## 4 Simulations  
In this section, the analysis of the simulation results for an ACV is described. The ACV parameters are provided from [11], which are listed in Table 1.  
To verify the effectiveness of the proposed controller, we approximate the nonlinear relationships between the actuator's input and output to constant values and adopt the backstepping controller for comparison. The backstepping controller is designed as:  
\[\theta_{backstepping}=0.5\bar{u}\]  
\[\bar{u}=-k_{2}u_{e}-mvr+d_{11}u+d_{11}v-\hat{W}_{u}H_{u}(x)+(z_{e}-\delta_{ \Delta})\cos\left(\psi_{e}\right)+m\dot{u}_{u}\]  
\[\dot{\hat{W}}_{u}=-\eta_{u}\hat{W}_{u}-u_{e}H_{u}(x)\]  
\[\delta_{backstepping}=0.07(t))\bar{r}\]  
\[\bar{r}=-k_{4}r_{e}+m\dot{u}_{r}-\left(k_{b}^{2}-r_{e}^{2}\right)\psi_{e}+d_{ 41}u+d_{42}v+d_{44}r-N_{res}-\hat{W}_{r}H_{r}(x)\]  
\[\dot{\hat{W}}_{r}=-\eta_{r}\hat{W}_{r}-(r_{e}/(k_{b}^{2}-r_{e}^{2}))H_{r}(x)\]  
The desired trajectory is composed with a straight line and a circular arc.The trajectory is generated by choosing \(u_{d}=30 knots\), \(v_{d}=0\), \(p_{d}=0\), \(r_{d}=0 rad/s\) for \(0s\leq t\leq 10s\) and \(r_{d}=0.1 rad/s\) for \(10s\leq t\leq 60s\). The ACV initial state variables are set as \(\eta(0)=[0,0,0,30 deg]^{T}\), \(v(0)=[30 knots,0,0,0]^{T}\), \(\eta_{d}(0)=[0,0,0,30 deg]^{T}\). The Nussbaum function \(N(s)=s^{2}cos(s)\) is introduced. The controller and update parameters are set as \(\sigma_{\Delta}=0.02\), \(k_{1}=0.35\), \(k_{2}=0.1\), \(k_{3}=1.5*Mass\), \(k_{4}=11\).  
The simulation results are shown in Figures 5-13. In this section, we refer to the controller designed in this article as the Nussbaum controller and the contrast controller as the backstepping controller. Figure 5 shows the trajectory tracking curves of the two controllers in the Earth coordinate system. The Nussbaum controller has a better tracking effect than the backstepping controller. The total position error \(z_{e}\) and yaw error \(\psi_{e}\) are presented in Figure 6. The position error curve illustrates that the Nussbaum controller converges faster and has a smaller amplitude. The partial enlarged view of the yaw error curve shows that the backstepping controller has periodic oscillations. The ACV position errors in the surge and sway directions are shown in Figure 7, which demonstrate that the proposed controller has a faster transient response and higher tracking precision compared to the backstepping controller. The virtual velocity tracking results are plotted in Figure 8. We can see that the surge velocity error under the Nussbaum controller can converge to near zero and remain stable with fast speed. Since both controllers introduce BLF to constrain the slew rate error, the virtual yaw rate errors are limited to a safe range, which can avoid side slip and tail swing phenomena caused by changes in the yaw rate that are too fast. Figures 9 and 10 show the evolution of the Nussbaum parameters about the pitch angle and rudder angle, respectively. Figure 11 is the roll angle curve. Due to the higher accuracy of the controller proposed in this paper, the control force is greater at the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Value** & **Variable** & **Value** \\ \hline \(m(kg)\) & 40,000 & \(I_{x}(kgm^{2})\) & \(1.8\times 10^{6}\) \\ \(I_{x}(kgm^{2})\) & \(2.5\times 10^{5}\) & \(S_{PP}(m^{2})\) & 45 \\ \(S_{LP}(m^{2})\) & 93 & \(S_{HP}(m^{2})\) & 260 \\ \(Q(m^{2}/s)\) & 140.8 & \(S_{c}(m^{2})\) & 212 \\ \(l_{sk}(m)\) & 65 & \(B_{c}(m)\) & 8.9 \\ \(h_{m}(m)\) & 2.4 & \(l_{c}(m)\) & 23.6 \\ \(h(m)\) & 1 & \(H_{hov}(m)\) & 5.9 \\ \(V_{w}(knot)\) & 5 & \(\beta_{w}(deg)\) & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main particulars of ACV.  
Figure 8: Surge speed error and yaw rate error of trajectory tracking.  
Figure 7: Position error of trajectory tracking.  
Figure 10: Nussbaum parameters.  
Figure 9: Nussbaum parameters.  
Figure 11: Roll angle of trajectory tracking.  
Figure 12: Pitch angle of the air propeller.  
Figure 13: Angle of the air rudder.