14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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Trajectory Optimization for a Generic HypersonicVehicle

Shahriar Keshmiri * and Richard Colgren §

The University of KansasDepartment of Aerospace Engineering

Maj Mirmirani ¥

California State University, Los AngelesCollege of Engineering, Computer Science, and Technology

California State University, Los Angeles

Abstract

This paper covers the trajectory optimization for a generic hypersonic vehicle (GHV). Theequations of motion for the GHV allow a trajectory to be designed for the vehicle by applyingendpoint constraints on position and heading. In this research, the optimum flight path angle formaximum range is investigated. The optimum altitude versus velocity to maximize the range isgenerated by applying MATLAB routines. The equations of motion are developed in the flat earthcoordinate system. A merged aerodynamic database is used for simulation purposes.

Nomenclature

CD Drag CoefficientCL Lift CoefficientDOF Degrees Of Freedomα Angle of Attackγ Flight Path AngleT Thrustw.r.t. With Respect To

Introduction

The calculus of variations is that branch of calculus in which extremal problems are investigatedunder more general constraints than ordinary maxima and minima conditions. More specifically,the calculus of variations is concerned with the maxima and minima of functional expressionswhere entire functions must be determined. Thus, the unknown in the calculus of variations is nota discrete set of points. Rather, it is a succession or assembly of an infinite set of points, allidentifying a curve, a surface, or a hypersurface, depending on the nature of the problem. Theoptimal control problem for a continuous system is a problem in the calculus of variation. Thecalculus of variations has served as an important tool in the performance analysis of airplanesand rockets over the past 60 years.

Generic Hypersonic Vehicle Description

The GHV mass model is based on the assumption of a rigid vehicle structure. The equations ofmotion used in the simulation account for the time varying center of mass, center of gravity, andmoments of inertia experienced during the flight of such a vehicle. The total mass of the vehicle,its c.g. location, and its products of inertia vary as fuel is consumed. It is assumed that the c.g.moves only along the body x-axis as the fuel consumed. Fuel slosh is not considered, and the

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA 2006-8157

Copyright © 2006 by Shahriar Keshmiri, Richard Colgren. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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products of inertia are assumed to be negligible. A sizing analysis discussed below for thegeneric hypersonic vehicle yielded a full-scale gross weight of 300,000 lbs and an overallfuselage length of 200 ft. A three dimensional drawing of the vehicle is given in Figure 1.Deflections of the elevons are measured with respect to the hinge line (perpendicular to thefuselage centerline). A fuselage-centerline-mounted vertical tail has a full span rudder with itshinge line at 25 percent chord from the trailing edge. Deflections of the rudder are measured withrespect to its hinge line. Positive deflections are trailing edge left. Small canards (each using a65 A series airfoil) are deployed at subsonic speeds for improved longitudinal stability and control[1].

Figure 1: The GHV 3D Model

Engine Model

The propulsion model for this research is developed using a 2-D forebody, inlet, and nozzle code.A 1-D isolator and burner code is used in this analysis. The MATLAB simulation written toanalyze the entire integrated vehicle including forebody, inlet, and nozzle flows assumes a 2-Dperfect gas. The burner performance characteristics are computed using a 1-D flow with liquidhydrogen combustion. The cycle analysis of the isolator and combustor is conducted usingRayleigh flow principles. The nozzle flow and dimensions are determined by applying the methodof characteristics.

Analytical Simulation of the Engine Performance

The engine performance is a function of several independent variables. In this research, it ismodeled as a function of flight Mach number and altitude. It is desired to find an approximaterelationship of the form:

( )1 2 3 mx , x , x ,..., xy y=(1)

The curve fitting procedure cannot be performed unless a linear or nonlinear relationship betweenthe independent variables is used. An example is the linear relationship as follows:

0 1 1 2 2 3 3 m+ x x x ... xmy a a a a a= + + + + (2)

The coefficients a0, a1, a2, …, am are determined using the data points available from theaerodynamic look-up tables. In the linear case, the sum of square errors (S) can be expressedas:

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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( )2

0 1 1,i 2 2,i 3 3,i m,i1

- x x x ... xn

i mi

S y a a a a a=

= − + + + +∑(3)

To minimize the sum of square errors (S), the partial derivatives of S with respect to theindependent variables are set equal to zero. This is shown in equation 3.

1, 1, 1, 02

1, 1, 1, 2, 1, , 1 1,2

2, 1, 2, 2, 2, 2, 2 2,

2, 1, , 2, , , ,

i i i i

i i i i i m i i i

i i i i i i i i

m i i m i i m i m i m m i i

n x x x a y

x x x x x x a x y

x x x x x x a x y

x x x x x x a x y

=

∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ∑

K

K

K

M M M M M M M

K(4)

For fast and efficient calculation of the coefficients, a MATLAB program is developed based on S.A mathematical model of the engine is developed by applying this technique.

The analytical model of engine is simulated as:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

22 2 2

2 3

-7 2 -4 -9 2

0

-2 -2 3 -3 4 4

-6 5 -1 5

T7.81 ach Alt + 3.71 10 Alt 1.08 10 Alt - 5.19 10 Alt

m

1.24 10 Alt 5.69 10 Alt 1.06 10 1.08 10 Alt 1.99

5.26 10 Alt 1.20 10

M Mach Mach Mach

Mach Mach Mach

Mach

= + × × × × × + × × × × × ×

− × × × − × × − × × − × × + ×

+ × × − × ×

&

The reduction in the air density is compensated for by either increasing the total velocity or bychanging the inlet area of the engine. This assumption leads to the modeling of the air mass flowrate as a constant parameter.

Analytical Simulation of the Aerodynamic Coefficients

The experimental investigation of the aerodynamic characteristics of the blunt body GHVconfiguration is used as the core of the aerodynamic model. The gaps in the wind tunnel datahave been filled using the best available CFD results.

The clean lift and drag coefficients of the GHV at hypersonic speeds are simulated as [4]:

-2 -2 -2 -4L-clean

-3 2 -4 2 -7 2

--4 3 -6 3 -5 4 -7

C = - 8.19 10 + 4.70 10 Mach )+ 1.86 10 - 4.73 10 ( Mach)

- 9.19 10 (Mach ) - 1.52 10 ( ) + 5.99 10 ( Mach)

+7.74 10 (M ) + 4.08 10 ( ) - 2.93 10 (M ) -3.91 10

(× × × × ×α × × α ×

× × × × α × × α×

× × × × α × × × × 4

-7 5 -8 5

( )

+4.12 10 (M )+1.30 10 ( )

α

× × × × α

( ) ( ) ( )( ) ( ) ( )

( )

-2 -2 -3 -4D-clean

2-3 2 -3 2 -7

-4 3 -4 3 -5 4

-5 4

C = + 8.717 10 - 3.307 10 Mach +3.179 10 - 1.25 10 Mach

+ 5.036 10 Mach - 1.10 10 + 1.40 10 Mach

- 3.65 10 (Mach )+ 3.17 10 + 1.27 10 (Mach )

- 2.98 10 ( ) -

× × × × × α × × α×

× × × × α × × α×

× × × × α × ×

× × α 7-7 5 51.70 10 (Mach ) +9.76 10 ( )−× × × × α

(5)

(6)

(7)

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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Trajectory Optimization Process

The most general of the problems in the calculus of variations in one independent variable are theproblems of Bolza, Mayer, and Lagrange [2]. A continuous dynamic system is described by thethree state vector X (t) at the time t. The state rate of change is shown as:

X f (X,u, t)=& (8)u(t) is the control vector.

In this study, the main objective is to find the optimum velocity direction to maximize the rangeat 0 ft t t≤ ≤ to maximize

( ) ( )f

0

t

f

t

J X t L X,u, t dt= φ + ∫ (9)

In a special case which is investigated in this study, the continuous dynamic system is presentedas:

( )D

1V T q S C m g sin

mx V cos

y V sin

= − ⋅ ⋅ − ⋅ ⋅ γ

= ⋅ γ= ⋅ γ

&

&

&

(10)

Here, “V” represents the velocity vector, “x” represents the range, altitude (alt) is represented by

“y”, and “γ” is the flight path angle. The state vector X is:

x

X y

V

=

The adjoint of the constraints X f (X, u, t)=& results in the adjoint of the performance index

( ) ( )f

0

t

f

t

J X t L X,u, t dt= φ + ∫ as given in equation 11. This is presented with a time-varying

Lagrange multiplier λ(t), by definition, included as follows:

( ) ( ) ( ){ }f

0

t

f

t

J X t L X(t), u(t), t (t) f X(t), u(t), t X dt = φ + + λ ⋅ − ∫ & (11)

The continuous dynamic system described by equation 10 is used to generate the optimaltrajectory by minimizing the performance index to maximize the range as given in equation 12.This equation is derived from the adjoint of the performance index provided as equation 11. Thisis solved in terms of the downrange velocity which is integrated from 0 to the final landing positiontf.

( ) ( )f

0

t

f

t

J X t L X,u, t dt x (range)= φ + = ∫ (12)

( ) ( )f

0

t tf

f

t 0

tf tf

0 0

J X t L X, u, t dt x x dt

J x dt J V cos dt

= φ + = = ⋅

= ⋅ ⇒ = ⋅ γ ⋅

∫ ∫

∫ ∫

&

&

(13)

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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A comparison of equation 13 with the original index performance ( ) ( )f

0

t

f

t

J X t L X,u, t dt= φ + ∫reveals that:

( )fX t 0φ = (14)

( )L X,u, t V cos X V &u= ⋅ γ ⇒ γ� � (15)

X V g sin f (X,u, t)= = ⋅ γ =& & (16)

( ) ( )y V DF V cos y V sin mV T q S C m g sin= ⋅ γ + λ − ⋅ γ + λ − + ⋅ ⋅ + ⋅ ⋅ γ&& (17)

( )y y

F F F F0 0

t y y t y y

∂ ∂ ∂ ∂ ∂ ∂− = ⇒ λ − = ⇒ λ = ∂ ∂ ∂ ∂ ∂ ∂

&&

(18)

( )V V

F F F F0 0

t V t y VV

∂ ∂ ∂ ∂ ∂ ∂ − = ⇒ λ − = ⇒ λ = ∂ ∂ ∂ ∂ ∂∂ &

&(19)

The boundary conditions are:

f

@ t=0 V=0 & =0

@t =t V=Vf

⇒ γ⇒

In order to calculate the optimized flight angle a MATLAB routine is develop (trajectory.m) tosolve the differential equation and the Lagrange multipliers.

Results

The following shows the results of the optimization process.

Figure 2: Optimized Flight Trajectory for GHV

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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Figure 3: Flight Path Angle versus Velocity

Figure 4: Variation of Flight Path Angle versus Velocity

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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Figure 5: Optimized Flight Trajectory for GHV

Figure 6: Flight Path Angle versus Velocity

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006- 8157

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Figure 7: Variation of Flight Path Angle versus Velocity

Summary

This paper covered the development of an optimized trajectory generated for different thrustspecific force values. The model and simulation are developed to support conceptual designstudies of hypersonic vehicles. The aerodynamic database for the Generic Hypersonic Vehicleuses experimental data and multiple CFD codes. The simulation includes optimized trajectory andflight path angle at every point of the flight to maximize the range of flight. A MATLAB routine(trajectory.m) is developed specifically for this application.

References

[1] W. Pelham Philips, Gregory J. Brauckmann, and William C. Woods, “Experimental Investigation of theAerodynamic Characteristics 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 GenericHypersonic Air Vehicle”, AIAA 2005-35352, August 2005.

[3] Shahriar Keshmiri, Maj D. Mirmirani, Richard Colgren, “Six-DOF Modeling and Simulation of a GenericHypersonic Vehicle for Conceptual Design Studies,” AIAA-2004-4805, August 2004.

[4] Sang Bum Choi, “Investigation of the Aerodynamic Characteristics of the Generic Hypersonic Vehicle,Winged-Cone Configuration, by STARS CFD Codes,” MFDCLAB, Los Angeles, California, 2005.

[5] Leitmann, G., “Optimization Techniques”, Academic Press, 1962.[6] Peter H. Zipfel, “Modeling and Simulation of Aerospace Vehicle Dynamics,” AIAA Educational Series,

2000.[7] Frank L. Lewis and Brian L. Stevens, “Aircraft Control and Simulation,” Wiley, 1992.[8] Jan Roskam, “Airplane Flight Dynamics and Automatic Flight Control, Part I,” DAR Corporation, 1997.[9] E. T. Curran and S. N. B. Murthy, “Scramjet Propulsion,” Department of the Air Force (Editor), Purdue

University.[10] “Conceptual Design of the OREAD EXPRESS: TransAtmospheric Cargo (TAC) Vehicle,” The University

of Kansas Propulsion Design Team, 1991-1992 AIAA Air Breathing Propulsion Competition, June 1992.[11] John H. Blakelock, “Automatic Control of Aircraft and Missiles,” Wiley, 1991.[12] Paul Dierckx, “Curve and Surface Fitting with Splines,” Oxford University, 1995.[13] Philip George, “Numerical Methods of Curve Fitting,” University Press, Cambridge, England, 1961.