## 7 Load Picking Case
### Simulation Results  
For the same load picking task, inverse kinematics was solved in MATLAB discretizing with a timestep \(\Delta t=1\times 10^{-3}\) s, firstly by using the generalized Jacobian and then using the extended generalized Jacobian. Resulting joint velocities were imported in the ADAMS model of the UAM to dynamically simulate the task. The results comparing the stroboscopic trajectories of load picking experiments for both algorithms are shown in Figure 22. The simulated joint angles, UAV translations, and UAV rotations and reaction torques are shown in Figure 23. For both the solution methods, the trajectory is successfully tracked with minimal position errors: for the generalized Jacobian with external forces, the end-effector position error is always smaller than \(12\times 10^{-5}\) m, and for the extended generalized Jacobian the end-effector position error is smaller than \(6\times 10^{-4}\) m at all times. Figure 21: Desired end-effector velocities for the load picking task. The vertical gray line indicates the start time of contact. Figure 22: Stroboscopic trajectories for the load picking task for the 3-DOF redundant UAM using the generalized Jacobian with external forces (**top**) and using the extended generalized Jacobian (**bottom**). The left column shows the forward motion until the start of the grasping, the central one shows the slow load lifting part, and the right one the fast part of the backward motion. 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; the black line shows the height of the undeformed contact spring (\(z=z_{S}\)). Figure 23: Comparison between generalized Jacobian with external forces (\(J_{\rm G}\)) and extended generalized Jacobian (\(J_{\rm ext}\)) solutions for the 3-DOF redundant UAM: manipulator motions (**top**), UAV translations (**middle**), UAV rotation and torque disturbance from the manipulator to the UAV (**bottom**). The vertical dark line indicates the start time of contact for the grasp. For the generalized Jacobian method, at the beginning of the task the manipulator exerts a torque of almost 0.6 Nm on the base and causes the UAV to rotate and translate. The base UAV also translates after the load grasping along \(x_{I}\) up to 10 cm, overtaking the initial position of the load (8 cm). Then, the UAV moves backward reaching an \(x_{I}\) of about 2.5 cm at the end of the task. In this movement, we can see oscillations on torque disturbance, UAV rotation, and horizontal translation since the UAV control torque requires time to stabilize the angle caused by the initial disturbance from the manipulator. For the extended generalized Jacobian method, we can see that the torque disturbance is reduced by two orders of magnitude compared to the previous case and is now smaller than \(1.5\times 10^{-3}\) Nm, the UAV rotation is now smaller than \(5\times 10^{-5}\) rad, negligible compared to the generalized Jacobian solution. Consequently, the UAV, except for some small oscillations of the millimeter magnitude due to the algorithm, keeps its initial horizontal position. Indeed, we are imposing a zero torque transferred to the UAV during the picking of a load not aligned with the CoG of the system, and this most probably causes these small oscillations. Please note that, in the inverse kinematics algorithm for the extended generalized Jacobian, the external torque in (34) is obtained from a previous timestep, so it is accounted for with a delay of \(\Delta t\). For both methods, we can see that the UAV altitude increases in the first second because of the inertia of the manipulator (moving downwards), then the UAV reduces its elevation (along \(z_{I}\)) until the load loses contact with the elastic ground; there are no substantial differences in vertical displacement between the two methods.