## 4 Simulation Results
(**a**) Trajectory of the USV using the traditional RC method; (**b**) Zoom view of the results from \(-10\) to \(40\) m in the X direction; (**c**) Zoom view of the results from \(790\) to \(805\) m in the X direction; (**d**) Zoom view of the results from \(225\) to \(298\) m in the X direction. Figure 5: Results of the NTTC method. (**a**) Trajectory of the USV using the NTTC method; (**b**) Zoom view of the results from \(-10\) to \(40\) m in the X direction; (**c**) Zoom view of the results from \(790\) to \(805\) m in the X direction; (**d**) Zoom view of the results from \(225\) to \(298\) m in the X direction. Figure 4a shows the region tracking results of the RC method. Figure 4b-d show the details of several parts of the tracking results. The red solid lines are the boundaries of the desired moving region. The dashed circle denotes the desired moving region, and the dashed line is the trajectory of the center of the moving-region. The blue dashed line is the trajectory of the USV that is controlled by the RC method. Figure 8: Position tracking errors in Y direction. Figure 6: Yaw angles of the USV under the two control methods. Figure 7: Position tracking errors in X direction. Figure 9: Yaw tracking errors. Figure 5a shows the trajectory tracking results of the NTTC method. Figure 5b-c are used to show the details of several parts of the tracking results. The red solid lines are the boundaries of the desired moving region. The dashed circle denotes the desired moving region, and the dashed line presents the trajectory of the center of the moving-region. The blue dashed line is the trajectory of the USV that is controlled by the NTTC method. Figures 4 and 5 indicate that the tracking trajectories of the USV fit the desired moving region. However, the USV converges to the boundaries of the desired moving region, as shown in Figure 4. Since the USV's motion is often influenced by winds, waves or currents, it will move out of the desired region by those disturbances. However, the USV converges into the desired moving region rather than the boundaries of the desired moving region under the proposed NTTC control scheme, which works better than the traditional RC control scheme (Figure 5). Figure 6 shows the yaw tracking results of the USV. From the figure, the yaw converges into the desired yaw moving region under the NTTC method. However, under the RC method, the USV's yaw converges to the bounds of the desired yaw moving region. Figures 7-9 show the tracking errors of the USV. In Figure 7, the black dashed lines are the boundaries of the desired region in the X direction, the red dashed line is the tracking error using the RC method, and the blue line shows the tracking error of the NTTC method. We can see that the errors are closed to zero, that is to say, under the RC and the NTTC method, the tracking errors in the X direction converge into \([-2,\ 2]\), which satisfies the desired position region. In Figure 8, the black dashed lines are the boundaries of the desired region in the Y direction, the red dashed line is the tracking error using the NTTC method, and the blue line shows the tracking error of the RC method. We can see that, under the RC method, the position tracking error in the Y direction converges to the position regional boundaries. However, when there are external disturbances, the error will be easy to move out of the desired region, which is not good for the USV's control. However, under the NTTC method, the position tracking error converges into the desired region \([-2,\ 2]\), and it satisfies the desired position region. Figure 9 shows the yaw tracking errors. The yaw error under the NTTC method converges into \([-5^{\circ},\ 5^{\circ}]\), which satisfies the desired yaw region. However, the yaw error under the RC method converges to the boundaries of the desired yaw region, and even out of the boundaries at some time points, such as 300 s to 400 s. Hence, the NTTC method is better than the RC method. Figure 10: Velocities of the USV.