## 2 Kinematics and Dynamics of Aerial Robotic Manipulator
The Lagrange-Euler approach [30, 32, 33] directly calculates the variations of the moment of inertia, vital for analysing the stability of controlling AEROM. The kinetic energy \(E_{k}\) and the potential energy \(E_{p}\) can be firstly obtained [32], formulated as:  
\[E_{k}=\frac{1}{2}\left[ \begin{array}{c}\mathbf{\dot{\chi}}_{\rm b}\\ \mathbf{\dot{q}}\end{array} \right]^{T}\left[ \begin{array}{c}\mathbf{H}_{\rm b}^{{}^ {\prime}},\mathbf{H}_{\rm bm}^{{}^{\prime}}\\ \mathbf{H}_{\rm bm}^{{}^{\prime}},\mathbf{H}_{\rm m}\end{array} \right]^{T}\left[ \begin{array}{c}\mathbf{\dot{\chi}}_{\rm b}\\ \mathbf{\dot{q}}\end{array} \right] \tag{11}\]  
\[E_{p}=-\sum_{i=1}^{n}m_{i}\mathbf{\dot{g}}^{T}\left[\mathbf{b}_{0}+\sum_{k=1}^{i} \left(\mathbf{A}\mathbf{\dot{C}}_{\rm j-1}-\mathbf{A}\mathbf{\dot{C}}_{\rm j}\right)\right] \tag{12}\]  
where \(\mathbf{\dot{\chi}}_{\rm b}=[\mathbf{v}_{\rm b};~{}\mathbf{\omega}_{\rm b}]\). \(\mathbf{H}_{\rm b}^{{}^{\prime}}\in\mathbb{R}^{6\times 6}\) and \(\mathbf{H}_{\rm m}\in\mathbb{R}^{n\times n}\) are respectively inertia matrices of UAV base and robotic manipulator. \(\mathbf{H}_{\rm bm}^{{}^{\prime}}\in\mathbb{R}^{6\times n}\) is the coupling inertia matrix of UAV and manipulator. \(\mathbf{A}\mathbf{\dot{C}}_{\rm j}\) are matrices related to \(\mathbf{b}_{i}\), \(\mathbf{p}_{i}\), \(\mathbf{r}_{i}\), \(\mathbf{k}_{i}\), etc. \(m_{i}\) denotes the mass of the \(i\)th body. Then, the dynamic equation of AEROM system can be derived through the Lagrange-Euler approach [33], shown as:  
\[\left[ \begin{array}{cc}\mathbf{H}_{\rm b}&\mathbf{H}_{\rm bm}\\ \mathbf{H}_{\rm bm}^{T}&\mathbf{H}_{\rm m}\end{array} \right]^{T}\left[ \begin{array}[] {c}\mathbf{\dot{\chi}}_{\rm b}\\ \mathbf{\ddot{q}}\end{array} \right]+\left[ \begin{array}{c}\mathbf{C}_{\rm b}\\ \mathbf{C}_{\rm m}\end{array} \right]+\left[ \begin{array}{c}\mathbf{G}_{\rm b}\\ \mathbf{G}_{\rm m}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}_{\rm p}\\ \mathbf{\tau}_{\rm m}\end{array} \right] \tag{13}\]  
where \(\mathbf{H}_{\rm b}\in\mathbb{R}^{6\times 6}\) and \(\mathbf{H}_{\rm m}\in\mathbb{R}^{n\times n}\) are, respectively, inertia matrices of UAV base and robotic manipulator. \(\mathbf{H}_{\rm bm}\in\mathbb{R}^{6\times n}\) is the coupling inertia matrix of UAV and manipulator. The work [32] describes the difference between \(\mathbf{H}_{\rm b}^{{}^{\prime}}\) and \(\mathbf{H}_{\rm b}\), and the difference between \(\mathbf{H}_{\rm bm}^{{}^{\prime}}\) and \(\mathbf{H}_{\rm m}\). \(\mathbf{C}_{\rm b}\in\mathbb{R}^{6\times 1}\) and \(\mathbf{C}_{\rm m}\in\mathbb{R}^{n\times 1}\) are respectively the Coriolis force and the centrifugal force. \(\mathbf{G}_{\rm b}\in\mathbb{R}^{6\times 1}\) and \(\mathbf{G}_{\rm m}\in\mathbb{R}^{n\times 1}\) are gravity items related to UAV base and robotic manipulator. \(\mathbf{F}_{\rm p}\in\mathbb{R}^{6\times 1}\) and \(\mathbf{\tau}_{\rm m}\in\mathbb{R}^{n\times 1}\) denote force/moment vectors of UAV and \(n\) joints, respectively. \(\mathbf{F}_{\rm p}\) and \(\mathbf{\tau}_{\rm m}\) both consist of two parts, the actuator force/moment and the external force/moment. The external force/moment denotes the atmosphere influence, air flow, etc. Since this paper focuses on AEROM trajectory planning through kinematics, the AEROM dynamics are not further analyzed.