AIAA 2016-1666
AIAA SciTech
4-8 January 2016, San Diego, California, USA
AIAA Modeling and Simulation Technologies Conference
Aerodynamic Model Update Using Parameter Identification
Supporting a Cessna Grand Caravan Engineering
Simulation
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
Alfonso Paris1 and Omeed Alaverdi2
AMERICAN SYSTEMS, Engineering & Analytical Solutions Directorate, Lexington Park, MD 20653
This paper outlines the initial nonlinear aerodynamic model development, and focuses on
the subsequent model updates based on flight test data, for an engineering simulation of the
Cessna Grand Caravan (C208B) aircraft. This modeling effort was in support of creating a
set of flight control laws and navigation algorithms capable of flying the aircraft in
autonomous mode. An initial simulation model was developed based on first principles and
empirical techniques. The aerodynamic model contained both airframe characteristics and
control surface hinge moments. A flight test program was carried out to gather data upon
which to further update and validate the simulation model. The resulting model was
instrumental during support of control system design, system flight test risk reduction
efforts, and overall system flight test preparation.
Nomenclature
ax,y,z
b
S
SH,V
TR,S
u,v,w
=
=
=
=
=
=
=
=
=
=
=
=
=
=
VH ,V
= horizontal and vertical tail volume coefficients
Vrw
x,y
XH,V
X,Y,Z
, 
E,A,R
Φ,θ,ψ
ηH,V
τ
ωP,SP,DR
ζP,SP,DR
Σ
=
=
=
=
=
=
=
=
=
=
=
=
c
CL,D,Y
Cm,n,l
f
Ixx,yy,zz
L,M,N
p,q,r
q
1
2
body axis linear accelerations along x, y, and z axes
reference wing span
reference planform mean aerodynamic chord
non-dimensional lift, drag, and sideforce aerodynamic coefficients
non-dimensional pitching, yawing, and rolling moment aerodynamic coefficients
generic function
body axis roll, pitch, and yaw moments of inertia
dimensional roll, pitch, and yaw aerodynamic coefficients
body axis roll, pitch, and yaw angular rates
dynamic pressure
wing reference planform area
horizontal and vertical tail reference planform area
roll mode and spiral mode time constants
body axis air relative velocities in x, y, and z axes
true airspeed
linear model state and output vectors
longitudinal distance from aircraft cg to horizontal and vertical tail aerodynamic centers
dimensional body axis x, y, and z aerodynamic coefficients
flow angle of attack and sideslip
elevator, differential aileron, and rudder aerodynamic control deflections
roll, pitch, and yaw Euler attitudes
horizontal and vertical tail dynamic pressure ratio
aerodynamic control surface effectiveness modeling term
phugoid, short period, and dutch roll system natural frequency
phugoid, short period, and dutch roll system damping
summation
Senior Aerospace Engineer, Engineering & Analytical Solutions Directorate, Senior Member AIAA.
Senior Aerospace Engineer, Engineering & Analytical Solutions Directorate, Senior Member AIAA.
1
American Institute of Aeronautics and Astronautics
Copyright © 2015 by AMERICAN SYSTEMS. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
I. Introduction
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
T
HE C208B is a high wing, single engine turboprop aircraft capable of carrying both passengers and/or cargo. A
fully nonlinear simulation of the aircraft was created to support development of autonomous flight controls,
navigation software and ground station software in the loop (SIL) testing. The simulation was used to support flight
test risk reduction and aided in flight test preparation efforts. The model included airframe mass characteristics,
ground reaction, airframe aerodynamics, control surface rigging and aerodynamic hinge moments, propulsion, and
system control actuators.
The focus of this paper is on the airframe aerodynamic model. An initial simulation model was developed based
on first principles and empirical techniques. A flight test program was then designed to collect data targeted at
updating and validating the nonlinear simulation model through application of Parameter Identification (PID)
techniques.
A series of ground and test flights were performed in late 2006 to early 2007 resulting in the collection of flight
test data suitable to support further model refinements as well as support PID efforts. Overall, data was collected in
Cruise (CR) Flaps 0, CR Flaps 10, Takeoff (TO) Flaps 20, and Landing (LD) Flaps 30 configurations. This paper
focuses on results obtained in the CR Flaps 0 and Flaps 10 configurations.
Data archival, preprocessing, and PID analysis were performed within the Integrated Data Evaluation and
Analysis System (IDEAS). IDEAS is a database management system and analysis software containing a full
complement of flight data preprocessing, calibration, simulation, model estimation, verification, and validation
tools.1,2
Flight test data collected for the purpose of aerodynamic parameter estimation typically consists of inertial and
air-relative sensor outputs. Prior to performing PID analyses, or traditional data reduction, it was important to
evaluate, and if necessary correct, the measured data to ensure kinematic consistency amongst them. Consequently,
a rigorous post-flight data calibration study was performed within IDEAS using the Navigation Identification
(NAVIDNT) tool. This tool couples an adaptive nonlinear Least-Squares Identification algorithm (LSIDNT) with a
set of rigid body navigation equations that model aircraft motion over an oblate, rotating Earth. 1,3,4 Sensor biases
and/or scale factors, as well as time invariant atmospheric winds, may be identified using this tool.
Basic airframe aerodynamic forces and moments were extracted from the collected flight test data, in addition to
control surface hinge moments, to support aerodynamic model update efforts. Aerodynamic model structures were
developed through analysis of these data using an equation error technique in IDEAS known as Athena employing
principal component axis regression. Athena expresses the overall aerodynamic forces and moments as linear
combinations of parameters such as stability derivatives and/or increments. In addition, to develop parameter
nonlinearities Athena supports the use of linear or cubic basis splines. Model refinements were installed in the
engineering simulation for the airframe and validation completed.
II. Aerodynamic Model Overview
Detailed engineering drawings and airfoil type definitions from the aircraft manufacturer provided the needed
data upon which to build a basis for an initial aerodynamic model. 5 Overall, the aerodynamic forces and moments
for the simulation are derived using a component buildup method. The forces and moments for each individual
aerodynamic component (wing, vertical tail, horizontal tail, and fuselage) are calculated and summed to produce the
overall aerodynamic forces and moments on the airframe.
The aircraft body axis velocities and angular rates are used to determine local velocities at the velocity reference
center defined for each specific component of the wing, fuselage, and tail sections. The wing and horizontal tail
sections are broken into two separate semi-span sections while the vertical tail is a single section. Each section has a
defined velocity reference center out along its span where local velocities and flow angles are determined to produce
a baseline local lift, drag, and pitching moment for the planform without aerodynamic controls deflected. Additional
sub-sections along each semi-span define control surface regions where local velocities and flow angles are used,
along with defined control deflections, to produce an incremental lift, drag, and pitching moment effect due to a
deflected surface. The sum of forces and moments for each major planform section, and its associated control
surface sub-sections, is made at an overall defined center of pressure for each planform section. The aerodynamic
contribution of each major planform component, and control surface sub-component, in the model has a locally
defined dihedral, sweep, and/or incidence relative to the aircraft body axis. A local lift, drag, and pitching moment
model is defined for each component.
The basic airframe lift, drag, and pitching moment for the wing, horizontal tail, and vertical tail planforms were
determined using XFOIL and AVL to obtain characteristics without control surfaces deflected. Aerodynamic
contributions resulting from deflected control surfaces were determined empirically using techniques for plain flap
2
American Institute of Aeronautics and Astronautics
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
and single slotted flap surfaces as outlined by McCormick.6 Control surface aerodynamic hinge moment models
were determined using empirical techniques for plain flaps as outlined by Perkins and Hage.7
The Caravan wing includes ailerons, aileron trim tabs, fowler flaps, and spoilers that are modeled in the
simulation. The wing is divided at the centerline into overall left and right sections. The wing sections, and each
sub-section (left and right ailerons, flaps, spoilers, and trim tabs), are individually analyzed as noted above. The left
and right horizontal tail sections contain both an elevator and elevator tab surface. The vertical tail contains only a
rudder control surface. The individual lift, drag and pitching moment contribution of each of these components are
calculated, transferred to the body axis, and then summed to form total forces and moments generated by the wing
about the overall defined aerodynamic reference center for the aircraft.
Main wing planform downwash is computed and affects the local flow angle at the horizontal tail. Thrust is
produced given a separate propeller model while momentum theory is employed to determine slipstream effects on
downstream surfaces such as the vertical and horizontal tail.8
The component buildup technique provides a resulting simulation with predictive capability based on physics. It
allows for a simulation that offers a wider range of application over those defined strictly on aircraft stability
derivatives. As an example, control surface off-nominal conditions can easily be modeled in such a system by
simply forcing the affected surface to the desired location. Propulsion wake effects on local components can also be
modeled readily. In addition, the requirement to initially define dynamic stability derivatives, such as C mq, Clp, and
Cnr, is removed as they now become a direct fall-out of the model as defined above. Classical stability derivatives
may be determined for the model using linear model extraction (LME) techniques as was often applied in this work.
The simulation was hosted within the IDEAS environment. This allowed the simulation model to be easily
accessible by the analysis tools available within IDEAS including LME analysis for providing linear models to the
control law development team. This also provided easy access to flight test data for validation and model
refinement purposes.
III. Available Flight Test Maneuvers
A flight test program was carried out to gather the data required to refine and validate the overall model. This
testing consisted of two ground tests and eleven flight tests. Ground testing collected propulsion static and dynamic
characteristics while flight testing collected aerodynamic flying qualities, hinge moment, and additional propulsion
static and dynamic performance. Data was gathered at representative operational configurations as follows:
•
•
•
•
CR 0 / 125 KIAS
CR 10 / 125 KIAS
TO 20 / 90 KIAS
LD 30 / 90 KIAS
Over three hundred individual test events were completed across the program within four weeks. Data from
various maneuver types were collected during this program to support model refinement and validation. The two
ground testing events supplied propulsion static and dynamic characteristics with the following maneuvers:
•
•
•
Ng maps
Engine dynamic response
Propeller governor dynamics
The eleven flight test events provided data related to aerodynamic flying qualities, aerodynamic hinge moments,
and propulsion system performance with the following maneuvers:
•
•
•
•
•
•
•
•
•
All axis doublets
All axis frequency sweeps
Pitch attitude captures
Wheel bank-to-bank roll attitude captures
Pedal bank-to-bank roll attitude captures
Longitudinal static stability
Lateral-directional static stability
Maneuvering stability
1-G wings level stalls
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American Institute of Aeronautics and Astronautics
•
•
•
•
Flap transitions
Power lever bodes
Level flight Acceleration-Deceleration
Ground effect
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
Throughout this testing the data gathered was inspected daily for quality and any necessary repeats were
scheduled into the subsequent flights. Having the data consumers, in this case the model and control law
developers, involved in the test data collection process proved beneficial in gathering the necessary data as
efficiently as possible, thus minimizing test event repeats. The quality of the test maneuvers, in addition to the data
itself, is of utmost importance when updating and validating a simulation model. Figure 1 presents an overview of
gathering data through flight test, data quality checking, and applying that data in support of model updates using
PID and validation. At any point in the process the requirement may arise to return to flight test to gather additional
data. The quality of the final model is only as good as the quality of the source data and its information content.
In this test program the aircraft was fully instrumented, including a test boom. Data of interest for this study
included, but was not limited to, the following:
•
•
•
•
•
•
•
•
•
•
Inertial position and velocities
Euler Attitudes
Body axis angular rates
Body axis linear accelerations
Flow angles (angle-of-attack and flank angle)
Air relative speed
Pressure Altitude
Ambient atmospheric conditions (OAT and Static Pressure)
Pilot inputs via control cable throw
Control surface deflections via control cable throw
IV. Dynamic Flight Test Data Pre-processing
Analysis within IDEAS examined pertinent channels from each maneuver for data dropouts and/or signal
wrapping. Appropriate utilities within the IDEAS Data Pre-processing And Reconstruction (DATPAR) toolbox
were employed to correct such anomalies when they occurred. 1,2 Specifically, the DATPAR UNWRAP utility was
used to generate continuous signals from discontinuous, wrapped, sensor outputs such as heading when necessary.
In addition, the DATPAR WILD_EDIT utility was used to discard “wild points” from the flight test data when
encountered.
Kinematic consistency evaluations were performed on the flight test data using NAVIDNT within IDEAS.
Recall that NAVIDNT couples the nonlinear least-squares algorithm LSIDNT with a set of navigation equations of
motion to form an output error sensor model identification scheme. 1,3,4 The navigation equations integrate flight
measured linear accelerations and angular rates to generate rigid body airframe responses. The goal of this process,
of course, is to ensure all accelerations, rates, angular orientations, flow angles, velocities (air-relative and inertial),
and inertial positions are consistent. Recall that NAVIDNT is used to estimate sensor biases and/or scale factors, as
well as time invariant atmospheric winds, to generate a kinematically consistent dataset. LSIDNT, the optimizer, is
a robust, adaptive nonlinear least-squares scheme based on the N2F family of algorithms encompassing the Newton
minimization scheme with an augmented version of the Gauss-Newton approximation.4 Details on the kinematic
consistency process, and algorithms within IDEAS, are outlined in detail in prior work by the authors. 3
NAVIDNT studies within IDEAS indicated the need for adjustments to be made to the locally measured flow
angles and air-relative velocity collected by the installed flight test boom. Note this boom was located on the
starboard wing tip. Both flow angles and velocity measurements needed to have the effect of body axis rates
removed in addition to accounting for necessary calibration adjustments. Figure 2 presents an example where test
boom true airspeed is transferred from its local measurement location to the actual aircraft CG. Body yaw rate
clearly is seen to affect the local airspeed measurement as expected during this pedal frequency sweep as the
starboard wing advances and retreats. The angle of attack measurement had to be adjusted for a boom installation
angle bias as well as account for local flow up-wash. Figure 3 presents an example of local angle of attack
measurement transferred to the aircraft CG and adjusted for installation bias and up-wash effects for a column
frequency sweep. Figure 4 presents a similar comparison but for a wheel frequency sweep. The removal of body
axis rate effects during the transfer to the CG is clear. The flank angle measurement had to be adjusted for the boom
4
American Institute of Aeronautics and Astronautics
installation angle as shown in Figure 5. Boom recorded air-relative speed required a calibration adjustment. All
axis doublets, frequency sweeps, and wind estimation turns were used to estimate the sensor calibration adjustments
mentioned using NAVIDNT within IDEAS.
The linear accelerations and inertial velocities were transferred from their local measurement location to the CG
in support of model development and evaluation efforts.
Overall, DATPAR tools were used to compute a variety of important histories for each maneuver including body
axis air relative velocities, sign convention adjustment, units book-keeping, and the transfer of locally measured
quantities to a common reference location to support the model update and validation processes. 1,2
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
V. Total Force and Moment Reconstruction
A variety of data is necessary to successfully extract total force and moment coefficients from flight data. These
data consist of body axis angular rates and accelerations, body axis linear accelerations, dynamic pressure, as well as
aircraft mass and inertia data. In addition, to facilitate transferring the overall moments to a specific reference point,
about which the aerodynamic model is to be developed, the center of gravity location for each maneuver must be
determined. Ideally, inflight thrust estimates, or those computed based on a validated propulsion deck model, would
be used to remove thrust effects from the overall total coefficients. In this case flight thrust was not available so the
total force and moment coefficients included both aerodynamic and thrust effects. Note during PID maneuvers the
power lever is maintained constant, and small amplitude airframe responses are targeted, thus resulting in thrust
contributing to an overall bias in the force and moment total coefficients. Here the slope of the coefficients with
respect to regressors such as angle of attack or sideslip are of greater significance and will be due primarily to the
basic airframe aerodynamic effects.
Angular rate, angular acceleration, linear acceleration, and dynamic pressure data were available in the flight test
data and subject to DATPAR processing as outlined in the previous section. Suitable information regarding aircraft
weight and balance characteristics such as mass, CG position, and moments of inertia were readily available from a
post-processing calculator developed for this aircraft making use of known starting fuel state and flight test recorded
fuel flow. All PID maneuvers were flown with throttles maintained at their trim position throughout each maneuver.
A variety of IDEAS DATPAR tools were employed to extract the total force and moment coefficients from the
flight data.1,2 This resulted in the overall body axis X, Y, and Z forces, as well as pitching moment, rolling moment,
and yawing moment coefficient histories with respect to the aircraft CG per PID maneuver. Overall, IDEAS
DATPAR utilities were used to back-calculate the aircraft body axis total forces and moments given pertinent
measured airframe linear and angular accelerations, airframe states, and weight and balance information. All body
axis total coefficients were reconstructed assuming a rigid body aircraft.
An additional IDEAS DATPAR utility was used to transfer all reconstructed body axis total moments to the
desired reference point about which the aerodynamic model would be developed.1,2 This transference involves the
incremental effects due to the reconstructed body axis aerodynamic forces about the aircraft CG being offset from
the desired aerodynamic reference center location.
VI. Output Error PID Technique
Initial control law design was being performed using the baseline nonlinear simulation model of the aircraft. This
included linear models extracted using LME techniques directly from the nonlinear simulation within IDEAS. The
longer term goal was to update the fidelity of the full nonlinear model given the available flight test data at hand.
However, in the short term it was desired to develop linear models from flight test data to provide directly to the
control law developers. This would allow for next step adjustments in the control law design while the full
nonlinear model was being updated and validated.
Both longitudinal and lateral-directional linear models were developed based on flight test data using nonlinear
least squares tools within MATLAB®. The linear model structures were written in C and compiled into a
MATLAB® mex-file format to reduce computational time during the iterative output error technique. The GaussNewton nonlinear least squares optimization algorithm was used to identify the linear model parameters of
interest.16 The iterative nature of the output error process is shown in Figure 6. In this analysis multiple
longitudinal or lateral-directional maneuvers were binned together at a given flight condition and analyzed. This
allowed for the identification of a single linear model given data from multiple test events. In this process, the flight
test data are used to define initial conditions for the linear models. Recorded flight test control inputs are applied
during propagation of the linear models and outputs then compared to those from the flight test event. The GaussNewton optimization algorithm works iteratively to update the model parameters to minimize this error until the
final parameters are determined.
5
American Institute of Aeronautics and Astronautics
The general state space model format was applied in both longitudinal and lateral-directional linear models for
all flap settings for which data was available:
x  Ax  Bu
(1)
y  Cx  Du
The longitudinal models were formulated as follows:
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
 u   X u
 w   Z
  u
 q   M u
  
   0
Xw
Zw
X
Z
Mw
0
u   1
w  0
  
q  0
   0
   
   0
  
a z   Z u
a  
 x   X U
 W0   g cos  0   u   X  E 

U 0   g sin  0   w  Z  E 
q
 

  q  M  E  E
Mq
0
  

1
0
    0 
q
0
0
1
0
0
1
0
1
U0
Zw
0
Zq
Xw
Xq
0
0
 0 
 0 
0


u


0
 0 



1  w   0  
 E
  
0  q   0 
 
  Z 
 E 
0

X 
0
 E 
(2)
(3)
The lateral-directional models were formulated as follows:
 v   Yv
 p   L
   v
 r    N v
  
   0
   0
Y
p
 W0 
Lp
Np
1
0
v  1
 p  0
  
r  0
   0
   
   0
   1
   U
a   0
 y   Yv
Yr
 U 0  g cos  0  0  v   Y A
0
0  p   L A
  
Nr
0
0  r    N  A
  
tan 0 
0
0     0
sec 0 
0
0    0
Lr
0
0
1
0
0
1
0
0
0
0
0
0
Yp
Yr
0 0
 0
0 0  v   0

0 0  p   0
 
1 0  r    0

0 1     0




0 0    0
 

Y
0 0
 A
Y R 
L R 
 A 
N R   
  R 
0 
0 
0 
0 
0 
  
0  A 

0  R 

0 
Y R 
(4)
(5)
In each flight condition, and flap configuration of interest, a group of pilot applied frequency sweeps was
presented to the optimization scheme simultaneously to generate a single linear model accounting for all the data
available at that condition. For example, in the CR Flaps 0 case the longitudinal model for 125 KIAS at 10k ft was
developed using a binned group of four longitudinal pilot applied frequency sweeps. The lateral-directional model
at the same flight condition was developed using a binned group of three wheel and two pedal frequency sweeps.
Sample linear model comparison plots are shown in Figure 7 for a longitudinal frequency sweep for CR Flaps 0.
The resulting linear model output is shown along with the recorded flight test data. Note the steady state values of
the states, controls, and responses have been removed as these are the perturbation linear model responses. Linear
model comparison to flight test data for a pedal frequency sweep at CR Flaps 0 configuration is shown in Figure 8.
Sample linear model comparison plots for CR Flaps 10 are shown for a longitudinal frequency sweep and
directional frequency sweep in Figure 9 and 10. In all cases the linear models perform well in emulating the aircraft
6
American Institute of Aeronautics and Astronautics
flight responses. The binned frequency sweeps provide good information content for the output error model
estimation process.
The longitudinal linear models for CR Flaps 0 and CR Flaps 10 configuration are shown below in Table 1. The
corresponding lateral-directional linear models are shown in Table 2. These models are representative of flight at
125 KIAS and 10k ft.
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
Table 1. Longitudinal linear model parameter estimates using output error.
Model
Parameter
Xu
Xw
Xq
Xe
Zu
Zw
Zq
Ze
Mu
Mw
Mq
Me
Uo
Wo
o
Parameter
Units
1/s
1/s
ft/(s rad)
ft/(s2 rad)
1/s
1/s
ft/(s rad)
ft/(s2 rad)
rad/(ft s)
rad/(ft s)
1/(s rad)
1/(s2 rad)
ft/s
ft/s
rad
CR
Flaps 0
-0.0191
0.166
0
0
-0.265
-1.412
-8.0897
-37.236
0
-0.0541
-2.999
-11.950
239.8
14.8
0.1141
CR
Flaps 10
-0.0444
0.126
0
0
-0.237
-1.518
-8.591
-37.482
0
-0.0373
-3.0751
-10.757
248.1
4.5
0.0092
Table 2. Lateral-directional linear model parameter estimates using output error.
Model
Parameter
CR
CR
Parameter
Units
Flaps 0
Flaps 10
Yv
1/s
-0.228
-0.220
Yp
ft/(s rad)
0.0748
0.276
Yr
ft/(s rad)
2.861
3.157
2
ft/(s
rad)
0
0
Ya
2
ft/(s rad)
12.943
13.253
Yr
Lv
rad/(ft s)
-0.0201
-0.0236
Lp
1/(s rad)
-2.671
-3.691
Lr
1/(s rad)
0.283
0.355
1/(s2 rad)
2.207
2.648
La
1/(s2 rad)
0.115
0.306
Lr
Nv
rad/(ft s)
0.0106
0.0115
Np
1/(s rad)
-0.394
-0.333
Nr
1/(s rad)
-0.650
-0.621
1/(s2 rad)
0.171
0.163
Na
2
1/(s
rad)
-2.862
-2.990
Nr
Uo
ft/s
233.5
244.8
Wo
ft/s
14.8
4.4
rad
0.0732
0.0157
o
In this linear model structure the aileron control is an effective overall roll control in that it is modeled as total
differential aileron (aL – aR). In addition, the upper wing surface roll control spoilers are geared to move with the
ailerons. As a result, their power is included in the effective aileron control power model terms.
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American Institute of Aeronautics and Astronautics
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The style of maneuvers used to develop the linear models were targeted at exciting the phugoid, short period,
dutch-roll, and roll modes. The natural frequencies, damping, roll mode, and spiral mode time constant were
determined for the output error identified linear models.
The initial full nonlinear model for this aircraft was examined using an LME facility to develop longitudinal and
lateral-directional linear models at similar operating conditions and configurations as tested in flight. The same
mode characteristics were determined from those models and are presented in Table 3 and Table 4 for comparison to
those obtained from flight.
Table 3. Comparison of linear model characteristic modes for CR Flaps 0 configuration.
Mode
CR Flaps 0
CR Flaps 0
Percent
Output
Initial
Difference
Error
Nonlinear
Model
Model LME
SP
4.12
6.63
60.92
(rad/s)
0.534
0.451
-15.54
SP
P
0.177
0.141
-20.34
(rad/s)
0.0789
0.179
126.87
P
DR
1.85
3.06
65.41
(rad/s)
0.18
0.155
-13.89
DR
TR
0.351
0.220
-37.30
(s)
TS
35.34
106.0
199.98
(s)
Table 4. Comparison of linear model characteristic modes for CR Flaps 10 configuration.
Mode CR Flaps 10 CR Flaps 10
Percent
Output
Initial
Difference
Error Model
Nonlinear
Model LME
SP
3.69
5.98
62.06
(rad/s)
0.623
0.469
-24.72
SP
P
0.154
0.151
-1.95
(rad/s)
0.148
0.168
13.51
P
DR
1.87
2.91
55.61
(rad/s)
0.187
0.154
-17.65
DR
TR
0.262
0.21
-19.99
(s)
TS
44.74
198.0
342.53
(s)
The initial non-linear model, based on first principles and empirical modeling, provided a starting point for
overall analysis. Based on first look results of Table 3 and Table 4 the model clearly required updates to better
match the aircraft mode characteristics. Best overall results with the original non-linear model were seen in the
short period (SP) and dutch-roll damping (DR) terms. The horizontal and vertical tail lift forces play a significant
role in these damping characteristics. The non-linear simulation horizontal and vertical tail planform lift curve
slopes were generated using CFD as previously mentioned. These results indicate a close first estimate for these lift
curve slopes. In the case of CR Flaps 0, SP was ~15% low while DR was ~ 14% low. The CR Flaps 10
configuration shows SP was ~25% low while DR was ~ 18% low.
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Given the linear model estimates for CR Flaps 0 of M q and Nr, a rapid update could be determined for the lift
curve slope linear region of the full model horizontal and vertical tails. For example, consider the relation for pitch
damping in the linear model:
M q  Cmq
q0 Sc 2
2 I yy 0U 0
(6)
This relation can be solved to determine Cmq given the known flight conditions, aircraft geometry, mass
characteristics, and overall linear model Mq estimate from flight data. The non-dimensional pitch damping
coefficient (Cmq) can also be defined as follows given Ref. 14:
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Cmq  2CL H  HVH
XH
c
(7)
Where the horizontal tail lift curve slope can be determined as follows:
CL 
H
Cmq
(8)
X
 2 HVH H
c
This relation provides a new overall estimate for the linear region of the horizontal tail planform lift curve slope that
can be blended directly into the full model supporting rapid turnaround. Where:
H 
VH 
qH
q

SH
X AC H  X CG
S
(9)

X H  X AC H  X CG
(10)
(11)
The dynamic pressure ratio  H models any increase in dynamic pressure experienced at the horizontal tail versus
free stream. In the full nonlinear simulation the term is defined given the modeled prop wake enveloping the tail
section. Once overall airframe drag was adjusted to get the proper throttle setting to match flight, this value could
be referenced for this particular flight condition and configuration. The dynamic pressure ratio could then be used to
support calculation of a new linear lift curve slope value for the horizontal tail. Given the available flight test
maneuvers in the CR Flaps 0 configuration it was determined that on overall increase in fuselage model drag of
~30% was necessary to capture the power setting matching flight. This is not unexpected as the fuselage initial
aerodynamic model was not based on CFD but was built upon empirical component representative shapes to model
the fuselage and strut aerodynamics.
An identical technique was employed for the vertical tail planform linear region lift curve slope. The relation for
yaw damping in the estimated linear model is as follows:
N r  Cnr
q0 Sb2
2 I zz 0U 0
(12)
This relation can be solved to determine Cnr given the known flight conditions, aircraft geometry, mass
characteristics, and overall linear model Nr estimate from flight data. The non-dimensional yaw damping coefficient
(Cnr) can also be defined as follows given Ref. 14:
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2 X V2
S
V V
2
b
S
Cnr  CL
V
(13)
Where the vertical tail lift curve slope can be determined as follows:
CL
V
b2 S
 Cnr
2 X V2V SV
(14)
This relation provides a new overall estimate for the linear region of the vertical tail planform lift curve slope that
can be blended directly into the full model supporting rapid turnaround. Where:
qV
q
(15)
XV  X ACV  X CG
(16)
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V 
Similarly, the dynamic pressure ratio
V models any increase in dynamic pressure experienced at the vertical tail
versus free stream.
Also analyzed for rapid model update for CR Flaps 0 configuration was the elevator and rudder control power
effectiveness parameters. Given the linear model structure identified, the elevator and rudder power force
coefficients are defined as follows:
Z E  CL E
Y R  CY R
qo S
m
qo S
m
(17)
(18)
The non-dimensional stability derivatives for lift due to elevator and side force due to rudder deflection are then
determined as follows:
CL   Z E
E
CY  Y R
R
m
qo S
m
qo S
(19)
(20)
Given the linear model estimates from flight, the known aircraft mass, geometry, and flight conditions, they can be
directly calculated. The non-dimensional elevator and rudder control power coefficients can also be defined as
follows given Ref. 14:
CL E  CL H  H
SH

S E
CY R  CLV V
SV

S R
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(21)
(22)
The elevator and rudder effectiveness model terms can then be calculated for direct blending into the non-linear
simulation model:
 
E
 
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R
CL S
E
CL H  H S H
CY S
R
CLV V SV
(23)
(24)
From these examples it is clear that other linear slope sections could be blended in a similar manner. However,
only the tail planform lift curve slope linear sections were updated in this fashion along with the elevator and rudder
control power. With this update the non-linear model overall pitch and yaw damping, along with pitch and yaw
control power, would provide representative flight characteristics.
The mode comparison results noted in Table 3 and 4 indicate the short period and dutch-roll frequencies of the
initial nonlinear model did not compare as well as the damping. In the case of CR Flaps 0, SP was ~61% high
while DR was ~ 65% high. The CR Flaps 10 configuration shows SP was ~62% high while DR was ~ 56% high.
Both the damping and static coefficient terms are known to play a significant role in the mode frequencies. For
example, Cmq and Cm shape the short period natural frequency together while Cnr and Cn shape the dutch-roll
frequency. The aerodynamic model terms with the greatest uncertainty in the full non-linear model were those
developed for the fuselage components. The best path forward was to blend in any necessary C m and Cn updates
through the fuselage model itself. The overall rate damping terms would be captured by the tail planform lift slope
corrections previously discussed. Any updates necessary for Cy and Clwould also be applied to the fuselage
section aerodynamics. If additional updates were necessary for C L it was decided to apply those to the main wing
planform. Any updates to the main wing planform lift curve slope would also effect the overall simulation roll
damping as a natural fall-out which in turn would alter the roll mode time constant (T R). Main wing lift curve slope
updates would also naturally alter the roll response with sideslip (Cl). Note Tables 3 and 4 indicate the non-linear
simulation, for both CR Flaps 0 and CR Flaps 10, have a roll mode time constant lower than what was identified in
flight by ~37% and ~20% respectively. This indicates the non-linear model overall roll damping is likely too high.
In both flap configurations the initial spiral mode time constant was high by ~200% and ~343% for CR Flaps 0 and
Flaps 10 respectively. Note the spiral mode root location is typically near zero on the root locus x-axis. Small
changes in the root location can make a notable difference in the resulting spiral time constant. The spiral mode
depends heavily on aircraft directional static stability (Cn) and dihedral effect (Cl). The prior discussion notes the
dutch-roll frequency of the original model needs adjustment and, as such, it is expected the spiral mode match will
improve as a direct fall-out.
In summary the updates required to the non-linear simulation in this phase of analysis are shown for CR Flaps 0:





Fuselage drag increase (~30%)
Horizontal tail lift curve slope (CLH)
Elevator control power (CLE)
Vertical tail lift curve slope (CLV)
Rudder control power (CYR)
At this phase in the analysis the only update applied for CR Flaps 10 was an adjustment in drag due to wing flap
deflection. The following section describes the equation error PID technique used to perform additional
aerodynamic model updates.
VII. Equation Error PID Technique
The next stage of analysis involved an equation error PID technique to extract aerodynamic model terms. The
equation error tool in IDEAS, known as Athena, is capable of determining the overall aerodynamic force and
moment models as linear combinations of parameters (typically stability derivatives and/or incremental coefficients)
using a principal component regression algorithm.1,2,9,10 This technique has been used with success in prior work by
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the authors.11,12 Athena allows these terms to be modeled with nonlinear dependencies as well. As an example,
consider a simple pitching moment buildup:
Cm  Cmbasic   Cme     e
(25)
In this example the overall model structure is a nonlinear function of angle-of-attack. To model the nonlinearities,
Athena supports the use of linear or cubic basis splines to estimate coefficients of the splines at the specified knot
locations. Eq. (25) is rewritten as:
K1
K2
i 1
i 1
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Cm   Cmbasic,i  f1,i     Cme,i  f 2,i    e
(26)
Cmbasic,i and Cme,i are the parameters to be estimated; K1 and K2 represent the number of angle-of-attack knot
locations defined for Cmbasic and Cme respectively; and f1,i() and f2,i() are basis functions. The basis functions are
defined such that each takes on a value of 1.0 at one knot location and a value of 0.0 at all other knot locations.
Athena assumes the model may be represented as a set of linearly combined, time-independent parameters with
the following structure:
y  Ap v
(27)
Vector y (size nx1) represents the total non-dimensional force or moment coefficient history vector under
investigation. The parameter vector p (size mx1) represents the stability and control derivatives under estimation,
and the regressor matrix A (size nxm) contains the independent variables. Vector v (size nx1) represents unmodeled aerodynamic responses and/or phenomena such as due to system/sensor noise. Consider the example
outlined by Eq. (25) and Eq. (26) above assuming two angle-of-attack knot locations are chosen for both C mbasic and
Cme. In this case, Eq. (27) becomes:
 C m 1   f 1,1  1
  


 
C m n   f 1,1  n 
f 1, 2  1
f 2,1  1   e 1


f 1, 2  n 
f 2,1  n    e n 
 C mbasic,1 
f 2, 2  1   e 1  
  v1 
 C mbasic, 2  


  
 C
 
me ,1
f 2, 2  n    e n  
 vn 
 C me, 2 
(28)
Athena uses a numerically robust singular value decomposition method to solve Eq. (27) and estimate the
parameter vector p. In general, Athena first determines an incremental response vector ys by removing the prior
model contributions (po):
y s  y - A po
(29)
Vector po consists of initial estimates of parameters as well as those that have been fixed and are not to be
estimated. The regressor matrix A is then thinned such that it contains only those columns that correspond to
parameters that are to be estimated. The thinned matrix As allows the identification statement of Eq. (27) to be reformulated as follows:
ys  A s p s  v
(30)
The thinned regressor matrix As is then decomposed into the following form:
A s  U SV T
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(31)
U and V are orthogonal and S is a diagonal matrix containing the singular values. Using Eq. (30) and Eq. (31)
together the free parameters are then estimated in principal component axes:
p s  VS UT y s
(32)
Note that vectors po and ps are combined to yield the final parameter estimates:
p  po  p s
(33)
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Output statistics for this technique are provided in the form of a fit percentage based upon the Theil inequality
coefficient statistic (U) defined as:14
1
N
U
1
N
N
  yˆ
i 1
N
  yˆ 
i 1
2
i
 yi 
2
i

(34)
N
 y 
1
N
i 1
2
i
N is the total number of points in the residual vector. This coefficient represents the ratio of the root mean square fit
error and the root mean square values of the estimated and actual signal summed together. The value of U always
falls between 0 and 1, with 0 indicating a perfect fit and 1 the worst fit. The Athena fit percentage (F), a measure of
signal fit quality, is defined as follows:
F  1001  U 
(35)
A 100% fit represents a perfect match with the measured data. Additionally, Athena breaks the fit error into bias
(UB), variance (UV), and covariance (UC) proportions as follows:13
yˆ  y 
2
UB 
1 N
 yˆi  yi 2

N i1
   

1
  yˆ  y 
N
(36)
2
UV
yˆ
y
N
i
i 1
UC 
(37)
2
i
21    yˆ y
1 N
 yˆi  yi 2

N i1
(38)
Where  and  represent the correlation coefficient and standard deviation respectively.

1
 y yˆ N
 yˆ
N
i 1
i

 yˆ  yi  y 
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(39)
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x 
1
N
N
 x
i 1
i
 x
2
(40)
The bias proportion presents the deviation of the average values of the simulated and measured data acting as a
measure of model systematic error. The variance proportion acts as a measure of the model’s ability to duplicate the
variability in the true system. The covariance proportion is a measure of non-systematic error (e.g. due to unmodelled random sensor/system noise). Note that these three proportions sum to 1, with the ideal fit having U B and
UV close to zero, with UC close to 1.
These fit statistics act as a measure of accuracy and/or certainty in the proposed model formulation under
investigation and provide clues into the effectiveness, or lack thereof, of adjustments introduced in the model
structure.
An additional strength of this algorithm is its ability to analyze multiple segments of information, in this case
PID maneuvers, at once. This allows for the extraction of global models from analysis groups of multiple PID
maneuvers. Overall, this procedure is a fast, single pass algorithm that results in good base model structure
determination.
This technique was used to analyze groups of flight test data, spanning angle of attack for example, in order to
develop both linear and nonlinear model coefficient updates. These updates were blended into the final
aerodynamic model. Recall each maneuver to be evaluated in the PID process had total forces and moments backcalculated from the flight test data and presented with respect to a common aerodynamic model reference center
matching that of the non-linear simulation model database.
A summary of updates required to the non-linear simulation using this technique is as shown for CR Flaps 0:






Main wing lift curve slope (CL)
Fuselage wing-body adjustment for Cm
Fuselage wing-body adjustment for Cy
Fuselage wing-body adjustment for Cn
Fuselage wing-body adjustment for Cl
Roll control power (Cla and Cna)
Updates required to the non-linear simulation for CR Flaps 10 were applied to the same terms as those above with
corrections implemented as a function of flap deflection.
As an example of this analysis consider the CR Flaps 0 configuration lift curve slope update. In this phase a
group of maneuvers were analyzed using this equation error technique to develop model updates for the main wing
lift curve slope (CL). This data was binned together and included the following maneuvers analyzed
simultaneously:
 Four column frequency sweep events
 Three 1-g wings level stall events
The angle-of-attack for these maneuvers collectively ranged from 1 o to 16o. The column frequency sweeps
remained in the linear region of the lift curve slope while the 1-g stalls spanned from the linear into the non-linear
region and stall break. The model structure considered in Athena was the following:
 qc 
  CL  E
CL  CL    CLq 
E
 2Vrw 
(41)
Due to the flight data coverage for angle-of-attack the model knot locations for Athena regarding the function C L()
were anchored at [1.0 5.0 9.0 11.0 13.0 16.0]. This instructs the algorithm to estimate six model entries for C L
anchored at the defined angle-of-attack values specified. Recall the prior analysis had already updated the tail
planform lift curve slopes as well as the elevator control power based on the linear model analysis. With those
updates having been blended into the non-linear simulation, a sweep utility was used to determine values of C Lq and
CLE from the non-linear simulation. The simulation sweep facility allows for configuring, and trimming, the
aircraft at a specified flight condition. Desired variables, such as pitch rate and elevator deflection for example, can
14
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then be individually perturbed over a specified range with the resulting change in overall aerodynamic coefficients
saved out. In this case the simulation based values for C Lq and CLE could then be set constant in the Athena model.
Athena was used to estimate the parameters for C L(). Lift coefficient model matches versus flight test data are
shown in Figure 11 for a sample column frequency sweep and 1-g stall maneuver. The angle-of-attack from flight is
also presented for each test case for reference. The resulting model indicates a good match against flight test data
with all seven maneuvers binned together resulting in an overall Theil fit percentage of 97.7% with U B = 0, UV =
0.004, and UC = 0.996. A comparison of the Athena identified lift curve slope with that obtained from the nonlinear model using the simulation sweep facility is shown in Figure 12. Overall it presents a close match below 9 o 
with the simulation slope being slightly higher. Above 9 o  the flight identified lift curve slope begins to develop a
non-linear, slope reducing, trend while the initial simulation predicted higher lift. Comparisons such as this were
used to re-shape simulation database coefficient characteristics such that they fair through the trends identified from
flight data.
This equation error technique was used to analyze the other parameters noted above with the same blending
technique employed to update the final model for both CR Flaps 0 and Flaps 10 configurations.
VIII. Simulation Validation
Having finalized and blended the aerodynamic model updates into the non-linear simulation a new set of linear
models were extracted. The simulation was configured to match the flight conditions of the linear models identified
previously from flight. The LME facility was then used again to generate corresponding linear models from the
simulation. The aerodynamic model characteristic comparisons are shown below in Table 5 and 6 for CR Flaps 0
and CR Flaps 10 respectively.
Table 5. Comparison of linear model characteristic modes for CR Flaps 0 after simulation update.
Mode
CR Flaps 0
CR Flaps 0
Percent
Output
Updated
Difference
Error
Nonlinear
Model
Model LME
SP
4.12
4.42
7.28
(rad/s)
0.534
0.54
1.12
SP
P
0.177
0.187
5.65
(rad/s)
0.0789
0.0965
22.31
P
DR
1.85
1.85
0.00
(rad/s)
0.18
0.217
20.56
DR
TR
0.351
0.308
-12.25
(s)
TS
35.34
26.7
-24.45
(s)
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Table 6. Comparison of linear model characteristic modes for CR Flaps 10 after simulation update.
Mode CR Flaps 10 CR Flaps 10
Percent
Output
Updated
Difference
Error Model
Nonlinear
Model LME
SP
3.69
4.01
8.67
(rad/s)
0.623
0.628
0.80
SP
P
0.154
0.148
-3.90
(rad/s)
0.148
0.0968
-34.59
P
DR
1.87
1.76
-5.88
(rad/s)
0.187
0.232
24.06
DR
TR
0.262
0.267
1.91
(s)
TS
44.74
45.3
1.25
(s)
Overall, the non-linear simulation mode characteristics show significant improvement following the PID model
updates. The initial non-linear simulation mode characteristics can be referenced in Table 3 and 4 for comparison.
Note the spiral mode time constants show vast improvement. In addition, the roll mode time constant has improved.
Another test of model fidelity entails the use of simulation propagation with outputs compared to flight data.
Flight test measurements for pilot column, wheel, pedal, and throttle inputs were used to override pilot control
inputs in the simulation. The atmospheric model within the simulation was initialized to flight test conditions.
Aircraft weight and balance information were initialized to data describing each flight test event examined. Prior to
 , p , q, r ) in the test
propagation, the simulation was trimmed to match any non-steady initial conditions ( u, v, w
data. In addition, the simulation was initialized with the appropriate airspeed, altitude, body axis angular rates, and
Euler attitudes as determined from the flight data.
In support of validation efforts a set of tolerance bounds were selected based upon information available in FAA
Advisory Circular 120-40B.15 Bounds are applied to the flight test data to display a region of acceptable match in
the validation plots. A simulation output propagation compared against flight test data can be seen in Figure 11.
This maneuver is a piloted longitudinal frequency sweep at CR Flaps 0, 125 KIAS, 10k ft. The flight test data was
used to create the bounds shown after applying tolerances from FAA AC-120-40B. Figure 11 presents a set of
output histories for this maneuver that indicate an accurate match in the longitudinal axis regarding both high and
low frequency portions of the event. The elevator control power, pitch damping, and region of short period mode
natural frequency excitation followed by the response attenuation are well emulated by the model for this frequency
sweep.
Figure 14 presents a sample roll rate validation match for a lateral frequency sweep at CR Flaps 0, 125 KIAS,
10k ft. Overall the model predicts the roll rate response well across the frequency ranges covered by this maneuver.
Roll axis control power and subsequent roll damping are captured by the model.
Figure 15 presents a sample yaw rate validation match for a directional frequency sweep at CR Flaps 0, 125
KIAS, 10k ft. Similarly, the updated model predicts the yaw rate response well across the frequency ranges covered
by this maneuver. The yaw axis control power, and subsequent rate damping, are captured by the model. In
addition, the model captures the excitation of the dutch-roll mode similarly to that of the flight test aircraft. Yaw
rate response attenuation as the pedal input frequency increases beyond the dutch-roll excitation frequency is also
emulated well by the model.
IX. Conclusion
An initial aerodynamic non-linear model of a Cessna Grand Caravan (C208B) aircraft was developed to support
engineering analysis and flight test studies. A flight test program was developed to collect the required data to
support the update, and subsequent validation, of the model. Both linear and non-linear model extraction techniques
were successfully applied to provide a more accurate flight model. The resulting product was instrumental in
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control system design, engineering analysis, off-nominal condition studies, and flight test preparation. Further detail
regarding the flight test build-up and final system demonstration for this aircraft can be found in Ref. 17.
References
1
Linse, D. J., "Aircraft System Identification Using Integrated Software Tools", RTA-SCI Symposium, Madrid,
Spain, May 1998.
2
Linse, D. J., "Improving Simulator Accuracy With Integrated Analysis of Flight Data", Interservice/Industry
Training, Simulation, and Education Conference, I/ITSEC Paper EC-046, November 2000.
3
Paris, A. C., and Alaverdi, O., "Post-Flight Inertial And Air-Data Sensor Calibration", AIAA Atmospheric Flight
Mechanics Conference, AIAA Paper 98-4450, Boston 1998.
Dennis, J. E. Jr., Gay, D. M., and Welsch, R. E., “An Adaptive Nonlinear Least-Squares Algorithm,” ACM
Transactions on Mathematical Software, Vol. 7, No. 3, September 1981, pp 348-368.
Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
4
5
Cessna Aircraft Company, “Substantiation and Critical Loads Summary Report S-208B-33”, Jan 2006.
6
McCormick, B. W., "Aerodynamics, Aeronautics, and Flight Mechanics", 1979.
7
Perkins, C. D., Hage, R. E., “Airplane Performance Stability and Control”, 1949.
8
McCormick, B. W., "Aerodynamics of VSTOL Flight", Academic Press, 1967.
9
Anderson, Laurence, C., "Robust Parameter Identification for Nonlinear Systems Using a Principal Components
Regression Algorithm", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 85-1766, August 1985.
Linse, D. J., “System Identification Software Design Document and User’s Manual for the Integrated Data
Evaluation and Analysis System (IDEAS)”, Science Applications International Corporation, SAIC Report No. 011393-2990-A005/A006, California, MD, November 1997.
10
11
Paris, A. C., and Alaverdi, O., "Nonlinear Aerodynamic Model Extraction From Flight Test Data for the S-3B
Viking", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2001-4015, Montreal, Canada 2001.
12
Paris, A. C., and Bonner, M., "Nonlinear Model Development From Flight Test Data for the F/A-18E Super
Hornet", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2003-5535, Austin, Texas 2003.
13
Pindyck, R. S., and Rubinfeld, D. L., "Econometric Models and Economic Forecasts", 3 rd Edition, McGraw-Hill,
Inc., 1991, pp 336-342.
Roskam, J., "Airplane Flight Dynamics and Automatic Flight Controls – Part I", Roskam Aviation and
Engineering Corporation, Ottawa, Kansas, 1982, pp 195-197, pp 203 - 205.
14
15
Federal Aviation Administration, "Airplane Simulator Qualification", U.S. Department of Transportation, AC-12040B, July 1991.
Maine, R.E., Iliff, K.W., "Application of Parameter Estimation to Aircraft Stability and Control – The Output
Error Approach", NASA-RP-1168, 1986.
16
17
Haider, M., Alaverdi, O., "Manned but Unpiloted: A Discussion of the Flight Testing Leading to a Demonstration
of the Caravan Optionally Piloted Aircraft (COPA)", Society of Experimental Test Pilots, 2008.
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Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666
Flight Test
Instrumentation
Correction
Onboard or Telemetry
Data
Kinematic Consistency
and Sensor Calibration
System
Identification
Model Structure
Determination and
Parameter Estimation
Model Update
and Validation
Simulation Update
and Validation
Analysis
New Simulation Model
Supporting
Engineering Analysis,
Flight Control Design,
Training, and Testing
Data
Problems
Insufficient
Identification
Data
Insufficient
Validation
Data
Figure 1. Overview of the flight test process
supporting System Identification based model
updates and final model validation.
Figure 2. Body axis rate effects removed during
transfer of test boom local true airspeed to actual
aircraft CG location for a pedal frequency sweep.
Figure 4. Local angle of attack measurement
transferred to the aircraft CG and adjusted for
installation bias and up-wash effects for a wheel
frequency sweep.
Figure 5. Flank angle measurement adjusted for
boom installation angle as shown in this pedal
frequency sweep.
Initial
Parameter
Estimates
Flight Test
Database
Binned Flight Test Events
Similar Flight Condition
& IC’s
Linear Model Structure
y
Updated
Parameter
Estimates
+
Figure 3. Local angle of attack measurement
transferred to the aircraft CG and adjusted for
installation bias and up-wash effects for a column
frequency sweep.
-
Nonlinear Least Squares
Optimizer
Figure 6. Output error parameter identification
technique.
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Figure 7. Longitudinal frequency sweep linear model response comparison against flight test data for CR
Flaps 0, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output]
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Figure 8. Directional frequency sweep linear model response comparison against flight test data for CR
Flaps 0, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output]
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Figure 9. Longitudinal frequency sweep linear model response comparison against flight test data for CR
Flaps 10, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output]
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Figure 10. Directional frequency sweep linear model response comparison against flight test data for CR
Flaps 10, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output]
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Column Frequency Sweep
1-g Stall
Figure 11. Equation error CL model response comparison against flight test data and corresponding angleof-attack envelope for CR Flaps 10, 125 KIAS, 10k ft maneuvers. [Solid Blue Line = Flight Data,
Red Dash Line = Model Output]
Figure 12. Equation error lift coefficient match from flight compared to initial simulation database for the
CR Flaps 0 configuration.
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Figure 13. Nonlinear simulation output compared to flight test data for a piloted longitudinal frequency
sweep at 125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied.
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Figure 14. Nonlinear simulation output compared to flight test data for a piloted lateral frequency sweep at
125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied.
Figure 15. Nonlinear simulation output compared to flight test data for a piloted directional frequency sweep
at 125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied.
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