## 6 Simulation
### UAM Hovering Attitude Control Under the Manipulator Operating State  
After the UAM reaches the target position under the constraint of position loop control, the manipulator runs for continuous operation in the air. The desired hovering attitude angle vector is \(\xi_{d}=[0,0,0]^{T}\). The inputs of the system are random rotation angles of three joints of the manipulator (shown in Figure 8), which simulates the real continuous aerial operation of a UAM. The system outputs are the helicopter's hovering attitude angles. The parameters of backstepping sliding mode control are as follows:  
\[k_{\xi}=\begin{bmatrix}10&0&0\\ 0&50&0\\ 0&0&18\end{bmatrix};k_{1}=\begin{bmatrix}10&0&0\\ 0&10&0\\ 0&0&10\end{bmatrix};k_{2}=\begin{bmatrix}10.5&0&0\\ 0&10.5&0\\ 0&0&10.5\end{bmatrix};\lambda=\begin{bmatrix}3&0&0\\ 0&3&0\\ 0&0&3\end{bmatrix} \tag{63}\]  
Figure 8: Angle curves of the manipulator joints.  
Figure 7: The servo’s rotation angle under the static manipulator state.  
Firstly, in order to evaluate the control effect, the step response of the attitude loop is presented in Figure 9, and the time-domain performances are confirmed in Table 6.  
The validity of the attitude loop controller is proved by Figure 10. In practical control, the desired helicopter hovering attitude angle vector is \(\xi_{d}=\left[0,0,0\right]^{T}\). Despite the influence generated by the operation of the manipulator to the entire system, the maximum amplitude is around 0.05 rad. However, the helicopter's hovering attitude angle fluctuation under the general backstepping controller is over 0.1 rad. The stronger robustness of the backstepping sliding mode controller is evidently proven.  
Combined with the analysis of the position control shown in Figure 11, the UAM is stabilized at the target point at about t = 3 s. Although the manipulator continuously  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\varphi\) & \(\theta\) & \(\psi\) \\ \hline Adjusting time/\(t_{\text{s}}\) & 8.2 s & 7.6 s & 7.8 s \\ Rise time/\(t_{\text{r}}\) & 2.1 s & 2.3 s & 2.1 s \\ Overshoot/\(\sigma\)\% & 22.2 & 16.7 & 18.3 \\ Steady state error (at 20 s)/\(e_{\text{ss}}\) & 0.05 & 0.04 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Time-domain performances in the attitude loop (with the error band of 5%).  
Figure 10: UAM’s hovering attitude under the running manipulator state.  
Figure 9: The step responses in the attitude loop.  
runs and influences the helicopter hovering, the hovering position of the UAM remains stable near the target point under the control of the position loop. The fine effect of the double-loop control strategy for UAM hovering position and attitude is proven.  
Moreover, based on the comparison shown in Figure 10, a further comparison is presented regarding the servo's rotation angle under two kinds of controllers. In Figure 12, the servo's rotation angle amplitudes under the control of the backstepping sliding mode are evidently smaller than the general backstepping control. The interval of the servo's rotation angle of [\(-\)1, 1] rad can be referred to by the model of D5830M. However, the rotation angle intervals of the servo under the general backstepping control are beyond the above range. Not only is the helicopter's hovering attitude stability poor under the general backstepping controller, but the feasibility of the actual control and the servo's amplitude limitation should be also further considered.