## 5 Numerical Simulation  
In this section, the main purpose is to verify the effectiveness of the proposed method. For the sake of comparison, the proposed scheme is compared with a sliding mode robust controller ([PERSON], et al., 2012) [28]. The underactuated USV is selected as a 1:70 CyberShip II [46,47], and the corresponding parameters of the dynamics are described in Table 1. Meanwhile, the control parameters and initial conditions are illustrated in Table 2.  
In the simulation, the desired trajectory is described as \(x_{d}=300\sin\left(0.03t\right)\), \(y_{d}=300\cos\left(0.03t\right)\), and the unmodeled dynamics are selected as  
\[\left[ \begin{array}{c}\Delta f_{u}\\ \Delta f_{v}\\ \Delta f_{r}\end{array} \right]=\left[ \begin{array}{c}-0.2d_{11}u^{2}-0.1d_{ 11}u^{3}\\ -0.2d_{22}v^{2}-0.1d_{22}v^{3}\\ -0.2d_{33}r^{2}-0.1d_{33}r^{3}\end{array} \right] \tag{51}\]  
Meanwhile, the time-varying ocean disturbances are chosen as  
\[\left[ \begin{array}{c}d_{u}\\ d_{v}\\ d_{r}\end{array} \right]=\left[ \begin{array}{c}1+1.5\sin\left(0.2t\right)+0.5 \cos\left(0.5t\right)\\ 1+1.2\sin\left(0.1t\right)+0.1\cos\left(0.4t\right)\\ 1+1.3\sin\left(0.5t\right)+0.2\cos\left(0.3t\right)\end{array} \right] \tag{52}\]  
In addition, in order to further verify the superiority of the proposed controller, MIAC index [27] is hired to quantify the tracking errors \(j_{e}\left(j=x,y\right)\), which can be expressed as  
\[MIAC=\frac{1}{t_{f}-t_{0}}\int_{t_{0}}^{t_{f}}|j_{e}\left(t\right)|du \tag{53}\]  
where \(t_{f}\) represents simulation time. The simulation results are depicted in Figures 3-10, and the performance indices are summarized in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \(m_{11}\) & 25.8 \\ \(m_{22}\) & 33.8 \\ \(m_{33}\) & 2.76 \\ \(d_{11}\) & 0.72 \\ \(d_{22}\) & 0.8896 \\ \(d_{33}\) & 1.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of CyberShip II. \begin{table}
\begin{tabular}{c c} \hline \hline
**Initial Condition** & \multicolumn{2}{c}{**Controller Parameters**} \\ \hline \(x\left(0\right)=0m\) & \(A_{u}=1\), \(A_{r}=1\), \(\tau_{\mathrm{t}\max}=250\), \(\tau_{\mathrm{t}\min}=-250\) \\ \(y\left(0\right)=0m\) & \(\tau_{\mathrm{t}\max}=450\), \(\tau_{\mathrm{t}\min}=-450\), \(K_{\mathrm{t}\mu}=4\), \(\sigma_{u}=0.1\) \\ \(\dot{\psi}\left(0\right)=0 rad\) & \(K_{\mathrm{\varphi}}=4\), \(\sigma_{r}=0.1\), \(\lambda=4.5\), \(M=2\), \(\gamma_{1}=0.1\) \\ \(u\left(0\right)=0m/s\) & \(\gamma_{2}=0.1\), \(\varepsilon_{1}=1\), \(\varepsilon_{2}=0.05\), \(\rho_{1}=0.1\), \(\rho_{2}=0.1\) \\ \(v\left(0\right)=0m/s\) & \(\Gamma_{1}=2.8\), \(\kappa_{1}=0.06\), \(\Gamma_{2}=2\), \(\kappa_{2}=0.005\), \(\varepsilon_{1}=10\), \(d_{\mathrm{\varphi}0}^{*}=0.1\) \\ \(u\left(0\right)=0 rad/s\) & \(\phi_{1}=0.8\), \(\varepsilon_{2}=5\), \(\phi_{2}=0.5\), \(k_{\mathrm{s}1}=1\), \(k_{\mathrm{s}2}=1\), \(d_{\mathrm{\varphi}0}^{*}=0.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial conditions and controller parameters. Figure 4: Position tracking results of circular trajectory. Figure 5: Tracking errors results of circular trajectory. Figure 3: Tracking performance results of circular trajectory. Figure 8: Disturbances and their bound estimates. Figure 6: Velocity errors results of circular trajectory. Figure 7: Control inputs of circular trajectory. Figure 3 shows the simulation results for circular tracking trajectory with the uncertain system dynamics and time-varying ocean disturbances. From Figure 3, it is obvious that two controllers can track the given trajectory accurately. However, the proposed scheme converges faster than that of ([PERSON] et al., 2012). Figure 4 demonstrates the position tracking results. It is evident that the proposed scheme provides better tracking performance than that of ([PERSON], et al., 2012). The results of the tracking errors and the velocity errors are presented in Figures 5 and 6, respectively.