varying ocean currents \(\mathbf{v}=[V_{x},V_{y}]^{T}\) are assumed to be irrotational and bounded, and there exist a positive constant \(M\), such that \(\|\dot{\mathbf{v}}\|\leqslant M,M>0\). The disturbances \(\tau_{wi}(i=u,v,r)\) are unknown time-varying and bounded, and the first derivative of them are also bounded such that \(|\dot{\tau}_{wi}|\leqslant\tau_{w}\), where \(\tau_{w}\) is unknown constant._  
**Remark 1**.: _Note that in (5) the off-diagonal terms of the inertia and damping are ignored. No matter a large scale surface vessel or a highly maneuverable unmanned surface vessel, these terms are relatively small than the main diagonal terms. Therefore, it is reasonable to omit these terms._  
**Remark 2**.: _The external disturbances \(\tau_{\text{twr}},\tau_{\text{twr}}\), and \(\tau_{\text{twr}}\) in Equation (5) represent the compound disturbances of the wind and wave disturbances. The ocean currents as the form of hydrodynamic terms with relative velocities are represented in Equation (4)._