## 4 Simulations and Discussion
### Circular Trajectory Simulation  
The following circular trajectory was chosen for the circle trajectory simulations:  
\[\left\{\begin{array}{l}x_{d}=\cos(0.125t)\\ y_{d}=\sin(0.125t)\end{array}\right. \tag{46}\]  
Figure 3: _Cont._  
Figure 2: Straight-line trajectory tracking in three cases: (**a**) trajectory tracking in the _xy_ plane; (**b**) trajectory tracking in the _x_-axis and the _y_-axis directions.  
Case 1 was without environment disturbances,  
Case 2 had constant environment disturbances \(\dot{\textbf{d}}_{1}\), chosen as described in Equation (44).  
Case 3 had time-varying environment disturbances \(\textbf{d}^{{}^{\prime}}_{2}\), as described in Equation (45).  
The initial conditions of the USV and the parameters of the disturbance observer were consistent with the parameters of the linear trajectory. The parameters used to design the NTSM controller based on Equations (37) and (42) are listed in Table 3.  
Figures 6-8 show the simulation curves for the circular trajectory. From the simulation results, it can be seen that for circular trajectories in three different cases, the designed control strategy enabled the USV to overcome the external disturbances to achieve tracking on circular trajectories. Figure 6a illustrates the overall trajectory tracking of the USV in the \(AXY\) plane. Figure 6b presents the trajectory tracking curves over time in the surge (\(X-\)axis) and sway (\(Y-\)axis) directions. The observation approximation effect and estimation errors are presented in Figure 3. Figure 7a shows the six state curves including position and velocity curves during the USV navigation. It can be observed in Figure 7b that, in the simulations with time-varying disturbances, the velocity curve has smaller fluctuations compared with the other two cases. This is because there is a certain error in the observed effect of the disturbance observer for the time-varying disturbances. From the position error curve and velocity error curve in Figure 7c,d, it can be seen that the error has a good convergence pattern, which means that the USV can achieve circular trajectory tracking. Figure 8 shows the variation curves for the surge force and yaw moment.  
Figure 3: Disturbance observer simulation results: (**a**) actual values and observed values for constant disturbances; (**b**) actual values and observed values for time-varying disturbances; (**c**) constant disturbance errors; (**d**) time-varying disturbance errors.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value Used in Case 1** & **Value Used in Case 2** & **Value Used in Case 3** \\ \hline \(\beta_{1}\) & 0.5 & 2.5 & 5 \\ \(\beta_{2}\) & 0.3 & 1.5 & 5 \\ \(\varepsilon_{1}\) & 1.4 & 0.3 & 0.01 \\ \(\varepsilon_{2}\) & 2.5 & 1 & 0.5 \\ \(\phi\) & 0.01 & 0.01 & 0.01 \\ \(p_{1}\) & 3 & 3 & 3 \\ \(p_{2}\) & 3 & 3 & 3 \\ \(q_{1}\) & 5 & 5 & 5 \\ \(q_{2}\) & 5 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Controller parameters for the circular trajectory.  
Figure 4: Position and velocity curves: (**a**) the actual position of USV in three cases; (**b**) the actual velocity of USV in three cases; (**c**) position tracking errors in three cases; (**d**) velocity tracking errors in three cases.  
Figure 5: Surge force and yaw moment in three cases.  
Figure 6: Circular trajectory tracking in three cases: (**a**) trajectory tracking in the \(x\)-\(y\) plane; (**b**) trajectory tracking in the \(x\)-axis and the \(y\)-axis directions.  
Figure 7: Position and velocity curves: (**a**) the actual position of USV in three cases; (**b**) the actual velocity of USV in three cases; (**c**) position tracking errors in three cases; (**d**) velocity tracking errors in three cases.  
The simulations for the straight-line and circular trajectories indicate that the designed control strategy is able to overcome external environmental disturbances in the process of trajectory tracking.