## 2 Kinematics and Dynamic Models of USV
### Desired Region Description and Error Dynamics  
In the region-based control framework, the desired moving target is specified by a region at the desired trajectory. Moreover, the USV must converge into the region. The desired region is specified by an inequality or by the objective function as  
\[f(\delta\eta_{\mathcal{L}})=\left[f(\delta\eta_{\mathcal{L}1}),\ f(\delta\eta_{ \mathcal{L}2}),\ldots,f(\delta\eta_{\mathcal{L}n})\right]^{T}\leq 0, \tag{3}\]  
where \(f(\delta\eta_{\mathcal{L}i})(i=1,2,\ldots,n)\) is a scalar function with continuous first partial derivative, \(n\) is the total number of objective function, and \(\delta\eta_{\mathcal{L}i}=\mathcal{L}(t)(\eta-\eta_{0i})=\mathcal{L}(t)\delta \eta_{i}\), \(\delta\eta_{i}=\eta-\eta_{0i}\). \(\eta_{0i}\) is the reference point of the desired region. \(f(\delta\eta_{\mathcal{L}i})(i=1,2,\ldots,n)\), where \(n\) is the total number of objective function. An illustration of the objective function is shown in Figure 3.  
\(\mathcal{L}(t)\) is a time varying and nonsingular scaling factor. \(\eta_{0i}\) and \(\mathcal{L}(t)\) are bounded functions of time. When the value of the scaling factor \(\mathcal{L}(t)\) changes, the size of the desired region will also change, which allows the USV to track a time-varying moving region. The USV moves in 3-DOF. Thus, we select \(n=3\). Then, the desired object function will be \(f(\delta\eta_{\mathcal{L}})=\left[f(\delta\eta_{\mathcal{L}1}),\ f(\delta\eta_{ \mathcal{L}2}),\ f(\delta\eta_{\mathcal{L}3})\right]^{T}\leq 0\). The desired region can be defined as  
\[\left\{ \begin{array}{c}f(\delta\eta_{\mathcal{L}1})=\left(x_{\mathcal{L}}-x _{0}\right)^{2}+\left(y_{\mathcal{L}}-y_{0}\right)^{2}-R_{p}^{2}\ \leq 0\\ f(\delta\eta_{\mathcal{L}2})=\left(x_{\mathcal{L}}-x_{0}\right)^{2}+\left(y_{ \mathcal{L}}-y_{0}\right)^{2}-R_{p}^{2}\ \leq 0\\ f(\delta\eta_{\mathcal{L}3})=\left(\psi_{\mathcal{L}}-\psi_{0}\right)^{2}-R_{ yaw}^{2}\leq 0\end{array} \right., \tag{4}\]  
where \(\boldsymbol{R}=\left[R_{p},R_{p},R_{yaw}\right]^{T}\) is the region boundary. A nonsingular matrix is defined as follows:  
\[\mathcal{L}(t)=\left[ \begin{array}{ccc}l_{x}&0&0\\ 0&l_{y}&0\\ 0&0&l_{\psi}\end{array} \right], \tag{5}\]  
Figure 3: Example of objective function. Figure 2: Closed loop diagram of the controller structure. where \(l_{x}\), \(l_{y}\), and \(l_{\psi}\) are the scaling factors of \(\delta\eta_{\mathcal{L}1}\), \(\delta\eta_{\mathcal{L}2}\), \(\delta\eta_{\mathcal{L}3}\), respectively, and  
\[\left[ \begin{array}{c}x_{\mathcal{L}}-x_{0}\\ y_{\mathcal{L}}-y_{0}\\ \psi_{\mathcal{L}}-\psi_{0}\end{array} \right]=\mathcal{L}(t)\left[ \begin{array} []{c}x-x_{0}\\ y-y_{0}\\ \psi-\psi_{0}\end{array} \right]. \tag{6}\]  
The desired region decreases with the increase of the scaling factors, and vice versa. The corresponding potential energy function for the desired region described in Equation (4) can be specified as  
\[P(\delta\eta_{\mathcal{L}i})=\frac{k_{pi}}{2}[\max(0,\,f(\delta\eta_{\mathcal{ L}i}))]^{2},\,i=1,\,2,\,3, \tag{7}\]  
where \(k_{pi}\) is a positive scalar. That is:  
\[P(\delta\eta_{\mathcal{L}i})=\left\{ \begin{array}{c}0,\quad f(\delta\eta_{ \mathcal{L}i})\leq 0\\ \frac{k_{pi}}{2}f(\delta\eta_{\mathcal{L}i})^{2},\quad f(\delta\eta_{ \mathcal{L}i})>0\end{array} \right.,\,i=1,\,2,\,3.