# Inverse Kinematics of a Serial Manipulator with a Free Joint for Aerial Manipulation
## 5 Simulation Results
### Manipulator on the UAV  
The method described in Section 3 was applied to the aerial manipulator described in Section 4.1 in the Matlab environment. The PID controller described in Section 2.1 was used to maintain the altitude of the UAV and to contrast its rotation. The torque applied by the manipulator on the UAV CoG during the tasks was analyzed, since it affects the UAV motion. First, the trajectory tracking task without the damper on the first joint was considered. The stroboscopic views for this task are shown in Figure 10, and resulting joint angles and torque applied by the manipulator on the UAV CoG are reported in Figure 11. The rotation of the UAV and its translation on the plane are shown in Figure 12. At half of the trajectory, when the end-effector motion is faster, there is an increase in the torque applied by the manipulator on the UAV CoG (note that this torque is due to the reaction forces acting on the non-actuated joint); however, this torque does not cause significant rotations of the UAV. Therefore, the translation of the UAV is mainly due to the inertia of the manipulator. As in previous simulations, joint limits (see Table 11) are satisfied and the position error of the end-effector is below \(2\cdot 10^{-4}\) m.  
Figure 8: Stroboscopic view of the 3-DOF manipulator attached to a fixed base with a damper while it executes the load picking task. **(left)** First stage (time \(<6\) s). **(center)** Second stage (time \(6\) s \(<\) time \(<16\) s). **(right)** Third stage (time \(>16\) s). The green marker represents the UAV’CoG, the red marker represents the end effector, and the black circles represent the manipulator’s revolute joints. Figure 9: Load picking for the 3-DOF manipulator attached to a fixed base with a damper. **(left)** Joint angles and **(right)** torque applied to the base during the task. Secondly, the 3-DOF UAM with a damper on the non-actuated joint performing the load picking task was considered. The corresponding stroboscopic views are shown in Figure 13, calculated joint angles and torque applied by the manipulator on the UAV CoG are shown in Figure 14, and UAV rotation and translation are shown in Figure 15.  
As for the case with fixed base, when the load is grasped, there is an increase in the torque applied by the manipulator on the UAV and damped vibrations are triggered. This torque peaks at 0.02 Nm. A similar trend is visible in the UAV rotation. Figure 11: Trajectory tracking for the 3-DOF manipulator on the UAV. (**left**) Joint angles and (**right**) torque applied to the UAV during the task. Figure 12: Trajectory tracking for the 3-DOF manipulator attached to the UAV. (**left**) Rotation and (**right**) translation of the UAV during the task. Figure 10: Stroboscopic view of the 3-DOF manipulator on the UAV executing the trajectory tracking task. (**left**) First stage (time \(<\) 5 s). (**right**) Second stage (time \(>\) 5 s). The green marker represents the UAV’CoG, the red marker represents the end effector, and the black circles represent the manipulator’s revolute joints. Due to the small rotation of the UAV rotation (maximum value is \(9\cdot 10^{-4}\) rad), the thrust force \(U_{1}\) is almost vertical throughout the simulation, and negligible external forces act on the UAM along the \(x\) direction. Therefore, in the first stage of the motion, while the end-effector moves to reach the load towards positive abscissa, the UAV translates towards negative abscissa. Moreover, the UAV translates towards positive abscissa while the load is lifted, and the final \(x\) position of the UAV is \(4.7\cdot 10^{-3}\) m. This is due to the absence of significant torque between the manipulator and the UAV, and similar behavior was observed in [6]. Figure 14: Load picking for the 3-DOF manipulator attached to the UAV with a damper. **(left)**hlhypen Joint angles and **(right)** torque applied to the UAV during the task. Figure 13: Stroboscopic view of the 3-DOF manipulator attached to the UAV with a damper while it executes the load picking task.