NASA/TM-2003-212145
A Collection of Nonlinear Aircraft Simulations
in MATLAB
Frederico R. Garza
George Washington University
Joint Institute for the Advancement of Flight Sciences
Langley Research Center, Hampton, Virginia
Eugene A. Morelli
Langley Research Center, Hampton, Virginia
January 2003
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NASA/TM-2003-212145
A Collection of Nonlinear Aircraft Simulations
in MATLAB
Frederico R. Garza
George Washington University
Joint Institute for the Advancement of Flight Sciences
Langley Research Center, Hampton, Virginia
Eugene A. Morelli
Langley Research Center, Hampton, Virginia
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23681-2199
January 2003
The use of trademarks or names of manufacturers in the 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.
Available from:
NASA Center for AeroSpace Information (CASI)
7121 Standard Drive
Hanover, MD 21076-1320
(301) 621-0390
National Technical Information Service (NTIS)
5285 Port Royal Road
Springfield, VA 22161-2171
(703) 605-6000
Table of Contents
TABLES iv
FIGURES V
ABSTRACT vi
NOMENCLATURE vii
SUPERSCRIPTS: x
SUBSCRIPTS: x
NON-DIMENSIONAL DERIVATIVES: xi
Longitudinal xi
Lateral / Directional xi
L INTRODUCTION 1
II. EQUATIONS OF MOTION 2
ASSUMPTIONS 2
COLLECTED EQUATIONS OF MOTION 2
ENGINE MODEL 6
AERODYNAMIC MODEL 7
ATMOSPHERIC PROPERTIES 7
AIRCRAFT STATES AND CONTROLS 8
III. GENERAL SIMULATION TOOLS 9
TRIM 9
LINEARIZATION II
Detailed Linearization Example 12
NUMERICAL INTEGRATION 15
Euler 's Method 16
Runge-Kutta Method of Order Two 16
Runge-Kutta Method of Order Four 17
Adams-Bashforth Method of Order Three 17
Detailed Numerical Integration Example 17
IV. AVDS/MATLAB INTERFACE 21
EQUIPMENT / REQUIREMENTS 21
SOFTWARE INSTALLATION 21
INTERACTIVE SIMULATION 22
Joystick Control Input 23
Initial States for Interactive Simulation 23
AVDS / MATLAB Piloted Simulation Software 24
Stick Gain Modification 26
Control Deflection Exaggeration Factor 27
Integration Routine 27
Starting the AVDS / MATLAB Simulation 27
V. SPECIFIC AIRCRAFT SIMULATION DETAILS 29
F- 16 SIMULATION 29
Aerodynamic Model and Database 30
Engine Model and Database 31
Mass Properties 33
F- 1 06B SIMULATION 33
Aerodynamic Model and Database 34
Engine Model and Database 35
Mass Properties 35
F- 14 SIMULATION 36
Aerodynamic Model and Database 37
Engine Model and Database 39
Mass Properties 39
F-4 SIMULATION 39
Aerodynamic Model and Database 40
Engine Model and Database 45
Mass Properties 45
FASER SIMULATION 45
Aerodynamic Model and Database 46
Engine Model and Database 48
Mass Properties 49
HL-20 SIMULATION 49
Aerodynamic Model and Database 50
Dynamic Equations 53
Mass Properties 53
11
X-31 SIMULATION 54
Aerodynamic Model and Database 55
Engine Model and Database 57
Mass Properties 57
A-7 SIMULATION 58
Aerodynamic Model and Database 59
Engine Model and Database 60
Mass Properties 60
VI. GENERATING 3-D MODELS FOR AVDS 62
MODEL CONSTRUCTION REQUIREMENTS 62
MODELING EXAMPLE OF A SIMPLE 3-D AIRCRAFT 62
IMPORTING A COMPLETED 3-D CAD MODEL INTO AVDS 66
VII. CONCLUDING REMARKS 68
Vm. ACKNOWLEDGEMENTS 69
IX. APPENDIX 70
X. REFERENCES 75
HI
Tables
Table 1 . Mass Properties of the Simulated F-16 33
Table 2. Mass Properties of the Simulated F-106B 36
Table 3. Mass Properties of the Simulated F-14 39
Table 4. Mass Properties of the Simulated F-4 45
Table 5. Mass Properties of the Simulated FASER 49
Table 6. Mass Properties of the Simulated HL-20 54
Table 7. Mass Properties of the Simulated X-3 1 58
Tables. Mass Properties of the Simulated A-7 61
Table A-1. F-16 Simulation Package (Directory name: F-16) 70
Table A-2. F-106B Simulation Package (Directory name: F-106B) 71
Table A-3. F-14 Simulation Package (Directory name: F-14) 72
Table A-4. F-4 Simulation Package (Directory name: F-4) 72
Table A-5. FASER Simulation Package (Directory name: FASER) 73
Table A-6. HL-20 Simulation Package (Directory name: HL-20) 73
Table A-7. X-31 Simulation Package (Directory name: X-31) 74
Table A-8. A-7 Simulation Package (Directory name: A-7) 74
IV
Figures
Figure 1. Flowchart for Numerical Linearization 12
Figure 2. Flowchart for Nonlinear Simulation 16
Figure 3. Elevator Perturbation 18
Figure 4. AVDS Aircraft State Initialization Window 24
Figure 5. Interactive Simulation Flowchart 25
Figure 6. Screen Shot of the F-16 AVDS / MATLAB Interactive Simulation .. 26
Figure 7. Three- View of the F-16 30
Figure 8. Three- View of the F-106B 34
Figure 9. Three-ViewoftheF-14 37
Figure 10. Three- View of the F-4 40
Figure U. Three- View of the FASER 46
Figure 12. Three- View of the HL-20 50
Figure 13. Three- View of the X-3 1 55
Figure 14. Three- View of the A-7 59
Figure 15. Three-ViewoftheMiG-15 Aircraft 63
Figure 16. Front and Rear Fuselage Openings 64
Figure 17. Modeled MiG- 15 Fuselage Section 64
Figure 18. 3-D Model with Wing, Tail, and Control Surfaces 65
Figure 19. Model with Wire Frame Canopy 65
Figure 20. Completed 3-D MiG- 15 Model 66
Figure 21. Input Screen for ^ICa^^me.^/m./m 67
ABSTRACT
Nonlinear six degree-of-freedom simulations for a variety of aircraft were created
using MATLAB. Data for aircraft geometry, aerodynamic characteristics, mass / inertia
properties, and engine characteristics were obtained from open literature publications
documenting wind tunnel experiments and flight tests. Each nonlinear simulation was
implemented within a common framework in MATLAB, and includes an interface with
another commercially-available program to read pilot inputs and produce a three-
dimensional (3-D) display of the simulated airplane motion. Aircraft simulations include
the General Dynamics F-16 Fighting Falcon, Convair F-106B Delta Dart, Grumman F-14
Tomcat, McDonnell Douglas F-4 Phantom, NASA Langley Free-Flying Aircraft for Sub-
scale Experimental Research (FASER), NASA HL-20 Lifting Body, NASA / DARPA X-
31 Enhanced Fighter Maneuverability Demonstrator, and the Vought A-7 Corsair II. All
nonlinear simulations and 3-D displays run in real time in response to pilot inputs, using
contemporary desktop personal computer hardware. The simulations can also be run in
batch mode. Each nonlinear simulation includes the full nonlinear dynamics of the bare
airframe, with a scaled direct connection from pilot inputs to control surface deflections
to provide adequate pilot control. Since all the nonlinear simulations are implemented
entirely in MATLAB, user-defined control laws can be added in a straightforward
fashion, and the simulations are portable across various computing platforms. Routines
for trim, linearization, and numerical integration are included. The general nonlinear
simulation framework and the specifics for each particular aircraft are documented.
VI
NOMENCLATURE
a angle of attack, deg or rad
/5 angle of sideslip, deg or rad
At numerical integration time step, s
^a, left Isft aileron control surface deflection, deg
^a, right ^ight ailcrou control surface deflection, deg
d^ aileron control surface deflection = ( d^ ,^^^1 " ^a. left )/^ . deg
d^ canard control surface deflection, deg
d^f differential flap deflection, deg
S^ elevator control surface deflection, deg
du- leading edge flap deflection, deg
di landing gear deflection, deg
d^f negative flap deflection, deg
dpf positive flap deflection, deg
d^ rudder control surface deflection, deg
d^ij speed brake deflection, deg
d symmetric wing spoiler deflection, deg
dff trailing edge flap deflection, deg
dff^ throttle deflection
^th left l^ft engine throttle deflection
^th.right right engine throttle deflection
Sj^ outboard thrust- vectoring paddle deflection, deg
Euler roll angle, rad
vn
Y flight path angle, rad
Euler pitch angle, rad
p atmospheric density, slug/ft^
r engine power level time constant, s
W Euler yaw angle, rad
ft>g„g angular velocity of the engine rotating mass about the X body-axis, rad/s
ax linear acceleration along X body-axis, g
ay linear acceleration along Y body-axis, g
a^ linear acceleration along Z body-axis, g
A system matrix
b wing span, ft
B input matrix
c mean aerodynamic chord, ft
C output matrix
c vector of mass / inertia properties
C1...C9 inertia constants
Ca non-dimensional axial force coefficient, C^ = -Cx
Cd drag coefficient
Cr, minimum drag coefficient
mill
Ci non-dimensional rolling moment coefficient
Cl lift coefficient
Ci lift-curve slope
Cm non-dimensional pitching moment coefficient
Cn non-dimensional normal force coefficient, C^ = -C^
Cn non-dimensional yawing moment coefficient
Cx non-dimensional X body-axis force coefficient
Cy non-dimensional Y body-axis force coefficient
viu
Cz non-dimensional Z body-axis force coefficient
D feed-through matrix
g earth gravitational acceleration = 32.174 ft/s
h altitude above mean sea level, ft
hg^g engine angular momentum about X body-axis, slug-ft^/s
/ X body-axis moment of inertia for the engine rotating mass, slug-ft
Ix X body-axis moment of inertia, slug-ft
Ixz X-Z body-axis product of inertia, slug-ft^
ly Y body-axis moment of inertia, slug-ft
1 2 Z body-axis moment of inertia, slug-ft
m aircraft mass, slugs
M Mach number
0[ ) order
p X body-axis angular velocity component, rad/s
P^ commanded engine power level, percent
P^ actual engine power level, percent
q Y body-axis angular velocity component, rad/s
q dynamic pressure, Ib/ft^
r Z body-axis angular velocity component, rad/s
S wing reference area, ft^
T engine thrust, lb
*
T temperature factor
Tidie idle engine thrust, lb
Ti rolling moment about X body-axis generated by thrust, ft-lb
Tm pitching moment about Y body-axis generated by thrust, ft-lb
Tmax maximum engine thrust, lb
Tmax.ab cuginc thrust with maximum afterburner, lb
IX
Tmii military engine thrust, lb
Tmin.ah engine thrust with minimum afterburner, lb
Tn yawing moment about Z body-axis generated by thrust, ft-lb
Tx X body-axis thrust component, lb
Ty Y body-axis thrust component, lb
Tz Z body-axis thrust component, lb
u X body-axis translational velocity component, ft/s
u control vector
Y body-axis translational velocity component, ft/s
Vf airspeed, ft/s
w Z body-axis translational velocity component, ft/s
X state vector
Xc.g. longitudinal center of gravity location, fraction of c
Xc.g.,ref reference longitudinal center of gravity location, fraction of c
%£ X earth-axis position, ft
y output vector
y^ Y earth-axis position, ft
V
SUPERSCRIPTS:
time derivative
SUBSCRIPTS:
basic or trim value
left
left
right
right
S
stability axes
E
earth axes
NON-DIMENSIONAL DERIVATIVES:
Longitudinal
Ci,
_dCx
di
Cx,
_dCx
di
c-,-
dCz
di
c„.
_^c„
di
where / = F^, a, S^, and/or other controls
Lateral / Directional
s=
dCj,
2F,
^-.
_dCx
2V,
^^.
^qc
s
^qc
2V,
J dj
2V,
^K
2V,
where j = JS.S^, S,. , and/or other controls and k = p,r, pg, r^
XI
I. INTRODUCTION
Nonlinear aircraft simulations are useflil for dynamic analysis, control law design and
validation, guidance and trajectory studies, air combat investigations, pilot training, and
many other tasks. Most real-time nonlinear aircraft simulations are implemented in
programming languages such as FORTRAN or C++, for historical reasons or to achieve
the required execution speed. Advances in desktop computing capabilities have made it
possible to implement real-time nonlinear aircraft simulations in an advanced computing
environment called MATLABi, using relatively inexpensive desktop personal computer
hardware. MATLAB is easy to learn and program, is portable across many computing
platforms without modification, and is a popular tool for scientific and engineering
computations. The speed of current desktop computing hardware is sufficient for real-
time execution of nonlinear aircraft simulations programmed in MATLAB, including
numerical integration of the nonlinear equations of motion, general nonlinear
aerodynamic and engine model computations, and interface with software to read pilot
inputs and display of a pilot view or a 3-D view of the aircraft in motion from an
arbitrary vantage point outside the aircraft.
In addition to these computational hardware and software capabilities, reports
available in the open literature contain aircraft geometry, mass / inertia properties, and
extensive wind tunnel and flight test aerodynamic data for a variety of different aircraft,
many of which are still in service. Data on engine performance is less common in the
open literature, but available nonetheless.
This report describes realistic and easy-to-use nonlinear aircraft simulations
implemented in MATLAB for a variety of different aircraft whose particular
characteristics were available in the open literature. The nonlinear simulations include a
general framework for the nonlinear equations of motion and a set of utility programs
that apply to all the aircraft. For each individual aircraft, routines that implement the
speciflc aerodynamic, propulsion, and mass / inertia characteristics of the aircraft were
developed and documented.
Aircraft simulations included in this report are the General Dynamics F-16 Fighting
Falcon, Convair F-106B Delta Dart, Grumman F-14 Tomcat, McDonnell Douglas F-4
Phantom, NASA Langley Free-Flying Aircraft for Sub-scale Experimental Research
(FASER), NASA HL-20 Lifting Body, NASA / DARPA X-31 Enhanced Fighter
Maneuverability Demonstrator, and the Vought A-7 Corsair IL Complete MATLAB
simulations for each individual aircraft were developed in independent software
packages. Each software package includes the capability to calculate aircraft trim
conditions, linearize the equations of motion, run batch simulations of linear and
nonlinear aircraft dynamics, and perform real-time interactive flight simulation. A
commercially-available program called Aviator Visual Design Simulator (AVDS)^ is
used to interface each MATLAB nonlinear aircraft simulation with a human pilot in real
time. Software to run each MATLAB simulation and exchange data with AVDS in real
time is included in the simulation packages, along with 3-D model deflnition files for
each aircraft. The 3-D model definition files are used by AVDS to create the 3-D display
of the aircraft in fiight.
Full documentation of each nonlinear aircraft simulation is included in this work,
along with instructions for implementing the simulation software. Examples illustrate the
steps required to execute the trim, linearization, and nonlinear simulation routines, and
run the MATLAB nonlinear aircraft simulations in real time using AVDS. A short
tutorial detailing the creation and importation of a 3-D Computer- Aided Design (CAD)
aircraft model into AVDS is also provided.
II. EQUATIONS OF MOTION
Aircraft nonlinear equations of motion were implemented in MATLAB. Nonlinear
aerodynamic and propulsion models were used to simulate the applied forces and
moments for each aircraft. Usually, the aerodynamic and propulsion models were
implemented as linear interpolations using look-up tables in a database. In some cases,
analytic polynomial models were used to model the aerodynamic and propulsion
dependencies on state and control variables. Some simplifying assumptions were made
in the development of the nonlinear aircraft simulations, as described next.
ASSUMPTIONS
Each modeled aircraft was assumed rigid with constant mass density and symmetry
about the X-Z plane in body axes. Forces and moments acting on the aircraft were from
aerodynamics, propulsion, and gravity. Thrust was assumed to act along the X body-axis
and through the center of gravity. A stationary atmosphere was assumed, with aircraft
fiight limited to altitudes less than 50,000 feet and subsonic speeds. The earth's
curvature was ignored and the earth was assumed to be fixed in inertial space, so that
earth axes were considered inertial axes. The gravity field was assumed to be uniform, so
that the center of mass and the center of gravity were coincident, and there were no
gravity moments or changes in gravity with altitude.
Full nonlinear six-degree-of-freedom rigid body translational and rotational aircraft
motion was modeled, neglecting relative motion of the aircraft internal components,
structural distortion, and sloshing of liquid fuel. The gyroscopic effects caused by
rotating engine turbomachinery were included for some aircraft. Airplane rotation
conventions follow the usual right-hand rule. Control surface defiections follow a
standard right-hand rule sign convention, in which the fingers curl in the direction of
positive control surface deflections when the thumb is pointed along the control surface
hinge line in the direction of a positive body axis.
COLLECTED EQUATIONS OF MOTION
The six nonlinear rigid-body equations of motion were used to model the nonlinear
aircraft dynamics in translational and rotational motion. These equations are^-^;
u = rv-qw- gsin0 +
qSCx + T
m
V = pw -ru + g cos sin +
qSCy
m
w = qu - pv + g cos cos +
qSCz
m
(1)
P^X - rlxz = ^SbCi -qr{lz-lY) + qplxz
qlj = qScC^-pr{lx-Iz)-[p^-r^]lxz-rKng
rlz - Pixz = qSbCn -pqih-Ix)- qrlxz + ^Kng
(2)
where /Zg„„ is the magnitude of the angular momentum vector for the rotating mass of
the engine, assumed to act along the positive X body-axis,
T
Kng = \_Kng O] = \_Ieng(Oeng O]
It is preferable to write the translational equations in terms of V^ , a, and P instead of
M,v, andw, because F'j,^, andy^ can be measured directly on real aircraft, and have a
more direct relationship to piloting and the aerodynamic forces and moments. The
relationships between V^,a,l5 and u,v,w are:
= Vm
V( =ylu +v +w
a = tan
P = sin
-1
(-]
[uj
f \
V
v^.y
and
u = Vi cos a cos P
v = V( sin /5
w = V( sin a cos /5
(3)
(4)
Differentiating the equations (3) with respect to time gives
■ UU + VV+ WW
uw-wu
"^ = —2 2 (5)
M +W
The translational states for the nonlinear aircraft simulations are Vj,a,and /^ . The
time derivatives of these translational states are found by computing ii, v, and w from (1),
then computing u,v,andw from (4) using current values of V^,a,and ^ , then
substituting these u, v, and w values and the current value of V^ into (5) to obtain
Vj,a,and ^ for use in the numerical integration routines. This approach avoids the
lengthy nonlinear calculations associated with the equations of motion written in wind
axes, and allows aircraft non-dimensional aerodynamic coefficient data to be
implemented in the body-axis coordinate system, with no need for converting the data to
wind axes components.
For the rotational motion, straightforward algebraic manipulation transforms the
equations (2) into a form suitable for numerical integration, with only one time derivative
on the left side of each equation,
P = {cir + C2P + c^Kng ) q + qSb{cj,Ci + C4C„ )
q = [c^p - c-jh^^g )r-C(,(p^-r^) + qSccjC^ (6)
r = (cgp - C2r + Cgh^^g )q + qSb {c^Q + CgC^ )
where
_ [h ^z)^z ^xz _ Vx h+^z)^xz _ Iz
^X^Z ~ ^XZ ^X^Z ~ ^xz ^x^z ~ ^xz
^xh-lxz ^y ^Y
_ 1 _\^x h)^x ^xz _ Ix
^^~T ^^~ 7~i T^ ^"^ ~ T~r 7~^
Y ^X^Z ^XZ ^X^Z ^XZ
The nonlinear aircraft simulation also includes rotational kinematic equations and
navigation equations. The rotational kinematic equations, which relate Euler angular
rates to body-axis angular rates, are given by:
= p + tan {q sin + r COS 0)
= qcos0-r sin (8)
■ _qsin0 + r cos
COS0
Equations (8) are nonlinear state equations for the Euler angles 0, 0, and IF , in a
form suitable for numerical integration. The rotational kinematic equations describe the
time evolution of the aircraft attitude angles, which are required to properly resolve the
gravity force along the aircraft body axes in (1). A singularity exists for the rotational
kinematic state equations at = ±90° . This is eliminated within the simulation code by
limiting to the ranges 0° to ±89.99° and ±90.01° to ±180°. This limits
(cos0)
to a
maximum value of approximately 5730, which allows sufficiently accurate calculations
of the kinematic state derivatives near the singularity. In the nonlinear simulation,
automatically skips from ±89.99° to ±90.01° and vice-versa depending on the direction of
aircraft pitch angle motion near = ±90° .
The navigation equations relate aircraft translational velocity components in body
axes to earth-axis components, neglecting wind effects. These differential equations
describe the time evolution of the position of the aircraft e.g. relative to earth axes:
x^ =u COS W COS + v[cos W sin sin - sin W cos0)
+w[cos W sin cos + sin W sin 0)
y^ =u sin W cos + v[sin¥ sin sin + cos W cos 0) (9)
+w[ sin W sin cos — cos W sin 0)
h = u sin - V cos sin0 - w cos cos
Assuming thrust acts along the X body-axis, body-axis accelerations Ux.ay, and a^
are calculated from
_ qSCx+T
ax
mg
aY = ^^^ (10)
mg
mg
Equations (1), (4), (5), (6), (7), (8), and (9) are the nonlinear aircraft equations of
motion for each aircraft simulation. These nonlinear dynamic equations are implemented
for each simulated aircraft in functions called ACname_deq.m, where ACname is the
aircraft name (e.g., F16_deq.m, F106B_deq.m, etc.). Body-axis accelerations
aj^, ay, and a2 are also calculated in each ACname_deq.m function, using equations
(10).
ENGINE MODEL
The engine power dynamic response was modeled with an additional state equation
as a simple first order lag in the actual power level response to commanded power level:
Pa=—{Pc-Pa) (11)
^eng
Commanded power level was computed as a flinction of throttle position,
Pc=Pc{^th)
and the engine power level time constant T was chosen at or near 1 sec. Engine thrust
force was typically computed by linear interpolation of an engine thrust database, as a
function of actual power level, altitude, and Mach number,
T = T{P^,h,M)
For each aircraft simulation, calculation of the engine thrust occurs in a function
called ACname_engine.m, where ACname is the aircraft name (e.g., F16_engine.m,
F106B_engine.m, etc.). The routine tgear.m implements the throttle gearing, which
translates the throttle deflection in the interval [0,1] to commanded power level in the
interval [0,100], but not always in a linear fashion. The routine rtau.m computes the
engine power level time constant T^^g , and pdot.m implements the engine thrust
dynamics in equation (11).
For many of the simulated aircraft, engine data was not available. In these cases, the
engine model for the F-16 aircraft was used, and the output thrust was scaled to match the
engine thrust of the specific aircraft.
AERODYNAMIC MODEL
Aerodynamic forces and moments acting on each aircraft were characterized in terms
of non-dimensional coefficients in body axes. Typically, the non-dimensional force and
moment coefficients were calculated from an expression that included separate build-up
functions for each non-dimensional coefficient. The value for each build-up function was
usually found by linear interpolation of aerodynamic data, using current values of the
states and controls. For values of states and controls outside the range of the available
data, the interpolation routines extrapolated linearly using the nearest data points. In a
few cases, build-up functions or complete non-dimensional coefficients were computed
using analytical polynomial expressions involving state and control variables. Once the
non-dimensional aerodynamic force and moment coefficients were calculated, these
values were used in equations (1) and (6) with thrust, mass / inertia properties, geometry,
and current states and controls to compute state time derivatives for the aircraft
dynamics. Subsequent sections describing the individual aircraft simulations contain
details of the aerodynamic model for each nonlinear aircraft simulation.
ATMOSPHERIC PROPERTIES
Air density and the speed of sound were calculated using relations modeling the U.S.
Standard Atmosphere^. Required quantities that depend on these atmospheric properties,
namely Mach number M, and dynamic pressure q , were also calculated. The
relationships are:
r*=l- 0.703x10"^ /z
/? = 0.002377 (r*)
4.14
M = \
Vt
^1.4(1716.3)390
V,
Jl.4(l716.3)(519r*)
/z> 35,000 ft
/z< 35,000 ft
(12)
^ = \pv?
The function atm.m was used to compute all the quantities from equations (12), given
h and Vt.
AIRCRAFT STATES AND CONTROLS
For a typical aircraft, the state vector was:
x = [V, a /5 W p q r Xe yE h P^\
(13)
The elements of the state vector are associated with equations (1), (4), (5), (6), (7), (8),
(9), and (1 1). A typical control vector was:
U = [Sth ^e ^a Sr\
(14)
The number of control inputs and engine power states varied depending on the number of
control surfaces and aircraft engines. The set of coupled, nonlinear, first-order ordinary
differential equations that comprise the simulation model can be represented by the
vector differential equation:
x = f(x,u)
(15)
The output equations can be represented by the vector equation:
y = h(x,u)
III. GENERAL SIMULATION TOOLS
Approximating the solution to the nonlinear equations of motion requires a suitable
numerical integration technique. Linearization of the nonlinear equations of motion
requires appropriate numerical gradient techniques, which are necessary to calculate
elements of the linearized system matrices. Numerical algorithms for both these tasks are
included in the software. Reference [8] provides background information on the
numerical techniques applied. Runge-Kutta and Adams-Bashforth numerical integration
routines are used to approximate solutions to the nonlinear equations of motion. Finite
difference numerical gradient procedures are used to determine the elements of the
linearized system matrices. The scripts called gen_ACname_model.m, where ACname is
the simulated aircraft name (e.g., gen_F16_model.m, genFl 06B model. m, etc.),
demonstrate the use of the trim and linearization tools for each simulated aircraft.
TRIM
A linearized form of the state-space model for the nonlinear equations of motion is
obtained using a first-order multivariable Taylor series expansion about an equilibrium
fiight condition known as trim. At the trim condition, all translational and rotational
accelerations are equal to zero. This condition is described by the equation:
Xo=0 = f(xQ,UQ)
(16)
where only the state equations for F^, a, /^, p, q, and r are included in equation (16). The
six force and moment equations are augmented with a rate of climb constraint, and a
coordinated turn constraint, for a total of eight trim equations^'^;
= f> =
= a =
UU + VV+ WW
uw — wii
2 , 2~
U +W
,=p= "•i-^"'
y.\\^-
f \
KVtJ
= p = (qr + C2P + c^h^ng ) q + qSb{c^Ci + c^C„ )
^ = q = (c5P-CiKng)f-Ce[p^-r^) + qScc-jC^ (17)
= r = (c^p - CjT + c<^\^g ) ^ + qSb{c^Ci + CgC^ )
^ ab + sinyJa -sin y+b ^ ^^„ r ^ ,. , . i
= to« & J ; 6' ;^ 90 [rate 01 climb constraint J
a - sin Y
= G cos /5 ( sin atan0 + cos a cos 0) ; 0^ 90° [coordinated turn constraint]
where
a = cos a cos /5
b = sin 0sin /3 + cos s in a cos P
WVt
G =
g
With eight trim equations to find the trim condition, any combination of eight state or
control surface defiection variables can be free to vary so that the trim equations can be
satisfied.
User-defined values for fixed states, control surface defiections, and trim values for y
and turn rate W are set in the fiinction ACnameJrm.m, where ACname is the simulated
aircraft name (e.g., F16_trm.m, F106B_trm.m, etc.), via code modification. The
ACnameJrm.m flinction calls the ACname_deq.m fiinction to implement the state
derivative equations included in the trim equations. Pitch angle is limited to the ranges
0° to ±89.99° and ±90.01° to ±180° in the function ^CnameJe^.m, as described above.
This eliminates the singularity at = ±90° within the rate of climb and coordinated turn
constraint equations. In addition, the engine power level dynamics in equation (11) are
bypassed for the trimming operation, so P^ is set equal to P^ inside ACname_trm.m.
Trim solutions are obtained by solving the eight nonlinear trim equations above,
using the function solve. m. Function solve.m uses a Newton-Raphson iterative solution
technique, where gradients are calculated in grad.m by central finite differences. Trim
solution convergence is determined using cnvrg.m. Convergence is achieved when the
absolute value of every trim equation is less than the solution tolerance, and the absolute
value of every change in the free states and controls for successive iterations has reduced
to less than the solution tolerance. The solution tolerance is specified as 1x10 within
the code. When the trim search is successful, the words CONVERGENCE CRITERIA
SATISFIED are output to the MATLAB command window.
10
LINEARIZATION
A linear, time-invariant (LTI) system representing aircraft dynamics for small
perturbations about a reference trim condition is given by the state and output equations:
X = Ax + Bu
y = Cx + Du
(18)
Elements of the constant-coefficient matrices in the linearized dynamic equations are
gradients evaluated at the trim values Xq and Uq :
A =
C =
df(x,u)
dx
dh(x,u)
dx
= nxn
matrix
*0
«0
= lxn
matrix
^0
•«o
B =
D =
df(x,u)
du
dh(x,u)
du
nxm matrix
Ixm matrix
(19)
where n is the number of elements in the state vector, m is the number of elements in the
control vector, and / is the number of elements in the output vector. The gradients are
obtained numerically by perturbing each state and control input independently and
recording the changes in the trimmed state and output equations. This is done using the
numerical technique of central finite differences:
df(x,u)
dx
df(x,u)
Xq.Mo
du
Xq.Uq
f(XQ+—,UQ)-f(XQ-—,UQ)
Ax
r/ ^u , ^ . Au ,
f(XQ , Uq + — ; -f(XQ,UQ-—)
Au
(20)
where Ax and Au are the perturbations of the state and control variables, respectively. In
the simulations, the output vector is the state vector [h[x,u) = x), so C is an identity
matrix, and Z) is a null matrix. A flowchart describing the numerical linearization
scheme is shown in figure 1 .
11
start
Initialize Finite
Difference Step
Size
[compute Partial Derivative|
Figure 1. Flowchart for Numerical Linearization
For a reference trim condition of straight and level flight, linearization results in two
decoupled sets of linear, constant-coefficient differential equations for longitudinal and
lateral / directional motion. The linearization is valid for small perturbations about the
reference trim condition.
Detailed Linearization Example
The trim state and control vectors (jcq and «o) are assembled in the fiinction icjirm.m,
for use in Inze.m, which computes the linear system matrices A, B, C, and D, using
central finite differences. Perturbations equal to 1% of the trim reference values are used
for each state and control variable during gradient calculation for the elements of the
system matrices. If the trim state or control is zero, the perturbation size is set to 0.01.
Setting the variable lonflg to 1 within the MATLAB workspace before executing
gen_ACname_model.m, where ACname is the simulated aircraft name (e.g.,
gen_F16_model.m, genFl 06B model. m, etc.), produces the longitudinal system
matrices. Setting lonflg to provides the lateral / directional system matrices.
Typical constant-coefficient longitudinal or lateral / directional system matrices A, B, C,
and D, for an aircraft with controls for throttle, elevator, aileron, and rudder deflection,
are created in the form:
12
For lonf Ig = 1:
~^^t~
\AVt'
Aa
Aa
Aq
= A
Aq
+ B
AS^
A0
A0
.^Pa_
_^Pa_
y = c
For lonf Ig ^ 1:
~A^~
-AP
Ap
= A
Ap
+ B
'^^a~
Ar
Ar
_AS,\
AO
A0
Aa
Aq
A0
A/5
Ap
Ar
A0
+ D
AS,
th
AS^
+ D
AS,
The following example illustrates the procedure used to trim the F-16 simulation and
calculate the longitudinal and lateral / directional linearized system matrices.
Fixed Quantities
V, = 500 ft/s X£ = ft
Free States / Controls
p = rad/s
q = rad/s
r = rad/s
!F = rad
_ =0.3
j£=Oft
A = 10, 000 ft
^ = 0.349
iF = 0.052 rad/s
a, /5, <P, 0, Sth, Se, Sa, and Sr
Step 1. Modify F16_trm.m to account for the desired trim condition, indicating the fixed
and free state and control variables. Within this file, the variables xf ree and
xinit control how the states are handled for trim. The definitions of xfree
and xinit are located on lines 117 and 118, respectively. Line 117 and 118
must be modified to read:
xfree = [0,1,1,1,1,0,0,0,0,0,0,0,0]';
xinit = [500,0,0,0,0,0,0,0,0,0,0,10000,0]';
In the vector xfree, a element indicates that the corresponding state is
fixed, while 1 specifies that the corresponding state is free to be chosen by the
trim solver. Values for the fixed states are input using the vector xinit. If the
state is fi'ee, the corresponding element of xinit contains the starting value
13
used to determine trim solution. If the state is fixed, the corresponding element
of xinit contains the fixed value.
The variables ufree and uinit contain information for fixed and free
control variables. They are located on lines 122 and 123, respectively. All
controls are free variables for this particular example. Lines 122 and 123 should
be changed to:
ufree = [1,1,1,1];
uinit = [0, 0, 0, 0] ;
Other starting values for the free state and control variables may need to be
tried in order to converge to the trim solution.
Flight path angle ^is input in radians on line 127:
gam=0.349;
which corresponds to an angle of 20 deg.
Turn rate is input in rad/s on line 131:
trate =0.052;
This corresponds to a turn rate of 3 deg/s. This completes the necessary
modifications to F16_trm.m. Save the modified file.
Step 2. Modify the file F16_massprop.m to account for an Xc.g. location at 30% of the
mean aerodynamic chord. Percentages must be normalized to values between
and 1. Change line 45 of the file to read:
xcg=0 . 3;
Step 3. Set the variable lonf Ig in the MATLAB workspace to either 1 or 0. Setting
lonflg to 1 will provide the longitudinal system matrices, while setting the
variable to will provide lateral / directional matrices.
Step 4. Run the file gen _F 16 model. m by typing the following on the MATLAB
command line:
gen_Fl 6_model;
This flinction computes the trim solution as well as the linear system matrices at
the desired flight condition and longitudinal center of gravity location. After
program execution, Xq and Uq are stored in the MATLAB workspace as
variables xO and uO, respectively. The linear system matrices are stored in the
MATLAB workspace as matrix variables A, B , C , and D.
The preceding procedure can be applied to the other supplied aircraft simulations with
little modification (e.g., replace F16 with F106B, F14, F4, etc. in the called functions).
14
NUMERICAL INTEGRATION
The equations of motion for a rigid aircraft are a set of coupled first-order nonlinear
ordinary differential equations. Experimental aerodynamic and thrust data are used to
model the applied aerodynamic and propulsion forces and moments for arbitrary states
and controls. There is no closed form solution to such problems, so the equations must
be solved using numerical integration. Techniques for solving this initial value problem
for ordinary differential equations are employed to obtain approximate solutions at
discrete points along the aircraft state trajectory.
The equations of motion have already been arranged into the form:
x = f(x,u)
(21)
where
x =
V, a /5 W p q r xe y^ h P,]'
(22)
u = [^th 4 d^ d,\
Control vector elements can be added or removed depending on the modeled aircraft.
A number of numerical integration techniques are available to approximate the
solutions of the well-posed equations of motion. The choice of which technique to use
depends on the desired accuracy of the solution and the computational effort to be
expended. Three types of numerical integration techniques are employed within the
software to approximate solutions to the equations of motion - Euler's method,
Runge-Kutta method of orders two and four, and Adams-Bashforth method of order
three. A fiowchart illustrating the procedure used during the nonlinear simulation is
given in figure 2.
15
(^^StarT)
t =0
Compute Discrete Quantities ^
Figure 2, Flowchart for Nonlinear Simulation
Euler's Method
Euler's method is a one-step integration routine that approximates solutions to the
initial value problem at discrete time steps ^t. Solutions to the state equations are
approximated starting from initial states given at t = Q with arbitrary control inputs
provided iox t >0. The local truncation error for this method is 0{M). A single state
derivative calculation is required for each time step, making this the least
computationally intensive of the three approximation methods used in the simulation.
However, it is also the least accurate. The algorithm is given in difference equation form
as:
x(t + At) = x(t) + Atf[x{t), u(t))
(23)
Runge-Kutta Method of Order Two
The second-order Runge-Kutta method is also a one-step integration routine but
provides for a more accurate solution approximation as compared to Euler's method.
However, two state derivative calculations per time step are required making it
comparatively slower than Euler's method. The local truncation error for this routine is
0{At^). The difference equation is given by:
x(t + At) = x(t) + Atf
VL
x(t) + ^f{x{t),u{t))
f
,u
At
t^
V 2 yy
(24)
16
Runge-Kutta Method of Order Four
The fourth-order Runge-Kutta method is a one-step routine with a local truncation
error of 0{/s.t '^). Solution approximations are more accurate than either the Euler or
second-order Runge-Kutta method, but the method runs one-fourth and one -half as fast,
respectively. The difference equation for this scheme is given by:
x(t + At) = x(t) + At
'^ h kj k-, kA^
(25)
where
k, = f{x{t),uit))
f
At
h = f x(t) +—ki,u
V 2
^ At
t + —
2
JJ
f
At
h = f x(t) +—k2,u
t + -
At
J J
h = f[x(t) + Atk^,u[t + At))
Adams-Bashforth Method of Order Three
The third-order Adams-Bashforth method is a multi-step routine that approximates
the solution of the initial value problem using three previous step calculations. The
approximation has a local truncation error of 0{/!d ^). Second-order Runge-Kutta is used
for the initial time steps before switching to the Adams-Bashforth integration method.
The difference equation for the third-order Adams-Bashforth method is given by
At
x[t + l)At) = x{t + 2At) + — \21>f(x{t + 2At),u[t + l)At))
12
-\6f(x{t + At),u{t + 2At)) + 5f(x{t),u{t + At))\
(26)
Detailed Numerical Integration Example
The state vector derivative x is integrated numerically in the function ACname.m,
where ACname is the simulated aircraft name (e.g., F16.m, F106B.m, F4.m, etc.) using
arbitrary control inputs for ^ > with initial states specified at t = 0. This produces a set
of state time histories x(t) and output time histories y(t) for t > 0. Inputs to ACname.m
are the control input time history u{t), time vector t, initial state vector jc at Z^ = 0, and the
vector of mass / inertia properties c. The function mksqw.m is provided to allow creation
of a u{t) comprised of arbitrary square waves with user-specified amplitudes, widths,
delay, step size, and total length in seconds. However, any u{t) can be used with the
17
nonlinear simulation. A user-specified integration routine, such as the Adams-Bashforth
method of order three (ab3.m), Runge-Kutta method of order two (rk2.m), or Runge-
Kutta method of order four (rk4.m), is called within the ACname.m function.
A sample procedure for calculating state and output trajectories is provided below for
the F-16 simulation. Procedures for the other aircraft simulations are similar.
Step 1. Create u{t) along with t, which will be inputs to the nonlinear simulation
function. The control vector must be stored in the MATLAB workspace as a
matrix with the individual controls making up the columns and the deflection of
each control over time comprising the rows. The time vector contains the
discrete time points for the control vector. An example using the supplied
function mksqw.m to create u{t) is shown below in figure 3. The elevator control
surface 4 will be deflected using a doublet with an amplitude of 1 degree that
will be applied after a 3 second delay, while keeping the remaining control
inputs {Sth, 5a, and <^) unperturbed from their trim values. Each pulse of the
doublet will have a width of 2 seconds. The entire length of the maneuver will
be 10 seconds with a sampling interval 0.025 seconds. The elevator perturbation
time history is shown in flgure 3.
Elevator Doublet Simulation
0)
Q
HI
1.5
0.5
-0.5
-1.5
4 5
Time (s)
10
Figure 3. Elevator Perturbation
a. Type the following command in MATLAB:
[usw, t]=mksqw(l, 2, [1 1], 3,0.025,10);
The first input to the function gives pulse amplitude, the second gives
the width of a single pulse in seconds, the third gives the vector of
18
integer pulse widths, the fourth gives the delay in seconds, the fifth
gives the sampling time, and the final input gives the length of the entire
maneuver in seconds. The output of the function is stored in the
MATLAB workspace under the variable names usw and t. The
variable usw is a column vector composed of 401 elements describing
the deflection history. The variable t is also a column vector composed
of 401 elements recording the sample times for usw.
b. Create the control matrix u within the MATLAB workspace by adding
usw to the second column of a 401 x 4 dimensional matrix, where the
first column corresponds to values of dth, the second to 4, the third to da,
and the last column to 5r. If the elements in the columns of u are
initially set to the trim values of the controls, then adding usw to
column 2 creates u{t) for an elevator doublet perturbation deflection,
leaving other control inputs unperturbed from trim values during the
maneuver.
Step 2. Define the initial state vector xO by creating a 13 x 1 vector composed of the
user-defined state initial conditions. This vector contains the same state
variables as the state vector x of the simulation model. To start from trim, use
the trim state vector output from icjirm.m or gen_ACname_model.m.
Step 3. Generate the mass property vector c by running the flinction F 1 6 massprop.m:
c=Fl 6_massprop;
The vector c will be available in the MATLAB workspace after execution of this
function. Center of gravity position Xc.g. can be adjusted within this file to
simulate various longitudinal e.g. locations.
Step 4. Run the nonlinear F-16 simulation using the following command on the
MATLAB command line:
[y,x]=F16(u,t,xO,c) ;
The output time history is stored in the MATLAB vector y, and the state output
time history is stored in the vector x.
By default, the nonlinear simulation implements the Runge-Kutta integration method
of order two. The user can modify the code to implement either the Runge-Kutta method
of order four or the Adams-Bashforth method of order three. This is done by modifying
the function Fid. m following the procedure below:
To use Fourth-Order Runge-Kutta
Change line 102 to read:
%x=rk2 ( 'F16_deq' ,u, t,xO, c) ;
line 101 to read:
x=rk4 ( 'F16_deq' ,u, t, xO, c) ;
19
and line 103 to read:
%x=ab3 ( 'F16_deq' ,u, t, xO, c) ;
To use Third-Order Adams -Bashf or th
Change line 102 to read:
%x=rk2 ( 'F16_deq' ,u, t,xO, c) ;
line 101 to read:
%x=rk4 ( 'F16_deq' ,u, t,xO, c) ;
and line 103 to read:
x=ab3 ( ' F16_deq' , u, t , xO, c) ;
20
IV. AVDS / MATLAB INTERFACE
A MATLAB software interface to the software package called AVDS was developed
to provide the capability to fly the MATLAB nonlinear aircraft simulations interactively
in real time, using AVDS to read pilot inputs and display the aircraft response. Pilot
inputs are measured by AVDS and used as inputs to the MATLAB simulations. Results
of the simulation numerical integration are sent to AVDS and used to draw 3-D images of
the aircraft in motion. Control deflections are also shown on the 3-D aircraft model
displayed by AVDS. The following material provides instructions for AVDS /
MATLAB software installation and usage.
EQUIPMENT / REQUIREMENTS
The nonlinear simulation software described here was developed and tested on
IBM-compatible personal computers (PC), using MATLAB version 6.1, and AVDS
version 1.2.14. System requirements for these commercially-available software tools are
specified in their documentation. A graphics accelerator card capable of hardware
acceleration using OpenGLT^ produces smoother-running piloted simulations.
Control inputs are supplied to AVDS via a mouse, joystick, and / or keyboard. A
joystick is recommended. A Microsoft® Side Winder® Precision 2 joystick has been used
exclusively during software development and is recommended for use; however, any
joystick can be used. This particular joystick contains a throttle control lever, hat switch,
rudder control, and eight programmable buttons, which provide for somewhat realistic
aircraft controls.
SOFTWARE INSTALLATION
AVDS is available from RasSimTech Ltd.® and is not part of the MATLAB aircraft
simulation packages. The AVDS software allowing data communication between
MATLAB and AVDS is available with version 1.2.14 of AVDS, available on the Internet
at http://www.rassimtech.com .
Once the AVDS software is installed, the 3-D aircraft images provided for each
simulated aircraft can be installed into AVDS. The 3-D image files are included in the
aircraft simulation packages. The provided 3-D aircraft models are installed into AVDS
using the following procedure:
Step 1. Copy the following aircraft 3-D files to the AVDS directory
\usr\local\A VDS\craft\:
\usr\local\F-l 6\F1 6_hires. txt
\usr\local\F-16\F16_lores.txt
\usr\local\F-106B\F106B. txt
\usr\local\F-14\F14.txt
21
\usr\local\F-4\F4.txt
\usr\local\FASER\FASER_hires.txt
\usr\local\FASER\FASERJores.txt
\usr\local\HL-20\HL20.txt
\usr\local\X-3 1\X3 1 .txt
\usr\local\A-7\A 7. txt
High and low resolution 3-D aircraft models are provided for the F-16 and
FASER aircraft. The user has the choice to use either during interactive piloted
simulations. Low -resolution models are provided for simulation on slower
computers.
Step 2. Add the following lines to the file craftcap.txt located in the directory
\usr\local\A VDS\craft\:
include "F16_lores.txt"
include "F16_hires.txt"
include "F106B.txt"
include "F14.txt"
include "F4.txt"
include "FASER_lores.txt"
include "FASER_hires.txt"
include "HL20.txt"
include "X31.txt"
include "A7.txt"
This file tells AVDS which 3-D aircraft models to make available for interactive
simulation.
INTERACTIVE SIMULATION
The user can interactively provide control inputs through AVDS using a mouse,
joystick, and/or keyboard. For control input device selection and configuration, consult
section 3.4.2.9 of the AVDS User's Manual^. The following will discuss providing
interactive control input with the Microsoft® SideWinder® Precision 2 joystick. Other
joysticks should work as well, but only this joystick was used in the development and
testing of the nonlinear simulations. Configuration of the initial aircraft states is also
described.
22
Joystick Control Input
Control inputs for pitch, roll, yaw, and thrust during piloted interactive simulations
can be provided by a Microsoft® Side Winder® Precision 2 joystick. For proper set-up of
the joystick, set the scale factors for X, Y, Z, and R joystick axes to 2, 2, 1, and -2,
respectively, within the AVDS joystick configuration screen (refer to section 3.4.2.9 of
the AVDS User's Manual^). This normalizes AVDS inputs to the ranges tol for thrust
control, and -1 to 1 for pitch, roll, and yaw. The normalized values are then converted to
actual control surface deflections, measured in degrees, within the function
AVDS_Matlab_ACname.m, where ACname is the simulated aircraft name (e.g.,
AVDS_Matlab_F16.m, AVDS_Matlab_F106B.m, AVDS_Matlab_F4.m, etc.). Other
joystick settings, such as dead-zone, bias, trim step size, etc., can also be conflgured
within the AVDS joystick conflguration screen.
Aircraft pitch control is achieved with the fore / aft movement of the joystick,
left / right movement of the joystick provides roll, and clockwise / counterclockwise twist
of the joystick supplies yaw control. Throttle deflection is actuated by means of a small
lever located at the base of the joystick. A hat switch located on the tip of the joystick is
used to set elevator and aileron control surface trim deflections. Trim elevator deflection
can be varied by pressing the top or bottom of the trim hat, while trim aileron can be set
by pressing the left and right sides of the trim hat. The eight programmable buttons of
the joystick have not been configured for use in AVDS / MATLAB interactive
simulations.
Initial States for Interactive Simulation
AVDS allows specification of aircraft initial conditions for interactive simulations.
The user can specify the aircraft starting altitude, velocity, Euler angles, and throttle
position. These are set in the Aircraft State Initialization pop-up window, shown in
figure 4, which is called within AVDS prior to simulation start-up. This is done as
follows:
Step 1. Select Initialize from the top menu
Step 2. Select Aircraft State
Initial altitude, given in feet above sea level, is adjusted by moving the corresponding
slider up or down. Initial body-axis velocity components u, v, and w (in ft/s), initial Euler
angles <P, 0, and W{m degrees), and initial throttle position dth (in percent) are adjusted
similarly. Starting xe and yg positions cannot be adjusted with these sliders. The default
starting positions for xe and yE are both set to ft. Before hitting the OK button to
complete initial state configuration, make certain that the Apply Settings button is
depressed so that a check appears in the box. This allows the initial states set with the
sliders to take effect in the interactive simulation. The button Apply Init Lat/Long has
no effect within the AVDS / MATLAB simulation since starting xe and yE aircraft
positions are set within the MATLAB simulation software.
23
FJMmiHMtiMM
- Latitude/Longitude
IE]
Apply Init Lat/Long | InitLat 1 0.00000000 InitLorg: 1 0.00000000
Cancel
-Aircraft State Settings-
Altitude u
Velocity
'Rotation Position Offset
~pS theta phi Throttle (K) X Y
42574 I 485 | ] | I f
r^^
ll^ i^.P.PJ^..^.^f.^in9^ I note: select the slider and use the f and i keys to make precise adiustments.
Figure 4. AVDS Aircraft State Initialization Window
A VDS /MA TLAB Piloted Simulation Software
Piloted simulations are executed using the function AVDS_Matlab_ACname.m
included in the aircraft simulation packages. This function establishes communication
between AVDS and MATLAB and implements the integration routine to numerically
solve the nonlinear equations of motion. Integration time steps during piloted
simulations are dynamically updated within the AVDS_Matlab_ACname.m file, to
prevent MATLAB from lagging behind AVDS simulation time. This allows
synchronization of the real-time simulation of the aircraft in MATLAB with the 3-D
display in AVDS. However, an upper limit (0.035 s) is imposed on the time steps to
prevent loss of model fidelity. For less powerfiil computers (e.g., 500 MHz and slower),
this can cause MATLAB to lag behind AVDS during simulation.
Control inputs at a given time step are sent from AVDS to MATLAB via a shared
memory connection. Solutions of the nonlinear equations of motion are computed in
MATLAB using the control inputs from AVDS. The solution is returned to AVDS by
MATLAB, using the same data pathway. Control inputs and aircraft initial states,
supplied by AVDS, are stored in the MATLAB workspace using the variable name
InputVector. The output vector, stored in the variable called OutputVector,
contains the translational and rotational states, angle of attack, sideslip angle, linear
accelerations in g's, airspeed, and Mach number, computed by the nonlinear simulation
in MATLAB. Elements within InputVector and OutputVector are identified to
AVDS via the file ACname.sim.ini, where ACname is the simulated aircraft name (e.g.,
F16.sim.ini, F106B.sim.ini, etc.). Refer to section 4.2 of the AVDS User's Manual^ for
modification instructions for this file. Typical elements of these vectors are shown
below:
24
InputVector = [//ag,,„ 5, 5^ 5, 5^^ t^YDS ^ ^ h (^ W V, P^ a fif
OuputVector = rx£' y^h00Wa/5gVfM [control surface deflections]]
where y?ag„,„ is a flag equal to one when AVDS is running, and zero otherwise, and
^AVDS ^^ ^^ simulation time in the AVDS program.
Figure 5 shows a flowchart illustrating the interactive simulation.
start
T
Initial State Input
Vector
f
--1
"
nteractive
Control Input
1
1
Control
nput Vector From
AVDS
MATLAB Nonlinear
Simulation
State Output Vector
From MATLAB
yes
y
y^^ Con
nue ^x^
v
Figure 5. Interactive Simulation Flowchart
The translational and rotational aircraft states calculated within MATLAB are used
by AVDS to display the 3-D aircraft model along the calculated trajectory. Quantities
important to the pilot (a, (5, M, Vt, h, 0, W, and a^ (g's)) are displayed to the user in the
AVDS Head-Up Display (HUD). A screen shot of the AVDS screen during interactive
simulation of the F-16 is shown in figure 6. Flight path of the aircraft is shown with
trailing ribbons generated on each wing tip and the aircraft center tail section.
25
I WT,<n.w I PMtf I I 3THIT I
Heading Angle (deg)
.f \
Airspeed (ft/s)
Angle of Sideslip (deg)
Angle of Attack (deg)
Mach Number
g's-
Pitch Angle Ladder (deg)
ir^-ii*rtt>Hir s^s^^^Ji^ Ks;*iriii^
Figure 6, Screen Shot of the F-16 AVDS / MATLAB Interactive Simulation
Stick Gain Modification
Stick gains for elevator, aileron, and rudder are applied within the function
AVDS_Matlab_ACname.m to allow for easier open-loop flying. All stick gains for the
simulated aircraft are set to 0.5 by default. The procedure used to modify the stick gains
for the F-16 simulation are given below (other simulation modifications are similar):
To change elevator stick gain
Modify line 123 within AVDS_Matlab_F16.m to read:
gain_el = xxx;
where xxx is the user-defined numerical value for the elevator stick gain.
To change aileron stick gain
Modify line 124 within AVDS_Matlab_F16.m to read:
gain_ail = xxx;
where xxx is the user-defined numerical value for the aileron stick gain.
To change rudder stick gain
Modify line 125 within AVDS_Matlab_F16.m to read:
gain_rdr = xxx;
where xxx is the user-defined numerical value for the rudder stick gain.
26
Control Deflection Exaggeration Factor
Deflections of aircraft control surfaces, shown in green on the AVDS aircraft 3-D
models, can be visually exaggerated for easier viewing. By default, aU deflections shown
in AVDS are exaggerated by a factor of two. The following provides an example of
modifying the exaggeration factor for the F-16. Other simulation modifications are
similar.
To adjust control surface deflection exaggeration factor
Modify line 130 within AVDS_Matlab_F16.m to read:
cdfac = xxx;
where xxx is the user-defined numerical value for the deflection exaggeration
factor. If cdfac is set to 2, the displayed control surface deflections will be 2
times the actual control surface deflections being used in the MATLAB
nonlinear aircraft simulation.
Integration Routine
Integration routines used to solve the equations of motion for the AVDS / MATLAB
interactive simulations are implemented within AVDS_Matlab_ACname.m as opposed to
abS.m, rk2.m, and rk4.m for the batch nonlinear simulations. The
AVDS_Matlab_ACname.m flinction can implement either the Adams-Bashforth
integration method of order three, the Runge-Kutta method of order two, or Euler's
method. The following gives an example for implementing particular integration
routines for the F-16 simulation (AVDS_Matlab_F16.m). Modifications for other
simulations are similar.
To implement Adams-Bashforth method of order three
Remove the % character from the beginning of lines 250-266 and add the %
character to the beginning of lines 270-274 and lines 278-280. MATLAB
ignores lines of code beginning with the % character during function execution.
To implement Runge-Kutta method of order two
Remove the % character from the beginning of lines 270-274 and add the %
character to the beginning of lines 250-266 and lines 278-280.
To implement Euler 's method
Remove the % character from the beginning of lines 278-280 and add the %
character to the beginning of lines 250-266 and lines 270-274.
Starting the A VDS/ MATLAB Simulation
The following provides an example of running the F-16 piloted simulation. A similar
procedure is used for the other provided simulations.
To start the piloted F-16 simulation
Step 1. Start AVDS
27
Step 2. Load the simulation initialization file Fl 6.sim.ini
a. Select File from the top menu
b. Select Simulation Init, then Open
c. Choose the file F16.sim.ini located in the F-16 directory of the simulation
software package
Step 3. Load the 3-D aircraft image F-16 (Lo-Res) or F-16 (Hi-Res)
a. Select Initialize from the top menu
b. Select Aircraft Image
c. Select either F- 1 6 (Lo-Res) or F- 1 6 (Hi-Res)
Step 4. In AVDS, switch to simulation mode and start the simulation
a. Press SIMULATION button on main toolbar
b. Press START button on main toolbar
Step 5. RunAVDS_Matlab_F16inMATLAB
a. Set the current directory to the F-16 directory of the simulation software package
b. Type AVDSMatlabF 1 6 on the MATLAB command line followed by ENTER
To end the simulation
Press the STOP button located on main toolbar in AVDS
28
V. SPECIFIC AIRCRAFT SIMULATION DETAILS
Details specific to each aircraft simulation package are given below. Unique
simulation characteristics such as aerodynamic and engine models, aircraft controls, and
valid ranges for the states and controls are given. Angle of attack, sideslip angle, and
control surface deflection inputs to the aerodynamic models are in degrees, unless
specified otherwise. Aircraft angular velocities are in radians per second, unless
specified otherwise. For a list of the files included in each package, refer to tables A-1
through A-8 in the Appendix.
F-16 SIMULATION
The General Dynamics F-16 is a single-seat, multi-role fighter with a blended
wing / body and a cropped delta wing planform with leading edge sweep of 40^". The
wing is fitted with leading edge fiaps and trailing edge flaperons (fiaps / ailerons). Tail
surfaces are swept and cantilevered. The horizontal stabilator is composed of two all-
moving tail plane halves, while the vertical tail is fitted with a trailing edge rudder.
Thrust is provided by one General Electric FllO-GE-100 or Pratt & Whitney FIOO-PW-
220 afterburning turbofan engine mounted in the rear fiiselage.
The aircraft was modeled with controls for dth, Se, 5a, and 4. Aerodynamic force and
moment data were derived from low-speed static and dynamic forced oscillation wind
tunnel tests conducted on a 16% scale model of the F-16 fiown out of ground effect, with
landing gear retracted, and no external stores^'^. Static aerodynamic data are in tabular
form as a flinction of angle of attack and sideslip over the ranges-10° < a< 45° and
-30° < y^< 30°. Dynamic data is provided in tabular form at zero sideslip angle over the
angle of attack range -10° < a < 45°. Each non-dimensional aerodynamic force and
moment coefficient is built up from component flinctions. Aerodynamic coefficients are
referenced to a center of gravity location at 0.35 c . Corrections to the flight center of
gravity position are made in the coefficient build-up equations.
The aerodynamic database was simplified slightly by dropping second order
dependencies (e.g., dependence of longitudinal aerodynamic forces on sideslip angle).
Throttle deflection is limited to the range <<?«/,< 1 , elevator deflection is limited to
-25° < 5e< 25°, aileron deflection is limited to -21.5° < 5a< 21.5°, and rudder deflection
is limited to -30° < (5;. < 30°.
Engine thrust data is in tabular form as a flinction of power level, altitude, and Mach
number over the ranges ft < /z < 50,000 ft and < M < 1, for idle, military, and
maximum power settings. Engine power level dynamics and gyroscopic effects^ are
included for this aircraft simulation, in the manner described previously. Figure 7 shows
a three-view of the F-16.
29
Figure?, Three- View of the F-16
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the F-16 vary
nonlinearly with flow angles (a, /5), aircraft angular velocities {p, q, r), and control
surface deflections (4, 4, Sr). Moment coefficients C„ and C„, include a correction for
the center of gravity position. The coefficients are computed as follows:
Cx = Cx^{ocA) +
qc
2V
C. {a)
'a
Cy=-0.02;5 + 0.021
Cz = Cz^ {ex)
v20,
2
+ 0.086
r;^\ r 6 ^
30
+
\j\jj
y^v.j
CY^{a)p + CY^{a)r^
1
Vl80y
f ^ \
0.19
v25.
+
qc
Cz {a)
30
Q=Qja,;5) + zlQ,
f ^ \
•^.=20°
V20y
+ zlC,
f^\ f l,\
'<J^=30°
V30y
+
v2f^.y
Q^(a);7 + Q^ (a)r
qc
'^.=20°
(•^c.g.,re/ ~^c.g.)Q
/^ ^
V20y
+ Z1C
^'^^ ^ 6 ^
'5^=30°
V30y
+
v2f^.y
C„^(a);7 + C„^(a)r
where
zlC,
■<y„=2o"
= C, ^(a,P)-CAa,l5)
'5^=W
AC,
'<y^=30'
= Q,^ Ja,l3)-Ci^{a,l3)
AC„
\=20°
'<5-^=30°
= C Ja,/5)-C,^{a,(5)
Data tables for the component functions of the non-dimensional coefficients are
defined in the file F16_aero_setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in F16_aero.m, using linear
interpolation in the flinctions cxo.m, czo.m, cmo.m, clo.m, cno.m, dlda.m, dldr.m, dnda.m,
dndr.m, and dampder.m.
Engine Model and Database
The F-16 is powered by a single afterburning turbofan jet engine, which is modeled
taking into account throttle gearing and engine power level lag. The engine angular
momentum heng is assumed to act along the aircraft X body-axis with a fixed value of
160 slug-ft^/s. Throttle gearing is implemented in the file tgear.m, which outputs
commanded engine power in percent of fiiU power [0,100] using throttle setting input
[0,1]. Commanded power Pc is a function of Sth, given by:
Pc{^th) =
\6A.9Adth
if 5,. < 0.11
■'th
217.38(5",, -117.38 ift?,, >0.77
■'th
th
Engine power level dynamic response is modeled using a first order lag with the time
constant computed by the function rtau.m. The module pdot.m calculates the rate of
31
change of power level with time using the difference between P^ and Pa- The actual
power level derivative P^ is given by:
where
P„ =
1
"eng
-{Pc-Pa)
c
60
40
^Pc
ifP > 50.0 and /'> 50.0
ifP > 50.0 and R< 50.0
ifR < 50.0 and /'> 50.0
if/'^< 50.0 and /'„< 50.0
•'eng
5.0
1
T
eng
5.0
1
T
eng
if4> 50.0 and /'„> 50.0
if/'^> 50.0 and P„ <50.0
ifP^< 50.0 and P„> 50.0
ifP^< 50.0 and P„< 50.0
"eng
1.0 if(P,-Pj<25.0
0.1 if(P,-Pj>50.0
1.9-0.036(P,-PJ if25.0<(P,-Pj<50.0
Engine thrust is modeled as a function of A, M, and Pa- Data tables for idle, military,
and maximum power as a function of h and M are defined in the file
F16_engine_setup.m- Linear interpolation and thrust calculation is performed in the
module F16_engine.m- Thrust is computed from:
32
T =
idle \ mil idle )
mil \ max mil J
a
V50y
50
if R < 50
ifP,>50
Mass Properties
The mass properties of the F-16 are given in the file F16_massprop.m. These
properties include the weight of the aircraft, Xc.g., Ix, h, h, and Ixz- This module also
calculates the constants Ci,C2,..-,c^ for use in the body-axis moment equations. The
longitudinal center of gravity location Xc.g. is given in fraction of mean aerodynamic
chord, and can be adjusted within this module to account for different longitudinal center
of gravity locations. Table 1 lists the mass properties used in the F-16 simulation.
Table 1. Mass Properties of the Simulated F-16
Parameter
Weight
(lb)
Ix
(slug-ft')
(slug-ft")
Iz
(slug-ft")
Ixz
(slug-ft")
Value
20,500
9,496
55,814
63,100
982
F-106B SIMULATION
The Convair F-106B is a two-seat all-weather interceptor incorporating a delta wing
planform with a single swept vertical tail surface. Elevon control surfaces, which
perform both elevator and aileron control fiinctions, are located at the trailing edges of
the wing. The rudder control surface is located at the trailing edge of the vertical tail.
Thrust is provided by a single Pratt & Whitney J75-P-17 afterburning jet engine mounted
in the rear fiaselage.
The aircraft simulation has controls for 5th, Se, 5a, and 5r. Low-speed static and
■7
dynamic forced oscillation wind tunnel tests on a 15% scale model of the aircraft
provided aerodynamic force and moment data. Tests were conducted in the NASA
Langley 30- by 60-foot wind tunnel, with the model flown out of ground effect, with
landing gear retracted, and no external storesio.n. Static and dynamic aerodynamic data
are in tabular form as a function of angle of attack over the range -5° < a< 50°. Each
non-dimensional aerodynamic force and moment coefficient is built up from component
functions. Aerodynamic coefficients are referenced to a center of gravity location at
0.275 c , and corrected to the flight center of gravity position in the coefficient build-up
equations.
33
Throttle deflection is limited to the range < J^/, < 1 , elevator deflection is limited to
-28° < Se< 28°, aileron deflection is limited to -7° < Sa< 7°, and rudder deflection is
limited to -25° <(5i< 25°.
The F-106B engine has maximum thrust values of 24,500 lb and 17,200 lb with and
without afterburner, respectively. A scaled version of the F-16 engine model was used
for this simulation. Idle, military, and maximum power settings were scaled using the
maximum thrust for the F-106B engine. Throttle gearing, gyroscopic effects, and engine
power level dynamics were carried over from the F-16 engine model. Figure 8 shows a
three-view of the F-106B aircraft.
Figures. Three- View of the F-106B
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the F-106B
vary nonlinearly with flow angles {a, P), aircraft angular velocities (p, q, r), and control
surface deflections (4, 4, Sr). Moment coefficients C„ and C„, include a correction for
the center of gravity position. The coefficients are computed as follows:
Cx = CL^{a,5^)sin
'' an^
Vl80y
CDAoc,Se)cos
^ an^
Vl80y
CY = C^p[cc)l5^
Cj.^(a);? + Cj.^(a)r)
C2=-Cl^[oc,5^)cos
'' ait"^
Vl80y
■CD^{.CC'Se)sin
'' ait"^
Vl80y
qc
Cm {a)
34
Ci=C,{a)/3 + ACi^
f ^ \
'P
+ AC,
rs\ f h^
^a=^° V ' J
'5 =15°
v25y
+
y^v,j
Ci^{a)p + Ci^{a)r^
qc
Cm = C [a, S,)+ ^ C [a) + (x,
2V, ■■■<i
^c.g.,ref e.g. I ^Z
)C2
\^' t J
C,=C,ia)/5 + AC,^
f ^ \
np
+ AC„
^.=7° V ' y
'^ =25°
25
+
V^jy
2V
(^C,^{a)p + C,^{a)r
^c.g..ref -^c.g. I '-F
)c.
where
ACi^^__^^=Ci^^_^Ja)-Ci^{a)
AC,
'^a--^' \=7°
'S^=25° 'S^=25° "
'0
zlC„, ^=C„, /a;-C (a)
',5^=25° 'S^=25°
Data tables for the component functions of the non-dimensional coefficients are
defined in the file F 106B aero setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in F106B_aero.m, using linear
interpolation in the functions cdo.m, clo.m, cmo.m, dldb.m, dlda.m, dldr.m, dndb.m,
dnda.m, dndr.m, dydb.m, dydr.m, and dampder.m.
Engine Model and Database
Thrust scaling for the F-106B engine is applied in the function
F106B engine setup.m. Linear interpolation and thrust calculation is performed in the
module F106B engine. m. Functions tgear.m, pdot.m, and rtau.m are unaltered from the
F-16 engine simulation.
Mass Properties
Aircraft mass properties are given in the module F106B_massprop.m. Table 2 lists
the mass properties used in the F-106B simulation.
35
Table 2, Mass Properties
of the Simulated F-106B
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^Xl
(slug-ft")
Value
29,776
18,634
177,858
191,236
5,539
r-14 SIMULATION
The Grumman F-14 is a two-seat, carrier-based, multi-role fighter with a variable
geometry wing and leading edge sweep varying between 20° and 68°. Small canard
surfaces extend from the leading edges of the fixed portion of the wing to provide extra
longitudinal stability. Spoilers are located on the upper surfaces of the wing and provide
aileron control during low speed flight. Trailing edge flaps are located at the rear portion
of the wing and extend over most of the span, while slats are fitted to the wing leading
edges. Two all-moving horizontal tail surfaces are located at the rear of the fuselage and
provide aileron and elevator control using differential and symmetric deflections,
respectively. Twin vertical tails, also located at the rear of the fuselage, are fitted with
trailing edge rudder control surfaces. Speed brakes deflect symmetrically from a location
at the rear fiaselage between the bases of the vertical tails. Two Pratt & Whitney TF30-P-
412A afterburning turbofan engines, located at the rear of the fuselage, provide thrust for
the aircraft.
The aircraft was modeled in the unswept wing configuration, with controls for 5th, 4,
5a, 5r, 5sp, and 5sb- Aircraft aerodynamic data was taken with the vehicle flown out of
ground effect, with landing gear retracted, and no external stores 1 2,13. Static
aerodynamic data are in tabular form as a function of angle of attack, sideslip angle, and
Mach number over the ranges -5° < a < 50°, -20° < /3 < 20°, and 0<M< 1. The
aerodynamic effects of angular rates are also in tabular form over the same angle of
attack range. Each non-dimensional aerodynamic force and moment coefficient is built
up from component fianctions. Aerodynamic coefficients are referenced to a center of
gravity location at 0.16 c , and corrected to the flight center of gravity position in the
coefficient build-up equations.
Throttle deflection is limited to the range < (Jj/, < 1 , elevator deflection is limited to
-30° < 5e < 12°, aileron deflection is limited to -21° < 5a < 21°, rudder deflection is
limited to -30° < 5r< 30°, wing spoiler deflection is limited to 0° < 5sp < 55°, and speed
brake deflection is limited to 0° < 5sh ^ 60°.
Tabular data for idle, military, and maximum thrust were used to model the thrust
produced by the F-14 engine. Engine dynamics and throttle gearing were carried over
from the F-16 engine model; however, engine angular momentum was neglected.
Figure 9 shows a three-view of the F-14.
36
Figure 9, Three- View of the F-14
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the F-14 model
vary nonlinearly with relative airflow quantities (a, P, M), aircraft angular velocities (p,
q, r), and control surface deflections (4, 4, 4-, ^sp, ^sh)- Moment coefficients C„ and Cm
include a correction for the center of gravity position. The coefficients are computed as
follows:
Cx = CxAcc,P) + ^Cx,^ +ACx,
'^st^
^b=(>'^°
V60y
+ AC
X
'^p=^^°
■^sp
Cy = Cyia,/5) + C. {a,/5)5, + Cy^ {a,/5)d, + C. {a)d,
sp
sp
^ b ^
K^VtJ
iCy (a)jO + Cy {a)r
Cz=Cz^{a,j3) + ACz,^ +ACz,
^b=60'
<^sb
V60y
+ ACy
^sp
"^p=^^°
V55y
+
qc
v2^y
Cz {a)
37
a r sp
+
^ b ^
[Ci^{a)p + Ci^{a)r
Ssb=(^^°
v60.
+ zlC„
'<y. =55°
sp
55
sp \ J
Vsp,m{M)+ ^ C (a)
"^ y^c.g.,ref ■'^c.g. ) ^Z
.)Cz
^ b ^
C,=C,^[a,p) + C {a,/3)S,+C {a,j3)S, + C {a) 3,^ +\ — \lc {a) p + C {a)
a r sp \ZVjJ^l^
[^c.g.,ref ^cgjCy
where
^Cx,, =Cx,^(aJ,SJ-Cj,^{aJ)
ACy =Cy (OC)-Cy («)
^■S =55° ^-d =55° ^0^ ^
sp sp
^Cz =Cz (a,/5,5J-Cz {a,/5)
e e
= €7 (oc)-Cy ia)
= €7 (oc)-Cy ia)
AC^
AC2
e e
AC„
= c
^sp^^^ ^sp=^^
7] I (M ) = wing spoiler rolling moment efficiency factor
given as a function of Mach number
VspM [^ ) = wing spoiler pitching moment efficiency factor
given as a function of Mach number
38
Data tables for the component functions of the non-dimensional coefficients are
defined in the file F14_aero_setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in F14_aero.m, using linear
interpolation in the functions etamsp.m, etalsp.m, dndr.m, dnda.m, dlda.m, dldr.m,
dczsp.m, dczsb.m, dczds.m, dcxsp.m, dcxsb.m, dcxds.m, dcmsp.m, dcmsb.m, dcmds.m,
czo.m, cyo.m, cydsp.m, cydr.m, cyda.m, cxo.m, cno.m, cndsp.m, cmo.m, clo.m, cldsp.m,
and dampder.m.
Engine Model and Database
Data tables for the F-14 engine are defined in the function F14_engine_setup.m.
Linear interpolation and thrust calculation is performed in the module F14_engine.m.
Functions tgear.m, pdot.m, and rtau.m are unaltered from the F-16 engine simulation.
Mass Properties
Mass properties for the aircraft are given in the module F14_massprop.m. Table 3
lists the mass properties used in the F-14 simulation.
Table 3. Mass Properties of the Simulated F-14
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^Xl
(slug-ft")
Value
48,669
58,500
227,000
276,000
-2,820
F-4 SIMULATION
The McDonnell Douglas F-4 is a two-seat all-weather fighter / bomber. The aircraft
is fitted with a low-mounted swept wing with wingtip dihedral. The tail section consists
of an all-moving horizontal stabilator placed in a cathedral configuration and a single
vertical tail. The trailing edge of the main wing houses control surfaces for the aileron
and flaps. The trailing edge of the vertical tail accommodates a rudder control surface.
Thrust is provided by two General Electric GE-J19-17 afterburning jet engines mounted
at the rear of the fuselage.
The simulation has controls for dth, de, da, and d,. Aerodynamic data were acquired
with the model flown out of ground effect, with landing gear retracted, and no external
storesi^. Static and dynamic aerodynamics are modeled in terms of polynomial
functions, which limit control surface effects to first order dependencies, angle of attack
static dependencies to second order, and sideslip angle static dependencies to third order.
Polynomial fits for each non-dimensional aerodynamic force and moment coefficient are
valid over an angle of attack range of -15° <a<55°. Aerodynamic coefficients are
referenced to a center of gravity location at 0.289 c , and corrected to the flight center of
gravity position in the coefficient build-up equations.
39
Throttle deflection is limited to the range < J^/, < 1 , elevator deflection is limited to
-21° < 5e< 1°, aileron deflection is limited to -15.5° < Sa< 15.5°, and rudder deflection
is limited to -30° < (Jr ^ 30°. A scaled version of the F-16 engine model is used with this
simulation. Thrust for idle, military, and maximum power settings were scaled according
to the maximum thrust values of the F-4 power plant. Throttle gearing and engine power
level dynamics were carried over from the F-16 engine model, neglecting engine angular
momentum effects. Figure 10 shows a three-view of the F-4 aircraft.
Figure 10. Three-View of the F-4
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the F-4 model
vary nonlinearly with flow angles {a, P), aircraft angular velocities (p, q, r), and control
surface deflections (4, ^a, S,)- Moment coefficients C„ and Cm include a correction for
the center of gravity position. The coefficients are computed as follows:
40
For a< 15°
C;^=-0.0434 + 2.39xl0"^a + 2.53xl0"V^-107xl0"^a;5^ +9.5x10"^ (5;
-%.5xl{)-^ 5J^ +
^ 180gc ^
(8.73x10"^ +0.001a-l. 75x10"^ a^)
Cy = -0.012;5 + 1. 55x10"^ (^■^ -8x10"^ (5;a
(2.25xl0"V + 0.0117r-3.67xl0"^ra +1.75xlO"V4)
+
^1806^
yTtlV.j
Cv = -0.131- 0.0538a- 4.76x10"' J -3.3x10"= (5'„a-7.5xlO"= J,
5 e 2
+
^ 180^c ^
7t2V,
(-0.111 + 5. 17xl0"^a-l. 1x10"^ a^)
Q=-5.98xlO"V- 2.83x10"^ aj5 + l. 51x10"^ aV
-J^ (6.1x10"^ +2.5xl0"^a- 2.6x10"^ a^)
-(^■^ (-2.3x10"^ +4.5x10"^ a)
+
^ 1806 ^
y7t2V,j
(-4.12x10"^/? -5.24xl0"V«+4.36xl0"^/?a^
+ 4.36xl0"V + 1.05xl0"Va + 5.24xl0"^r(5;)
Q =-6.61x10"^ -2.67xl0"^a-6.48xl0"V^
-2.65xl0"^a;5^-6.54xl0"^4-8.49xl0"^4a
+3.74x10"^ (5"^y^2 -3. 5x10"^ (^'Z
+
^ 180gc ^
7t2V,
(-0.0473-1.57x10 ^a) + (x,
c.g.,ref -^c.g. j '^Z
)C2
41
C„=2.28xlO"V + l-79xlO"V^ +1-4x10"^ J„
+7.0xlO"^J^a-9.0xlO"^J,. + 4.0x10"^ J^a
+
(-6.63xlO"V-l-92xlO"Va +5.06x10"^^^"
-6.06xl0"V-8.73xl0"Vj^+8.7xl0"Vj^a)
^c.g.,ref ■^c.g.j'^Y
)q
For 15° < a<30'
Cjf =0.141-0.0154a + 2.96xl0"V-3.72xl0"V^
+ A.\Ax\0~^al3^ - 9.12xlO"Vy^^+1.82xlO"^(5'e
-7.3x10 ^5^a+
^ 180gc ^
(-0.0602 + 2.04xl0"'a)
Cy=-2.08xl0"2;5 + 6.07xl0"^a;5 + 2.37xl0"V^
-3.64xl0"^aj5^+ 2.3x10"^ J^ -5.9x10"^ J^G
+
^ 1806 ^
(-1.62xlO"V + 3. 32x10"^ /?a
+ 0.031 lr-1.4xlO"^ra + 1.75xlO"VjJ
Q =-0.608- 0.022a - 6.77x10"^ (5; + 9.7x10"^ (5'ea-7.5xlO"^(5'/
+
^ 180^c ^
3^,2^
(1.136 -0.1418a + 3.11xlO"'a^)
42
Q = -1. 29x10"^ ;5 + l. 04x10"^ a;5-2.02xl0"^a^y?
+ 1.36xl0"V^-113xl0"^a;5^ + 2.01xl0"^a^;5^
-J„(7.74xl0"^-1.9xl0"^a)-J^ (-2.0x10"^ +5.0xl0"^a)
+
(2.78xl0"^/?-2.79xl0"V«-6.81xl0"V
+ 6.46xlO"Va + 5.24x10"^ rJJ
C^ = 0.0549 - 6.08 xlO"^a-l. 69 xl0"^y5^ + 5.64xlO"V;5^
-8.14xl0"^4 + l.lxl0"^4a-3.5xl0"^(5'/
+
^ 180gc ^
v^2^y
(-0.0951 + lAxlO-'a) + (x,g_,,f-x,g)Cz
C„=1.02xl0"2;5- 5. 12xl0"^a;5- 5.27x10" V^
+ 3.79xl0"^aj5^ + 9.1xl0"^J^+3.0xl0"^J„a
-1.37xlO"^J^ + 3.8xlO"^J^a
+
(0.0236p-2.5xl0 ^;7a + 6.25xl0 ^ pa^
+ 6.2xl0"V-4.89xl0"Va-8.73xl0"^r4 + 8.7x10" V^a)
''c.g.,ref ■^c.g.l'-^Y
)Cy
43
For a> 30°
C;f= -0.0326-2. 16xl0"^a + 4.89xlO"V - 1.24xlO"V^
-5 „/32
7 „2 o2
+ 1.076xl0"=a;5 -1.54 X 10"' a^;^^ + 7.5x1 0"^(5;
-3.9xlO"^(5'„a+
^ 180gc ^
(-0.026 + 8.73xlO"V)
Cy=-2.095xl0"V-6.36xl0"^a;5-2.15xl0"V^
+5.42x10"^ ay5^ + l. 4x10"^ J^- 2.6x10"^ J^a
+
(0.196/?-9.27xlO"Vc^ + 101xlO"V«^
+0.0326r-2.55xl0"^ra + 3.26x10"^ ra^ + 1.75xlO"VjJ
C2=-0.891-0.01146a+6.2xl0"^4 - 5Ax\{)~^5^a
+ 6.2xl0"^4a2 -7.5x10"^ j/
+
^ 180gc ^
■4^,2^
(0.589 -0.0494a+ 6.11xlO"V^)
Q=1.18xl0"V-5.29xl0"^a;5 + 4.88xl0"^a^;5
-2.2xlO"V^ +9.05xl0"^a;g^ -9.08xlO"'^aV
-5.0x10"^ J„-J^('-9.0xl0"^+l. 8x10"^ «;
+
^ 1806 ^
(-0.0428/? + 1.82xlO"V«-l-94xlO"V«^
+0.073r-3.02xl0"Va + 3.14xl0"Va2+5.24xl0"VjJ
C„ = 7.3x10"^ -5.5xl0"^a-7.93xl0"^4 + 8.23x10"^ 4a
-3.5xlO"^j/ +
180^c
A 2^
(0.16-0.0101a + 1.05x10" a")
'^\-^c.g.,ref ^c.g.j^Z
)c.
44
C„ = -9.23xlO"V + 1-52x10"^ a;5 + 1.62x10"^^
-3.46xl0"^a;5^ + 1. 5x10"^ J^ -6.8x10"^ J^
+1.2xl0"^J^a-Jg(1.67xl0"V-5.56xl0"^aj5
-3.81x10"^^^ +1.27xl0"^a;5^)
+
^1806^
(0.03S5p-lJ3xlO~^ pa + I.92xl0~^ pa^
-0.0202r + 3. 67xlO"Va + 2.58x10"^ rJ„
-1.31xlO"Vj^a + l. 69x10"^ rJ^a^) \(xc.g.,ref '^c.g^Cy
The non-dimensional aerodynamic force and moment coefficients are computed for
this simulation in F4_aero.m.
Engine Model and Database
Scaled thrust values for the F-4 power plant are applied in the function
F4_engine_setup.m. Linear interpolation and thrust calculation are performed in the
module F4_engine.m. Functions tgear.m, pdot.m, and rtau.m are unaltered from the F-16
engine simulation.
Mass Properties
Aircraft mass properties are given in the module F4_massprop.m. Table 4 lists the
mass properties used in the F-4 simulation.
Table 4. Mass Properties of the Simulated F-4
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^xz
(slug-ft")
Value
38,924
24,970
122,190
139,800
1,175
FASER SIMULATION
A radio-controlled aircraft, called the NASA Langley FASER, has the form of a
high-wing general aviation aircraft. The aircraft was modeled with controls for 5th, Se, 5a,
and 5r. Non-dimensional static and dynamic aerodynamic data were estimated^^ using
airfoil data and approximations based on aircraft geometry, for the aircraft out of ground
effect. The aerodynamic data is in tabular form as a function of angle of attack, Mach
45
number, lift coefficient, and thrust over the ranges -10° < a < 30°, 0.05 <M< 0.3,
-l<Ci <1.6, and lb < T < 10 lb. Each non-dimensional aerodynamic force and
moment coefficient is built up from component functions, which are provided in the
stability-axis frame. Aerodynamic coefficients are referenced to a center of gravity
location at 0.273 c , and corrected to the flight center of gravity position in the coefficient
build-up equations.
Throttle deflection is limited to the range < Sth^ I, elevator deflection is limited to
-25° < Se< 25°, aileron deflection is limited to -25° < Sa< 25°, and rudder deflection is
limited to -30° < Sr < 30°. FASER is fitted with a single electric motor that drives a
propeller to generate thrust. Figure 1 1 shows a three-view of the FASER.
Figure 11. Three- View of the FASER
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the FASER
vary nonlinearly with relative airflow quantities (a, P, M), lift coefficient, (Cl ), thrust
(7), aircraft angular velocities (p, q, r), and control surface deflections (4, 4, Sr).
Moment coefficients C„ and Cm include a correction for the center of gravity position.
The coefficients are computed as follows:
Cx = Cl sin
180
Cj^cos
180
46
Cy=Cy^{M)
Vl80y
+ C. (M)
^5.7t^
vlSOy
+
' b ^
K^VtJ
C2 = -Ci COS
[Cy {a,M)ps + Cy {a,M)rs
^COT^
Vl80y
■Q^^m
C^) sin
^ an'^
Vl80y
^COT^
Vl80y
qc
C™ (a,M) + C_ (a,M)
^cy„;r^
+ ^^m,^ + \^c.g.,ref ~ ^c.g. ) '-Z
vlSOy
.)C.
C„ = C/ e 5m + C^ c C05
i^c.g.,re/ ^c.g.)<^y
where
C^=Cr{aM) + Cr {a,M)
^d^Tt^
Vl80y
+ zlC, +
qc
C,Ja,M)
Cd=CdAoc>M) + ACo
Ci,s=Ci^{M,C,)
vlSOy
^d.7t~^
+ C, (M) ^^ +C, (a,M)
^S.7t^
vlSOy
+
^ b^
2V
C, {M)ps+C, {a,M,CL)rs
Ps '5
Cn,S=Cn^{0:'M)
Vl80y
( d 7t\
+ C„ {M,Cj) -^— +C„ (a,M)
^cy.;r^
vl80;
+
v2^.y
(c (a,M,Cjp^+C {a,M,CL)rs\
\ Ps 's I
^CL,^=CL,^{a,M,T)-CL^{a,M)
^Cn,j.=Co,^{a,M,T)-Cj,^{a,M)
47
Ps=pcos
^ an^
Vl80y
+ rsm
^ an^
Vl80y
rj = —p sin
^ an^
Vl80y
-\-rcos
^ an^
Vl80y
Data tables for the component functions of the non-dimensional coefficients are
defined in the file F ASER aero setup. m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in FASER_aero.m, using linear
interpolation in one-, two-, and three-dimensional interpolation fianctions intl.m, int2.m,
and intS.m.
Engine Model and Database
Propulsion for the FASER is modeled neglecting engine angular momentum.
Throttle gearing is implemented in the file tgear.m, which outputs commanded engine
power in percent of full power [0,100] using throttle setting input [0,1]. Commanded
power Pc is a linear fianction of Sth, given by:
p,{s,,) = ms,,
Engine power level dynamic response is modeled using a first order lag with time
constant computed by the function rtau.m. The module pdot.m calculates the rate of
change of power level with time using the difference between Pc and Pa- The actual
power level derivative P^ is given by:
1
"eng
[Pc-Pa)
where
"eng
1.0 if(P,-Pj<25.0
0.2 if(P,-Pj>50.0
l-0-(^^j[(/'.-/'.)-25.0] if25.0<(P,-P,)<50.0
Engine thrust is modeled as a linear fianction of Pa- Thrust calculation is performed
in the module FASER_engine.m using the equation:
r = 0. IR
48
Mass Properties
Mass properties of the aircraft are given in the module FASERjnassprop.m. Table 5
lists the mass properties used in the FASER simulation.
Table 5.
Mass Properties
of the Simulated FASER
Parameter
Weight
(lb)
Ix
(slug-ft')
Iy
(slug-ft")
(slug-ft")
Ixz
(slug-ft")
Value
16.5
1.324
1.946
0.627
0.317
HL-20 SIMULATION
The NASA HL-20 is a manned lifting-body spacecraft. It is designed to be placed
into earth orbit by an expendable launch vehicle and re-enter the atmosphere for an
unpowered approach and landing. Aircraft control surfaces consist of an all moving
vertical tail (rudder), two upper and lower body flaps, and two trailing edge wing flaps.
Elevator control is provided by symmetric deflection of the wing flaps, aileron control by
differential deflection of the wing flaps, positive flap control by symmetric deflection of
the lower body flaps, negative flap control by symmetric deflection of the upper body
flaps, and differential flap control by differential deflection of the upper and lower body
flaps.
The HL-20 was modeled with controls for 4, Sa, Sr, Spf, dnf, Sdf, and dig. Low-speed
(M <0.6) wind tunnel data for a scale model of the aircraft flown out of ground effect
with landing gear retracted and no external stores were taken in the Langley 30- by 60-
foot tunnel and Calspan 8-foot transonic tunneli^'i^. Landing gear effects were based on
scaled Space Shuttle data, and dynamic derivative data were found using estimation
techniques. Static and dynamic aerodynamic data are modeled with polynomial fits as a
function of angle of attack and sideslip angle over the ranges -10° < or < 30° and
-10° < y5< 10°. Each non-dimensional aerodynamic force and moment coefficient is built
up from component fianctions. Aerodynamic coefficients are referenced to a center of
gravity location at 0.54 c , and corrected to the flight center of gravity position in the
coefficient build-up equations.
Elevator deflection is limited to -30° < 5e < 30°, aileron deflection is limited to
-30° <d^< 30° , rudder deflection is limited to -30° < dr< 30°, positive flap deflection
is limited to 0° < dpf < 60°, negative flap deflection is limited to -60° < dnf < 0°,
differential flap deflection is limited to -60° < 5df < 60°, and landing gear deflection is
limited to 0° < 5ig < 98°. An engine model is not included in this simulation since the
HL-20 uses no propulsion during atmospheric flight after re-entry. Figure 12 shows a
three-view of the HL-20.
49
Figure 12, Three- View of the HL-20
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the HL-20
model vary nonlinearly with flow angles (a, JS), aircraft angular velocities (p, q, r), and
control deflections {de, Sa, Sr, Spf, dnf, Sdf, Sig). Moment coefficients C„ and Cm include a
correction for the center of gravity position. The coefficients are computed as follows:
Cy=Cy +Cy^S„+Cy^ <5'„+Cy„ (^^+Cy„ S„f+Cy„ S„f + C y ^
'df
+AC
X.8,,
Cj = -0.01242/? + Cj. S, + Cy S, + Cy S^f
Cy=Cy+CyS^+Cy^ S „r + C y . 5„f ^ AC
'^s, '
'pf
pf
"nf
V
^Z,5,,
50
^ b^
Ci = -0.00787 A + C, S^ + C, 5, + C, 5, if A f Q ;? + Q r )
^m i^Q "^S '^ '"S P^ '"S "/ '"'^Is
^ b^
cq
^m '^Y^c.g.,ref ■'^c.g.j^Z
Cn=Cn^{a,/5) + C^^ da+C^gS,+C^^ 5df+\ — \[cp + C^A-
a r dj I zKj i^ P I
[b
where
[^c.g..ref ^cgjCy
Cx - -(7.362x10"^ -2.56xl0"V-2.208xl0~V-2.262xl0"V+2.966xl0"^cir^
-3.64xl0"V+9.388xl0"^V -5.299x10"^ |y5|-4.709xlO"V^ +8.572x10"^ p'^
-4.199xlO"V^ +1.295 xlO"V|y5|)
Cx --(-1.854x10"^ + 2. 83xl0"^a-6.966xl0"^a^ + l. 323x10"^ cir^-2.758xlO"V)
Cx =-(9.776xl0"^-2.703xl0"^a-8.303xl0"V+6.645xl0"V-1.273xl0"V)
a
Cx = -(5.812x10"^ + 1.41xlO"^a-2.585xlO"V +3.051x10"^ or^ -8. 161xlO"V)
Cy^ =3.357x10"^ -1.661xl0"^a-3.28xl0"^a2 +5.526x10"^ a^-3.269xl0"^°a^
Cy = 1.855x10"^ +1.128xl0"^a + 6.069xl0"^a^-1.78xl0"V^-1.886xl0"^^a^
Cy =2.672x10"^ -3. 849xl0"^a+4.564xl0"V^ +1.798x10"^ a^-4.099xl0"^°a^
Cy =-(-9.025xl0"H4.07xl0"^a + 3.094xl0"^aHl.564xl0"^a^ -1.386x10"^ a^
-10 ^,6
-4 /o2
+2.545x10"^ a^ -1.189xlO"'V +2.564x10"' L5 +8.501x10"^;^
-1.156x10"
-6 o4
/3' +3.416xlO"V -4.862xlO"VL5;
Q =-(5.140x10"^ +3.683xl0"^a-6.092xl0"^aH2.818xl0"V^-2.459xl0"V^)
51
Cx =-(1.31x10"^ +1.565xl0"V-1.542xl0"V)
Cx =-(- 4.415 X 10"^ -4.056xl0"^a- 4.657 xlO"V)
V
Cx =-(- 6.043x10"^ -1.858xl0"^a+8xl0"V-4.848xl0"V + 1.36xl0"V^)
^Cx,5i^=Cx,5i^[aAs]-Cx^{cc)
Cz =-(3.7779x10"^ -7.017xlO"VHl.4xlO"^°a^)
Cv =-(3.711x10"^ -3.547xl0"^a-2.706xl0"^aH2.938xl0"V^
-5.552 xlO"V^)
AC,.=C,.(aA,)-Cz^{a)
C, = 2.538x10"^ +1.963xl0"^a-3.725xl0"V+3.539xl0"V -1.778x10"^ V
C, =2.26x10"^ -1.299xl0"^a + 5.565xl0"V-3.382xl0"V+6.461xl0"V^
C, =-(7.453x10"^ -1.811xl0"^a-1.264xl0"V+9.972xl0"V-2.684xl0"V)
% ^ '
C, =-0.5261357 + 0.029749438a-0.006960559a2 + 0.00053564a^-1.35043xl0"V
p
C, =0.497793859-0.028742272a+0.016480243a2-0.001439044a^+3.56179xl0"V
V
52
C„ =2.632x10"^ -2.226x10"^ 6^-1.8594x10"^ aH6.001xlO"V^+1.828xlO"V^
-9.733xlO"V^+1.71xlO"^V^ -5.233x10"^ |y^| + 6.795xlO"V^
-1.993x10"^
?3
6/34 . r Ari.,1A-5.
/3' +1.341x10"° y^^+6.061xlO"^aL^
r = -1.903x10"^ -1.593xlO"^a + 2.611xlO"V^ +5. 116x10"^ a^-1.626xlO"V^
r = -9.896x10"^ -1.494xl0"V^+6.303xl0""a^
5 /•
r =-1.086x10"^ +1.57xl0"^a + 4.174xl0"V^-1.133xl0"V^+2.723xl0"V
^C„,<j. = C^. (a, Sjg ) - C (a)
C^ =-0.195552417-0.006776550a + .00285193a^-0.000184146a^
+2.45653 X 10"^ a^
C„ =-2.769x10"^ -4.377xl0"^a + 9.952xl0"V -3.642xlO"V +4.692xlO"V
C =-1.278x10"^ +1.32xl0"^a-4.72xl0"V+2.371xl0"V-3.340xl0"V^
r
C„ =-5.107x10"^ +1.108xl0"^a-1.547xl0"V-1.552xl0"V+1.413xl0"^V
C„ =0.385147809-0.001356061a-0.00209489a2+8.52445xl0"V+1.02309xl0"V
p
r =-0.790855552-0.010770449a-0.00095452a2+0.000151569a^-5.13541xl0"V
'V
Data tables required for the component functions of the non-dimensional coefficients
are defined in the file HL20_aero_setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in HL20_aero.m. Linear
interpolation is done in the fimctions cno.m, cmlgear.m, cxlgear.m, and czlgear.m.
Dynamic Equations
Thrust effects and the power level state derivative were removed from the dynamic
equations for the HL-20 simulation.
Mass Properties
Aircraft mass properties are given in the module HL20 massprop.m. Table 6 lists the
mass properties used in the HL-20 simulation.
53
Table 6. Mass Properties of the Simulated HL-20
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^xz
(slug-ft")
Value
19,100
7,512
33,594
35,644
X-31 SIMULATION
The NASA / DARPA X-3 1 is a thrust-vectored, single-seat research aircraft designed
to explore highly maneuverable flight. The aircraft has a cranked delta-wing planform,
all-moving canard surfaces mounted near the nose, and a single vertical tail. The wing
houses leading and trailing edge flaps. Trailing edge flaps act as elevon control surfaces,
which function as separate elevator and aileron control surfaces within the simulation.
Speed brakes are attached to the rear section of the fuselage and deflect symmetrically
outward when actuated. A single General Electric F404-GE-400 non- afterburning
turbofan engine provides thrust for the aircraft. Three paddles are symmetrically
attached at the rear exit of the engine nozzle to provide thrust- vectoring capabilities.
However, thrust vectoring is not implemented in this simulation. The effect of symmetric
outboard deflection of the three thrust-vectoring paddles is modeled as an additional
speed-braking device.
The aircraft was modeled with controls for dth, 4, 4, S,, 5c, S^h, and dtv Low-speed
static wind tunnel tests provided force and moment aerodynamic data for a 19% scale
model of the aircrafti^. Tests were conducted in the Langley 14- by 22-foot subsonic
tunnel with the model flown out of ground effect, with landing gear retracted, and no
external stores. Dynamic data were obtained from forced oscillation wind tunnel
testingi9. Static aerodynamic data are in tabular form as a fianction of angle of attack and
sideslip angle over the ranges -5° < a< 55° and -18° < y^< 18°. Each non-dimensional
aerodynamic force and moment coefficient is built up from component functions.
Aerodynamic coefficients are referenced to a center of gravity location at 0.5286 c , and
corrected to the flight center of gravity position in the coefficient build-up equations.
Throttle deflection is limited to the range Q < dth'^l, elevator deflection is limited to
-40° < 4 < 30°, aileron deflection is limited to -30° < da < 30°, rudder deflection is
limited to -30° < dr< 30°, canard deflection is limited to -20° < 4 < 60°, speed brake
deflection is hmited to 0° < d^h < 50°, and thrust vectoring paddle deflection is hmited to
0° < 5tv ^ 50°. Data from the F-16 engine model was scaled using the maximum thrust
values for the X-31 engine. Throttle gearing and engine power level dynamics were
carried over from the F-16 engine model; however, engine angular momentum was
ignored. Figure 13 shows a three-view of the X-31.
54
Figure 13. Three- View of the X-31
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the X-3 1 model
vary nonlinearly with flow angles {a, P), aircraft angular velocities (/?, q, r), and control
surface deflections (Se, da, 5r, 5c, Ssh, Stv). Moment coefficients C„ and C„, include a
correction for center of gravity position. The coefficients are computed as follows:
Cx =
CaMA) + C. {a,S,)S, + AC^^
'^.i=50°
v50.
r^ \
+ AC,
'<y,^=50°
v50.
Cy = CyM + ACy,
r ^ \
S,-'iO°
V30y
r ^ \
+ ACy
'<y«=3o°
V30y
+ ACy
'^st^
''^.^=50°
V50y
r^ \
+ACy
■<y,^=5o°
50
Cz =
'^st^
^b-^^°
V50y
+ AC
N
'S^^SG'
(d \
I 50 J
qc
Cn {a)
55
z' s: \
30
r^ \
+ AC,
Vjuy
'^.=30°
Ci=Ci (iB) + ACi
v30y
+ AC,
'^b-5^°
V50y
+ AC,
(5 ^
'<y,^=5o° 1 50 ,
so
Ssb=^^°
v50.
+ AC„
'<y,.=5o°
f^ ^
50
Vv--^" V^uy
+
qc
Cn=CAP)^AC,^
Cm {^) + {^c.g.,ref ^cgj^Z
<y^=30°U0y
+ ^C„
''^.=30°
V30y
+ /ic„
^-^..^
''^.^=50
i^^=Du V ->" y
50
+ /1C„
Xv=50°l50,
+
v2^ry
(C„^(a);; + C„^(a)r)- I (
^c.g.,ref -^c.g.j^Y
)c.
where
^sb=50 ^sb=50 ^
AC^ = C^
^sb=50 ^sb=50'
^V30^=%.0^(^)-^^o(/^)
^S^^^,=%^^3^J/^)-C,(/^)
ZlCy = Cy
(«)-Cy„(A)
56
^S,^3,=S,^30^(/^)-C„„(/^)
^S.O^=S.O^(^)-^"o(/^)
Data tables for the component functions of the non-dimensional coefficients are
defined in the file X3 1 _aero_setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in X31_aero.m, using linear
interpolation in the functions ca_can.m, dca_dele.m, dca_dsb.m, dca_dtv.m,
cnorm_can.m, dcnorm_dele.m, dcnorm_dsb.m, dcnorm_dtv.m, cy_beta.m, dcy_drdr.m,
dcy_dail.m, dcy_dsb.m, dcy_dtv.m, cm_can.m, dcm_dele.m, dcm_dsb.m, dcm_dtv.m,
cljbeta.m, dcl_drdr.m, dcl_dail.m, dcl_dsb.m, dcl_dtv.m, cn_beta.m, dcn_drdr.m,
dcn_dail.m, dcn_dsb.m, dcn_dtv.m, caq.m, cyr.m, cyp.m, cnormq.m, clr.m, clp.m, cmq.m,
cnr.m, and cnp.m.
Engine Model and Database
Thrust scaling for the X-31 engine is applied in the function X3 1 _engine_setup.m.
Linear interpolation and thrust calculation is performed in the module X31_engine.m.
Functions tgear.m, pdot.m, and rtau.m are unaltered from the F-16 engine simulation.
Mass Properties
Aircraft mass properties are given in the module X3 1 jnassprop.m. Table 7 lists the
mass properties used in the X-3 1 simulation.
57
Table 7. Mass Properties of the Simulated X-31
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^Xl
(slug-ft")
Value
16,000
3,553
50,645
49,367
-156
A-7 SIMULATION
The Vought A-7 Corsair II is a single-seat tactical attack fighter based on the F-8
Crusader platform. The aircraft is controlled with ailerons located on the trailing edge of
the main wing, a rudder control surface located at the trailing edge of the vertical tail, and
an all-moving horizontal stabilator located at the rear of the fuselage. Thrust is provided
by a single non-afterburning Pratt & Whitney TF30-P-408 turbofan engine mounted in
the rear fuselage.
Aircraft controls are provided for 5th, 4, 4, and 5r. Low-speed static wind tunnel
tests and dynamic forced oscillation wind tunnel tests provided force and moment data
for a scale model of the aircraft flown out of ground effect, with landing gear retracted,
and no external stores^". Static and dynamic aerodynamic data are in tabular form as a
function of angle of attack and sideslip over the ranges 0° < a< 90° and 0° < P< 90°.
Each non-dimensional aerodynamic force and moment coefficient is built up from
component fianctions. Aerodynamic coefficients are referenced to a center of gravity
location at 0.35 c , and corrected to the flight center of gravity position in the coefficient
build-up equations.
Throttle deflection is limited to the range < (Jj/, < 1 , elevator deflection is limited to
-25° < 5e< 15°, aileron deflection is limited to -25° < 5a< 25°, and rudder deflection is
limited to -30° < 5r< 30°. The A-7 engine has a maximum thrust of 13,400 lb. A scaled
version of the F-16 engine model was used for this simulation. Idle and military power
thrust data were scaled according to the maximum thrust for the A-7 engine. Throttle
gearing, engine gyroscopic effects, and engine power level dynamics were carried over
from the F-16 engine model. Figure 14 shows a three-view of the A-7 aircraft.
58
Figure 14. Three- View of the A-7
Aerodynamic Model and Database
The non-dimensional aerodynamic force and moment coefficients for the A-7 vary
nonlinearly with flow angles (a, /5), aircraft angular velocities (p, q, r), and control
surface deflections (4, 4, Sr). Moment coefficients C„ and C„, include a correction for
the center of gravity position. The coefficients are computed as follows:
Cy. = C^ [a,d^)sin
U80
C^{a,d^)cos
an:
180
Cy=Cy^(a,\l3\)sign{p) + Cy^{a)d^ + Cy^ («,<^„,„,..)<^„,„,A.-Q^ (cc,S,j,fl)5^
a, right
a Jefi
left
Cy,=-C,cos\
( an
U80
CjySin
'' an^
Vl80y
59
C, = C (a,\p\)sign{^) + C {a,S^^„^,,)-C (aAMft) + C {a)5,
a, right
a, left
+
f 1 \
2V
C, (a)p + C, (a)r
C =C (a,S) +
qc
C {a,d) + C {a,d) +C . (a, 5 .A
+C . (a,S , A + ix ,--x )C7
"• a,fe/( V a,te/( / V e.g., re/ e.g./ Z
^Z,^
, 2F ,
Data tables for the component functions of the non-dimensional coefficients are
defined in the file A7 _aero_setup.m. The non-dimensional aerodynamic force and
moment coefficients are computed for this simulation in A7_aero.m, using linear
interpolation in one- and two-dimensional interpolation functions intl.m and intl.m.
Engine Model and Database
Thrust scaling for the A-7 engine is applied in the function ^7_e«gme_5eZ^M/?.m using
the F-16 engine data, omitting afterburner data. Throttle gearing, implemented in
tgear.m, outputs commanded engine power in percent of fiall power [0,100] using throttle
setting input [0,1]. Commanded power Pc is a linear fianction of Sth, given by
P,(c^J = 100^,,
Functions /((ioZ^.m and rtau.m are unaltered from the F-16 engine simulation.
Linear interpolation and thrust calculation is performed in the module A7 _engine.m,
scaling F-16 engine data to the maximum thrust of the A-7 engine. Scaled thrust is
computed from
T =
13,400
vl2,680y
(?;,,, +0.01(7;, -?;.,„)/'„)
Mass Properties
Aircraft mass properties are given in the module A? massprop.m. Table 8 lists the
mass properties used in the A-7 simulation.
60
Table 8.
Mass Properties of the Simulated A-7
Parameter
Weight
(lb)
Ix
(slug-ft")
(slug-ft")
(slug-ft")
^Xl
(slug-ft")
Value
22,699
16,970
65,430
76,130
4,030
61
VI. GENERATING 3-D MODELS FOR AVDS
Three-dimensional aircraft models used within AVDS are constructed using a
separate CAD drafting program. Models created with CAD are converted to an input
format suitable for AVDS. AVDS supplies a utility that allows conversion of Drawing
Interchange Format (DXF) 3-D aircraft models into AVDS format. The following
sections provide a tutorial for creating and adding a 3-D aircraft model into AVDS for
use in AVDS / MATLAB piloted simulations.
MODEL CONSTRUCTION REQUIREMENTS
Three-dimensional aircraft models must be constructed in AutoCAD" DXF format
using surface modeling techniques. The AVDS conversion program does not recognize
models created using solid modeling techniques. Surface models have exteriors
composed of meshes made of three and four sided polygons whose vertices are located in
three-dimensional space. The total number of polygons in a surface mesh determines
model complexity. AVDS / MATLAB interactive simulations are computationally
intensive; therefore, it is necessary to create 3-D models that are as simple as possible in
order to minimize graphics rendering requirements. Complex models with dense surface
meshes can slow AVDS / MATLAB interactive simulations considerably on less
powerful computers.
Many 3-D CAD programs support both surface modeling and the DXF format. Ruled
surfaces, tabulated surfaces, and edge surfaces are used extensively during surface model
creation. A surface created using any of these tools is recognized by the AVDS
conversion utility. However, revolved surfaces and two-dimensional lines are not
recognized during conversion.
Aircraft 3-D models should be created using two-dimensional surfaces for the wings
and horizontal / vertical tails. This reduces the complexity of the aircraft surface mesh
while not excessively detracting from aircraft reahsm. The fuselage, canopy, and other
significant aircraft structures should be created as simply as possible using low-density
surface meshes. Following these guidelines will simplify model making as well as
prevent any unnecessary slow-downs during piloted simulations run on less powerfial
computers.
MODELING EXAMPLE OF A SIMPLE 3-D AIRCRAFT
An example is given to illustrate the surface model creation of a simple 3-D aircraft,
shown in figure 15. It is assumed that the reader has familiarity with using and drawing
in AutoCAD Release 14 (or higher) or an equivalent 3-D CAD program. For more
information on CAD design or surface modeling consult any AutoCAD Release 14 (or
higher) reference book and surface-modeling tutorial such as Reference [21]. Aileron,
rudder, and elevator control surfaces will be included in the model, while wheels and any
external stores are ignored.
62
Figure 15. Three- View of the MiG-15 Aircraft
It is convenient to model aircraft using available three-views such as the one shown
in figure 15. These drawings may be found on various internet sites, and in references
such as [22]. Dimensions used during initial modeling of the aircraft should be those of
the drawing rather than the actual airplane. This simplifies model-making by eliminating
the need to scale the dimension of every point on the model. Once modeling is complete,
the finished 3-D aircraft must be scaled to the dimensions of the actual aircraft.
Dimensions must be in feet for proper implementation into AVDS. The scaling feature is
available in most CAD programs. The scaled model must also be aligned along the
world coordinate system (WCS), prior to conversion into the AVDS format, with the
nose of the model pointing along the positive X-axis and positive Z-axis pointing out the
top of the model. The origin of the WCS must be positioned at the location of the aircraft
center of gravity. AVDS uses this origin location as placement for the body, earth, and
stability axes on the model during interactive simulations.
The fiaselage for this particular aircraft is a relatively simple shape and can be
modeled as a cylinder with the front and back openings used as the intake and exhaust.
Figure 16 shows the prehminary front and rear openings of the aircraft.
63
'; sees ♦jc'c^^'^ Hfcy-'
1D(]I« ]4.3H.D(»n
^KtP GRD CRTHO I
i
Figure 16. Front and Rear Fuselage Openings
The AutoCAD command RULESURF is used to create a surface mesh between the
intake and exhaust. The completed fuselage is shown in figure 17.
D tf i a Jl ':=i * *! IB rf --'
r. « -
.101 x|
B^ p-rf'isL:" dl" ■■
..,. ,(|
,.,.^ ^1 ,,.^ ^|,
J
^^.
1I.K44.S4€£a[«0a
Figure 17. Modeled MiG-15 Fuselage Section
The wing, tail, and control surfaces of the aircraft are modeled using planar surfaces.
This is accomplished using the AutoCAD command 3DFACE. Figure 18 shows the
64
aircraft with the various surfaces added. RULESURF was used to create surface meshes
for the modeled intake and exhaust.
©fir E* S" S™- F*«« Irf. C^ D^EC* 'S^l, l;S^ l^rft- [**
-.^l.='J
s B a J«t J as o rf ■■' " * - : ih w < ^Ej • ^t ct q a 3 ?■■ i-i
7
BQ|J^*'«-» d|i:..«. Jl M^ -1,1 la^ -1|...
J
f^Mara
ah»a) ing:s5tf».- In.
Jit:
I 'v3:dQJ iJ»*^^f^l!3(3K[*
Figure 18. 3-D Model with Wing, Tail, and Control Surfaces
The canopy can be modeled as an elongated dome. A wire frame is constructed
outlining the general shape of the canopy, shown in figure 19. Surface meshes are drawn
over the wire frame using the RULESURF command.
jFfe. E* iV- imc- Fama lo* B- CWeckp >!«drr Es»m IbfrA- K*
y
--.ajii
B'dM-^Sw' dl- ■■■■■ dl '>'■-< J:l — "■<■'-' dl
_l
^^3,
«;W7. -SUH.EK
S.n°P GR<[i DItTMltfSiA lUtu/ldlFlWx LWT
Figure 19. Model with Wire Frame Canopy
65
The finished canopy is shown in Figure 20. The final drawing must be scaled (in
feet) to the dimensions of the actual aircraft and aligned accordingly with the WCS
specified previously.
Ql Hr Ed< '£^m ^Bor F^ntf 1.tJs Q^ Dmt^m HfFHf ftfim ^Hr^tw |M -IBl xl
Figure 20. Completed 3-D MiG-15 Model
IMPORTING A COMPLETED 3-D CAD MODEL INTO AVDS
A completed model saved in DXF format is converted to the AVDS format with the
DXF to AVDS conversion utility. For usage instructions, refer to Appendix E of the
AVDS User's Manual^. The box marked Two sided polygons should be checked in the
interface window of the conversion utility prior to conversion, in order to create surface
meshes composed of front and back sided polygons. If this option is not chosen, meshes
composed of either front or back sided polygons will be shown in AVDS. This would
cause surfaces to disappear unexpectedly when viewing the aircraft from certain angles.
The DXF to AVDS conversion utility has difficulty implementing entire aircraft
models into the AVDS format. Therefore, it is necessary to separate each major model
element (e.g., elements that will share the same color, control surfaces elements, canopy,
etc.) into individually saved DXF files. These individual files are converted separately
and assembled into a single AVDS craft file by the user. The user must identify section
color, wing and tail ribbon location, pilot view, control surfaces, and control surface
rotation axes for each newly compiled AVDS craft file, following the format given in
Appendix A of the AVDS User's Manual^.
Control surfaces for new aircraft models must also be identified within the files
ACname.sim.ini and AVDS_Matlab_ACname.m, where ACname is the aircraft name (e.g.
F16.sim.ini, AVDS_Matlab_F16, etc.), using the same control surface category
66
designations as were used in the AVDS craft file. Refer to Section 4.2 in the AVDS
User's Manual for instructions on creating / modifying ACname.sim.ini files. Figure 21
shows the input screen for files of this type. Within this configuration window, FCS
must be set to Basic FCS, AIRCRAFT should be set to MATLAB AC, and Simulation
Input Type should be set to Positions. Elements within the MATLAB vectors
InputVector and OutputVector, which have been declared by the user in
AVDS_Matlab_ACname.m, must also be declared to AVDS via the ACname.sim.ini file.
The user must declare InputVector elements under the MATLAB AC sub-heading
section of INPUTS, while OutputVector elements must be declared under the
Environment sub-heading section under INPUTS. Control surfaces are identified in
AVDS as sfcOl to sfcl5 under the Environment sub-heading section under INPUTS.
UMim.m«H!M^i.BMBWJt.-«imi'HJ.ii:Hjmpwiami^i»
Description: JDefault FCS
EG Tif. Environment
B^tj: Basic FCS
dE levator
d^ileron
dR udder
dThrottle
\ MATLAB AC
' (3duges
' Record Data To File
' Basic FCS
' MATLAB AC
Not Used
Simulation Input T^ipg
Accelerations) Positi
ilipns ■ '
— Jmatlabac [3
HI. Aircraft Data File
h
Record Filename
RecordData.sa'/e.tKt
Record Rate Hz
■1 ^=
SaveLat/Longs
Figure 21. Input Screen for ACname.sim.ini
67
VII. CONCLUDING REMARKS
Software has been developed in MATLAB for six-degree-of-freedom nonlinear
simulations of various aircraft. Trim, linearization, numerical integration, batch
simulation, 3-D visualization, and real-time piloted simulation software are included for
all simulations.
Aerodynamic models were developed using data gathered from open literature reports
documenting wind tunnel tests, flight tests, and analytic estimation techniques. Most of
the engine models contained in the aircraft simulation packages were based on the F-16
engine model, using scaled values for thrust. Other models were used for aircraft engines
when suitable data was available or when the jet engine model did not apply (e.g.,
FASER and the HL-20 aircraft).
Interactive piloted simulations can be run using the nonlinear aircraft simulation
packages and visual flight simulator software. The nonlinear simulations described in
this report can be used for education, control law development, dynamic analysis, and
many other purposes.
The structure of the simulations was designed for easy use and to allow modifications
with relatively little effort. The collection of aircraft simulations gives the user a range of
aircraft dynamics to explore. This compilation is also a testament of the applicability of
the basic simulation structure to a variety of aircraft.
Future improvements to this work, outside of adding more aircraft, could include
such changes as adding landing gear dynamics, allowing for more general aircraft e.g.
locations, including a more realistic gravitational model that accounts for earth geodesy,
including a non-stationary atmosphere (e.g., turbulence, micro-bursts, winds, etc.), and
adding aeroelastic effects. Since AVDS currently allows only four control inputs, an
improvement that allows the user to command a larger number of controls would be
desirable for interactive simulation. It is anticipated that network capability will be built
into fiature editions of the simulations, allowing multiple users to fly interactively with
(or against) one another in real time. Finally, an obvious improvement for nearly all the
aircraft simulations would be the addition of a flight control system. This was
intentionally omitted in the current work, to encourage the use of these aircraft nonlinear
simulations for control system design and analysis.
68
VIII. ACKNOWLEDGEMENTS
The original versions of the aerodynamic and engine models for some of the aircraft
simulations described in this report were developed by students in the George
Washington University graduate course entitled Stability and Control of Aircraft, taught
by the second author. The authors gratefially acknowledge the work of Javier Velez on
the F-14, Bo Trieu on the F-4, Byron Monzon on the FASER, Pat Hinchy on the HL-20,
Mike Lensi on the X-31, and Brian Stewart and Tony Tyler on the A-7. Thanks also to
Steve Rasmussen for technical work on the MATLAB/AVDS interface software.
69
IX. APPENDIX
Table A-1. F-16 Simulation Package (Directory name: F-16)
F-16 Directory
abS.m
F16_deq.nn
atm.m
F16 engine. m
AVDS Matlab F16.m
F16 engine setup. m
clo.m
F16 hires.txt
cmo.m
F16 lores.txt
cno.m
F16 massprop.m
cnvrg.m
F16 trm.m
cxo.m
gen F16 model. m
czo.m
grad.m
dampder.m
ic ftrm.m
dlda.m
Inze.m
dldr.m
ml^sqw.nn
dnda.m
pdot.m
dndr.m
rl<2.nn
F16.m
rl<4.nn
F16.sinn.ini
rtau.m
F16 aero.m
solve. m
F16 aero setup. m
tgear.m
70
Table A-2. F-106B Simulation Package (Directory name: F-106B)
F-106B Directory
abS.m
F106B aero.m
atm.m
F106B aero setup. m
AVDS Matlab F106B.m
F106B deq.m
cdo.m
F106B_engine.nn
clo.m
F106B engine setup. m
cmo.m
F106B massprop.m
cnvrg.m
F106B trm.m
dampder.m
gen_F1 06B_nnodel.nn
dlda.m
grad.m
didb.m
ic ftrm.m
dldr.m
Inze.m
dnda.m
mksqw.m
dndb.m
pdot.m
dndr.m
rk2.nn
dydb.m
rk4.nn
dydr.m
rtau.m
F106B.m
solve. m
F106B.sinn.ini
tgear.m
F106B.txt
71
Table A-3. F-14 Simulation Package (Directory name: F-14)
F-14 Directory
abS.m
dcmsp.m
F14 deq.m
atm.m
dcxds.m
F14 engine. m
AVDS Matlab F14.m
dcxsb.m
F14 engine setup. m
cldsp.m
dcxsp.m
F14_nnassprop.nn
clo.m
dczds.m
F14 trm.m
cmo.m
dczsb.m
gen F14 model. m
cndsp.m
dczsp.m
grad.m
cno.m
dlda.m
ic ftrm.m
cnvrg.m
dldr.m
Inze.m
cxo.m
dnda.m
ml^sqw.nn
cyda.m
dndr.m
pdot.m
cydr.m
etalsp.m
rl<2.nn
cydsp.m
etamsp.m
rl<4.nn
cyo.m
F14.m
rtau.m
czo.m
F14.sinn.ini
solve. m
dampder.m
F14.txt
tgear.m
dcmds.m
F14 aero.m
dcmsb.m
F14 aero setup. m
Table A-4. F-4 Simulation Package (Directory name: F-4)
F-4 Directory
abS.m
F4 trm.M
atm.m
gen F4 model. m
AVDS Matlab F4.m
grad.m
cnvrg.m
ic ftrm.m
F4.m
Inze.m
F4.sim.ini
ml<sqw.m
F4.txt
pdot.m
F4 aero.m
rl<2.m
F4 deq.m
rl<4.m
F4 engine. m
rtau.m
F4_engine_setup.m
solve. m
F4 massprop.m
tgear.m
72
Table A-5. FASER Simulation Package (Directory name: FASER)
FASER Directory
abS.m
grad.m
atm.m
ic ftrm.m
AVDS Matlab FASER.m
intl.m
cnvrg.m
int2.m
FASER.m
intS.m
FASER. Sim. ini
lat aero tables. mat
FASER aero.m
Inze.m
FASER_aero_setup.m
Ion aero tables. mat
FASER deq.m
mksqw.m
FASER engine. m
pdot.m
FASER hires.txt
rk2.m
FASER lores.txt
rk4.m
FASER massprop.m
rtau.m
FASER trm.m
solve. m
gen FASER model. m
tgear.m
Table A-6. HL-20 Simulation Package (Directory name: HL-20)
HL-20 Directory
abS.m
HL20.txt
atm.m
HL20 aero.m
AVDS Matlab HL20.m
HL20 aero setup. m
cmlgear.m
HL20 deq.m
cno.m
HL20 massprop.m
cnvrg.m
HL20 trm.m
cxigear.m
ic ftrm.m
czlgear.m
Inze.m
gen HL20 model. m
mksqw.m
grad.m
rk2.m
HL20.m
rk4.m
HL20.sim.ini
solve. m
73
Table A-7. X-31 Simulation Package (Directory name: X-31)
X-31 Directory
abS.m
dca dtv.m
ie ftrm.m
atm.m
del dail.m
Inze.m
AVDS Matlab X31.m
del drdr.m
ml<sqw.m
caq.m
del dsb.m
pdot.m
ca can.m
del dtv.m
rl<2.m
clp.m
dem dele.m
rl<4.m
clr.m
dem dsb.m
rtau.m
cl beta.m
dem dtv.m
solve. m
cmq.m
denorm dele.m
tgear.m
cm can.m
denorm dsb.m
X31.m
cnormq.m
denorm dtv.m
X31.sim.ini
cnorm can.m
den dail.m
X31.txt
cnp.m
den drdr.m
X31 aero.m
cnr.m
den dsb.m
X31 aero setup. m
cnvrg.m
den dtv.m
X31 deq.m
en beta.m
dey_dail.m
X31_engine.m
cyp.m
dey drdr.m
X31 engine setup. m
cyr.m
dey dsb.m
X31 massprop.m
cy beta.m
dey dtv.m
X31 trm.m
dca dele.m
gen_X31_model.m
dca dsb.m
grad.m
Table A-8. A-7 Simulation Package (Directory name: A-7)
A-7 Directory
ab3.m
gen A7 model. m
atm.m
grad.m
AVDS Matlab A7.m
ie ftrm.m
A7.txt
intl.m
A7.m
int2.m
A7.sim.ini
Inze.m
A7 aero.m
ml<sqw.m
A7_aero_setup.m
pdot.m
A7 deq.m
rl<2.m
A7 engine. m
rl<4.m
A7 engine setup. m
rtau.m
A7 massprop.m
solve. m
A7 trm.m
tgear.m
cnvrg.m
74
X. REFERENCES
1 . Using MATLAB, Version 6, The MathWorks, Inc., Natick, MA, November 2000.
2. AVDS User's Manual, RasSim Tech Ltd., Columbus, OH, October 2000.
3. Stevens, B.L. and Lewis, F.L., Aircraft Control and Simulation, John Wiley & Sons,
Inc., New York, NY, 1992.
4. Morelli, E.A., Airplane Flight Mechanics - Stability and Control, Class Notes,
December 2000.
5. Schmidt, L.V. Introduction to Aircraft Flight Dynamics, American Institute of
Aeronautics and Astronautics, Inc., Reston, VA, 1998.
6. Cook, M.V. Flight Dynamics Principles, John Wiley & Sons Inc., New York, NY,
1997.
7. Etkin, B. and Reid, L.D. Dynamics of Flight, Stability and Control, "y Ed., John
Wiley & Sons, Inc., New York, NY, 1996.
8. Burden, R.L. and Faires, J.D. Numerical Analysis 6' Ed., Brooks / Cole Publishing
Company, Pacific Grove, CA. 1997.
F-16
9. Nguyen, L.T., Ogburn, M.E., Gilbert, W.P., Kibler, K.S., Brown, P.W., and Deal,
P.L., "Simulator Study of Stall / Post-Stall Characteristics of a Fighter Airplane With
Relaxed Longitudinal Static Stability", NASA TP 1538, December 1979.
F-106B
10. Teper, G.L., "Aircraft Stability and Control Data", NASA CR 96008, April 1969.
11. Yip, L.P., "Wind-Tunnel Free-Flight Investigation of a 0.15-Scale Model of the
F-106B Airplane With Vortex Flaps", NASA TP 2700, May 1987.
F-14
12. Freudinger, L.C., and Kehoe, M.W., "Flutter Clearance of the F-14A Variable-Sweep
Transition Flight Experiment Airplane-Phase 2", NASA TM I0I7I7, July 1990.
13. Gilbert, W.P., Nguyen, L.T., and Van Gunst, R.W. "Simulator Study of Applications
of Automatic Departure and Spin Prevention Concepts to a Variable Sweep Fighter
Airplane," NASA TM X-2928, November 1973.
F-4
14. Eulrich, B.J., and Weingrarten, N.C., "Identification and Correlation of the F-4E Stall
/ Post-Stall Aerodynamic Stability and Control Characteristics from Existing Test
Data", AFFDL-TR-73-125, May 1974.
75
FASER
15. Monzon, B.R., "Non-Linear Simulation Development For a Sub-Scale Research
Airplane", M.S. Thesis, The George Washington University, January 2002.
HL-20
16. Dutton, K.E., "Optimal Control Theory Determination of Feasible Retum-to-Launch-
Site Aborts for the HL-20 Personnel Launch System Vehicle", NASA TP 3449, July
1994.
17. Jackson, E.B., and Cruz, C.I., "Preliminary Subsonic Aerodynamic Model for
Simulation Studies of the HL-20 Lifting Body", NASA TM 4302, August 1992.
X-31
18. Banks, D.W., Gatlin, G.M., and Paulson Jr., J.W., "Low-Speed Longitudinal and
Lateral-Directional Aerodynamic Characteristics of the X-31 Configuration", NASA
TM 4351, October 1992.
19. Smith, M.S., "Analysis of Wind Tunnel Oscillatory Data of the X-31 A Aircraft",
NASA CR-1999-208725, February 1999.
A-7
20. Johnston, D.E., Hogge, J.R., and Teper, G.L. "Investigation of Flying Qualities of
Military Aircraft at High Angles of Attack, Vol. 11. Appendices", AFFDL-TR-74-61,
June 1974.
General
21 . McFarlane, B. Modelling with AutoCAD R14for Windows NT and Windows 95, John
Wiley & Sons Inc., New York, New York. 1999.
22. Taylor, J.W.R. ed. Jane's All the World's Aircraft 1989-90, Jane's Information Group
Ltd., Coulsdon, Surrey, UK. 1989.
76
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1. AGENCY USE ONLY {Leave blank)
2. REPORT DATE
January 2003
3. REPORT TYPE AND DATES COVERED
Technical Memorandum
4. TITLE AND SUBTITLE
A Collection of Nonlinear Aircraft Simulations in MATLAB
6. AUTHOR(S)
Garza, Frederico R.; and Morelli, Eugene A.
5. FUNDING NUMBERS
WU 728-30-30-03
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
NASA Langley Research Center
Hampton, VA 23681-2199
8. PERFORMING ORGANIZATION
REPORT NUMBER
L-18259
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, DC 20546-0001
10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
NASA/TM-2003-212145
11. SUPPLEMENTARY NOTES
12a. DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified-Unlimited
Subject Category 05 Distribution: Nonstandard
AvailabiUty: NASA CASI (301) 621-0390
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
Nonlinear six degree-of- freedom simulations for a variety of aircraft were created using MATLAB. Data for aircraft
geometry, aerodynamic characteristics, mass / inertia properties, and engine characteristics were obtained from open
literature publications documenting wind tunnel experiments and flight tests. Each nonlinear simulation was implemented
within a common framework in MATLAB, and includes an interface with another commercially-available program to read
pilot inputs and produce a three-dimensional (3-D) display of the simulated airplane motion. Aircraft simulations include the
General Dynamics F-16 Fighting Falcon, Convair F-106B Delta Dart, Grumman F-14 Tomcat, McDonnell Douglas F-4
Phantom, NASA Langley Free-Flying Aircraft for Sub-scale Experimental Research (FASER), NASA HL-20 Lifting Body,
NASA / DARPA X-3 1 Enhanced Fighter Maneuverability Demonstrator, and the Vought A-7 Corsair II. All nonlinear
simulations and 3-D displays run in real time in response to pilot inputs, using contemporary desktop personal computer
hardware. The simulations can also be run in batch mode. Each nonlinear simulation includes the full nonlinear dynamics of
the bare airframe, with a scaled direct connection from pilot inputs to control surface deflections to provide adequate pilot
control. Since all the nonlinear simulations are implemented entirely in MATLAB, user-defined control laws can be added
in a straightforward fashion, and the simulations are portable across various computing platforms. Routines for trim,
linearization, and numerical integration are included. The general nonlinear simulation framework and the specifics for each
particular aircraft are documented.
14. SUBJECT TERMS
Real-time desktop nonlinear aircraft simulations
15. NUMBER OF PAGES
92
16. PRICE CODE
17. SECURITY CLASSIFICATION
OF REPORT
Unclassified
18. SECURITY CLASSIFICATION
OF THIS PAGE
Unclassified
19. SECURITY CLASSIFICATION
OF ABSTRACT
Unclassified
20. LIMITATION
OF ABSTRACT
UL
NSN 7540-01-280-5500
Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. Z-39-18
298-102
