## 4 Simulation and Discussion
### Working Case 1 (Spiral Climbing Trajectory)  
The trajectory of selected working case 1 is shown in the following equation:  
\[\left\{ \begin{array}{l}x_{d}=2\sin(0.1\pi t)\\ y_{d}=2\cos(0.1\pi t)\\ z_{d}=0.1t-2\\ \varphi_{d}=0.01\end{array} \right. \tag{67}\]  
The main purpose of the trajectory design is to test the ability of the controller to track the trajectory in 3D space. Figures 4 and 5 show the tracking response of the position loop with the proposed controller and basic controller. It can be clearly seen that the proposed composite robust controller performs well under time-varying system model parameters and external disturbances. Figure 4 shows the tracking evolution results of the three channels of translation motion over time. Although a good tracking effect can be obtained for each channel under the action of two controllers, it can be seen from local magnification that the HAUV can track the desired trajectory more accurately under the proposed controller. It can be seen in Figure 5 that the HAUV trajectory starts from underwater through the wave surface and spirals up into the air, showing a good 3D tracking effect in the whole process. Figure 6 shows the effect diagram of the proposed observer's observation results for the unknown disturbance and the set value comparison. It can be seen in the estimation results that although the initial value of the observer is quite different from the observed disturbance value, and the accurate observation value is quickly obtained. The estimation error can be quickly reduced, which proves that the proposed observer can provide reliable disturbance estimation values for the controller design. In the local magnification image, the observer produces large fluctuations in the initial stage, and the actual observation error is not strictly 0 in the steady state stage. Here, the adaptive parameter evolution results after removing DO are also tested. Figure 7 shows the time evolution of the designed adaptive law, and it can be observed that the adaptive parameters of the position loop converge to a bounded constant after initial growth. When the adaptive law and the disturbance observer work together, the uncertainty compensation in the controller is assumed by the observer, the adaptive law plays an auxiliary role, and the adaptive parameter estimation is small. When the observer is removed, the adaptive law plays the main role, the estimated value is large, and obvious chattering phenomena may occur. Figure 4: Trajectory tracking evolution of translational motion in working case 1. Figure 5: Evolution of 3D spiral climb trajectory in working case 1. Figure 6: Observation effect of disturbance observer in working case 1. (**a**) Position loop disturbance observation effect. (**b**) Attitude loop disturbance observation effect. Figures 8 and 9 show the attitude-tracking effect of the proposed inner loop controller. It can be observed in Figure 8 that the controller can drive the HAUV to accurately track the desired attitude angle. It can be seen in the locally enlarged image that the attitude angle tracking curve fluctuates more obviously under the action of the controller in reference [16]. Figure 9 shows that the four parameters of the quaternion can accurately track the expected quaternion value. Figure 10 shows the tracking error of the quaternion. When the inner loop subsystem is disturbed, the DO-enhanced quaternion PD controller can obtain a better tracking effect in both convergence speed and steady-state performance under the DO compensation effect. Figure 11 shows the evolution process of the learning rate parameter. It can be observed that the learning rate parameter increases rapidly in the initial stage, which is also the reason for the faster convergence rate of error. The analysis-based reason is that the error is the largest in the initial stage, and the error change rate is also large under the drive of the controller, which puts the learning rate parameter in a large state.