## 2 Kinematics and Dynamics of Aerial Robotic Manipulator
\right]+\mathbf{I}_{\text{m}}\dot{\mathbf{q}} \tag{7}\]  
where \(\mathbf{I}_{\text{b}}\in\mathbb{R}^{6\times 6}\) and \(\mathbf{I}_{\text{m}}\in\mathbb{R}^{6\times n}\) are, respectively, Jacobian matrices of the UAV base and the robotic manipulator. \(\mathbf{q}=[q_{1},\ldots,q_{n}]\in\mathbb{R}^{n}\) denotes joint configuration, where \(q_{i}(i=1,\ldots,n)\) denotes the \(i\)th joint angle. \(\dot{\mathbf{q}}\) denotes the velocity vector of \(n\) joints. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Representation** \\ \hline \(C_{\text{gy}}\) & Center of mass of AEROM system \\ \(B_{i}\) (\(i=0,\ldots,n\)) & Body \(i\), where \(B_{0}\) denotes UAV base and \(B_{i}\) (\(i=1,\ldots,n\)) denotes \(i\)th link of robotic manipulator \\ \(C_{i}\) (\(i=0,\ldots,n\)) & Center of mass of \(B_{i}\) \\ \(J_{i}\) (\(i=0,\ldots,n\)) & Joint \(i\), connecting \(B_{i-1}\) and \(B_{i}\) \\ \(\mathbf{a}_{i}\in\mathbb{R}^{3}\) (\(i=1,\ldots,n\)) & Vector from \(J_{i}\) to \(C_{i}\) \\ \(\mathbf{b}_{i}\in\mathbb{R}^{3}\) (\(i=0,\ldots,n\)) & Vector from \(C_{i}\) to \(J_{i+1}\) \\ \(\mathbf{r}_{i}\in\mathbb{R}^{3}\) (\(i=0,\ldots,n\)) & Position of \(C_{i}\) in inertial coordinate system \\ \(\mathbf{p}_{i}\in\mathbb{R}^{3}\) (\(i=1,\ldots,n\)) & Position of \(J_{i}\) in inertial coordinate system \\ \(\mathbf{r}_{\text{g}}\in\mathbb{R}^{3}\) & Position of \(C_{\text{gy}}\) in inertial coordinate system \\ \(\mathbf{p}_{\text{e}}\in\mathbb{R}^{3}\) & Position of end effector in inertial coordinate system \\ \(\sum_{i}\) (\(i=0,\ldots,n\)) & Coordinate system established in \(B_{i}\) with origin \(O_{i}\), axes \(X_{i}\), \(Y_{i}\) and \(Z_{i}\) \\ \(\sum_{\text{I}}\) & Inertial coordinate system with origin \(O_{\text{I}}\), axes \(X_{\text{H}}\), \(Y_{\text{I}}\) and \(Z_{\text{I}}\) \\  & Coordinate system established in end effector, with origin \(O_{\text{e}}\), axes \(X_{\text{e}}\), \(Y_{\text{e}}\) \\ \(\sum_{\text{e}}\) & and \(Z_{\text{e}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Representation of symbols in Figure 2.  
In the first case, the pose of UAV base is invariable (i.e., position and attitude of UAV are controlled). The first term of right-hand side in Equation (7) is \(\mathbf{0}\), shown as:  
\[\dot{\mathbf{\chi}}_{\rm e}=\mathbf{I}_{\rm m}\dot{\mathbf{q}} \tag{8}\]  
In the second case, pose of UAV base is free (i.e., position and attitude of UAV are not controlled). Assume gravity of each component of AEROM system is exactly compensated by corresponding rotors or servo-motors, the resultant of external force acting on the AEROM system is \(\mathbf{0}\). Then, the AEROM system satisfies the conservation of momentum, shown as follows:  
\[\mathbf{H}_{\rm b}\mathbf{\dot{\chi}}_{\rm b}+\mathbf{H}_{\rm bm}\dot{\mathbf{q}}=\mathbf{C} \tag{9}\]  
where \(\mathbf{\dot{\chi}}_{\rm b}=[\mathbf{v}_{\rm b};~{}\mathbf{\omega}_{\rm b}]\). \(\mathbf{H}_{\rm b}\in\mathbb{R}^{6\times 6}\) is inertia matrix of UAV base and has the ortho-symmetric property. \(\mathbf{H}_{\rm bm}\in\mathbb{R}^{6\times n}\) is coupling inertia matrix of UAV base and robotic manipulator, and \(\mathbf{C}\) is a constant vector. The constant \(\mathbf{C}\) is usually set as \(\mathbf{0}\)[7, 8, 31]; the same goes for this case. Then, the UAV velocity is formulated as: \(\mathbf{\dot{\chi}}_{\rm b}=\mathbf{H}_{\rm b}^{-1}\mathbf{H}_{\rm bm}\dot{\mathbf{q}}\). According to the Equation (7) and the Conservation of Momentum (9), the EE velocity is formulated as:  
\[\mathbf{\dot{\chi}}_{\rm e}=\left[ \begin{array}{c}\mathbf{v}_{\rm e}\\ \mathbf{\omega}_{\rm e}\end{array} \right]=\left(\mathbf{I}_{m}-\mathbf{I}_{\rm b}\mathbf{H}_{ \rm bm}\right)\dot{\mathbf{q}}=\mathbf{I}_{\rm g}\dot{\mathbf{q}} \tag{10}\]  
The Equations (8) and (10) are, namely, differential kinematic equations of AEROM system in the above two cases. The AEROM dynamics are necessary to control the EE of robotic manipulator.