## 3 Models of UAV
### Dynamic Model  
In order to obtain the dynamic model of the quadrotor UAV manipulator system, the Euler-Lagrange equation is introduced. \[\frac{d}{dt}\frac{\partial L}{\partial\mathrm{H_{k}}}-\frac{\partial L}{ \partial\mathrm{H_{k}}}=u_{k}+d_{k},L=K-U \tag{15}\]  
where \(k=(1,...,8)\). \(L\): the Lagrangian with kinetic energy \(K\) and potential energy \(U\) of the integrated system. \(u_{k}\): the generalized driving force. \(d_{k}\): external disturbance applied to the system. Additionally, \(K\) could be given in detail as  
\[\left\{ \begin{array}{l}K=K_{b}+\sum\limits_{i=1}^{2}K_{i}\\ K_{b}=\frac{1}{2}m_{b}(\dot{\boldsymbol{p}}^{I})^{T}\dot{\boldsymbol{p}}^{I}+ \frac{1}{2}(\boldsymbol{\omega^{I}})^{T}\mathbf{\mathit{R_{b}}}\mathbf{ \mathit{I_{b}}}\mathbf{\mathit{R_{b}^{T}}}\boldsymbol{\omega^{I}}\\ K_{i}=\frac{1}{2}m_{i}(\boldsymbol{p_{i}^{I}})^{T}\boldsymbol{p}_{i}^{I}+ \frac{1}{2}(\boldsymbol{\omega_{i}^{I}})^{T}\mathbf{\mathit{R_{b}}}\mathbf{ \mathit{R_{i}^{I}}}\mathbf{\mathit{R_{b}^{T}}}\boldsymbol{\omega_{i}^{I}} \end{array} \right. \tag{16}\]  
where \(m_{b}\) is the quadrotor base mass, \(m_{i}\) is the \(i\)-th robotic arm mass (\(i=(1,2)\)), \(\boldsymbol{I}\) is the inertia matrix, and \(\mathbf{\mathit{R_{i}^{b}}}\) the rotation matrix between the frame fixed to the center of mass of the \(i\)-th link and \(\mathbf{O_{b}}\). Then, the total potential is provided as  
\[\left\{ \begin{array}{l}U=U_{b}+\sum\limits_{i=1}^{2}U_{i}\\ U_{b}=m_{b}g\epsilon_{3}^{c}p^{I}\\ U_{i}=m_{i}g\epsilon_{3}^{c}p_{i}^{I}\end{array} \right. \tag{17}\]where \(\mathbf{e_{3}}\) denotes unit vector \([0,0,1]^{T}\) and \(g\) represents the gravity constant. By substituting Equations (16) and (17) into Equation (15), the dynamic model of the quadrotor UAV manipulator system could be obtained as  
\[M(\mathbf{H})\ddot{\mathbf{H}}+\mathcal{C}(\mathbf{H},\dot{\mathbf{H}})+G( \mathbf{H})=\mathbf{u}+\mathbf{d} \tag{18}\]  
Moreover, the total kinetic energy can be expressed:  
\[K=\frac{1}{2}\dot{\mathbf{H}}^{T}M(\mathbf{H})\dot{\mathbf{H}} \tag{19}\]  
where  
\[M(\mathbf{H})=\left[ \begin{array}{ccc}\mathbf{M_{11}}&\mathbf{M_{12}}&\mathbf{M_{13}}\\ \mathbf{M_{21}}&\mathbf{M_{22}}&\mathbf{M_{23}}\\ \mathbf{M_{31}}&\mathbf{M_{32}}&\mathbf{M_{33}}\end{array} \right] \tag{20}\]  
Details of the inertia matrix \(M(\mathbf{H})\) and Coriolis matrix \(\mathcal{C}(\mathbf{H},\dot{\mathbf{H}})\) can be found in [27]. The \(G(\mathbf{H})\) could be obtained via partial derivative Equation (21):  
\[G(\mathbf{H})=\frac{\partial U}{\partial\mathbf{H}} \tag{21}\]  
Express Equation (18) in detail as  
\[\left[ \begin{array}{ccc}\mathbf{M_{11}}&\mathbf{M_{12}}&\mathbf{M_{13}}\\ \mathbf{M_{21}}&\mathbf{M_{22}}&\mathbf{M_{23}}\\ \mathbf{M_{31}}&\mathbf{M_{32}}&\mathbf{M_{33}}\end{array} \right]\left[ \begin{array}{c} \mathbf{\dot{p}^{I}}\\ \mathbf{\dot{\Psi}}\\ \mathbf{\dot{\eta}}\end{array} \right]+\left[ \begin{array}{c}\mathbf{G_{1}}\\ \mathbf{G_{2}}\\ \mathbf{G_{3}}\end{array} \right]+\left[ \begin{array}{ccc}\mathbf{G_{1}}\\ \mathbf{G_{2}}\\ \mathbf{G_{3}}\end{array} \right]+\left[ \begin{array}{ccc}\mathbf{G_{1}}\\ \mathbf{G_{2}}\\ \mathbf{G_{3}}\end{array} \right]\left[ \begin{array}{c}\mathbf{\dot{p}^{I}}\\ \mathbf{\dot{\Psi}}\\ \mathbf{\dot{\eta}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{u_{t}}\\ \mathbf{u_{r}}\\ \mathbf{u_{l}}\end{array} \right]+\left[ \begin{array}{c}\mathbf{d_{t}}\\ \mathbf{d_{r}}\\ \mathbf{d_{l}}\end{array} \right] \tag{22}\]  
where \(\mathbf{u_{t}},\mathbf{u_{r}},\mathbf{u_{l}}\) represent the generalized control inputs corresponding to \(\mathbf{p^{I}},\mathbf{\dot{\Psi}},\mathbf{\eta}\). \(\mathbf{u_{l}}\) is not used, as we do not consider a dynamic robotic arm in this work. Vector \(\mathbf{d}=[d_{t},d_{t},d_{r}]^{T}\) are the lumped external disturbance.