## 4 Simulation Results
(**a**) Surge velocities of the USV using the RC and NTTC method; (**b**) Sway velocities of the USV using the RC and NTTC method; (**c**) Yaw velocities of the USV using the RC and NTTC method. Figure 10 shows the velocities of the USV using the RC method and the NTTC method. In Figure 10, Figure 10a shows the \(u\) velocities using the NTTC method and the RC method, Figure 10b shows the \(v\) velocities and Figure 10c shows the yaw velocities. We can see that, at \(t=200\) s, \(600\) s and \(1400\) s, there are overshoots for the velocities using the RC method, however, the velocities using the NTTC method are smoother than using the RC method. Figures 11 and 12 are the control inputs during the trajectory tracking. In Figure 11, the control inputs \(\tau_{u}\), \(\tau_{v}\), \(\tau_{v}\) using the RC method are chattering during some time point. However, in Figure 12, the control inputs \(\tau_{u}\), \(\  
In Figure 13, the desired trajectory of the moving region is a straight line. The USV tracks the moving region using the traditional RC method. The trajectory tracking simulation results at \(t=0\) s, 10 s, 20 s, 80 s, 120 s, 200 s are also presented in Figure 13. The circle is the desired moving region, the blue dashed line is the trajectory of the desired moving region's center. The red dashed line is the trajectory of the USV under the RC control method and the blue triangle shows the USV. We can see that because of influenced by the wave disturbances, the trajectory of the USV is chattering. After  
Figure 4:0 Position errors of the USV under the traditional RC method. Figure 4:1 Control inputs of the USV under the traditional the method. about \(t=20\) s, the USV converges to the boundary of the desired moving region, which is the same as the results in Figure 4:2 However, Figure 14 shows that the position error in the X direction converges to the boundary of the desired error region. Under wave disturbances, the USV will move in and out of the desired moving-region frequently, which will cause frequently control inputs as shown in Figure 4:3 Figure 15 is the results of control inputs \(\tau_{u},\ \tau_{v},\tau_{r}\) of the USV using the RC control method. We can see that the control inputs chattering frequently under wave disturbances, which is bad for the USV's propellers. Nevertheless, in Figure 16, the USV converges into the moving region other than the boundary of the desired moving region under the NTTC method. To be consistent with the simulation in Figure 13, the trajectory tracking simulation results at \(t=0\) s, \(10\) s, \(20\) s, \(80\) s, \(120\) s, and \(200\) s are also shown in Figure 4:4 The circle is the desired moving region, the blue dashed line is the trajectory of the desired moving region's center. The red dashed line is the trajectory of the USV under NTTC control method and the blue triangle shows the USV. We can see that the trajectory is influenced by the wave disturbances, so it isn't a straight line. After about \(t=20\) s, the USV converges into the desired moving region, which is the same as the results in Figure 4:5 Figure 17 show the position errors using NTTC method. The difference with Figure 14 is that the position error in X direction converges into the error region other than the boundary of the error region. Figure 18 is the results of control inputs \(\tau_{u},\ \tau_{v},\tau_{r}\) of the USV using NTTC method, we can see that the NTTC method gives more smooth control inputs for the USV when there are disturbances. To compare the control inputs of the two methods, we use the Root-Mean-Square (RMS) value of the control inputs to measure their performance. The comparisons are shown in Figure 4:6 The RMSs of control inputs \(\tau_{u},\ \tau_{v},\tau_{r}\) using NTTC method are smaller than that using the RC method. The comparison of the simulation results of Figures 15 and 18 indicates that the NTTC method works better under disturbances than the RC method.