Improving hovering performance of tethered unmannedhelicopters with nonlinear control strategies

Luis A. Sandino1, Manuel Bejar2, Konstantin Kondak3 and Anibal Ollero1,4

Abstract— Hovering capabilities of unmanned heli-copters can be seriously affected by wind effects. Onepossible solution for improving hovering performanceunder such circumstances is the use of a tethered setupthat takes advantage of the tension exerted on the cablethat links the helicopter to the ground. This paperpresents a more elaborated strategy for helicopter controlin this augmented setup that extends previous work onthe subject by the authors. Particularly, a combination ofclassical PID control laws together with model inversionblocks constitutes the base of the new controller. Addi-tionally, feed-forward action for counteracting rotationalcouplings is also accounted for. The resulting nonlinearcontrol structure considers the complex and nonlinearnature of the tethered system in a better way. Severaldemonstrating simulations under artificially generatedwind influences are presented to endorse the validity ofthe new proposed controller.

Keywords—Unmanned aerial vehicles, helicopter, teth-ered systems, modeling, model-based control, PID, sta-bility augmentation

I. INTRODUCTION

Helicopters and other rotorcraft-based aircrafthave flight capabilities such as hover, VerticalTake-Off and Landing (VTOL) and pirouette,which cannot be achieved by conventional fixed-wing aircraft. These features are consequence oftheir functional controllability in lateral, longitu-dinal and vertical directions with almost constantyaw-attitude.

The aforementioned hovering capability allowsremotely piloted and autonomous helicopters to beextensively used nowadays for applications such as

1L.A. Sandino and A. Ollero are with GRVC, University ofSeville, Seville, SPAIN (B) lsandino at us.es

2M. Bejar is with GRVC, University Pablo de Olavide, Seville,SPAIN mbejdom at upo.es

3K. Kondak is with German Aerospace Agency (DLR), Oberp-faffenhofen, GERMANY konstantin.kondak at dlr.de

4A. Ollero is with Center for Advanced Aerospace Technologies(CATEC), Seville, SPAIN aollero at catec.aero

inspection, accurate measurement and other aerialrobotic applications. However, the performance ofthis valuable feature can be seriously affected byexternal disturbances such as wind effects. Thelatter could be even more significant when dealingwith small-size helicopters.

In order to address the performance degrada-tion on hovering capabilities under external dis-turbances, an augmented setup that consists ofthe unmanned helicopter itself, a tether connectingthe helicopter to the ground, and a ground devicethat is in charge of fixing certain tension valueson the tether during the operation of the systemwas proposed in [1]. The justification for thisaugmented setup is the stabilizing action of thetether tension. To authors knowledge, this use ofthe tether tension to increase stability of hoveringmaneuvers in helicopters is almost unexplored inthe literature. The only related precedents con-sidering the tethered configuration for rotorcraftprototypes different from helicopters are thoseof [2], where the linearized equations describingthe perturbed longitudinal motion of a tetheredrotorcraft are presented and [3], where a discussionof the control and stabilization problems involvedin the tethered configuration for a rotor platformprototype is presented, with special emphasis ontether dynamics. The authors of that work also pro-pose two approaches for the design of an automatichover controller for the tethered system. Finally, itis also worth mentioning the contributions of [4].Although it corresponds to the different applicationscenario of landing a helicopter, it is referencedhere since it also makes use of a tether as anadditional resource for helicopter control. In thoseworks, the benefit of the use of the tether isimproving the controllability of the system ratherthan the stability.

In the augmented configuration proposed in [1]

2013 International Conference on Unmanned Aircraft Systems (ICUAS)May 28-31, 2013, Grand Hyatt Atlanta, Atlanta, GA

978-1-4799-0817-2/13/$31.00 ©2013 IEEE 443

the tether effect on the system has been provedto be two-fold. On the one hand, it providesrobustness against external perturbations due tothe stabilizing properties of the tether tension inthe translational dynamics. On the other hand, themoment induced by the existing offset betweenthe tether tension application point and the centerof mass, produces undesired coupling betweenrotational and translational dynamics that makesmore difficult the controllability of the system. Thevalue of the moment caused by the tether could besimilar or even larger than the moments required tocontrol the rotation of the helicopter in free flightwithout any tethering device. The control strategyproposed in [1] for taking advantage of the stabi-lizing tethered setup considers the operation of thesystem under a constant tether tension, which is en-sured by the ground device. Simulations performeddemonstrated that even a basic linear approach forcontrolling the helicopter was enough to produce asignificant improvement of hovering performancein presence of wind disturbances.

This paper presents a more elaborated strategyfor controlling the helicopter, while adopting thesame control paradigm of ensuring a constanttether tension by means of the ground device.The aim is to better consider the nonlinear natureof the system. The proposed controller combinesclassical PID, PI and P control laws with modelinversion. In addition to this, feed-forward action isalso considered to counteract the moment inducedby the tether.

This paper begins with a detailed descriptionof the tethered configuration. To this end, SectionII presents the dynamical model correspondingto the augmented system: small-size helicopter,tether and ground device. Later in Section III,the improved strategy for helicopter control isexplained in detail. Finally, in Section IV, severaldemonstrating simulations under artificially gener-ated wind influence are presented to endorse thevalidity of the proposed approach. Section V isdevoted to conclusions and future work.

II. TETHERED CONFIGURATION MODEL

This section presents the model correspondingto the tethered configuration. The objective is notto present very elaborated equations for high-fidelity simulations but rather to derive the best

1

NO

FO

TRO

MRO

n3

n2

n1

HO

O

dO

-MR

,3

dO

-F,3

dO-TR,1

H

N

P

q9

F

f3f2

f1

mr3

mr2

mr1

c3

dO

-P,3

q7

q8

n'2

TC

MR

Fig. 1: Tethered configuration modeling scheme:reference frames, centers of mass, and dimensionsof interest for modeling purposes

model for control design purposes. The formercriterion demands a simple and manageable modelfor easing the analysis to derive the control laws,but capable to accurately reproduce the main be-havior of the real system at the same time.

According to the authors of [5], the dynamicsof a small-size helicopter with a stiff main rotor ismainly described by its mechanical model. Theirattempt to include an elaborated model for aero-dynamics of the main rotor in controller designdid not show significant improvements in controlexperiments performance. This could be consid-ered as evidence for the fact that the approach toanalyze the helicopter behavior by means of itsmechanical model is suitable for practical appli-cations. This paper follows the same assumption.Concerning the modeling approach, this paperfollows Kane’s methodology [6], like in otherprevious works of the authors [7].

A. Tethered helicopter

The system is depicted in Fig. 1. In line with [5]the mechanical characterization of the helicopteraccounts for two separated rigid bodies, fuselageF and stiff main rotor MR (modeled as a thinsolid disk with constant angular speed) whereas thetail rotor TR will only act as a force applicationpoint on the fuselage. This characterization arisesfrom the fact that for most commercially availablesmall-size helicopters the inertial effects of themain rotor (gyroscopic effects) become the main

444

TABLE I: Direction cosine matrix

f1 f2 f3n1 c5c6 −c5s6 s5n2 c4s6 + s4s5c6 c4c6 − s4s5s6 −s4c5n3 s4s6 − c4s5c6 s4c6 + c4s5s6 c4c5

where si = sin(qi) and ci = cos(qi)

component influencing the rotational dynamics ofthe whole mechanical system whilst the tail rotorinertial influence is negligible.

The position of center of mass HO in the inertialreference frame N is described by generalizedcoordinates qi (i = 1, 2, 3):

pNO→HO

= q1n1 + q2n2 + q3n3 (1)

Generalized coordinates qi (i = 4, 5, 6) are theEuler-angles (roll, pitch and yaw) correspondingto successive rotations (Body1-2-3 order [6]) thatdescribe the orientation of F in the inertial refer-ence frame N . Thus, dextral set of orthogonal unitvectors fi (i = 1, 2, 3) fixed in F and vectors ni

are geometrically related by the direction cosinematrix shown in Table I.

Generalized speeds ui (i = 1, · · · , 6) are definedas:

NvHO= u1n1 + u2n2 + u3n3 (2)

NωF = u4f1 + u5f2 + u6f3 (3)

which leads to the following kinematic differentialequations for translation:

q̇i = ui (i = 1, 2, 3) (4)

as well as for rotation:

q̇4 = − (s6u5 − c6u4) /c5

q̇5 = s6u4 + c6u5

q̇6 = u6 + s5 (s6u5 − c6u4) /c5 (5)

Concerning forces and moments applied to thehelicopter (see Fig. 2), main rotor generates aforce FMR = fMR,3f3 applied at point MRO andmoments MMR,i = tMR,ifi (i = 1, 2, 3) appliedto rigid body MR, whereas tail rotor generatesa force FTR = fTR,2f2 applied at point TRO

and a moment MF = tTR,2f2 applied to rigidbody F . Force of gravity Wj = −mjgn3 (j =F,MR) applied at centers of mass FO and MRO

is also considered, where g is the acceleration of

1

FOTRO

MRO

O

MMR,2

FMR

MF

FTR

MMR,3

MMR,1

WF

WMR

P

TC

Fig. 2: Forces and moments applied to the heli-copter

gravity. The application of Kane’s method leads tothe following dynamical differential equations fortranslation:

mH u̇1 = fMR,3s5 − fTR,2c5s6

mH u̇2 = fTR,2(c4c6 − s4s5s6)− fMR,3s4c5

mH u̇3 = fMR,3c4c5 −mHg+

+ fTR,2(s4c6 + c4s5s6) (6)

as well as for rotation:

K4p4u̇4 = tMR,1 + dO−HO,3fTR,2+

+ (K456u6 +K45)u5

K5p5u̇5 = tMR,2 + tTR,2+

+ (K546u6 +K54)u4

K6p6u̇6 = tMR,3 + dO−TRO,1fTR,2+

+K645u4u5 (7)

where mH is the total mass and parameters Kxxx

are functions of the inertial characterization of thehelicopter according to the expressions given inTable II.

Since only the mechanical model will be usedfor modeling analysis and subsequent derivation ofthe controller, as was discussed at the beginningof this section, the forces and moments exertedon the system will be considered as system inputsfor control design. More precisely, the subset ofthese forces and moments whose values can befixed independently of each other by the controlleris given by fMR,3, tMR,1, tMR,2 and fTR,2. In theimplementation of controllers for real helicoptersthese inputs will be transformed to servo positionsusing simple linear functions with only three un-known constants: the first constant describes therelation between the main rotor collective pitch and

445

the lifting force fMR,3, the second one the relationbetween the main rotor cyclic pitches and momentstMR,1 and tMR,2 and the third one the relationbetween the tail rotor collective pitch and theforce fTR,2. These constants can be identified inexperiments. In [5] this approach for characterizinglinearly the aerodynamics was verified successfullyin flight experiments.

When accounting for the tether effect into themodel, it is assumed for the sake of simplicity thatthe tether end at ground coincides with the originof the inertial frame N . On the other end, the tetheris connected to helicopter point P as depicted inFig. 1.

Since the elastic model adopted for tether ten-sion in next steps will depend on the instantlongitude of the cable, it is more convenient for themanageability of the resulting dynamic model thathelicopter position in inertial frame N is definedby means of the generalized spherical coordinatesq7,8,9 represented in Fig. 1 and Fig. 5a. As canbe seen, angular variables q7,8 correspond to twosuccessive rotations at point NO that align vectorn3 with the tether direction, defined by vector c3.The remaining configuration coordinate q9 definesthe instant length of the tether. Hence, position ofpoint P is given by:

pNO→P = q9c3 (8)

Since the tether is modeled as an elastic element,the tension acting at point P (See Fig. 2) is thengiven by:

TC = −TCc3 = −KC(q9 − LN)c3

KC

{= 0 for q9 < LN

> 0 for q9 > LN

(9)

where LN and KC are the natural length andelasticity constants of the tether, respectively.

The use of spherical variables as the configura-tion variables for helicopter position at the sametime that the motion variables are still Cartesian,makes it necessary a new analysis on the kine-matics of the system. To this end,

NdpNO→HO

dtis

compared against (2), taking into account thatpNO→HO

= pNO→P + pP→O + pO→HO . The re-sulting equations are solved for q̇7,8,9, which yields

the following:[q̇7 q̇8 q̇9

]T= M ·

[u1 · · · u6

]T (10)

where matrix M is function of generalized coor-dinates qi (i = 4, · · · 9) and parameters dO−HO,3

and dO−P,3. While it is true that kinematical equa-tions (10) are more complex than those of (4)(M is a dense matrix), the real advantage of us-ing spherical configuration variables together withCartesian motion variables is given by the resultingdynamical differential equations, since they aremuch more compact for the tethered configurationthan those corresponding to Cartesian coordinatesfor both configuration and motion variables. Finalexpressions for the dynamics of the tethered con-figuration are given by:

mH u̇1 = RHS1 − TCs8

mH u̇2 = RHS2 + TCs7c8

mH u̇3 = RHS3 − TCc7c8 (11)

for translation, whereas rotational equations aregiven by:

K4p4u̇4 = RHS4 + TC(dO−P,3 − dO−HO,3)·· (c7c8(s4c6 + s5s6c4)−− s7c8(c4c6 − s4s5s6)− s6s8c5)

K5p5u̇5 = RHS5 + TC(dO−P,3 − dO−HO,3)·· (s7c8(s6c4 + s4s5c6)−− c7c8(s4s6 − s5c4c6)− s8c5c6)

K6p6u̇6 = RHS6 (12)

where RHSi is the right hand side of the cor-responding equations in (6) and (5). The valuesof the model parameters that have been used forsimulation purposes in this work are shown inTable II.

B. Ground device for tether tension controlThe ground device is in charge of controlling

the tether tension during the operation of thesystem. Since tether tension was modeled as anelastic element in (9), a natural way to controlthe tether tension by the ground device wouldbe varying the natural length LN , that is, reelingin/out the tether. Then, the model correspondingto the ground device must define a relationshipbetween the variation in the natural length that is

446

TABLE II: Parameters of the helicopter mechani-cal model

Parameter Value Units

Gravity acceleration g 9.81 ms2

Total mass mH 12.67 kg

Prin. moments of inertia IH11 0.7642 Nms2

- IH22 1.1642 Nms2

- IH33 1.2318 Nms2

MR disk inertia IMR11 0.1159 Nms2

MR angular speed ωMR −141.37 1s

Geometry dO−FO,3 −0.11 m

- dO−MRO,3 0.166 m

- dO−TRO,1 −1.08 m

- dO−P,3 −0.3 m

dO−HO,3 −0.0954 m

Model constants K4p4 IH11 Nms2

K456 IH22 − IH33 Nms2

K45 −2IMR11ωMR Nms

K5p5 IH22 Nms2

K546 IH33 − IH11 Nms2

K54 2IMR11ωMR Nms

K6p6 IH33 Nms2

K645 IH11 − IH22 Nms2

Init. T. natural length L0N 9.375 m

T. elasticity constant KC 40 Nm

being reeled in/out and certain system input thatallows to control such variation. For the purpose ofthis paper the following simple model shall suffice:

dLN

dt= UC (13)

where UC is the control signal that allows to windor unwind the tether.

III. CONTROL STRATEGY

The controller proposed for the tethered config-uration is based on the control scheme for freeflight presented in [5] and depicted in Fig. 3.The gray blocks denote the helicopter dynamicswhilst the white blocks denote the controller itself.Although the resulting controller is nonlinear, the

underlying approach is quite simple since it isbased on linearization through model inversion andPID control.

The main idea is to adjust the orientation of themain rotor plane and therefore of the lifting forcein order to generate the translational accelerationsrequired to reduce the position error. The inputsfor the control scheme are given by the desiredposition q∗1,2,3 and the desired yaw angle q∗6 . Ascan be seen in Fig. 3 the complete scheme consistsof three control loops, presented in more detail inFig. 4.

The outer loop controller CTRAS calculates fromposition errors the required translational accelera-tions u̇∗1,2,3 to reduce such deviations, and this isimplemented by means of a PID control law:

u̇∗i = KqiP (q∗i − qi) +Kqi

I

ˆ(q∗i − qi) dt−Kqi

Dui

i = 1, 2, 3 (14)These accelerations are converted into the de-

sired orientation for the main rotor plane q∗4,5,and the desired lifting force fMR,3, through theinversion of translational dynamics in block D−1123

For the purpose of this inversion, input fTR,2 isconsidered a disturbance and hence neglected. Re-sulting expressions after some additional algebraicmanipulations are given by:

fMR,3 = mH

√(u̇∗1)

2 + (u̇∗2)2 + (u̇∗3 + g)2

q∗5 = arcsin

(mH u̇

∗1

fMR,3

)q∗4 = arcsin

(− mH u̇

∗2

fMR,3c∗5

)(15)

The desired orientation for the main rotor planeq∗4,5 given by CTRAS is controlled in the inner loopCROT through a proper calculation of momentstMR,1 and tMR,2. To this end, desired Euler anglederivatives q̇∗4,5 are calculated from angular errorsusing a Proportional gain:

q̇∗j = KqjP

(q∗j − qj

)j = 4, 5 (16)

These Euler angle derivatives are then convertedinto the desired angular speeds u∗4,5 in block K−1456

by means of the inversion of rotational kinematics(5):

u∗4 = c5c6q̇∗4 + s6q̇

∗5

u∗5 = −c5s6q̇∗4 + c6q̇∗5 (17)

447

123456 456ROTTRASq1,2,3* q1,2,3

fMR,3

q4,5* tMR,1,2 u’4,5,6 u4,5,6 q’4,5,6 q4,5,6 u’1,2,3 q’1,2,3

TAIL

q6* fTR,2

u4,5q4,5,6

tMR,3 tTR,2

q1,2,3

q6 u6

fMR,3

fTR,2

Fig. 3: General control scheme for helicopter in free flight

Then, desired angular accelerations u̇∗4,5 are cal-culated from deviations in angular speed usinganother Proportional gain:

u̇∗j = Kuj

P

(u∗j − uj

)j = 4, 5 (18)

These resulting angular accelerations are con-verted into moments tMR,1 and tMR,2 by thecorresponding inversion of rotational dynamics inblock D−1456. Again considering fTR,2 and tTR,2

as disturbances, and assuming u6 = 0 (tail rotorcontrolled dynamics several times faster than an-gular dynamics in longitudinal and lateral axes),algebraic manipulation of (7) yields the following:

tMR,1 = K4p4u̇∗4 −K45

ˆu̇∗5dt

tMR,2 = K5p5u̇5 −K54

ˆu̇∗4dt (19)

Finally, the desired yaw angle q∗6 is controlledseparately with the third loop controller CTAIL.To this end, desired angular speed u∗6 is calculatedfrom angular error using a PI control law. Then,tail rotor force fTR,2 is calculated from angularspeed error using a Proportional gain:

u∗6 = Kq6P (q∗6 − q6) +Kq6

I

ˆ(q∗6 − q6) dt

fTR,2 = Ku6P (u∗6 − u6) (20)

Feed-forward actionAs was mentioned before, tether effect on the

system is two-fold. On the one hand, it providesrobustness against external perturbations due to the

-1456

-1456

-1123

1,2,3

1,2,3

1,2,3

MR,3

4,5

TRAS

4,5

4,5 4,5

4,5

4,5 MR,1,2

ROT

6

66

6 TR,2

TAIL

5,6

Fig. 4: Detail of the controller blocks

stabilizing properties of the tether tension in thetranslational dynamics (11). On the other hand, themoment induced by the existing offset betweenthe tether tension application point P and thecenter of mass HO, produces undesired coupling in(12) between rotational and translational dynamicsthat makes more difficult the controllability of thesystem. It is important to notice that the valueof the moment caused by the tether could besimilar or even larger than the torques requiredto control the rotation of the helicopter in freeflight without any tethering device. In order toaccount for that undesired rotational influence, afeed-forward action based on estimating the tethertension is needed. To this end, a force sensor (loadcell) for measuring the tension magnitude is used.Additionally, in order to establish the orientationof the tension vector, the device holding the tetheris an universal joint that includes two opticalencoders for measuring the angles, as illustrated

448

n1

n2

n3

c3

NO

7

8

n2'

(a) Universal joint

q4,5*

q4,5,6 u4,5

tMR,1,2ROT

F-fwdTC

pP->HO

+-

(b) Modified rotational controller

Fig. 5: Feed-forward scheme

in Fig. 5a. The setup of these sensors is similar tothe load transportation device described in [8].

Once the estimation for the tether tension vectoris available, a compensator block CF−fwd calculat-ing the tether moment

ttetheri =(pP→HO ×TC

est)· fi i = 1, 2 (21)

is added to the orientation controller in order tosubtract its value from moments calculated in theorientation controller as shown in Fig 5b.

Control objective for tether tension

As was mentioned before, the ground deviceis in charge of fixing a certain tension value onthe tether during the operation of the system. Tothat end, the measured magnitude of the tethertension T est

C is compared against the referencetension for the tether T ref

C . For this purpose, itis necessary to define the desired temporal profilefor this value. The objective is to maximize thebenefits of the stabilizing effect in translationaldynamics of the tether, while at the same time theundesired rotational influence remains under con-trol. The complexity of satisfying both objectivessimultaneously suggests the definition of a simpleprofile that ease the design process of the controllerfor the helicopter. Accordingly, the selected profilefor the tether tension is given by a constant valueof T ref

C . While it is true that high values for thisprofile would reinforce tether stabilizing action intranslational dynamics, it should also be accountedthat the higher this value is, the higher the con-trol degradation for helicopter rotational dynamicscould be. Then, a trade-off criterion suggests thatthe maximum value for tether tension should bedefined in such a way that the induced tethermoment is always less than the maximum moment

exerted by the main rotor control action, whichcorresponds to the saturation of the cyclic pitch.By using an estimation of the previous limit forcyclic saturation for a typical commercial small-size helicopter, it was concluded in [1] that themagnitude of the tether tension should not exceed20% of the lifting force at hovering.

The error between measured tension and refer-ence value is used to generate the actuation signalUC in (13) by means of a PI type control law:

UC = KCP · (T est

C − T refC )+

+KCI

ˆ(T est

C − T refC )dt (22)

where the controller gains are set to KCP = 1 and

KCI = 0.1.

Tuning controller parameters

The parameters of the helicopter controller havebeen tuned by means of the classical pole as-signment method, as described in [5]. After abasic analysis of the mechanical limitations of thesystem by means of experiments providing stepinputs, it is concluded that all the poles shouldbe set around -1 for translational and rotationaldynamics and -10 for tail rotor dynamics. This isa trade-off value that guarantees a proper dynamicrange at the same time that the mechanical limitsof the system are not reached.

Concerning translational and rotational con-trollers CTRAS and CROT , the expressions thatrelate the control parameters with the desired polesp1,··· ,9 for the corresponding close-loop system areshown below:

Kuj

P = − (p1 + p2 + p3 + p4 + p5)

KqjP = p4p5 + p3 (p4 + p5)+

+ p2 (p3 + p4 + p5)+

+ p1 (p2 + p3 + p4 + p5)

KqiD = − (p3p4p5 + p2 (p3 (p4 + p5) + p4p5)+

+ p1 (p2 (p3 + p4 + p5)+

+ p3 (p4 + p5) + p4p5)) /KqjP

KqiP = (p2p3p4p5 + p1 (p3p4p5+

+ p2 (p3 (p4 + p5) + p4p5))) /KqjP

KqiI = −p1p2p3p4p5/K

qjP (23)

449

The same equivalence but for the tail controllerCTAIL is the following:

Ku6P = p7 + p8 + p9

Kq6P = − (p7p8 + p9 (p7 + p8)) /K

u6P

Kq6I = p7p8p9/K

u6P (24)

All these expressions have been obtained afterderiving the transfer functions of the closed-looplinear systems that result from the application ofmodel inversion and the different PID, PI and Pcontrol laws:

qiq∗i

=(K

qjP (Kqi

I +KqiP s))/(

KqiI K

qjP +Kqi

P KqjP s+Kqi

DKqjP s

2+

+KqjP s

3 +Kuj

P s4 + s5

)i = 1, 2, 3

q6q∗6

= Ku6P (Kq6

I +Kq6P s) /(

Kq6I K

u6P +Kq6

P Ku6P s+Ku6

P s2 + s3

)(25)

IV. SIMULATION RESULTS

The control approach presented in previoussection have been tested in simulation with thecomplete set of non-linear equations derived forthe extended tethered configuration (5), (10), (11)and (12). These simulations show the responseof the system in presence of artificially generatedwind influence. Additionally, in order to betterillustrate the advantages of the improved controllerproposed in this paper over the work presentedin [1], simulations corresponding to such linearcontrol approach have been also considered forcomparison purposes. Likewise, the behavior ofthe unmanned helicopter in free flight is alsoshown in some graphs for recalling the benefit ofthe tether configuration.

The reference value adopted for tether tension isassumed to be approximately 20% of the helicoptertotal weight, i.e. ≈ 25N . Initial conditions forstate variables correspond to hovering equilibrium.Two different simulations have been performedto test the performance of the proposed controlstrategy under the effect of wind disturbances.In the first one, a longitudinal disturbance in thetranslational dynamics is simulated by means of aforce Fw1 applied at the helicopter center of mass.More precisely, it consists of the combination of

a pulse force Fw1 = 20n1 at instant t = 10 sand a sinusoidal force Fw1 = 20 sin(2π · 0.1 · t)n1

starting at time t = 30 s. In a similar way, thesecond simulation under study consists of a lateraldisturbance in the translational dynamics simulatedby means of the combination of a pulse forceFw2 = 20n2 at instant t = 10 s and a sinusoidalforce Fw2 = 20 sin(2π · 0.1 · t)n2 starting at timet = 30 s applied at helicopter center of mass. Fig.6 illustrates the response of the state variablesmainly affected by such perturbations, q1 and q2respectively. As can be seen, the behavior of theimproved controller is better than in the free case.Furthermore, when comparing to the linear con-troller presented in [1], a significant improvementcan be seen in q2.

For completing the evaluation of the improvednonlinear control structure presented in this paper,a more exhaustive comparison of the completeset of variables for simulation 2 is shown inFig. 7. These graphs include the evolution of allstate variables as well as the tether tension TC ,tether natural length LN (tether length deployedby ground device) and the helicopter control inputsfMR,3, tMR,1, tMR,2, fTR,2 (including feed-forwardaction based on T est

C ). Clearly, the response of thenonlinear controller is more precise due to thenonlinear nature of the system. 3D animations ofboth simulations can be seen in [9].

V. CONCLUSIONS

This paper extends previous contributions of theauthors about the use of tethered configurationsfor improving performance in hovering maneuversperformed by unmanned helicopters. More pre-cisely, this work presents a more elaborated strat-egy for controlling the helicopter, while adoptingthe same control paradigm of ensuring a constanttether tension by means of the ground device.

The aim was to better consider the complexand nonlinear nature of the system. To this end,instead of the basic linear scheme employed for thefirst proof of concept, a combination of classicalPID control laws together with model inversionblocks constitutes the base of the new controller.Additionally, feed-forward action for counteractingrotational couplings are also accounted for.

The analysis of the simulations confirms thebenefits of the new controller. On the one hand, the

450

−20

−10

0

10

20

Fw

1 [N

]

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

4

5

q1 [

m]

t [s]

≈ 28%

(a) Evolution of state variable q1 in simulation 1

−20

−10

0

10

20

Fw

2 [N

]

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

4

5

q2 [

m]

t [s]

≈ 29%

(b) Evolution of state variable q2 in simulation 2

Fig. 6: Response of the state variables mainly affected by disturbance in both simulations. Dash-dot forcontroller reference, dotted for free helicopter, dashed for tethered helicopter (linear controller) [1] andsolid for tethered helicopter (nonlinear controller)

response of the state variables directly related tothe perturbation signals is better than in the linearcase. On other hand, when analyzing the completeset of variables, it is evident that the behavior ofthe nonlinear controller is more precise due to thecomplex nature of the system.

Although previous analysis concluded that thenew approach yields better performance, thereseems to be still room for improvement. To thisend, more advanced techniques belonging to thecontrol branch specifically devoted to nonlinearsystems will be investigated in future work.

Last but not least, in order to endorse the validityof the conclusions obtained in these simulationworks, next step will be the realization of fieldexperiments. There is already some preliminarywork carried out in this line, like the design ofthe ground platform required for the experimentalsetup.

ACKNOWLEDGMENTS

This work is supported by European Commis-sion Project EC-SAFEMOBIL (FP7-ICT-2011-7),Spanish Science and Innovation Ministry RD&INational Plan Project CLEAR (DPI2011-28937-C02-01) and Excellence Project of Junta de An-

dalucía RURBAN (P09-TIC-5121). Partially sup-ported by FEDER funds.

REFERENCES

[1] L. Sandino, M. Bejar, K. Kondak, and A. Ollero, “On the useof tethered configurations for augmenting hovering stability insmall-size autonomous helicopters,” Journal of Intelligent &Robotic Systems, vol. 70, pp. 509–525, 2013.

[2] D. Rye, “Longitudinal stability of a hovering, tethered rotor-craft.” Journal of Guidance, Control, and Dynamics, vol. 8,no. 6, pp. 743–752, 1985.

[3] G. Schmidt and R. Swik, “Automatic hover control of anunmanned tethered rotorplatform,” Automatica, vol. 10, no. 4,pp. 393–403, 1974.

[4] B. Ahmed and H. Pota, “Backstepping-based landing controlof a RUAV using tether incorporating flapping correctiondynamics.” Proceedings of the American Control Conference,2008, pp. 2728–2733.

[5] K. Kondak, M. Bernard, N. Losse, and G. Hommel, “Elab-orated modeling and control for autonomous small size heli-copters,” VDI Berichte, vol. 1956, pp. 207–216, 2006.

[6] T. R. Kane and D. A. Levinson, Dynamics. Theory andApplications. McGraw-Hill, 1985.

[7] L. Sandino, M. Bejar, and A. Ollero, “A survey on methodsfor elaborated modeling of the mechanics of a small-sizehelicopter. analysis and comparison,” Journal of Intelligent &Robotic Systems, 2013.

[8] M. Bernard, K. Kondak, I. Maza, and A. Ollero, “Autonomoustransportation and deployment with aerial robots for search andrescue missions,” Journal of Field Robotics, vol. 28, no. 6, pp.914–931, 2011.

[9] http://grvc.us.es/staff/sandino/videos/ICUAS2013.

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−20

−10

0

10

20F

w2 [N

]

−0.15

−0.1

−0.05

0

0.05

0.1

q1 [

m]

−4

−2

0

2

4

q2 [

m]

9.5

10

10.5

11

q3 [

m]

−10

0

10

20

q4 [

º]

−1

−0.5

0

0.5

q5 [

º]

0 10 20 30 40 50 60−10

−5

0

5

q6 [

º]

t [s]

(a) Evolution of state variables q1,2,3 (linear position) and q4,5,6(Euler angles)

−20

−10

0

10

20

Fw

2 [N

]

24.4

24.6

24.8

25

25.2

25.4

TC [

N]

9

9.2

9.4

9.6

9.8

10

10.2

LN [

m]

146

148

150

152

f MR

,3 [

N]

−10

−5

0

5

t MR

,2 [

N·m

]

−2

0

2

4

6

t MR

,1 [

N·m

]

0 10 20 30 40 50 6013.4

13.6

13.8

14

f TR

,2 [

N]

t [s]

(b) Evolution of the tether tension TC , tether natural length LN andthe control inputs fMR,3, tMR,1, tMR,2, fTR,2

Fig. 7: Simulation 2 results. Dash-dot for controller reference, dashed for tethered helicopter (linearcontroller) [1] and solid for tethered helicopter (nonlinear controller)

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