## 5 Generalized Jacobian with External Forces
### Simulation Results  
In order to validate the method proposed in Section 5.1, the inverse kinematics was solved in MATLAB using a timestep of \(\Delta t=1\times 10^{-3}\) s and then the resulting joint velocities were used to perform the dynamic simulation of the task in ADAMS. We first considered the case of the 2-DOF nonredundant manipulator tracking the line trajectory: as opposed to the results obtained with the 2-DOF trolley model with PID control feedback (Section 4.3), the position error of the end-effector is now smaller than \(4\times 10^{-5}\) m at all times even if the base has horizontal translations bigger than 10 cm. During the operation, the torque transmitted by the manipulator to the base makes the UAV rotate, as a result \(U_{1}\) has a horizontal component and the base translates along \(x_{I}\). At the end  
Figure 9: External forces on the UAM for the generalized Jacobian formulation. of the operation, the control torque \(U_{2}\) stabilizes the rotation of the base, which tends to return to its initial position. The stroboscopic trajectories, joint angles, UAV translations and rotation, and torque disturbance transmitted from the manipulator to the UAV are shown in Figures 10 and 11. The computational time to perform the inverse kinematics for tracking the above-mentioned linear trajectory (5 s) with the 2-DOF manipulator is 2.51 s.  
Figure 11: Manipulator (**top**) and UAV motions and torque disturbance (**middle** and **bottom**) for the 2-DOF UAM using the generalized Jacobian with external forces. Figure 10: Stroboscopic trajectories for the first half (**left**) and second half (**right**) of the linear trajectory task for the 2-DOF UAM using the generalized Jacobian with external forces. The UAV base is the large green marker, joints are the black circles, links are the blue lines, and end-effector positions tracked are the red crosses. We also tracked the same trajectory with a 3-DOF redundant manipulator using the same approach, where the right pseudo-inverse of the generalized Jacobian (8) was used to find the minimum joint velocities solution. The end-effector position error is smaller than \(~{}4\times 10^{-5}\) m at all times. As in the 2-DOF manipulator case, the base rotates and translates due to the torque disturbances from the manipulator. The stroboscopic trajectories, joint angles, UAV translations and rotation, and torque disturbance transmitted from the manipulator to the UAV are shown in Figures 12 and 13. Figure 12: Stroboscopic trajectories for the first half (**left**) and second half (**right**) of the linear trajectory task for the 3-DOF redundant UAM using the generalized Jacobian with external forces. The UAV base is the large green marker, joints are the black circles, links are the blue lines, and end-effector positions tracked are the red crosses. Figure 13: Manipulator (**top**) and UAV motions and torque disturbance (**middle** and **bottom**) for the 3-DOF UAM using the generalized Jacobian with external forces performing a linear trajectory task. Finally, we tested the method on the circular trajectory with the redundant 3-DOF manipulator. The results in Figures 14 and 15 show that, at the beginning of the task, the UAV translates horizontally following the motion of the end-effector, reaching a maximum displacement of about 9 cm from its initial position, and that in the last part of the task the UAV tends to return to the initial position. In this case, the UAV also shows a visible vertical translation bigger than 1 cm, however, the end-effector position error is always smaller than \(12\times 10^{-5}\) m.  
Figure 14: Stroboscopic trajectories for the first half (**left**) and second half (**right**) of the circular trajectory task for the 3-DOF redundant UAM using the generalized Jacobian with external forces. The UAV base is the large green marker, joints are the black circles, links are the blue lines, and end-effector positions tracked are the red crosses.