## 4 Simulation and Discussion
### Simulation of the Vehicle Crossing the Water Surface for Takeoff  
In the following simulation, the initial position and attitude of the HAUV are, respectively, selected to be \(x_{0}=0.3\), \(y_{0}=0.4\), \(z_{0}=-2\), and \(\phi_{0}=0,\theta_{0}=0,\psi_{0}=0\). To verify the robustness of the designed controller, the lumped uncertainty is reset as follows:  
\[ \begin{split}& d_{x}=0.3\times(cos(0.4t)+k_{s}),d_{y}=0.3\times( sin(0.5t)+k_{s}),d_{z}=0.3\times(sin(0.7t)+k_{s})\\ & d_{\phi}=0.3\times(cos(0.4t)+k_{s}),d_{\theta}=0.3\times(sin(0.5t )+k_{s}),d_{\psi}=0.3\times(sin(0.7t)+k_{s})\end{split}  \tag{44}\]  
The desired water-exit trajectory is determined using Equation (45):  
\[x_{d}= \begin{cases}1,t\in[0,20)\\ sin(0.1\pi(t-20)),t\in[20,50]\end{cases} ,y_{d}= \begin{cases}1,t\in[0,20)\\ cos(0.1\pi(t-20)),t\in[20,50]\end{cases} ,z_{d}= \begin{cases}0.15t-2,t\in[0,20) \\ 0.25t-4,t\in[20,50]\end{cases}  \tag{45}\]  
Autonomous takeoff and landing control remains a challenge for our designed coaxial HAUV because a water surface is usually not calm and complex sea conditions often stimulate compound disturbances to the vehicle. To reduce the difficulty faced by the HAUV when crossing disturbed water, we designed a piecewise trajectory. In the process of breaking through the trans-medium water surface, our trajectory is vertical, and the vertical climbing speed is selected to be small, at 0.15 m/s. After completely leaving the water surface, in order to prevent falls caused by the beating of wind and waves, we choose a higher climbing speed of 0.25 m/s to allow the HAUV to quickly escape from the near-water surface with a spiral trajectory. In the position loop, the trajectory tracking performance of the HAUV based on the design control strategy is shown in Figures 4 and 5. The simulation results show that the proposed method has strong robustness under complex interference. Meanwhile, the feedforward PD controller lacks robustness to disturbance, resulting in large fluctuations in trajectory tracking. The three subgraphs shown in Figure 4 describe the effect of position tracking in the \(x\), \(y\), and \(z\) directions, respectively. As shown in Figure 4, we can see that the proposed position controller enables the HAUV to track the desired value quickly and accurately, even if the desired position changes rapidly. Figures 4 and 5, respectively, show the effect of three-dimensional trajectory tracking and the effect of three sections. According to the effect of the coaxial HAUV tracking the desired trajectory, we can observe that the HAUV maneuvers from its initial point to the desired trajectory from under water, rises vertically through the water into the air, and, finally, performs an upward spiral maneuver in the air. The results show that the proposed controller can accurately track the specified trajectory instruction under conditions of parameter uncertainty and disturbance. Figure 6 shows the results of 2D planar trajectory tracking. It can be seen from the figure that there is a large deviation between the position of the vehicle and the expected position at the initial moment, but the position tracking approaches the expected trajectory within a very short time, which shows the good control accuracy of the ANFTSMC. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(b_{i}\) & 0.2 & \(b_{i+1}\) & 0.8 \\ \(c_{i}\) & 1.21 & \(\delta\) & 0.5 \\ \(\alpha_{i}\) & 2 & \(\beta_{i}\) & \(5/3\) \\ \(\mu_{0i}\) & 0.5 & \(\mu_{1i}\) & 0.001 \\ \(\mu_{2i}\) & 0.01 & \(\varepsilon\) & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The controller and adaptive law parameters used in the simulation. Figure 4: Effect of the coaxial HAUV’s position tracking. Figure 5: Effect of the coaxial HAUV’s 3D trajectory tracking when exiting water. Figure 6: Exiting-water tracking responses of the coaxial HAUV in the 2D plane. Figure 7 shows the attitude angle tracking effect of the HAUV.