On the use of tethered configurations for augmentinghovering stability in small-size autonomous helicopters

Luis A. Sandino1, Manuel Bejar2, Konstantin Kondak3 and Anibal Ollero4

Abstract— Helicopters are well-known by their ho-vering capabilities. However, the performance of thisvaluable feature can be seriously affected by externaldisturbances such as wind effects. The latter could beeven more significant when dealing with small-size heli-copters, which are commonly adopted as base platformsfor developing unmanned aerial vehicles. Motivated bythis context, this work proposes an augmented configu-ration for performing more stable hovering maneuversthat consists of the unmanned helicopter itself, a tetherconnecting the helicopter to the ground, and a device onground adjusting the tether tension. A modeling analysison the inherent benefits to the proposed configuration aswell as the control guidelines to exploit such potentialitiesare presented in this paper. As a proof a concept,a first basic implementation of the control structurefor the entire system is also included. Finally, severaldemonstrating simulations under artificially generatedwind influences are presented to endorse the validity ofthe proposed approach.

Keywords—Unmanned aerial vehicles, helicopter, mo-deling, tethered systems, stability augmentation, control.

I. INTRODUCTION

In the last few years Unmanned Aerial Vehicles(UAVs) have attracted significant interest. On theone hand, the higher mobility and maneuverabilityof UAVs with respect to ground vehicles makesthem a natural approach for tasks like informationgathering or even the deployment of instrumenta-tion. On the other hand, UAVs avoid the humanrisk inherent in human-piloted aerial vehicles, par-ticularly on missions in hostile environments, andthey can be smaller and more maneuverable. The

1L.A. Sandino is with University of Seville, Seville, SPAINlsandino at us.es

2M. Bejar is with University Pablo de Olavide, Seville, SPAINmbejdom 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

operational costs can be also lower than those ofmanned aircraft.

When comparing different UAV configurations,it is observed that helicopters and other rotorcraft-based unmanned aerial vehicles have flight ca-pabilities such as hover, Vertical Take-Off andLanding (VTOL) and pirouette, which cannotbe achieved by conventional fixed-wing aircraft.These features are consequence of their functionalcontrollability in lateral, longitudinal and verticaldirections with almost constant yaw-attitude.

The aforementioned hovering capability allowsremotely piloted and autonomous helicopters tobe extensively used nowadays for applications in-volving aerial and lateral views, including aerialphotography, cinematography, inspection and otheraerial robotic applications. However, the perfor-mance of this valuable feature can be seriouslyaffected by external disturbances such as windeffects. The latter could be even more significantwhen dealing with small-size helicopters.

Although many advances concerning helicoptercontrol have been proposed in the recent literature,not many of them pay attention to this specificissue of hovering performance. Most of them arebased on robust control techniques, like [1] and [2].A more recent contribution that deserves specialmention is [3] since it specifically addresses theproblem of stabilization in a hover configurationsubjected to parametric uncertainty and externaldisturbances. To this end, a multi-input multi-output proportional-integral-derivative control lawis reformulated into a full-state feedback controllaw to synthesize the controller by using robustH∞ control theory. Model simulations were carriedout to verify the performance of the proposedcontroller when suppressing external disturbances.A work along the same line, but using a diffe-rent variant of H∞ control is that of [4]. In this

case, a robust controller for hovering control usingextended H∞ loop shaping design techniques ispresented. Then, a simulation study is conductedto evaluate its performance in terms of disturbancerejection and other relevant indexes. The obtainedresults were very satisfying since disturbance windgust attenuation up to 95 percent were achieved.

In order to address the performance degrada-tion on hovering capabilities under external dis-turbances, this work suggests a totally differentapproach to those sophisticated control methodspreviously mentioned. More precisely, the paperproposes an augmented setup that consists of theunmanned helicopter itself, a tether connectingthe helicopter to the ground, and a device onground that is in charge of fixing certain tensionvalues on the tether during the operation of thesystem. The justification for this augmented setupis the stabilizing action of the tether tension.To authors knowledge, this usage of the tethertension to increase stability of hovering maneu-vers in helicopters is almost unexplored in theliterature. The only related precedents consideredthe tethered configuration for rotorcraft prototypesdifferent from helicopters are those of [5], wherethe linearized equations describing the perturbedlongitudinal motion of a tethered rotorcraft werepresented and [6], where a discussion of the controland stabilization problems involved in the tetheredconfiguration for a rotor platform prototype ispresented, with special emphasis on tether dyna-mics. The authors of that work also propose twoapproaches for the design of an automatic hovercontroller for the tethered system. Finally, it isalso worth mentioning the contributions of [7]and [8]. Although both corresponds to a differentapplication scenario, landing a helicopter on a shipdeck, they are cited here since also make use ofa tether as an additional resource for helicoptercontrol. In those works, the benefit of the usageof the tether is improving the controllability of thesystem instead of the stability.

In addition to the previously introduced tetheradvantages in terms of its stabilizing properties,the usage of the tether offers some other interest-ing potential benefits. More precisely, the tethercould provide an alternative measurement for thetranslational position of the helicopter. To this end,

proper sensors should be added to the tether setup.On the one hand, angular encoders to know therelative orientation between helicopter and tether.On the other hand, an altimeter of high precisionto know the helicopter altitude with respect to theground. Indeed, these information together withthe helicopter attitude would yield an alternativeestimation for the translational position of thesystem, whose reliability would not be affected bytypical degradation factors of GPS systems.

This paper begins with a modeling analysis onthe tethered configuration in order to justify itsinherent stabilizing properties. To this end, SectionII presents the model considered for a free small-size helicopter. Then, Section III extends the modelto include the effects of the tethered configuration.Later on, in Section IV a discussion on the designguidelines for the control strategy of the entiresystem is presented. According to such guidelines,a first basic implementation is proposed in SectionV to demonstrate the feasibility of the approach.More advanced control techniques following thesame guidelines will be considered in the closefuture to improve the performance. That first exem-plifying implementation includes a position con-troller on board the helicopter capable of dealingwith the tether influence, as well as a simpletension controller for the device on ground. Finally,in Section VI, several demonstrating simulationsunder artificially generated wind influences arepresented to endorse the validity of the proposedapproach. Section VII is devoted to conclusionsand future work.

II. HELICOPTER MODEL

This section introduces the model adopted forthe non-tethered configuration, that is, for the freehelicopter. The airframe corresponds to a small-size helicopter with a stiff main rotor. The ob-jective is not to present very elaborated equationsfor high-fidelity simulations but rather to derivethe best model for control design purposes. Theformer criterion demands a simple and manageablemodel for easing the analysis to derive the controllaws, but at the same time capable to accuratelyreproduce the main behaviors of the real system.

According to the authors of [9], the dynamicsof a small size helicopter with a stiff main rotor is

Fig. 1. Helicopter description: reference frames, centers of mass,and dimensions of interest for modeling purposes

mainly described by its mechanical model. Theirattempt to include an elaborated model for aero-dynamics of the main rotor in controller designdid not show any improvements in control expe-riments performance. This could be considered asevidence for the fact that the approach to analyzethe helicopter behavior by means of its mechanicalmodel is suitable for practical applications. Thispaper follows the same assumption and hence themodeling analysis for the free helicopter as wellas for the tethered configuration will be focusedon the mechanics.

The system is depicted in Fig. 1 and in line with[9] the mechanical characterization of the systemaccounts for two separated rigid bodies, fuselageand stiff main rotor, whereas the tail rotor will onlyact as a force application point on the fuselage.This characterization arises from the fact that formost commercially available small-size helicoptersthe inertial effects of the main rotor (gyroscopiceffects) become the main component influencingthe rotational dynamics of the whole mechanicalsystem whilst the tail rotor inertial influence isnegligible.

There are several methods in the field of classi-cal mechanics for producing equations of motion.Although the equations produced by the differentalternatives are equivalent in terms of numericalresults in simulation, Kane’s method has provedto hold some unique advantages when comparedto other traditional approaches [10]. On the onehand, the usage of generalized coordinates allowsto embed configuration constraints. On the otherhand, the adoption of generalized speeds enablesthe derivation of a compact model in the form ofa first order differential equations that are uncou-

pled in the generalized speed derivatives. Anotherremarkable advantage is that constraint forces aredisregarded from the outset of the analysis. Finallyit should be also noted that equations of motiongenerated by Kane’s method are characterized bytheir easy computerization and computational effi-ciency.

A. Geometry and mass/inertia distribution

Helicopter motion is described in an inertial re-ference frame N where a dextral set of orthogonalunit vectors ni (i = 1, 2, 3) is fixed. Fuselage isdenoted by F , whereas its mass and center of massare given by mF and FO respectively. Main rotoris denoted by MR and is modeled as a thin soliddisk with constant angular speed ωMR. The samenomenclature criterion for fuselage stands for thiscase, that is, mass and center of mass are denotedrespectively by mMR and MRO.

A dextral set of orthogonal unit vectors fi (i =1, 2, 3) fixed in F , and similarly, a dextral set oforthogonal unit vectors mri (i = 1, 2, 3) fixed inMR, allows the definition of the central inertiadyadic of fuselage and main rotor as:

IMR/MRO

= IMR11mr1mr1 + IMR11mr2mr2+

+ 2 · IMR11mr3mr3

IF/FO

= IF11f1f1 + IF22f2f2 + IF33f3f3 (1)

where mr3 = f3 corresponds to the rotation axisfixed in F . Scalar constants IF11, IF22, IF33 andIMR11 are the principal moments of inertia of thecorresponding bodies.

An additional reference point O fixed in F isused to describe the location of points FO, MRO ,TRO (center of mass of tail rotor) and P (attachingpoint of the tether in the helicopter) by means ofthe corresponding dimensions defined in Fig. 1:

pO→FO

= dO−FO,3f3

pO→MRO

= dO−MRO,3f3

pO→TRO

= dO−TRO,1f1

pO→P = dO−P,3f3 (2)

Assumption that FO is located on the rotationaxis of the main rotor will be achieved in the realsystem by an appropriate placement of the equip-ment on the fuselage. Under all these definitions

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).

and assumptions, the center of mass HO of thewhole system relative to O is given by:

pO→HO

=mFp

O→FO+mMRp

O→MRO

mF +mMR

=mF · dO−FO,3 +mMR · dO−MRO,3

mF +mMR

f3

= dO−HO,3f3 (3)

B. Configuration variables

Once system geometry and mass distributionhave been specified, next step is the definition ofthe configuration variables that describe its posi-tion and orientation. The position of center of massHO in the inertial reference frame N is describedby generalized coordinates qi (i = 1, 2, 3):

pNO→HO

= q1n1 + q2n2 + q3n3 (4)

Generalized coordinates qi (i = 4, 5, 6) are theEuler-angles (roll, pitch and yaw) correspondingto successive rotations (body123 order, see [11])that describe the orientation of F in the inertialreference frame N . Thus, unit vectors ni and fiare geometrically related by the direction cosinematrix shown in Table I.

C. Motion variables and kinematics

Once generalized coordinates are defined, kine-matic equations can be formulated as follows:

NvHO

,NdpO→HO

dt= q1n1 + q2n2 + q3n3 (5)

NωF , f1Ndf2dt· f3 + f2

Ndf3dt· f1 + f3

Ndf1dt· f2

= (s6q5 + c5c6q4)f1 + (c6q5 − s6c5q4)f2++ (q6 + s5q4)f3 (6)

Generalized speeds ui (i = 1, · · · , 6) are definedin such a way that (5) and (6) can be written in a

Fig. 2. Forces and torques applied to the helicopter. Tension forceTC is only present in the tethered configuration.

more compact way:

ui , qi (i = 1, 2, 3) ⇒NvHO

= u1n1 + u2n2 + u3n3 (7)

u4 , s6q5 + c5c6q4

u5 , c6q5 − s6c5q4u6 , q6 + s5q4

⇒NωF = u4f1 + u5f2 + u6f3 (8)

The former leads to the following kinematicdifferential equations:

qi = ui (i = 1, 2, 3) (9)q4 = − (s6u5 − c6u4) /c5 (10)q5 = s6u4 + c6u5 (11)q6 = u6 + s5 (s6u5 − c6u4) /c5 (12)

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

D. DynamicsConcerning forces and torques applied to the

system (see Fig. 2), main rotor generates a forceFMR = fMR,3f3 applied at point MRO and torquesMMR,i = tMR,ifi (i = 1, 2, 3) applied to rigidbody MR, whereas tail rotor generates a forceFTR = fTR,2f2 applied at point TRO and a torqueMF = tTR,2f2 applied to rigid body F . Force ofgravity Wj = −mjgn3 (j = F,MR) applied atcenters of mass FO and MRO is also considered,where g is the acceleration of gravity.

Next step according to Kane’s methodologyis the derivation of partial velocities and partialangular velocities for all the force application

points and rigid bodies belonging to the systemunder study. After that, generalized active/inertiaforces can be defined as the dot product of par-tial velocities and active/inertia forces, as well asthe dot product of partial angular velocities andactive/inertia torques. Then, the so-called Kane’sequations are a straightforward extension that ag-gregates for each generalized speed the projectionsof all the active/inertia forces onto the variationdirection of the corresponding generalized speed.The application of previous steps can be easilyperformed by means of Autolev (version 4 ofMotionGenesis software [12]), an advanced sym-bolic manipulator for mechanical systems that im-plements Newton-Euler principles through Kane’smethod [13]. The former leads to the followingdynamical differential equations:

(mF +mMR)u1 = fMR,3s5 − fTR,2c5s6 (13)(mF +mMR)u2 = fTR,2(c4c6 − s4s5s6)−

− fMR,3s4c5 (14)(mF +mMR)u3 = fMR,3c4c5+

+ fTR,2(s4c6 + c4s5s6)−− (mF +mMR)g (15)

K4p4u4 = tMR,1 + dO−HO,3fTR,2+

+ (K456u6 +K45)u5 (16)K5p5u5 = tMR,2 + tTR,2+

+ (K546u6 +K54)u4 (17)K6p6u6 = tMR,3 + dO−TRO,1fTR,2+

+K645u4u5 (18)

where parameters Kxxx are given by:

K4p4 =mF ·mMR · (dO−FO,3 − dO−MRO,3)

2

mF +mMR

+

+ IF11 + IMR11 (19)

K456 =mF ·mMR · (dO−FO,3 − dO−MRO,3)

2

mF +mMR

+

+ IF22 − IF33 − IMR11+ (20)K45 = −2IMR11ωMR (21)

K5p5 =mF ·mMR · (dO−FO,3 − dO−MRO,3)

2

mF +mMR

+

+ IF22 + IMR11 (22)

K546 = −mF ·mMR · (dO−FO,3 − dO−MRO,3)

2

mF +mMR

+

+ IF33 − IF11 + IMR11 (23)K54 = 2IMR11ωMR (24)K6p6 = IF33 + 2 · IMR11 (25)K645 = IF11 − IF22 (26)

The complete set of motion equations for thehelicopter mechanical model is then given by (9)-(12) together with (13)-(18). It is remarkable thatthese equations are uncoupled in the generalizedspeed derivatives thank to the choice of generalizedspeeds made in (7) and (8). This fact makes themodel suitable for numerical integration. Note alsothat constraint forces and torques (i.e. interactionforces and torques between main rotor and fuse-lage) were not present during the formulation. Thereason is that the vector space used to projectNewton-Euler’s equations to derive Kane’s equa-tions is orthogonal to those forces and torques,which allows to disregard them from the outsetof the analysis.

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 torques exerted onthe system will be considered as system inputsfor control design. More precisely, the subset ofthese forces and torques whose values can be fixedindependently of each other by the controller isgiven 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 andthe lifting force fMR,3, the second one the relationbetween the main rotor cyclic pitches and torquestMR,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 [9] this approach for characterizinglinearly the aerodynamics was verified successfullyin flight experiments.

Finally, the values of model parameters used forsimulation in this work are shown in Table II.

TABLE II

PARAMETERS OF THE HELICOPTER MECHANICAL MODEL

Parameter Value Units

Gravity acceleration g 9.81 ms2

Fuselage mass mF 12 kg

Main rotor mass mMR 0.67 kg

Fuselage inertia IF11 0.6 Nms2

- IF22 1 Nms2

- IF33 1 Nms2

Main rotor inertia IMR11 0.1159 Nms2

Main rotor 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

III. EXTENDED MODEL FOR THETETHERED CONFIGURATION

This section extends the free helicopter model toinclude the remaining parts of the proposed systemfor augmenting hovering stability in small-sizeautonomous helicopters: a tether connecting thehelicopter to the ground, and a device on groundin charge of fixing certain tension values on thetether during the operation of the system. Again theemphasis will be on characterizing mechanicallythe system to provide a manageable model forcontrol analysis and design purposes.

A. Tethered helicopter

For the sake of simplicity, it is assumed thattether extreme on ground coincides with the originof the inertial frame N . On the other extreme,the tether is connected to helicopter point P asdepicted in Fig. 3.

In this new setup, helicopter position in inertialframe N is re-defined in a more convenient wayby the new generalized spherical coordinates q7,8,9represented in Fig. 3 and Fig. 4. As can be seen,angular variables q7,8 correspond to two successiverotations at point NO that align vector n3 withtether direction, defined by vector c3 . The remain-ing configuration coordinate q9 defines the instantlength of the tether. Hence, position of point P is

Fig. 3. Helicopter and tether modeling scheme

TABLE III

PARAMETERS OF THE TETHER USED IN SIMULATION

Parameter Value Units

Initial tether natural length L0N 9.375 m

Tether elasticity constant KC 40 Nm

given by:pNO→P = q9c3 (27)

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

(28)

where LN and KC are the natural length andelasticity constants of the tether, respectively. Thevalues of these tether parameters that will be usedfor simulation are shown in Table III.

This new definition of the spherical configu-ration variables at the same time that Cartesianmotion variables are maintained, requires a newanalysis on the kinematics of the system. To thisend, (7) is compared against

NdpNO→HO

dt, taking

into account that pNO→HO= pNO→P + pP→O +

pO→HO . The resulting equations are solved forq7,8,9, which yields the following:[

q7 q8 q9]T

= M ·[u1 · · · u6

]T (29)

where matrix M is function of generalized coordi-nates qi (i = 4, · · · 9) and parameters dO−HO,3 anddO−P,3. While it is true that kinematical equations(29) are more complex than those of (9) (M is adense matrix), the real advantage of using spheri-cal configuration variables together with Cartesianmotion variables is given by the resulting dyna-mical differential equations, since they are muchmore compact in this new tethered configurationthan those corresponding to the standard Carte-sian coordinates for both configuration and motionvariables. The final expressions for the dynamicsof the tethered configuration are obtained againby means of Kane’s methodology, and are shownbelow:

(mF +mMR)u1 = RHS1 − TCs8 (30)(mF +mMR)u2 = RHS2 + TCs7c8 (31)(mF +mMR)u3 = RHS3 − TCc7c8 (32)

K4p4u4 = RHS4+

+ TC(dO−P,3 − dO−HO,3)·· (c7c8(s4c6 + s5s6c4)−− s7c8(c4c6 − s4s5s6)−− s6s8c5) (33)

K5p5u5 = RHS5+

+ TC(dO−P,3 − dO−HO,3)·· (s7c8(s6c4 + s4s5c6)−− c7c8(s4s6 − s5c4c6)−− s8c5c6) (34)

K6p6u6 = RHS6 (35)

where RHSi is the right hand side of the corres-ponding equations in (13)-(18).

B. Device on ground for tether tension control

Tether tension was modeled as an elastic ele-ment in (28), rewritten here for the sake of clarityof the subsequent development:

TC = −TCc3 = −KC(q9 − LN)c3

KC

{= 0 for q9 < LN

> 0 for q9 > LN

(36)

This implies that the natural way to controlthe tether tension by the ground device would bevarying the natural length LN , that is, reeling in

or out the tether. Then, the model correspondingto the ground device must define a relationshipbetween the variation in the natural length thatis being reeled out and certain system input thatallows to control such variation. For the purpose ofthis paper, establishing a first proof of concept ofthe tethered configuration for augmenting stabilityin hover maneuvers, the following simple modelshall suffice:

LN = RC (37)

where RC is the control signal that allows towind or unwind the tether. The analysis on thedesired patterns for tether tension T ref

C that shouldbe enforced by this ground device in order toachieve the best performance for the entire tetheredconfiguration, will be addressed in next section.

IV. MODELING ANALYSIS ANDCONTROL GUIDELINES

A. On the tether influence in system dynamicsBy a careful analysis of the tethered confi-

guration dynamics given by (30)-(35), it can beconcluded that the tether effect on the system istwo-fold. On the one hand, it provides robustnessagainst external perturbations due to the stabilizingproperties of the tether tension in the translationaldynamics (30)-(32). On the other hand, the mo-ment induced by the existing offset between thetether tension application point P and the centerof mass HO, produces undesired coupling in (33)-(35) between rotational and translational dynamicsthat difficult the controllability of the system. Atthis point it is important to notice that the value ofthe moment caused by the tether could be similaror even larger than the torques required to controlthe rotation of the free helicopter without anytethering device.

B. Relationship between translational and rota-tional dynamics

When analyzing the dynamics of the uncons-trained configuration given by (13)-(18), it is easyto note that rotational dynamics are not coupledwith translational dynamics, which means thatrotational equations only depend on the angularspeeds of the helicopter and the control torques.Conversely, translational accelerations are always

given by the absolute value of the lift force exertedby the main rotor, but also by the orientation ofthe helicopter. Hence the relationship between ro-tational and translational dynamics for a helicopterin free flight can be expressed as an unidirectionalinfluence rotation⇒ translation, which in terms ofthe state variables is equivalent to:

u456 → q456 → u123 (38)

In contrast to previous configuration, in thedynamics of the extended tethered model givenby (30)-(35) there appears an additional forceacting on the helicopter due to tether tension. Thisadditional force produces in turn a moment actingon the rotational dynamics of the fuselage since theapplication point of the force will not correspondexactly with the center of mass of the system. Thismoment depends on orientation and translationalmotion of the helicopter in the inertial frame N . Asa result of this chain of influences, the relationshipbetween rotational and translational dynamics ofthe system is now more complex and given bya bidirectional influence rotation ⇔ translation,which corresponds to:

u456 → q456 → u123 → q789

q456 ← u456 ← q789 ← u123 (39)

C. Control objective for tether tensionAs was advanced in the section devoted to the

ground device for controlling the tether tension,it is necessary to define the desired pattern forT ref

C , the commanded tension on the tether. Theobjective is to maximize the benefits of the stabi-lizing effect in translational dynamics of the tether,while at the same time the undesired rotationalinfluence remains under control. The complexity ofsatisfying both objectives simultaneously suggeststhe definition of a simple pattern that ease thedesign process of the controller for the helicopter.Accordingly, the selected pattern for the tethertension is given by a constant profile on T ref

C . Thisalso allows to conclude that the system will bealways operating around certain work point, whichpaves the way for considering also linear controltechniques for this complex set-up.

As a consequence of previous discussion, thecontrol philosophy for the tethered configurationis to actively compensate for the relative motion

between the helicopter and the tether extreme onground by reeling in or out the tether, dependingon the tension deviation from its reference value.Next point in the discussion is the establishment ofsome operational ranges for those constant valuesto be imposed for the tether tension. While it istrue that high values would reinforce its stabilizingaction in translational dynamics, it must also beaccounted that the higher the value is the higherthe control degradation for helicopter rotationaldynamics could be. Then, a trade-off criterionsuggests that the maximum value for tether tensionshould be defined in such a way that the inducedtether torque is always less than the maximummoment exerted by the main rotor control action,which corresponds to the saturation of the cyclicpitch. By using an estimation of previous limit forcyclic saturation for a typical commercial small-size helicopter, tmax

MR1,2 in (40), it can be concludedthat the magnitude of the tether tension should notexceed 20% of the pilot control authority in thelifting force action at hover:∣∣∣pP→HO ×TC

∣∣∣ < tmaxMR1,2 ⇒∣∣dO−P,3 − dO−HO,3

∣∣TC < tmaxMR1,2

tmaxMR1,2∣∣dO−P,3 − dO−HO,3

∣∣ ≈ 0.2 · fhoverMR,3 (40)

Finally, in order to prevent the tether frombreaking due to abrupt tension increments causedby wind gusts or any other perturbation source,additional safety devices should be included in thesystem set-up to allow the release of the tether insuch cases.

D. Feed-forward to counteract tether torqueIn order to account for that undesired rotational

influence described in previous paragraphs, a feed-forward action based on estimating the tether ten-sion vector is proposed. To this end, a force sensorfor measuring the tension magnitude T est

C will beinstalled in the tether. Commercial devices likeload cells are available in the market for thesepurposes. Additionally, in order to establish theorientation qest7,8 of the tension vector, the deviceholding the tether will be an universal joint thatincludes two optical encoders for measuring theangles. The set-up of these sensors is illustrated inFig. (4).

Fig. 4. Universal joint

Once the estimation TestC for the tether tension

vector is available, the influence of the tether set-up on the rotational dynamics of the helicoptercan be expressed by the torque pP→HO × TC

est.Thus, the resulting orientation controller for thetethered configuration is composed of the orien-tation controller for the free helicopter and acompensator block Cf−fwd calculating that tethertorque influencing the rotational dynamics of thefree helicopter, in order to subtract its value fromtorques calculated in the orientation controller forthe free helicopter.

The usage of the compensator Cf−fwd has twomain advantages, as can be seen in [14]. First, theorientation controller for such complex systemsbecomes quite simple. Second, the closed loopsystem becomes very robust against variation ofsystem parameters and disturbances. Another fac-tor that endorses the robustness of the approach isthat the calculated compensation torque is alwaysin the correct phase, as long as the orientation ofthe helicopter is known.

E. Tuning of helicopter controller

Since the purpose of the tethered configurationis to enhance the responsiveness of the free he-licopter in the presence of external disturbances,the values of the parameters for the helicoptercontroller in the tethered configuration will be thesame than those calculated for the free helicopter.In this way, the tether will really add an additionalstabilizing effect over the close-loop dynamics ofthe free system.

F. Tether as sensing deviceIn addition to the tether advantages already

introduced in terms of its stabilizing properties,the usage of the tether offers some other interest-ing potential benefits. More precisely, the tethercould provide an alternative way of measuring thetranslational position of the helicopter. Indeed, theorientation information coming from the angularencoders in the tether mechanical joint and theIMU on board the helicopter, together with thehelicopter altitude, would yield such an alternativeestimation whose reliability would not be affectedby typical degradation factors of GPS systems. Tothis end, a altimeter of high precision should beadded to the sensor setup of the tethered system.

V. DESIGN OF THE COMPLETECONTROL SCHEME

According to the guidelines for control designpreviously discussed, this section presents a firstbasic control structure in order to validate theusage of the tethered configuration for hoveringstabilization. More advanced control techniquesfollowing the same guidelines will be considered infuture developments to improve the performance.More precisely, in this first proof of concept asimple LQI controller together with the alreadyintroduced feed-forward action will suffice for theposition controller on board the helicopter. In thesame way, a PI tension controller is proposed forthe device on ground.

A. Position controller for the helicopter1) LQI algorithm: This subsection is devoted to

the regulation control problem for linear positionand yaw orientation of the helicopter, given byvariables q1,2,3 and q6 respectively. Expressionsgiven by (10)-(12), (29) and (30)-(35) define thefollowing non-linear model for the tethered confi-guration :

x = f(x,u)

y = h(x) (41)

where state, input, and output vectors are givenrespectively by:

x = [q4, q5, q6, q7, q8, q9, u1, u2, u3, u4, u5, u6]T

u = [fMR,3, tMR,1, tMR,2, fTR,2]T

y = [q1, q2, q3, q6]T (42)

and the relationship y = h(x) between outputvariables q1,2,3 and state variables q4,5,7,8,9 is for-mulated as:

q1 = q9s8 − (dO−P,3 − dO−HO,3)c5 (43)q2 = −q9s7c8 + (dO−P,3 − dO−HO,3)s4c5 (44)q3 = q9c7c8 − (dO−P,3 − dO−HO,3)c4c5 (45)

As was advanced in previous sections, the firststep in the process for designing the helicoptercontroller is the calculation of the control parame-ters corresponding to the free configuration. Forthis purpose, a linearized discrete-time state spacerepresentation is obtained by linearizing the tethe-red equations (10)-(12), (29), (30)-(35) and (43)-(45) around a helicopter hover condition whereTC = 0, which is indeed equivalent to linearizethe free helicopter model. The values for state,output and input variables corresponding to thisequilibrium point are shown in table IV. Thislinearization process yields the following represen-tation of the free helicopter system:

xk+1 = Gxk +Huk

yk = Cxk (46)

where xk , uk and yk denote the variations aroundthe equilibrium point adopted in the linearization.

The proposed control approach is the LinearQuadratic Integral control (LQI) with outputweighting, based on the well-known LinearQuadratic Regulator (LQR) [15]. This control al-gorithm aims at optimizing the following objectivefunction:

J(uk) =1

2

∞∑k=0

(yTkQyyk + uT

kRuuk) (47)

where Qy and Ru are appropriately chosen asconstant and positive-definite weighting matricesthat penalize the quadratic error in output andcontrol input variations around the linearizationpoint, respectively. The solution to this optimiza-tion problem can be written in terms of the fol-lowing simple state feedback law:

uk = −KLQR · xk

KLQR = (Ru +HTPH)−1HTPG (48)

TABLE IV

VALUES FOR VARIABLES AT HOVER FOR FREE CONFIGURATION

(TC = 0)

Variable Value Units

q4 5.29 o

q5 = q7 = q8 0 o

x0|F q9 10 m

u1 = u2 = u3 0 ms

u4 = u5 = u6 0 rads

q1 0 m

y0|F q2 −0.0189 m

q3 10.2037 m

q6 0 o

fMR,3 123.7633 N

u0|F tMR,1 1.0933 Nm

tMR,2 0 Nm

fTR,2 11.4596 N

where matrix P is the unique positive definitesolution to the algebraic Riccatti equation:

P = −GTPH(Ru +HTPH)−1HTPG+

+GTPG+Qx (49)

and where Qx = CTQyC.In addition to this standard development corres-

ponding to the classical LQR, LQI adds integralaction to the control scheme for achieving zerosteady-state error. To this end, the state vector isaugmented to include the time integration of thedifference between the reference signal rk and thevariation of the system output yk. The integratoroutput zi,k+1 is computed using the classical for-ward Euler formula:

zi,k+1 = zi,k +4t · (ri,k − yi,k) i = 1, 2, 3, 6 (50)

where 4t is the sample time. Thus, the augmentedstate vector for this LQI approach is:

xLQIk =

[xk

zk

](51)

and model equations (46) for the free helicoptersystem can be rewritten consequently in the fol-

TABLE V

VALUES FOR VARIABLES AT HOVER FOR TETHERED

CONFIGURATION (TC = T refC )

Variable Value Units

q4 5.29 o

q5 = q7 = q8 0 o

x0|T q9 10 m

u1 = u2 = u3 0 ms

u4 = u5 = u6 0 rads

q1 0 m

y0|T q2 −0.0189 m

q3 10.2037 m

q6 0 o

fMR,3 148.6568 N

u0|T tMR,1 1.3132 Nm

tMR,2 0 Nm

fTR,2 13.7645 N

lowing manner:

xLQIk+1 = GLQIxLQI

k +HLQIuk

yLQIk = CLQIxLQI

k (52)

with

GLQI =

[G 0T I4×4

]HLQI =

[H0

]CLQI =

[C 00 I4×4

](53)

and where I4×4 is a 4 × 4 identity matrix andT = −4t ·C. The application of the standarddesign steps (47), (48) and (49) for the classicalLQR approach, to the new system (52) and (53) ge-nerated for addressing the LQI design, yields a newstate feedback law uLQI

k = −KLQI · xLQIk that al-

lows the tracking of certain profile references in theoutput variables q1, q2, q3, and q6 with zero steady-sate error. Matrices Qy and Ru were selectedby a rule of thumb as diagonal matrices whoseelements are

[1 1 1 1 10 10 10 10

]and[

1 1 1 1], respectively.

2) Control law for the complete nonlinearmodel of the tethered configuration: Once the con-troller gain matrix KLQI |F for the free configura-tion is calculated, the input vector for the complete

Fig. 5. Control law for the complete nonlinear model of thetethered configuration

nonlinear model of the tethered configuration in(41) is obtained as:

u|T = − KLQI |F · xLQIk

∣∣∣T + u0|T −Cf−fwd (54)

where u0|T denotes the input vector correspondingto the tethered helicopter hover condition withTC = T ref

C . The values for this equilibrium pointare shown in table V. Furthermore, the termCf−fwd is the estimated tether torque correspond-ing to the feed-forward action:

Cf−fwd =[0 C1 C2 0

]TC1 = T est

C (dO−P,3 − dO−HO,3)·· (c7c8(s4c6 + s5s6c4)−− s7c8(c4c6 − s4s5s6)−− s6s8c5)

C2 = T estC (dO−P,3 − dO−HO,3)·· (s7c8(s6c4 + s4s5c6)−− c7c8(s4s6 − s5c4c6)−− s8c5c6) (55)

Finally, the reference vector rk for yk , the varia-tion around the trim point of the output variablesq1,2,3,6, is set to zero in order to keep the stateand output vectors x and y at the equilibriumpoint given by x0|T and y0|T respectively. Anillustrative block diagram of the complete non-linear law is depicted in Fig. 5.

B. Tether tension controller

The second control objective is regulation forthe tether tension TC . As explained before, thedevice on the ground is in charge of fixing certaintension values on the tether during the operation ofthe system. To that end, the measured magnitude

of the tether tension T estC is compared against the

constant reference value T refC fixed according to

(40). The error is used to generate the actuationsignal RC in (37) by means of a PI controller:

RC = KP ·(TC−T refC )+KI

ˆ(TC−T ref

C )dt (56)

where the controller gains are set to KP = 1 andKI = 0.1.

VI. SIMULATIONS FOR THE TETHEREDCONFIGURATION

The control strategies presented in previous sec-tion have been tested in simulation with the com-plete set of non-linear equations derived for theextended tethered configuration, (10)-(12), (29),(30)-(35) and (43)-(45). These simulations willshow the response of the system in the presence ofartificially generated wind influences. Additionally,in order to better illustrate the stabilizing propertiesof the tether set-up, simulations corresponding tothe free helicopter have been also considered forcomparison purposes.

As was advanced in previous sections, the re-ference value adopted for tension controller isassumed to be approximately 20% of the helicoptertotal weight, i.e. 25N . Initial conditions for statevariables in both simulations, tethered and freeconfigurations, correspond to the equilibrium pointused for the linearization (see Tables IV and V).Finally, the reference for output variables q1,2,3,6 isalso given by the values corresponding to the trimpoint.

Two different simulations have been performedto illustrate the disturbance rejection capabilities ofthe tether configuration. In the first one, a longi-tudinal disturbance in the translational dynamicsis simulated by means of a force Fw1 appliedat the helicopter center of mass. More precisely,it consists of the combination of a pulse forceFw1 = 20n1 at instant t = 10 s and a sinusoidalforce Fw1 = 20 sin(2π · 0.1 · t)n1 starting at timet = 30 s, as can be seen in the first graph of Fig.6. The evolution of the controlled variables q1, q2and q3 together with the state variables q4 and q5is shown in Fig. 6 whereas the evolution of thetether tension TC and the control inputs fMR,3,tMR,1 and tMR,2 generated by the controller can

be seen in Fig. 7. A video with a 3D animation ofthis simulation can be seen in [16].

Likewise, in the second simulation, a lateraldisturbance in the translational dynamics is simu-lated by means of the combination of a pulseforce Fw2 = 20n2 at instant t = 10 s and asinusoidal force Fw2 = 20 sin(2π·0.1·t)n2 startingat time t = 30 s applied at helicopter center ofmass, as can be seen in the first graph of Fig. 8.The evolution of the variables of interest is shownin Figs. 8 and 9, following the same distributionpattern of the first simulation. A video with a 3Danimation of this simulation can be seen in [16].

As can be observed, the tethered set-up signifi-cantly improves the performance when rejectingtypical patterns of disturbances in longitudinaland lateral axis. More precisely, the disturbanceeffect in the controlled variables q1 and q2 wherethe suggested perturbation profiles mainly maketheir influence, is reduced up to a 30% from theresponse of the free system. Furthermore, boththe control action and the angular behavior of thehelicopter are similar to those of the free configura-tion. Additional discussions on going further withthe performance improvement through alternativecontrol techniques are out of the scope of this firstproof of concept and will be addressed in futurework.

VII. CONCLUSIONSThis paper has presented the benefits of a te-

thered configuration for improving hovering sta-bility of autonomous small-size helicopters in thepresence of disturbances. The augmented set-upconsists of the unmanned helicopter itself, a tetherconnecting the helicopter to the ground, and adevice on ground adjusting the tether tension.

To that end, a modeling analysis on the me-chanical model of the tethered system was con-ducted. This study revealed that the tether effecton the system is two-fold: first, the tether tensionprovides robustness against external perturbationsin the translational dynamics, and second, thetether moment induced in the helicopter producesundesired couplings between rotational and trans-lational dynamics that difficult the controllabilityof the system.

After that first analysis, a discussion on con-trol guidelines to exploit the potentialities of this

Fig. 6. Simulation 1. Evolution of output variables q1, q2 and q3and state variables q4 and q5

Fig. 7. Simulation 1. Evolution of tether tension TC and controlinputs fMR,3, tMR,1 and tMR,2 (including feed-forward actionbased on T est

C in the tether case)

Fig. 8. Simulation 2. Evolution of output variables q1, q2 and q3and state variables q4 and q5

Fig. 9. Simulation 2. Evolution of tether tension TC and controlinputs fMR,3, tMR,1 and tMR,2 (including feed-forward actionbased on T est

C in the tether case)

augmented set-up is also included in the paper.The former leads to define the control objectivefor the tether tension influencing the helicopteras well as to propose methods for bounding theundesired couplings in the system. More precisely,the selected profile for the tension value is constantand as high as possible to maximize its stabilizingeffect, provided that it does not reach the criticalvalue that would produce the saturation in therotational control action of the helicopter whencounteracting the moment induced by the tether.Additionally, a feed-forward action is added tothe control scheme to minimize the effect of theaforementioned couplings.

As a proof a concept, a first basic implementa-tion of the control structure for the entire systemis also depicted. With this purpose, a classicalLQI controller together with the already introducedfeed-forward action sufficed for the position con-troller on board the helicopter. In the same way,a PI tension controller is proposed for the deviceon ground. More advanced control techniques fol-lowing the same control guidelines suggested inthis paper will be addressed in the close future toimprove the performance.

Finally, several simulations under artificially ge-nerated wind influences have been presented toendorse the validity of the proposed approach. Theconclusion is clear, the tethered set-up significantlyimproves the performance when rejecting typicalpatterns of disturbance in longitudinal and lateralaxis. Furthermore, both the control action and theangular behavior of the helicopter are similar tothose of the free configuration.

Last but not least, some other potential ad-vantages of the tether apart from its stabilizingproperties have been also highlighted. For instance,it could provide an alternative way of estimatingthe translational position of the helicopter whosereliability would not be affected by typical degra-dation factors of GPS systems.

ACKNOWLEDGMENTS

This work is supported by Excellence Projectof Junta de Andalucia (P09-TEP-5120), Euro-

pean Commission Projects ARCAS and EC-SAFEMOBIL (FP7-ICT-2011-7) and Spanish Sci-ence and Innovation Ministry (MICINN) RD&INational Plan Project CLEAR (DPI2011-28937-C02-01), partially supported by FEDER funds.

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