October 2012

NASA/TM–2012-217778

Nonlinear Aerodynamic Modeling From Flight 

Data Using Advanced Piloted Maneuvers and 

Fuzzy Logic

Jay M. Brandon and Eugene A. Morelli

Langley Research Center, Hampton, Virginia
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October 2012

NASA/TM–2012-217778

Nonlinear Aerodynamic Modeling From Flight 

Data Using Advanced Piloted Maneuvers and 

Fuzzy Logic

Jay M. Brandon and Eugene A. Morelli

Langley Research Center, Hampton, Virginia
Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802

Acknowledgements

This project was made possible by the close work, cooperation, and resources provided

by the National Test Pilot School, Mojave, CA. Thanks to the engineering, piloting, and 

technical staff who facilitated obtaining the flight data.

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an 

official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics 

and Space Administration.
Abstract

Results of the Aeronautics Research Mission Directorate Seedling 

Project Phase I research project entitled “Nonlinear Aerodynamics 

Modeling using Fuzzy Logic” are presented. Efficient and rapid flight 

test capabilities were developed for estimating highly nonlinear models 

of airplane aerodynamics over a large flight envelope. Results showed 

that the flight maneuvers developed, used in conjunction with the fuzzylogic system identification algorithms, produced very good model fits of 

the data, with no model structure inputs required, for flight conditions 

ranging from cruise to departure and spin conditions.

Introduction

This report summarizes the results from a NASA 

Aeronautics Research Mission (ARMD) Seedling 

Phase I project entitled, “Nonlinear 

Aerodynamics Modeling using Fuzzy Logic”.

Mathematical models are required for analysis 

and understanding of any physical system. One 

challenge with designing and testing aircraft that 

operate over large flight envelopes or are 

unconventional in design is the identification of 

proper aerodynamic models and model structures. 

The aerodynamic characteristics may be functions 

of many variables and can vary in nonlinear and 

time-dependent ways. 

The current state-of-the-art for nonlinear 

aerodynamic model structure determination 

involves selection of terms from a pool of 

postulated modeling terms, sometimes in an 

iterative fashion, based on data transformations 

and statistical modeling metrics computed from 

the data. In this research, advanced modeling 

techniques using fuzzy logic are used to advance 

the state-of-the-art in nonlinear aerodynamic 

modeling in a way that does not require 

specification of candidate modeling terms other 

than selection of a set of explanatory variables to 

use in the model development. The ability to 

model the nonlinear behaviors that exist due to 

separated aerodynamic flow, structural / 

propulsive / aerodynamic / flight control 

interactions, mass properties changes, etc., is 

absolutely critical to the successful and efficient 

development of future novel airplanes that will be 

needed to fulfill the goals of increased 

performance for future aircraft. This is a key 

component in the NASA “Learn-to-Fly” concept, 

where the intent is to autonomously develop 

vehicle characterization and control strategies, up 

through the ability to fly a vehicle, with minimum 

human interaction and time. The ability to 

rapidly update simulation tools (either 

aerodynamic models, or flight response models) 

based on flight test can be improved with the 

results of this work, and the results are applicable 

to both piloted and autonomous vehicles.

The focus of the research was to enable the 

identification of nonlinear aerodynamic models 

over a large flight envelope with very little flight 

test time required. Novel, continuously-varying 

manual control inputs were applied as flight 

conditions changed, resulting in rich data content 

from the flight test for model development across 

large flight envelopes – including post-stall 

conditions. The use of fuzzy logic system 

identification allowed for aerodynamics modeling 

without need for formal model structure 

determination by the analyst. This work builds 

on previous studies using fuzzy logic for system 

identification1

, and on maneuver design concepts 

using uncorrelated multi-axis flight test inputs2,3.

Symbols

ax, ay, az body-axis translational 

accelerations, g
2

A(x) membership function value for 

variable x

b reference wing span, ft

c̅ reference mean aerodynamic 

chord, ft

Cl, Cm, Cn nondimensional aerodynamic 

rolling, pitching, and yawing 

moment

Cx, Cy, Cz nondimensional body-axis 

aerodynamic force coefficients

g graviational acceleration, 32.174 

ft/sec2

M Mach number

Ns model search stage number

p coefficient of membership 

function 

P value of membership function

p, q, r body-axis roll, pitch, and yaw 

rates, deg/s

p̂or phat nondimensional pitch rate, p̂= pb

2V

q� dynamic pressure, psf

q� or qhat nondimensional pitch rate, q� = qc̅

2V

r̂or rhat nondimensional pitch rate, r̂= rb

2V

R2 multiple correlation coefficient, 

R2 = 1 − ∑ (y�j−yj) m 2 j=1

∑ (y�j−yj) m 2 j=1

y output to be modeled

y� mean of the output to be modeled

y� output estimate

α angle of attack, deg

β angle of sideslip, deg

δ control deflection, deg

ω angular rate

Subcripts:

a aileron

e elevator

r rudder

dot time rate of change

Abbreviations:

CG Center of gravity

Hp Pressure altitude, ft

KCAS Calibrated airspeed, kt

NTPS National Test Pilot School

SSE Sum of squared errors, 

SSE = ∑ (y�j − yj) m 2

j=1

Approach

Flight maneuver development was an important 

aspect of the project. The data from the flight 

tests must have enough information content to be 

suitable for identifying the aerodynamic effects. 

Therefore, several input types were used in flight 

during this research, ranging from traditional 

doublet, single-axis inputs, multi-step inputs, and 

a range of small and larger amplitude “fuzzy” 

(semi-random, variable frequency and amplitude) 

maneuvers – both single axis and simultaneous 

multi-axis control inputs. Addtionally, inputs 

were superimposed on gradually changing flight 

conditions (decelerations or accelerations to 

change nominal angle of attack), and also through 

stall and post-stall flight conditions.

After obtaining flight data, preprocessing of the 

data was done to reduce sensor systematic errors, 

reduce noise effects, generate necessary 

computed parameters from flight data such as 

dynamic pressure, and compute overall 

nondimensional forces and moments. This 

preprocessing was required before modeling work 

began. After the data preprocessing steps were

completed, fuzzy logic system identification 

algorithms were used to develop nonlinear 

mathematical models of the aerodynamics of the 

airplane over a wide range of flight conditions. 

Fuzzy Logic Model Development

An objective of the Phase I project was to 

develop a system identification approach without 

a requirement for prior specification of model 

structure for nonlinear and multivariable 

aerodynamic responses of an aircraft in flight. 

Several tasks had to be accomplished to 

successfully complete this research. 

First, the development of the fuzzy logic 

algorithm was completed based on work done 

previously1 with improvements in the underlying 

membership functions and in several other 

aspects. A schematic of the fuzzy logic algorithm 

and procedures used to analyze flight data are 
3

shown in Figure 1. The process steps are outlined 

below:

1. Pre-processing: Obtain flight data and make 

the required corrections to be able to compute 

overall forces and moments acting on the 

aircraft and the aircraft states. This includes 

such things as airspeed and altitude 

corrections, conversion to pressures, 

computation of ambient air temperature from 

measured total temperature, correction of 

angle of attack and sideslip vane data for 

sensor location, etc.

2. Selection of Explanatory Variables: Flight 

variables are selected that the algorithm will 

use to develop mathematical models for the 

nondimensional aerodynamic coefficients. 

Although no functional form of the models is 

needed to be specified (such as interactions 

between variables, etc. – those come from the 

fuzzy logic algorithm), the variables that will 

be used in a resulting model need to be 

defined. The data is partitioned into training 

and testing databases.

3. Calculation of Membership Functions: 

Membership functions are used to divide 

explanatory variables into various ranges. 

The interpretation of membership function 

values is as follows: the membership function 

value is 1 over the range of values of the 

associated explanatory variable where that 

explanatory variable is very important. For a 

membership function value of 0, the 

explanatory variable is not important over that 

range of values. When only one membership 

function is selected, it has a constant value of 

one over the entire range of the explanatory 

variable. With only one membership 

function for a selected explanatory variable, 

the fuzzy model is identical to a standard 

linear model. An example of selection of 1, 

2, or 3 membership functions for a particular 

explanatory variable, x, is shown in Figure 2. 

A set of membership functions is graphically 

depicted in Figure 1 for a case with 6 

membership functions for a given explanatory 

variable, x. All explanatory variable inputs 

into the fuzzy algorithm are scaled to be 

between 0-1.

4. Calculation of Internal Function Coefficients 

for all Fuzzy Cells: The fuzzification of the 

data is accomplished by the use of many 

fuzzy cells, each comprised of a linear 

equation in the explanatory variables through 

the use of their associated membership 

functions. Each fuzzy cell internal function 

includes a combination of one membership 

function for each explanatory variable. A 

coefficient for each of the membership 

function variables in each fuzzy cell is 

determined by gradient optimization or least 

squares procedures to minimize the 

summation of squared errors using the set of 

data partitioned for training. Currently an 

exhaustive set of all combinations of 

membership functions for each explanatory 

variable is used to populate the fuzzy cells. 

Therefore, the models can become large.

5. Calculate Model Output: With the fuzzy cells 

defined, the estimates of the model are 

computed using a weighted output of the 

fuzzy cells obtained by multiplication of the 

membership functions to determine the subset 

of fuzzy cells that will be important in the 

model estimate.

6. Check for Adequacy of Model: The multiple 

correlation coefficient (R2

) of the fit of the 

model to the training data is evaluated. If it is 

above a chosen threshold (indicating a good 

model fit to the data), then that fuzzy model is 

applied to the set of data for testing. If the fit 

of the training data is not good enough, or if 

the testing data results are continuing to show

a better fit of the data as more membership 

functions are added, more membership 

functions are added to the model, and the 

training and testing processes are re-done.

7. Compute Final Fuzzy Model: After reaching 

an acceptable fit of the flight data with the 

training data, and when the testing data R2

value starts to decrease, the fit is considered 

complete and all of the data (training and 

testing) is used in a final estimation procedure 

to compute the final coefficients of all the 

internal functions in the fuzzy cells.
4

 

Figure 1 Fuzzy logic system identification flow diagram

Figure 2 Example of membership function orders.

Flight Test Maneuvers

A piloted flight test technique was developed, 

called “fuzzy inputs”, that resulted in rich 

dynamic content from the flight test over a large 

range of flight conditions with a small number of 

maneuvers. These maneuvers were hand-flown 

approximations to optimized inputs already tested 

on other aircraft using programmed maneuvers2,3, 

applied over a very large flight envelope range 

including conditions representing cruise, 

maneuvering, stall, departure, and post-stall 

departure dynamics. The intent of the maneuvers 

was to apply uncorrelated, multi-axis inputs to 

produce uncorrelated multi-axis aircraft responses

with signal-to-noise ratios high enough for 

accurate aerodynamics modeling. The pilot’s 

task was to input commands without the 

maneuvers prescribed in detail before-hand, to 

attempt as large of a variation combinations of 
5

airplane states with as little correlation as possible 

between the piloted inputs across axes.

Test Aircraft Description

The test airplane was an Aermacchi Impala MB326M operated at the National Test Pilot School 

in Mojave CA, shown in Figure 3. The Impala is 

a 2-seat, tandem single-engine turbojet, and is 

used as a trainer or light attack aircraft. This 

particular airplane is fitted with a flight test 

noseboom with angle-of-attack and sideslip angle 

measurement vanes. Airdata is provided through 

the production system. The primary flight control 

system is completely reversible, with control 

pushrods connecting the stick and pedals with the 

ailerons, elevator, and rudder. All control 

surfaces are statically and dynamically balanced. 

The aileron aerodynamic balancing is obtained by 

Irving diaphragms and balance tabs on the trailing 

edge. The elevator is aerodynamically balanced 

by two tabs on the trailing edge. Trim for roll 

and pitch forces is obtained though a coolie-hat 

switch on the control sticks. Trim tabs are 

actuated by electromechanical servos. Trim 

indications are available in both cockpits. No 

springs or bobweights are included in the control 

system. Secondary flight controls are powered by 

the hydraulic system. The speedbrake is on the 

bottom surface of the fuselage, and the deflection 

is automatically limited to approximately half 

travel when the gear is extended to preclude the 

speed brake from striking the ground. Flaps are 

selectable in 3 positions: up, take-off, and 

landing. Steering is accomplished on the ground 

by differential braking and a castoring nosewheel.

The airplane is powered by a single Rolls Royce 

Viper turbojet engine. The engine produces 

approximately 2500 lb of thrust at sea level static 

conditions. The maximum gross weight is 9600 

lb (basic airplane with tiptanks). For this test, the 

weight of the fueled airplane with crew was 

approximately 8185 lb. 

Each cockpit is equipped with a Martin-Baker 

MK-AS.06A/M ejection seat that allows ejection 

from the airplane at all speeds and flight altitudes 

down to zero speed and altitude. Flight controls 

and flight instruments are repeated in both 

cockpits. A cabin pressurization system 

maintains cabin pressure and provides heat and 

air conditioning. A single canopy covers both 

cockpits.

Figure 3 Test Aircraft.

Reference geometric characteristics of the 

airplane are listed in Table 1.

Table 1 Airplane Geometry

Wing Area (includes 

projected area to centerline)

204.5 ft2

Wing Span 34.659 ft

c̅ 6.233 ft

FS leading-edge of c̅ 327.72 in

Height 12.20 ft

Distance between main 

wheels

7.78 ft

Distance from main gear to 

nose gear

13.64 ft

A 3-view sketch of the airplane is shown in

Figure 4, although the sketch does not include the 

nose boom which is fitted onto the test airplane. 

The FS, WL, and BL stations are all in inches. 

Positive numbers are: FS aft, WL up, BL out the 

right wing.
6

Figure 4 Three-view sketch of airplane.

The maximum gross weight of the airplane is 

9,600 lb. The allowable CG range is 22% - 30% 

c̅. The weight and balance limitations can not be 

exceeded by normal operating or loading 

conditions. Table 2 shows the typical CG burn 

diagram for the airplane as tested, and Table 3

lists the inertia estimates available for the test 

airplane.

Measurements recorded with the onboard data 

system and telemetered to the ground included: α, 

β, ax, ay, az, p, q, r, δa, δe, δr, KCAS, and Hp. 

Table 4 shows the location of the inertial sensors 

and α/β vanes. Additionally, total temperature, 

fuel used, and engine RPM were recorded by 

hand for each flight maneuver. A second set of 

inertial data was recorded with a tablet computer 

mounted in the aft cockpit (Figure 5) including 

additional measurements of ax, ay, az, p, q, r, and 

GPS position and velocities.

Table 2 Weight and Balance

Table 3 Inertia Estimates

Table 4 Sensor Locations

Figure 5 Tablet computer installation in rear 

cockpit.

Ixx Iyy Izz

Full Fuselage Fuel, Tip Tanks 

Full, 2 Crew 14515 7880 21900

300 lb Fuselage Fuel, Tip Tanks 

Empty, 2 Crew 4785 7830 12075

Configuration Moments of Inertia, slug-ft2

FS WL BL

AHRS (rate gyros and 

accelerometers) 334.98 28 1.25

α vane 124.04 10 6.25

β vane 120.42 16.25 0

Location, in Sensor
7

Results from Phase I

All milestones and plans proposed for Phase I 

ARMD Seedling Fund research, which funded 

this work, have been completed. The 12 planned 

data flights were conducted to test the 

development of nonlinear aerodynamic models 

from flight test using fuzzy logic and more 

conventional analysis tools. 

The aerodynamic forces and moments were 

nondimensionalized to the standard coefficients 

of body-axis forces (Cx, Cy, Cz) and moments (Cl, 

Cm, Cn). One of the most significant aspects of 

the project was the development and 

demonstration of test and analysis techniques that 

allowed for development of models 

encompassing a large range of flight conditions 

using very little flight test time. The flight testing 

used “fuzzy maneuvers”, which ideally were 

airplane excitations implemented by the pilot in 

which quantities used for the aerodynamic 

modeling (control surface positions and airplane 

responses) were varied in an uncorrelated manner 

over a large range. An example comparison of a 

traditional multi-axis doublet maneuver and a 

multi-axis fuzzy input maneuver is shown in 

Figure 6. Note that fuzzy maneuvering 

decorrelated the explanatory variables by having 

the pilot fly the aircraft to enforce that condition, 

whereas conventional doublet maneuvers rely on 

moving controls one at a time to bring about 

decorrelation through time sequencing. 

Figure 6 Comparison of traditional doublet inputs and fuzzy inputs.

With practice, the multi-axis combined inputs 

were achieved by the pilot with a crosscorrelation coefficient of about 0.3 – indicating a 

good level of independence between the inputs in 

each of the axes. Using maneuver designs such 

as orthogonal sum-of-sines control inputs that are 

programmed and executed directly by a 
8

computer, even better independence of inputs can

be achieved2. 

Using one fuzzy maneuver, a nonlinear 

aerodynamic model can be developed that covers 

a large range of flight conditions from low angle 

of attack cruise up through stall angles of attack. 

Figure 6 shows that the fuzzy maneuver evokes 

more dynamic response from the airplane than a 

conventional doublet maneuver, which is 

important for accurate and efficient modeling. 

Furthermore, the fuzzy maneuver covered a range 

of flight conditions with a single maneuver. 

Twelve data flights were conducted during this 

research project, see Table 5. After analysis of 

the first 6 flights, data inconsistencies were 

observed that led to an extensive effort to correct 

and recalibrate the flight test instrumentation. It 

was found that several of the instrumentation 

channels were affected very significantly by

changes in temperarture, and there were 

indications of some data channels interacting with 

other data channels. Because of the data system 

problems, models developed with data from one 

maneuver and flight would not produce consistent 

results with models developed from similar 

maneuvers later in the flight or from other flights. 

As a result, improvements in the data system 

were made, and the first 6 flights were largely 

ignored for detailed model work. Due to time 

and funding constraints, the instrumentation 

system was not fully corrected and calibrated;

however, the second set of 6 flights were 

conducted with a much more consistent system

with better data quality.

Table 5 Data Flight Summary

Flight Primary Maneuvers / Objectives

1 Systems checks and model verification

2 Systems checks and model verification

3 Multi-step and fuzzy step inputs

4 Decels and larger amplitude fuzzy inputs

5 Air data cal and small amplitude highspeed flight inputs

6 Stall and post-stall inputs

7 Conventional doublet maneuvers and low 

amplitude inputs across α range

8 Fuzzy inputs at discrete flight conditions

9 Large amplitude fuzzy maneuvers and 

thrust modeling maneuvers

10 Departures - spins

11 Small amplitude maneuvers across 

envelope

12 Thrust modeling maneuvers, steady 

sideslips, small and moderate fuzzy 

maneuvers

Using the fuzzy inputs and the resultant flight 

data, the model fit to nondimensional pitching 

moment coefficient using the fuzzy algorithm 

described above is shown in Figure 7, for a 

maneuver from Flight 9. Explanatory variables 

selected for use by the fuzzy logic algorithm 

were: α, β, δe, δa, δr, p̂, q�, r. The selection of 

explanatory variables was made to include 

possible effects of coupling of lateral-directional 

and longitudinal effects. The correlation 

coefficient for the fit is high (0.96), indicating a 

good fit, as can be seen in the plot of the 

identified model output and flight data. This 

example is one model for flight conditions 

corresponding to cruise flight all the way to stall.
9

Figure 7 Example fuzzy model fit to pitching moment data.

Another example of the fuzzy logic model fit to 

the flight data, using the same expanatory 

variables from Flight 9, is shown in Figure 8. 

This demonstrates the hysteresis and overshoots 

in the vertical force coefficient (similar to the lift 

coefficient) seen in dynamic maneuvers through 

the stall angle of attack using different angle of 

attack rates. The fuzzy logic modeling accurately 

characterizes these nonlinear effects in the flight 

data. 

0 20 40 60 80 100 120 140 160

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Cm

Time, sec

 

 

Data

Model, R2

 = 0.9649
10

Figure 8 Model fit to Normal force coefficient.

An example of a model from the fuzzy maneuver,

showing values of some standard stability 

derivatives linearized along the normal trim 

conditions, is shown in Figure 9. Also shown in 

Figure 9 is a comparison of results from flight test 

data obtained with doublet inputs and a leastsquares equation error modeling method. The 

doublet data were obtained from 20 maneuvers 

over 6 different data flights. The fuzzy model 

was obtained with one maneuver during one 

flight and used the explanatory variables listed in 

Table 6. The expalnatory variable, α̇, was added 

because of results including departure and spin 

data, it was seen to provide better predictions of 

flight data not used in the model development –

some examples of the predictive capability of the 

models will be presented later. The comparisons 

with standard doublet techniques illustrate that 

the flight test technique employing fuzzy inputs 

provided high information content in an highly 

efficient manner, compared to conventional trimbased approaches. Of even more significance, the 

conventional results yield no modeling results for 

angle of attack above stall (about α = 12°) due to 

constraints of the employed method to be 

perturbations about a trim condition, whereas the 

fuzzy modeling results go through the stall and 

departed regions of flight. 

Comparison of the results shown in Figure 9 

shows excellent correlation of the static terms

from the two different methods. The 

conventional method shows more scatter for the 

dynamic derivatives and less damping predicted. 

The reduced pitch rate damping compared to the 

fuzzy model is likely due to the lack of the 

explanatory variable of angle-of-attack rate in the 

conventional results that were part of the fuzzy 

-10 0 10 20 30 40

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α, deg

-C

Z

 

 

Data

Model, R2

 = 0.99625
11

logic model. Although the fuzzy logic modeling 

developed the complex nonlinear aerodynamic 

model without an analyst specificying a model 

structure beyond providing a set of candidate 

explanatory variables, other modeling approaches 

could use the same fuzzy input data and provide 

the improvement in test efficiency demonstrated 

by this test approach.

Figure 9 Linearized fuzzy model results from one maneuver and comparison with traditional 

results.

Combining several maneuvers from Flight 10 so 

that characteristics at extreme flight conditions 

could be included in the model was done by 

including a deceleration maneuver such as shown 

on the right side of Figure 6 with maneuvers 

including departures that resulted in right and left 

upright spins, and a left inverted spin. Fuzzy 

logic models were then developed from this 

-5 0 5 10 15 20 25 30 35

-0.04

-0.02

0

Cm

α, per deg

DF9C4

-5 0 5 10 15 20 25 30 35

-0.03

-0.02

-0.01

0

Cmδe, per deg

-5 0 5 10 15 20 25 30 35

0

2

4 x 10-3

Cm

α-dot, per deg/sec

-5 0 5 10 15 20 25 30 35

-20

-10

0

Cmqhat

α, deg

Fuzzy

LESQ
12

combined set of maneuvers (all obtained during 

one flight) resulting in data with large angular 

rates and large attitude angles of approximately 

-40 ≤ α ≤ 40 and -20 ≤ β ≤ 20. Due to limitations 

of time and instrumentation, an engine thrust 

model was not developed, so the axial coefficient 

includes some thrust effects, although to 

minimize that effect, all combined runs had the 

power set to idle.

Model fit quality for the 6 components of the 

aerodynamics with the set of combined 

maneuvers covering the extended flight envelope 

are shown in Table 6. One set of explanatory 

variables was used in all the fits (across the flight 

envelope), so engineering judgement was made in 

selecting the state variables most likely to 

contribute to the aerodynamics. The chosen 

variables are also listed in Table 6 for each axis.

Table 6 Fuzzy Model Fit Quality with 

Combined Maneuvers

Coefficient Correlation 

Coefficient, R2

Explanatory 

Variables

Cx 0.983 α, β, δe, δa, δr, α̇, 

p̂, q�, r̂

Cy 0.967 α, β, δe, δa, δr, β̇

, 

p̂, q�, r̂

Cz 0.997 α, β, δe, δa, δr, α̇, 

p̂, q�, r̂

Cl 0.950 α, β, δe, δa, δr, β̇

, 

p̂, q�, r̂

Cm 0.971 α, β, δe, δa, δr, α̇, 

p̂, q�, r̂

Cn 0.964 α, β, δe, δa, δr, β̇

, 

p̂, q�, r̂

A test of model adequacy is whether or not the 

model can predict data that were not used in the 

development of the model. There were many 

instrumentation issues that made flight-to-flight 

data comparisons difficult in the present study;

however, some examples of the model predictive 

capability are presented below. A maneuver from 

Flight 8 (deceleration from cruise to stall 

conditions with fuzzy inputs) was used for an 

example case. The data from the Flight 8 

maneuver (explanatory variables) were input into 

the fuzzy logic models developed from the 

combined maneuvers from Flight 10, and the 

model outputs were compared to the calculations 

of force and moment coefficients directly from 

Flight 8 data. Figures 10 through 15 show the 

comparisons of the model and flight data, and the 

correlation coefficient for predictions on all 6 

components of aerodynamic forces and moments.
13

Figure 10 Model prediction of axial force.

0 20 40 60 80 100 120 140 160

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

CX

Time, sec

Data

Model, R2

 = 0.89658
14

Figure 11 Model prediction of side force.

0 20 40 60 80 100 120 140 160

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

CY

Time, sec

Data

Model, R2 = 0.94982
15

Figure 12 Model prediction of normal force.

0 20 40 60 80 100 120 140 160

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

CZ

Time, sec

Data

Model, R2 = 0.95651
16

Figure 13 Model prediction of rolling moment.

0 20 40 60 80 100 120 140 160

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Cl

Time, sec

Data

Model, R2 = 0.94666
17

Figure 14 Model prediction of pitching moment.

0 20 40 60 80 100 120 140 160

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Cm

Time, sec

Data

Model, R2 = 0.91229
18

Figure 15 Model prediction of yawing moment.

These results show good model comparisons with 

the flight data for this prediction case. The axial 

force coefficient shows the least accurate fit, with 

the correlation coefficient just under R2 = 0.9, 

which is likely due to the unmodeled thrust 

effects mentioned previously. All aerodynamic 

coefficient models exhibited a very high 

correlation coefficient for prediction over a large 

portion of the aircraft flight envelope. 

In summary, the Phase I ARMD Seedling Fund 

project experienced many of the typical issues 

and problems found in flight testing projects. 

Instrumentation issues including faulty electronic 

components and unknown sensitivities to changes 

in temperature resulted in inconsistent data over 

the first six flights. Project funding and time 

constraints did not allow for comprehensive data 

system development and checkout, but the data 

system was substantially improved for the second 

set of 6 data flights. An alternate inertial 

instrumentation system was installed (tablet 

computer) and was used with the ship data system 

to improve the accuracy of the flight test 

measurementes. Results show promise of 

substantial improvements in the capability for

developing models of airplane aerodynamics in 

flight regimes that traditional methods can not 

even obtain, and in the development of very 

nonlinear models rapidly with significantly 

reduced flight test time. Maturation of this 

technology could result in substantial cost and 

time savings in flight test programs, and can be 

an enabler for self-learning airplanes, resulting in 

reduced development cost of new vehicles and 

more robust and safe flight operations.

0 20 40 60 80 100 120 140 160

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Cn

Time, sec

Data

Model, R2 = 0.95421
19

Lessons Learned

The Phase I research project was an initial step to 

investigate methods to develop models of highly 

nonlinear aerodynamics from flight without 

needing to define model structure. Multi-axis

input techniques for piloted maneuver execution 

were developed to provide rich data with small 

amounts of flight time. During the testing, 

several issues were encountered that will need to 

be addressed in future research of this type. 

Data Parameters. Several parameters were 

recorded by hand due to the cost of adding 

additional data parameters to the existing data 

stream. This resulted in difficulties in reducing 

the flight data. Future tests should include engine 

(RPM, fuel used) and air temperature 

measurements in the automated data system. 

Instrumentation system checkout and calibration. 

Calibration and resolution of the data were 

problematic for many of the parameters. It was 

found after the sixth flight that several parameters 

had very large variations due to temperature 

changes, and cross-talk between channels was 

discovered. Pre-flight comprehensive data 

checks need to be conducted to develop 

confidence in the data system – including 

calibration checks with multiple-inputs-at-a-time, 

and checks for temperature sensitivities. 

Calibration for temperature effects can be made if 

appropriate temperature measurements are added.

Pre and post flight checks. All flights should use

procedures to identify if data values have drifted 

during flight. Starting and stopping the data 

system recording at the same location on the 

ramp, conducting control deflection checks on the 

ground, and including repeatable flight conditions 

on each flight can help identify problems. 

Implementing real-time modeling could also help 

to diagnose these conditions. 

Fuzzy logic model results can be sensitive to 

explanatory variable selection and can also 

generate very nonlinear results. Care must be 

taken to not over-parameterize the model, and 

future work is needed to develop criteria for 

optimizing the models and to assess the

magnitude of uncertainties of the model. 

Extrapolation of the models outside of data used 

to develop the model can result in unexpected 

results due to nonlinearities, so care needs to be 

taken in the development of the input ranges and 

use of the models.

Next Steps

The technology readiness level (TRL) of this 

technology is currently estimated at level 4. The 

concepts have been applied to flight data and

preliminary results have been obtained; however,

many refinements need to be completed to make 

the fuzzy logic algorithm ready for wide 

application, and real-time analysis needs to be 

included to support the “Learn-To-Fly” 

objectives. Next steps should include:

1. Advancement of fuzzy logic algorithms

a. Improvements in speed of model 

determination

b. Optimization of model size

c. Quantification of uncertainties

2. Development of real-time inflight maneuver 

assessment and pilot guidance

3. Demonstration of inflight maneuver 

assessment and pilot guidance

4. Demonstration of inflight aerodynamic model 

development

5. Demonstration of inflight model verification 

using flight maneuvers from current flight

6. Demonstration of development of 

aerodynamic models suitable for analysis and 

simulation post flight.

Concluding Remarks

A research study was conducted to develop 

efficient flight test and analysis methods for 

estimation of highly nonlinear aerodynamic 

models of a jet trainer aircraft. Flight data were 

obtained using simultaneous multi-axis 

perturbation inputs throughout large flight 

envelope excursions, resulting in flight data with 

rich information content for aerodynamic 

modeling over large portions of the flight 

envelope. A fuzzy logic algorithm was used to 

identify a model of the aerodynamics without any 
20

specification of the model structure beyond 

selection of explanatory variables. Excellent 

matching of flight data – even through stall and 

post-stall gyrations - was achieved with the flight 

test methods and fuzzy logic algorithms, showing 

promise for practical, rapid simulation model 

development directly from flight test data. 

References

1. Wang, Zhongjun; Lan, C. Edward; and 

Brandon, Jay M.: “Fuzzy Logic Modeling of 

Nonlinear Unsteady Aerodynamics”, AIAA 

98-4351.

2. Morelli, E.A. “Flight Test Maneuvers for 

Efficient Aerodynamic Modeling,” 

AIAA-2011-6672, AIAA Atmospheric Flight 

Mechanics Conference, Portland, OR, August 

2011. 

3. Morelli, E.A. “Efficient Global Aerodynamic 

Modeling from Flight Data,” AIAA-2012-

1050, 50th AIAA Aerospace Sciences Meeting, 

Nashville, TN, January 2012. 
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Nonlinear Aerodynamic Modeling From Flight Data Using Advanced 

Piloted Maneuvers and Fuzzy Logic 

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Brandon, Jay M.; Morelli, Eugene A.

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NASA Langley Research Center

Hampton, VA 23681-2199

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14. ABSTRACT

Results of the Aeronautics Research Mission Directorate Seedling Project Phase I research project entitled “Nonlinear 

Aerodynamics Modeling using Fuzzy Logic” are presented. Efficient and rapid flight test capabilities were developed for 

estimating highly nonlinear models of airplane aerodynamics over a large flight envelope. Results showed that the flight 

maneuvers developed, used in conjunction with the fuzzy-logic system identification algorithms, produced very good model 

fits of the data, with no model structure inputs required, for flight conditions ranging from cruise to departure and spin 

conditions.

15. SUBJECT TERMS

Aerodynamics; Flight tests; Fuzzy logic; System identification

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