## 3 Trajectory Planning of Aerial Robotic Manipulator  
### Description of Trajectory Planning Problem  
The trajectory planning problem for AEROM is studied here. The pose of UAV base is free, i.e., the second case of the aforementioned kinematics. The AEROM system has the non-holonomic characteristic, implying that the pose of AEROM relies on both inverse kinematics and coupling between UAV base and robotic manipulator. The initial, desired and actually planned poses of EE are respectively denoted as \(\mathbf{\chi}_{\rm e}^{\rm in}\), \(\mathbf{\chi}_{\rm e}^{\rm de}\) and \(\mathbf{\chi}_{\rm e}^{\rm ac}\). The initial, desired and actually planned poses of UAV base are respectively denoted as \(\mathbf{\chi}_{\rm b}^{\rm in}\), \(\mathbf{\chi}_{\rm b}^{\rm de}\) and \(\mathbf{\chi}_{\rm b}^{\rm ac}\). The initial, desired and actually planned joint configurations are respectively denoted as \(\mathbf{q}^{\rm in}\), \(\mathbf{q}^{\rm de}\) and \(\mathbf{q}^{\rm ac}\). On the basis of aforementioned AEROM kinematics, the actually planned poses of EE and UAV base are respectively calculated as:  
\[\mathbf{\chi}_{\rm e}^{\rm ac}=\mathbf{\chi}_{\rm e}^{\rm in}+\int_{t_{0}}^{t_{f}}\mathbf{ J}_{\rm g}\mathbf{\dot{q}}\,\mathrm{dt} \tag{14}\]  
\[\mathbf{\chi}_{\rm b}^{\rm ac}=\mathbf{\chi}_{\rm b}^{\rm in}+\int_{t_{0}}^{t_{f}}\bm {H}_{\rm b}^{-1}\mathbf{H}_{\rm bm}\mathbf{\dot{q}}\,\mathrm{dt} \tag{15}\]  
The differences between the actually planned pose and desired pose of EE and UAV base are respectively given by Equations (16) and (17). The differences between actually planned joint configuration and desired joint configuration is given by Equation (18). \[\delta\mathbf{\chi}_{\rm e}=\mathbf{\chi}_{\rm e}^{\rm ac}-\mathbf{\chi}_{\rm e}^{\rm de} \tag{16}\]  
\[\delta\mathbf{\chi}_{\rm b}=\mathbf{\chi}_{\rm b}^{\rm ac}-\mathbf{\chi}_{\rm b}^{\rm de} \tag{17}\]  
\[\delta\mathbf{q}=\mathbf{q}^{\rm ac}-\mathbf{q}^{\rm de} \tag{18}\]  
The trajectory planning of AEROM aims to obtain appropriate joint trajectories yielding initial values and constraints:  
\[\left\{ \begin{array}{c}\mathbf{\chi}_{\rm b}(t_{0})=\mathbf{\chi}_{\rm b}^{0},\ \mathbf{\chi}_{\rm e}(t_{0})=\mathbf{\chi}_{\rm e}^{0}\mathbf{\chi}_{\rm e}(t_{f})=\mathbf{ \chi}_{\rm e}^{f}\\ \mathbf{q}(t_{0})=\mathbf{q}^{0},\ \mathbf{\dot{q}}(t_{0})=\mathbf{\ddot{q}}(t_{0})=\mathbf{0}\\ \mathbf{q}(t_{f})=\mathbf{q}^{f},\ \mathbf{\dot{q}}(t_{f})=\mathbf{\ddot{q}}(t_{f})=\mathbf{0}\\ \mathbf{q}_{\rm min}\leqslant\mathbf{q}\leqslant\mathbf{q}_{\rm max}\\ \mathbf{\ddot{q}}_{\rm min}\leqslant\mathbf{\ddot{q}}\leqslant\mathbf{\ddot{q}}_{\rm max} \\ \mathbf{\ddot{q}}_{\rm min}\leqslant\mathbf{\ddot{q}}\leqslant\mathbf{\ddot{q}}_{\rm max} \end{array} \right. \tag{19}\]  
In Equation (19), \(\mathbf{\chi}_{\rm b}^{0}\) is the initial pose of UAV base. \(\mathbf{\chi}_{\rm e}^{0}\) and \(\mathbf{\chi}_{\rm e}^{f}\) are respectively the initial pose and the final pose of EE. \(\mathbf{q}^{0}\) and \(\mathbf{q}^{f}\) are the initial and the final joint configurations, respectively. In this paper, \(\mathbf{q}_{\rm min}=-\mathbf{q}_{\rm max}\), \(\mathbf{\dot{q}}_{\rm min}=-\mathbf{\dot{q}}_{\rm max}\), and \(\mathbf{\ddot{q}}_{\rm min}=-\mathbf{\ddot{q}}_{\rm max}\).