Estimation of Longitudinal Derivatives of Hansa-3
Aircraft
Sanjay Singh
Amity Institute of Aerospace Engineering, Amity University, Noida-201303
ssingh10@amity.edu
ABSTRACT
A new Feed Forward Neural Network (FFNN) based method
is proposed to extract aerodynamic parameters from flight
data of test aircraft. The proposed method (Modified Delta
method) draws its inspiration from FFNN based the Delta
method for estimating stability and control derivatives. The
neural network is trained using differential variation of
aircraft motion/control variables and coefficients, as the
network inputs and outputs respectively. The trained neural
network is then presented with a suitably modified input file
and the corresponding predicted output file of aerodynamic
coefficients is obtained. An appropriate interpretation and
manipulation of such input-output files yields the estimates of
the parameter. The method is applied on real flights data of
HANSA-3 aircraft. Further, a new method based on FFNN to
validate the extracted aerodynamic model has been also
proposed, which by passes the requirement of solving
equations of motion.
Keywords
Longitudinal,
Aerodynamic,
Parameters, Validation.
Mathematical
Model,
INTRODUCTION
A new thrust area is emerging in the area of aircraft
aerodynamic
modeling
and
parameter
estimation:
development of techniques using artificial neural networks
(ANN) for flight vehicle identification. Recently artificial
neural networks modeling has been attempted for aircraft
dynamics where aircraft motion variables and control inputs
are mapped to predict the total aerodynamic coefficients [1-4].
In the past, the most widely used parameter estimation
methods have been equation error method, output error
method and filter error method. Application of these methods
requires a priori postulations of an aircraft model. On the
other hand, a class of neural networks called the feed forward
neural networks (FFNNs) work as a general function
approximators, and are capable of approximating any
continuous function to any desired accuracy by an appropriate
network structure [5]. This ability of FFNNs has been utilized
to model aircraft dynamics. However, the FFNNs lead to a
black-box type of modeling wherein no physical significance
can be attached to either the network structure or the network
weights [2]. The recent interest in, and fascinating with the
evolving applications of ANNs to diverse fields such as signal
processing, pattern recognition, system identification and
control have led many researchers to explore their capabilities
for aircraft aerodynamic modeling and estimation of
aerodynamic coefficients (stability and control derivatives).
Significant contributions have been made in this direction by
Hess [1], Linse and Stengel [3], Youseff and Juang [4], and
Raol and Jategaonkar [6].
Raisinghani, Ghosh and Kalra [7] proposed two new methods
for explicitly estimating aircraft parameters from the flight
data using FFNNs. The results obtained for simulated flight
data and real flight data have shown the success and the
potential of the proposed methods. For real flight data, in
addition to training being less than perfect, the parameters
may not be strictly constant, i.e., the parameters may vary
slightly with other motion and control variables. Furthermore,
all of the corrections and axes transformations done on the
data would introduce their own uncertainties. All these
factors, contribute toward different estimates at different time
points [7]. The scheme proposed in [7] to calculate confidence
level in the estimates does not work always, specially, if the
distribution of the numerical values of the estimated
parameters is skewed. Further a careful look into the Delta
method proposed in [7] reveals that it does not suggest any
procedure for validating the estimates by comparing estimated
response (with the help of estimated parameters) with the
flight generated response (real flight data) for a known control
input, other than used for generating real flight data for
parameter estimation purpose. The motivation to pursue this
work lies in improving the Delta method so that the estimated
parameters have larger confidence bound (lesser scatter) and
validating a methodology [8] to validate the extracted model
by comparing the estimated response with the flight response
generated by a control input not used for estimation purpose.
It is in this context that the present work explores the
suitability of the newly proposed Modified Delta (MD)
method [8] by applying it on real flight data obtained by
executing different longitudinal maneuvers using Hansa-III
aircraft. Further the FFNN based scheme to validate the
estimated model using time histories of measured
motion/control variables of the airplane for a given control
input excitation has also been demonstrated. FFNN based
scheme which bypasses the requirement of solving equations
of motion, to validate extracted model has been demonstrated
using real flight data generated by different types of control
inputs.
MODIFIED DELTA METHOD
The proposed Modified Delta method is based on interpreting
the stability and control derivatives as follows: If we could
obtain variation in the value of an aerodynamic coefficient
due to variation in only one of the motion/control variables
while the variation in other motion/control variables are
identically zero, then the ratio of the variation of the
aerodynamic coefficient to variation of the non-zero
motion/control variable will yield the corresponding
stability/control derivative. Let us say that the FFNN is
trained to map differential variations in input variables,
(  , q , and  e ) to the network output variable
(variation), CL . Now one input (say  ) at a time is
chosen to be at its original value while the rest of the network
inputs ( q , and  e ) are set to zero. The predicted value of
the aerodynamic coefficient CL corresponding to such a
modified file is divided by the non-zero variation in motion
variable (  ) to yield the corresponding stability/control
derivatives, CL . Similarly, all the parameters can be

estimated by suitably modifying the input file. Figure 1
schematically represents the training strategy for application
of the Modified Delta method using FFNN.
Input
s
∑ f
(qc 2V )
C L
∑ f
or
Cm
Output Layer
 e
Hidden Layer
Fig 1 Schematic of FFNN for proposed aerodynamic
modeling
SCHEME TO VALIDATE ESTIMATED
MODEL USING FFNN BASED METHOD
One of the procedures to validate aircraft parameter
estimation method is to compare the estimated response
(generated with estimated parameters) with the flight
measured response generated with control input other than
used for estimation purpose. Only way to validate by
comparing motion variables, generated using new control
input, would require solving of equations of motion. In this
paper a scheme using FFNN to validate the estimated model
by comparing flight measured variables generated using
different control inputs (not used for estimation) is presented.
This proposed scheme does not require solving of equations
of motion for validation. As a first step, the longitudinal
e
(2)
Once the training has been established, a new set of real flight
data generated through flight maneuver using a different
control input, say  e (other than used for estimation
2
purposes) is used for validation. The time histories of motion
variables corresponding to this control input designated as
 2 , q2 , and e are fed as input to already trained neural
2
network for validation. The weights estimated in the first step
are kept fixed for the prediction purposes. However, the
e
estimated numerical value of C Le , and C m
to be used as
2
2
input to neural network along with  2 , q2 , and
computed by plugging the numerical values of the estimated
parameters into the following equations:
CLe  CLe  CLe 2  CLe  q2 c / 2V   CLe  e2
2
0

q
e
(3)
e
e
e
Cm
 Cm
 Cm
  Cme  q2 c / 2V   Cme e2
2
0
 2
q
e
(4)
The already FFNN trained neural model is now used to
predict estimated motion variables (e )t 1 , and ( qe )t 1
corresponding to new control input containing time histories
e
)t as shown in Fig.
of (2 )t , ( q2 )t , ( e )t , (C Le ) t , (Cm
2
2
2
e
generated using control inputs  e .
2
Inputs
∑ 
(1 )t
∑ 
( q1 )t
( e1 )t
(CL1 ) t
(Cm1 ) t
Inputs Layer
∑ 
(1)t 1
∑ 
( q1 )t 1
Output Layer
e
Inputs
∑ 
e
and Cm are estimated by applying Modified Delta method
e
( 2 )t
∑ 
on flight data generated by a known control inputs (say
 e1 ).
For training, a neural mapping between input vector
containing time histories (at time, t , s) of (1 )t , ( q1 )t ,
( e )t , (CL )t , (Cm )t , and the output vector containing
1
∑ 
∑ 
Hidden Layer
e
e
parameters C Le , C Le , CL , C L , C m
, Cm
, Cm ,

0

0
q
e
q
1
( q2 )t
( e2 )t
1
time histories (at time, t  1 , s) of (1 )t 1 , ( q1 )t 1 , is
established using back propagation algorithm (BPA) as shown
in Fig. 2a. For the case of real flight data, the (CL )t , and
1
(Cm1 )t would be computed using the measured value of
acceleration a z and q through the following equations:
CL1   2 maz  V 2 S
e2 are
2b. The estimated responses of (e )t 1 , ( qe )t 1 are then
compared with the measured responses of (2 )t 1 , ( q2 )t 1
∑ f
Inputs Layer
Cm1  2 q I y  V 2 S c
(1)
(CLe2 ) t
e
(Cm
)
2 t
Inputs Layer
∑ 
∑ 
( e )t 1
∑ 
( qe )t 1
Output Layer
∑ 
Hidden Layer
Fig 2 Validation scheme for longitudinal estimates using
FFNN; a) training, b) validation
GENERATION OF REAL FLIGHT DATA
AND ESTIMATION OF PARAMETERS
At the Indian Institute of Technology, Kanpur, India,
during last couple of years, a technology testing aircraft
system is being developed by modifying Hansa-III aircraft, in
collaboration with the aircraft manufacturer, National
Aerospace Laboratories, India, a single engine, two seated
trainer airplane and instrumenting the same for the research
purposes with a wide range of sensors for flight data
acquisition. An onboard measurement system installed in the
test aircraft Hansa-III provides measurement of a large
number of signals such as aircraft motion variables,
atmospheric conditions, control surface positions etc. The
measurements made in flight are recorded onboard using
suitable interface with standard laptop.
Various longitudinal flight maneuvers were carried out and
flight data containing information about motion/control
variables were acquired on board. The flight data containing
numerical values of angle of attack (  ), linear accelerations
( ax , a y , az ), angular rates ( p, q, r ), aircraft orientation
( ,  ,  ), airspeed ( V ), and height (h) etc. were processed
first for data compatibility check and then used for parameter
estimation purposes. The flight data generated with multistep
3-2-1-1 elevator inputs applied
separately during 0 to 11.94 s and 22.36 to 28.5 s as shown in
Fig. 3a are designated as FLT1P and FLT1V respectively. The
attempt was to feed typical 3-2-1-1 elevator input to excite the
short period dynamics of the airplane. However, it was not
possible to exactly duplicate 3-2-1-1 type of maneuver.
Finally, a series of doublets elevator inputs (one in reverse
order) were fed to aircraft for the purpose of excitation of
short period dynamics. Flight data FLT2V had almost similar
multistep elevator form however, flight data FLT2P was
generated with a two sets doublet elevator inputs having
reverse order of elevator deflection. The elevator input
FLT2V was applied during early phase of the cruise (0 to 3.6
s), where as FLT2P was applied during later phase of cruise
(19.02 to 36.64 s) as shown in Fig. 3b.
Fig 3b Measured flight data of flight 2
DATA COMPATIBILITY CHECK
In practice, it is very often found that biases, scale factors and
time shifts are usually present in recorded real flight data [9,
10]. For conventional methods of aircraft parameter
estimation, it is well known that data compatibility checks
prior to estimation of parameters helps to improve the
accuracy of the estimates [10]. Thus, data compatibility check
was carried out before using the data for aerodynamic
modeling and parameter estimation. The Maximum
Likelihood method was applied to get estimates of biases,
scale factors and zero shifts in various recorded motion and
input variables. Thus, the flight data of Hansa-III were
corrected using the estimated values of bias errors in linear
accelerations and angular rate, scale factors and zero shifts in
angle of attack and pitch angle. Hereforth any reference of
real flight data would assume that the flight data had all
correction incorporated after and through data compatibility
check.
PARAMETER ESTIMATION
FLIGHT DATA
Fig 3a Measured flight data of flight 1
FROM
The flight data FLT1P, FLT1V, FLT2P, and FLT2V of
Hansa-III aircraft obtained after data compatibility check are
used for purpose of parameter estimation and validation of the
estimated model. The flight data FLT1P and FLT2P were
used for extracting aerodynamic model from flight data.
However it may be mentioned that control input from duration
0.0 to 11.94 sec of FLT1P and 19.02 to 36.64 sec of FLT2P
were used for the purpose of parameter estimation only. It was
decided to use flight data generated using control inputs other
than used for parameter estimation, for validation of extracted
aerodynamic model. For validation of the estimated
aerodynamic model, the flight data corresponding to control
inputs from duration 22.36 to 28.50 sec of FLT1V and 0.0 to
3.60 sec of FLT2V were used. Longitudinal stability and
control derivatives were then estimated using the Maximum
Likelihood method, the Delta method, and the Modified Delta
method. The numerical values of longitudinal estimated
parameters along with Cramer-Rao bound and standard
deviation are listed in Table 1.
As can be seen from the Table 1, that all the strong derivatives
obtain using Maximum Likelihood method are in close
agreement with those obtained from the neural network based
methods. However, the weak derivatives C Lq and C L
e
have not been estimated very well either by Maximum
Likelihood method or the neural network based methods. The
parameters estimated via the Modified Delta method are well
estimated with less standard deviation as compared to the
Delta method estimates.
The next step is to validate the estimates obtained from ML
method and neural network based methods. The standard
procedure to validate aircraft parameter estimation method is
to compare estimated response with the flight measured
response generated with control input other than used for
estimation purpose. For validation of estimated parameters
obtained from the Delta and Modified Delta method, a FFNN
based scheme given in Fig. 2 was used. As a first step, a
neural mapping between input vector containing time histories
of ( )t , ( q)t , ( e )t , (CL )t , (Cm )t , and the output vector
containing time histories of ( )t 1 , ( q)t 1 , was established
for flight data FLT1P (0.0 to 11.94 sec) and FLT2P (19.02 to
36.64 sec), using scheme as shown in Fig. 2a. The lift
coefficient (CL )t and the moment coefficient (Cm )t
corresponding to tth sec for known values of measured normal
and pitch accelerations were computed using Eqs. (1) and (2)
respectively. Once the training was completed, new sets of
real flight data containing the time histories of  2 , q2 , and
e2 generated by elevator control input of FLT1V (22.36 to
28.50 sec) and FLT2V (0.0 to 3.60 sec), are used as input to
already trained neural network for validation. However, the
e
estimated numerical value of C Le , and C m
to be used as
2
2
input to neural network along with  2 , q2 , and
e2 are
computed by plugging the numerical values of the estimated
parameters (obtained from Delta and Modified Delta method)
for flight data FLT1P-FLT2P in the Eqs. (3) and (4). The
already FFNN trained neural model was then used to predict
estimated motion variables (e )t 1 , and ( qe )t 1
corresponding to new control input containing time histories
of (2 )t , ( q2 )t , ( e )t ,
2
e
(C Le )t , (Cm
)t as shown in
2
2
Fig. 2b. A comparison between the predicted and the
measured response is graphically presented in Fig. 4 typically
for FLT2V (0.0 to 3.60 sec). Excellent matching among the
estimated and measured variables was observed for the case
of Modified Delta method. It is interesting to observe that the
estimated response using ML estimates had inferior matching
after 3 sec. Based on this result, it can be concluded that the
Modified Delta method can advantageous be apply on real
flight data for estimation of aerodynamic parameters.
Fig. 4 Validation of aerodynamic model using flight data
FLT2V
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
CONCLUSION
An improved Delta method, the Modified Delta method has
been proposed for estimating the aircraft parameters from
flight data using the feed forward neural networks. The results
obtained for real flight data have shown the success and
potential of the proposed methods. As compared to the Delta
method, the proposed Modified Delta method yields estimates
with lesser standard deviation. The results suggest that the
Modified Delta method can be used advantageously to
estimate parameters of an aircraft from flight data.
[10]
Hess, R.A. On the use of back propagation with feed
forward neural networks for the aerodynamic
estimation problem, AIAA Paper 93 3639, 1993.
Basappa; Jategaonkar, R.V. Aspects of feed forward
neural network modeling and its application to lateraldirectional flight-data, DLR IB-111-95/30, 1995.
Linse, D.J.; Stengel, R.F. Identification of
aerodynamic coefficients using computational neural
networks, Journal of Guidance Control and Dynamics,
Vol. 16, No. 6, 1993, pp 1018–1025.
Youseff, H. M. Estimation of aerodynamic
coefficients using neural networks, AIAA Paper 933639, 1993.
Hornik, K.; Stinchcombe, M.; White, H. Multi layer
feed forward neural networks are universal
approximator, Neural networks, Vol. 2, No. 5, 1989,
pp. 359-366.
Raol, J.R.; Jategaonkar, R.V. Aircraft parameter
estimation using recurrent neural networks -- a critical
appraisal, AIAA Paper 95-3004, 1995.
Raisinghani, S.C.; Ghosh, A.K.; Kalra, P.K. Two new
techniques for aircraft parameter estimation using
neural networks, The Aeronautical Journal, Vol. 102,
No. 1011, 1998, pp. 25-29.
Singh, S.; Ghosh, A.K. Improved delta method for
parameter estimation from real flight data of an
aircraft using neural networks, 17th IFAC Symposium
on Automatic Control in Aerospace, Toulouse, France,
2007.
Klein, V.; Schies, J.R. Compatibility check of
measured flight aircraft responses using kinematics
equations and extended kalman filter, NASA TN
D8514, 1977.
Doherr, K.F.; Hamel, P.; Jategaonkar, R.V.
Identification of the aerodynamic model of the DLR
research aircraft ATTAS from flight test data, DLR,
DLR-FB 90-40, Braunschweig, Germany, 1990.
Table 1 Estimates of longitudinal derivatives from flight data
FLT1P
Parameters
FLT2P
ML
0.295(0.459)*
DM
0.261(0.110)+
MDM
0.281(0.035)+
ML
0.337(0.507)
DM
0.300(0.156)
MDM
0.314(0.095)
CL
5.263(0.294)
4.271(0.822)
4.314(0.154)
5.036(0.268)
4.049(0.522)
4.359(0.466)
CLq
-54.839(14.95)
21.465(8.752)
26.595(4.577)
-56.63(15.5)
22.798(14.52)
23.767(9.893)
CL
-3.040(5.484)
0.218(0.661)
0.299(0.290)
-2.84(5.624)
0.293(1.081)
0.281(0.471)
Cm0
0.072(0.014)
0.066(0.056)
0.069(0.033)
0.071(0.014)
0.064(0.077)
0.066(0.060)
Cm
-0.3418(0.150)
-0.328(0.341)
-0.392(0.074)
-0.302(0.13)
-0.311(0.025)
-0.327(0.021)
Cmq
-7.090(0.5738)
-5.910(5.376)
-5.514(2.856)
-8.492(5.53)
-6.448(0.046)
-7.448(0.039)
Cm
-0.554(0.188)
-0.519(0.185)
-0.538(0.053)
-0.608(0.17)
-0.551(0.050)
-0.53(0.035)
CL0
e
e