## 1 Introduction  
Since 1970s, towed cable systems have been analyzed for various applications of payload aerial transportation including payload delivery, kites, aerial refueling systems, brick transportation and rescue missions [1]. Recently, unmanned aerial vehicles (UAVs) capabilities for the transportation and manipulation of objects have caught the attention of researchers for the transportation of diverse types of objects, for inspection and maintenance of industrial elements and surfaces and for other industrial- and emergency-related applications [2, 3, 4]. Researchers have invested a big effort in last years in developing different control models, vision systems and grasping or contact mechanisms in order to cope with all the difficulties that these systems find [5]. In particular, multi-rotors have become increasingly affordable for e by industries for delivery and inspection, with some start-ups becoming successful companies. Cooperative teams of quadrotors have great potential for some applications such as suspended object transportation [6, 7]. More specifically, cooperative tasks of quadrotor transportation of deformable linear objects (DLO) (i.e., cables or hoses) is being proved to be very useful in emergencies and in hardly accessible areas [8], fire extinction [9], windmill turbine cleaning [10], liquid spraying [11, 12] or transportation of payloads suspended from cables attached to the UAVs [13, 14]. In the latter case, most e approaches assume that cables are rigid links connecting the UAV and the payload without intrinsic dynamics. However, in the other cases, accurate DLO geometrical and dynamical modeling is essential in order to achieve precise and robust control of the entire system encompassing the DLO and the transporting quadrotors. Recent works deal with DLO modeling by catenaries [13], while others decompose the DLO into a sequence of connected rigid links [15] in order to build up the control system coping with DLO transportation and manipulation. Catenary modeling has been applied to design proportional derivative (PD) controllers [16] that achieve the task of DLO transportation by a team of quadrotors [17; 18]; however, catenary models cannot cope with aggressive maneuvers, involving sharp changes in direction and an acceleration of the drones desired trajectories. Sudden changes in the relative positions of the drones can abruptly change the shape of the DLO segments pushing their geometrical models off limits. Such sagging conditions appear due to distances that are too short or too long between the drones, making the DLO catenary model fall into singularities so that the entire control system fails. Alternatively, the parabola may be used as an approximation for the catenary [19; 20; 21] that does not suffer from sagging conditions. In previous studies [18], we have developed the online adaptation of a fuzzy logic rule system that sets the parameters of the drone PD controllers. This adaptation is carried out independently for each drone in the team transporting the DLO. Figure 1 presents an overall block diagram of the system. We have shown the effectiveness of the fuzzy logic adaptive control system to achieve some aggressive maneuvers with sharp direction changes, compromising UAV team formations during DLO transportation. The enhanced control system permits UAVs to follow the mission path and to retain a distance between them in a stable and smooth manner. The main contribution of this paper is as follows: We present a hybrid modeling system switching between catenary and parabola models of DLO segments hanging between pairs of drones in order to achieve robust control in the presence of wind disturbances and aggressive maneuvers and overcoming sagging conditions. We provide the decision threshold based on the relation between the distance among drones and the actual length of the DLO. The article structure is as follows: Section 2, reviews related works in the literature. Section 3 presents the hybrid DLO modeling switching between the catenary and parabola.