# Aerial Grasping of an Object in the Strong Wind: Robust Control of an Aerial Manipulator
## 2 System Model
### Dynamic of the UAV  
The aerial manipulator can be taken as a special aerial platform whose mass distribution could be changed due to the movement of the robotic arm. The disturbance that the manipulator imposes on the hex-rotor can be partially represented by the motion of the CoM offset. Thus, the CoM offset in the UAV's dynamic model can partially reflect influence of the robotic arm, as in a previous work [27]. The dynamics of the hex-rotor with the CoM offset, \(r_{oc}\), is expressed as follows [28]:  
\[\dot{v}_{b} =-\frac{F_{t}}{m_{s}}R_{B}e_{3}+ge_{3}+{}^{I}R_{B}({}^{B}\omega_{b} \times({}^{B}\omega_{b}\times{}^{B}r_{oc}(q)))+\frac{\Delta_{F}}{m_{s}} \tag{6}\] \[{}^{B}\dot{\omega}_{b} =I_{b}^{-1}(\tau-{}^{B}\omega_{b}\times({}^{I_{B}}\omega_{b})+{}^{ B}r_{oc}(q)\times{}^{I}R_{B}^{-1}m_{s}ge_{3})+I_{b}^{-1}\Delta_{\tau} \tag{7}\]  
where \(m_{s}\) is the total mass of the aerial manipulator system. \(g\) is the gravity acceleration. \(e_{1}=\left[\begin{array}{ccc}1&0&0\end{array}\right]^{T}\), \(e_{2}=\left[\begin{array}{ccc}0&1&0\end{array}\right]^{T}\) and \(e_{3}=\left[\begin{array}{ccc}0&0&1\end{array}\right]^{T}\) are the unit vectors with three dimensions. \(I_{b}\) is the inertia matrix of the hex-rotor. \(F_{t}\) and \(\tau\) are the thrust and torque acting on the hex-rotor, respectively, and they are generated by the rotors of the hex-rotor. \({}^{B}r_{oc}(q)\) is CoM offset with respect to \(\Sigma_{B}\). \({}^{B}r_{oc}(q)\) is a function of the \(q\) that is joint angle vector of the robotic arm. Thus, based on the kinematics of the robotic arm [26], \({}^{B}r_{\infty}(q)\) can be calculated through the measurement of the joint angles. \(\Delta_{F}\) and \(\Delta_{\tau}\) are force and torque disturbance, respectively, and they include wind disturbance and disturbance of the robotic arm excepting CoM offset terms.  
**Remark I**: The CoM offset terms in the hex-rotor dynamics, Equations (6) and (7), contain the partial static disturbance of the robotic arm, and don't contain the disturbance caused by the inertia matrix changing, and don't contain the dynamic disturbance that is caused by joint velocity and acceleration of the robotic arm. We consider other parts of the disturbance in \(\Delta_{F}\) and \(\Delta_{\tau}\).  
As we know, the driving thrust and torque of the hex-rotor come from the aerodynamic force of high speed rotating rotors. The hex-rotor totally has six rotors, and serial number and rotation direction of the rotors are shown in Figure 2. The angular velocity of rotor \(i\) is denoted by \(\omega_{i}\) (\(i=1;2;\)\(\ldots;6\)). Thrust and torque are related to the rotor speed of the rotors as follow:  
\[\left[\begin{array}{c}F_{t}\\ \tau\end{array}\right]=\left[\begin{array}{cccccc}c_{T}&c_{T}&c_{T}&c_{T}&c_ {T}&c_{T}\\ -dc_{T}&dc_{T}&\frac{1}{2}dc_{T}&-\frac{1}{2}dc_{T}&-\frac{1}{2}dc_{T}&\frac{ 1}{2}dc_{T}\\ 0&0&\frac{\sqrt{3}}{2}dc_{T}&-\frac{\sqrt{3}}{2}dc_{T}&\frac{\sqrt{3}}{2}dc_{T }&-\frac{\sqrt{3}}{2}dc_{T}\\ -c_{\tau}&c_{\tau}&-c_{\tau}&c_{\tau}&c_{\tau}&-c_{\tau}\end{array}\right]=M \overline{\omega} \tag{8}\]  
where \(c_{T}\) is thrust coefficient. \(c_{\tau}\) is torque coefficient. \(d\) is the distance from center of rotor to the geometrical center of the hex-rotor. \(M\) is the mixer matrix of the hex-tor and \(\overline{\omega}\) is the vector of square of rotor speed.