This paper covers the development of a Six Degrees of Freedom simulation of a generic hypersonic vehicle (GHV) [1]based on two different aerodynamic models and the aerodynamic database developed in reference [2] for control andnavigation purposes. The results from APAS, which is an engineering level CFD program, from a high fidelity CFDcode, STARS, and from wind tunnel experiments are used in this research. For the GHV model a combined cycleengine including a turbojet, ramjet-scramjet, and rocket engines are designed to cover subsonic, supersonic, andhypersonic speeds. Using numerical linearization techniques including the Jacobian method, the LTI state equationswere developed. This work is supported partially by an Air Force grant, in support of their emphasis on newaerospace vehicle concepts and hypersonic technologies.
NOMENCLATURE
α = angle of attack, deg.
β = sideslip angle, red.δa = deflection angle of the right elevon, deg.δe = deflection angle of the left elevon, deg.δr = deflection angle of the rudder, deg.φ = fuel equivalence ratio, n.d. (Phi)alt. = altitude, fta. n. = normal accelerationA.O.A = angle of attack, deg.B = body frame coordinateb = lateral-directional reference length, span, ftc = longitudinal reference length, mean aerodynamic chord, ftCT = thrust coefficientc.g. = vehicle center of gravityDOF = degrees of freedomfa,p = aerodynamic and propulsive forcesGHV = Generic Hypersonic VehicleIsp = engine specific impulse, sec.IXX, IYY, IZZ =roll, pitch, and yaw moments of inertia respectively, slug-ft2
M = mach number, n.d.n.d. = nondimensional (~)PLA = pilot lever angle, (0% to 100%)p, q, r = body rateq0 = dynamic pressureSref. = reference area, theoretical wing area , ft2
thr = throttle angleVt = speed of the vehicle
I. INTRODUCTION
his paper presents the development of a Six Degrees of Freedom (6-DOF) simulation of a generic hypersonic vehicle(GHV) [1] based on three different aerodynamic models. It was developed to support Air Force funded conceptual
design studies on hypersonic flight vehicles at the Flight Research Laboratory at the University of Kansas, and theMultidisciplinary Flight Dynamics and Control Laboratory at California State University, Los Angeles. The aerodynamic
Six-DOF Modeling and Simulation of a Generic Hypersonic Vehiclefor Control and Navigation Purposes
Shahriar Keshmiri and Richard ColgrenThe University of Kansas, Lawrence, KS 66045
Maj MirmiraniCalifornia State University, Los Angeles, CA 90032
T
AIAA Guidance, Navigation, and Control Conference and Exhibit21 - 24 August 2006, Keystone, Colorado
characteristics of the vehicle were developed using CFD studies conducted at NASA Langley, Rockwell International [1], andthe California State University, Los Angeles [2, 3]. These were compared with wind tunnel results for a similar configurationtested at NASA Langley. These results were digitized and organized into more than 80 lookup tables for this simulation. Inaddition, analytical expressions up to seventh order polynomials were developed based on the values of Sum of Squares dueto Error (SSE) were implemented. A multiple cycle engine model was developed which includes both air breathing and rocketpropulsion modes to cover subsonic, supersonic, and hypersonic speed ranges. Both MATLAB/Simulink and Fortran codeswere developed as the implementation of the nonlinear simulation. Results from straight and level flight, as well as ascent anddescent, are presented.
II. MODELING AND SIMULATION PROCESS
A. Vehicle Descriptionhe three-view drawing of the Generic Hypersonic Vehicle (GHV) is given in Figure 1. Deflections of the elevons aremeasured with respect to the hinge line, which is perpendicular to the fuselage centerline. A fuselage, centerline-mounted
vertical tail has a full span rudder with its hinge line at 25 percent of the chord from the trailing edge. Deflections of therudder are measured with respect to its hinge line. Positive deflections are tailing edge left. Small canards (65 A seriesairfoil) are deployed at subsonic speeds for improved longitudinal stability and control. A sizing analysis of the vehicleyielded an estimated full-scale gross weight of 300,000 lbs. The equations of motion account for the time varying center ofmass, center of gravity, and moments of inertia as experienced during flight. It is assumed that the c.g. moves only along thebody x-axis as fuel is consumed; vertical changes are not modeled. Fuel slosh is not considered, and the products of inertiaare assumed negligible.
B. Aerodynamic Modeln the development of the aerodynamic database for the GHV, the aerodynamic characteristics for the blunt body of theGHV have been used as the core of the simulation model. The gaps in the wind tunnel data have been filled using the best
available CFD results. The CFD results are compared with the equivalent wind tunnel data for authenticity. The expressionsfor the aerodynamic forces and the aerodynamic coefficients acting on the GHV are developed. The aerodynamic databasecovers the range of flight Mach numbers, angles of attack (A. O. A) or (α), sideslip angles (β), and control surface deflections(δa, δe, and δr) [2]. The aerodynamic model is then used within the simulation of the GHV, and for comparison purposes theCFD results also been used too. More than 80 look-up tables were developed based on wind tunnel, APAS, and STARSresults.
T
I
Figure 1: Three View of the Generic Hypersonic Air Vehicle
3
Table 1: Example APAS, STARS, and Wind Tunnel Results
C. Engine Modeln order to operate through all Mach regimes, a combined-cycle propulsion system has been developed for the GHV. Thisengine is a combination of a hypothetical turbojet, ramjet-scramjet, and rocket motor. Turbojets are particularly suited for
the low-speed portions of the mission and have adequate performance up to Mach 3.0 although; in this research, it is limitedto Mach 2.0. The ramjet engine is used to cover the main portion of the supersonic flight. The ramjet-scramjet engine coverssupersonic to low hypersonic speeds. The rocket engine is designed in order to reach a maximum of Mach 24. The followingfigures show how the required thrust force and the CT changes versus Mach number for different speed regimes.
Figure 2: Required Thrust Force (Subsonic Pull-up)
Requierd Thrust ForceThe Generic Hypersonic Air Vehicle
0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
6.0E+05
7.0E+05
8.0E+05
0 5 10 15
Angle of Attack (deg.)
Th
rust
Fo
rce
(lb
f)
M=2.00
M=2.50
M=3.00
M=3.50
M=4.00
Figure 3: Thrust Force versus A. O. A.
Figure 4: Ramjet Engine (Thrust Coefficient versus Mach Number)
Figure 5: CT versus Mach Number
5
Figure 6: CT versus Mach Number
In order to use the engine model within the simulation of the GHV, analytical expressions are developed for each engine asfollows:Turbojet Engine: 0.00 < Mach < 2.0
For altitude greater or equal 57,000 ftPLA)(21295PLA)(3.24e05PLA)(3.24e05-1.64e04Thrust ×+×+×+=
D. Flight Profile
Using the method discussed in the reference [2], the flight trajectory of the GHV is chosen within a narrow range of
dynamic pressures of from 500.00 to 2000.00 psf. This helps us to avoid large forces and drag values on the GHV.Considering that airbreathing engines generate thrust in direct proportion to the mass flow rate they are able to capture fromthe atmosphere, if the GHV flies faster at a constant dynamic pressure, the available free stream mass flow rate per unit areareduces. To resolve this problem, when the trajectory of the vehicle is designed to maintain the maximum mass flow ratewhile increasing Mach Number, it is also designed to fly at a constantly increasing dynamic pressure. This is shown in Figure7. A simulated flight trajectory is shown in Figure 8.
Figure 7: Mass Flow Results
6
III. Flat-Earth Equations of Motion
The transitional equations of a flight vehicle subjected to the aerodynamic and thrust forces fa,p, and the gravitational
acceleration g under the approximations of a “flat-Earth”, are discussed in this section. Newton’s second law with respect tothe inertial frame I states that the time rate of change of linear momentum equals the externally applied forces:
[ ][ ] mgfvvDm
vvDmvmD
mgfvmD
paEB
BEEB
B
EB
BEEB
BEB
E
paEB
E
+=Ω+
Ω+=
+=
,
,
To generate the ordinary time derivative, the rotational derivative must be expressed in the coordinate system B] associated
with the reference frame B.
[ ] [ ] [ ] [ ]BBpa
BEB
BBE
BEB gmfvm
dt
vdm +=Ω+
,
The aerodynamic and propulsive forces, the linear velocity of vehicle with respect to Earth, and the angular velocity of thevehicle with respect to the Earth are expressed in the body axis coordinate system. In the flat Earth approximation the gravityvector, simplifies to: [ ] [ ]gg
L00= .
The transitional equation in matrix form is:
[ ] [ ] [ ] [ ] [ ]LBLBpa
BEB
BBEEB gTmfvm
dt
vdm +=Ω+
,
[ ]L
BL
B
pa
pa
pa
BBB
mg
T
f
f
f
w
v
u
pq
pr
qr
dtdwdt
dvdt
du
m
+
=
−−
−+
0
0
0
0
0
3,
2,
1,
[ ]BL
BL
ttt
ttt
ttt
T
=
333231
232221
131211
Where
The equations are expressed in scalar form as:
gtm
fpvqu
dt
dw
gtm
frupw
dt
dv
gtm
fqwrv
dt
du
pa
pa
pa
333,
232,
131,
++−=
++−=
++−=
Trajectory
0
20000
40000
60000
80000
100000
120000
0 5 10 15 20
Flight Mach NumberA
ltit
ud
e Ascent
Level flight
Descent
Figure 8: Trajectory for the GHV
7
The integration of the above three differential equations yields the velocity vector. It is integrated once more to obtain thelocation of the air vehicle c. g. with respect to an Earth-fixed reference point. The integration is carried out in the local-levelcoordinate system:
[ ] [ ] [ ]BEB
BLL
BEE vTSD =
In coordinate form:
[ ]B
BL
L
BE
BE
BE
w
v
u
T
dtds
dtds
dtds
=
3
2
1
We have six differential equations that describe the transitional motion of the air vehicle with the Earth as the inertialreference frame. All six equations are nonlinear and coupled by the body rates p, q, and r and the directional cosinematrix [ ]BLT . Euler’s law is adequate for modeling the rotational degrees of freedom. Euler’s law states that the time rate of
change of angular momentum equals the externally applied moments. It governs the rotational degrees of freedom. E ispicked as an inertial frame as it confirms the inertial frame, which was picked for modeling transnational motions.
BB
BBEBEBBB
BEBB
B
BBEB
BBEBEB
BB
BBEB
BE
IDDIID
MIID
MID
ωωω
ωω
ω
+=
=Ω+
=
)(
For the rigid body the last term, BB
B ID , is zero. The equation reduces to:
BBEB
BBEBEBB
B MIDI =Ω+ ωω
Solving for the time derivative
[ ]( ) [ ] [ ] [ ] [ ]( )BB
BBEBBB
BBEBBB
BBE
MIIdt
dw +Ω−=
−ω
1
The above set of equations represents the three rotational degrees of freedom. This set is coupled with the transnational
equations through the aerodynamic moments [ ]BBM [8], [16].
IV. NUMERICAL LINEARIZATION PROCESS
A MATLAB program (LIN.m) was written in order to calculate the Jacobian matrices for the set of nonlinear state
equations. The linearization algorithm chooses smaller and smaller perturbations in the independent variable and thencompares three successive approximations to the particular partial derivatives. If these approximations agree within a certaintolerance, then the algorithm would be terminated. Otherwise, the size of perturbation would be set smaller. The linearizedmodel is derived for straight and level flight at the specified velocity and altitude with zero banking angle. In this case, thelongitudinal and the lateral subsystems are decoupled. The linearized model is:
DUCXY
BUAXX
+=+=&
8
Also, the state variables are defined as:
=
r
p
Thrust
q
Alt
Vt
X
φβ
θα
The control variables are defined as:
]r
e
a
PLA[U
δδδ
=
The output variables are defined as:
]
q
n.a.[y
α
=
The flow chart of the GHV Simulation is shown in Figure 9.
Nonlinear State-Space Model
),(
),(
uXgY
uXfX
==&
Steady-StateTrim
NonlinearSimulation
Initializationof Data
Initializationof Data
Linear State-Space Model
DuCXY
BuAXX
+=+=&
Linear DesignTechniques
LinearizedModel
Control SystemModel
CoefficientData
TrialDesign
Figure 9: Simulation Flow Chart
9
The following are example A, B, C, and D matrices for the GHV model trimmed at Mach = 5.00 and Alt = 65,000 ft. instraight level flight.
In this paper, the 6-DOF equations of motion for the GHV are developed, and the linearized model of the GHV for LTIcontrol purposes simulated. The linear model may be used for any control and navigation research problem, and can be usedto generate different trim points throughout the trajectory of the vehicle. Also, the nonlinear equations of motion can be usedfor more advance control law development and verification efforts. These efforts would include the verification and validationof robust or nonlinear controllers. Our sincere thanks go to US Air Force for their support.
REFERENCES
[1] W. Pelham Philips, Gregory J. Brauckmann,and William C. Woods, “Experimental Investigation of the AerodynamicCharacteristics for a Winged-Cone Concept,” AIAA 87-49098, 1987.
[2] S. Keshmiri, R. D. Colgren, and M. Mirmirani, “Development of an Aerodynamic Database for a Generic Hypersonic AirVehicle”, AIAA 2005-35352, 2005.
[3] Sang Bum Choi, “Investigation of the aerodynamic characteristics of the Generic Hypersonic Vehicle, Winged-ConeConfiguration, by STARS CFD codes,” MFDCLAB, Los Angeles, California, 2005.
[4] W. Pelham Phillips, Gregory J. Brauckmann, John R. Mico and William C. Woods, “Experimental Investigation of theAerodynamic Characteristics for a Winged-Cone Concept,” NASA Langley Research Center, 1991.
[5] Charles E. Dole, James E. Lewis, “Flight Theory and Aerodynamics, Second Edition,” 2000.[6] John D. Shaughnessy, S. Zane Pinckney, John D. McMinn, Christopher I. Cruz, and Marie-Louise Kelley, “Hypersonic
Vehicle Simulation Model, Winged-Cone Configuration” NASA Langley, 1991.[7] Banu N Pamadi, “Performance, Stability, Dynamics, and Control of the Airplane,” AIAA Educational Series, 1998.[8] Peter H. Zipfel, “Modeling and Simulation of Aerospace Vehicle Dynamics,” AIAA Educational series, 2000.[9] Frank L. Lewis and Brian L. Stevens, “Aircraft Control and Simulation,” Wiley, 1992.[10] Jan Roskam, “Airplane Flight Dynamics and Automatic Flight Control part I,” DAR Corporation, 1997.[11] E. T. Curran and S. N. B. Murthy, “Scramjet Propulsion,” Department of the Air Force (Editor), Purdue University.[12] “Conceptual Design of the OREAD EXPRESS: TransAtmospheric Cargo (TAC) Vehicle,” The University of Kansas
Propulsion Design Team June-1992 (1991/ 1992 AIAA/ AIR BREATHING PROPULSION competition).[13] John H. Blakelock, “Automatic Control of Aircraft and Missiles,” Wiley, 1991.
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[14] Paul Dierckx, “Curve and Surface Fitting with Splines,” Oxford University, 1995.[15] Philip George, “Numerical Methods of Curve Fitting,” Cambridge [Eng.] University Press, 1961.[16] Shahriar Keshmiri, Maj D. Mirmirani, Richard Colgren, “Six-DOF Modeling and Simulation of a Generic Hypersonic
Vehicle for Conceptual Design Studies,” AIAA-2004-4805, August 2004.
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