Hybrid Modeling of Deformable Linear Objects for Their Cooperative Transportation by Teams of Quadrotors

  Citation: Estevez, J.; Lopez-Guede, J.M.; Garate, G.; Graña, M. Hybrid Modeling of Deformable Linear Objects for Their Cooperative Transportation by Teams of Quadrotors. Appl. Sci. 2022,12, 5253. https://doi.org/10.3390/ app12105253 Academic Editor: Alessandro Gasparetto Received: 9 March 2022 Accepted: 17 May 2022 Published: 23 May 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Hybrid Modeling of Deformable Linear Objects for Their Cooperative Transportation by Teams of Quadrotors Julian Estevez 1,* , Jose Manuel Lopez-Guede 2, Gorka Garate 1and Manuel Graña 3 1Faculty of Engineering of Gipuzkoa, University of the Basque Country, 20018 San Sebastian, Spain; [email protected] 2Faculty of Engineering of Vitoria, University of the Basque Country, 01006 Vitoria, Spain; [email protected] 3Faculty of Computer Science, University of the Basque Country, 20018 San Sebastian, Spain; [email protected] *Correspondence: [email protected] Abstract: This paper deals with the control of a team of unmanned air vehicles (UAVs), specifically quadrotors, for which their mission is the transportation of a deformable linear object (DLO), i.e., a cable, hose or similar object in quasi-stationary state, while cruising towards destination. Such missions have strong industrial applications in the transportation of hoses or power cables to specific locations, such as the emergency power or water supply in hazard situations such as fires or earthquake damaged structures. This control must be robust to withstand strong and sudden wind disturbances and remain stable after aggressive maneuvers, i.e., sharp changes of direction or acceleration. To cope with these, we have previously developed the online adaptation of the proportional derivative (PD) controllers of the quadrotors thrusters, implemented by a fuzzy logic rule system that experienced adaptation by a stochastic gradient rule. However, sagging conditions appearing when the transporting drones are too close or too far away induce singularities in the DLO catenary models, breaking apart the control system. The paper’s main contribution is the formulation of the hybrid selective model of the DLO sections as either catenaries or parabolas, which allows us to overcome these sagging conditions. We provide the specific decision rule to shift between DLO models. Simulation results demonstrate the performance of the proposed approach under stringent conditions. Keywords: quadorotor; deformable linear objects; payload transportation 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 industrialand emergency-related applications [ 2 – 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 Appl. Sci. 2022,12, 5253. https://doi.org/10.3390/app12105253 https://www.mdpi.com/journal/applsci Appl. Sci. 2022,12, 5253 2 of 17 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 [19–21] 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 1presents 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. Figure 1. Overall block diagram. 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 3presents the hybrid DLO modeling switching between the catenary and parabola. Section 4presents the follow-the-leader formation of the quadrotor team that will be followed in the experiment. Section 5describes the quadrotor control system, including the novel online adaptive PD tuning based on a fuzzy adaptive gradient descent rule. Section 6describes the experimental settings. Section 7reports the results that demonstrate the effectiveness of the proposed control system. Finally, Section 8 provides our conclusions and lines of future work. Appl. Sci. 2022,12, 5253 3 of 17 2. Related Works Table 1offers a summary categorization of the relevant literature. Discrete DLO models tackle the problem by breaking down the structure into a number of rigid rod elements of finite length, which can be physically simulated as pendulums, curve segments or by a network of masses and springs, so called lumped-mass models. These models require the representation of forces and moments at each element so that it is possible to model the motion of each and all of them. A study comparing several modeling methods concluded that the lumped-mass representation is the most versatile method, despite the large amount of computational resources required for its implementation [ 22 ]. Alternatively, some approaches propose the modeling of the DLO as a chain of rigid links allowing the differentially flat control [ 23 ] of the team carrying the DLO transportation under quasi-stationary conditions [ 15 ]. Nevertheless, comparative studies on cable structure modeling show the superior numerical efficiency of catenaries for the study of different force situations, vibrations and torsions [24–26]. Table 1. Summary comparison of DLO related research. DvC = Discrete versus continuous model. Ref. Year DvC Paradigm Applications [27] 1971 continuous catenary cable towed by aircraft [22] 1973 both Analytical methods (survey) Ocean Science [28] 1981 continuous catenary cable structures [29] 1995 continuous catenary cable towed by aircraft [30] 1999 both catenary vs. rod elements – [31] 2000 continous catenary underwater towing [32] 2001 continuous catenary ocean sciences, mooring [33] 2001 continuous catenary underwater towing [24] 2006 continuous catenary designs of nets of cables [34] 2007 continuoys rigid link tethered UAV [35] 2008 continuous parabola cable structures [36] 2008 continuous catenary ocean sciences, mooring [37] 2009 continuous rigid link cooperating UAV payload transport [38] 2010 continuous rigid link cooperating UAV payload transport [39] 2012 continuous catenary cable-driven parallel robot [40] 2012 continuous rigid link cooperating UAV payload transport [41] 2013 continuous rigid link tethered UAV [42] 2013 continuous catenary cable-driven parallel robot [43] 2015 continous rigid link tethered survillance UAV [44] 2015 discrete series of rigid links cooperating UAV payload transport [18,45] 2015, 2017 continuous catenary DLO transportation by n≥2 UAVs [46] 2016 continuous catenary tethered UAV [47] 2017 continuous catenary tethered UAVs [48] 2017 continuous catenary cable transportation by 2 UAVs [25] 2018 continuous catenary suspension bridges [26] 2018 continuous catenary suspension bridges [49] 2018 continuous catenary and parabola cable vibrations [15] 2020 discrete chain of rigid links hose transportation by 2 UAVs [13] 2021 continuous catenary cable transportation by 2 UAVs Catenaries are widely accepted as accurate cable models, for instance, in mooring cable simulations [ 32 , 36 , 39 ], and in the design of civil cable structures [ 28 , 50 ]. Continuous cable Appl. Sci. 2022,12, 5253 4 of 17 modeling by catenaries is more accurate and less computationally demanding [ 30 ] than discretization approaches. 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. During recent years, there has been a large effort devoted to the development of a flexible dynamic model of low computational cost of DLO payloads, because, despite their passive nature, payload configurations might affect the performance of the control of the robot carrying out the transportation task. Taut cables modeled as a metal bar are valid only for a small spectrum of applications. Discrete cable modeling remains computationally too costly. Catenary models emerged as a possible representation model with promising results and have already been tested in simulations of simple robotic experiments. In order to capture the best possible reality, by taking into account the bibliography on cable Appl. Sci. 2022,12, 5253 5 of 17 structures [ 22 , 24 , 30 , 53 ], we propose a hybrid catenary–parabola cable model so that the control systems of teams of drones for aerial transportation of long cables can cope with demanding maneuvers. As far as the authors know, this type of switching model has never been applied in cable transportation tasks with quadrotors. 3. Parabola–Catenary Hybrid DLO Geometrical Model The catenary equation y=acosh x a is derived from well-known fundamental equations of applied mechanics as the shape that takes a flexible but non-elastic DLO hanging from two extremes under its own weight. Parabola equation y=ax2 has been used as a surrogate geometric model approximation of the shape of hanging cables for both static or kinematics analysis [ 30 , 54 , 55 ], because it is more robust to extreme conditions that induce singularities in the catenary equation. In general applications, such as modeling the dynamic behavior of cables or bridge structures, the dynamic simulations of objects modeled alternatively as a parabola or the catenary are very similar, except under very heavy payloads where the differences among these functions might introduce substantial differences on analysis results [ 24 ]. Moreover, the lower computational cost of a parabola is another reason for its use in the mathematical modeling of cables [ 25 ]. In the case of robotics, this simplification has been contested in some applications [ 47 ]. Figure 2visualizes the approximation of a catenary by a parabola, which may be good enough for some applications [55–58], especially in the development of cable-driven robots [59]. Figure 2. Visual comparison of the catenary and parabola curves with parameter a. This article proposes a hybrid between catenary and parabola geometric models for a DLO section hanging between two drones. Automated switching from one model to the other occurs when the Euclidean distance between the drones is too short relative to the actual length of the DLO section. In this situation, the catenary is no longer a good approximation to the shape of the DLO. Heuristically, we have set the threshold for the shift between models at d<L/ 3, where L is the length of the DLO section and d the Euclidean horizontal distance between the drones supporting it. In drone team operation modeling, there are some previous studies on hybrid modeling for control strategies and their formation [ 60 – 62 ], but there are no previous studies on hybrid payload modeling. 4. Quadrotor Team Formation Strategy The quadrotor team transporting the DLO is a follow-the-leader column platoon formation [ 63 ] that offers advantages for obstacle avoidance and needs only the specification of the trajectory of the leader to guide the entire team. In aerial transport, this configuration represents a novelty, as most of the published research studies study the collaborative transportation of a heavy load using different approaches to calculate the payload’s position relative to the the quadrotors at any moment, such as the Udwadia–Kalaba method for modeling [ 6 ] focused on the estimation of the position of each UAV with respect to the payload in a dynamic equation minimization method. Other studies’ use geometric criteria for calculating the UAVs’ desired position to accomplish the task. For instance, ref. [ 64 ] sets the formation with Delaunay triangles, and [65] uses vectorial conditions. Appl. Sci. 2022,12, 5253 6 of 17 Our approach is inspired in ground robotics [ 63 ], adding the extra constraint of maintaining a horizontal Euclidean distance between robots. Orientation and position of each robot are calculated at each moment. The graphical representation of the team configuration over the (X,Y) plane can be seen in Figure 3, where ρ corresponds to the desired distance between robots, and the L and F subindices denote the leader’s and follower’s variables, respectively. Vectors VL and VF are the motion directions of the leader and follower drones, respectively. Figure 3. Follow-the-leader platoon model. Finally, the equations for the position and orientation of the follower UAV relative to the position of the leader UAV [64,66], are as follows.    xF=xL−ρcos(α+ψF) yF=yL+ρsin(α+ψF) ψF=ϕ+ψL−π . (1) 5. System Control The control system of each quadrotor in the team is composed of an inner and outer loop with proportional derivative (PD) controllers for each degree of freedom. Figure 4 depicts the structure of the system. The inner control loop is in charge of controlling the attitude of the quadrotor by providing rotor commands to achieve desired attitude angles. The outer control loop is in charge of following the desired trajectory by providing the desired attitude angles to the inner control loop. Tuning of the controller parameters by an offline Particle Swarm Optimization algorithm achieved the vertical equiload configuration in the inner loop [ 17 , 45 ] minimizing the final height adjustment overshot and proving to be a scalable system. Figure 4. Control system for each UAV in the system. Appl. Sci. 2022,12, 5253 7 of 17 Online Adaptation of PD Controllers Offline tuning of the PD controllers [ 17 ] in the outer loop is unable to cope with aggressive maneuvers, such as short radius curves and sharp changes of direction, and neither provides adaptations to different lengths and weights of DLOs. Therefore, we proposed [ 18 ] and their online adaptive tuning following an Adaptive Fuzzy Modulation (AFM) approach, combining a gradient descent adaptation rule and a fuzzy membership function activation [ 67 , 68 ]. The membership functions act on the PD controller parameters if a fuzzy logic expression is satisfied, following the Takagi–Sugeno controller design paradigm [69]. The perceived error Pe (cf. Equation (2)) measures the relative error between the real position of the UAV Yreal and its reference Yre f at each moment. Pe =Yre f −Yreal Yre f ·100, (2) The proposed fuzzy tuning rules contemplate four error conditions dependent on Pe value modeled by corresponding four triangle-shaped membership functions {µi(Pe)}4 i=1 , for which its membership supports are provided by the following. Dµ1=(−2, −9),Dµ2=(−1, −5),Dµ3=(1, 5),Dµ4=(4, 9). (3) The adaptation of parameter Kp is modulated by functions µ1(Pe(t)) and µ4(Pe(t)) through Equation (4), while the adaptation of parameter Kd is modulated by functions µ2(Pe(t)) and µ3(Pe(t)) with Equation (5): Kp(t+1)=Kp(t)+αe(t)(µ1(Pe(t)) +µ4(Pe(t))) (4) Kd(t+1) = Kd(t)+αe(t)(µ2(Pe(t)) +µ3(Pe(t))) (5) where α is the adaptation factor, which takes a constant value between 0 and 1 during the entire experiment. The e(t) functions compute the instantaneous error relative to the desired values θd and φd of the angles that determine the motion in the XY plane. The online adaptation Equations (4) and (5) follow a stochastic gradient descent algorithm The convergence of the continuously adaptive process of the fuzzy logic control approach has been proven [18]. 6. Experiments We have carried out three computational simulation experiments that require a team of three quadrotors transporting a DLO attached to them in a follow-the-leader strategy, where followers try to mimic the motion of the leader, i.e., following parallel paths to the one of the leader trying to preserve the distance among quadrotors. In both experimental simulations, time is discretized in steps of 0.1 s, the DLO is modeled by the catenaryparabola approximation and we apply a fuzzy logic approach for the adaptive tuning of the PD controller. Both experiments feature sharp path changes that were unmanageable with previous versions of the controller [17]. Experiment 1: In this experiment, the nominal path set for the leader quadrotor has sudden changes of direction, as shown in Figure 5. The objective of the experiment is to check whether the drone team’s formation remains stable and is able to cope with the different path corners, particularly with the sharp angle located at x= 400 cm. For the experiment, both with and without wind disturbance conditions in the X direction are used to model the following dynamic equation d(t) = 5+5 sin(π 2t). Experiment 2: In this experiment, the three quadrotors transport the DLO in a straight line path. When the leader drone has traversed a distance of x= 350 cm, it suffers a sudden lateral disturbance consisting of a push displacing it 80 cm in the Y positive direction, as shown in Figure 6. The experiment aims to observe how the leader and followers recover the nominal path after the disturbance. Experiment 3: Now, the three quadrotors must follow a spiral 3D ascending path specified by x= 100 sin(t) , y= 100 cos(t) , z= 5 (t) , with t= [ 0 : 3 π] , as seen in Figure 7. Appl. Sci. 2022,12, 5253 8 of 17 This test aims to check the capacity of the drones control system to cope with the three direction paths at the same time, with no-wind conditions, and considering only the hybrid DLO model of catenary and parabola. The leader drone’s starting position is at (0, 0, 0). Figure 5. Nominal path for the drone leader in Experiment 1, featuring sharp changes in direction. Figure 6. Experiment 2. Nominal path followed by the leader drone suffering a sudden lateral disturbance. Figure 7. Spiral 3D ascending path in Experiment 3. In the three experiments, the initial distance in the horizontal axis between drones at the extremes and the central drone is the same and is set as ρd=70 cm. Moreover, in order to ensure balanced energy consumption, the system is in equiload conditions [ 18 , 45 ]. As a consequence, no further correction of altitude is applied, although the horizontal Euclidean distance between robots might change. In platoon formation, the maximum vertical thrust was limited for each quadrotor to 20 N following standard hardware specifications. The length of the catenaries was set to L0= 240 cm; the mass density of the DLO was set to w= 0.005 [kg/cm] . For online fuzzy tuning of the PD controller, the adaptation factor was set to α= 0.5. Dynamic parameters of each quadrotor appear in Table 2. We set the initial PD parameter values as follows: Kpx =Kpy =0.22; Kdx =Kdy =0.76. Appl. Sci. 2022,12, 5253 9 of 17 Experiments have been coded in house in Scilab 5.4. No other public or private software solutions have been used. The code of the implementation has been published in an online repository (https://github.com/Julestevez/Quadrotor-simulator/tree/master/ catenary%20and%20parabola%20hybrid%20modelling, accessed on 10 May 2022). Table 2. Quadrotor structural and dynamic parameters. Parameter Value mass, m0.5 kg arm length, l25 cm inertia moments, Ixx =Iyy 5×10−3[Nms2] inertia moment, Izz 1×10−2[Nms2] propeller thrust coefficient, b3×10−6[Ns2] drag, d1×10−7[Nms2] 7. Results 7.1. Experiment 1 Figure 8shows the paths followed by the team of quadrotos under wind conditions in a simulation lasting 60 s of simulated time. We found that the drones followed the same path in the repetitions without wind disturbances. Figure 9shows the position for the three quadrotors and the DLO at different moments of the simulation, where we can observe how the follower drones attempt to keep their linear formation by preserving, as much as possible, the DLO configuration and the distance among quadrotors. In the following, let us denote D1 , D2 and D3 as the leader, mid and rear drones, respectively. Figures 10 and 11 show the plot in time of the thrust of D1 without and with wind disturbances, respectively. 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