[american institute of aeronautics and astronautics 44th aiaa aerospace sciences meeting and exhibit...

Effects of Hypersonic Cruise Trajectory Optimization

Coupled with Airbreathing Vehicle Design

Ryan P. Starkey∗ Falcon Rankins †

Darryll J. Pines ‡

University of Maryland, College Park, MD, 20742, USA

The effects of vehicle and propulsion system configuration on trajectory optimization forhypersonic cruise are explored for computational efficiency, solution accuracy, design spacesensitivity, and overall problem feasibility. A systematic exploration of the design spacefor a single optimized, Mach 10 configuration for both optimized trajectories and optimaldesign and fidelity of trajectory code interpolation matrices is performed and comparedto the expense of running a real-time vehicle performance solver. Periodic trajectoriesare optimized for minimum fuel consumption (maximum range) using a parallelized Dif-ferential Evolutionary Scheme. Results show that: 1) flying similar vehicles along a giventrajectory produces non-optimal performance, 2) optimal trajectories for similar vehiclesexploit non-linearities in the system, 3) interpolation tables with as few as 108 points cangenerate results within 3% of the real-time calculations, and 4) there is a cross-over point atwhich coupled vehicle/trajectory optimization is more efficient using real-time performancecalculations versus interpolation table generation.

Nomenclature

p vehicle inertial position, mq quaternion vectorV vehicle inertial velocity, m/sm mass flow rate, kg/sC CoefficientM Mach numberq dynamic pressure, N/m2

R Range, kmS Area, m2

V Velocity, m/sZ Altitude, m

Subscripts

af airframecap capturecooling cooling fueld dragf fuelkl keel-linel lift

∗Faculty Research Scientist, Department of Aerospace Engineering, University of Maryland, Senior Member AIAA,[email protected]

†Graduate Research Assistant, Department of Aerospace Engineering, University of Maryland, Student Member AIAA,[email protected]

‡Professor, Department of Aerospace Engineering, University of Maryland, Associate Fellow AIAA, [email protected]

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American Institute of Aeronautics and Astronautics

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-1036

Copyright © 2006 by University of Maryland. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

plan planformt thrust

Symbols

α angle of attackφ fuel-to-air equivalence ratio

I. Introduction

The coupled multi-disciplinary problem of hypersonic air-breathing vehicle and trajectory optimization isa topic of ongoing interest. The question of what single design point along a given trajectory will provide

overall optimal engine and aerodynamic performance is often raised. Also associated with this design processis the question of what form the optimal trajectory will take. The trade-offs involved in this optimizationprocess are explored and discussed for optimal trajectories for a given vehicle design, as well as interpolationtable density as a function of error and computation time. Also, vehicle design-point effects are exploredthrough a parametric study of combustor flow path and airframe scaling.

In order for this type of optimization process to be possible the computational expense for each vehicledesign must be kept to a minimum. Analytical approximations, as well as reduced order equation sets areused whenever possible to speed up the design process with minimal sacrifice in accuracy. Even with theseconcessions, the computational time for the vehicle complexity prohibits real-time trajectory optimization(as will be shown), but results indicate that improvements may be made to make it possible. First, a vehicleconfiguration is optimized for maximum lift-to-weight ratio (L/W ) and thrust-to-drag ratio (T/D) at anequivalence ratio of φ = 1 and dynamic pressure of q = 1000 psf. Look-up tables for coefficients of lift, drag,and thrust (Cl, Cd, and Ct, respectively), and fuel mass flow rates for the combustor and minimum cooling(mf and mcooling, respectively) as functions of dynamic pressure (q), Mach number (M), angle-of-attack(α), and equivalence ratio (φ) are then generated for input into the trajectory code. Furthermore, to permitvehicle refinement during trajectory optimization (done parametrically for this abstract, but soon to beincorporated into the optimization code) the vehicle components are split into keel-line components, as wellas the contribution of the remainder of the airframe.

II. Modeling

A. Aerodynamic/Propulsive/Geometric Modeling

Figure 1. Example optimized vehicle configuration.

An example vehicle design is shown in Fig. 1 andcan be referred to for understanding (the flowpathis from left to right). As shown in the examplevehicle, the keel-line geometry consists of: part ofthe waverider forebody, a multiple planar-ramp in-let external compression system, a scramjet engineincluding isolator and multiple diverging sections,an internal nozzle section, and an external half-plugnozzle. A total of 22 design variables are used todefine the vehicle geometry (6 forebody, 2 inlet, 6combustor, 2 nozzle, 2 airframe) and the flight con-ditions (Mach number, dynamic pressure, angle ofattack, equivalence ratio).

The airframe (everything excluding the en-gine flowpath) is analyzed using a modified shock-expansion method, which was developed for a hy-personic missile study study.1, 2 For simplicity,the vehicle is analyzed in a series of assumedtwo-dimensional streamlines. The modified shock-expansion method corrects the flowfield propertiesin areas where the cross-stream radius of curvatureis small (large pressure gradients).

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In order to capture all of the desired variations in the combustor geometry and their associated effectson the chemistry, a quasi-1D model was developed by O’Brien et al.3 with partial basis on derivations givenby Shapiro4 and Turns.5 Allowances were made to accommodate for variable cross-sectional area, includethe effects of friction using a hydraulic diameter approximation, include the effects of heat transfer with thewall, and allow for the inclusion of a mixing profile which increments the amount of combustible fuel. Thismethod has been used in many air-breathing vehicle design studies, including a hypersonic missile study6

and a rocket-based combined-cycle vehicle study.7 Recently, modifications have been made to account forvariable-geometry combustor modeling for higher fidelity off-design performance calculations.8

B. Trajectory Modeling

The current work examines trajectory types for hypersonic cruise.9–14 In all cases, only a portion of thetotal mission trajectory is examined; i.e., although the vehicle may be intended for missions up to 20000km (global reach), the current work looks only at 1000 km sections of this trajectory. Therefore, one of therequirements of the current work is that the vehicle states (Z, V , α, θ) at the end of the examined trajectoriesmatch the initial states. The optimized trajectories could then conceivably be repeated as necessary to reachthe desired destination.

The point-mass equations of motion for flight in a vertical plane over a non-rotating spherical earth aregiven by:

dp

dt= V (1)

dV

dt= B

1

mF B + g (2)

dm

dt= −max

[

(

dm

dt

)

f

,

(

dm

dt

)

cooling

]

(3)

dq

dt= −1

2Ωqq (4)

where the position, velocity, and gravity vectors are specified in an inertial reference frame. Quaternionsare used to track vehicle orientation using the standard update matrix Ωq, which is dependent on vehiclerotational rates. Further details can be found in standard vehicle dynamics texts.15 B is the direction cosinematrix that rotates the external force vector, F B, from the body axes to the inertial reference frame. F B

consists of the vehicle lift, drag and thrust, which are defined in terms of their respective force coefficients inequations similar to L = qCLS, where S is the appropriate reference area. The aerodynamic coefficients anddm/dt values are obtained by linear interpolation of tables given as a function of equivalence ratio, dynamicpressure, Mach number and angle-of-attack.

A 1976 U.S. Standard Atmosphere model16 was used in the simulation. Speed of sound is given bya =

√γRT where γ and R are assumed constant and equal to 1.4 and 287 J/(kg K), respectively. The effects

of wind, as well as rotation and oblateness of the earth were neglected.Trajectories are defined by initial velocity, initial altitude, initial flight path angle and a set of control

points that define the vehicle pitch and pitch derivative with respect to time. The simulation can define atrajectory either through prescribing discrete altitude or vehicle pitch control points and their respective firstderivatives. Piecewise cubic polynomials are then fit between the control points to establish a correspondingprofile with a continuous second derivative.

The trajectories were optimized for range-averaged fuel usage using 18 design variables: 5 vehicle pitchpoints, 5 pitch profile slopes, initial altitude, initial velocity, initial flight path angle, and 5 engine controlvariables. The engine control information includes two movable switch points that allow the fuel-to-air equiv-alence ratio to vary between 0 and 1.2. The trajectory optimizations were also subject to four constraints:Zfinal = Zinitial ± 10 m, Vfinal = Vinitial ± 5 m/s, γfinal = γinitial ± 0.1, and αfinal = αinitial.

III. Optimization Methodology

A popular alternative to gradient-based methods is a family of evolutionary optimization methods. Mostevolutionary methods share a similar framework of starting with a random initial population of design

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variables that interact through some form of mutation, recombination, and selection to progress to thenext generation. The current work employs a Differential Evolution Scheme (DES)17–20 to optimize thevehicle trajectory parameters. DES is a simple scheme that uses straightforward methods for mutation,recombination, and selection. Given a set of N parameter vectors that define the population membersat a particular generation X = [x1, x2, . . . , xN ] a trial vector can be developed for each xn where tn =xa + F (xb −xc) using random values for b and c, such that a 6= b 6= c. Although Price and Storn17 proposea = n, Rai19 suggests a also be a random value. Further, Qing,20 replaces xa with the x vector producingthe best fitness function. The current work suggests that the latter method converges the population to alocal minimum at a faster rate than using a random a. Therefore, the optimization method employed hereswitches from the method used by Qing to Rai’s method once the population average nears the fitness ofthe best population member. F is a user defined scaling constant that Price empirically determined to liein the range (0 < F ≤ 1.2). For the current work, F = 0.9. Finally, in the current work, if an element of thetrial vector lies outside of the specified variable bounds, the element is reset to the nearest bound.

Each trial vector then passes through the recombination stage, wherein the population of children vectors,C = [c1, c2, . . . , cN ] is developed. Each element in a particular child vector is determined by a simplerecombination operator:

cni =

ti if ri ≤ R

xni if ri > R(5)

where ri is a random number (0 ≤ ri < 1) and R is user defined (0 < R < 1).In the selection stage, every member of the parent population is compared to its corresponding child

vector. If the child vector produces a better fitness function, it replaces the parent vector. Otherwise, theparent vector remains a member of the population. This technique provides the optimization framework forthe University of Maryland Parallel Trajectory Optimization Program (UPTOP) which has been utilized forthis paper.

The coupled optimization of these vehicle geometries and trajectories is difficult due to the inherentlynon-linear nature of high speed aerodynamics, the sensitivity of scramjet engines to their inlet conditions,and the enormous number of function calls necessary for trajectory optimization. Varying the magnitudesof many of the design variables by small amounts often changes an ideal configuration into one which isgeometrically non-feasible, non-realistic, or non-functional (i.e., insufficient propulsion or choked flow).

IV. Results

A. Baseline Vehicle Geometry and Design Space

The baseline goal was to design a Mach 10 cruiser that had sufficient excess lift ( weight) and thrust ( drag)to fly periodic cruise trajectories in order to maximize range by taking advantage of the natural phugoidmode of the vehicle. The baseline vehicle configuration (which most of the results in this paper are basedon) is shown in Fig. 2 on the next page. This vehicle has a volume of V = 1575 m3, a planform area ofSplan = 773 m2, a capture area of Scap = 28.8 m2 and was optimized for a Mach 10, q = 1000 psf design pointassuming a combustor mixing and burning efficiency of 90%. For simplicity an average density estimate of100 kg/m3 was used to determine the vehicle mass (159530 kg). The keel-line section (colored orange) inFig. 2 on the following page shows the surfaces affected by the streamtube passing through the engine.

Once the vehicle optimization has been performed, off-design performance is calculated and tabulatedinto input tables for the trajectory optimizer. Generally, this disjoint manner is how vehicle and trajectoriesoptimizations are performed. Part of the goal of this paper is to explore the effects of code coupling as thefirst step to creating a fully-integrated vehicle aerodynamic, propulsive, and trajectory optimization softwarepackage. The flight envelope bounds and resolutions for dynamic pressure, Mach number, equivalence ratio,and angle-of-attack are: 23940 N/m2 < q < 95761 N/m2 (500 psf < q < 2000 psf), 8 < M∞ < 12, 0.0 < φ <1.2, and -4 < α < 4.

A thrust-to-drag ratio contour map for the 1000 psf flight condition is shown in Fig. 3 on the next pagefor the baseline vehicle configuration. The thrust-to-drag ratio scales with both the contour color and thenode size. The black line indicate the condition where thrust is equal to drag. This vehicle data-set isrepresentative of what is passed to the trajectory code for each configuration, with each node requiringsolution of the entire vehicle aerodynamic and propulsive performance. As the desired flight design spacebounds or nodal resolution increased the number of total calculated nodes can increase dramatically.

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Figure 2. Waverider based vehicle optimized for Mach 10 flight at 1000 psf.

Figure 3. Waverider-based vehicle T/D ratio contourswith symbols of relative size and color. The areas belowthe black lines indicate portions of the flight envelopewhere drag is greater than thrust. Missing sectionsindicate choked flow in the combustor.

As seen in the plot, the vehicle has been de-signed with large performance margins around thedesign point. By designing the geometry at a lowerMach number and then allowing the trajectory codeto overspeed it, the inlet can take advantage of alarger capture area, especially at positive angles-of-attack. This translated into increased inlet pressureand temperature and hence, increased thrust.

The trajectory for the baseline vehicle was thenoptimized for minimum range averaged fuel con-sumption. In order to determine the vehicle perfor-mance sensitivities, the trajectory code input valueswere each varied through the design space range.Figure 4 on the following page shows the coefficientsof keel-line and airframe lift (column 1), keel-linenet thrust and airframe drag (column 2), and fuelmass flow rate and minimum cooling fuel mass flowrate (column 3) for sweeps in angle-of-attack (row1), Mach number (row 2), fuel-to-air equivalence ra-tio (row 3), and dynamic pressure (row 4). Thebaseline point for these sensitivities was chosen asthe initial flight condition from the optimized cruisetrajectory (M = 11.64, α = -0.5084, φ = 0.5853,and q = 34557 N/m2).

Figure 4 on the next page shows that the co-efficients are most sensitive to the angle-of-attack,whereas the other inputs of M , φ and q result in mostly linear trends. The only other major sensitivityof note are the Ct,kl and Cd,af for changing dynamic pressure, as shown in Fig. 4(k) on the following page.Although only taken at a single point, there are a number important conclusions which can be drawn fromthese figures:

1. Coefficient scaling by the dynamic pressure is only reliable very close to the actual data point.

2. Linear data interpolation will require the largest number of points for angle-of-attack to minimizeerrors.

3. Finite-rate chemistry, variable geometry engine operation results in localized data non-linearities.

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Angle of Attack

Lift

Coe

ffici

ent(

Cl)

-4 -2 0 2 40

0.02

0.04

0.06

0.08

keel-line

airframe

(a) Cl,kl and Cl,af vs. α.

Angle of Attack

Thr

ustC

oeffi

cien

t(C

t,kl)

Dra

gC

oeffi

cien

t(C

d,af

)

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.006

0.008

0.01

0.012

0.014

0.016

0.018

(b) Ct,kl and Cd,af vs. α.

Angle of Attack

Fue

lMas

sF

low

Rat

e(m

f)[k

g/s]

-4 -2 0 2 4

6

8

10

12

Actual

Cooling

(c) mf and mcooling vs. α.

Mach Number

Lift

Coe

ffici

ent(

Cl)

10 10.5 11 11.5 12 12.5 13

0.04

0.05

0.06

keel-line

airframe

(d) Cl,kl and Cl,af vs. M .

Mach Number

Thr

ustC

oeffi

cien

t(C

t,kl)

Dra

gC

oeffi

cien

t(C

d,af

)

10 10.5 11 11.5 12 12.5 13

0.15

0.2

0.25

0.0077

0.0078

0.0079

0.008

0.0081

0.0082

0.0083

0.0084

0.0085

(e) Ct,kl and Cd,af vs. M .

Mach Number

Fue

lMas

sF

low

Rat

e(m

f)[k

g/s]

10 10.5 11 11.5 12 12.5 13

6

8

10

12

14

Actual

Cooling

(f) mf and mcooling vs. M .

Equivalence Ratio

Lift

Coe

ffici

ent(

Cl)

0 0.2 0.4 0.6 0.8 1 1.2

0.03

0.04

0.05

0.06

0.07

keel-line

airframe

(g) Cl,kl and Cl,af vs. φ.

Equivalence Ratio

Thr

ustC

oeffi

cien

t(C

t,kl)

Dra

gC

oeffi

cien

t(C

d,af

)

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.00805

0.00806

0.00807

0.00808

0.00809

0.0081

0.00811

(h) Ct,kl and Cd,af vs φ.

Equivalence Ratio

Fue

lMas

sF

low

Rat

e(m

f)[k

g/s]

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

20

25

actual

cooling

(i) mf and mcooling vs. φ.

Dynamic Pressure (q) [Pa]

Lift

Coe

ffici

ent(

Cl)

20000 40000 60000 80000 100000 120000

0.04

0.045

0.05

0.055

keel-line

airframe

(j) Cl,kl and Cl,af vs. q.


Thr

ustC

oeffi

cien

t(C

t,kl)

Dra

gC

oeffi

cien

t(C

d,af

)

50000 1000000.21

0.22

0.23

0.24

0.25

0.26

0.27

0.0074

0.0076

0.0078

0.008

0.0082

0.0084

(k) Ct,kl and Cd,af vs. q.


Fue

lMas

sF

low

Rat

e(m

f)[k

g/s]

20000 40000 60000 80000 100000 1200005

10

15

20

25

30

35

40

45

actual

cooling

(l) mf and mcooling vs. q.

Figure 4. Coefficient sensitivities to changes in angle-of-attack, Mach number, equivalence ratio, and dynamicpressure, taken around the optimized cruise trajectory initial flight point.

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The effects of these insights on coupled vehicle/trajectory optimizations will be shown in results subsection Con the next page.

B. Parametric Scaling Study

In order to demonstrate the necessity for coupled vehicle/trajectory optimizations, a simple two variableparametric study of the effects of changes in both the airframe planform area and scramjet capture arearelative to the design point was performed. To do this simply, the vehicle lift coefficient was split into akeel-line component and an airframe component (remainder of the vehicle), as was shown by the colors inFig. 2 on page 5. The keel-line is also characterized by a net coefficient of thrust relative to the capture areaand the airframe with a drag coefficient relative to the airframe planform area. The vehicle height and lengthwere held constant throughout all scaling to keep the engine flowpath unaltered, but he vehicle volume andmass were scaled by the appropriate amounts.

By decoupling the airframe and keel-line components the relative scaling effects on range averaged fuelconsumption and estimated range were studied for simple perturbations. The values of Splan,af and Scap

were scaled up and down by 20% to represent the perturbations which may be utilized for optimizing avehicle configuration along a given trajectory. First, 1000 km trajectory segments were optimized for eachof the vehicles, with the results for mf/R shown in table 1. Based on this fuel usage value and predictedfuel volume, an estimated cruise range is calculated (table 2), neglecting changes in vehicle performancewith downrange mass changes. These tables show a wide variance in the vehicle performance for changesin planform and capture areas. What was not expected was the large magnitude differences and that theconfiguration with the best range averaged fuel use and estimated range was the vehicle with the smallerplanform area and larger engine capture area. The other interesting point is that mf/R for Splan,af = 1.0 andScap = 1.2 increased from the baseline case by 3.32%, but the estimated range increased by 5.73% (due to theincreased vehicle volume). Clearly the non-linear behavior exhibited by this simple parametric investigationinto the coupling between optimal trajectories for simple vehicle perturbations shows the necessity of vehicledependent trajectory optimization.

Secondly, each of the vehicles were flown along the optimized baseline trajectory (Splan,af = 1.0, Scap =1.0), with the results shown in Fig. 5 on the next page. This indicates that trying to fit all vehicles in anoptimization to a common trajectory is actually penalizing the performance of all but the optimal vehicle forthe particular trajectory. Commonly, air-breathing vehicles are optimized for a constant dynamic pressuretrajectory, but clearly that is over constraining the problem resulting in a non-optimal solution.

Table 1. Range averaged fuel consumption [mf /R] (kg/km) for optimized 1000 km trajectory segments forrelative changes in airframe planform size and capture area.

Relative Splan

Relative Scap 0.8 1.0 1.2

0.8 2.33 3.14 3.29

1.0 2.30 2.71 3.24

1.2 2.24 2.80 3.15

Table 2. Estimated range for optimized 1000 km trajectory segments for relative changes in airframe planformsize and capture area.

Relative Splan

Relative Scap 0.8 1.0 1.2

0.8 20460 18649 18619

1.0 22940 21834 20503

1.2 25805 23085 22708

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Range (km)

Mac

hN

umbe

r

0 200 400 600 800

11

11.1

11.2

11.3

11.4

Baseline0.8 x af, 0.8 x en0.8 x af, 1.2 x en1.2 x af, 0.8 x en1.2 x af, 1.2 x en

(a) Mach number.

Range (km)

Alti

tude

(km

)

0 200 400 600 800 1000

37.5

38

38.5

39

39.5

40

Baseline0.8 x af, 0.8 x en0.8 x af, 1.2 x en1.2 x af, 0.8 x en1.2 x af, 1.2 x en

(b) Altitude.

Figure 5. Profiles for the scaling study vehicles flying the baseline optimal trajectory.

C. Accuracy Versus Computational Time

The preceeding two sections have demonstrated the difficulties encountered when trying to determine anoptimal vehicle and the resulting trajectory. This section investigates the magnitudes of errors versus com-putational time for optimizing with various trajectory input table densities versus real-time computations ofthe vehicle performance. In order to determine the computational efficiency of various levels of optimizationand code integration, trajectory input table density with selective variable specific refinement was investi-gated with the results shown in Table 3. To establish a baseline for means of comparison, the vehicle andtrajectory codes were coupled, and a representative trajectory was flown with the vehicle performance coef-ficients calculated in “real-time” at each trajectory point. Using this method, the resulting range-averagedfuel usage and estimated range values of 3.725 kg/km and 16195 km, respectively, were obtained. As in theprevious section, the range is an estimated value based on the calculated 1000 km trajectory and estimatedfuel volume.

Table 3. Errors for various trajectory input table densities compared to a real time vehicle calculation for asingle trajectory.

Table Info Fuel Usage % Error Est. Range Table Gen. Traj Opt Joint Veh/Traj

∆q ∆M ∆φ ∆α total pts (kg/km) (km) Time (s) Time (hrs) Opt Time (yrs)

real-time vehicle calculation 3.725 - 16195 - 37010.12 4.22

- - 0.6 4.0 6 4.627 24.22521 13037 1.94 1.13 0.02

1500.0 4.0 0.6 8.0 24 3.409 8.47127 17694 11.6 1.13 0.09

1500.0 4.0 0.6 2.0 60 3.778 1.44459 15964 29.1 1.14 0.23

1500.0 4.0 0.4 4.0 64 3.784 1.59163 15941 46.5 1.13 0.12

1500.0 4.0 0.6 1.0 108 3.788 1.69276 15925 52.3 1.14 0.41

1000.0 2.0 0.2 2.0 504 3.750 0.69434 16083 244 1.14 0.37

500.0 1.0 0.1 1.0 4004 3.736 0.29602 16147 1940 1.2 1.94

250.0 0.5 0.05 0.5 64064 3.723 0.04109 16201 31039 1.67 15.38

125.0 0.025 0.25 0.25 1025024 3.725 0.00408 16194 496620 9.75 246.06

75.0 0.0125 0.125 0.125 16400384 3.724 0.00465 16196 7945922 2208.33 62990.89

The first five columns of Table 3 show the various variable refinements and total trajectory input tablesizes used for each row. Each data set was run for the same baseline trajectory to determine the errors

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associated with the input table resolution as compared to the real-time calculation shown in the first row.The error in the fuel usage quickly drops below 2% for input table with more than 60 points. It must benoted that the input tables used in the last 3 cases of the study were built using the resolution shown inTable 3 on the preceding page, but with relatively narrow bounds for the independent variables, based onresults from the real-time case. Although the extrapolated table size is reported here, the tables actuallyused were of a manageable size. Care was taken to ensure the resolution of these tables doubled within thestencils of the next coarser table, thereby providing the same result for these truncated tables as would beseen if the full size tables were used.

The aerodynamic and propulsion analysis code runs an average of 2.06 cases/second resulting in linearlyincreasing table creation time for increasing table density. Using an evolutionary based optimization method,and assuming approximately 5000 generations are required to converge a solution with a population size of50 results in the calculation of 250000 different trajectories to obtain an optimal solution for a particularvehicle. For a 1000 km trajectory solved using a 1 sec time step and a 4th order Runge-Kutta routine(requiring 4 function calls/timestep), a trajectory optimization requires 275 million function calls. Using theinterpolation tables for input, the trajectory optimizer makes approximately 67700 function calls per second,and a trajectory optimization on a single processor takes 4060 seconds (67 minutes). When added to thetime required to generate the lookup table for a single vehicle, the single vehicle trajectory optimization timebecomes a function of table density, as shown in the second to last column of Table 3 on the previous page.Figure 6 shows similar results for single-vehicle trajectory optimization with varying trajectory lengths andvarying interpolation table density (showing tables with 30, 96, 504, 4004, 64064, 1025024, and 16400384points). The results show that for table sizes of less than 4004 points the trajectory optimization time islargely dependent on the trajectory optimization, whereas above 4004 points it becomes dependent on thetable computation time. For the table densities shown, no case intersects the time required to perform thetrajectory optimization by calling the vehicle performance code at each step to determine the aerodynamicand propulsive coefficients. It is always more efficient time-wise to generate an interpolation table first andthen optimize the vehicle. It should be noted that these processor times are based on a single processor andparallel computation could improve the values to a reasonable time frame.

102

103

104

102

103

104

105

106

107

108

109

1010

Total Trajectory Time (s)

App

rox.

Pro

cess

or T

ime

for

Opt

imiz

atio

n (s

)

Real Time Coeff. Computation

Increasing InterpolationTable Density

Figure 6. Computational time versus vehicle input table density for asingle vehicle trajectory optimization.

The last column of Table 3 onthe previous page shows the timerequired in cpu years to performa joint vehicle-trajectory optimiza-tion, wherein all vehicle and tra-jectory parameters are included inan optimization. Such an optimiza-tion requires interpolation tables fora new vehicle to be generated foreach of the estimated 250000 tra-jectories required for the optimiza-tion (assuming an optimal solutioncan still be obtained using 5000 gen-erations). The actual optimizationtime quickly becomes trivially smallin comparison to generating the in-put tables for each vehicle. Theseresults are shown in Fig. 7 on thefollowing page with contours of de-creasing interpolation table density(showing tables with 30, 96, 504,4004, 64064, 1025024, and 16400384points) along with the estimatedtime for real-time vehicle computa-

tions. An interesting crossover point is the intersection of a 504 point interpolation table with the real-timecomputation for a 500 second trajectory optimization at about 5×107 seconds (≈ 5 cpu years).

With such large computational times, these optimizations will undoubtedly require the use of parallelprocessing, to which DES lends itself particularly well. To better understand the benefits that may be

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found using multiple processors, the results of a simple study of the time required to run UPTOP usingthe table interpolation method with between 1 and 15 processors is shown in Table 4. UPTOP is writtenusing MPI21 protocols and implements a “head” process that uses a self-scheduling method to send jobsto available nodes. The results show the average overall time required to calculate 1000 generations of 50population members and the time to calculate a single generation. The final two columns of Table 4 showthe speed of the parallel computations relative to the single processor speed and the efficiency of the parallelcomputation (i.e., for an efficiency of 100%, the computational speed would reduce by a factor of n whenusing n processors). The results show decreased efficiency for an increasing number of processors, whichsuggests that careful, thorough attention must be paid to the development of an effective parallel code.In the case of UPTOP, further development is required for the inter-generational processes (sorting of thepopulation, comparing the children and parent solutions, and rebuilding the new parent population), whichare currently handled on a single processor. Of note, because these inter-generation processes do not coincidewith the trajectory computations, the “head” node is not included in the parallel results of Table 4, andthe number of processors represents the number of nodes available to accept trajectory calculations. Accessto a larger number of processors is required to see if the loss in efficiency stabilizes with a larger numberof nodes, and further research is required to understand where gains may be found when using the massivecomputational arrays required to solve the problems presented in the current work.

Table 4. Study of time required to solve 1000 DES generations using UPTOP with parallel processing.

# proc Time/1000 gen (s) Time/gen (s) Rel. Speed % Efficiency

1 531.73 1.88 1.00 100.0

2 304.74 3.28 1.74 87.2

3 204.11 4.90 2.61 86.8

5 123.30 8.11 4.31 86.2

10 63.76 15.68 8.34 83.4

15 47.13 21.22 11.28 75.2

102

103

104

106

107

108

109

1010

1011

1012

1013

Total Trajectory Time (s)

App

rox.

Pro

cess

or T

ime

for

Opt

imiz

atio

n (s

)

Real Time Coeff. Computation

Decreasing Interpolation Table Density

Figure 7. Computational time versus vehicle input table density for acoupled vehicle/trajectory optimization.

The effects of using a widelybounded interpolation table for ini-tial trajectory optimization (as op-posed to building a table for aknown trajectory) as a function oftable density is shown in Table 5on the following page. The errorsshown are relative to the most densetable. The effects of table densityshow that tables above 108 pointscan converge within 3% of the re-sults of the 4004 point table. Thevariations in the results are likelydue to the effects of the linear in-terpolation scheme used, as well asthe proximity of the nodes to thevalues required. One powerful waywhich may lead to a drastic reduc-tion in the required interpolationtable density is the use of a higherorder interpolation scheme, such asa response surface map or neuralnetwork.

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Table 5. Optimal range-averaged fuel consumption (kg/km) for varying interpolation table density, including% variation from most dense table.

Table Info Opt Fuel Usage % Variation

∆q ∆M ∆φ ∆α total pts (kg/km)

- 4.0 0.6 2.0 30 3.237 21.62

2000.0 4.0 0.4 4.0 64 3.028 13.76

1500.0 4.0 0.6 1.0 108 2.679 0.66

1500.0 4.0 0.3 1.0 180 2.719 2.16

1500.0 2.0 0.3 1.0 270 2.730 2.59

750.0 4.0 0.3 1.0 270 2.705 1.64

1000.0 2.0 0.2 2.0 504 2.723 2.31

500.0 1.0 0.1 1.0 4004 2.661 -

V. Conclusions

A study into the coupled nature of vehicle and trajectory design and optimization has been performed forperiodic trajectories. By coupling the vehicle design and analysis with trajectory optimization the true vehicleperformance advantages may be uncovered. To highlight this requirement a simple parametric investigation ofthe effects of two degree of freedom vehicle perturbations on optimal trajectories was performed indicating thenecessity of vehicle dependent trajectory calculations. Also, the effects of flying a vehicle along a preselectedtrajectory was shown to be clearly non-optimal.

The sensitivities of the baseline vehicle design space showed the highest dependence on angle-of-attackresolution which led to intelligently selected ranges of interpolation table generation. For tables generatedto fly a given trajectory bound, as few as 60 points resulted in less than 2% error when compared to thereal time results. For tables generated to optimize a trajectory as few as 108 points allowed for less than 3%error. The effects of these table densities on trajectory and vehicle/trajectory coupled optimization showedthat with parallel processing, optimal results are possible, although still quite costly to achieve.

There are a number of methods which could improve the computational expense of this effort:

1. Higher order interpolation (response surface map, neural network) to reduce the required interpolationtable density.

2. Improved computational efficiency of vehicle performance to speed the real-time calculation of aerody-namic and propulsive coefficients while solving a trajectory.

3. Hybridization of the DES with a gradient-based optimization method to improve optimization efficiencyin terms of function calls.

4. Improved handling of error points in the interpolation tables (caused by thermal choking of the engine),which currently result in an invalid trajectory when encountered at any step by the optimizer.

5. Hybridization of the real-time and interpolation techniques, or use of advanced data-basing methodsto populate interpolation tables only around required points.

6. Improved problem description for enhanced parallel processing capabilities.

VI. Acknowledgments

This research was supported by the Space Vehicle Technology Institute (SVTI), one of the NASA Univer-sity Institutes, under grant NCC3-989, with joint sponsorship from the Department of Defense. Appreciationis expressed to Dr. Ken Yu, director of the SVT Institute at the University of Maryland, Claudia Meyer ofthe NASA Glenn Research Center, program manager of the University Institute activity, and to Dr. JohnSchmisseur and Dr. Walter Jones of the Air Force Office of Scientific Research, the support of whom isgreatly appreciated.

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References

1Starkey, R. P., Investigation of Air-Breathing, Hypersonic Missile Configurations Within External Box Constraints,Ph.D. thesis, Department of Aerospace Engineering, University of Maryland, College Park, Oct. 2000.

2Starkey, R. P. and Lewis, M. J., “A Shock-Expansion Method for Determining Surface Properties On Irregular Geome-tries,” AIAA Paper 2002-0547 , Jan. 2002.

3O’Brien, T., Starkey, R. P., and Lewis, M. J., “A Quasi-One-Dimensional High-Speed Engine Model with Finite-RateChemistry,” AIAA Journal of Propulsion and Power , Vol. 17, No. 6, Nov.-Dec. 2001, pp. 1366–1374.

4Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow , Vol. 1, McGraw-Hill Book Company,New York, 1953.

5Turns, S. R., An Introduction to Combustion, McGraw-Hill Book Company, New York, 1996.6Starkey, R. P. and Lewis, M. J., “Sensitivity of Hydrocarbon Combustion Modeling for Hypersonic Missile Design,” AIAA

Journal of Propulsion and Power , Vol. 19, No. 1, Jan.-Feb. 2003, pp. 89–97.7O’Brien, T., RBCC Engine-Airframe Integration on an Osculating Cone Waverider Vehicle, Ph.D. thesis, University of

Maryland, 2001.8Starkey, R. P., “Off-Design Performance Characterization of a Variable Geometry Scramjet,” Tech. rep., ISABE-2005-111,

September 2005.9Speyer, J. L., “On the Fuel Optimality of Cruise,” Journal of Aircraft , Vol. 10, No. 12, 1973, pp. 763–765.

10Speyer, J. L., “Periodic Optimal Flight,” Journal of Guidance, Control and Dynamics, Vol. 19, No. 4, 1996, pp. 745–753.11Chuang, C. H. and Morimoto, H., “Periodic Optimal Cruise for a Hypersonic Vehicle with Constraints,” Journal of

Spacecraft and Rockets, Vol. 34, No. 2, 1997, pp. 165–171.12Carter, P. H., Pines, D. J., and von Eggers Rudd, L., “Approximate Performance of Periodic Hypersonic Cruise Trajec-

tories for Global Reach,” Journal of Aircraft , Vol. 35, No. 6, 1998, pp. 857–867.13von Eggers Rudd, L., Pines, D. J., and Carter, P. H., “Suboptimal Damped Periodic Cruise Trajectories for Hypersonic

Flight,” Journal of Aircraft , Vol. 36, No. 2, 1999, pp. 405–412.14Rankins, F., “Suboptimal Periodic Hypersonic Cruise Trajectories for Increased Heating Performance,” Department of

Aerospace Engineering Scholarly Paper, University of Maryland, College Park.15Stevens, B. L. and Lewis, F. L., Aircraft Control and Simulation, Wiley, New York, 1992.16U.S. Standard Atmosphere, 1976 , National Oceanic and Atmospheric Administration, National Aeronautics and Space

Administration, United States Air Force, Washington, DC, 1976.17Price, K. and Storn, R., “Differential Evolution,” Dr. Dobb’s Journal , April 1997, pp. 18–24.18Madavan, N. K., “Aerodynamic Shape Optimization Using Hybridized Differential Evolution,” Tech. rep., AIAA Paper

2003-3792, June 2003.19Rai, M. M., “Robust Optimal Aerodynamic Design Using Evolutionary Methods and Neural Networks,” Tech. rep., AIAA

Paper 2004-778, January 2004.20Qing, A., “Electromagnetic Inverse Scattering of Multiple Two-Dimensional Perfectly Conducting Objects by the Differ-

ential Evolution Strategy,” IEEE Transactions on Antennas and Propogation, Vol. 51, No. 6, 2003, pp. 1251–1262.21Snir, M., Otto, S., Huss-Lederman, S., Wilker, D., and Dongarra, J., MPI: The Complete Reference, The MIT Press,

Cambridge, MA, 1996.

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