# A Hybrid Control Approach for the Swing Free Transportation of a Double Pendulum with a Quadrotor
## 1 Introduction
In [17], the authors use a Hamiltonian approach for the control of the nonlinear system, where the swinging oscillations are controlled by an interconnection and damping assignment-passivity based control. Geometric control is utilized in [18] in the model of cable decomposed in a series of linear linked pieces. Adaptive control and slung load pose estimation for small size quadrotors has been also introduced [19]. In the last years, artificial neural network (ANN) based algorithms are being tested for the control of quadrotors, despite their limitations in size, weight, power consumption, and response to the changing environments and flying conditions, which require a continuous retraining process of the ANN architecture [20], even more so in payload transportation applications. Besides the application of deep learning for quadrotors artificial vision navigation systems [21], classical control strategies have been proposed, like MPC for the elimination of the overshoot at load release time [22], or a reinforcement learning strategy based on the minimization of the payload oscillations for an automated cargo delivery [23]. Conventional PD controllers have proven to be an efficient solution for inner control loop in payload transportation task by quadrotors, due to their simplicity, robustness and good results [6; 19; 24; 25; 26]. Adaptive PID controller can be employed for coping with wind disturbances and main external forces that affect the performance of the system [5; 25]. On the other hand, the MPC is designed to solve an optimization problem by defining and minimizing an objective function which is related to both the optimal state and control inputs. Although the interest towards this configuration gained the attention of researchers in recent years [27; 28; 29], they still did not solve completely all the problems or explore the possibilities related to it. The state of the art in transportation of slung loads with quadrotors can be broadly classified into two categories [4]: (a) Feedback control for tracking desired payload trajectories [17; 18]. (b) Trajectory planning for a quadrotor with a slung load [30; 31; 32; 33]. This paper follows the second approach. As in most of the prior works, the following dynamic simplifications were made without losing precision in the results: (1) The aerodynamic effects of the quadrotor and the payload are neglected. (2) The dynamics and perturbations of the payload are considered completely decoupled from quadrotors, considering that the the payload motion is bounded [33]. (3) The cable model is taut and non-elastic. (4) We consider the 2D plane XZ for the modeling of the double pendulum [34; 35]. These assumptions are considered to be realistic enough for the dynamic modeling of the quadrotor + slung load system, except when the quadrotor enters aggressive maneuvers [36]. According to [37], the discretization of the pendulum in two mass particles and massless and taut links represents a valid approach for the current study purpose, and it requires lower computational cost and offers very similar results. Moreover, the usage of the pendulum mathematical model as the DLO physical model widens the research possibilities, as it permits an easier dynamic calculus of taking off and landing, which is not a fully studied field [7]. Very recent works deal with the control of the transportation by an quadrotor of a payload modeled as a double pendulum [38; 39; 40]. The contributions of this paper are as follows: We present a control configuration with a PD controller embedded in a MPC controller to transport a double pendulum swinging free and smoothly. The double pendulum is a highly nonlinear and unstable multi-body solid. The PD controller is used for quadrotor attitude control while the MPC pursues the precise translational control that tries to minimize the oscillations of the payload. The considerations of the shocks of the two mass particles represent a big challenge for the motion control of the quadrotor.