NASA Technical Memorandum 101721

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Effects of Simplifying Assumptions

on Optimal Trajectory Estimation

for a High-Performance Aircraft

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Lura E. Kern, Steve D. Belle, and Eugene L. Duke

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April 1990

(NASA-TM-101721) EFFrCTS OF SIMPLIFYING

ASS_!HPITONS ON OPTI_AL T_AJ_CTGRY ESTT_ATTON

_OR A _TGH-PEREO_MANC£ AIRCRAFI (NASA)

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National Aeronautics and

Space Administration

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NASA Technical Memorandum 101721

Effects of Simplifying Assumptions

on Optimal Trajectory Estimation

for a High-Performance Aircraft

Lura E. Kern and Eugene L. Duke

Ames Research Center, Dryden Flight Research Facility, Edwards, California

Steve D. Belle

PRC Systems Services, Edwards, California

_ 1990

National Aeronautics and

Space Administration

Ames Research Center

Dryden Flight Research Facility

Edwards, California 93523-0273
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CONTENTS

INTRODUCTION

NOMENCLATURE 1

2

Subscripts .................................................

ENERGY STATE APPROXIMATION 2

2

Background ................................................

Flight Envelope Calculations ........................................ 4

Trajectory Calculations ........................................... 5

6 Aircraft Model ...............................................

MODELING ASSUMPTIONS 6

Atmospheric Models ............................................ 6

Gravitational Model Equations ....................................... 7

7

Equations of Motion ............................................

Constraint Conditions 8 ...........................................

RESULTS 9

9

Effects of Atmospheric Models ......................................

Effects 9 of Gravitational Models ......................................

Effects of Equations of Motion ....................................... 9

Effects of Constraint Conditions on Trajectory Calculations ........................ 10

CONCLUSIONS 10

REFERENCES 11

FIGURES 11

PRECEDING PAGE BLANK NOT FILMED

!11
2 _
INTRODUCTION

When analyzing the performance of an aircraft, certain simplifying assumptions, which decrease the complexity

of the problem, can often be made. The degree of accuracy required in the solution may determine the extent

to which these simplifying assumptions are incorporated. A complex model may yield more accurate results if it

describes the real situation more thoroughly. However, a complex model usually involves more computation time,

makes the analysis more difficult, and often requires more information to do the analysis. Therefore, to choose

the simplifying assumptions intelligently, it is important to know what effects the assumptions may have on the

calculated performance of a vehicle. This paper examines several simplifying assumptions, compares the effects of

simplified models to those of the more complex ones, and draws conclusions about the impact of these assumptions

on flight envelope generation and optimal trajectory calculation. Models which affect an aircraft are analyzed, but the

implications of simplifying the model of the aircraft itself are not studied. The examples in this paper are atmospheric

models, gravitational models, different models for equations of motion, and constraint conditions.

The results are calculated using the energy state approximation (Bryson, Desai, and Hoffman, 1969). In the

energy state approximation, the aircraft is modeled as a point mass in the vertical plane, and specific energy is used

as a state variable. The results are for a high-performance aircraft in minimum time-to-energy trajectories.

NOMENCLATURE

a speed of sound, ft

D drag, lb

Es specific energy, ft

f function, or forces, lb

fd

G gravitational constant,

g acceleration due to gravity, sef--_tc

h geopotential altitude, ft

J cost function

L lift, lb

L,, temperature lapse rate,

M Mach number

Me mass of the Earth, lb

Mo mean molecular weight

m mass of the vehicle, slug

ib

P pressure, f7

PLA power lever angle

P_ specific power, ft

?/ dynamic pressure,

/i_ range, ft

/_ radius of the Earth, ft
_

T

V

%u

z

o_

P

fl lb

universal gas constant,

thrust, lb, or temperature, °R

time, sec

ft

velocity, se-TCrc

fuel weight, lb

fuel flow, _r

geometric altitude, ft ......

angle of attack, rad

flightpath angle, rad, or ratio of specific heats

density of air, ft"

first derivative

Subscripts

ave average

b base

idI idle

h in the horizontal plane

tim limit

max maximum

min minimum

re f reference

v in the vertical plane

z x-body axis of the vehicle

z z-body axis of the ychiclc

ENERGY STATE APPROXIMATION

Background

The trajectories discussed in this paper are calculated using the energy state approximation theory (Bryson,

Desai, and Hoffman, 1969). This theory models the aircraft as a point-mass and assumes a fiat, nonrotating Earth,

which is reflected in the equations of motion used. The general equations of motion for a point-mass in the vertical plane are derived by Micle (1962). They can be given by, first, summing the forces in thedirection of the

velocity vector

Txcos ot - Tz sin _ - D -mg sin'7 = m_ r _ (1)

where T is thrust, o_is angle of attack, D is drag, mg is the vehicle's weight, ,/is the flightpath angle, m is the

vehicle's mass, and V is the first derivative of velocity. By summing the forces perpend{cular to the vcqoc]ty vector

T_ sin ot + Tz cos _ + L - mg cos "y= mV7 (2)

2
whereL is lift, V is velocity, and _ is the first derivative of the flightpath angle. These equations assume that

the vehicle thrust vector is fixed with respect to the vehicle body axes and acts in both the a: and z body axes of

the vehicle.

In the energy state approximation, the total energy per unit mass, or specific energy, (E_) is considered a state

variable and is derived from the energy equation (Bryson, Desai, and Hoffman, 1969)

1

E = rngz + _-rnV 2

(3)

where E is energy, and z is geometric altitude. This is divided by the vehicle weight to give the specific energy

V 2

Es = z+ -- (4)

2g

When this quantity is differentiated with respect to time, the equation for specific power (Ps) is given

vf"

v,= -- (5)

g

An expression for V" can be derived from equation 1 resulting in

_. = T_ cos o_- T_ sin ¢x- D -mg sin ?

An expression for _?can be derived from the definition of'/

so that

(6)

= V sin '/ (8)

Substituting equations 6 and 8 into equation 5 results in:

Ps= Vsin'/+ v(Tzc°s°e-Tzsin°t-D-mgsin'/'_ (9)

\ mg /

By subtracting out the V sin '/terms this equation gives an expression for Psp.= V(Tzc°sa-Tzsin°_-D) (lO)

Equation 10 gives Ps as a function of velocity, thrust, angle of attack, and drag. Drag, however, is a function of

Mach number, altitude, and angle of attack. Thrust is a function of power lever angle (PLA). Therefore, it can be

seen that P_ is a function of altitude, Mach number, angle of attack, and PLA and is independent of flightpath angle.

Given the expression for Ps, it is possible to derive the cost function for a minimum time-to-energy path. The

minimum time-to-energy cost function is defined as (Erzbcrger, Barman, and McLean, 1975)

J = /: (ll)

where J is a cost function and t is time.

,7 = sin -J --z (7)

V
From equation 5, specific power is defined as

dE_

P, - (12) dt

Substituting equation 12 into equation 11 gives

] = fo E, dEo (13)

To find the minimum for equation 13, it is sufficient to minimize the quantity inside the integral {d_._E'_ Since kP,/"

the energy state approximation assumes that cost functions arc computed along curves of constant specific energy,

it is sufficient to minimize the quantity I_. This is the same as maximizing Ps- The cost function for a minimum

time-to-energy trajectory, then, can be given by:

maximize J = P_ (14)

Flight Envelope Calculations

The flight envelope can be calculated by solving the general equations ofmotion (eqs. 1 and 2) for trim in straight

and level flight. For this case, '7 = 0, V = 0, and '_ = 0. Therefore, the general equations of motion for a point

mass in the vertical plane are reduced to:

T_ cos c_ - T. sin oe - D = 0 (15)

T_ sin oe + T, cos a+ L -mg = 0 (16)

For actual implementation in a numerical nonlinear equation solver, the above equations can be written as:

fh(oe, PLA, z,M) = T_cos or- Tz sin ae _ 1............ (I7)

D

L + T_ sinot- T_ cosot _

fv(oe, PLA, z, M) = - 1 (18)

rng

The trim equations of motion, then, are dcfined by setting f_(_,PLA, z, M) = 0 and fh(oe, PLA, z, M) = 0. Note

that these equations make no attempt to force the pitching moment equation to zero. While this is consistent with

the assumption of the vehicle as a point mass, the effects of the vehicle's longitudinal surfaces are neglected:

The flight envelope is calculated in four parts (fig. 1): the angle-of-attack boundary, the thrust boundary, the

Mach boundary, and the dynamic pressure boundary. The angle-of-attack boundary is defined by setting the angle of

attack equal to the maximum angle of attack of the vehicle. An altitude is picked, and the trim equations of motion

are solved for that altitude. This gives

fh(OZlim, PLA, z,-_f, M) = 0 (19)

and

f,,( Cqim, PLA, z,._f , M) = 0 (20)

This gives two equations and two unknown variables, which can be solved using a numerical nonlinear

equation solver. : .....

4

=
The thrust boundary is calculated using the same method. The PLA is set to its limiting value and a Mach number

is chosen for the calculations. The two trim equations of motion are then solved for the corresponding angle of attack

and altitude:

fh(o_, PLA/im, z, M_e/) = 0 (21)

and

fv(o_, PLAt/,,,, z, M_,I) = 0 (22)

The Mach boundary is given by setting the Mach number equal to the vehicle's Mach limit.

The dynamic pressure boundary is calculated by solving the equation relating dynamic pressure and velocity,

using the dynamic pressure limit of the vehicle

V,._I = (23)

whei'e ?/is the dynamic pressure and p is the density of air. The Mach number is found from the resulting velocity,

and the trim equations of motion are solved using a reference altitude

fh(o_,PLA, z_,y, M_ef) = 0 (24)

and

fo(o_, PLA, z,e/, M,.e/) = 0

(25)

Trajectory Calculations

To calculate trajectories, the cost function is maximized along curves of constant specific energy within the flight

envelope of the vehicle. For the numerical results in this paper, this was done in the following way:

1. An array of specific energy curves encompassing the vehicle's flight envelope was determined (fig. 1).

2. A point along the first curve, representing the lowest value of specific energy, was chosen, yielding an altitude

and a corresponding Mach number.

3. This altitude and Mach number point was entered into an optimization routine that varied angle of attack and

PLA to determine a maximum value for the cost function, specific power, at that point.

4. A diffcrent altitude and Mach number point on the first curve was then chosen, and the optimization routine

calculated a maximum value for the cost function at the new point.

5. These two maximum Values for the Cost function were compared and the higher of the two was stored.

6. The maximum COStfunction value was calculated at other points along the first energy curve and the highest

value found along the curve was stored. This value identified the optimum point on that energy curve.

7. The optimum point was then calculated along the remaining energy curves within the flight envelope. The

altitude and Mach number path connecting these points defined the optimal trajectory. An optimal trajectory

is shown in figure 1.

Parameters such as the time, range, and fuel used along the trajectory can be calculated between energy curves.

This is not the same as the analysis at each point along an energy curve, which assumes steady, horizontal flight. Information for the total trajectory can be gained by summing the lime, range, and fuel used between each

cncrgy curve.
Thetimeit takesto gofromone energy curve to another is given by

Range (R) is then calculated based on the time

at = AEs (26)

Psa_¢

A R = V cos ,,/At (27)

where 3' is given in equation 7. Finally, fuel used is a product of the fuel flow (w) multiplied by the time increment

Aw = _bAt (28)

where w is fucl wcight.

Aircraft Model

As stated previously, the energy state approximation models the aircraft as a point mass in the vertical plane.

This paper does not assess the assumptions made about the aircraft, but concentrates instead on assumptiQn_made

about outside parameters which will affect the trajectory calculations. These trajectory calculations were based on a

mathematical model of a typical modem, supersonic fighter aircraft. This model supplied lift, drag, thrust, and fuel

flow as a function of Mach numberl ahitude,angie of attack_, and PEA ihroughout the flight envelope.

.. 2__ . __2 .

MODELING ASSUMPTIONS

Atmospheric Models _ : ! _ _ _:_=_i:_ _ i: _,: : .... =::

Models of the atmosphere are used to determine altitude-dependent parameters such as air density, pressure,

temperature, and speed of sound. This paper looks at the 1962 standard atmosphere (NASA/U.S. Air Force/U.S.

Weather Bureau, 1962), as well as models for a standard day, cold day, and hot day for Edwards Air Force Base,

Califomia (Johnson, 1975). The equations used to determine atmospheric-dependent parameters are given here.

The general equation for temperature is (NASA/U.S. Air Force/U.S. Weather Bureau, 1962)

T = Tb+ L.,(h - hb) (29)

Here, Lm is the gradient of the temperature With geopotcntial altitude, hb IS the ge0potcntial altitude ai the base of

a particular layer by a specific value of Lm, and Tb is the value oft at altitude hb. Temperature profiles for the four

day types are given in figure 2.

The general equation for pressure (P) is dependent on the value of the gradient (L,,). If L,n _/0, then P is

given by

-_b \ Tb + L,nH ]

and if Lm = 0, P is given by

_ ( MoH, P = exp

Pb R*T ,/

Here, H = h - hb, Mo is the mean molccular wcight, and R* is the univcrsal gas constant.

(30)

(31)

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6
MrDensity is calculated from the pressure and temperature:

MoP

P= R*T

(32)

and, finally, the speed of sound is calculated from the temperature:

_R*T

VM o

(33)

Gravitational Model Equations

Acceleration due to gravity can be calculated several ways. The two ways that make sense for a model that

assumes a fiat, nonrotating Earth are: 1) calculating gravity as a function of distance from the surface of the Earth,

and 2) giving gravity as a constant value, assuming no variation with altitude.

The acceleration due to gravity as a function of distance from the Earth is (NASA/U.S. Air ForcefO.S. Weather

Bureau, 1962) GMe (34)

g = (R_ + z) 2

where G is a gravitational constant, Me is the mass of the Earth, and Re is the radius of the Earth.

A simpler means of determining the acceleration duc to gravity is to assume that it does not vary with distance

from the Earth. The value for g, then, is given by the constant

ft

g = 32.174 _ (35)

sec 2

Equations of Motion

The general equations of motion for a point mass in the vertical plane are given by equations 1 and 2. Several

assumptions can be made which affect the flight envelope and trajectory calculations in two distinct ways. First,

assumptions for the equations of motion will affect the calculations of the flight envelope. Second, the equations of

motion are used in calculating specific power, which is used as the cost function for calculating minimum time-toenergy trajectories.

To calculate the flight envelope, the aircraft is assumed to have straight-and-level flight at trim conditions, giving

equations 15 and 16:

Tz cos c_ - Tz sin oL- D = 0

Tz sin a + Tz cos o_+ L -mg = 0

These give the most general method of modeling the trim equations of motion used in calculating the vehicle's

flight envelope.

If the thrust vector is assumed to be aligned with the x-body axis of the vehicle, then there is no thrust along the

z-body axis, and the equations simplify to

T cos a - D = 0 (36)

T sin ol + L - mg = 0 (37)
To further simplify the equations, it can be assumed that the angle of attack is small, in which case cos o_ _ 1

and sin a _ 0. This gives

T- D= 0 (38)

L - rag = 0 (39)

These same assumptions can be used in calculating specific power. The most general equation for specific power

is given by equation 10:

ps= V(Tzc°s°_-Tzsin°_-D)

If it is assumed that the thrust lies along the body axis of the vehicle, then 7", is zero and the equation for specific

power becomes:

P, = V ( T cos __ D) (40) \ mg

Finally, assuming that the angle of attack is small gives:

ps=v T-D (41)

Constraint Conditions

Constraint conditions limit the possible solutions to an optimization problem. The more constraints put on the

problem, the smaller the range of solutions becomes. Because constraint conditions are applied to the optimization,

the choice of the constraint conditions affects only the optimal trajectory and has no affect on the flight envelope.

The simplest case in this paper is an unconstrained trajectory. This means that there are no constraints on the

cost function, and only necessary limits are put on angle of attack and PLA to keep them within realistic values. The

maximization problem for optimizing unconstrained trajectory is

maximize J(a, PLA, h, M)

subject to cr_;,_ _< ol _< a,_

PLAidt < PLA < PLA,na_

Thc quantity J(a, PLA, h, M) is thc cost function being optimizcd, specific powcr.

Bccausc the cncrgy state approximation assumcs that thc aircraft is in straight and lcvel flight at each point, this

can be rcflcctcd in the constraint conditions. To do this, a constraint equation that balanccs the equ_a_fionspf, motion

in the vcrtical direction is addcd. Thc maximization problem for optimizing trajcctory constraincd to level flight is:

maximize J(a, PLA, h, M)

subject to f,,(a, PLA, h, M) = 0

O_rnin < Ol < Otrnaz

PLAidt _< PLA < PLAma_

Finally, the trajectory can be constrained to realistic flightpath angles. This can also be thought of as constraining

the velocity in the vertical direction, so that the rate of climb (k,) can never exceed the total velocity. This can be

sccn from the definition of the fllghtpath angle

q,= sin -1

V

"v
The maximization problem for optimizing a trajectory constrained to realistic flightpath angles is

maximize

subject to

3"(or, PLA, h, M)

fv( o_,PLA, h, M) = 0

-i_<,y___i

Otmi n < Ot < Otma z

PLA_ctt < PLA < PLA,_

RESULTS

Effects of Atmospheric Models

The effects of the atmospheric models on the trajectory and flight envelope are shown as a result of aerodynamic

effects only; the effects of atmosphere on engine performance were not considered. The results show that the effects

on aerodynamics are significant. Figure 3 shows trajectories for the four different atmospheric models analyzed:

standard day, Edwards cold day, Edwards standard day, and Edwards hot day. The flight envelope was calculated

for standard day and Edwards cold day only.

The choice of atmospheric model had an effect on the optimal trajectory, especially in the high energy region,

(fig. 3). At the higher energies, the maximum P, occurs at a higher altitude for the warmer days. The differences

are due to the differences in densities at those altitudes between the atmospheric models. The temperature affects

the density, which affects the calculated lift and drag. This is also reflected in the flight envelope, which shows the

effects of atmospheric model on the dynamic pressure boundary, which is a function of the density of the air, as well

as the thrust boundary at high energies.

Figures 4(a), (b), and (c) show the effects of the different atmosphere models on the performance of the vehicle.

In agreement with the data shown on figure 3, figure 4(a) shows that the vehicle must climb higher on warmer days

in order to maximize its cost function. These higher altitude trajectories take more time and range (fig. 4(b)) and

use more fuel (fig. 4(c)) than the lower altitude trajectories to achieve the same energy. They are, therefore, less

efficient. If a large degree of accuracy is required in the calculations, the atmosphere must be modeled as close to

the actual conditions as possible.

Effects of Gravitational Models

*4¢

The effect o_fthe two different gravitational models on flight envelope and trajectory is shown in figure 5. The

two gravitational models are the altitude dependent model, based on a simple Newtonian force equation (eq. 34),

and the simpler model in which gravity is assumed constant (eq. 35). Figure 5 shows that the flight envelope and

optimum trajectory are overlaid for the two gravity models.

Effects of Equations of Motion

Figure 6 shows the effects of the differences in the equations of motion models on the flight envelope and optimal

trajectory. For the most general equations, the angle between the thrust vector and the body axis was increased until

a noticeable effect was seen on the trajectory calculations. This angle was determined to be approximately 6 °. At

angles up to 6°, there was only a small effect on the flight envelope and trajectory. The only difference is in the

angle-of-attack boundary of the flight envelope. The case with the thrust aligned with body axis overlays the case

with the 6° angle between thrust and body axis, but both are slightly different from the zero angle-of-attack case.
Figures7(a)and(b)showthatsimplifyingtheequationsof motionhadlittleeffectonthecalculatedperformance

of thevehicle.It tookathrustmisalignmentangleof 6° to showaneffectonperformance, inwhichcaseaslight

increaseinrangeandfuelusedwereneededtoachievethesameenergystateasthesimplercases.

Effects of Constraint Conditions on Trajectory Calculations

Constraint conditions had some effect on the trajectory, as figure 8(a) shows. For the constrained to level flight

and the flightpath constrained cases, a condition of the constraints was that the optimization pick a point where the

forces in the vertical direction are balanced. Because of this, limiting the flightpath to realistic angles had the same

effect as constraining the trajectory to level flight. The unconstrained case, however, shows that a different optimal

trajectory was picked.

There is no significant difference in aircraft performance between the constrained to level flight case and the

flightpath constrained case. However, both differed from the unconstrained case. The constraint conditions caused

a less favorable but more realistic trajectory to be chosen. Figures 8(b) and (c) show that the constrained cases take

more range and fuel to achieve the same energy state than the unconstrained case does.

CONCLUSIONS

This paper examined the effects of various modeling assumptions on flight envelope generation and optimal

trajectory estimation. The parameters investigated included atmospheric models, gravitational models, equations of

motion, and constraint conditions.

The different types of atmospheric models showed a definite effect on the trajectory, especially at high energies.

Of the four examined, the colder days had better performance characteristics. The warmer day models took more

time and distance to achieve the same energy state. If a large degree of accuracy is required in calculations, the

atmosphere should be modeled as close to the actual conditions as possible.

The two gravitational models examined were gravity as a function of altitude and a constant valued gravity force.

Thcy showed no difference in effect on flight envelope or trajectory calculations.

Three different sets of equations of motion were examined. The assumption of zero angle of attack showed

a noticeable but small effcct on the angle-of-attack boundary of the flight envelope, but showed no effect on the

trajectory. For small angles between the thrust vector and the body axes, simplifying the equations of motion in the

vertical plane had very little effect on calculated performance.

The different constraint conditions had an effect on the trajectory. Constraining the trajectory to level flight

limited the possible points to be chosen for theoptimal path, and so made a difference in the optimal trajectory. The

further constraint of limiting the flightpath to reasonable limits had no effect compared to the constrained to level

flight case. ......

Ames Research Center

Dryden Flight Research Facility

National Aeronautics and Space Administration

Edwards, California, January 3, 1990

L

v

10
REFERENCES

Bryson, Arthur E., Jr., Mukund N. Desai, and William C. Hoffman, "Energy-State Approximation in Performance

Optimization of Supersonic Aircraft," J. Aircraft, vol. 6, no. 6, Nov. 1969. Also available as AIAA 68-877,

Aug. 1968.

Erzberger, Heinz, John E Barman, and John D. McLean, "Optimum Flight Profiles for Short Haul Missions," AIAA

75-1124, Aug. 1975.

Johnson, D.L., Hot, Cold, and Annual Reference Atmospheres for Edwards Air Force Base, California (1975

Version), NASA TM-X-64970, 1975.

Miele, Angelo, Flight Mechanics 1: Theory of Flight Paths, Addison-Wesley, 1962.

NASA/U.S. Air Force/U.S. Weather Bureau, U.S. Standard Atmosphere, 1962, Washington, D.C., Dec. 1962.

Altitude,

It

70 x 1,000

60

5O /

/

Thrust

Constant specific

energy curves

Flight envelope

.... Optimal trajectory

Mach

boundary

4O

3O

20

/

/

I

I

I

I

| Optimal

I

_'_ attack

10 bounds_

Optimal

trajectory

Dynamic

pressure

boundary

0 .5 1.0 1.5 2.0 2.5

Mach number 10188

Figure 1. Sample flight envelope and optimal trajectory for a high-performance vehicle.

11
Altitude,

11

70 x 1000

6O

5O

4O

3O

2O

10

0

-120

¢

I

!

\

-9O -60 -3O 0 3O 6O

Temperature, °F

Figure 2. Temperature profilcs for different day types.

Standard day

Edwards cold day

Edwards standard day

Edwards hot day t _

12
70 x 1000

Constan! specific

energy curves

Standard day

Edwards cold day

Edwards standard day

Edwards hot day

6O

50

Altitude,

ft

40

3O

Optimal

trajectories

2O ht

envelopes

10

0 ,5 1.0 1,5 2.0 2.5

Mach number lOlgO

Figure 3. Effect of atmospheric model on flight envelope and trajectory. Flight envelope calculated for standard day

,and Edwards cold day only.

]3
Altitude,

It

7O

6O

5O

4O

3O

20i

10

1000

B Standard day

-- Edwards cold day

.... Edwards standard day

Edwards hot day

w

50 100 150 200 250 300 350

Time, sec IOlgl

Figure 4. Effects of ammspheric model.

(a) Altitude as a function of time.

14
Range,

fl

70 x 1000

60 --

50 --

40

30

20 --

10

Standard day

Edwards cold day

Edwards standard day

Edwards hot day

m

,50

J

js

I I I I I I

100 150 200 250 300 350

Time, sec 101GQ

(b) Range as a function of time.

Figure 4. Continued.

15
8x 1000

p Standard dly

I EEd::rrdd: :ta°ldddra_ dlY .,.J

,L f

=

=

I ! !

0 50 100 150 200 250 300

Time. iec

(c) Fuel used as a function of time.

Fi_/e 4. Concluded.

I

35O

10193

16
xl_O

Constant specific

energy curves

Constant gravity

-- -- -- Newlon gravity

6O

5O

Altitude,

ft

4O

3O

Optimal

trajectory

20

hi

envelopes

10

.5 1.0 1.5 2.0 2.5

Mach number 10194

Figure 5. Effect of gravitational model on flight envelope and trajectory.

17
70 x 1000

Constant specific

energy curves

Zero angle of attack

Thrust aligned

Thrust not aligned

6O

111

5O

Altitude,

11

4O

3O

trajectory

envelope

10

Figure 6.

18
v

350 x 1000

F Zero angle of attack

| Thrust aligned

300 t- ----_ Thrust not aligned _4'

250

F

0 20 40 60 80 1O0 120 140 160 180 200

"rime, $ec Ioig6

(a) Range as a function of time.

Figure 7. Effects of equations of motion.

?

19
500 x 1000

40O

Zero angle of attack

Thrust aligned

Thrust not aligned

V

11

350

300

Fuel used, 250

Ib

2OO

150

100

5O

0 20 40 60 80 i00 120

Time, see

(b) Fucl uscd as a function of time.

Figure 7. Concluded.

140 160 180 200

lo197

2O
70 x 1000

60

Constant specific

energy curves

Unconstrained

Level flight constraint

Flight path constraint

5o

Altitude,

ft

4O

3O

2O

Optimal

trajectory

Flight

envelope

10

.5 1.0 1.5

Mech number

(a) Trajectory.

Figure 8. Effects of constraint conditions.

2.0 2.5

10198

L

21
400:

350

30O

250

Range,

fl _O

150

100

5O

1000

/

/

,, Unconstr,.ined

mm, m_mm

.... Level fl|gh( constraint

..... FIIghl palh conltra}n!

I I I ! ! I I

50 100 150 200 250 300 350

Time,-sec I01_

(b) Range as a fun-cti6n of time.

Figure 8. Continued.

I

22!_-: i_

!
7x1000

Fuel used,

Ib

F Unconstrained

I Level flight constraint

6 _ Flight path constraint

5

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0 50 100 150 200

- Time, 8ec

(c) Fuel used as a_t'unction of time.

Figure 8. Concluded.

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Nat=y_ Aeronautics

I

1. Report No. ]

NASA TM-101721 1

4. Title and Subtitle

Report Documentation Page

2. Government Accession No.

Effects of Simplifying Assumptions on Optimal Trajectory Estimation

for a High-Performance Aircraft

7. Author(s)

Lura E. Kern, Steve D. Belle, and Eugene L. Duke

9, Performing Organization Name and Address

NASA Ames Research Center

Dryden Flight Research Facility

P.O. Box 273, Edwards, CA 93523-0273

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546

3. Recipient's Catalog No.

5. Report Date

April 1990

6. Performing Organization Code

8. Performing Organization Report No.

H- 1597

10. Work Unit No,

RTOP 505-66-71

11. Contract or Grant No,

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15, Supplementary Notes

16. Abstract

When analyzing the performance of an aircraft, certain simplifying assumptions, which decrease the complexity of the problem, can often be made. The degree of accuracy required in the solution may determine

the extent to which these simplifying assumptions are incorporated. A complex model may yield more accurate results if it describes the real situation more thoroughly. However, a complex model usually involves

more computation time, makes the analysis more difficult, and often requires more information to do the

analysis. Therefore, to choose the simplifying assumptions intelligently, it is important to know what effects

the assumptions may have on the calculated performance of a vehicle. This paper examines several simplifying assumptions;compares the effects of simplified models to those of the more complex ones, and draws

conclusions about the impact of these assumptions on flight envelope generation and optimal trajectory calculation. Models which affect an aircraft are analyzed, but the implications of simplifying the model of

the aircraft itself are not studied. The examples in this paper are atmospheric models, gravitational models,

different models for equations of motion, and constraint conditions.

17. Key Words (Suggested by Author(s))

Energy state approximation

Simplifying assumptions

Trajectory optimization

19. Security Classif, (of this report)

Unclassified

20, Security Classif. (of this page)

18. Distribution Statement

Unclassified -- Unlimited

Subject category 08

Unclassified F No. 26 of pages [22. Price A03

¢

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