# A Hybrid Control Approach for the Swing Free Transportation of a Double Pendulum with a Quadrotor
## 1 Introduction  
The research community has studied different approaches for the aerial transportation of payloads task and since the 70s. Specifically, towed cable systems have been analyzed in various applications like payload delivery, kites, aerial refueling systems, brick transportation, and rescue missions. The enhancements of these tasks would permit a wider spectrum of possibilities for unmanned aerial vehicles (quadrotors), including transportation of bigger and different types of objects, inspection and maintenance of elements and surfaces, and emergency services related applications. In recent years, there has been extensive interest in developing solutions for delivering a payload by connecting it to a multirotor quadrotor with cables [1, 2, 3, 4]. In these approaches the payload is often considered to be a particle mass at the end of the cable. We call this suspended load, which is also known as _slung load_. Despite the different alternative quadrotors considered in the literature to transport such payload, in this article we will focus on quadrotors. Quadrotors deployment of cable-suspended payloads can save time and energy, comparing to other methods, and thus, this configuration provides potential benefits that may extend the application possibilities of these aerial systems [5]. However, its passive dynamics affect the performance of the quadrotor, by the introduction of a new set of inputs, such as tensions and residual vibrations. Quadrotors are a nonlinear, unstable and underactuated system, and coupled dynamics with the attached payload adds an extra complexity and alters the performance of precise positioning and path-following, and thusit is required a careful control design in order to cope with them. The problem is far from being solved [6; 7]. Research on payload stabilization suspended by a cable started decades ago, when dealing with the control of cranes whose payload was modeled by a lumped-mass [8]. This model is simple, but it captures the complex dynamics of payload motion. Classic equations of applied mechanics were used to model a suspended load in cranes [9], and later extended in [10]. Nowadays, the stabilization of the slung load is pursued through the minimization of the particle mass swing, which enables the reduction of the vibrations and sudden shock in the quadrotor dynamic system. It is known that the transportation speed and the removal of the payload oscillations are contradictory control goals [11], therefore, the fulfillment of both objectives represents a challenging task for control engineering. First research works on payload transportation by quadrotors appeared in the late 1990s [12; 13]. Early attempts involved quasi-static motions to position and orient a suspended load [1]. Since then, the most common dynamic modeling is the lift vector of a payload to be fixed along the z-direction in the frame fixed to be quadrotor body, so that quadrotors can only generate their translational motion by tilting the body frame by the propellers in a certain direction. Usually, for these systems a cascade control configuration is used, where an outer-loop controller is in charge of the translational control and then, an inner loop is used to obtain the tilt angle calculated by the outer loop. For instance, [14] use a Model Predictive Controller (MPC) strategy and linear-quadratic regulator (LQR) control for slung load operations in hover and aggressive maneuvers, while [15] use the swinging load angles as an error input for a proportional derivative (PD) controller that corrects the translational controller during paths and maneuvers. An alternative way to deal with the dynamics of the suspended load is to incorporate it directly into the quadrotor control dynamics. In [16], the authors present a multi-rotor with a hybrid system, interchangeable between a taut or a flexible cable. Depending on the maneuvers to be done and obstacles found, the quadrotor control switches between the two configurations thanks to a mixed-integer quadratic programming.