## 4 Simulation and Discussion
### Working Case 2 (Vertical Climbing Trajectory)  
In the previous section, we tested the performance comparison between the designed controller and controller in [16]. The performance of the controller after the disturbance observer is removed is tested here, and other control parameters are the same as the controller designed in this paper. The trajectory of working case 2 is shown in the following equation. The vertical trajectory was chosen to test the ability of the coaxial HAUV to perform the fastest transmedium maneuvering in a disturbed environment. \[\left\{ \begin{array}{l}x_{d}=0\\ y_{d}=0\\ z_{d}=0.1t-2\\ \vartheta_{d}=0.01\end{array} \right. \tag{68}\]  
Some simulation results for case 2 are provided in Figures 14-23. Considering that the water surface of the real ocean environment is undulating, the spiral ascent under the wave water condition may cause the body to hit the wave and lead to the failure of take-off. Therefore, a cross-domain maneuver strategy of vertical climbing is adopted for condition 2. Figure 14 shows the tracking evolution results of fixed-point climbing, which shows that the translation subsystem can still obtain better tracking results with DO enhancement. It can be seen in Figure 15 that the HAUV starts from underwater and passes through the wave surface vertically, showing a good 3D tracking effect. Figure 16 illustrates the rendering of the proposed observer for the unknown disturbance compared with the set value. In the results of disturbance estimation, the designed observer can quickly obtain the exact observation values. In particular, compared with the position loop, the observation value of the attitude loop observer fluctuates greatly in the initial stage. In future work, the optimal design should be carried out to avoid such situations. Figure 17 shows the time evolution results of the designed adaptive law, and the adaptive parameter estimates are stable and bounded. At the same time, the adaptive parameter values of the two controllers are obviously different, which means that the position controller adopted in this study can reduce chattering. Figure 13: Evolution of control input signal in working case 1. Figures 18 and 19 show the attitude-tracking effect of the proposed inner loop controller; it can be observed that the HAUV tracks the desired Euler angle quickly, and the four parameters of the quaternions can accurately track the expected quaternions. At the same time, Figure 20 shows the tracking error of the quaternion and the fluctuation of the error in a small neighborhood of 0. The above results show that the controller based on observer enhancement technology can obtain a better tracking effect in both steady-state error and convergence characteristics. Figure 14: Trajectory tracking evolution of fixed-point ascent motion for case 2. Figure 15: Three-dimensional trajectory evolution of vertical climb in case 2. Figure 21 shows the change process of the learning rate parameter under working case 2. In addition to the rapid increase in parameters in the initial stage, the learning rate parameters in the whole process are stable near the initial value, indicating that the control error of the controller is very small. Figure 22 shows the evolutionary effects of position and attitude-tracking errors. It is obvious that the controller with DO has better error convergence characteristics. Figure 23 shows the change process of the control signal after calculation. It can be seen that in the process of cross-media simulation, the changes in the four control inputs are stable and can be changed in real time according to the control target, and the control signal output of the adopted controller is more accurate. In general, vertical cross-domain maneuvering is a more conservative and robust strategy, which can be realized with a small attitude change. The horizontal maneuvering ability can be abandoned in bad ocean conditions to prioritize the successful completion of cross-domain processes. Figure 19: Evolution of quaternion tracking control effect in case 2.