## 4 Simulation Results  
In this section, a numerical simulation of an HAUQ is presented to verify the effectiveness of the proposed position and attitude control algorithm and the observer strategy using the MATLAB platform. Design a water-air crossing flight scene in which the center of gravity of the HAUQ comes out of the water from point \((0,0,-0.5\ m)\) and climbs to point \((0,0,0.5\ m)\) to drain the water inside the body. Then, the control effect of the proposed position and attitude control algorithm and the disturbance observer are verified. The design parameters of the HAUQ are given by  
\[M=7\ Kg,\ g=9.8\ m/s^{2},\ l=0.5\ m,\ C=1 \tag{67}\]  
and the fundamental moment of inertia matrix is  
\[J= \begin{bmatrix}0.325&0&0\\ 0&0.285&0\\ 0&0&0.181\end{bmatrix}  \tag{68}\]Assume that the disturbance change caused by a large amount of water in the slender body is  
\[M_{\Delta}=2\times 0.005^{t},\;J_{\Delta}= \begin{bmatrix}0.01\times 0.005^{t}&0. 01\times 0.005^{t}&0.01\times 0.005^{t}\\ 0.01\times 0.005^{t}&0.01\times 0.005^{t}&0.01\times 0.005^{t}\\ 0.01\times 0.005^{t}&0.01\times 0.005^{t}&0.01\times 0.005^{t}\end{bmatrix} \] \[L_{\Delta}= \begin{bmatrix}F_{e}^{t}(t)&F_{e}^{t}(t)&-F_{e}^{t}(t)& -F_{e}^{t}(t)\\ F_{e}^{t}(t)&F_{e}^{t}(t)&-F_{e}^{t}(t)\\ 0&0&0&0\end{bmatrix} ,\;F_{e}^{t}(t)=0.02\times 0.005^{t} \tag{69}\]  
The unmodeled items of the complex dynamics of the HAUQ are composed of state variable feedback, the wave disturbance signal, and the random noise which are given by  
\[A_{P}(t,\eta)= \begin{bmatrix}0.25&1.00&3.00\\ 1.00&5.00&1.00\\ 1.00&1.00&0.10\end{bmatrix} \hat{P}+ \begin{bmatrix}0.25 sin\big{(}\frac{3}{4} \pi t\big{)}\\ 0.25 cos\big{(}\frac{3}{4}\pi t\big{)}\\ 0.25 cos\big{(}\frac{3}{4}\pi t\big{)}\end{bmatrix} + \begin{bmatrix}\sqrt{2}rand (1)\\ \sqrt{2}rand(1)\\ \sqrt{2}rand(1)\end{bmatrix} \] \[R_{\Theta}J^{-1}B_{\omega}(\omega^{b})= \begin{bmatrix}0.35&0. 00&0.00\\ 0.00&0.10&0.00\\ 0.00&0.00&0.50\end{bmatrix} \hat{\Theta}+ \begin{bmatrix}\sqrt{0.000001}rand(1) \\ \sqrt{0.0000001}rand(1)\\ \sqrt{0.0000001}rand(1)\end{bmatrix}  \tag{70}\]  
where \(\sqrt{2}rand(1)\) is a Gaussian random signal with the standard deviation \(\sqrt{2}\), the mean value \(0\), and the variance \(1\). \(\sqrt{0.0000001}rand(1)\) is also a Gaussian random signal with the standard deviation \(\sqrt{0.0000001}\), the mean value \(0\), and the variance \(1\). For the unknown uncertainty caused by complex dynamics, the polynomial regression method is adopted and the adaptive law is used to estimate the unknown weight. Define \(\eta_{1}/1000=[\eta_{1}^{x},\eta_{1}^{y},\eta_{1}^{z}]^{T}\), \(\eta_{2}=[\eta_{2}^{x},\eta_{2}^{y},\eta_{2}^{z}]^{T}\), \(\xi_{1}=[\xi_{1}^{y},\xi_{1}^{y},\xi_{1}^{y}]^{T}\), and \(\xi_{2}=[\xi_{2}^{y},\xi_{2}^{y},\xi_{2}^{y}]^{T}\), where the position feedback estimation \(\eta_{1}\) after dividing by \(1000\) is used to estimate the uncertainty to avoid the large initial values. The basis functions are given by  
\[\Phi(\eta)= [(\eta_{1}^{x})^{2},(\eta_{1}^{y})^{2},(\eta_{1}^{z})^{2},(\eta_{2 }^{x})^{2},(\eta_{2}^{y})^{2},(\eta_{2}^{z})^{2},\eta_{1}^{x}\eta_{1}^{y},\eta _{1}^{z}\eta_{1}^{z},\eta_{2}^{x},\eta_{1}^{x}\eta_{2}^{y},\eta_{1}^{x}\eta_{2 }^{z}  
Figure 4: Disturbance observer state variable of the hybrid aerial underwater quadrotor. Figure 5: The unknown uncertainty and its estimate of the hybrid aerial underwater quadrotor. Figure 2 shows the position and the velocity curves in the \((x,y,z)\) directions. Figure 2a,b illustrate that the position changes of the HAUQ in the two directions of the \(x-\)axis and \(y-\)axis are less than \(10\times 10^{-4}\)\(m\) and \(4\times 10^{-4}\)\(m\), which shows that the HAUQ climbs. After \(t>0.6\) seconds, Figure 2c illustrates that the HAUQ reaches the fixed\(-\)point hover drainage position. Figure 2d gives the velocity curves of \((x,y,z)\). The velocity in the \(x-\)axis and \(y-\)axis is less than \(0.02\)\(m/s\), and that in the \(z-\)axis is less than \(8\)\(m/s\).