# Aerial Grasping of an Object in the Strong Wind: Robust Control of an Aerial Manipulator
## 2 System Model
### System Kinematics  
The aerial manipulator presented in this paper is composed of a hex-rotor and a 7-DoF manipulator. The coordinate frames of the whole system are defined in Figure 1. The North-East-Down (NED) inertial frame, the body fixed frame of the hex-rotor and the end-effector frame are denoted by \(\Sigma_{I}\), \(\Sigma_{B}\) and \(\Sigma_{E}\), respectively. Point \(O\) is the coordinate origin of \(\Sigma_{B}\) and it coincides with the CoM of the hex-rotor. The CoM of the whole aerial manipulator system is denoted by point \(C\). When the manipulator moves, the position of point \(C\) relative to point \(O\) will change. Hence, we use \(r_{oc}\) to denote the relative position between point \(O\) and point \(C\). The position and attitude of the hex-rotor and the end-effector are represented by those of \(\Sigma_{B}\) and \(\Sigma_{E}\), respectively. The absolute position of the hex-rotor with respect to \(\Sigma_{I}\) is denoted by \(p_{b}\), and the attitude is described by the \(Z\)-\(Y\)-\(X\) Euler angles denoted by \(\Phi_{b}=\left[ \begin{array}{ccc}\varphi&\theta&\psi\end{array} \right]^{T}\), where \(\varphi\), \(\theta\) and \(\psi\) are roll, pitch and yaw angle, respectively. The rotation matrix from \(\Sigma_{B}\) to \(\Sigma_{I}\) is denoted by \({}^{I}R_{B}\). The detail of \({}^{I}R_{B}\) is as follow:  
\[{}^{I}R_{B}=\left[ \begin{array}{ccc}c\theta c\psi&s\varphi s\theta c\psi-c \varphi s\psi&c\varphi s\theta c\psi+s\varphi s\psi\\ c\theta s\psi&s\varphi s\theta s\psi+c\varphi c\varphi&c\varphi s\theta s\psi-s \varphi c\psi\\ -s\theta&s\varphi c\theta&c\varphi c\theta\end{array} \right. \tag{1}\]  
Figure 1: Frames in aerial manipulator system. where \(c\) and \(s\) denote trigonometric functions \(cos()\) and \(sin()\), respectively. The absolute position of the end-effector with respect to \(\Sigma_{I}\) is denoted by \(p_{e}\). The absolute attitude of the end-effector with respect to \(\Sigma_{I}\) is denoted by \(\Phi_{e}\). The attitude of the end-effector relative to hex-rotor with respect to \(\Sigma_{B}\) is denoted by \({}^{B}\Phi_{be}\). As in Equation (1), \(\Phi_{e}\) determines the rotation matrix from \(\Sigma_{E}\) to \(\Sigma_{I}\), denoted by \({}^{I}R_{E}\), and \({}^{B}\Phi_{be}\) determines the rotation matrix from \(\Sigma_{E}\) to \(\Sigma_{B}\), denoted by \({}^{B}R_{E}\). They have relation with the position and attitude of the hex-rotor as follows:  
\[\left\{ \begin{array}{l}p_{e}=p_{b}+{}^{I}R_{B}{}^{B}p_{be}\\ {}^{I}R_{E}={}^{I}R_{B}{}^{B}R_{E}\end{array} \right. \tag{2}\]  
where \({}^{B}p_{be}\) is position of the end-effector relative to hex-rotor with respect to \(\Sigma_{B}\). For \({}^{B}p_{be}\) and \({}^{B}R_{E}\), they can be got through forward kinematics of the manipulator with the joint angle [26].\(v_{b}\) denotes the absolute velocity of the hex-rotor with respect to \(\Sigma_{I}\). \({}^{B}\omega_{b}\) denotes the absolute angular velocity of hex-rotor with respect to \(\Sigma_{B}\). They have a relationship with \(p_{b}\) and \(\Phi_{b}\) as follows:  
\[\left\{ \begin{array}{l}\dot{p}_{b}=v_{b}\\ \dot{\Phi}_{b}=T(\Phi_{b})^{B}\omega_{b}\end{array} \right. \tag{3}\]  
where \(T(\Phi_{b})=\left[ \begin{array}{ll}1&s\varphi\tan\theta&c\varphi\tan\theta \\ 0&c\varphi&-s\varphi\\ 0&s\varphi\sec\theta&c\varphi\sec\theta\end{array} \right]\). The absolute velocity and angular velocity of end-effector with respect to \(\Sigma_{I}\) are denoted by \(v_{e}\) and \(\omega_{e}\), respectively. They have a relationship with \(v_{b}\) and \({}^{B}\omega_{b}\) as follows:  
\[\left\{ \begin{array}{l}v_{e}=v_{b}+{}^{I}R_{B}({}^{B}\omega_{b}\times{}^{B} p_{be}+{}^{B}v_{be})\\ \omega_{e}={}^{I}R_{B}({}^{B}\omega_{b}+{}^{B}\omega_{be})\end{array} \right. \tag{4}\]  
where \({}^{B}v_{be}\) and \({}^{B}\omega_{be}\) are the velocity and angular velocity of the end-effector relative to the UAV body fixed frame with respect to \(\Sigma_{B}\). \({}^{B}v_{be}\) and \({}^{B}v_{be}\) can be got through velocity kinematics of the robotic arm, as follow:  
\[\left[