The Catenary Robot: Design and Control of a Cable Propelled by Two Quadrotors
IIEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021 1
The Catenary Robot: Design and Control of a CablePropelled by Two Quadrotors
Diego S. D’antonio, Gustavo A. Cardona, and David Salda˜na
Abstract —Transporting objects using aerial robots has beenwidely studied in the literature. Still, those approaches alwaysassume that the connection between the quadrotor and the loadis made in a previous stage. However, that previous stage usuallyrequires human intervention, and autonomous procedures tolocate and attach the object are not considered. Additionally, mostof the approaches assume cables as rigid links, but manipulatingcables requires considering the state when the cables are hanging.In this work, we design and control a catenary robot . Our robotis able to transport hook-shaped objects in the environment. Therobotic system is composed of two quadrotors attached to the twoends of a cable. By defining the catenary curve with five degreesof freedom, position in 3-D, orientation in the z-axis, and span,we can drive the two quadrotors to track a given trajectory.We validate our approach with simulations and real robots. Wepresent four different scenarios of experiments. Our numericalsolution is computationally fast and can be executed in real-time.
Index Terms —Aerial Systems: Applications; Mobile Manipu-lation; Cellular and Modular Robots
I. I
NTRODUCTION I N recent years, aerial robots have become popular inindustry and academia because of their low cost and a largenumber of applications. Especially in object transportation,aerial vehicles such as quadrotors have demonstrated to beeffective as a solution. Some of the main applications ofquadrotors are: delivering last-mile packages for retail andwholesale companies [1]; transporting supplies to disasterareas [2]; accessing dangerous areas for humans such as forestfires [3], and delivering medicines and food such as needed inremote regions.Quadrotors can overcome payload restrictions by cooperat-ing with others to manipulate and transport objects that areeither suspended or attached. In the aerial robotics literature,there is a large number of works that tackle the problem ofcable-suspended load transportation, using either one quadro-tor [4], [5] or multiple [6], [7], [8]. In the case of multiplequadrotors, there are two well-known approaches, the point-mass load, and multiple contact points. In the point massapproach all the cables go to the same contact point in the mass[8]. In the multiple contact approach, the robots are attachedto different places on the load, which makes the problem morechallenging since it involves rotational dynamics in addition.On the other hand, rigid cables [7] and flexible cables [9]. Withrigid cables, quadrotors are more susceptible to disturbancesgenerated by other linked quadrotors. However, it is easier to
Manuscript received: October, 15, 2020; Revised December, 13, 2020;Accepted February, 8, 2021.This paper was recommended for publication by Editor Pauline Poundsupon evaluation of the Associate Editor and Reviewers’ comments.D. Salazar-D’antonio, G. A. Cardona, and D. Salda˜na are withthe Autonomous and Intelligent Robotics Laboratory (AIRLab) atLehigh University, PA, USA: { diego.s.dantonio, gcardona,saldana } @lehigh.edu Digital Object Identifier (DOI): see top of this page.
Fig. 1: The catenary robot , composed of a pair of quadrotorsattached to the two ends of a cable. Controlling the lowestpoint, span and orientation of the curve, the catenary robot canbe used to interact and pull objects, in this case, an umbrella.Video available at: https://youtu.be/3SaKKjl6os0 localize where the load is due to the length is fixed. Flexiblecables allow stretching, thus small disturbances do not affectthe performance of linked quadrotors too much. Nevertheless,localizing the load might suffer errors caused by vibrations.Despite the work that has been done extensively in thesuspended load transportation field, all the approaches assumethat the connection between the quadrotor and the load ismade in a previous stage. However, that previous stage usuallyrequires human intervention, and autonomous procedures tolocate and attach the object are not considered. Additionally,most of the approaches assume cables are rigid links, butmanipulating cables requires to consider the case when thecables are hanging. In this work, we design and control a catenary robot as in Fig. 1, where a cable is attached totwo quadrotors, allowing it to take its natural form causedby gravity and describing a catenary curve [10]. Catenarydynamics has been studied before [11], including applicationsin robotics such as obstacle avoidance with a hanging cableor servoing visual approaches [12], [13], [14]. There are alsoapplications of lifting a hose with multiple quadrotors [15].However, those approaches were not developed with cablemanipulation in mind. It is crucial to find a reference pointfor the catenary to attach to objects and manipulate them. Wechoose the lowest point of the curve as the reference because itcan be used to attach objects. In this work, we consider objectsthat have embedded hooks in their shape, making them easyto be pulled without knotting or tightening. Some examplesof these objects are shown in Fig. 2.The main contributions of this paper are twofold. First, we a r X i v : . [ c s . R O ] M a r IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021
Fig. 2: Objects that naturally have one or multiple hooks intheir shape. a) an umbrella has a hook shape in its handle. b)A chair has narrow parts allowing a cable to go through them.c) A saxophone has a hook shape in its neck. d) A bicycle canbe pulled passing a cable through its handlebar or its saddle.propose a versatile robot, called catenary robot, that is able tonavigate in the environment and to get attached to objectswith a hook-like shape. Second, we develop a trajectory-tracking method that controls the pose and the shape of thecatenary curve. In the state of the art, the cable is modeled asa chain based on many links and joints but controlling such amechanism is highly expensive since each join increases thedimensions of the configuration space. In contrast, our methodoffers a fast numerical solution.II. T HE C ATENARY R OBOT
Cables offer a versatile way to interact and transport objects,but usually it is challenging to find a way to attach cablesto objects in an autonomous way. In this work, we want tomanipulate a cable using quadrotors in a way that it can beattached to hooks or objects with hook shapes, some of theseobjects can be seen in Fig. 2. Once the cable is attachedto the object, other methods in the literature can be used fortransportation [6], [7], [8]. The cable, hanging from the twoquadrotors, adopts the form of the catenary curve, which variesdepending on the position of the quadrotors, mostly in theiraltitude difference. This lead us to the problem of finding away to control the pose of the lowest point of the cable and theshape of the catenary curve. Our robot is defined as follows.
Definition 1 ( Catenary robot ) . A catenary robot or flyingcatenary is a mechanical system composed of a cable withtwo quadrotors attached to its ends. Note that when the cableis hanging from the two quadrotors and there are no objectscolliding with the cable, this naturally adopts a catenary curveform. The cable is flexible and non-stretchable. The catenary robot is illustrated in Fig. 3. The Worldframe, {W} , is a fixed coordinate frame with its z -axispointing upwards. The reference point is the lowest point ofthe catenary in {W} , denoted by x C ∈ R . The quadrotors areindexed as A and B , and its locations in { W } are denotedby x A ∈ R and x B ∈ R , respectively. We assume thatthe quadrotors are always at the same altitude. In the caseof perturbations, the controller has to drive them back to thesame high. Both quadrotors have same mass m and inertiatensor J . The cable has a length (cid:96) and a mass m C . Eachquadrotor has a frame, denoted by {A} and {B} ; the origin ofeach frame is located at its center of mass, the x -axis points tothe front of the robot, and the z -axis points in the direction ofthe rotors. The orientation of quadrotors A and B with respectto the world frame are described by the rotation matrices W R A and W R B in SO(3), respectively. The desired x -axis of eachquadrotor is parallel to the x -axis of the catenary frame. Fig. 3: Coordinate frames of a catenary robot.The catenary frame {C} , has its origin at the lowest pointof the catenary curve. Assuming that the catenary is alwayson a vertical plane, called the catenary plane. The x -axis of {C} is normal to the catenary plane, and y -axis of {C} isdefined by the unitarian tangent vector of the catenary curveat its lowest point. The orientation of the catenary frame withrespect to {W} is described by the rotation matrix W R C .In this paper, we assume that the robot moves slow enoughsuch that the cable does not swing. Therefore, the catenarycan only rotate with respect to its z -axis. We denote the yawangle of the catenary by ψ C , and therefore its rotation matrixis W R C = Rot Z ( ψ C ) , where Rot Z is a function that returnsthe rotation matrix of an angle in the z -axis. The quadrotorscan keep a static orientation for the yaw angle but rotationsaround the center of the reference point can create torsion onthe cable. Therefore, we want the yaw angle of the quadrotorsto be the same as the yaw angle of the catenary.The cable hangs from its two end points forming a catenarycurve [10], starting at point x A and ending at point x B . Theequation of the curve in the catenary frame is α ( r ) = ⎡⎢⎢⎢⎢⎢⎣ ra ( cosh ra − ) ⎤⎥⎥⎥⎥⎥⎦ , (1)where r ∈ [− s, s ] is the parameter of the curve; the variable s is equal to half of the span of the catenary (see Fig. 3); andthe value a ∈ R ≥ can be obtained by using the equation ofthe length of the catenary as (cid:96) = a sinh ( sa ) . (2)Here we know the length of the cable (cid:96) and the distance s that comes from the location of the quadrotors. Since, thisequation is transcendental, meaning that it is not possible tosolve a analytically, we will have to use a numerical solution.In the catenary frame, the lowest point of the curve is α ( ) = in {C} , and the location of the robots are α (− s ) = C x A , and α ( s ) = C x B . Each quadrotor has four propellers that generate a totalthrust f i and a torque vector τ i . The translational and ro-tational dynamics of each quadrotor i ∈ { A, B } are describedby the Newton-Euler equations, m ¨x i = − mg e + W R i f i e + W R C t i , (3) J ˙ ω i = − ω i × J ω i + τ i , (4) . D’ANTONIO et al. : THE CATENARY ROBOT: DESIGN AND CONTROL OF A CABLE PROPELLED BY TWO QUADROTORS 3 where g is the gravity constant, e = [ , , ] ⊺ , ω i is theangular velocity, and the vector t i is the tension force thatthe cable generates on the i th quadrotor.Using a geometric controller [16], we can drive the robotsto a desired attitude W R di , and thrust f di . In this way, thethrust in the z -axis of the quadrotor can be used to pull thecable in any direction. The force vector in {W} generated byeach quadrotor is f i = f i W R i e , the thrust vector is obtained by multiplying the rotation matrixof its attitude, i.e. W R A , and the thrust generated by themotors, i.e. f A , in the z -axis of the body frame.The configuration space of the catenary curve is associatedwith its position, orientation, and span. The sag depends onthe length of the cable and the span, but the length is fixed,making the sag a function of the span. Each quadrotor offersfour inputs, and the catenary curve is described using fivevariables, satisfying the condition of the system to be fullyactuated. The objective of this work is to control the catenaryin its configuration space.III. T RAJECTORY T RACKING AND C ONTROL
Given a desired trajectory for the catenary robot, specifiedby its reference point, x C ( t ) , orientation ψ ( t ) , and span s ( t ) ,we design a controller to track the trajectory using the controlinputs of the quadrotors f i and W R i , i ∈ { A, B } . An overviewof the control architecture is illustrated in Fig. 4. The firstblock receives the trajectory of the catenary robot and convertsit into trajectories for the position of each quadrotor. Thesecond block tracks the trajectory based on the attitude ofthe quadrotors. Finally, an attitude controller in SO(3) in thethird block. A. Trajectory of the quadrotors
Our first step is to convert the trajectory of the catenary,defined by x C ( t ) , ψ ( t ) , and s ( t ) , into the trajectory of thequadrotors, x dA ( t ) , and x dB ( t ) , including its first, and secondderivatives. The desired location, velocity and acceleration ofeach quadrotor i ∈ { A, B } in the world frame with respect tothe catenary frame are x di = x dC + W R C ( ψ ) C x i , (5) ˙x di = ˙x dC + W ˙R C ( ψ ) C x i + W R C ( ψ ) C ˙x i , (6) ¨x di = ¨x dC + W ¨R C ( ψ ) C x i + W R C ( ψ ) C ¨x i + W ˙R C ( ψ ) C ˙x i . (7)The point C x A and its derivatives can be computed using thespan of the catenary that comes from the function s ( t ) . Byevaluating the catenary function α at − s , i.e., α (− s ) = C x A , we can obtain the point C x A and compute its derivatives C x A = ⎡⎢⎢⎢⎢⎢⎣ , − s,a ( cosh ( sa ) − ) ⎤⎥⎥⎥⎥⎥⎦ , (8) C ˙x A = ⎡⎢⎢⎢⎢⎢⎣ , − ˙ s, ˙ a ( cosh ( sa ) − ) + ( ˙ s − s ˙ aa ) sinh sa ⎤⎥⎥⎥⎥⎥⎦ , (9) C ¨x A = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ , − ¨ s, a ( ˙ sa − s ˙ aa ) sinh ( sa )+ a ( ˙ sa − s ˙ aa ) cosh ( sa )+ a ( s ˙ a a − a ˙ sa − s ¨ aa + ¨ sa ) sinh ( sa )+ ¨ a ( cosh ( sa ) − ) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (10)In a similar way, the position and derivatives of the point C x B are obtained by evaluating (1) in s , i.e., α ( s ) = C x B . In theseequations, we still need to compute the variable of the catenary a and its derivatives. Derivating (2), = ( ˙ s − s ˙ aa ) cosh ( sa ) + ˙ a sinh ( sa ) , (11) = a ( ˙ sa − s ˙ aa ) sinh ( sa ) + a ( ˙ sa − s ˙ aa ) cosh ( sa ) (12) + ( s ˙ a a − a ˙ sa − s ¨ aa + ¨ s ) cosh ( sa ) + ¨ a sinh ( sa ) . The equations (2), (11) and (12) are transcendental and wecannot analytically solve our variables of interest a and itsderivatives. However, we need those values to control the cate-nary, so in each control iteration, we compute them by solvingthese equations numerically using the bisection method [17].In order to reduce the torsion of the cable, we want to makethe quadrotors to always point in the direction of the normalvector of the catenary plane. Therefore, ψ i = ψ d . In this way,we convert the inputs of the catenary in trajectories for thequadrotors.
B. Tracking controller
Using the desired position, velocity and acceleration of eachquadrotor, we can compute the errors in the trajectory as e p = x di − x i , and e v = ˙x di − ˙x i . Using a geometric controller [16], each quadrotor can bedriven to generate a thrust force vector f i . Our desired forcevector drives the tracking errors to zero, and compensates forthe gravity force and the tension generated by the cable f di = K p e p + K v e v + m ¨x di − mg e + W R C t i , u (13)where K p and K v are positive proportional matrices that driveposition and velocity errors to zero as time increases. t i isthe tension in the catenary frame in A , B and its directionis tangent to the catenary curve. A well-known result fromthe catenary curve [10] is that tension at any point is t =[ , ± w a, w z ] ⊺ , where w = m / (cid:96) is weight of rope per unitlength. Then, evaluating in the two ends, we obtain t A = [ , − w a, w C z A ] ⊺ , t B = [ , w a, w C z B ] ⊺ . It is important to highlight that the tension increases with themass of the cable; requiring to increase the tilting angle of thequadrotors to compensate the tension.In order to drive the robot to generate the force vector, wecan control its attitude in SO(3). The desired rotation matrixis defined by the unitarian vectors z di = f di ∥ f di ∥ , y di = z di × x i ∥ z di × x i ∥ , and x di = y di × z di ∥ y di × z di ∥ , IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021
Fig. 4: Control architecture for the catenary robot.where the vector x i comes from the desired orientation ofthe quadrotor, which is the same as the catenary, i.e. x i = Rot z ( ψ ) e . Then the desired rotation matrix is W R di = [ x di , y di , z di ] . The thrust is then computed as f i = f di ⋅ W R di e . The proof of the stability properties of this controller to gen-erate the desired force f di can be found in [16]. By generatingthe desired force in (13), we can track the desired positionand compensate for the gravity force and the cable’s tension.Our desired points x di , i ∈ { A, B } , and their derivativescome from the numerical solution. Although there is an errorassociated with the approximation, the feedback controller cancompensate for it during running time. The important partis that the error does not accumulate over time since thenumerical approximation has to be performed in each controlloop. IV. E XPERIMENTS
To test the catenary robot, we designed four differentexperiments . First, we track a trajectory where the catenarypoint x C is driven to x dC , and then it remains static whilethe span and yaw orientation oscillate. We demonstrate thateven when the robots are moving around, the lowest point willremain at the same location once that this point is reached.This experiment is performed in both simulation and actualrobots. Additionally, we change the cables to demonstrate theeffect of increasing the weight of the cable. When the weight isbig enough it affects the motion of the catenary robot and it isnecessary to consider it in the model so the thrust compensatesthe extra force. Second, we define a trajectory of the catenaryto move through obstacles that require accurate motion. Eventhough we do not have direct measurements of the curve, weare able to track a trajectory, based on the catenary equationsfrom Section III. In our third experiment, we interact withand attach to an umbrella handle (an object with a naturallyhook-shaped attach point) by controlling the lowest pointo the catenary curve. Finally, the last experiment considersthe transportation of a hook-shaped object. We generate atrajectory that allows the catenary robot to self-attach to theobject and lift it to take it to another place in the environment. A. Simulation: Varying span and yaw
In this simulation, we implemented the dynamic equationsfrom (3) and (4), and our controller in Matlab. We tested witha trajectory that describes a flower-like shape by var ying thespan and yaw of the trajectory while maintaining the point x C The source code for simulations and actual robots is available at https://github.com/swarmslab/Catenary_Robot
Fig. 5: 3D trajectories of the quadrotors and the catenarytracking a span s ( t ) and yaw ψ ( t ) trajectories while x dC isstatic.fixed. We set the length of the cable as (cid:96) = m , its mass as m C = . gr , in the quadrotor mass m = g. The trajectoryis defined by x dC = [ , , . ] , ψ ( t ) = t / , and s ( t ) = . + .
15 cos ( t ) . The result of the simulation is shown in Fig. 5. The desiredtrajectories are denoted by the red dashed lines, the lowestpoint is in green solid line and the path of the quadrotors { A, B } are in blue and black solid lines, respectively. It canbe seen the catenary robot is able to lead the lowest pointfrom x C to x dC in finite time and remain there for the rest ofthe simulation. In the same way, the desired trajectory for thespan is perfectly tracked by the controller. Our next step istracking the same trajectory with an actual catenary robot. B. Catenary robot design
In our experimental testbed, we used the Crazyflie-ROSframework [18] to command the robots. For the localizationof the quadrotors, we use the motion capture system (Opti-track) operating at 120 Hz. The quadrotors internally measuretheir angular velocities using their IMU sensors. The originalgeometric controller in the firmware of the Crazyflie robotextended to include the tension of the cable. Since the originalCrazyflie has a low payload ( < g), we designed a quadrotorwith brushless motors based on the crazybolt controller; itsweight is . g and its payload is g. C. Experiments with actual robots
The experiments for the catenary robot are the following. . D’ANTONIO et al. : THE CATENARY ROBOT: DESIGN AND CONTROL OF A CABLE PROPELLED BY TWO QUADROTORS 5 − . . . x ( m ) ref C x A x B x C − . . . y ( m ) . . z ( m ) ψ ( R a d ) t ( s ) . . s ( m ) x ( m ) − . − . . . . y ( m ) − . − . . . . z ( m ) . . . . . . x A x A desiredx B x B desiredx C x C desired Fig. 6: Results of Experiment 1: the position of quadrotors x A , x B and the catenary point x C with respect to each axis.Additionally the evolution of yaw-angle ψ and span s withtheir respecting desired trajectories that are in dashed red lines. Experiment 1.1
Varying span and yaw:
Performing thesame experiment of the simulation in Fig. 5 but now using theactual robot. We present the results of the trajectory trackingin Fig. 6 for in x -, y -, and z -axis, as well as the yaw angle andspan. Here is possible to see how the real implementation hasa larger error in comparison with the simulation that assumesperfect conditions. It can be seen that there is an error whilemaintaining the lowest point of the catenary in a static location,which is close to zero in the x - and y -axis, but the z -axishas a small oscillating error. The average error in position is µ x = e − , µ y = e − , and µ z = e − , and its standarddeviations σ x = e − , σ y = e − , σ z = . . The errors foryaw-angle ψ ( t ) and span s ( t ) are µ ψ = . e − , µ s = . e − ,and their standard deviation σ ψ = e − , and σ s = . . Asa result, we can say that the z -coordinate and the span s arethe most sensitive variables. . . . . . . . . − x ( m ) ref C x A x B x C . . . . . . . . − . . . y ( m ) . . . . . . . . . . . z ( m ) . . . . . . . . − . . . ψ ( r a d ) . . . . . . . . t ( s ) . . . . s ( m ) Fig. 7: Results of Experiment 2: position of quadrotors { A, B } , x A , x B , and the lowest point of the catenary x C areshown in light blue, olive, and blue respectively. All desiredtrajectories are in dashed red lines. We show the signal in x -axis, y -axis, z -axis, yaw-angle ψ , and s span of the catenary. Experiment 1.2
Flying with different types of cables:
Weperform the previous experiment with different types of cables.A rope, a steel cable, and a plastic chain with a weight of6.23, 14.17, and 56.39 g respectively. For the first and secondcable, their weight is so low that the tension that it generatesis neglectable. For the third cable, it is necessary to includetension term t i in the controller for stabilizing the flight. As aresult, the quadrotors will fly tilting outwards to compensatethe tension from the cable. Experiment 2.
Trajectory tracking:
In this experiment, thegoal is to maintain a constant altitude of the lowest catenarypoint while moving in the x -axis. The trajectory is x dc ( t ) =[ t, , . ] , ψ d = , and s ( t ) = ⎧⎪⎪⎨⎪⎪⎩ . ≤ t < π . + . ( t ) π ≤ t < π . π ≤ t < ∞ . (14)As shown in Fig. 7, the lowest point of the catenary starts tofollow the trajectory having an error in the y -axis and x -axisclose to zero. When the point starts to change as expectedby x dc ( t ) , the quadrotors change the position making the spangreater to maintain the lowest point at the same altitude. Theaverage error in position is µ x = . , µ y = e − , and µ z = . , and its standard deviations σ x = e − , σ y = e − , σ z = . . Similar to the previous experiment z and s are themost sensitive variables and they are mainly affected by thevariation in span after 12.5 seconds. Experiment 3.
Pulling an umbrella:
We want to use thecatenary robot to interact with objects. By driving the robot inthe proper manner, it can be used to pull objects with hook-like shapes. In this case, we design a trajectory that allows
IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021 − x ( m ) ref C x A x B x C y ( m ) z ( m ) − . . . ψ ( r a d ) t ( s ) . . . . s ( m ) Fig. 8: Results of Experiment 3: Attaching to the handleof an umbrella. the position of quadrotors x A , x B and thecatenary point x C with respect to each axis. Additionally theevolution of yaw-angle ψ and span s with their respectingdesired trajectories that are in dashed red lines.the catenary robot to pull the handle of an umbrella. Theprocedure has three stages. First, the robot takes off to beplaced in an initial location above ground. Second, the robotmoves from the initial location to the front of the umbrella.Third, the lowest point of the catenary is right bellow thehook of the umbrella handle and then the catenary robot pullsthe umbrella. We generate the minimum snap trajectory [19]for following waypoints ([[− . , − . , . ] , [ . , − . , . ] , [ . , . , . ] , [ . , . , . ]]) and fixing the desired spanat s = . . It is noticeable that the catenary robot is able tofollow the generated trajectories by following the waypoints.This is illustrated in Fig. 8 through the evolution in time of theposition of quadrotors x A , x B , catenary point x C , span s , andyaw-angle ψ . The average error in position is µ x = e − , µ y = e − , and µ z = . , and its standard deviations σ x = . , σ y = . , σ z = . . The errors for yaw-angle ψ ( t ) and span s ( t ) µ ψ = e − , µ s = e − , and their standarddeviation σ ψ = . , and σ s = . e − . Notice that aroundtime s the catenary robot starts to slightly deviate from thedesired trajectory, this is given by the extra tension generatedby the umbrella which is not considered in the model. Experiment 4.
Transporting a hook-shaped object:
Weinteract with an object that has a hook-shape. The catenaryrobot follows a trajectory that allows it to pull the object,lift it, and place it in a different location. Here is important tonote that it is necessary to include the mass of the object in themodel to generate the proper control input that accomplishesthe transportation task. V. C
ONCLUSIONS AND F UTURE W ORK
In this work, we proposed a robotic system, called the cate-nary robot, composed of a cable propelled by two quadrotors.We designed a controller to track trajectories for the five de-grees of freedom of the catenary: position in three dimensions,yaw orientation, and span. Each degree of freedom can becontrolled independently based on the forces generated by thequadrotors. By estimating and controlling the catenary, weshowed that it is possible to interact with objects that havehook-like shapes, e.g., an umbrella that has a hook shape inits handle. We have demonstrated the successful functionalityof our system in simulation and actual robots. In a future work,we want to use the catenary robot to manipulate object withouthook-shapes. R
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