## 6 Simulation
### UAM Hovering Position Control Under the Stationary Manipulator State  
In this condition, the UAM system's center of gravity and moment of inertia are constant. The initial position of the UAM in the inertial frame is \((X,Y,Z)=(0,0,0)\), and the experiment commands the UAM to fly from an initial position with a stationary state to the desired position of \((P_{xx},P_{yr},P_{2r})=(2,5,3)\). Given that the total mass of the UAM is 0.94 kg, the disturbance functions in Equation (24) are set as \(n_{pi}(t)=0.2\sin 5t(N)\), where \(i=x,y,z\). This ensures a model perturbation over 20% after normalization. The parameters of sliding mode control are shown in Table 4.  
Firstly, the time-domain performances in the position loop are analyzed according to Figure 4. Since the disturbance of the system is considered, there is a slight chattering in the steady state. The control performances are shown in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(k_{z}\) & 0.9 & \(k_{y}\) & 5.5 \\ \(\beta_{z}\) & 1.6 & \(\beta_{y}\) & 2.8 \\ \(q_{z}\) & 1.1 & \(k_{\psi}\) & 3.8 \\ \(k_{x}\) & 2 & \(\beta_{\psi}\) & 4.8 \\ \(\beta_{x}\) & 2.5 & \(\tilde{k}_{x,y,z},\tilde{k}_{\psi}\) & 0.22, 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of sliding mode control. Figure 4: The step responses in the position loop. Then, considering the desired position \((P_{xx},P_{yx},P_{zr})=(2,5,3)\), in Figure 5, the UAM can reach there in about 3 s, and it can hover at the destination after that. Under the sliding mode controller, the position curves of the helicopter can converge near the expected value. It can be proven that the position loop is able to ensure the UAM completes the hovering at the destination. Furthermore, the error analysis is as shown in Figure 6. Under the persistent disturbance with an amplitude of 0.2 N, the amplitude of the error curve in the steady state is about 0.01 m, and the steady state error is less than 2%. The validity of the disturbance resistance is proven; in addition, the high precision and strong robustness of the system are testified. Moreover, considering the chattering problem of the ordinary sliding mode controller, we add the robust items \(\overline{k}_{x,y,z}\) in the progression laws to suppress chattering and disturbance. By comparing the hovering position errors of the general sliding mode controller (red curves) and improved sliding mode controller (blue curves), the proposed sliding mode controller with robust items can effectively suppress disturbance and weaken chattering. Regarding the calculation of the servo's rotation angle, since Formula (6) contains the item of induced velocity, harboring a strong coupling relationship with the helicopter's  
Figure 5: UAM’s hovering position under the static manipulator state: (**a**) position curve of UAM; (**b**) 3D position of UAM. Figure 6: UAM’s hovering position error. \begin{table}
\begin{tabular}{c c c c} \hline  & **X** & **Y** & **Z** \\ \hline Adjusting time/\(\mathsf{t_{s}}\) & 2.8 s & 2.0 s & 2.7 s \\ Rise time/\(\mathsf{t_{s}}\) & 1.8 s & 1.5 s & 1.9 s \\ Overshoot/\(\sigma\)\% & 0 & 0 & 0 \\ Steady state error (at 10 s)/\(\mathsf{e_{ss}}\) & 0.01 & 0.02 & 0.02 \\ \hline \end{tabular}
\end{table} 
Table 5: Time-domain performances in position loop (with the error band of 2%). flight velocity and \(T_{mr}\), solving the servo's rotation angles will become difficult. However, according to the previous measurement on flight experiments (m = 0.94 kg), the induced velocity is \(v_{i,mr}\in[4.12,4.32]\). Due to the small interval of this item, the average of \(v_{i,mr}=4.2\) is adopted for approximation. The rotation angles of the servo for controlling the helicopter's flight state are shown in Figure 7. Because of the continuous disturbance, there are still slight fluctuations during the late period.