## 7 Results  
### Experiment 1  
Figure 8 shows the paths followed by the team of quadrotos under wind conditions in a simulation lasting 60 s of simulated time. We found that the drones followed the same path in the repetitions without wind disturbances. Figure 9 shows the position for the three quadrotors and the DLO at different moments of the simulation, where we can observe how the follower drones attempt to keep their linear formation by preserving, as much as possible, the DLO configuration and the distance among quadrotors. In the following, let us denote \(D_{1}\), \(D_{2}\) and \(D_{3}\) as the leader, mid and rear drones, respectively. Figures 10 and 11 show the plot in time of the thrust of \(D_{1}\) without and with wind disturbances, respectively. It can be appreciated that the response to the wind perturbations introduces some changes in the thrust profile in order to follow the nominal path as close as possible, thanks to the online tuning of the PD controller by the AFM algorithm.  
Figure 12 plots the evolution in time of the Euclidean distance between each pair of drones without wind disturbances. Variations around the nominal distance remain bounded. The plots of the temporal evolution of the PD parameters tuned online by the AFM are shown in Figure 13 in both \(X\) and \(Y\) axes for the simulation under wind disturbances.  
At different moments of the simulation, the DLO model switched from a catenary to a parabola model during some short intervals. In particular, in Figure 12, we detect two milestones when switching happens. The first DLO model transition occurs around second 90, when the distance between the robots becomes too big. On the contrary, the second DLO model transition around second 170 occurs because the distance between \(D_{2}\) and \(D_{3}\) is too small.  
Figure 10: Plot of the instantaneous thrust of leader drone (\(D_{1}\)) during the simulation without wind disturbances.  
Figure 9: Position of the drones and the DLO at different moments of Experiment 1. Color code: red, green and blue correspond to leader, mid and rear drones, respectively.  
Finally, at the end of the simulation, the cable horizontal distance becomes big again. These instants of transition between DLO models correspond to the first two changes of direction and the final straight trajectory section in Figure 5. However, as we can see, thanks to the proposed hybrid cable modeling, the changes in thrust were bounded in those critical moments, as can be seen in Figures 10 and 11. Finally, Figure 14 shows the  
Figure 11: Plot of the instantaneous thrust of leader drone (\(D_{1}\)) during the simulation under wind disturbances.  
Figure 12: Euclidean distance along time between each pair of drones transporting the DLO in Experiment 1 without wind perturbations.  
trajectories of the three quadrotors when we do not use the DLO model transition from catenary to parabola, nor do we apply the AFM online tuning of the PD controllers. As we can observe, the error of the nominal path following by the drones is large compared to Figure 8, and the drone must follow alternative and wider path curves in order to retain the catenary shape of the DLO. Moreover, the nonadaptive PD control is not able to cope with the sharp corners in the path during transportation.  
Figure 14: Trajectoriesof the quadrotors in Experiment 1 when neither the catenary-parabola approximation nor the online AFM tuning of the PD controllers were applied. Color code: red, green and blue correspond to leader, mid and rear drones, respectively.  
Figure 13: Instantaneous values of the AFM online-tuned parameters of the PD controller of \(D_{1}\) along both \(X\) and \(Y\) axes.