## 2 Kinematics and Dynamic Models of USV
### Control Law Formulation  
The control object is to control the USV tracking the moving target region, and to make the USV's position \((x,y)\) and yaw \(\psi\) converge into the desired regions. That is, the desired objective function (Equation (4)) always holds. Based on the region error \(\Xi_{e}\), a useful reference vector \(\
u_{re}\) is defined as:  
\[\
u_{re}=\mathcal{I}^{-1}(\psi)\left(\dot{\eta}_{0}-\mathcal{L}(t)^{-1}\dot{ \mathcal{L}}(t)\delta\eta\right)-\epsilon\mathcal{I}^{-1}(\psi)\mathcal{L}(t) ^{-1}\Xi_{e}, \tag{12}\]  
where \(\epsilon\) is a positive constant, and \(\
u_{re}=\left[u_{re},\,v_{re},\,r_{re}\right]^{T}\). The matrix \(\mathcal{I}^{-1}(\psi)\) is the inverse of the rotation matrix \(\mathcal{I}(\psi)\). \(\dot{\mathcal{L}}(t)\) and \(\mathcal{L}(t)^{-1}\) are the time derivative and inverse of the scaling matrix \(\mathcal{L}(t)\), respectively. On the basis of \(\
u_{re}\), \(\Xi_{e}\), and the subsequent stability analysis, a filtered tracking error vector for the USV is defined as  
\[\mathbf{r}_{e}=\mathbf{v}-\mathbf{v}_{re}=\mathbf{v}-\mathbf{J}^{-1}(\psi)\left(\dot{\mathbf{\eta}}_{0} -\mathcal{L}(t)^{-1}\dot{\mathcal{L}}(t)\delta\eta\right)+\epsilon\mathcal{I} ^{-1}(\psi)\mathcal{L}(t)^{-1}\Xi_{e}. \tag{13}\]  
The time derivative of \(\mathbf{r}_{e}\) is calculated as  
\[\dot{\mathbf{r}}_{e}=\dot{\mathbf{J}}^{-1}(\psi)(\dot{\mathbf{\eta}}_{0}- \mathcal{L}(t)^{-1}\dot{\mathcal{L}}(t)\delta\eta)+\mathbf{J}^{-1}(\psi)[\dot\mathbf{{\eta}}_{0}-\mathcal{L}(t)^{-1}\dot{\mathcal{L}}(t)\dot{\delta\mathbf{\eta}}-( \mathcal{L}(t)^{-1}\ddot{\mathcal{L}}(t)+\] \[\dot{\mathcal{L}}(t)^{-1}\dot{\mathcal{L}}(t))\delta\eta]-\epsilon \dot{\mathbf{J}}^{-1}(\psi)\mathcal{L}(t)^{-1}\Xi_{e}-\epsilon\mathcal{I}^{-1}( \psi)(\mathcal{L}(t)^{-1}\dot{\Xi}_{e}+\dot{\mathcal{L}}(t)^{-1}\Xi_{e}),\]where \(\dot{\delta\mathbf{\eta}}=\dot{\mathbf{\eta}}-\dot{\mathbf{\eta}}_{0}\). Then, the open loop error of the USV is expressed as follows:  
\[\dot{\mathbf{M}}\dot{\mathbf{r}}_{e}+\mathbf{C}(\mathbf{\
u})\mathbf{r}_{e}+\mathbf{D}(\mathbf{\
u})\mathbf{r}_{ e}+\mathbf{g}(\cdot)+\Phi(\cdot)\mathbf{\theta}=\mathbf{\tau}, \tag{14}\]  
where \(\mathbf{g}(\cdot)+\Phi(\cdot)\mathbf{\theta}=\mathbf{M}\dot{\mathbf{v}}_{re}+\mathbf{C}(\mathbf{\
u}) \mathbf{v}_{re}+\mathbf{D}(\mathbf{\
u})\mathbf{v}_{re}\), \(\mathbf{\theta}\) is a vector of unknown parameters, and \(\Phi(\cdot)\) is a known regression matrix [27]. \(g(\cdot)\) is the known part of \(\mathbf{M}\dot{\mathbf{v}}_{re}+\mathbf{C}(\mathbf{\
u})\mathbf{v}_{re}+\mathbf{D}(\mathbf{\
u})\mathbf{v}_{re}\). Furthermore,  
\[\theta\Big{[}Y_{|r|p},\ Y_{r},\ Y_{|v|r},\ Y_{|r|r},\ N_{|r|p},\ N_{r},\ N_{|v|r}, \ N_{|r|r}\Big{]}^{T},\]  
\[\Phi(\cdot)\left[ \begin{array}{cccccc}0&0&0&0&0&0&0\\ -|r|v_{re}&-r_{re}&-|v|r_{re}&-|r|r_{re}&0&0&0\\ 0&0&0&0&-|r|v_{re}&-r_{re}&-|v|r_{re}&-|r|r_{re}\end{array} \right],\]  
\[g(\cdot)=\left[ \begin{array}{cccc}m_{11}\dot{u}_{re}+C_{13}r_{re}-X_{u}u_{r} -X_{|u|u|}u|u|u_{re}-X_{uuuu}u^{2}u_{re}\\ m_{22}\dot{v}_{re}+m_{23}\dot{r}_{re}+C_{23}r_{re}-Y_{v}v_{re}-Y_{|v|v|}v|v_{re} \\ m_{32}\dot{v}_{re}+m_{33}\dot{r}_{re}-C_{13}u_{re}-C_{23}v_{re}-N_{v}v_{re}-N_ {|v|v|}v|v_{re}\end{array} \right].\]  
The NTTC law can be proposed as:  
\[\mathbf{\tau}=-\mathbf{J}^{\mathbf{T}}(\psi)\mathcal{L}(t)\mathbf{\Xi}_{e}-\mathbf{\beta}_{v}\cdot \mathbf{r}_{e}+\Phi(\cdot)\cdot\dot{\mathbf{\theta}}+\mathbf{g}(\cdot)-\mathbf{J}^{\mathbf{T}}( \psi)\mathcal{L}(t)\big{(}\overline{\mathbf{K}}_{s}-\mathbf{\varpi}\big{)}\text{sgn}( \delta\mathbf{\eta}), \tag{15}\]  
where \(\mathbf{\beta}_{v}\) is a positive constant matrix. The estimated parameters \(\dot{\mathbf{\theta}}\) are updated using the following update law:  
\[\dot{\mathbf{\theta}}=-\Gamma\cdot\Phi(\cdot)^{T}\cdot\mathbf{r}_{e}, \tag{16}\]  
where \(\Gamma\) is a symmetric positive definite. The last term \(\mathbf{J}^{\mathbf{T}}(\psi)\mathcal{L}(t)\big{(}\overline{\mathbf{K}}_{s}-\mathbf{\varpi} \big{)}\text{sgn}(\delta\mathbf{\eta})\) in Equation (15) is the switching gain control term.