## 2 Related Works
The cable modeling by a catenary relies on the following assumptions and simplifications [51]: The mass per unit length of cable is constant, there is no torsion, the cable cannot increase its length and the cross-section of the cable is much smaller than the longitudinal dimension, corresponding to a 2D solid at any time, hence resulting in the general term of deformable linear object (DLO) that we use in this paper. In some studies, the geometry of cable segments has been approximated by parabolas [35] and other second degree polynomials [24], which allow faster computations and are robust enough to accurately model the sagged cables [49]. Following this background, we propose our hybrid DLO modeling methods, as discussed below. Tethered UAVs, also known as taut tethers, are a special case where the UAV has its center of the mass attached to the global coordinate frame origin by a DLO, which is a tense cable with negligible mass. Most research studies consider that, in this configuration, the cables are rigid links. Different variations and evolutions of this model have been proposed in the last years for different tasks in ground robotics or ship operations [34; 41; 43; 52]. Catenaries usage for tethered UAVs have not been deeply studied despite early promising results [29; 46]. For UAV transportation of payloads hanging from a cable [13], cable dynamics modeling is a key factor for understanding the system's dynamics and permit the computation of forces exerted on the quadrotor, where the motion of the entire cable is represented as a continuous structure with appropriate boundary conditions. Their main advantage is that the simultaneous consideration of each point in the material permits the calculus of more accurate cable dynamics [27; 29; 31; 33]. In the cooperative transportation of a load by a team of UAVs, the cables are often modeled as a rigid link [37; 38]. Cooperative aerial towing problem is similar to the problem of controlling cable-actuated parallel manipulators in three dimensions. In these systems, the variation of the lengths of cable attachments determines the payload orientation and position. Following this line of research, [44] modeled the system as a serially connected links system for the cooperative transportation and orientation of a rigid two-dimensional payload. The transportation of a DLO attached by cables to the UAVs under severe dynamic limitations and quasi-static conditions was also achieved [40]. In previous studies dealing with the cooperative transportation of DLOs by teams of UAVs, DLO sections were modeled as catenaries in an equiload vertical configuration [18; 45]. Additional sources confirm that catenaries are a good modeling alternative for the cooperative aerial transportation of DLOs [13], including visual servoing approaches [47], and collision avoidance [39; 48]. However, catenary curves are hyperbolic functions that suffer from numerical singularities, which may lead the control system to collapse when the distances between quadrotors are too short or too large relative to the DLO's section length due to cable sagging [42]. This situation occurs when the team of quadrotors must perform aggressive maneuvers consisting of sudden sharp changes of direction and/or accelerations. In order to deal with these extreme conditions, we propose the hybrid modeling of DLOs that shifts between catenary and parabola models according to the system state. Finally, the transportation of a cable by pairs of UAVs has been proposed for the grasping and transportation of objects featuring some kind of hook, such as umbrellas [13]. After the hooking maneuver, the shape of the cable can be modeled by straight sections, and the entire system can be treated as cooperative payload transportation from suspended cables.