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154 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 1, JANUARY 2012

Brief Papers

Robust Nonlinear Controls of Model-Scale Helicopters Under Lateral and

Vertical Wind GustsFranois Lonard, Adnan Martini, and Gabriel Abba, Member, IEEE

AbstractA helicopter maneuvers naturally in an environ-ment where the execution of the task can easily be affected byatmospheric turbulence, which leads to variations of its modelparameters. This paper discusses the nature of the disturbancesacting on the helicopter and proposes an approach to counterthe effects. The disturbance consists of vertical and lateral windgusts. A 7-degrees-of-freedom (DOF) nonlinear Lagrangian modelwith unknown disturbances is used. The model presents quiteinteresting control challenges due to nonlinearities, aerodynamic

forces, underactuation, and its non-minimum phase dynamics.Two approaches of robust control are compared via simulationswith a Tiny CP3 helicopter model: an approximate feedbacklinearization and an active disturbance rejection control usingthe approximate feedback linearization procedure. Several sim-ulations show that adding an observer can compensate the effectof disturbances. The proposed controller has been tested in areal-time application to control the yaw angular displacement of aTiny CP3 mini-helicopter mounted on an experiment platform.

Index TermsAutonomous helicopter, disturbance rejection,identification, nonlinear control, nonlinear model.

I. INTRODUCTION

HIGH levels of agility, maneuverability, and the capabilityof operating in degraded visual environments and ad-

verse weather conditions are the current trends of helicoptercontrol design. Helicopter flight control systems should makethese performance requirements achievable by improvingtracking performance and disturbance rejection capabilities.Robustness is one of the critical issues, which must be con-sidered in the control system design for high-performanceautonomous helicopter, since any mathematical helicoptermodel, especially those covering large flight envelopes, will un-avoidably have uncertainty due to the empirical representation

of aerodynamic forces and moments.Control design of autonomous flying systems has now be-come a very challenging area of research, as indicated in the

Manuscript received January 28, 2010; revised June 16, 2010 and October01, 2010; accepted December 02, 2010. Manuscript received in final form De-cember 17, 2010. Date of publication January 24, 2011; date of current versionDecember 14, 2011. Recommended by Associate Editor Y. Bestaoui.

A. Martini is with the Faculty of Mechanical Engineering, Aleppo University,Aleppo 16351, Syria (e-mail: [email protected]).

F. Lonard and G. Abba are with Laboratoire de Conception FabricationCommandeENSAM ParisTech Metz, cole Nationale dIngnieurs de Metz,57070 METZ, France (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2010.2102023

literature [1], [2]. Many previous works focus on (linear, non-linear, and/or robust, etc.) control, and give particular attentionto the analysis of the stability [3], but very few studies [4 ][6]have been made on the influence of wind gust acting on theflying system, whereas it is a crucial problem for outdoor ap-plications, especially in urban environment. As a matter of fact,if the autonomous flying system, i.e., unmanned aerial vehicles(UAVs) (especially, when this system is relativelyslight) crosses

a threshold, it can be disturbed by wind gusts and leave its tra-jectory, which could be critical in a highly dense urban context.

In recent papers, feedback linearization techniques have beenapplied to helicopter controls. The main difficulty in the ap-plication of such an approach is the fact that, for any mean-ingful selection of outputs, the helicopter dynamics are non-minimum phase, and hence are not directly inputoutput lin-earizable. However, it is possible to find good approximationsof the helicopter dynamics [7], [8] such that the approximatesystem is inputoutput linearizable, and bounded tracking canbe achieved.

In [5] and [9], two controllers [nonlinear feedback and an

active disturbance rejection control (ADRC) based on a non-linear extended state observer (ESO)] are designed for a non-linear reduced-order model of a 3-DOF helicopter. For 7-DOFhelicopter, ADRC is carried out to deal with model error [10]and vertical wind gust [11 ]. In [6], a control strategy stabilizesthe position of the flying vehicle in wind gust environment, inspite of unknown aerodynamic efforts. This control strategy isbased on robust backstepping approach [12] and estimation ofthe unknown aerodynamic forces.

In this paper, to keep stability and performance of a 7-DOFautonomous helicopter in the presence of uncertainties and ex-ogenous disturbances, two robust control techniques are pro-posed and tested on a real helicopter. The first control is an ap-proximate feedback linearization. The second robust control isan ADRC [13], [14] using the approximate feedback lineariza-tion procedure and based on a nonlinear approximate ESO.

In Section II, a model of a Tiny CP3 disturbed helicopteris presented. In Section III, the design and the application oftwo approaches of robust control for the approximate modelare proposed. In Section IV, several simulations of helicoptersunder vertical or lateral wind gusts show the relevance of thetwo controls, which are described in the paper. We present inSection V-D, the experimental validation of the ADRC on theyaw angle dynamics for the Tiny CP3 autonomous helicopter.Finally, some conclusions are drawn in Section VI.

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LONARD et al.: ROBUST NONLINEAR CONTROLS OF MODEL-SCALE HELICOPTERS UNDER LATERAL AND VERTICAL WIND GUSTS 155

Fig. 1. Reference frames of helicopter.

II. MODEL OF THE 7-DOF TINY CP3 DISTURBED HELICOPTER

This section presents the nonlinear model of Tiny CP3 heli-copter disturbed by wind gusts. This model is obtained using amodel of nondisturbed Vario Benzin Trainer [15], [16], wherethe physical parameters of Tiny CP3 are used. Moreover, thewind gust influences are introduced by adding some aerodynam-

ical additional forces and torques. More precisely, the dynamicmodel of Tiny CP3 helicopter is based on the EulerLagrangeformalism, where a vector of generalized coordinatesis defined as in [16]: (seeFig. 1), where denotes the position vectorof the center of mass cm of the helicopter relative to thenavigation frame for aircraft attitude problems (see Fig. 1).

is the main rotor azimuth angle, and ( ) denotethe three Euler angles (yaw, pitch, and roll angles) ex-pressed in the body fixed frame .In addition, we define: , and let

be the rotation

matrix representing the orientation of the body fixed framewith respect to the inertial frame , where is anorthogonal matrix. The vectordenotes the angular velocity of the helicopter in the body-fixedframe, which can also be written as: , wheredenote the generalized velocity vector and is a matrix

(1)

This allows to define . The vector of control inputs of Tiny CP3 helicopter is given by:

, where is the collective pitchangle (swashplate displacement) of the main rotor and the motorpower, is the collective pitch angle (swashplate displace-ment) of the tail rotor, and and are the longitudinal andlateral cyclic pitch angles of the main rotor. We can now applythe EulerLagrange method with and obtain themotion equations of the helicopter

(2)

where is the potential energy of the helicopter, is kineticenergy, is the aerodynamic forces and torques,

applied to the helicopter at the center of mass (see Table I), whilerepresents the external aerodynamical forces

TABLE ICOMPONENTS OF SIMPLIFIED EXTERNAL FORCES VECTOR [15 ]

and torques produced by the wind gust. and are, respec-tively, the main and tail rotor thrust. Here, stands for mainrotor and for tail rotor and is the amplitude of the dragforce created by induced velocity in the disc of the main rotor,

and are the main and tail rotor drag torque, respectively.is the motor torque, which is assumed to be

proportional to the first control input. and are the longi-tudinal and lateral flapping angles of the main rotor blades andis the main rotor blade stiffness. and represent the main

and tail rotor center localization with respect to the center ofmass, respectively.

The development of (2) makes it possible to obtain the fol-lowing equations [11], [16 ]:

(3)

with is the inertia matrix, is theCoriolis and centrifugal forces matrix, and represents thevector of conservative forces. isthe vector of generalized forces and

. Expressions for , and can be found in [15] and [16]. The induced gust velocity [17] is denoted as . We can then

decompose the dynamics of (3) into slow translational dynamicsand fast rotational dynamics

(4)

where is the total mass of the helicopter and is the gravita-tion constant. Expressions for and can be found in [9]and [11 ] in case of vertical wind gusts, and in Section V-D forlateral wind gusts.

III. DESIGN AND IMPLEMENTATION OF THE CONTROL

A. Approximate Feedback Linearization Control

The expressions of forces and torques (which contain fourcontrols ) are very complex and have strong non-linearities. Therefore, it is appropriate to consider the main rotorthrust and torques as the new vector of con-trol inputs. Then, the real controls can be calculated. The ob-

jective of the flight control is to design an autopilot systemfor the miniature helicopter to let the vertical,

lateral, and longitudinal dynamics track a desired smooth tra-jectories and the desired yaw angle . Thetracking errors and should convergeasymptotically to zero in the presence of wind gusts. The calcu-

lation of the relative degrees gives: asit could be seen in relation (4). The standard helicopter model

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have a dimension and, which implies the existence of an internal dynamic char-

acterized by a 6-D zero dynamics [18]. A simulation study ofthe model (3) shows that the zero dynamics, parameterized by

, are not asymptotically stable. These zero dy-namics are obtained by solving (4) with

. More precisely,induces that andinduces that, which enables us to find . Now reportingthis in the three last equations of (4) provides the zeros dy-namics equations of helicopter. It can also be shown that theexact state-space linearization fails to transform the system intoa linear and controllable system. Hence, it is impossible to fullylinearize the nonlinear system [8]. Neglecting the couplings be-tween moments and forces, an approximated full-state lineariz-able system with dynamic decoupling can be obtained [8].

Starting from (4), and neglecting the small forces acting onhelicopter and for , an

approximate model of the dynamic of translation is obtained

(5)

In order to make the approximate model (5) linearizable, weapply the dynamic extension procedure by adding two integra-tors for the thrust control input . To simplify the expressions,we propose the change of input variables

that gives .We thus consider as the input vector control

, where . Using theinputoutput feedback linearization procedure of the posi-tion , we take the third-time derivative of (5)

(6)

The decoupling matrix has rank 2 only, and therefore is notinvertible. Hereafter, we propose to use dynamic decoupling al-

gorithm, and continue differentiating the position . At last, theiteration ends,as the decouplingmatrix hasfullrank andis invertible (if ). The extended system can be writtenin the following form:

(7)

for which thevectorrelativedegree is . Thestatevector of our extended system can be written as following:

. Its order is 16 and itscontrol vector is . We can rewrite the true system in normal

form using , where is such thatthe transformation define a coordinate changewith the particularity that depends only on and [19].

Define , and, we have a representation of the

full-state model helicopter [see (4)]:

(8)

in which represents the small forces acting onhelicopter and the wind gust forces and torques

and .It appears that the sum of relative degree of our extended

system is 14, while its size is . There is adifference of 2, which corresponds to the dynamics of the mainrotor, which are free and which create dynamics of order 2, butthey arestable. It maybe noted that this persistent zero dynamicsdoes not exist in the helicopter studied in [1 ], [3], and [8] be-cause we consider the helicopter with 7 DOF.

We use the approximate system (7) to find the new controls, which linearize it. We obtain the following equations:

, where . We can thenapply the following tracking control law for the approximatesystem (7), and for the true system (8)

(9)

where and are gain values calculatedby pole placement.

B. Active and Approached Disturbance Rejection Control

Disturbance in a system can be estimated using derivative of

state and control [20]. Numerical differentiation is then nec-essary and can be obtained using feedback stabilization withsliding-mode control [21], high-gain observer [22], [23], or viaintegrations [20]. An other approach consists of using nonlinearobserver [24], [25] to build the states and the disturbances [26],and then, this disturbance estimate can be used by the con-troller to suppress the real disturbance [13], [27]. In this paper,a methodology of generic design is adapted to treat the dis-turbance: the ADRC, which has been originally proposed forsecond-order systems, is described by the following [13], [14]:

(10)

where represents the dynamics of the model and the dis-turbance, is the input of unknown disturbance, is the control

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input, and is the measured output. The system (10) is initiallyincreased

(11)

where and . is treated asan increased state. Here, and are unknown. By considering

as a state, it can be estimated with a state estimator.Han [13] proposed a nonlinear observer for (11)

(12)

where

The observer error is and:

The observer is reduced to the following set of state equations,and is called ESO:

(13)

Active disturbance rejection [13] consist in using directlyin the control to compensate uncertainty or disturbance ofsystem (10): , where is a new control input.The system (10) becomes , and for a pro-portional derivative (PD) controller, ,where is the reference input to track.

Let us now carry out such a disturbance rejection approachto our helicopter model (8) . Reporting (9) in (8) shows that theobserver (13) can be used for dynamic

(14)

[here and in (10)] and the observer in is as

follows:

(15)

where , and for control, an ordinary activecontrol rejection is proposed with a PD controller

(16)

On the other hand, the translation dynamic has an order four

and disturbance does not affect last , but the secondone . To overcome this problem, it

is proposed hereafter, an approximate new active disturbancerejection (AADRC) based on an approximate observer definedas following:

(17)

where . Comparing (17) to (8), it can be noticedthat does not r epresent a n estimation o f , w hichis unknown. That is why the observer (17) is called approachedobserver.Nevertheless, for control law, is used to compensateinfluence of disturbances

(18)

The stability of the perturbed helicopter (8) controlled using ob-server-based control law (16) and (18) in case of smooth trajec-tories and not too large vertical wind gusts has been demon-strated in [11]. A similar demonstration can be done when a lat-eral wind gust is applied. Moreover, the observer parametersare typically tuned by pole placement with and isset to be approximately 10% of the variation range of its inputsignal [5], [11 ].

IV. SIMULATION RESULTS

With Vario helicopter models, good stabilization and trajec-tory tracking with ADRC controls have been proved when ver-tical wind gusts are applied [9], [11]. Hereafter, lateral windgusts are used to test the two proposed robust controls on a Tinyhelicopter. This helicopter is described in next section. The de-sired trajectories were chosen as two set points in hover and avertical helix [28] between 30 and 55 s (see Fig. 2). The initialconditions are rd, and

. The initial value adopted for the main rotor thrust forceis N. For translational dynamics, pole placement is

used so as to fix, for each axis, 4 poles in for approximatefeedback linearization control (AFLC) control and 4 poles infor AADRC control. For rotational dynamics, a pole placementis also designed to fix 2 poles in for AFLC and AADRCcontrol. Moreover, a linear observer is used (i.e., )for AADRC. It is designed by pole placement in order to fix 5poles in for the observers of translational dynamics and 3poles in for the observers of rotational dynamics.

At time s until s, a lateral wind gust is appliedby adding to . is generated as the output of asecond-order filter of natural frequency 1 rad/s to an impulseinput of magnitude N and duration of 4 s (see Fig. 4).This lateral wind gust simulates the effect of a specific fan when

it is manually switched ON and OFF, as it could be seen in nextsection.

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Fig. 2. Desired trajectories.

Fig. 3. Tracking errors: AFLC control and - AADRC control.

Now we show some simulation results. As seen in Figs. 3 and4, both controls (with or without an observer) manage to stabi-lize , and . The difference between the two proposedcontrols appears in Fig. 3, where the tracking errors are less sig-nificant by using the AADRC compared to AFLC. The AADRCcompensates quickly the effect of the disturbance and the smallforces that destabilize the system. One can also observe withthe AFLC static errors in hover ( m in x, m in y,and m in z). These errors are larger than one obtainedwith Vario helicopter [9], [11] because for Vario helicopter thebandwidths of AFLC controllers are higher, which reduce theseerrors. On the other hand, these static errors are canceled whenAADRC is used.

In Fig. 4, it appears that and have larger magnitudeswith AFLC. This fact can be explained because in hover flight,

the steady state of and are greater. As a consequence, tocompensate helicopter weight, a larger is necessary.

V. EXPERIMENT VALIDATION

Our experimental plant (see Figs. 5 and 6) is based on NI6229 (National Instruments) acquisition board used to transmitthe control signals and to get the main rotor angular rotor an-gular speed for the control loop. The flying machine is a minirotorcraft having two rotors (Tiny CP3). Its physical character-istics are given in Table II. The helicopter can move in fourdirections corresponding to ( , and ) axes and it has 5DOF: is considered as the fifth degree of freedom. Movementin , and are possible due to a spherical joint which en-

ables small rotations in and of about and unboundedone in . In -direction, a variation of 50 cm is possible if the

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Fig. 4. Variations of ( AFLC control and - AADRC control), and .

Fig. 5. Tiny CP3 autonomous helicopter experimental platform.

Tiny CP3 is not too overloaded. The measure of yaw angle isobtained using an 3DM-GX2 IMU (Inertial Sensor Unit) fromMicro Strain Inc. This IMU provides yaw, pitch, roll angles,three rates, and three-axis accelerations too. It is fixed on themoving platform and sends its information to the PC computerthrough an RS232C link at 115 200 baud. The whole systemis controlled by a program written in Labview, which calculatesandsends control signals each 20ms tothe 40MHz ESky 6channel radio transmitter. The helicopter has a dc motor, whichenables us to determine by the measure of its dc supply voltagedenoted : , where r/min/V.

A. Identification Experiments

The problem addressed in this section is identification ex-periments to estimate [29][31], the parameters of the dy-namics model. The model of the Tiny CP3 dynamics was de-rived using least-square identification techniques [32][34]. If

denotes the yaw angular velocity of Tiny CP3 helicopter, i.e.,

, then a realistic model of the yaw dynamic can be pro-posed [2], [35], [36]

(19)

where the terms: (1) is the yaw torque divided by , the yaw inertia

moment term around the -axis of the platform;

(2) is the friction provided by the gyro yaw feedback ofTiny CP3 divided by ;

(3) is the main rotor drag torque divided by ;with and . From (19), it can be noticed cou-plings between main rotor drag torque and tail rotor thrust. Thehelicopter control is refreshed each sampling time ms,so as a numerical model of (19) should be introduced. UsingEuler approximation of a derivative [37][39], one can replace

by

(20)

with . A numerical model of (19) can then be obtained

with

(21)

so as

(22)

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Fig. 6. Interaction between the system and the external devices.

Fig. 7. Variations of , identified , and measured in open loop, V, V, and V.

TABLE IISIZE OF THE TINY CP3 AUTONOMOUS HELICOPTER

If , one can write

(23)

with , such as its th row is as follows:

Least-square solution of (23) is then

(24)

For identification of vector , an adequate input has beenchosen: a pseudorandom binary sequence (PRBS) of order 5with a maximum length of 31 (see Fig. 7, where 1 V correspondsto 340 r/min for ).

The parameter vector is then evaluated using (24) and afterthe parameter vector: is deduced from(22)

% % %

Model response and helicopter response can be compared inFig. 7. A small error between the model and helicopter can be

seen. In fact, a relative rms error of 4.3% has been calculated,which is a very satisfying error, since the proposed nonlinear

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Fig. 8. Practical performances of AADRC control. V, V, and V).

TABLE IIICOMPARISON OF VARIO BENZIN TRAINER AND TINY CP3SCALED HELICOPTER CHARACTERISTICS

model (19) is quite simple. One can also notice that is neg-

ative providing a stabilizing effect for yaw dynamic. Moreover,and are similar in amplitude but of opposite sign so as an

equilibrium point can be found when . Moreover, only anaccuracy of about 17% is obtained for parameter , which isnot a problem, as a robust controller will be used to close theloop.

B. Helicopter Parameter Analysis

In Section II, a Tiny CP3 helicopter model is proposed basedon a Vario helicopter model. As a matter of fact, both of themhave two bladed teetering rotors equipped with Bell stabilizerbar. Moreover, as proposed in [35] for the R50 and UH-1H he-

licopters, one can scale the main characteristics using, for in-stance, the ratio between rotor radius of our two helicopters.Such a scaling has been successfully tested on Vario and TinyCP3, as illustrated in Table III. A ratio of 3 is obtained or thscale. This ratio precisely scales the rotor speed and the tail rotorthrust but an inaccuracy of about 30% is obtained for weight,yaw torque, and main rotor thrust. Nevertheless, this impreci-sion of the model scale is comparable with one found in [35]for R50 and UH-1H helicopter, where a ratio of wasobtained.

C. Control of Yaw Angle Using ADRC Technique

As mentioned previously, the developed mathematical model

is intended for the design of control systems. First, we neglectthe friction and the main rotor drag torque terms of (19), since

they are considered here as perturbation for the control loop. Wethen introduce a new control , such as

(25)

A linear perturbed system is obtained

(26)

where represents the disturbance. A numericalobserver is built using (13) with (i.e., a linearobserver is implemented) and Euler approximation of a deriva-tive

(27)

A PD controller is used in order to attenuate the effects ofdisturbance

(28)

The control signal takes account of the terms, which de-pend on the observer . The fourth part , which

also comes from the observer, is added to eliminate the effect ofdisturbance in this system.

D. Experiment Results

In this section, experiment results are presented to illustratethe performance and robustness of the proposed control lawswhen applied on theyawdynamics of Tiny CP3 helicopter underthe influence of lateral wind gust generated by a fan (see Fig. 6).The chosen bandwidth of the yaw closed loop is rad/sand for the observer gains, three poles in are fixed.

A large lateral wind gust velocity is chosen: if velocity in frontand behind tail rotor are, respectively, denoted by and , thenthe induced velocity at the tail rotor plane is

and the tail rotor thrust , where is the density ofair, and is the area of tail rotor disc [17]. In the case, where

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Fig. 9. Observer outputstrajectory and observer errors.

the fan is switched OFF, the measure of air velocities (m/s and m/s) enable us to get m/s and

N.Whileinthecase,wherethefanisswitchedON( m/s and m/s) give us velocity m/sand force N. At last, the difference betweenand provides N as an estimation of rotortail thrust induced by fan when it is switched ON. To generatelateral wind gust, the fan is manually switched ON at sfor a duration of 4 s. As a matter of fact, represents 29%of , which means that this lateral wind gust acts as a signifi-

cant disturbance for Tiny CP3 helicopter. On the other hand, asimple trajectory is proposed to validate the designed control,as shown in Fig. 8. In this figure, one can also see that AADRCcontrol manages to quickly stabilize in presence of wind gust.Moreover, it compensates the effect of the disturbance. Afterthe wind gust is compensated, comes back to its equilibriumvalue.

Fig. 9 shows that the observer succeeds in accurately buildingat one, and the same time, the angle and the disturbance (see

). One can also notice that yaw trajectory errors induced bylateral wind gust are quite similar to one obtained in simulationfor AADRC control, as illustrated in Fig. 3.

VI. CONCLUSION

A perturbed Tiny CP3 helicopter model is developed based ona Vario helicopter model. As a feedback control, a dynamic de-coupling method obtained with an approximate minimum phasemodel named AFLC is proposed. To deal with uncertainty, ver-tical and lateral wind gusts, an approximate disturbance ob-server is added (AADRC). Simulations show that theAADRC ismore effective than the AFLC, i.e., the tracking error are less im-portant in presence of disturbance (small forces, air resistance,vertical, and lateral wind gust). However, in the presence of non-linear disturbances, the system after linearization remains non-

linear. The observer used here overcomes easily these nonlin-earities by an inner estimation of the external disturbances to

impose desired stability and robustness properties on the globalclosed-loop system. The zero dynamics stabilizes quickly withthe two controls.

In order to validate our disturbance observer approach for he-licopter, the yaw control of a Tiny CP3 mounted on an experi-ment platform has been proposed. After identification of somenonlinear model parameters of the yaw axis, the Tiny CP3 is sta-bilized with an AADRC. A lateral wind gust is also used withoutlosing the stability of the whole closed loop. Moreover, in spiteof the use of a quite simple model for yaw axis, identified model

and real helicopter have a similar velocity behavior.

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