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Aerodynamic Model Validation of Eurofighter Aircraft

E. Özger1 EADS Military Air Systems, 85077 Manching, Germany

Aerodynamic model validation in the Eurofighter project aims at providing the best possible aerodynamic description of the aircraft. All aspects of aerodynamic model validation from maneuver set-up to parameter estimation and derivation of matching criteria as well as transfer of the estimation results into a consolidated model update are dealt with in this investigation. These aspects form the elements of a chain where the weakest one determines the quality of the overall results. A sensible maneuver set-up should aim for decreasing correlations among the signals. When the parameter estimation process accounts also for existing knowledge coming from wind tunnel data estimations get a firmer basis where also consciousness for matching criteria is a must. Transferring estimation results into a subsequent correction of the existing aerodynamic model should account for both the nature of the estimation model and the applicability area of the estimates. This work concludes with the application of the aerodynamic validation process to Eurofighter data.

Nomenclature bv

= overall correction parameter ADMbv

= aerodynamic model correction parameter

priorbv

= a-priori overall correction parameter

Statebv

= state correction parameter cv,w = correlation coefficient of v and w signals C = correlation coefficient matrix

cvΔ = equation error in roll, pitch and yaw cl = rolling moment coefficient Δclx = rolling moment correction w.r.t. x cm = pitching moment coefficient Δcmx = pitching moment correction w.r.t. x cn = yawing moment coefficient Δcnx = yawing moment correction w.r.t. x dv

= measurement vector fv

= functional of correction function FCD = frequency biased canard doublet FFD = frequency biased flaperon doublet FRD = frequency biased rudder doublet I = inertia matrix lμ = reference cord Ma = Mach number M = fisher information matrix J,N = maximum number of samples N = covariance matrix of measurement p = probability density function P = covariance matrix of parameters

RatioCl,n = residual error in cl,n based on error bound RD = roll doublet RR = rapid rolling s = half span S = reference area SHSS = steady-heading sideslip t = time v,w = arbitrary signals WUT = wind-up turn xcg,ycg = x and y position of centre of gravity yv = measurement α,α& = angle of attack and its rate β = angle of sideslip δ = measurement and model error δ, δ& = flaperon deflection and rate ε = residual error or uncertainty

Boundεv = error bound Δ = difference between actual and trim value η,η& = canard deflection and rate κ = ratio of covariances ωv = rate vector ξ = aileron deflection ζ = rudder deflection ℑ = functional Subscripts/Superscripts 0 = static offset ADM = aerodynamic model ctrl = control surface deflections

1 Flight Test Engineer, Flight Test, Rechliner Str., 85077 Manching, Germany, and AIAA Member.

AIAA Atmospheric Flight Mechanics Conference and Exhibit20 - 23 August 2007, Hilton Head, South Carolina

AIAA 2007-6718

Copyright © 2007 by EADS Military Air Systems. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Figure 1. Eurofighter aircraft

p,q,r = roll, pitch, and yaw rate rqp &&& ,, = roll, pitch, and yaw acceleration

PD = pitch doublet PUSH = push-over q = dynamic pressure

decorr = de-correlated FT = flight test G = gravity related i,j,n = i-th, j-th, n-th sample prior/post = before/after estimation

I. Introduction HE Eurofighter Typhoon aircraft is a fighter of the 4th generation. Its high agility is guaranteed by a complex control system since the configuration with respect to flight mechanics is naturally unstable. The control laws

are based upon the aerodynamic model of the aircraft that is mainly derived by wind tunnel investigations but also verified and corrected due to findings in flight testing. Only a concise description of the aerodynamic characteristics will lead to a high flight performance throughout the flight envelope without compromising safe operations at any flight regime. Therefore the aerodynamic model validation plays a crucial role in the Eurofighter development.

The beginnings of aerodynamic model validation in the Eurofighter project reflected the knowledge and experience of preceding aircraft, namely Tornado in case of EADS Military Air Systems. Therefore, the processes and methods were tailored for the requirements of a stable configuration in terms of flight mechanics. But both the flight envelope and the quality requirements of the Tornado are inferior to the Eurofighter. It took some time to realize the deficiencies in the approach and to take efficient measures to cope with the new challenge posed by Eurofighter. Thus, this paper is the sum of the actual knowledge and reflects the up to date approach.

There is extensive literature in the field of parameter estimation1,2,3,4,5 reflecting the importance of these methods in the development of civil and military aircraft. But aerodynamic model validation encompasses more than just parameter estimation. Maneuvers must be set-up such that the parameter estimation method at hand can be utilized in the most efficient way, namely quantifying aerodynamics unambiguously. Ambiguity in estimation results is produced by correlated flight test signals. Thus, correlations among the excitation signals must be avoided, monitored and controlled if possible. Parameter estimation leads inevitably to an improved matching between the flight test response and the corrected model prediction and therefore to a reduced residual error between them. But who judges whether a certain residual error is tolerable or not? The question can be reduced to finding a reasonable measure for the residual error in order to decide if the parameter estimation produces an acceptable matching. The next challenge is to transfer parameter estimation results of various maneuvers into a model update.

Thus, this paper touches upon the above mentioned aspects of aerodynamic model validation, that are: - maneuver set-up, - parameter estimation, - correlation of signals, - elaboration of matching criteria, - transferring estimation results into an aerodynamic model update.

If any of these non-trivial aspects are not accounted for properly estimation results may not fulfill the quality standards required in the development process of such an aircraft which leads to a waste of highly expensive flight tests.

Time dependent effects pose a further complexity in aerodynamic model validation which is not treated here (see Greenwell's reviews6,7 for introduction). But the paper focuses on the validation of time-invariant aerodynamics since it represents more than 95% of the flight envelope of Eurofighter aircraft and nearly 100% of the flight envelope of typical civil aircraft.

II. Aircraft under Investigation The Eurofighter aircraft (see Fig. 1) is a delta canard

configuration primarily designed for air superiority with strong emphasis for air to surface roles. Its unstable flight mechanical design guarantees high performance which needs control laws based on a concise description of aerodynamics. Since the flight envelope comprises low subsonic up to high supersonic Mach numbers as well as a large range with respect to angle of attack and high dynamics there are a multitude of different aerodynamic effects to be modeled precisely.

T

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The aerodynamic model of Eurofighter is mainly based on wind tunnel data enriched by some flight test based corrections, as it is the case for most of civil and military aircraft. The difference with respect to other aircraft is the requirement for a high quality standard posed to aerodynamic modeling throughout the large flight envelope. Therefore the validation of the aerodynamic model by means of flight testing is a vital issue in the development process.

III. Aspects of Aerodynamic Model Validation The following section reviews all aspects of aerodynamic model validation that proved well in the frame of the

Eurofighter project.

A. Maneuver Set-up In order to extract the aerodynamic characteristics of an aircraft by means of parameter estimation the aircraft

must be excited correspondingly. A variety of maneuvers can be enumerated, such as: - Wind-up turn (WUT): Increasing angle of attack from trimmed Mach number condition up to target

angle of attack in a slow manner at constant Mach number - Push over (PUSH): Decreasing angle of attack from trimmed Mach number condition down to target

angle of attack in a slow manner at constant Mach number - Steady heading sideslip (SHSS): Applying maximum pedal from 1g trimmed condition in both

directions successively to obtain maximum and minimum angle of sideslip in a slow manner at constant Mach number and angle of attack

- Pitch doublet (PD): Applying sinusoidal pitch stick input (half stick, duration approximately 2 sec) at test angle of attack and Mach number

- Frequency biased canard doublet (FCD): Applying sinusoidal canard input (performed automatically on request by flight control system, duration approximately 2 sec) at test angle of attack and Mach number

- Frequency biased flaperon doublet (FFD): Applying sinusoidal symmetric trailing edge flap input (performed automatically on request by flight control system, duration approximately 2 sec) at test angle of attack and Mach number

- Roll doublet (RD): Applying sinusoidal roll stick input (half stick, duration approximately 2 sec) at test angle of attack and Mach number

- Rapid rolling (RR): Applying full roll stick input at test angle of attack and Mach number such that one full roll is accomplished

- Frequency biased rudder doublet (FRD): Applying sinusoidal rudder input (performed automatically on request by flight control system, duration approximately 2 sec) at test angle of attack and Mach number

The first three of the list belong to the group of quasi-stationary maneuvers which give a quick overview on the matching between model and flight test over a large range of angle of attack and of angle of sideslip. But their use for parameter estimation is limited due to the low excitation level, and due to the high correlation between angle of attack/sideslip and control surface deflection signals leading to biased estimates. The risk of correlated signals may also apply partly for the rest of the maneuvers whose excitation level is far larger.

Therefore special attention must be given beforehand whether the relevant signals for parameter estimation become correlated. Here, the correlation coefficient cv,w between two signals v,w over N samples is defined as

( )( )

( ) ( )∑∑

∑

==

=

−⋅−

−−=

N

ii

N

ii

N

iii

wv

wwvv

wwvvc

1

2

1

2

1, with ∑

==

N

iivNv

1

1 and ∑=

=N

iiwNw

1

1 as mean values. (1)

The correlation coefficient matrix of the above presented maneuvers excluding SHSS flown at flight conditions of high altitude, Mach=1.05 and angle of attack α=10° is shown in Table 1 where highly correlated parameter pairs in the various maneuvers with values |cv,w|>0.7 are shown.

For the longitudinal maneuvers (WUT, PUSH, PD, FCD, FFD, FCD+PD, FFD+PD, FCD+FFD) it becomes evident that the correlation between pitch rate q and angle of attack rate α& are always very high making it impossible to separate steady cmq and unsteady α&mc effects. Moreover, correlation between angle of attack α and control surface deflection signals η,δ occur that biases and spoils corresponding estimates. Correlations between the control surface deflection rates can also occur and lead to difficulties in separating the corresponding estimates.

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Maneuver Type |cv,w| > 0.7 WUT (α,δ) (q,α& ) ( δη &&, ) PUSH (α,δ) (q,α& )

PD (α,η) (q,α& ) FCD (α,η) (α,δ) (η,δ) (q,α& ) ( δ&,q ) ( δα && , ) FFD (α,η) (q,α& ) ( δδ &, ) ( δ&,q ) ( δα && , )

FCD+PD (α,η) (q,α& ) FFD+PD (α,η) (q,α& )

FCD+FFD (q,α& ) FRD (p,β) (p,ξ) (β,ξ) RD (p,r) (p,β) (β,ζ) (ξ,ζ) RR (p,r) (β,ζ) (ξ,ζ)

FRD+RD - Table 1. Highly correlated parameters in the various maneuvers

Combining FCD, FFD and PD maneuvers produces better results in terms of correlations where the FCD+FFD in quick succession results in the lowest correlations guaranteeing the best results in terms of unbiased estimates.

The picture is similar for the lateral directional maneuvers (FRD, RD, RR, FRD+RD). Correlations are apparent between the rates p,r , angle of sideslip β and control surface deflections ξ,ζ that makes it difficult, if not impossible, to separate corresponding estimates. Combining the FRD and RD maneuvers in quick succession improves significantly the correlation matrix so that all estimates may become unbiased.

It must be born in mind that aircraft behavior with respect to correlated signals may change depending on different flight conditions (e.g. low altitude-high altitude, subsonic-supersonic) so that such an correlation analysis with offline simulations should be performed before starting individual flight test tasks. Even if the offline simulations utilize a-priori knowledge of the aerodynamic model that is to be validated the correlation analysis gives a good overview of what information content can be expected in the flight data.

In this section the focus is laid upon the issue of correlated signals since it is a prerequisite in designing reasonable and efficient flight testing. There is further research in optimal input design (e.g. see Morelli8) that aims at the best possible information content and excitation level for flight test maneuvers which would go beyond the boundaries of this paper.

B. Bayesian Parameter Estimation 1. Formulation of the Estimation Problem The starting point for the correction of the aerodynamic model is the evaluation of the equations of motion

( ) GIntakeNozzleFT MMMMIdtd vvvvv +++=ω with (2)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅⋅⋅⋅⋅⋅⋅⋅⋅

=sSqclSqcsSqc

M

nFT

mFT

lFT

FT μ

v (3)

where the aircraft inertia I is crudely modelled by a so-called load sheet. The load sheet is a table where the aircraft inertia is given as a function of the measured fuel content. The aircraft motion ωv is measured during flight. The intake momentum IntakeM

v and nozzle moments NozzleM

v of the EJ200 engines are modelled by a thrust-in-flight

deck provided by Eurojet. The thrust-in-flight deck needs measured input parameters such as compressor and turbine rpm as well as pressures and temperatures. The output of the equations of motion are the inertia based moments FTM

v around the aircraft centre of gravity.

In parallel, the existing aerodynamic model (ADM) is fed by the measured parameters such as angle of attack α, angle of sideslip β, Mach number, control deflections δ etc to give predicted forces and moments around the aerodynamic reference centre (see equation 3).

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( )( )( )

ℑ=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ℑℑℑ

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

vv

,...,,,,...,,,,...,,,

δβαδβαδβα

MaMaMa

ccc

c

n

m

l

nADM

mADM

lADM

ADM (4)

The difference between predicted moment coefficients coming from the ADM and “measured” moment coefficients coming from the equations of motion is assumed to be covered by a linear correction model

ζξβ ζξβ ΔΔ+ΔΔ+ΔΔ+ΔΔ+ΔΔ+Δ=− llllrlpllADMlFT cccrcpcccc 0 (5a)

δηαδηα δηαδηα&&& &&& ΔΔ+ΔΔ+ΔΔ+ΔΔ+ΔΔ+ΔΔ+ΔΔ+Δ=− mmmmqmmmmmADMmFT cccqccccccc 0 (5b)

ζξβ ζξβ ΔΔ+ΔΔ+ΔΔ+ΔΔ+ΔΔ+Δ=− nnnnrnpnnADMnFT cccrcpcccc 0 (5c)

with Δα = α(t) - αTrim (same procedure for the other measured values). Equation 5 can also be written as

( )ADMADMFT

ADMnFT

mADMmFT

lADMlFTbfccc

cccccc

vvvvv=Δ=−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

with (6)

,...,,,,,( 0 ζξβ llllrlplADM ccccccb ΔΔΔΔΔΔ=v

,...,,,,,,,... 0 δηαδηα &&& mmmmqmmmm cccccccc ΔΔΔΔΔΔΔΔ (7)

Tnnnnrnpn cccccc ),,,,,... 0 ζξβ ΔΔΔΔΔΔ

Additionally, measured and modelled parameters such as α, β, Ma, xcg, ycg, thrust are corrected by δα, δβ, δMa, δxcg, δ ycg, δthrust estimated in parallel with ADMb

v to account for measurement errors and modelling deficiencies.

These parameters form the state correction vector Statebv

( )TcgcgState ThrustyxMab δδδδδβδα ,,,,,=v

. (8)

Together with the parameters of the correction model ADMbv

state correction parameters Statebv

form the overall parameter vector

⎟⎟⎠

⎞⎜⎜⎝

⎛=

State

ADMbb

b v

vv

. (9)

to be estimated. 2. Introducing the Estimator Parameter estimation can be regarded as the process to determine the most probable parameter value to minimise

the error between the real world and a model of it. Thus, the estimation problem can be defined as the maximisation of a probability density function of the parameter to be estimated )|( ybp vv

with bv

being the overall parameter vector and yv defining the measurement.

Applying Bayes’ rule )|( ybp vv can be further split to

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Probability Density Function

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

-6 -4 -2 0 2 4 6

Deviation from Expected Value in sigma

Prob

abili

ty D

ensi

ty

2σ Value2σ Value

Lower/Upper Tolerance

Figure 2. Probability density function of an element of, for example, the parameter vector b

v and its related

tolerances according to its 2σ value

)(

)()|()|(yp

bpbypybp v

vvvvv ⋅= (10)

with )|( bypvv being the probability density function of the measurement yv given the parameter vector b

v and

)(bpv

being the probability density function of the parameter estimate. The probability density function of the measurement )(yp v plays no role since it is not a function of the parameter vector b

vto be estimated. It is therefore

omitted in the following. The probability density functions are assumed to be Gaussian distributions that are further described in the following:

The probability density function of the measurement

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−Δ−Δ−= − )()(2

1exp2

1)|( 1 bfcNbfcN

byp T vvvvvvvv

π (11)

with cvΔ being the difference between the moment coefficients calculated by the equations of motion and the predicted ones of the ADM and f

r being the linear correction model in terms of the parameter vector b

v. The width

of the Gaussian distribution is determined by the covariance matrix N which is evaluated by the covariance values of all measured parameters contributing to the equations of motion and the ADM (see Soijer9 and Özger10,11 for details).

The probability density function of the parameter

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= −

priorT

prior bbPbbP

bpvvvvv 1

21exp

2

1)(π

(12)

with priorbv

being the a-priori mean values of the parameter vector bv

to be estimated that are, due to the difference approach chosen here, equivalent to zero. Here also, the width of the Gaussian distribution is defined by the covariance matrix P that is determined by the tolerances (= a-priori cov( b

v)) imposed on each parameter of the

vector bv

to be estimated (see Soijer9 and Özger10,11 for details). The tolerances of the elements of the parameter vector b

v are equivalent to the 2σ values of their probability density functions, denoting the probability of

exceedance to 0.044 (see also Fig. 2). These tolerances form the basis of the a-priori knowledge on the aerodynamic model.

It is clear that the combined probability density function )|( ybp vv

consists of two parts, namely the contribution of the measurement )|( byp

vv and the contribution of the a-priori knowledge )(bp

v.

The probability density function of the measurement has its maximum values where the error between the measured value cvΔ and the correction model f

r has

its minimum whereas the probability density function )(bp

v has its maximum where the

difference between a-priori value and estimation is zero. Any parameter estimate will be an optimal trade-off between the two contributions, namely measurement and a-priori knowledge. This fact is graphically displayed in Fig. 3 where the combined probability density function is shown as a 3D hill structure. The parameter axis is denoted by b and the measurement axis by y. Determining the most probable value in terms of measurement and a-priori values is shown as the evaluation of the maximum value along the measurement line.

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Figure 3. Combined probability density function p(b|y) for Bayesian estimation

Figure 4. Combined probability density function p(b|y) for Maximum Likelihood estimation

The difference to Maximum Likelihood is evident since there exists no a-priori knowledge on the parameter to

be estimated so that )(bpv

is a constant and the parameter estimation is reduced to the optimisation of )|( bypvv .

Therefore any parameter estimate is only optimal with respect to the actual measurement discarding current knowledge on the model. This can also be shown graphically with a 2D hill structure representing the “combined” probability function (see Fig. 4).

Here again, the most probable value in terms of measurement and a-priori probabilities is determined as the evaluation of the maximum value along the measurement line.

The difference between MLE and BE is evident when considering measurements with low information content. Then the measurement line cutting the hill is more or less parallel to the parameter axis b. In case of MLE the corresponding parameter b

v is a very large value since the cutting point of ridge of the hill and the measurement line

lies far downstream. The picture is different in case of BE. The 3D hill structure guarantees that the most probable parameter b

v is bounded and equals more or less to the a-priori value since, in case of a parallel measurement line,

the parameter estimate is more determined by the a-priori probability function. 3. A-posteriori Covariances Starting from the fisher information matrix

iTii dNdM

vv 1−= with the measurement vector (13)

( )iiiiiiii qd δηαδηα &&&v

ΔΔΔΔΔΔΔ= 1 in case of longitudinals and (14a)

( )iiiiii rpd ζξβ ΔΔΔΔΔ= 1v

in case of lateral/directionals and (14b)

where i denotes the i-th sample. Thus, the a-posteriori covariances can be determined according to

a-posteriori1

1

1)cov(−

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑

N

iiADM PMb

vover all N samples. (15)

The above determined covariances define minimum optimistic parameter bounds. But since the relative values of the covariances of the estimated parameters among each other are utilized in the following (see chapter E) this approach can be justified. Morelli12 proposes a method to determine covariances of the estimated parameters giving more accurate parameter bounds.

C. Dealing with Correlations In section A and B of this chapter it is shown that a proper maneuver set-up will guarantee maximum

information output in terms of nearly unbiased parameter estimates. Due to constraints coming from various fields like clearance issues or lack of information in the past flight test data may not be generated in the optimal way with respect to the issue of correlations.

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Therefore a new approach has been elaborated to recover correlated flight test data. Assume two signals Δv=v-vTrim and Δw=w-wTrim within a maneuver to be highly correlated. Their corresponding estimates cv and cw cannot be determined since any combination of cv and cw produces a similar acceptable matching. The situation can be characterized as having two unknowns but just one equation. If a second maneuver, at the same conditions, is added that has also a high correlation of Δv and Δw, but dissimilar to the previous one, the correlation can be dissolved and the estimates will get unbiased.

This can be expressed in mathematical terms. For both maneuvers where there is a high correlation between Δv and Δw a functional relationship can be determined, namely:

Maneuver 1: Δv1 = a ⋅ Δw1 (16a)

Maneuver 2: Δv2 = b ⋅ Δw2 (16b)

by estimating a and b. The correction model approach turns out to be

Maneuver 1: cv1 Δv1 + cw1 Δw1 = (cv1 a + cw1) Δw1 = cvw1 Δw1 (17a)

Maneuver 2: cv2 Δv2 + cw2 Δw2 = (cv2 b + cw2) Δw2 = cvw2 Δw2 (17b)

The estimator has determined "biased" values of cv and cw for both maneuvers from which cvw1 and cvw2 can be computed easily together with a and b according to eqn. 17. In order to dissolve the correlations a linear equation system is formed

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⇔⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⇔

=+=+

2

1

2

1

2

111

vw

vw

w

v

vw

vw

w

v

vwwv

vwwvcc

cc

Acc

cc

ba

ccbcccac

. (18)

that can be solved for cv and cw. Mathematically, eqn. 18 can be understood that, under the assumption of highly correlated signals Δv and Δw, the solution for cv and cw lies for both maneuvers on a line with different gradients a and b. If there is just one maneuver any combination of cv and cw lying on that line will solve the estimation problem. Thus, the solution cv and cw should comply with both maneuvers meaning that the crossing point of both lines determines the "unbiased" estimates cv and cw.

The condition of the matrix A gets better if the correlation of Δv and Δw in the two maneuvers are as dissimilar as possible. If the correlations within both maneuvers are similar the equation becomes ill conditioned and the crossing point cannot be well determined. The best results can be obtained if during one maneuver the signals are positively correlated and during the other one negatively, or vice versa. This approach can be readily enlarged to more than two dimensional problems, though it will get more and more difficult to have a well conditioned matrix A .

-3

-2

-1

0

1

2

3

4

-14 -12 -10 -8 -6 -4 -2 0 2

canard [deg]

flape

ron

[deg

]

Figure 5. Cross-plot of canard and flaperon signals for maneuver 1 (PD, corr(η,δ)=-0.80), blue dots representing samples, black solid line estimated line with gradient a

-2

-1

0

1

2

3

4

-12 -10 -8 -6 -4 -2 0 2 4

canard [deg]

flape

ron

[deg

]

Figure 6. Cross-plot of canard and flaperon signals for maneuver 2 (FCD, corr(η,δ)=0.77), blue dots representing samples, black solid line estimated line with gradient b

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man#1 man#2 corr(η,δ) -0.80 0.77 Δcmη -0.0043 0.0017 Δcmδ 0.0013 0.0046 εm 0.0019 0.0018

Δcmη,decorr -0.0002 -0.0002 Δcmδ,decorr 0.0091 0.0091 εm,decorr 0.0020 0.0018

Table 2. Correlation, estimation and residual error of maneuver 1 and 2 before and after dissolving correlations

The following example of Eurofighter flight test data will show how the approach applies. Two maneuvers at similar conditions (high altitude, Mach≈0.83, α≈14°) are considered where maneuver 1 is a PD and maneuver 2 is a FCD. The correlation analysis for both maneuvers shows clearly that canard and flaperon signals are strongly correlated with each other (see cross-plots in Fig. 5 and 6).

Table 2 summarizes the relevant data of maneuver 1 and 2 (correlation coefficient, corresponding estimates and residual error) as well as the recovered data by means of above presented approach (subscript decorr). It becomes clear that the residual error changes only slightly by the de-correlation process.

Although this approach is convincing in its results the requirement for maneuvers flown at very similar conditions having large dissimilar correlations in the same signals poses a heavy constraint, particularly for more than two correlated signals. Moreover, this approach is just a fall-back solution in case the maneuver set-up has failed. Therefore it does not replace a sensitive maneuver set-up.

D. Elaboration of Matching Criteria How good the model predicts the aerodynamics of the aircraft can be

seen by means of the equation error cvΔ during the maneuvers. Their root mean square value, defined as

∑=Δ=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

N

ii

n

m

lc

N 1

21 vv

εεε

ε (19)

is a well-known value to measure the residual error. But how much error is tolerable? The question can be reduced to finding a scale with which the residual error can be compared and judged. To obtain this, the uncertainties in the measurement of the parameters such as the rates, rate accelerations and control surface deflections will be accounted for as are summarized below:

- maximum measurement error in control surface deflections δctrl = ±0.5° - maximum measurement error in pitch and yaw rate δq=δr=±0.4°/sec - maximum measurement error in roll rate δp=±1.0°/sec - maximum measurement error in pitch and yaw acceleration q&δ = r&δ =±1.0°/sec2 - maximum measurement error in roll acceleration p&δ =±1.6°/sec2

Three main error contributions are identified, namely coming from the measurement uncertainty in control surface deflections, rates and accelerations. Together with the inertia modelling and knowledge of the aerodynamic model they are determined in the following way.

Contribution of the control surface deflection measurement uncertainty

ctrl

nn

mm

ll

ctrl

cc

cc

cc

δε

ζξ

δη

ζξ

⋅

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+

+

+

=v evaluated at trim conditions. (20)

Contribution of the rate measurement uncertainty

( )( ) ⎟

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⋅+

⋅+

⋅

=

VsrccVl

qcc

Vspc

nrnp

mmq

lp

rate

δ

δ

δ

ε μα&

v evaluated at trim conditions. (21)

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Contribution of the acceleration measurement uncertainty

qS

srIlqIspI

zz

yy

xx

accel1

⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅⋅⋅⋅⋅⋅

=&

&

&v

δδδ

ε μ evaluated at trim conditions. (22)

From these contributions the error bound is determined according to

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

++

++

++

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=2,

2,

2,

2,

2,

2,

2,

2,

2,

,

,

,

accelnratenctrln

accelmratemctrlm

accellratelctrll

Boundn

Boundm

Boundl

Bound

εεε

εεε

εεε

εεε

εv . (23)

Comparing the residual error εv with the corresponding error bound Boundεv will yield a better judgment with

respect to an acceptable or unacceptable matching by attributing a certain meaning to the residual error εv .

E. Updating the Aerodynamic Model After having produced flight test in best possible quality (see section A) and determined the estimates (see

sections B and C) that produce an acceptable matching between flight and model (see section D) the results will be cast into an update of the model. The procedure to accomplish this task is presented in this section.

Starting from eqn. 5 non-linear functions ℑ for each maneuver and each estimate are generated along the corresponding maneuver path. This is shown for the rolling moment modeling. The approaches for pitching and yawing moment modeling are similar.

( ) 00 ,,, lnnnnjcl cMach Δ=ℑ δηα (24a)

( ) lpnnnnnjclp cpMach Δ=ℑ ,,,, δηα (24b)

( ) lrnnnnnjclr crMach Δ=ℑ ,,,, δηα (24c)

( ) ββ βδηα lnnnnnjcl cMach Δ=ℑ ,,,, (24d)

( ) ξξ ξδηα lnnnnnjcl cMach Δ=ℑ ,,,, (24e)

( ) ζζ ζδηα lnnnnnjcl cMach Δ=ℑ ,,,, with (24f)

j denoting the j-th maneuver (maximum number of maneuvers being J) and αn, Mach n, η n, δ n, p n, r n, ξ n, ζ n the n-th sample (of maximum N samples) along the path of the j-th maneuver. First, hyper-cubes are formed with respect to the relevant parameters (α, Mach, η, δ in case of ℑcl0) over the entire flown parameter space. They can be regular with equal step size in the relevant parameters or unstructured. The data of the functions ℑj are each re-assigned to the hyper-cubes accounting additionally for the ratio of prior to post covariance that serves as weighting. The calculation for each hyper-cube is shown in the following for ℑcl0

( )( )

∑

∑ ∑

=

= =ℑ⋅

=ℑJ

jjj

J

j

N

nnnnn

jclj

iiiicl

N

Mach

Mach

j

1

1 10

0

,,,

,,,κ

δηακ

δηα with (25a)

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11

( )( ) jl

jlj cposterioria

cprioria

0

0

cov

cov

Δ−

Δ−=κ (25b)

where κj is the ratio of the covariances (a-priori/a-posteriori) of the estimate before (meaning the tolerances comprised in the covariance matrix P as defined in eqn. 12) and after estimation (as defined in eqn. 15) of the j-th maneuver. The summation in eqn. 25a spans over J maneuvers and Nj samples within the j-th maneuver where only the samples are considered that lie within the i-th hyper-cube. This procedure is repeated accordingly for the rest of the functions.

The result of the above presented procedure are functions ℑ describing hyper-surfaces in terms of their arguments. These hyper-surfaces can be smoothed if needed.

IV. Application to Flight Test Maneuvers The above described approach is readily applied to Eurofighter flight test data in order to present the efficiency

of the method. For this, lateral-directional maneuvers, namely FRD+RD, are used which are flown at high altitude, transonic Mach number and low to medium angles of attack, see Fig. 7. It is the aim to improve aerodynamic modeling in the roll and yaw axis at these conditions.

The linear correction model approach applied to the correction derivatives Δcl0, Δclp, Δclr, Δclβ, Δclξ, Δclζ, Δcn0, Δcnp, Δcnr, Δcnβ, Δcnξ, Δcnζ (eqn. 5) together with the Bayesian estimator produced estimates that improved the matching between flight test and model correction significantly. This can be seen in Fig. 8 where the residual error of rolling and yawing moment before and after estimation is referred to the error bounds in order to non-dimensionalize it (see eqn. 26 for derivation).

RatioClprior = εl,prior/εl,Bound and RatioClpost = εl,post/εl,Bound (26a)

RatioCnprior = εn,prior/εn,Bound and RatioCnpost = εn,post/εn,Bound (26b)

-5

0

5

10

15

20

0,9 0,95 1 1,05 1,1 1,15 1,2 1,25

Mach

Ang

le o

f Atta

ck

Figure 7. Testmatrix of investigated maneuvers in terms of angle of attack and Mach number

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

RatioPrior

Rat

ioPo

st

RatioClRatioCn

Figure 8. Residual error in rolling and yawing moment referred to the error bounds before and after estimation

Before estimation the residual error referred to the error bound averaged over all maneuvers has a value of Average(RatioClprior)=1.3 and Average(RatioCnprior)=1.4 meaning that the model error before estimation is 30% in case of the rolling moment, and 40% in case of the yawing moment above the error bounds. After estimation these values reduce to Average(RatioClpost)=0.73 and Average(RatioCnpost)=0.75.

Concerning the parameter estimates only the lateral stability correction derivative Δclβ and the rudder efficiency correction Δcnζ are shown in Fig. 9 and 10 with respect to angle of attack and Mach number, as an example. Moreover the ratio of post to prior covariances (κ-1) of these estimates are shown in Fig. 11 and 12 in terms of angle of attack and Mach number. It becomes clear that the covariances are reduced significantly to less than 30% of the prior covariance value meaning that the a-posteriori knowledge of the parameters is increased with respect to the a-priori knowledge.

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-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

-5 0 5 10 15 20

AoA

DC

lbet

aMach=0.95Mach=0.98Mach=1.05Mach=1.1Mach=1.2

Figure 9. Lateral stability correction derivative Δclβ versus angle of attack and Mach number

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

-5 0 5 10 15 20

AoA

DC

nzet

a

Mach=0.95Mach=0.98Mach=1.05Mach=1.1Mach=1.2

Figure 10. Rudder efficiency correction Δcnζ versus angle of attack and Mach number

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

-5 0 5 10 15 20

AoA

1/ka

ppa(

DC

lbet

a)

Mach=0.95Mach=0.98Mach=1.05Mach=1.1Mach=1.2

Figure 11. Ratio of post to prior covariance of lateral stability correction derivative Δclβ versus angle of attack and Mach number

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

-5 0 5 10 15 20

AoA

1/ka

ppa(

DC

nzet

a)

Mach=0.95Mach=0.98Mach=1.05Mach=1.1Mach=1.2

Figure 12. Ratio of post to prior covariance rudder efficiency correction Δcnζ versus angle of attack and Mach number

The procedure for updating the aerodynamic model presented in section III,E is applied in the following to all investigated derivatives (Δcl0, Δclp, Δclr, Δclβ, Δclξ, Δclζ, Δcn0, Δcnp, Δcnr, Δcnβ, Δcnξ, Δcnζ) producing corresponding correction functions ℑ in terms of angle of attack and Mach number. Only for the lateral stability Δclβ and rudder efficiency Δcnζ correction the corresponding correction functions are shown in Fig. 13 and 14, as an example.

Angle of Attack

-20

0

20

40

Mach

00.5

11.5

2

Fclb

-0.15

-0.1

-0.05

0

0.05

0.1

X

Y

Z

Figure 13. Correction function on lateral stability ℑclβ(α, Mach) derived according to procedure in section III,E

Angle of Attack

-20

-10

0

10

20

30

40

Mach

00.5

11.5

2

F cnz

-0.04

-0.02

0

0.02

X Y

Z

Figure 14. Correction function on rudder efficiency ℑcnζ(α, Mach) derived according to procedure in section III,E

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The error statistics of the correction functions reveal that the matching between flight test and model after correction approach can be significantly improved. After generating the correction functions, the residual error referred to the error bound averaged over all maneuvers has a value of Average(RatioClcorr)=1.01 and Average(RatioCncorr)=0.97 meaning that the model can be corrected to an acceptable level in terms of the error bounds. Of course, these values are still higher than the values after estimation Average(RatioClpost)=0.73 and Average(RatioCnpost)=0.75. But it has to be born in mind that the estimation values are optimal with respect to each individual maneuver, producing the smallest residual error, and the correction function must comply with all maneuvers so that it is just an optimal compromise.

V. Conclusion Aerodynamic model validation is a core issue in the development of civil and military aircraft. Therefore special

attention must be drawn on how maneuvers are set-up, how the aerodynamic characteristics are estimated, when the matching has an acceptable level, and how these results are transferred to a sensible aerodynamic model update. That these issues are not trivial engineers have experienced during the Eurofighter project. Thus, a systematic and sound approach has been developed at EADS Military Air Systems addressing them appropriately. This approach is presented in detail in the frame of this paper.

The main outcomes of this investigation can be summarized in the following points. The avoidance of correlations among the signals serving for estimation play a major role in the design of the maneuver set-up. The correlations may spoil and bias the parameter estimation results leading to false conclusions.

The estimation procedure for extracting aerodynamic characteristics should not only determine the corresponding values but also account for existing a-priori knowledge and be able to measure the increase in the a-posteriori knowledge of the derivatives.

The decision whether a residual error between flight test and model (before/after estimation) is acceptable is obtained by accounting for the so-called error bounds. These error bounds are determined by means of the existing aerodynamic model, the inertia modeling and the knowledge on the uncertainty of the flight test instrumentation.

The transfer of the estimation results into an update of the aerodynamic model can only be an optimal compromise with respect to the performance of the individual estimation results for each maneuver. Nonetheless, the approach proposed in this investigation and applied to Eurofighter flight data showed a significant improvement in the matching between flight test and corrected model and an acceptable level of the residual error with respect of the corresponding error bounds.

Acknowledgments The author would like to thank Charalambos Kifonidis of EADS Military Air Systems for providing offline

simulations for maneuver correlation analysis.

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presented in the AIAA Atmospheric Flight Mechanics Conference and Exhibit, August 2004, AIAA-2004-5276 7Greenwell, D. I., "Comparative Evaluation of Unsteady Aerodynamic Modelling Approaches", presented in the AIAA

Atmospheric Flight Mechanics Conference and Exhibit, August 2004, AIAA-2004-5272 8Morelli, E. A., "Flight Test Validation of Optimal Input Design and Comparison to Conventional Inputs", presented in the

AIAA Atmospheric Flight Mechanics Conference and Exhibit, August 1997, AIAA-1997-3711 9Soijer, M. W., "Bayesian Equation-Error Aerodynamic Model Validation and Refinement", Journal of Aircraft, to be

published 10Özger, E., and Meyer, E., "Aerodynamic Model Validation at EADS-M", DGLR Conference Paper 2005-220, September

2005 11Özger, E., "Investigation on the Influence of Time Shifts of Measured Input Signals on Parameter Estimation Results",

DGLR Conference Paper 2006-072, November 2006 12Morelli, E. A., "Practical Aspects of the Equation-Error Method for Aircraft Parameter Estimation", presented in the AIAA

Atmospheric Flight Mechanics Conference and Exhibit, August 2006, AIAA-2006-6144