# Aerial Tele-Manipulation with Passive Tool via Parallel Position/Force Control +
## 5 Experimental Results
### Results  
#### 5.2.1 Stationary Object Experiment  
Initially the quadrotor was located at the origin of the global coordinate, while the object, which was a 0.15 m \(\times\) 0.15 m plate, was located at [1.0 m, 0.0 m and \(-\)0.85 m] with downward pointing \(\mathbf{z}_{w}\). The human operator drove the quadrotor towards the object, and once the robot reached the object, the driver commanded a variable continuous force vector, by means of the haptics device. Finally, the human operator commanded the robot to leave the object, and brought it back to free flight. Figure 7-top shows the snapshots of different moments of the experiment, while Figures 8-left, 9-left, and 10-left show the results of the experiment. Figure 8a shows the position tracking during the experiment, while the position error is explicitly reported in Figure 9-left, for the reader's convenience. As can be seen, before the contact event, and after the quadrotor leaves the object, the position error is low, that is, bounded below 7 cm, while during the contact there is significant position error, that is, \(\approx\)35 cm, specifically in \(x\)-direction which corresponds to the normal component of force. It is worth underlining the fact that, during the interaction, the position error does not provide relevant information for evaluating the controller's performance, as during that phase the platform is force-controlled, and the position error represents a necessary condition to guarantee the force tracking. As a matter of fact, a higher position error during the contact with stationary object means a higher thrust and a higher pitch angle is demanded, which results in larger applied forces. Figure 8b shows the contact maintenance function output, which keeps the desired force vector inside the friction cone. It can bee seen that, when the commanded force  
Figure 7: Snapshots of stationary object (**top**) and movable object (**bottom**) experiments. The experiments aimed to dock and apply a commanded time-varying force to the objects. (**a**,**a**) In free flight, (**b**,**b**) establishing a contact with the object, (**c**,**c**) applying force vectors, (**d**,**d**) releasing the contact, and (**e**,**e**) in free flight again. violates the friction constraints of (10), the normal component of the desired commanded force is increased while the other two tangential components are decreased, and the desired norms of the input (\(\mathbf{f^{\star}_{d}}\)) and output (\(\mathbf{f_{d}}\)) are equal. Figure 8c shows the force tracking control result during the experiment. As is evident, the force tracking performance is very good (average absolute error of 0.11 N, which represents less than 10% relative error, and a maximum absolute error of 0.51 N). The contact transition is shown in a separate window inside Figure 8c, which introduces a very smooth contact transition with only a small single bouncing event. Note that in the transition phase of the robot force control with a rigid environment, having a small amount of bouncing and inadvertent losses of contact are common, even for grounded robotic arm manipulators with non-zero reaching velocity [33]. The haptic feedback components are shown in Figure 10-left. The master position (the position of the haptic device's handle with respect to the center of its workspace) is shown in creating the spring-damper force \(f_{PD}\) that brings back the master device's handle to the center of its workspace (Figure 10a). The position tracking error of the aerial manipulator in the free flight condition along with the force applied to the object in contact create the second constitutive component of the haptic feedback \(f_{pf}\), which is shown in Figure 10b. The haptic feedback in the contact condition is equal to the measured force from the force sensor with the opposite direction.