## 3 Models of UAV
### Kinematic Model  
Figure 1 shows several coordinates in UAV that we consider. \(O_{I}\): the world-fixed inertial reference frame. \(O_{\mathbf{b}}\): the body-fixed reference frame centered in the quadrotor Center of Gravity (CoG). \(O_{i}\): the coordinate of robotic arm \(i\), where \((i=1,2)\) denotes the link number. Moreover, several parameters definition are given as follows. \(p^{I}=[x,y,z]^{T}\): the absolute position of \(O_{\mathbf{b}}\) with respect to \(O_{I}\). \(\mathbf{\
abla}=[\varphi,\theta,\psi]^{T}\): Euler angles used for representing the UAV attitude angle. \(\mathbf{\eta}=[\eta_{1},\eta_{2}]^{T}\): the joint angles of the robotic arm relative to the zero position. \(\mathbf{H}=[(p^{I})^{T},\mathbf{\
abla}^{T},\mathbf{\eta}^{T}]^{T}\in R^{8 \times 1}\): a vector used for including all the generalized variables. \(\dot{p}^{I}\): the absolute linear velocity of UAV described in \(O_{\mathbf{I}}\). \(\dot{p}^{b}\): the linear velocity of UAV with respect to \(O_{\mathbf{b}}\). \(\omega^{I}\)=\([\omega_{x}^{I},\omega_{y}^{I},\omega_{z}^{I}]^{T}\): UAV's absolute rotational velocity described in \(O_{\mathbf{I}}\). \(\omega^{b}=[\omega_{x^{\prime}}^{b},\omega_{y^{\prime}}^{b},\omega_{z}^{b}]^{T}\): the rotational velocity of UAV relative to \(O_{\mathbf{b}}\). Then we could get:  
\[\left\{ \begin{array}{l}\dot{p}^{I}=R_{b}\dot{p}^{b}\\ \omega^{I}=R_{t}\Psi\\ \omega^{b}=(R_{b})^{T}\omega^{I}=(R_{b})^{T}R_{t}\Psi=R_{r}\dot{\Psi}\end{array} \right. \tag{10}\]  
where \(R_{t}\in R^{3\times 3}\) represents the transformation matrix for converting Euler angle rates \(\dot{\mathbf{\Psi}}\) into \(\omega^{I}\). \(R_{b}\in R^{3\times 3}\) denotes the rotation matrix representing the orientation of \(O_{\mathbf{b}}\) relative to \(O_{I}\), and \(R_{r}\)=\((R_{b})^{T}R_{t}\in R^{3\times 3}\) is used to map the derivative of \(\mathbf{\
abla}\) into UAV angular rate expressed in \(O_{\mathbf{b}}\). Figure 1: Quadrotor and robotic arm system with coordinate reference frames. Where  
\[R_{b}=\left[ \begin{array}{ccc}c\varrho c\psi&sq\beta c\psi-c\varphi s\psi&c \varphi s\partial c\psi+sg\psi\\ c\theta\psi&sq\beta s\theta\psi+c\varrho c\varphi&c\varphi s\theta s\psi-s \varrho c\psi\\ -s\theta&s\varrho c\theta&c\varphi c\theta\end{array} \right] \tag{11}\]  
\(c(\cdot)\) and \(s(\cdot)\) denote \(cos(\cdot)\) and \(sin(\cdot)\), respectively. Moreover, the position and angular rate of the frame \(\mathbf{O_{i}}\), which is fixed to the robotic arm, with respect to \(\mathbf{O_{I}}\) are provided by  
\[\left\{ \begin{array}{l}\boldsymbol{p_{i}^{I}}=\boldsymbol{p^{I}}+\mathbf{R_ {b}}\boldsymbol{p_{i}^{b}}\\ \boldsymbol{\omega_{i}^{I}}=\boldsymbol{\omega^{I}}+\mathbf{R_{b}} \boldsymbol{\omega_{i}^{b}}\end{array} \right. \tag{12}\]  
where \(i=(x,y,z)\). The vectors \(\boldsymbol{p_{i}^{b}}\), \(\boldsymbol{\omega_{i}^{I}}\), which are expressed in \(\mathbf{O_{b}}\), represent the position of \(\mathbf{O_{i}}\) and the angular rate of the i-th robotic arm frame with respect to \(\mathbf{O_{b}}\), respectively. Additionally, more relationships are provided,  
\[\left\{ \begin{array}{l}\dot{\boldsymbol{p_{i}^{b}}}{=}\boldsymbol{J_{pi}} \dot{\boldsymbol{\eta}}\\ \boldsymbol{\omega_{i}^{b}}{=}\boldsymbol{J_{ri}}\dot{\boldsymbol{\eta}} \end{array} \right. \tag{13}\]  
where \(\boldsymbol{J_{pi}}\in\mathbf{\mathit{R}}^{2\times 2}\) and \(\boldsymbol{J_{ri}}\in\mathbf{\mathit{R}}^{2\times 2}\) are Jacobian matrices representing the translational and angular velocities of each robotic arm link to the \(\dot{\boldsymbol{\eta}}\), respectively. According to Equations (12) and (13), we could get the translational and angular velocity of \(\mathbf{O_{i}}\) with respect to \(\mathbf{O_{I}}\) as  
\[\left\{