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Experimental Wind Field Estimation and AircraftIdentification

Jean-Philippe Condomines, Murat Bronz, Gautier Hattenberger,Jean-François Erdelyi

To cite this version:Jean-Philippe Condomines, Murat Bronz, Gautier Hattenberger, Jean-François Erdelyi. ExperimentalWind Field Estimation and Aircraft Identification. IMAV 2015: International Micro Air VehiclesConference and Flight Competition, Sep 2015, Aachen, Germany. <hal-01204624>

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ExperimentalWindFieldEstimationandAircraftIdentification

CONFERENCEPAPER·SEPTEMBER2015

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Jean-PhilippeCondomines

EcoleNationaledel'AviationCivile

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MuratBronz

EcoleNationaledel'AviationCivile

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GautierHattenberger

EcoleNationaledel'AviationCivile

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Jean-FrançoisErdelyi

PaulSabatierUniversity-ToulouseIII

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Experimental Wind Field Estimationand Aircraft Identification

Jean-Philippe Condomines∗, Murat Bronz∗, Gautier Hattenberger∗, Jean-Francois ErdelyiENAC; UAV Lab ; 7 avenue Edouard-Belin, F-31055 Toulouse, France

Universite de Toulouse ; ENAC; F-31077 Toulouse, France

ABSTRACT

The presented work is focusing on the windestimation and airframe identification based onreal flight experiments as part of the SkyScannerproject. The overall objective of this project isto estimate the local wind field in order to studythe formation of cumulus-type clouds with a fleetof autonomous mini-UAVs involving many as-pects including flight control and energy harvest-ing. For this purpose, a small UAV has beenequipped with airspeed and angle of attack sen-sors. Flight data are recorded on-board at highspeed for post-analyses. An approach based onUnscented Kalman Filter (UKF) is proposed fornonlinear wind estimation. As a first result, windupdraft estimation is highlighted by exploitingrecorded flight test data. In addition to this, amotor test bench has been developed in order toestablish a model of the propulsion system fromwind tunnel experiments. It will be combinedwith a classical aerodynamic model for airframeidentification Preliminary flight results are pre-sented.

1 INTRODUCTION

Small Unmanned Aerial Vehicle (UAV) are now widelyused for atmospheric and meteorological researches [1, 2].They can easily carry compact sensors, but when it comes tomore important payloads like particles and aerosols detectors,the remaining available weight can be limited, unless a biggerairframe is chosen. An important parameter for atmosphericstudies, but also for long endurance autonomous flights [3], isthe observation of the local wind field. Such information canbe used for autonomous soaring for instance. The SkyScan-ner1 project aims at studying and experimenting a fleet ofcoordinated Mini UAVs which adaptively samples cumulus-type clouds. This is involving many aspects including flightcontrol and energy harvesting. The main tackled challengesare a better understanding of the clouds micro-physics, smallairframe optimization, and optimized fleet control.

The presented work, as part of the SkyScanner project, isfocusing on the wind estimation and airframe identification∗firstname.lastname@enac.fr

1 http://www.laas.fr/projects/skyscanner

based on real flight experiments. For this purpose, a smallUAV has been equipped with airspeed and angle of attacksensors. Flight data are recorded on-board at high speed forpost-analyses. Several solutions have been used for wind es-timation [4, 5, 6] and system identification [7, 8]. The pro-posed solution will be based on Unscented Kalman Filter(UKF) [9, 10], widely used in its square-root version form[11, 12, 13].

In addition to the flight data, a model of the propulsionsystem is required in order to evaluate the propulsive powerof the plane based on the flight speed and the consumed elec-trical power directly measured on the battery. A motor testbench have been designed for this purpose, with an automaticrecording sequence controlled by a computer.

In the sequel, first section presents the models and theproblem formulation. Then, the theoretical basis of the esti-mation and identification algorithms are presented. The nextsection details the experimental setup and the instruments em-bedded on the UAV. To conclude, the final part gathers all themotor test bench and the experimental results obtained fromthe flights.

2 PROBLEM FORMULATION

2.1 Wind fieldSmall UAVs are very sensitive to relative high wind gusts

because of their size, hence satisfying the real-life demandsbecomes difficult. In general wind speed is assumed to be aspatially and temporally varying vector field s.t.

w(x,y,z, t) =

wix(x,y,z, t)

wiy(x,y,z, t)

wiz(x,y,z, t)

where the superscript i denotes components expressed in

the inertial frame. A small vehicle flying through this field isinfluenced by three components of the wind field and gradi-ents of the wind field such that :

ddt

w(x,y,z, t) =

wix

wiy

wiz

=

wiζ→x

wiζ→y

wiζ→z

+∇w(x,y,z)

xyz

(1)

where the subscript ζ denotes the time rate of change ofwind velocity at the point (x,y,z). Three component x, y andz represent the velocity of the vehicle with respect to inertial

frame and ∇w(x,y,z) is the gradient of the wind field. As-suming a constant wind field as seen by the vehicle, the lastterm of Eq.1 becomes to zero, however this approximationis only applicable when vehicle velocity is large compared tothe “point” rates of change of wind velocity (e.g, wing span issignificantly larger than tail height, so vertical gradient of thelateral airmass velocity has negligible effect on roll rate). Thispaper considers the simultaneous nonlinear state estimationof aircraft body-axis velocity component and wind velocitycomponent in the North-East-Down (NED) inertial referenceframe using this assumption.

2.2 Vehicle Dynamics and KinematicsIn order to tackle a wide range of applications, various im-

plementations of flight dynamics models, in terms of assump-tions and numerical techniques, therefore exist. To overcomethe difficulty for an UAV to derive a reliable representativeaerodynamic model, they are commonly represented using a6 Degrees of Freedom (DoF) kinematic model (3 DoF corre-spond to the translational motion (VN ,VE ,VD) and the 3 re-maining DoF to the rotational motion (ϕ,θ ,ψ)).

L

D

mg

Tα

β

zb

yb

xb

B

w

B

zb

yb

xb

xi

yi

zi

O

r

va

q−1 ∗ . . . ∗ q

Fig. 1: Reference frames.

Assuming a flat non-rotating Earth the flying rigid bodymotion in turbulent conditions can be located at r with ve-locity vi having components VN ,VE ,VD in an inertial frameI, where xi,yi and zi define unit vector (see figure 1). Usinga standard body-fixed coordinate frame with airmass-relativevelocity va having components u,v,w in the body xb, yb, zb di-rections, respectively, the acceleration of the aircraft in theinertial frame can be mathematically described s.t:

r = vi + w

Developing inertial velocity

r =ddt

va +ω×va +ddt

w

Hence

ddt

va =

uvw

−ω×va−ddt

w (2)

Using angular velocity ω =(

p q r)T and aerodynam-

ics forces X,Y,Z depending on functions of trust T, drag Dand lift L expressed in the body x,y,z directions, the funda-mental principle of dynamics now becomes :

XYZ

+

00

mg

=

m

uvw

− 0 −w v

w 0 −u−v u 0

pqr

− d

dt wixddt wiyddt wiz

(3)

In Eq.3, the latter quantity ddt w is expressed in the inertial

frame and can be converted in the body frame through a Di-rection Cosine Matrix (DCM) Ra

i which is defined by succes-sive rotation of the roll, pitch and yaw angles of the aircrafts.t :

Rai =(Ri

a)T =

cθcψ cθsψ −sθ

sϕsθcψ− cϕsψ sϕsθsψ + cϕcψ sϕcθ

cϕsθcψ + sϕsψ cϕsθsψ− sϕcψ cϕcθ

In the next sections, Galilean transformations will be madeby using a standard quaternionial form, i.e, Ra

i · r = q−1 ∗ r∗q, Ri

a · r = q ∗ r ∗ q−1. Note that symbol ∗ corresponds tothe quaternion product. Using the aforementioned standardquaternional form provides at the same time :

• a global parametrization;

• avoids the mathematical singularities inherent to Eulerangles;

• and is convenient for calculations and simulations.

For more details about formulas used on quaternion in thispaper, see Appendix A. Finally, the state dynamics for thebody-axis velocity states are given by

u =Xm−gsinθ −qw+ rv− wix cosθ cosψ

−wiy cosθ sinψ + wiz sinθ

v =Ym−gsinϕ cosθ + pw− ru

− wix(sinϕ sinθ cosψ− cosϕ sinψ)

− wiy(sinϕ sinθ sinψ + cosϕ cosψ)

− wiz sinϕ cosθ

w =Zm−gcosϕ cosθ +qu− pv

− wix(cosϕ sinθ cosψ + sinϕ sinψ)

− wiy(cosϕ sinθ sinψ− sinϕ cosψ)

− wiz cosϕ cosθ

with wi(·) denotes the rate of change of a component of thewind velocity expressed in the inertial frame.

2.3 Aerodynamic and propulsionAerodynamic lift and drag forces in stability axes can be

written as follow:

L =12.ρ.V 2

a .SrefCL (4)

D =12.ρ.V 2

a .SrefCD (5)

with,

CL =CL0 +CLα .α +CLβ .β +1

2Va

bCLp

cCLq

bCLr

.

pqr

+∑

sfcCLsfc.δsfc

CD =CD0 +CDk.C2L +∑

sfcCDsfc.δsfc

(6)

where α is the angle of attack, β the side-slip angle andδsfc the elevator deflection. Since we are mostly interestedin steady flight conditions, close to straight line without side-slip, and that the effect of the elevator on lift and drag forces issmall, the equations can be reduced for performance analysisto:

L =12.ρ.V 2

a .Sref(CL0 +CLα .α) (7)

D =12.ρ.V 2

a .Sref(CD0 +CDk.C2L) (8)

In steady level flight, the weight is equilibrated by the lift,and the drag by the thrust. As a first approximation, the thrust,or more precisely the propulsive power P (the product of thethrust T by the airspeed Va) can be expressed as:

P = η .Pelec (9)

where, Pelec is the electrical power drown from the batteriesand η is an efficiency coefficient, function of the advanceratio, the propeller and motor characteristics, the Reynoldsnumber and the electrical efficiency of the global propulsivesystem.

A more complex propulsion model [14], with a first orderDC motor model, might be used in later studies in order todefine an analytic description of η .

3 WIND FIELD ESTIMATION AND AIRCRAFT MODELIDENTIFICATION

3.1 Wind field estimationThis problem considers simultaneous estimation of air-

craft body-axis velocity (u,v,w) and wind velocity compo-nents (wix,wiy,wiz). Both process and measurement equa-tions are not dependent on aerodynamic force describedabove. The estimation is performed through a fusion algo-rithm of low-cost inertial sensors used for UAV navigation[12]. The navigation quality is limited by inertial sensors per-formance specifies by the size, power and cost constraintsof the UAV. To recover navigation accuracy using low-costaided-INS (Inertial Navigation System), it is necessary touse, if possible, additional instruments (e.g. magnetometers,barometer, which are used to improve the heading and posi-tion accuracies) and/or nonlinear estimation algorithms to im-prove the flight handling qualities of the aerial vehicle. Thenonlinear state estimation makes use of 2 triaxial sensors plusboth GPS and Pitot tube sensor units which deliver a total of10 scalar measurement signals:

• 3 gyroscopes providing a measurement of the instanta-neous angular velocity vector denoted by ωm ∈ R3 s.t.ωm = [pm, qm, rm]

T ;

• 3 accelerometers giving a measurement of the spe-cific acceleration denoted by am ∈ R3 s.t. am =[amx, amy, amz]

T ;

• 1 GPS unit measuring both position (not used) andvelocity vectors denoted by yV = vi ∈ R3 s.t. vectorvi = [VN ,VE ,VD]

T is used in the observation equations;

• 1 pitot tube sensor providing a scalar measurement ofthe air data denoted by yVa.

All the sensors embedded are low-cost, and therefore haveimperfections. The major error sources in the navigation sys-tem are due to: - all of the disturbances (noises) that affect allthe instruments; - the potential incorrect navigation systeminitialization (e.g. on magnetometers or barometric sensor);- and the inadequacy between the real local Earth’s gravityvalue and the one used for computation. The largest error isusually a bias instability (expressed respectively in deg/hr forgyros and µg for the accelerometers).

All these measurements are obviously corrupted by addi-tive noises for which it appears reasonable to assimilate theirstochastic properties to the ones of Gaussian processes. Theircovariances have been identified in [15] from logged sensordata using the Allan variance method [16]. Moreover, theseerrors correspond to the random nature of wind evolution nec-essary in Eq.2.

Using these values, the state space representation corre-sponding to Ms can be described in a compact form such as:

x = f (x,u) and y = h(x,u)

where: x = [u,v,w,wix,wiy,wiz]T ,u = [ωT

m ,aTm]

T and y =[yT

V ,yVa]T are the state, input and output vectors respectively.

Ms

u = amx−gsinθ + rm · v−qm ·wv = amy +gcosθ sinϕ + pm ·w− rm ·uw = amz +gcosθ sinϕ +qm ·u− pm · vwix = 0 (process)wiy = 0wiz = 0

yV N

yV E

yV D

= q∗

uvw

∗q−1 (measurement)

yVa =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣u

vw

−q−1 ∗

wix

wiy

wiz

∗q

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Obviously, to implement these equation in a discrete-timefilter, a first order discretization is used [17].{

xk = xk−1 +Ts. fc(xk−1,uk−1)+νk

yk = h(xk)+µk

where f is the discrete-time state transition, h is the nonlinearobservation function which depend on DCM through quater-nion and Ts is the sampling time of the system. ν ,µ are thezero-mean Gaussian process noise vectors with covariancematrix, Q, R. Using these relationships, the angle of attackand side-slip are calculated from the body axis velocity com-ponents by

α = tan−1(w

u

)β = sin−1

( v√u2 + v2 +w2

)Since the measurement equation formulation contains non-linear function, a nonlinear state estimation technique suchas EKF [18] or UKF is required. The SR-UKF (Square-RootUKF) was selected for this work due to its ease of implemen-tation and outperforms relative to EKF. Identification of both

sigma points

Y = f(X )weighted sample mean

and covariance

meanSPKF

SPKFcovariance

transformedsigma points

Fig. 2: Sigma point approach.

aerodynamic coefficient CL and CD can be lead by changingthe process equation which is ongoing work.

This study uses flight data collected with a small UAVwhich is susceptible to perform some trajectories which leadsto reduce maneuverability and so unobservability on twocomponents of wind speed. Wind speed unobservabilitypoints out a particular behavior of UKF which is shown by aslow drift on state vector due to using sigmas-points for pre-diction and correction steps. Indeed, in case of strong nonlin-earity, predictions are obtained by weighting the sigma-pointpredictions whose results differs from the direct calculation ofthe prediction from the estimated state (see figure 2). Firstly,effects can be observed in the calculation of the predictedstate where the successive accumulation of these differencescan lead to significant drifts, and secondly, in the correctionof the state from the prediction measure. These effects can beeliminated by performing the calculation of prediction of thestate and measures from the estimated average state, whileretaining the sigma points for the calculation of covariance.

3.2 Aircraft model identificationThe aircraft model identification is using the equation

from section 2.3 during steady flight phases, when the air-speed is almost constant, thus the acceleration is zero. As aresults, lift equals weight and thrust equals drag. Since thepropulsion model was still under investigation at the timeof the first flight experiments, the methodology have beenadapted in order to estimate the drag. The procedure de-scribed in [19] have been used. It consists of performing sev-eral gliding phases at different airspeed and angle of attack inorder to estimate the drag from the glide flight path γ:

tanγ =−DL

=−CD

CL(10)

The identification of the lift coefficient is done usingEq. 7. For each flight phase, the airspeed Va and the angleof attack α are averaged. Then a linear regression is used toestimate the two parameters CL0 and CLα .

In order to identify the drag coefficient, only the glidingphases are considered as stated above and equations 8 and 10are used. Three parameters are then needed, the lift coeffi-cient and the angle of attack that are already computed, andthe flight path angle γ . Since this angle can’t be directly mea-sured, two methods have been evaluated. The first method isusing an angle of attack installed on the UAV and the pitch

angle θ estimated using the IMU sensor. With the kinematicrelation θ = α + γ , the path angle is estimated by averagingover the complete phase. The second method is using the re-lation that the lift over drag ratio is also the ratio between thehorizontal distance and altitude lost during a glide. The mainconstraint is that the experiment needs to be done with almostno wind so that the ground and aerodynamic flight paths arethe same. After estimating the parameter γ , the drag coeffi-cient is computed with Eq. 10, and second order polynomialregression is done between CD and α in order to estimate CD0,CDk.

Experimental results are presented in section 5.1 and theyare showing a good correlation between the two methods.

For a futur work, the UKF estimation algorithm presentedat the previous section will be applied for these parametersidentification.

4 EXPERIMENTAL SETUP

4.1 UAV instrumentationAs mention above, a mini UAV has been equipped with

several sensors in order to make in-flight measurements. Theframe itself is a commercial foam plane Solius from Multi-plex2, a 2.16 meter wingspan motorized glider. The autopi-lot is an Apogee3 board using the Paparazzi system [20, 21],which includes a SD card slot for high speed logging.

It would have been possible to connect all the requiredsensors to the main autopilot, but due to electrical problemswith long cables, it has been decided to split the Data Acqui-sition System (referred as DAQ board) from the flight control(referred as AP board). Hence, a second Apogee board wasused to record the sensors, which is possible since there is nofeedback of the wind estimation to the flight control at thisstage of the project. The DAQ board is already equipped with3-axis gyroscopes and accelerometers, and a low resolutionbarometer. The INS filter [13] used to estimate the positionand orientation of the plane also requires a magnetometer anda GPS, that have been externally mounted.

The SkyScanner project aims at studying the formationof clouds. The meteorological parameters will be measuredusing a dedicated board Meteo-Stick. This board has high res-olution absolute pressure sensor, differential pressure sensor,temperature sensor and humidity sensor. For this study, onlythe differential pressure sensor was used connected to a Pitottube in order to measure the airspeed of the plane.

The figure 3 is showing the final integration of the mea-surement system, with the DAQ and Meteo-Stick stacked, theGPS and the magnetometer at the back.

The main sensor addition is an angle of attack sensor,mounted on the wing close to the Pitot tube. This sensor ismade of a US DIGITAL MA3-P12-125-B4 angular sensor.It is an absolute position sensor using hall effect with 12-bit

2 http://www.multiplex-rc.de3 http://wiki.paparazziuav.org/wiki/Apogee/v1.004 http://www.usdigital.com/products/encoders/absolute/rotary/shaft/ma3

Fig. 3: Integration of the data acquisition system in the noseof the UAV.

internal converter giving less than 0.09◦ of resolution with avery low noise. The vane has been 3D-printed and mounteddirectly on the shaft of the sensor. The figure 4 shows thefinal integration of this two sensors on the wing. The pieceholding them has also been made with a 3D printer.

Fig. 4: Integration of the Pitot tube and the angle of attacksensor on the leading edge of the wing.

The final setup for the SkyScanner project will also in-clude a current sensor in order to measure the electrical powerdrown by the motor that will be used in the aerodynamicand propulsion models, and some additional scientific sen-sors dedicated to the atmospheric research part, such as Liq-uid Water Content sensors, not directly related to the windestimation.

4.2 Motor test bench

In order to integrate the propulsion model to the estima-tion process, it is necessary to establish the relation betweenthe electrical current consumed by the motor, the rotationspeed, the flight speed and the resulting propulsion forces andtorque generated. Since it is not possible to embedded thenecessary sensors to estimate this late parameters in flight,a motor test bench provides the propulsion model based onwind tunnel experiments.

The figure 5 shows the bench. Two force sensors areused to measure the propulsive force and the motor torque.The bench itself is an assembly of 3D-printed pieces and alu-minum bars.

Fig. 5: Final version of motor test bench.

In addition to the mechanical mounting of the motor andits propeller, an electronic board is required for the sensors’signal conditioning. Finally, a myRIO data acquisition boardfrom National Instruments connects them to the lab computer.A graphical interface developed using LabView allows to con-trol the motor PWM command and synchronize all the mea-surements, making the process almost fully automated (thewind tunnel speed is currently control by hand). The figure 6presents the global architecture and wiring of the motor testbench.

Fig. 6: Overview of the motor test bench experimental setup.

5 EXPERIMENTAL RESULTS

5.1 Flight testsA few flight tests using the experimental setup described

in the previous section have already been conducted (see fig-

ure 7).

Fig. 7: Solius glider fully equipped for experimental flight.

Some preliminary results have been analyzed in order toaces the quality of the measurements, especially the angle ofattack sensor since the Meteo-Sick sensors have already beenvalidated in a previous project. The figure 8 shows the goodcorrelation between the variation of the airspeed and the angleof attack (one increasing when the other decrease). Note thatstudy uses angle of attack data collected with a constant offsetwhich can be removed from raw data.

-5

0

5

10

15

20

0 100 200

ang

le o

f att

ack

(d

eg

ree)

or

air

speed

(m

/s)

time (s)

angle of attack (degree)aispeed (m/s)

Fig. 8: Comparison of the angle of attack (red, plain line) andthe airspeed (green, dashed line).

Flight plans In order to perform a correct estimation ofthe wind or the aerodynamic model, it is necessary to per-form appropriate flight patterns. Concerning the wind esti-mation, observability is achieved by changing the flight di-rection. Hence, the chosen flight patterns are circles or smallvariation along a given segment. Figure 9 shows the horizon-tal wind field estimation along the aircraft trajectory. Furtherflight data analysis will be conducted in order to correctlytune the algorithm. The extraction of the vertical componentof the wind will be possible by including the angle of attackmeasurements and not only the airspeed norm as it is cur-rently the case.

An other type of flight pattern have been used in orderto estimate the lift and drag coefficients as a function of theangle of attack. Several gliding phases are done at different

Fig. 9: Estimation of a wind updraft (red) during a glidingphase with confidence bounds (green).

airspeed following the same protocol than [19]. The figure 10is showing four of these flight phases extracted from the com-plete experimental flight.

From the complete flight, four gliding phases and threecruise level flight phases have been selected for their stableairspeed and away from stall point. The figure 11 is a plot ofthe lift coefficient CL over the angle of attack α , using bothcruise and gliding phases. Figure 12 shows the drag coeffi-cient CD over α computed with the two methods as describedsection 3.2. Both methods are giving very similar results,which is validating the identification methodology.

The table 1 summarizes the estimated aerodynamic coef-ficients (with α in degree):

5.2 Motors analysis

The motor test bench have been placed in a wind tunneland the parameters have been recorded at different airspeedfrom 0 (static thrust) up to 22 m/s. The resulting thrust versusRPM is shown on figure 13. We can see that the motor isgenerating a fair amount of static thrust (up to 10 N for a1.5 kg plane) allowing easy take-off. But at higher speed,

Fig. 10: Different gliding flight path.

CL0 CLα

0.2831 0.04119

CD0 CDkmethod 1 0.01848 0.2034method 2 0.01839 0.1912

Tab. 1: Aerodynamic coefficients identification for lift anddrag.

alpha

-4 -2 0 2 4 6 8 10 12

CL

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

gliding phase

level flight

curve fit

Fig. 11: Lift coefficient versus angle of attack.

the motor is not generating positive thrust at low RPM (at 22m/s it needs at least 80% of throttle) since the glider was notoriginally designed for high speed.

The main interest for improving the wind estimation andaircraft identification is to find a simple relation between thepropulsive power and a measurable parameter. The figure 14is showing this propulsive power versus the electrical powerdrown from the battery. This later value can be measured on-board from voltage and current sensors. As we can see, forthe useful flight speeds from 10 to 15 m/s, their is a simplelinear dependency of these two parameters.

CL²

0.1 0.2 0.3 0.4 0.5 0.6

CD

0.02

0.04

0.06

0.08

0.1

0.12

0.14

method1

method2

curve fit

curve fit

Fig. 12: Drag coefficient versus squared lift coefficient, esti-mated with two different methods.

-4

-2

0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000 8000

Thr

ust [

N]

RPM [rev/min]

0 m/s5 m/s

10 m/s13 m/s15 m/s18 m/s22 m/s

Fig. 13: Thrust versus RPM at different airspeed from windtunnel experiment.

6 CONCLUSION

This article has presented the theoretical basis of a windestimation algorithm based on Unscented Kalman Filter. Ex-perimental flights have already been conducted in order toget real data. A foam glider have been equipped with air-speed and angle of attack sensors in addition to the tradi-tional GPS+IMU units needed for the autonomous flight. Thepropulsion model have been identified using a motor testbench in a wind tunnel. Further developments will integratethis model to the aircraft aerodynamic identification processand to the wind estimation.

This work is only a first step in a larger project aiming

-20

0

20

40

60

80

100

-50 0 50 100 150 200 250

Aer

o po

wer

[W]

Electrical power [W]

10 m/s13 m/s15 m/s

Fig. 14: Propulsive power versus electrical power input foruseful flight speeds.

at conducting atmospheric research using a fleet of coopera-tive UAVs. The estimation of the local wind field is thereforean important part of the project, but can be reused in manyother applications such as long endurance flights based onautonomous soaring.

ACKNOWLEDGEMENTS

The SkyScanner project is founded by the RTRA-STAEfoundation.

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APPENDIX A: QUATERNIONS AND ROTATIONS

An unit quaternion provide a convenient mathematicalnotation and a global parametrization for representing orien-tation and rotation of a rigid body in three dimensions. In-deed, for any unit quaternion

q = q0 +q = cosθ

2+usin

θ

2

and for any vector p ∈ R3 the action of the operator

q−1 ∗p∗q = Rq ·p

is associated to a rotation matrix Rq ∈ S0(3) which is a rota-tion of the coordinate frame about axis u = q

||q|| through anangle θ .Note that a vector p ∈ R3 can be viewed as a pure quaternion

whose real part is zero(

0p

). Thus, when the real part is a

scalar denoted p0 ∈ R the quaternion p is given as :

p =

(p0p

)