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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

Chinese Journal of Aeronautics 25 (2012) 361-371

Aerodynamic Modeling and Parameter Estimation from QAR Data of an Airplane Approaching a High-altitude Airport

WANG Qinga,b,*, WU Kaiyuanb,c, ZHANG Tianjiaoa,b, KONG Yi’nana,b, QIAN Weiqia,b aState Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China bComputational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

cScientific Research Base of Civil Aviation Flight Technology and Safety, Guanghan 618307, China

Received 23 April 2011; revised 24 August 2011; accepted 16 February 2012

Abstract

Aerodynamic modeling and parameter estimation from quick accesses recorder (QAR) data is an important technical way to analyze the effects of highland weather conditions upon aerodynamic characteristics of airplane. It is also an essential content of flight accident analysis. The related techniques are developed in the present paper, including the geometric method for angle of attack and sideslip angle estimation, the extended Kalman filter associated with modified Bryson-Frazier smoother (EKF-MBF) method for aerodynamic coefficient identification, the radial basis function (RBF) neural network method for aerodynamic mod-eling, and the Delta method for stability/control derivative estimation. As an application example, the QAR data of a civil air-plane approaching a high-altitude airport are processed and the aerodynamic coefficient and derivative estimates are obtained. The estimation results are reasonable, which shows that the developed techniques are feasible. The causes for the distribution of aerodynamic derivative estimates are analyzed. Accordingly, several measures to improve estimation accuracy are put forward.

Keywords: civil airplane; aerodynamics; QAR data; aerodynamic modeling; aerodynamic parameter estimation; flight safety;

EKF-MBF method; neural network

1. Introduction1

Since most of civil airplanes are equipped with quick accesses recorder (QAR), a larger number of flight data are recorded and stored. Aerodynamic mod-eling and stability/control derivative estimation using these data is an important technical way to obtain the aerodynamic characteristics of airplane at flight states. Especially for take-off and landing at high-altitude airports, where complex weather conditions endanger the flight safety, QAR data can provide the scientific basis for analyzing the effects of highland weather

*Corresponding author. Tel.: +86-816-2463065.

E-mail address: [email protected]

Foundation item: National Natural Science Foundation of China (60832012)

1000-9361 © 2012 Elsevier Ltd. doi: 10.1016/S1000-9361(11)60397-X

condition upon the aerodynamic characteristics of air-plane. Once flight accident happens, the QAR data can be utilized to check the aerodynamic characteristics of airplane via aerodynamic modeling and parameter es-timation.

In general, aerodynamic parameter estimation of air-plane is performed via special flight tests [1-3]. In these flight tests, the pilots control the airplanes to execute the designed maneuvers, with flight states measured by the onboard sensor systems. In recent years, much at-tention is paid to aerodynamic characteristics identifi-cation and analysis using the flight data recorder (FDR) data, especially in flight accident analysis [4]. Aerody-namic parameter estimation using the QAR data has not been published in literatures up to now. However, many researchers express their interest in this work. It is anticipated that the aerodynamic parameter estima-tion using the QAR data, as a supplementary means for flight accident analysis, becomes an important applica-Open access under CC BY-NC-ND license.

http://creativecommons.org/licenses/by-nc-nd/4.0/

· 362 · WANG Qing et al. / Chinese Journal of Aeronautics 25(2012) 361-371 No.3

tion field, with which researchers can obtain QAR data more conveniently.

Compared to the measurement system used in regu-lar flight tests, the QAR provides less number of dy-namic variables with poorer accuracy and lower sam-pling rate. This brings about many difficulties in aero-dynamic modeling and parameter estimation. It seems inadequate to adopt the traditional aerodynamic mod-els [5-6] (such as the polynomial model and the spline model) and parameter estimation methods [7-9] (such as the maximum likelihood method and the extended Kalman filter method). The present research develops the techniques of aerodynamic modeling and parameter estimation from QAR data, including the geometric method for angle of attack and sideslip angle estima-tion, the extended Kalman filter associated with modi-fied Bryson-Frazier smoother (EKF-MBF) method for aerodynamic coefficient identification, the radial basis function (RBF) neural network method for aerody-namic modeling, and the Delta method for stability and control derivative estimation. The QAR data of some civil airplane approaching a high-altitude airport are

processed, and the aerodynamic coefficients and stability and control derivatives of the airplane at flight states are obtained.

2. QAR Data

QAR, used by civil airplanes for flight inspect, be-comes a scientific means to guarantee flight safety and to improve the running efficiency. The inspect results provide the basis for flight technique check, safety evaluation, accident analysis and airplane maintenance. Since QAR data contain the variables related to aero-dynamic characteristics of airplane, such as kinematical and dynamic variables and control surface deflections, they can be utilized for aerodynamic modeling and parameter estimation.

The QAR data of some airplane contain 360 vari-ables. The variables needed for aerodynamic analysis are as follows: mass characteristics, engine characteris-tics, kinematical and dynamic variables, control surface deflections, atmospheric parameters, and wind field, as listed in Table 1 (In the following data processing, the

Table 1 Variables used for aerodynamic analysis in the QAR data

QAR variables

Type Title of data table Physical meaning SymbolUnit

Resolving power

Sampling rate/

s�1

Mass characteristics Gross weight Mass of the airplane m kg 1 1/64

SELECTED EGT (T495) #1 Temperature of the left engine TEng,l �C 1 1

SELECTED EGT (T495) #2 Temperature of the right engine TEng,r �C 1 1

SELECTED FUEL FLOW #1 Fuel flow of the left engine fuel,lm� kg/h 8 1

SELECTED FUEL FLOW #2 Fuel flow of the right engine fuel,rm� kg/h 8 1

LEFT ENG N2 TACHOMETER Rev of the left engine Eng,lp r/min 0.1 1/4

Engine characteris-tics

RIGHT ENG N2 TACHOMETER Rev of the right engine Eng,rp r/min 0.1 1/4

LONGITUDINAL ACCELERATION Longitudinal acceleration nx 0.002 1

LATERAL ACCELERATION Lateral acceleration ny 0.002 1

VERTICAL ACCELERATION Vertical acceleration nz 0.01 1

CAPT DISPLAY PITCH ATT Pitch angle � (�) 0.1 1

CAPTAIN'S DISPLAY HEADING Yaw angle � (�) 0.3 1

CAPT DISPLAY ROLL ATT Roll angle � (�) 0.1 1

CAPT DISPLAY GROUND SPEED Ground speed V n mile/h 0.5 1

FLIGHT PATH ANGLE Angle of climb �k (�) 0.35 1

TRACK ANGLE TRUE Flight-path azimuth angle �k (�) 1 1

PRES POSN LAT Latitude B (�) 0.001 1

PRES POSN LONG Longitude � (�) 0.001 1

FLIGHT PATH x POS x coordinate of flight path x ft 1/4

FLIGHT PATH y POS y coordinate of flight path y ft 1/4

Kinematical and dynamic variables

ALTITUDE Altitude h ft 1 1

ELEVATOR POSN-LEFT Deflection of the left elevator �e,l (�) 0.1 1

ELEVATOR POSN-RIGHT Deflection of the right elevator �e,r (�) 0.1 1

T.E. FLAP POSN-LEFT Deflection of the left flap �f,l (�) 0.1 1/2

T.E. FLAP POSN-RIGHT Deflection of the right flap �f,r (�) 0.1 1/2

RUDDER POSITION Rudder deflection �r (�) 0.1 1

AILERON POSN-LEFT Deflection of the left aileron �a,l (�) 0.1 1

Control deflections

AILERON POSN-RIGHT Deflection of the right aileron �a,r (�) 0.1 1

TOTAL AIR TEMP Total air temperature T0 �C 0.25 1

BARO COR ALT Corrected baro-altitude hbaro ft 32 1/64

COMPUTED AIRSPEED Computed airspeed Vc n mile/h 0.25 1

CALCULATED MACH Computed Mach number Mac 0.002 1/36

Atmospheric pa-rameters

ANGLE OF ATTACK Angle of attack (�) 1/2

WIND SPEED Wind speed Vw n mile/h 0.1 1/4 Wind field

WIND DIRECTION TRUE Wind direction �w (�) 0.5 1/4

Note: 1 ft=0.304 8 m

No.3 WANG Qing et al. / Chinese Journal of Aeronautics 25(2012) 361-371 · 363 ·

dimension of all variables will be converted to the met-ric system). The 6th and 7th columns list the resolving powers and sampling rates of the variables respectively, which are inferred just from the data.

Aerodynamic modeling and parameter estimation using the present QAR data faces several problems:

1) Different from the measurement system in flight tests, the QAR does not record angular rate data. Aerodynamic moment identification will be based on none but attitude angle data. Furthermore, the meas-urement data of certain variables, such as angle of at-tack and coordinates of flight path, are faulty.

2) The resolving powers of the data are low, com-pared to the variation ranges of variables. The low re-solving powers destroy the measurement accuracy consequentially.

3) In general, the aerodynamic parameter estimation requires that the sampling rates of dynamic variables should be at least 25 times larger than the mode fre-quencies of airplane. The QAR data do not satisfy this requirement. For instance, the longitudinal short-period frequency of the airplane at Mach number Ma=0.3 and

altitude h=4 km is about 0.05 Hz, and the Dutch fre-quency is about 0.12 Hz. The sampling rate of 3 Hz is required, while that of the QAR is no higher than 1 Hz.

4) The aerodynamic modeling and parameter estima-tion requires that the motion modes of airplane are mo-tivated sufficiently through control inputs. But for a civil airplane, the pilot always tries to keep the airplane smooth and steady in flight. For the present QAR data, the modes of the airplane are motivated to some degree due to the disturbance of wind field near the high-alti-tude airport, though insufficiently.

3. Theoretical Development

3.1. Angle of attack and sideslip angle estimation

Angle of attack and sideslip angle are two of most important variables for aerodynamic analysis [10-11]. The QAR records angle of attack, but the measurement data are incorrect. It is necessary to estimate angle of attack and sideslip angle from the related variables’ meas-urement.

In the present research, angle of attack and sideslip angle are calculated from flight path angles, attitude angles and wind field parameters, according to the geometric relation. The problem is formulated in detail as follows:

Step 1 Project the flight velocity (ground speed observation) to the airplane-carried normal earth axis system by

g k k

g k k

kg

cos cos

cos sin

sin

u Vv V

Vw

� ��

�

� ��

(1)

where ug, vg and wg are the flight velocity components

in the airplane-carried normal earth axis system; V is

the flight velocity; �k the angle of climb and �k the flight-path azimuth angle.

Step 2 Project the cross wind speed to the same axis system by

g,w w w

g,w w w

g,w

cos

sin

0

u Vv V

w

��

� ��

(2)

where ug,w, vg,w and wg,w are the wind speed compo-nents in the airplane-carried normal earth axis system;

Vw is the cross wind speed and �w the cross wind di-rection.

Step 3 Compute the relative velocity components of airplane to air by

g g g,w

g g g,w

g g g,w

u u u

v v v

w w w

� ��

� ��

�

�

�

(3)

where gu� , gv� and gw� are the relative velocity com-

ponents of airplane to air in the airplane-carried normal earth axis system.

Step 4 Translate the relative velocity from the air-plane-carried normal earth axis system to the body axis system by

g

, 3 3 g

g

[ ]i j

uuv c vw w

�

� ��

��

(4)

where u� , v� and w� are the relative velocity com-

ponents of airplane to air in the body axis system. The translation matrix is as follows:

cos cos cos sin sin

sin cos sin sin sin sincos sin

[ ] sin cos cos cos3 3

sin cos cos sin sin coscos cos

sin sin cos sin

cij

� � � � �

� � � � � ��

� � � �

� � � � � ��

� � � �

��

� ��

(5)

where � is the pith angle, � the yaw angle, and � the roll angle.

Step 4 Compute angle of attack and sideslip angle by

2 2 2

arctan( / )

arcsin( / )

w uv V

V u v w

�

� ��

� � ��

� ��

� � � �

(6)

where is the angle of attack and � the sideslip an-gle.

It is noted that Eq. (2) does not include the vertical wind effects which are not known. Therefore, the


above algorithm applies only to the cases with slight vertical wind.

3.2. Aerodynamic coefficient identification

The EKF-MBF method [12] is employed to estimate the aerodynamic coefficients.

Each of the six-component aerodynamic coefficients is assumed as a 3rd order Gauss-Markov. The state equations of the problem are composed of the 6-DOF equations and the 3rd order Gauss-Markov.

0 13 1

0 23 2

0 33 3

211 12 13 4

2 221 22 5

231 32 33 6

7

( ) /

/

/

( / ) ( / )

( ) /

( / ) ( / )

( sin cos ) / cos

cos

z x

y

z x

u rv qw X P m g cv pw ru Y m g cw qu pv Z m g c

p A pq A qr L J J A N J J

q A pr A p r M J

r A pq A qr A L J J N J Jq r

q

��

�

�

��

� �

� � � � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � �

�

��

�

�

��

8

9

13 23 33 10

1 11 1 2 12 2 13

1 14 1 2 15 2 16

1 17 1 2 18 2 19

1 20 1 2 21 2 22

1 23 1 2 24 2 25

sin

( sin cos ) tan

( )

, ,

, ,

, ,

, ,

, ,

r

p q r

h uc vc wc

X X X X X

Y Y Y Y Y

Z Z Z Z Z

L L L L L

M M M M M

N

� �

� � � � �

�

� � �

� � �

� � �

� � �

� � �

� �

� � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

�

�

� � ��

1 26 1 2 27 2 28, ,N N N N� � �

��

(7)

where

211

2 212

13

21

22

2 231

232

33

2 2

( ) /

[ ( ) ] /

/

( ) /

/

[ ( ) ] /

( ) /

/

x y z xz

z y z xz

xz x

x z y

xz y

x x y xz

y z x xz

xz z

x z xz

A J J J J J

A J J J J J

A J JA J J J

A J J

A J J J J J

A J J J J J

A J J

J J J J

� � � ��

� ��

� � ��

� � ��

u, v and w are the fight velocity components in the

body axis system; p, q and r the body angular rates; Jx,

Jy and Jz the moment of inertia referred to the body

axes x, y and z respectively; Jxz is the product of inertia

referred to the symmetric plane xOz; P the thrust of

engines; X, Y and Z are the aerodynamic, and L, M and

N the aerodynamic moments in the body axis system;

g0 the acceleration due to gravity; �i (i=1,2,�,28) are

the process noises, and the top mark “ � ” denotes the differential with respect to time.

The observation equations are as follows:

m 0 1

m 0 2

m 0 3

m 4

m 5

m 6

2 2 2m 7

m 8

/( )

/( )

/( )

x

y

z

n X mg

n Y mg

n Z mg

V u v wh h

�

�

�

� � ��

��

� ��

� ��

� ��

(8)

where nx, ny and nz are axial, side and normal load fac-

tor respectively; h is the height; �i (i=1,2,�,8) are measurement noises, and the subscript “m” denotes the measurement.

The aerodynamic force and moment coefficients are then calculated from the state estimation results by

A

/( )

/( )

/( )

/( )

/( )

/( )

A

Y

N

l

m

n

C X q SC Y q SC Z q SC L q SbC M q ScC N q Sb

�

�

�

�

�

�

� ��

��

(9)

where CA, CY and CN are the coefficients of axial, side

and normal forces respectively; Cl, Cm and Cn the coef-

ficients of roll, pitch and yaw moments; m is the mass

of airplane; S the wing reference area; cA the mean

aerodynamic chord, b the wing span; q� the dynamic pressure.

In EKF-MBF method, EKF is implemented in the forward passage for state filtration and MBF is imple-mented in the backward passage for state smoothing. The EKF-MBF algorithm is formulated as follows.

3.2.1. Extended Kalman filter

1) Covariance propagation

T

| 1 1| 1k k k k� � �� P �P � Q (10)

where

1| 1

( )k k

t� �

��fF

xx (11a)

21

( ) ( ( ) )2

t t t t� � � � �� I F F

(11b)

Pk|k-1 is the filter covariance matrix propagated from

instant k�1 to k, Pk�1|k�1 the filter covariance matrix at

instant k�1, Q the process noise covariance matrix, f the function vector in the right of state equations, x the


state vector, xk�1|k�1 the estimated value of state vector

at instant k�1, I the unity matrix, �t the time interval between samples.

2) State propagation

1

| 1 ( ( ), ) dk

k

t

k kt

t t t�

� � �x f x

(12)

where xk|k�1 is the propagated value of state vector from

instant k�1 to k. 3) Kalman gain matrix

T T 1

| 1 | 1 | 1 | 1 | 1( )k k k k k k k k k k k�

� � � � �� K P H H P H R (13)

where

| 1| 1

k kk k

��

��hH

xx (14)

Kk is the matrix of Kalman filter gain, R the measure-

ment noise covariance matrix, h the function vector in the right of observation equations.

4) State update

| | 1 ( )k k k k k k k�� x x K z h

(15)

where xk|k is the estimated value of state vector at in-

stant k, zk the observation vector measured at instant k,

hk the prediction value of function vector h at instant k. 5) Covariance update

| | 1 | 1( )k k k k k k k� �� P I K H P

(16)

where Pk|k is the filter covariance matrix at instant k.

3.2.2. Modified Bryson-Frazier smoother

1) Initialization

The adjoint variables, vector � and matrix , are ini-tialized at t=tm, the last iteration of the EKF by

0

T 1| | 1 | 1 ,mm m m m m m m t t

�� H D r

(17a)

0

T 1| | | 1 | 1 ,mm m m m m m m m t t

�� H D H

(17b)

where

1 T

| 1 | 1 | 1 | 1m m m m m m m m�

� � � �� D H P H R

(18)

rm is the residual generated by the EKF and � denotes

the Kronecker delta function. �=0 if tm is not an ob-servation time.

2) Adjoint variable propagation The adjoint variables are evaluated at time tk by

backward propagation of the following equations from time tk+1.

T

1| 1k k� �� F �

(19a)

T T T

1| 1 1| 1( )k k k k� � � �� F � F �

(19b)

Here F is given by Eq. (11a), however it is calculated for xk+1|k+1. The numerical integration of Eq. (19a) and

Eq. (19b) produces �k|k+1 and k|k+1.

1

| 1 1| 1 dk

k

t

k k k kt

t�

� � ��

(20a)

1

| 1 1| 1 dk

k

t

k k k kt

t�

� � ��

(20b)

3) Adjoint variable update The matrices of adjoint variables are updated at time

tk by evaluation of the following equations.

)( 1|

T

1|

1

1|

T

1|1|| ��

�� kkkkkkkkkkkkkk �KDrDH��

(21)

T| | 1 | 1 | 1

T 1| 1 | 1 | 1

( ) ( )

k k k k k k k k k k

k k k k k k

� � �

��

� � � �� I K H � I K H

H D H (22)

4) State vector smoothing The vector of smoothed state estimates is obtained

by correcting the EKF filter state estimates. The vector of smoothed state variable estimates is given by

kkkkkkkk |EKF

|EKF

|smoother

|ˆˆ �Pxx ��

(23)

and the corresponding covariance matrix is given by

EKF||

EKF|

EKF|

smoother| kkkkkkkkkk P�PPP ��

(24)

3.3. Aerodynamic modeling using RBF network

3.3.1. RBF neural network

In the region of function approximation, the back propagation (BP) network, a kind of multi-layer feed forward neural network, is extensively adopted earlier. In recent years, RBF neural network is introduced, which provides a more efficient tool for function ap-proximating. The RBF network is superior to the BP network in many aspects, such as approximating ca-pacity, taxonomy ability and learning velocity.

The RBF network is a kind of 3-layer feed forward neural network, as shown in Fig. 1. The input layer is composed of source knots. The number of knots in the hidden layer depends upon the object to be modeled. The activation functions of the hidden knots are taken as radial basis functions. For the output layer, the knots are activated by linear functions.

Fig. 1 Schematic of RBF neural network.


For a RBF network with m input knots, n hidden knots and l output knots, the outputs are given by

!�

�n

jjjkk Rwy

1

)(ˆ x (25)

where x is the input vector, ky the kth output, wjk the

link weight between the jth hidden knot and the kth output knot, and Rj the jth radial basis function. Rj is usually taken as the following Gaussian function.

2|| ||( ) exp

jj

jR

"

# $�% &� �% &' (

x cx

(26)

where cj is the center of the jth radial basis function, "j

the width of the j th radial basis function, and || � || the

Euclidian distance between x and cj. The values of ra-dial basis functions are located in the interval (0,1). The closer an input is to a center, the greater the value of the radial basis function is. For a given input, only those centers close to it are activated.

3.3.2. RBF network model of aerodynamics

In quasi-steady cases, the aerodynamic coefficients are nonlinear functions of the variables such as Mach

number Ma, angle of attack , sideslip angle �, roll

rate p, pitch rate q, yaw rate r, elevator deflection �e,

rudder deflection �r, aileron deflection �a, flap deflec-

tion �f.

e f

a r

( , , , , , ) for , ,

( , , , , , , ) for , ,

i i A N mi

i i Y l n

C Ma q C C C CC

C Ma p r C C C C � � � � � �

��

��

(27)

Each of the aerodynamic coefficients is modeled by a multi-input single-output RBF network individually.

The input knots of the RBF network model are (Ma,

, �, q, �e, �f) for a longitudinal aerodynamic coeffi-

cient, whereas (Ma, , �, p, r, �a, �r) for a lateral aero-dynamic coefficient. The output knot is Ci.

3.3.3. Learning algorithm of RBF network

The number of RBFs n, the centers cj and the widths

"j are the governing parameters for a RBF network. If n is too small, the RBF network cannot achieve a sat-isfying fitting precision. On the contrary, the sample data will be over-fitted and accordingly the prediction performance will decrease.

In the present learning algorithm of RBF network,

the number of RBFs n, the centers cj and the widths "j are obtained by clustering all the samples for network

training. The widths "j are then modified by a one- dimensional optimization algorithm [13].

1) Determination of the centers For a RBF network, each input variable may have a

different value region. All the samples for network training must be normalized. Set the set of normalized

samples to be {(xi, yi), i=1,2, ,N}. The number of

RBFs n, the centers cj and the widths "j are determined by the following clustering algorithm.

Step 1 Initially, assign an appropriate value to the radius of clusters d, and mark all samples by “0”, which indicates that the corresponding sample is not included in any super-ellipsoid cluster.

Step 2 Select arbitrarily a sample from those marked by “0” as a new cluster, the center of which is the sample itself. The sample is then marked by “1”, which indicates that the corresponding sample is in-cluded in a super-ellipsoid cluster.

Step 3 Suppose the index of the current cluster is j, and its center is cj. For each of the samples marked by

“0”, calculate its distance to the center cj.

|| ||i i jd � �x x

(28)

If di<d (d is the given radius of clusters), the sample xi is then incorporated into the jth cluster, and is marked by “1”. At the same time, the center of the current clus-ter is updated by

( )

( 1)

1

kj ik

j

kk

� ��

�

c xc

(29)

where k is the number of the samples included up to now in the jth cluster. Repeat this process until no sample can be incorporated into the jth cluster. Then return to Step 2.

Step 4 When all samples are marked by “1”, the clustering process is completed, and the number of

clusters n and their centers cj (j=1,2,�,n) are obtained.

The widths "j=�d, where � is an adjustable parameter. 2) Determination of the widths

As stated above, the widths of RBFs "j=�d. The

adjustable parameter� is optimized by minimizing the following cost function.

2

1 1

ˆ( )N l

ki kii k

E y y� �

� �!! (30)

A one-dimensional searching algorithm is employed to search the optimal solution. In each step of the opti-mization algorithm, the weights wjk need to be deter-mined.

3) Determination of the weights Rewrite Eq. (25) in the form of matrix

RWY ��

(31)

The weight matrix W can be calculated by the least square method as follows:

T1T )( RRRYW ��

(32)

3.4. Stability and control derivative estimation

In Refs. [14]-[15], the Delta method for aerody-namic derivative estimation has been developed. In the Delta method, the stability and control derivatives are defined as the variation in aerodynamic force or mo-ment caused by a small variation in one of the mo-tion/control variables about its nominal value, whereas


all of the other variables are held constant. For exam-ple, if the pitch moment coefficient Cm is the nonlinear

function of Mach number Ma, angle of attack , side-

slip angle �, pitch rate q, elevator deflection �e and flap

deflection �f, as expressed by Eq. (27), the stability

derivative Cm represents variation in Cm with respect

to , whereas all the other variables like Ma, �, q, �e

and �f are held constant. With central difference

adopted, Cm is represented by

e f

e f

[ ( , , , , , )

( , , , , , )]/(2 )

m m

m

C C Ma qC Ma q � � �

� � � � � � �

�� (33)

To estimate Cm via the Delta method, the RBF neu-ral network is first trained to map the network input

file variables Ma, , �, q, �e, and �f to the output file variable Cm . Next, a modified network input file is

prepared wherein values at all time points are per-

turbed by ) � while all the other variables retain their original values. This modified file is now presented to the trained RBF network and the corresponding pre-dicted values of the perturbed Cm are obtained at the

output node. The estimate for the derivative Cm is given by Eq. (33). Each aerodynamic derivative is es-timated at all the time points, thus yielding estimated values of each derivative equal to the number of in-put-output pairs used in training.

4. Results and Discussion

The QAR data of some civil airplane approaching a high-altitude airport are taken as an example. As shown in Fig. 2, the airplane descends from 5 822 m to 3 949 m and decelerates from 172.9 m/s to 87.7 m/s, while

Fig. 2 Flight velocity, altitude and control surface deflec-tion angles.

the tail edge flap deflects to 30� via three successive deflections. The airplane performs the turning maneu-ver by aileron handling.

The present QAR data provide the measurement of angle of attack. However, angle of attack at each time point takes the value of either 109.70 or 108.30, which

is obviously faulty. The variables and � can only be estimated using the method described in Section 3.1. The estimation results are shown in Fig. 3. The solid lines indicate the estimates with the effects of cross wind corrected, while the dashed lines denote the esti-mates without wind correction. The figure shows that the wind field has great effects upon the estimates. Es-pecially for sideslip angle, the maximum correction is

greater than 5�. Therefore, it is absolutely necessary to

correct the effects of wind field in the process of and

� estimation.

Fig. 3 Estimated angle of attack and sideslip angle.

The aerodynamic coefficient identification is im-plemented via the EKF-MBF method, with the obser-vation variables taken as

y=[nx ny nz � � � V h]T

The identification results are shown in Fig. 4, de-noted by the solid lines. The axial force coefficient is not provided because the thrust force of engine cannot


Fig. 4 Results of aerodynamic identification and RBF net-work prediction.

be reckoned accurately from the QAR data. In order to estimate stability and control derivatives,

the RBF neural network models of aerodynamics are first constructed. The network input knots are taken as

the variables Ma, , �, q, �e and �f for longitudinal

aerodynamic coefficients, while Ma, , �, p, r, �a and

�r for lateral aerodynamic coefficients. The flight state variables and the aerodynamic coefficient at each time point form a training sample. The RBF network is trained utilizing all the training samples, and the RBF network models of aerodynamics are then yielded. The dashed lines in Fig. 4 denote the predictions of the RBF network models, which fit the identification re-sults well. It is noted that the fitting degree of aerody-

namic moments is lower than that of aerodynamic forces. The line accelerations reflect the action of aerodynamic forces directly, while the attitude angles reflect the action of aerodynamic moments indirectly. Consequently, the aerodynamic forces are estimated with higher accuracy than the aerodynamic moments.

The stability and control derivatives are estimated at all the time points via Delta method, using the RBF network models of aerodynamics. Figures 5-9 show the estimation results. The following conclusions are drawn from the figures.

1)The signs of the estimates are correct and the quantities are reasonable in total.

2) The estimates of each aerodynamic derivative are distributed in a certain region. The distribution of esti-mates is caused mainly by three factors: the errors in the estimated aerodynamic coefficients, the inadequate motivation of motion modes, and the small amount of available flight data.

3) The relative distribution range of dynamic stability derivative estimates is much larger than that of static stability and control derivative estimates. Of all the components of each aerodynamic moment, the dy-namic derivative contributes the least. As a result, the dynamic derivative estimates are affected more seri- ously by the errors in QAR data than the static stability/control derivative estimates.

4) The aerodynamic derivatives vary slightly with Mach number for Ma<0.55.

Fig. 5 Estimated normal force derivative with respect to angle of attack.

Fig. 6 Estimated side force derivative with respect to side-slip angle.

In order to validate the aerodynamic derivative esti-mation results, numerical flight simulation is carried out using the 6-DOF dynamics equations. The linear


Fig. 7 Estimated pitch moment derivatives.

Fig. 8 Estimated roll moment derivatives.

Fig. 9 Estimated yaw moment derivatives.

aerodynamic models are adopted, where all the aerodynamic derivatives taking their average values over the simulation period. The simulation is performed only for the period of 2 820-2 940 s, when the flight states vary significantly. If the period is lengthened, the integration trends to diverge due to accumulation errors. The initial flight states take the measurement values and/or estimation values. Figure 10 shows the com-parison between the simulation results and the QAR data. The simulation results have the same time-varying trends as the QAR data, though the dis-crepancy is obvious.


Fig. 10 Validation of aerodynamic derivative estimation

results.

5. Conclusions

1) The techniques of aerodynamic modeling and pa-

rameter estimation from QAR data are developed in the present research, including the geometric method for angle of attack and sideslip angle estimation, the EKF-MBF method for aerodynamic coefficient identi-fication, the RBF neural network method for aerody-namic modeling, and the Delta method for stability and control derivative estimation. The application example shows the feasibility of the methods. However, the angle of attack and sideslip angle estimation method applies only to the cases with slight vertical wind.

2) The aerodynamic coefficients and stability and control derivatives of a civil airplane approaching a high-altitude airport are estimated from QAR data. The estimates are reasonable in total, though the aerody-namic derivative estimates are distributed in certain regions.

3) The developed techniques of aerodynamic mod-eling and parameter estimation are also applicable to the analysis of flight test data and flight data recorder data. In extended applications, the methods might be modified appropriately according to the flight data to be processed. For example, the angular rates, when measured, should be supplemented into the observation vector of aerodynamic coefficient identification.

4) The following measures are effective to improve the accuracy of aerodynamic coefficient and derivative estimates: adding variables to be measured (such as angular rates, angle of attack, and sideslip angle), adopting sensors of higher accuracy and sampling rate, motivating the motion modes of airplane via control inputs, and processing as many groups of flight data simultaneously as possible.

Acknowledgement

The authors would like to thank Prof. Yu Jiang in Civil Aviation Flight University of China for providing the QAR data and for his help in data analysis. We would also like to thank Prof. C. Edward Lan in the University of Kansas for the instructive discussion we had about the QAR data and the present paper.

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Biographies:

WANG Qing received his B.S. degree in application mathe-matics from the PLA Information Engineering University in 1987, M.S. degree in aerodynamics from China Aerodynam-ics Research and Development Center in 1990, and Ph.D. degree in flight dynamics from Northwestern Polytechnical University in 1995, and then became a researcher in China Aerodynamics Research and Development Center. His main research interest is in system identification, flight dynamics, and aerodynamics. E-mail: [email protected] WU Kaiyuan received B.S. degree from Northwestern Poly-technical University in 1967, and then became a researcher in China Aerodynamics Research and Development Center. His main research interest are in high speed aerodynamics and, aerodynamics at high angles of attack and flight safety. E-mail: [email protected]

ZHANG Tianjiao received her B.S. degree in aircraft design from Beijing University of Aeronautics and Astronautics in 2007, M.S. degree in aircraft design from China Aerody-namics Research and Development Center in 2010. She is currently a Ph.D. candidate in China Aerodynamics Research and Development Center. Her main research interest is in aerodynamic parameter identification of Flight vehicle. E-mail: [email protected]

Aerodynamic Modeling and Parameter Estimation from QAR Data … · 2016. 12. 2. · civil airplane...

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