Towards a heterogeneous cable-connected team of UAVsfor aerial manipulation

Vishal Abhishek, Vaibhav Srivastava, and Ranjan Mukherjee

Abstract— We study control of a heterogeneous team oftwo UAVs connected through a flexible cable such that thesmaller UAV provides more accessibility in terms of reachingconstrained locations, the larger UAV provides the base whereheavier equipment can be installed, and the flexible cable actsas the conduit to carry electrical or communication cables,or for transporting fluids between the two UAVs. Our controldesign leverages the catenary model to compute the quasi-staticcable tension and uses it to feedback linearize the system. Anextended high-gain observer is used to estimate system statesand to refine the estimate of cable tension. The estimates areused in the output feedback controller and the stability of theoverall system is established. We illustrate the effectiveness ofour methods using numerical simulations.

I. INTRODUCTION

Unmanned Aerial Vehicles (UAVs) provide a very usefulsolution to perform tasks at remote locations such as inspec-tion, surveillance, tool operation, or package delivery. Withthe advances in usage and technology, multiple UAVs arerequired to work cooperatively to perform a task.

In this paper, we consider a heterogeneous team of twoUAVs connected together with a flexible cable such thatthe smaller (tool) UAV provides more accessibility in termsto reaching constrained locations, the larger (base) UAVprovides the base where heavier equipment can be installed,and the flexible cable acts as the conduit to carry electricalor communication cables, or for transporting fluids betweenthe two UAVs. Applications of such a system include highbandwidth sensing in which the smaller UAV carries sensorsand tracks a trajectory to achieve desired spatio-temporalsensing footprint, and the collected high resolution datais sent to the data processing station on the larger UAVthrough the high bandwidth communication cable. Anotherexample concerns cleaning windows of high-rise buildings orremotely located solar panels, wherein the larger UAV carriescleaning liquid reservoir, the cable transfers the liquid to thesmaller UAV, which cleans the surface.

Several UAV-based manipulation systems have been de-signed, for example, for aerial pick and place [1], avian in-spired grasping and perching [2, 3], mobile manipulation [4–6], assembly [7], valve turning [8]; see [9] for an extensivereview. In [10], the authors consider a rigid rod-like objectsuspended at one end from a UAV through a spherical joint

This work was supported in part by ARO grant W911NF-18-1-0325.V. Abhishek and R. Mukherjee are with the Department of Mechanical

Engineering, Michigan State University, East Lansing, MI 48824-1226, USAabhishe3,mukherji at egr.msu.edu

V. Srivastava is with the Electrical and Computer Engineering, MichiganState University, East Lansing, MI 48824-1226, USA vaibhav ategr.msu.edu

and study scenarios with both torque-actuated and actuation-free joint. In [11], the authors design a spherically connectedmultiquadrotor (SmQ) platform in which multiple quadrotorsare used as rotating thrust generators for the platform. In[12], the authors design a dual arm connected at the tipof a flexible link attached to the UAV. In [13] and [14],a manipulator arm is connected at the end of a rod and aplatform, respectively, which in turn are attached to the UAVthrough a passive spherical joint and act as a pendulum. Ina companion paper [15], we design an extended high gainobserver based output feedback control for multi-body multi-rotor based manipulation, where a birotor connected to aUAV through a rigid rod is considered.

Using cables to connect a load from a UAV has beenconsidered extensively. Transporting a load using suspendedcables has been considered both in single UAV [16–19] aswell as multi-UAV scenarios [20–24]. Control of quadrotorswith cable suspended payload using flexible cables has beenstudied and nonlinear geometric control has been designedin [25–27]. Differential flatness and geometric control ofquadrotors with a payload suspended through flexible cableshas been considered in [28]. Control of a UAV tethered toa taut cable fixed at one end is studied and an observerbased control is discussed in [29, 30]. Here, the cable isassumed to be always taut and any flexibility is ignored. In[31] a geometric control of a tethered quadrotor is consideredfor tracking a desired trajectory while a linearization basedcontroller stabilizes the flexible tether. In [32] multiplequadrotors carrying a flexible hose are considered. The cableis modeled as a series of lumped mass system and differentialflatness of this system is studied. A linear time-varying LQRis designed to track desired trajectories assuming variationbased linearized system. In contrast, our feedback linearizingcontrol design relies on directly estimating cable tensionusing catenary model and subsequently refining it using theextended high gain observer to incorporate the influence ofcable dynamics.

In this paper, we present the control of a cable connecteddual-UAV system. The cable is assumed to be constant inlength while flexible in its bending. We design an extendedhigh gain observer (EHGO) based controller to track adesired trajectory of the tool and base UAVs. We use quasi-static tension in a cable under gravity obtained using catenarymodel as an estimate of cable tension. The estimate isrefined using extended high gain observer to incorporate theinfluence of the cable dynamics. The state and cable tensionestimates are used to design an output feedback controller,which is systematically analyzed. The proposed controlleractively estimates and compensates cable tension leading to

decoupled dynamics for the base and the tool UAV . Thus,the controller can be independently applied to both UAVs.For simplicity of exposition, in the following we restrict thepresentation to the case where the base UAV is hoveringwhile the tool UAV tracks a desired trajectory.

The remainder of the paper is organized in the followingway. Section II presents the design and modeling of thedual-UAV system. The EHGO-based control design andassociated stability analysis are presented in Sections IIIand IV, respectively. Numerical simulations illustrating theeffectiveness of the proposed controller are presented inSection V. Finally, Section VI concludes the paper.

II. SYSTEM MODELING

We consider a team of two heterogeneous UAVs connectedthrough a flexible cable such that one of the UAVs (referredas the tool UAV) tracks a desired trajectory while the otherUAV (referred as the base UAV) provides the required thrust.The resulting system referred as cable connected dual-UAVsystem is shown in Fig. 1. We focus on the case where thebase UAV is hovering. Our controller design decouples thetwo UAVs by canceling the estimated tension due to thecable at both UAVs. Therefore, we focus on control of thetool UAV only. A similar controller can be designed for thebase UAV as well. For further analysis, we restrict to thecase of planar motion in a vertical plane. We use catenaryequations which yield the tension in the cable assuming staticequilibrium under gravity.

𝑓

𝜏 𝛽

𝑌

𝑋

(𝑥, 𝑦)

𝑂

cable

𝑇2𝑀𝑔

base UAV

tool UAV

(𝑥0, 𝑦0)

Fig. 1. Cable connected dual-UAV system

A. Catenary Model

To compute the tension in the cable we use the catenarymodel [33, chapter 10] which provides the shape as wellas the tension in the cable fixed at the two ends assumingstatic equilibrium under gravity. Let λ be the mass per unitlength of the cable, g acceleration due to gravity, s be thedistance along the length of the cable, varying from s = 0at one end point P0 = (x0, y0) and s = L at the other endpoint P = (x, y). The tension in the cable at any point isgiven in terms of three parameters of the curve: a, xc and yc.These parameters can be determined by using the coordinatesof the two end points. Below we give the expressions for

these parameters. The parameter a is obtained by solvingthe following transcendental equation√

L2 − (y − y0)2 = 2a sinh|x− x0|

2a.

The expressions for xc and yc are given by

xc = x0 + sign(x− x0) a log(D ∓√D2 − 4E

2

),

yc = y0 − a coshx− xca

,

where, in the above expression for xc, the sign in theargument of log(·) is selected negative if y − y0 ≥ 0 andpositive if y − y0 < 0. The expressions for D and E usedabove are given by

D = 2Lexp(|x− x0|/a)

a(exp(|x− x0|/a)− 1); E = exp(|x− x0|/a).

The tension at the end P for the case when x − x0 6= 0 isgiven by

T cx = sign(x− x0) λga, T cy = λg√

(y − yc)2 − a2,

while for the case when x− x0 = 0, we have

T cx = 0, T cy = λgL+ (y − y0)

2.

B. Dynamic Model of the tool UAV

We assume that the cable is connected to the two UAVsat their respective centers of mass. The system alongwiththe forces at the tool UAV can be seen in Fig. 1. We adoptthe following notation

M mass of the tool UAVI moment of inertia of the tool UAV about Z

axis passing through its center of massm mass of the cableL length of the cable(x, y) position coordinates of the tool UAV

in the inertial frameβ angular position of the tool UAVf rotor thrust on the tool UAVT2 cable tension at the tool UAV with components

Tx and Ty in x and y direction, respectivelyτ rotor moment on the tool UAV about its centerg acceleration due to gravity

The equations of motion of the tool UAV are

Mx = −f sinβ − Tx,My = f cosβ − Ty −Mg,

Iβ = τ.

(2)

Let σx := 1M (Tx−T cx), σy := 1

M (Ty−T cy ) be the differencebetween the tension computed using quasi-static catenaryequations and the actual cable tension. Define q1 = [x, y]>,q2 = q1, σq := [σx, σy]> and q = [q>1 , q

>2 ]>. Let σq :=

ϕq(t, q), where ϕq(t, q) is an unknown function which iscontinuous and bounded on any compact subset of R≥0×R4.

III. CONTROL DESIGN

We design an EHGO based output feedback control totrack a desired trajectory for the tool UAV position coordi-nates. Let the desired trajectory be qd1(t) = [xd(t), yd(t)]>

and define tracking error e1 := qd1−q1 = [ex, ey]>, e2 := e1and e := [(e1)>, (e2)>]>. We assume feedback available inposition and orientation of the tool UAV, namely, outputs q1and β. We begin with state feedback control design.

A. State Feedback Control

The position error dynamics can be written using (2) as

e1 = qd1 −

[− fM sinβ − T c

x

M − σxfM cosβ − T c

y

M − g − σy

].

Here we have written Tx as T cx+Mσx and Ty as T cy +Mσy .Now, assuming the state as well as disturbance signals σxand σy are known, it can be seen that designing a controllaw such that[−f sinβf cosβ

]= M(qd1+K1e1+K2e1)+

[T cx +Mσx

T cy +Mg +Mσy

],

linearizes the position tracking error dynamics. Thus,

f =

∥∥∥∥M(qd1 +K1e1 +K2e1) +

[T cx +Mσx

T cy +Mg +Mσy

] ∥∥∥∥,(3a)

βc = atan2

(−M(xd +K1ex +K2ex)− T cx −Mσx,

M(yd +K1ey +K2ey) + T cy +Mg +Mσy

),

(3b)

where atan2(·) is the 2-argument arc tangent function. Thecommanded value βc will be used as the desired trajectoryfor the orientation dynamics. The orientation dynamics of theUAV are written through the following change of variables

β1 = βc − β, β2 =˙β1 = βc − β, β = [β1, β2]>,

leading to

˙β1 = β2, (4a)˙β2 =

−τI

+ βc. (4b)

We set the control input f as given by (3a) and τ as

τ = I(K3β1 +K4β2 + βc). (5)

The closed-loop system dynamics then becomes

e = Aee+

[02

eβ(t, e, β1)

], (6a)

˙β = Aββ, (6b)

where

Ae =

[02×2 I2×2

−K1I2×2 −K2I2×2

], Aβ =

[0 1−K3 −K4

],

eβ(t, e, β1) =f

M

[sin (βc − β1)− sinβc

cosβc − cos (βc − β1)

].

Note that eβ(t, e, 0) = 0 for all t > 0 . In the state feedbackcontrol law as designed above we need states e1, β andfeedforward term qd1 which are known through measurementor by design along with states e2 and β2, disturbance signalsσx and σy , and feedforward term βc which we assumeunknown and estimate using extended high gain observerin the next section.

B. Extended High-Gain Observer Design

We design an extended high-gain observer (EHGO) toestimate unmeasured states as well as uncertainties in theform of modeling error and external disturbances. Theseestimates will be used to cancel out the uncertainties inthe control design, resulting in improved performance ofthe feedback linearizing controllers. We begin by notingfrom the previous subsection that the control input τ fora state feedback based linearizing controller requires thecomputation of βc. In order to avoid this, we lump thisquantity with any disturbance in the right hand side of (4b)to arrive at

˙β1 = β2, (7a)˙β2 =

−τI

+ σβ . (7b)

The new state feedback linearizing control law for rotationaldynamics will be τ = I(K3β1 + K4β2 + σβ). We treat σβas a state and assume σβ = ϕβ(t, β), where ϕβ(t, β) is anunknown function that is continuous and bounded on anycompact subset of R≥0 × (−π2 ,

π2 ).

Define χ = [(e1)>, (e2)>, (σq)>, β1, β2, σβ ]> ∈ R9,

A =

[02×2 I2×2 02×2

02×2 02×2 I2×2

02×2 02×2 02×2

]⊕[0 1 00 0 10 0 0

], H =

[ρ1/εI2ρ2/ε

2I2ρ3/ε

3I2

]⊕[ρ1/ε

ρ2/ε2

ρ3/ε3

],

and C = [ I2×2 02×2 02×2 ]⊕ [ 1 0 0 ] ,

where ⊕ denotes matrix direct sum, ε > 0 is some suffi-ciently small constant and ρi’s are selected such that

s3 + ρ1s2 + ρ2s+ ρ3 = 0,

is Hurwitz. Then, the combined system dynamics can bewritten with states χ as

χ = Aχ+ g(t, β, f, τ, T cx , Tcy ) +

04×1ϕq

02×1ϕβ

, (8a)

y = Cχ, (8b)

where

g(t, β, f, τ, T cx , Tcy ) =

02xd(t)− 1

M (−f sinβ − T cx)yd(t)− 1

M (f cosβ − T cy −Mg)020− τI0

.

The EHGO for the above combined system dynamics cannow be written as

˙χ = Aχ+ g(t, β, f, τ, T cx , Tcy ) +Hye, (9a)

ye = C(χ− χ). (9b)

where χ = [(e1)>, (e2)>, σq,ˆβ1,

ˆβ2, σβ ]T ∈ R9 is the

estimate of χ. We can now define an output-feedback con-troller to ensure tracking of the trajectory qd(t). The output-feedback controller is designed using the estimates from theEHGO as

f =

∥∥∥∥M(qd1 +K1e+K2e2) +

[T cx +Mσx

T cy +Mg +Mσy

] ∥∥∥∥,(10a)

τ = I(K3β1 +K4ˆβ2 + σβ), (10b)

with commanded angle βc now given by

βc = atan2

(−M(xd +K1ex +K2

ˆex)− T cx −Mσx,

M(yd +K1ey +K2ˆey) + T cy +Mg +Mσy

),

where [ˆex, ˆey]> = e2 and [σx, σy]> = σq .

IV. STABILITY ANALYSIS

The stability of the state feedback control, observer esti-mates, and output feedback controller will now be proven.We begin by restricting the domain of operation by establish-ing a compact positively invariant set. We then prove stabilityof the closed-loop system under state feedback, convergenceof the observer estimates, and finally prove stability of theclosed-loop system under output feedback.

A. Restricting Domain of Operation

Note that when the cable is taut and not in a verticalalignment, i.e. x 6= x0, the catenary model no longer holdsand there is a singularity in the motion of the UAVs. Also,at such configurations the cable tension as per the catenarymodel becomes unbounded. In order to prevent the cablefrom being taut, the distance between the two UAVs mustremain striclty smaller than the cable length L. To restrict thedomain of operation such that (x− x0)2 + (y − y0)2 ≤ L∗,where L∗ < L, we make the following assumption about thedesired trajectory of the tool UAV

(xd − x0)2 + (yd − y0)2 < L∗ − δe. (11)

If the position tracking error ‖e1‖ ≤ δe, then the aboveassumption ensures that the tool UAV remains in the desireddomain of operation. Further to restrict the rotational angleβ such that −π/2 < β < π/2, we assume the followingrestriction on the commanded angle |βc| < π/2 − δ,which means if the rotational tracking error ‖β‖ ≤ δ, thenthe requirement on β is satisfied. For the state feedbackcontrolled system (6), we show below that there exist aconsistent choice of δe and δ, and a positively invariantcompact set Ω = Ωe × Ωβ , such that if the initial trackingerror lie in Ω, then ‖e1(t)‖ ≤ δe and ‖β(t)‖ ≤ δ.

Consider the following Lyapunov function for the rota-tional error dynamics

Vβ = β>Pββ, where PβAβ +A>

βPβ = −I2. (12)

Since Aβ is Hurwitz, a symmetric positive definite Pβ asdefined above exists. Further,

λmin(Pβ)‖β‖2 ≤ Vβ ≤ λmax(Pβ)‖β‖2, Vβ = −‖β‖2.

Define Ωβ = Vβ ≤ cβ. It can be seen from above that Ωβis a positively invariant set. Now, taking cβ = λmin(Pβ)δ2,implies ‖β‖ ≤ δ and hence ‖β‖ ≤ δ as required.

Similar to the rotational dynamics, consider the followingLyapunov function for the position dynamics

Ve = e>Pee, where PeAe +A>e Pe = −I4. (13)

Taking derivative,

Ve = −‖e‖2 + 2[0>2 , e>β ]Pee

≤ −‖e‖2 + 2λmax(Pe)‖eβ‖‖e‖.(14)

Since eβ(t, β), written as function of time t and β, iscontinous and uniformly bounded in t (as all signals remainbounded under bounded desired trajectory assumption) andits partial derivative with respect to β is continous, eβ(t, β)is Lipschitz in β on Ωβ . Hence, in Ωβ

‖eβ(β)− eβ(0)‖ ≤ L‖β‖ ≤ Lδ, (15)

which yields

Ve ≤ −‖e‖2 + 2λmax(Pe)Lδ‖e‖. (16)

Therefore, for ‖e‖ > 2λmax(Pe)Lδ, Ve < 0. Using this withthe fact that λmin(Pe)‖e‖2 ≤ Ve ≤ λmax(Pe)‖e‖2 makesΩe = Ve ≤ ce positively invariant with a choice of cesatisfying

ce ≥ λmax(Pe)(2λmax(Pe)Lδ)2. (17)

Therefore, the domain of operation defined as Ω = Ωe×Ωβ,is compact and positively invariant under state feedback.

Note that choosing ce = λmin(Pe)δ2e implies ‖e1‖ ≤ δe.

Therefore, the choice of two error tolerances δe and δ mustsatisfy

λmin(Pe)δ2e ≥ λmax(Pe)(2λmax(Pe)Lδ)

2.

Hence, for a given choice of error tolerancessatisfying the above, if the initial tracking errorssatisfy ‖e(0)‖ ≤

√λmin(Pe)/λmax(Pe)δe and

‖β(0)‖ ≤√λmin(Pβ)/λmax(Pβ)δ, then (e(0), β(0)) ∈ Ω,

and hence ‖e1(t)‖ ≤ δe and ‖β(t)‖ ≤ δ.

B. Stability under State Feedback

Lemma 1 (Stability under State Feedback): For theclosed-loop system under state feedback (6), with initialtracking error (e(0), β(0)) in Ω, the system states(e(t), β(t)) remain in Ω for all t > 0 and exponentiallyconverge to the origin.

Proof: Taking a composite Lyapunov function

V = dVe + Vβ , d > 0 (18)

and following the generalized proof for cascaded systemstability in the Appendix of [34], it can be shown that theclosed-loop state feedback system given by (6) convergesexponentially to the origin for d chosen small enough.

C. Stability of Output Feedback

The system under output feedback is a singularly perturbedsystem which can be split into two time-scales. The systemdynamics and control reside in the slow time-scale while theobserver resides in the fast time-scale. The observer estimatescan be shown to converge to an O(ε) neighborhood of thetrue estimates.

Lemma 2 (Convergence of EHGO Estimates): Theobserver estimates χ converge exponentially to an O(ε)neighborhood of the true states χ, for any ε ∈ (0, ε∗) forsufficiently small ε∗ > 0.

The lemma can be proved following the standard high gainobserver proof given in [35].

Theorem 1 (Stability under Output Feedback): Theclosed-loop system under output feedback, with initialconditions in the interior of Ω, exponentially converges toan O(ε) neighborhood of the origin when ε is chosen smallenough.

Proof: The proof follows the standard arguments usedin the stability analysis of extended high gain observerbased output feedback control [35] and utilises the factsthat state-feedback controlled error system is exponentiallystable over a compact invariant domain of operation (Lemma1), the observer dynamics reside in a faster timescale, andthe observer estimates exponentially converge to an O(ε)neighborhood of the true states (Lemma 2).

V. SIMULATIONS

We perform simulations where we track a desired trajec-tory for the tool UAV position coordinates x, y. To computethe tension which is canceled in the control laws, we usethe quasi-static catenary equations presented in II-A. Theseequations estimate the tension at both the ends of thecable assuming these ends to be fixed and use the positioncoordinates of the two ends. In these simulations, we keepthe base UAV end fixed and treat it as the origin, however,a disturbance on the base UAV position is added to accountfor more realistic conditions in which oscillations aroundthe hovering position occur. The position of the base UAVis known to the tool UAV either through a central unit orthrough a sensor. Note that in the quasi-static calculations ofcable tension, only the relative position of the two ends ofthe cable matters. In order to simulate the dynamics of theflexible cable, we use the discretized lumped mass model of

[32] with one end of the cable attached to the base UAVand the other attached to the tool UAV. The cable length istaken as 10 m with mass per unit length λ = 0.5 kg/m. Wehave considered equidistant discretization into 10 segmentswhere the mass of the segment is considered lumped atthe discretization points. We select a desired trajectory inposition coordinates from time t = 0 to 10 s, with 0−5 s as9th order polynomial in time such that the tool end coversa displacement of 2 m in each of the coordinates and havederivatives upto 5th order as 0 at initial time t = 0 s andtime t = 5 s. We keep the desired coordinates fixed at 2 mfrom the start position for time 5 − 10 s. The trajectory ischosen so as to have the smoothness requirement of uptofifth order derivatives since the control law involves secondorder derivatives of βc which further involves fourth orderderivatives of the desired trajectory. The initial coordinatesof the tool UAV is set as x = 5 m, y = 0 m relative to thebase UAV. We also add a disturbance in the position of thebase UAV as a sinusoidal signal: 0.1 sin t in both x and ycoordinates. We begin with simulations of full state feedbackcontrol using control law as per (3a), (3b) and (5). Here, wehave assumed the difference in cable tension due to cabledynamics, i.e., σx and σy as zero, while for computation ofβc we have used numerical differentiation using backwarddifference algorithm. The plots for trajectory tracking for thecase of state feedback (SF) as well as EHGO based controlare shown in Fig. 2. As can be seen from the plots, thesystem is able to closely track the desired trajectory and thetracking error goes to zero. Figure 2(b) and Fig. 2(c) showthe plots of tracking error for SF based control as well asEHGO based control. The tracking in EHGO case can beseen to be better than the state feedback control.

VI. CONCLUSIONS

We consider control of a team of two UAVs connectedthrough a flexible cable. We design an extended high gainobserver based output feedback control where the outputavailable through sensing are the UAV position and angularcoordinates. We use the catenary model to compute the quasi-static cable tension and cancel it in our control law. Thedifference between the quasi-static cable tension and theactual tension is assumed to be a disturbance and is estimatedby the extended high gain observer. We numerically illustratethe effectiveness of the proposed approach in the presenceof external disturbances. In this work, we have consideredthe case where the system is restricted to move in a plane.The future work includes extension of these results to three-dimensional environments.

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