## 5 Hovering Attitude Control of the Unmanned Helicopter
### Controller Design of UAM's Hovering Attitude  
Regarding the problem of the UAM's hovering attitude stability with the manipulator continuous operation, the pure backstepping controller has a weak disturbance resistance. Thus, a backstepping integral sliding mode controller with higher robustness for the stability of UAM's hovering attitude is proposed in this section. Firstly, the second-order dynamic model of the UAM's attitude is divided into two subsystems: the angle vector \(\xi\) and angular velocity vector \(\omega\). The helicopter's attitude angle vector is defined as \(\xi=[\varphi,\theta,\psi]^{T}\), and the yield is \(\dot{\xi}=\omega=[p,q,r]^{T}\). The attitude angle error vector is calculated as \(e_{\xi}=\xi_{d}-\xi\), where \(\xi_{d}\) denotes the desired attitude angle vector, and it takes a time-derivative yield \(\dot{e}_{\xi}=\dot{\xi}_{d}-\omega\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{i}\) & \(\mathbf{\alpha_{i-1}}^{\text{\tiny{T}}}\) & \(\mathbf{a_{i-1}}\)/m & \(\mathbf{d_{i}}\)/m & \(\mathbf{\theta_{i}}^{\text{\tiny{T}}}\) & \(\mathbf{\theta_{i0}}^{\text{\tiny{T}}}\) \\ \hline
1 & 0 & 0 & 0.05 & \(\theta_{1}\) & 180 \\
2 & 90 & 0 & 0 & \(\theta_{2}\) & 90 \\
3 & 0 & 0.1 & 0 & \(\theta_{3}\) & 90 \\ \(E\) & — & 0.1 & — & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table of DH parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Link** & \(\mathbf{m_{i}}\)**/**kg** & \(\mathbf{j_{i}}\)**/**kg**-\(\mathbf{m^{2}}\)**= **diag**\(\{\mathbf{j_{i_{j_{j}}}},\mathbf{j_{i_{k}}}\}\) & \(\mathbf{O_{i}C_{i}/m}\) \\ \hline
0-B & 0.58 & \(\{6.50\times 10^{-2},13.82\times 10^{-2},21.05\times 10^{-2}\}\) & \([0,0,0]^{T}\) \\
1 & 0.09 & \(\{2.65\times 10^{-2},2.65\times 10^{-5},1.13\times 10^{-5}\}\) & \([0,0,0]^{T}\) \\
2 & 0.12 & \(\{1.95\times 10^{-5},1.18\times 10^{-4},1.18\times 10^{-4}\}\) & \([0.05,0,0]^{T}\) \\
3-E & 0.15 & \(\{1.80\times 10^{-5},1.09\times 10^{-4},1.09\times 10^{-4}\}\) & \([0.10,0,0]^{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of links. Regarding the subsystem of \(\xi\), the first Lyapunov function is assumed as  
\[V_{1}=\frac{1}{2}e_{\xi}^{2} \tag{53}\]  
and it takes the time derivative as  
\[\dot{V}_{1}=e_{\xi}\dot{e}_{\xi}=e_{\xi}(\dot{\xi}_{d}-\omega) \tag{54}\]  
In order to make the value of \(\dot{V}_{1}\) become negative definite, the virtual control quantity is set as \(\alpha=\dot{\xi}_{d}+k_{\xi}e_{\xi}\), satisfying \(k_{\xi}>0\). Thus, the value of \(\dot{V}_{1}\) is expressed as  
\[\dot{V}_{1}=-k_{\xi}e_{\xi}+e_{\xi}(\alpha-\omega) \tag{55}\]  
when \(\alpha-\omega=0\), \(\dot{V}_{1}\) is negative definite. Furthermore, regarding the subsystem of \(\omega\), the integral slide mode surface is designed as  
\[s=(\alpha-\omega)+\lambda\int_{0}^{t}(\alpha-\omega)dt \tag{56}\]  
where \(\lambda>0\) is the constant coefficient and \(s\) takes the time derivative as  
\[\dot{s}=\dot{\alpha}-\dot{\omega}+\lambda(\alpha-\omega) \tag{57}\]  
Then, according to the progression law of \(\dot{s}=-k_{1}s-k_{2}\text{sgn}(s)\), the control law of the system is designed as  
\[U_{\omega}=C^{-1}\Big{\{}l_{l}[\ddot{\xi}_{d}+k_{\xi}\dot{e}_{\xi}+\lambda( \dot{\xi}_{d}+k_{\xi}e_{\xi}-\omega)+k_{1}s+k_{2}\text{sgn}(s)]+\omega\times(I _{l}\omega)-T\Big{\}} \tag{58}\]  
where \(k_{1}\) and \(k_{2}\) are diagonal matrices with all the positive constant coefficient elements. Proof. : The Lyapunov function on the entire second-order system is considered as  
\[V=V_{1}+V_{2}=\frac{1}{2}e_{\xi}^{2}+\frac{1}{2}s^{2} \tag{59}\]  
According to the previous analysis, when the second subsystem regarding \(\omega\) is stable, the first subsystem of \(\xi\) is stable. Regarding the second subsystem, \(V_{2}\) is positive definite, following  
\[