# A Hybrid Control Approach for the Swing Free Transportation of a Double Pendulum with a Quadrotor
## 5 Conclusions  
A system composed of a quadrotor and a double pendulum is a highly nonlinear dynamical system which represents a big challenge to achieve a smooth and precise control performance. The proposed MPC-PC control configuration approach aims to dampen the swing of the double pendulum by the precise and smooth control of the drone orientation. Transporting the pendulum with the least possible oscillations, we can avoid large perturbations affecting the quadrotor and the aggressive increase or decrease of the horizontal and vertical positions of the loads relative to the quadrotor.  
The experiment reported in this article validates the proposed control approach for the transportation of a driven double pendulum. Including a comparison against a PD-PD control system configuration. The MPC-PD proposal achieves a three fold reduction of the double pendulum swinging angle. The experiment proved that the quadrotor under MPC-PD control is capable of following the assigned path, despite the variability of \(T_{1Z}\) and \(T_{1X}\). The navigation control is robust enough to stick close to the parabolic planned trajectory (Figure 5). We considered that the most easily perturbable pendulum mass was \(m_{2}\), and thus, in Figure 7 we can see that the maximum oscillation has a maximum of about 0.23 rads, and rapidly converges to 0. This stability of the driven double pendulum can also be observed in Figure 9, where \(T_{1Z}\) is plotted. As each pendulum mass weights 0.3 kg, we see that the vertical component of tension of the cable attached to the drone holds \(6N\) along the whole trajectory, which means that \(T_{1X}\) force is almost null and therefore, it encourages the navigation stability of the drone along the path. Therefore, the design of the MPC described in Section 3 proves to be capable of achieving both the swing-free trajectory and the precise navigation along the planned path. Finally, in Figure 10, we can see that the quadrotor manages the double pendulum swing rejection by some compensations in the heading angle, \(\theta\). Since we start with the pendulum masses vertically alienated, i.e., \(\theta_{1}=\theta_{2}=0\), the beginning of the advance of the quadrotor forces the pendulum swing backwards due to inertia, which is compensated by the quadrotor by reducing its heading angle till it overcomes the transient starting regime and thus, making the two pendulum masses swing slowly forward.  
In the recent bibliography suggested by the reviewers, we found a quite recent work dealing with the control of a double pendulum transportation, while trying to amortiguate the double pendulum swing [38; 39; 40]. Specifically, they use classical PID controllers to handle this problem. Contrary to our proposed MPC-PD approach, PIDs can neither  
Figure 10: Quadrotor \(\theta\) angle along time under MPC-PD control.  
include system constraints, nor they can benefit from the robustness of Kalmann filtering. Experimental results in [38; 39; 40] are quite interesting and we plan future work in order to reproduce their with our MPC-PD control approach. Further work would be directed to test the adaptive behaviour in aggressive maneuvers and include interactions with physical walls for the payload, such as collisions or taking offs.  
Conceptualization, J.E. and M.G.; methodology, J.M.L.-G. and M.G.; software, J.E. and G.G.; validation, J.E.; writing--original draft preparation, J.E.; writing--review and editing, All; funding acquisition, M.G. All authors have read and agreed to the published version of the manuscript.  
This work has been partially supported by FEDER funds through MINECO project TIN2017-85827-P, and project KK-20200044 of the Elkartek 2020 funding program of the Basque Government. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the [PERSON] grant agreement No 777720.  
Not applicable.  
The authors declare no conflict of interest.