## 4 Controller Design for Unmanned Helicopter Hovering Position  
Combined with Equations (3)-(5), Equations (1) and (2) can be concretely expressed as Formula (23).  
\[\left\{\begin{array}{l}m\tilde{P}_{x}=-(\sin\varphi\sin\psi+\cos\varphi\sin \theta\cos\psi)T_{mr}+n_{Px}\\ m\tilde{P}_{y}=(\sin\varphi\cos\psi-\cos\varphi\sin\theta\sin\psi)T_{mr}+n_{Py }\\ m\tilde{P}_{z}=-T_{mr}(\cos\varphi\cos\theta)+mg+n_{p_{z}}\\ l_{x}\tilde{p}=(H_{mr}T_{mr}+k_{\beta})\sin b+T_{tr}H_{tr}\\ l_{y}\tilde{q}=(H_{mr}T_{mr}+k_{\beta})\sin a\\ l_{z}\tilde{r}=N_{mr}+T_{tr}D_{tr}\end{array}\right. \tag{23}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(m\) & 0.94 kg & \(\Omega_{t}/\Omega_{mr}\) & 4.73 \\ \(\rho\) & 1.29 kg/m\({}^{3}\) & \(I_{x}\) & 0.0650 kg\(\cdot\)m\({}^{2}\) \\ \(g\) & 9.81 N/kg & \(I_{y}\) & 0.1382 kg\(\cdot\)m\({}^{2}\) \\ \(R_{mr}\) & 0.248 m & \(I_{z}\) & 0.2105 kg\(\cdot\)m\({}^{2}\) \\ \(c_{mr}\) & 0.03 m & \(C_{la,mr}\) & 4.411 rad\({}^{-1}\) \\ \(R_{tr}\) & 0.051 m & \(C_{la,hr}\) & 2.729 rad\({}^{-1}\) \\ \(c_{tr}\) & 0.016 m & \(k_{\beta}\) & 45.96 N\(\cdot\)m \\ \(D_{tr}\) & 0.320 m & \(\gamma_{m}\) & 1.72 \\ \(H_{tr}\) & 0.012 m & \(I_{\beta}\) & \(3.75\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ \(H_{mr}\) & 0.077 m & \(P_{mr}\) & 41 W \\ \(\Omega_{tr}\) & 165 rad/s & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relevant parameters of small unmanned helicopter.  
where the conversion from the inertial coordinate frame to the body coordinate frame is involved. In addition, according to [17], the roll and pitch as well as the flapping angles are close to 0 rad when the helicopter is hovering, and thus there are the following approximations: \(sina=a\), \(simb=b\), \(sinq=\varphi\), \(sinq=\theta\) and \(cosa=csob=cos\varphi=cos\theta=1\).  
Then, Formula (23) is simplified as follows:  
\[\left\{\begin{array}{l}m\ddot{P}_{x}=-(\varphi\sin\psi+\theta\cos\psi)T_{mr }+n_{Px}\\ m\ddot{P}_{y}=(\varphi\cos\psi-\theta\sin\psi)T_{mr}+n_{Py}\\ m\ddot{P}_{z}=-T_{mr}+m_{S}+n_{P2}\\ I_{x}\dot{p}=(H_{mr}T_{mr}+\dot{k}_{\beta})b+T_{tr}H_{tr}\\ I_{y}\dot{q}=(H_{mr}T_{mr}+\dot{k}_{\beta})a\\ I_{z}\dot{r}=N_{mr}+T_{tr}D_{tr}\end{array}\right. \tag{24}\]  
The disturbance on the position orientation is considered. And the relations of the attitude angles are as follows:  
\[\begin{array}{l}\dot{\varphi}=p\\ \dot{\theta}=q\\ \dot{\psi}=r\end{array} \tag{25}\]  
According to Equation (13), since \(P_{mr}\) and \(\Omega_{mr}\) are constant, \(T_{tr}\) is consequently also constant. After trimming for the helicopter, the tail rotor torque will not change during the height variation, so there is no need to separately consider the yaw angle control. When the helicopter moves along the inertia coordinate frame \(X\)-axis direction and \(Y\)-axis direction, this accompanies changes in \(\varphi\) and \(\theta\). So, it is necessary to establish the relationship between attitude angles through attitude decoupling. We define  
\[\begin{array}{l}U_{x}=-(\varphi\sin\psi+\theta\cos\psi)\\ U_{y}=\varphi\cos\psi-\theta\sin\psi\end{array} \tag{26}\]  
then  
\[\begin{array}{l}\varphi=-U_{x}\sin\psi+U_{y}\cos\psi\\ \theta=-U_{x}\cos\psi-U_{y}\sin\psi\end{array} \tag{27}\]  
In addition, during the helicopter flights, the manipulator under the helicopter should stay contracted and stationary, ensuring that the UAM overall center of gravity and moment of inertia are not changed.  
In the outer loop command generator, the attitude angles can be calculated from Equation (27). Combining the angular velocity (\(p\), \(q\)) with Equations (13) and (22), the lateral cyclic pitch angle and the longitudinal cyclic pitch angle can be calculated. Finally, the servo's rotation angles \(\delta_{lat}\) and \(\delta_{lon}\) can be calculated. According to the height control law, the servo's rotation angle \(\dot{\delta}_{col}\) can be calculated.