# Aerial Grasping of an Object in the Strong Wind: Robust Control of an Aerial Manipulator
## 3 System Control
### Hex-rotor Control
#### 3.1.1 Translational and Rotational Dynamic Decoupling  
The system dynamic models in Equations (6) and (7) are the translational dynamic and rotational dynamic, respectively. In the hierarchical inner-outer loop control structure, the UAV is seen as a cascade system composed of two subsystems, that is, the translational dynamic subsystem and rotational dynamic subsystem, respectively. From Equation (6), we can see that the translational dynamic couples with the rotational dynamic through the rotation matrix \({}^{I}R_{B}\). In order to decouple them, a virtual control input, \(\
u_{1}\), is introduced [29], and it is defined as follows:  
\[\
u_{1}=-\frac{F_{t}}{m_{s}}I\ R^{B,d}e_{3}+ge_{3}+{}^{I}R_{B}(^{B}\omega_{b} \times(^{B}\omega_{b}\times{}^{B}r_{\infty}(q))) \tag{9}\]  
where \({}^{I}R_{B,d}\) is the rotation matrix determined by the desired attitude angle, \(\Phi_{b,d}=\left[\begin{array}{cccc}\varphi_{d}&\theta_{d}&\psi_{d}\end{array} \right]^{T}\), which is needed in the position control loop. Substituting Equation (9) into Equation (6), the translational dynamic of the hex-rotor can be expressed as follows:  
\[\dot{v}_{b}=\
u_{1}+\frac{\Delta_{F}}{m_{s}}+\delta(e_{\Phi_{b}}) \tag{10}\]  
\[\delta(e_{\Phi_{b}})=\frac{F_{t}}{m_{s}}({}^{I}R^{B,d}-{}^{I}R^{B})e_{3} \tag{11}\]  
where \(e_{\Phi_{b}}\) is the attitude error defined as \(e_{\Phi_{b}}=\Phi_{b}-\Phi_{b,d}\). \(\delta(e_{\Phi_{b}})\) is the interconnection term between the translational and rotational dynamic after decoupled by the virtual input \(\
u_{1}\). This virtual input is given by the \(H\infty\) position controller, which will be presented in the next subsection. Then, in order to actuate the outer loop actually, we need to translate the virtual input into the thrust and desired attitude angle, which can be tracked fast enough in the attitude control loop [29]. We define the virtual input vector as \(\
u_{1}=\left[\begin{array}{cccc}\
u_{1x}&\
u_{1y}&\
u_{1z}\end{array} \right]^{T}\). Combining Equation (1) and Equation (9), we get the thrust, desired roll angle and desired pitch angle as follows:  
\[\left\{\begin{array}{l}F_{t}=\left\|m_{s}\
u_{1}-m_{s}ge_{3}-m_{s}{}^{I}R_{B }(^{B}\omega_{b}\times(^{B}\omega_{b}\times{}^{B}r_{\infty}(q)))\right\|\\ \varphi_{d}=\arcsin(\frac{m_{s}}{F_{t}}(\
u_{1x}\psi_{d}-\
u_{1y}c\psi_{d})) \\ \theta_{d}=\arctan(\frac{m_{s}}{F_{t}}(\
u_{1x}\psi_{d}-\
u_{1y}s\psi_{d})) \end{array}\right. \tag{12}\]  
Figure 3: Control structure of the aerial manipulator system.  
Linearization of the rotational dynamic is essential to design the \(H\infty\) controller. Based on the feedback linearization theory of the MIMO (Multiple Input Multiple Output) system [30], the rotational dynamic in Equation (7) is feedback linearizable at \(\theta\
eq\pi/2\) through the following equation:  
\[\tau= I_{b}T^{-1}(\Phi_{b})(-\tilde{T}(\Phi_{b})^{B}\omega_{b}+T(\Phi_{b})I_{b} ^{-1}(^{B}\omega_{b}\times(I_{b}^{B}\omega_{b})+\
u_{2}))-{}^{B}r_{\infty}(q) \times{}^{I}R_{B}^{-1}m_{s}ge_{3} \tag{13}\]  
where \(\
u_{2}\) is the virtual input of the linearized system, and it is given by the \(H\infty\) attitude controller. Thus, after decoupling and linearizing, the system dynamic will be as follows:  
\[\left\{\begin{array}{l}\dot{v}_{b}=\
u_{1}+\delta(e_{\Phi_{b}})+\frac{\Delta _{\tau}}{m_{s}}\\ \ddot{\Phi}_{b}=\
u_{2}+T(\[PERSON]_{b})I_{b}^{-1}\Delta_{\tau}\end{array}\right. \tag{14}\]