## 2 Kinematics and Dynamics of Aerial Robotic Manipulator  
Figure 2 shows the 2-D sketch of an AEROM system composed of \(n+1\) bodies. The UAV mobile platform denotes the base and is remarked as body \(B_{0}\). The robotic manipulator is a single arm with \(n\) bodies and an EE. Adjacent bodies are connected with a revolute joint with one degree of freedom (DOF). Table 1 lists meanings of symbols and variables in Figure 2. Axes of \(\Sigma_{0}\) are respectively parallel to axes of \(\Sigma_{1}\) at the initial time of the system maneuver. The axes of \(\Sigma_{e}\) are also parallel to the axes of \(\Sigma_{n}\) at the initial time. The frames \(\Sigma_{i}\) (\(i=1,\ldots,n\)) are built using the Denavit-Hartenberg (D-H) approach [29, 30]. The detailed procedure is summarized as follows:  
1. Align the axis \(Z_{i}\) and the origin \(O_{i}\). \(Z_{i}\) is located at the (\(i+1\))th joint, and \(O_{i}\) denotes the crosspoint of \(Z_{i}\) and common perpendicular of \(Z_{i}\) and \(Z_{i-1}\). 2. Align the axis \(X_{i}\). The \(X_{i}\)-axis is along the common perpendicular of \(Z_{i}\) and \(Z_{i-1}\), and the direction is from \(J_{i}\) to \(J_{i+1}\). 3. Align the axis \(Y_{i}\) according to the right-hand rule. Along with establishment of frames \(\Sigma_{i}\), four categories of D-H parameters are constructed as follows. * \(a_{i}\) -- The distance between \(Z_{i}\) and \(Z_{i+1}\) along \(X_{i}\). * \(\alpha_{i}\) -- The angle from \(Z_{i}\) to \(Z_{i+1}\) around \(X_{i}\). * \(d_{i}\) -- The distance between \(X_{i-1}\) and \(X_{i}\) along \(Z_{i}\). * \(\theta_{i}\) -- The angle from \(X_{i-1}\) to \(X_{i}\) around \(Z_{i}\). Figure 2: 2-D sketch of an AEROM system, with symbols employed for the derivation of AEROM mathematical models. In this paper, the AEROM system is considered as a unique entity. Each component \(B_{i}\) (\(i=0,\ldots,n\)) of AEROM is assumed to be a rigid body with uniform mass distribution. The center of mass of \(B_{i}\) is located at the geometric center \(C_{i}\). On the basis of Figure 2, positions of \(i\)th body centre and EE are depicted as:  
\[\mathbf{r}_{i}=\mathbf{r}_{0}+\mathbf{b}_{0}+\sum_{k=1}^{i-1}(\mathbf{a}_{k}+\mathbf{b}_{k})+\mathbf{a} _{i} \tag{1}\]  
\[\mathbf{p}_{\text{e}}=\mathbf{r}_{0}+\mathbf{b}_{0}+\sum_{k=1}^{n}(\mathbf{a}_{k}+\mathbf{b}_{k}) \tag{2}\]  
Taking the derivatives of \(\mathbf{r}_{i}\) and \(\mathbf{p}_{\text{e}}\), the linear velocities of \(i\)th body centre and EE are then formulated as:  
\[\mathbf{v}_{i}=\dot{\mathbf{r}}_{i}=\mathbf{v}_{\text{b}}+\mathbf{\omega}_{\text{b}}\times(\bm {r}_{i}-\mathbf{r}_{0})+\sum_{k=1}^{i}[\mathbf{k}_{k}\times(\mathbf{r}_{i}-\mathbf{p}_{k})]\dot {q}_{k} \tag{3}\]  
\[\mathbf{v}_{\text{e}}=\dot{\mathbf{p}}_{\text{e}}=\mathbf{v}_{\text{b}}+\mathbf{\omega}_{\text {b}}\times(\mathbf{p}_{\text{e}}-\mathbf{r}_{0})+\sum_{k=1}^{n}[\mathbf{k}_{k}\times(\bm {p}_{\text{e}}-\mathbf{p}_{k})]\dot{q}_{k} \tag{4}\]  
where \(\mathbf{v}_{i}\in\mathbb{R}^{3}\), \(\mathbf{v}_{\text{e}}\in\mathbb{R}^{3}\), \(\mathbf{v}_{\text{b}}\in\mathbb{R}^{3}\) and \(\mathbf{\omega}_{\text{b}}\in\mathbb{R}^{3}\). \(\mathbf{k}_{i}\in\mathbb{R}^{3}\) denotes the unit rotational velocity of \(i\)th joint. \(\mathbf{v}_{\text{b}}\) and \(\mathbf{\omega}_{\text{b}}\) are, respectively, the linear velocity and angular velocity of the UAV base. Angular velocities of \(i\)th body centre and EE are calculated as:  
\[\mathbf{\omega}_{i}=\mathbf{\omega}_{\text{b}}+\sum_{k=1}^{i}\mathbf{k}_{k}\dot{q}_{k} \tag{5}\]  
\[\mathbf{\omega}_{\text{e}}=\mathbf{\omega}_{\text{b}}+\sum_{k=1}^{n}\mathbf{k}_{k}\dot{q} _{k} \tag{6}\]  
where \(\mathbf{\omega}_{i}\in\mathbb{R}^{3}\) and \(\mathbf{\omega}_{\text{e}}\in\mathbb{R}^{3}\). Combining Equations (4) and (6), EE velocity is given as follows:  
\[\mathbf{\dot{\chi}}_{\text{e}}=\left[ \begin{array}{c}\mathbf{v}_{\text{e}}\\ \mathbf{\omega}_{\text{e}}\end{array} \right]=\mathbf{I}_{\text{b}}\left[ \begin{array}[ ]{c}\mathbf{v}_{\text{b}}\\ \mathbf{\omega}_{\text{b}}\end{array}