COMPUTERIZATION OF AIRCRAFTNAVAL AIR TRAINING AND OPERATINGPROCEDURES STANDARDIZATION (NATOPS)

FLIGHT PERFORMANCE CHARTS

Johnny Dean Restivo

NAVAL POSTGRADUATE

Monterey, California

COMPUTERIZATION OF AIRCR.\FTNAVAL AIR TRAINING AND OPERATING

PROCEDURES STANDARDIZATION (NATOPS)FLIGHT PERFORMANCE CHARTS

by

Johnny Dean Restivo

June 1978

Thesis Advisor D. M. Lay ton

Approved for public release; distribution unlimited.

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Computerization of Aircraft Naval AirTraining and Operating ProceduresStandardization (NATOPS) FlightPerformance Charts

7. AUTHORr»J

Johnny Dean Restivo

9. PERFORMING ORGANIZATION NAME ANO AOORESS

Naval Postgraduate SchoolMonterey, California 93940

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Naval Postgraduate SchoolMonterey, California 93940

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19. KEY WORDS (Continu* on ravaraa »id» H nmeasmary mnd IdanlKy by bloek niaKbar)

NATOPS ComputerizationPerformance ChartsLeast Squares MethodA-6E Aircraft

Aviation SafetyOperational Readiness-AircraftHand-Held CalculatorsAircraft Systems Computers

20. ABSTRrACT. CConl/nua on /avar** aidm H nmeoammrr and Idantltr by hloek mtmbor)Ihis thesis computerizes aircraft Naval Air Training and

Operating Procedures Standardization Program (NATOPS) flightperformance charts and shows it to be feasible with a highdegree of accuracy. The computer programs developed are'^NormalTake-off and Cruise Performance for the A-6E aircraft and areadaptable to hand-held calculators, desk calculators, or existingaircraft systems computers. The feasibility, procedures, and

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techniques shown in this thesis are applicable to any aircraft,both fixed and rotary wing.

A much improved and more accurate method of missionplanning is needed to reduce NATOPS chart errors during pre-flight planning and to reduce the accident potential of aircrews who make performance decisions during flight from pastexperience or old information, without accounting for allvariables which affect aircraft performance. A large per-centage of Navy/Marine Corps aircraft accidents has beendirectly or indirectly attributed to misinterpretation ofNATOPS charts and/or poor performance decisions. ComputerizedNATOPS performance charts will significantly reduce humanerrors inherent to visual chart interpretation and, whencoupled with aircrews having computer access to make airborneaircraft performance decisions based on ail performance vari-ables, will greatly enhance safety of flight in completingany or all mission phases, thereby increasing operationalreadiness of all Navy/Marine Corps fleet units.

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Approved for public release; distribution unlimited

Computerization of Aircraft Naval Air Training andOperating Procedures Standardization (NATOPS)

Flight Performance Charts

by

Johnny Dean RestivoMajor, United States Marine CorpsB.A., Texas A§M University, 1966

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

June 1978

ABSTRACT

This thesis computerizes aircraft Naval Air Training and

Operating Procedures Standardization Program (NATOPS) flight

performance charts and shows it to be feasible with a high

degree of accuracy. The computer programs developed are

Normal Take-off and Cruise Performance for the A-6E aircraft

and are adaptable to hand-held calculators, desk calculators,

or existing aircraft systems computers. The feasibility,

procedures, and techniques shown in this thesis are applicable

to any aircraft, both fixed and rotary wing.

A much improved and more accurate method of mission plan-

ning is needed to reduce NATOPS chart errors during pre-flight

planning and to reduce the accident potential of air crews who

make performance decisions during flight from past experience

or old information, without accounting for all variables which

affect aircraft performance. A large percentage of Navy/

Marine Corps aircraft accidents has been directly or indirectly

attributed to misinterpretation of NATOPS charts and/or poor

performance decisions. Computerized NATOPS performance charts

will significantly reduce human errors inherent to visual

chart interpretation and, when coupled with aircrews having

computer access to make airborne aircraft performance deci-

sions based on all performance variables, will greatly enhance

safety of flight in completing any or all mission phases,

thereby increasing operational readiness of all Navy/Marine

Corps fleet units.

TABLE OF CONTENTS

I. INTRODUCTION 8

II. EXPERIMENTAL PROCEDURE 11

A. GENERAL ORGANIZATION 11

B. TAKE-OFF PERFORMANCE 13

C. CRUISE PERFORMANCE 16

III. RESULTS AND DISCUSSION 21

A. TAKE-OFF PERFORMANCE 21

B. CRUISE PERFORMANCE 25

IV. CONCLUSIONS AND RECOMMENDATIONS 30

APPENDIX A: Least Squares Method o£ PolynomialApproximation 32

APPENDIX B: Take-off Performance Computer Program 38

APPENDIX C: Cruise Performance Phase Charts 47

Phase I - Clean Aircraft Transfer Scale 47

Phase II - Aircraft Reference Number 48

Phase III - Pounds of Fuel per Nautical Mile-49

Phase IV - Fuel Flow 50

APPENDIX D: Cruise Performance Computer Program 51

APPENDIX E: Sample Program Results 59

1. Take-off Performance 59

2. Cruise Performance 60

BIBLIOGRAPHY 61

INITIAL DISTRIBUTION LIST 62

LIST OF TABLES

I. TAKE-OFF PERFORMANCE SUB -CHARTS 14

II. CRUISE PERFORMANCE SUB-CHARTS 17

III. TAKE-OFF DISTANCE ACCURACY 25

IV. PHASE I - CRUISE PERFORMANCE ACCUR^^CY 29

V. CURVE FIT DATA 35

LIST OF FIGURES

1. Family of Curves 12

2. Mach Comparison and Decremation 18

3. A-6E Normal Take-off Distance 22

4. Take-off Sub-Chart II Approximated Curves 23

5. Cruise Performance, Phase I 26

6. Error Measurement 32

7. Curve Fit Data Plot 36

I

I. INTRODUCTION

The objective of this thesis was to investigate the feas-

ibility of computerizing aircraft Naval Air Training and

Operating Procedures Standardization Program (NATOPS) flight

performance charts with a high degree of accuracy and to

develop computer programs which are adaptable to hand-held

calculators, desk calculators, or existing internal aircraft

systems computers. The NATOPS performance charts of opera-

tional aircraft used in the Navy and Marine Corps are a

conglomeration of graphs, charts, curves, and tables which

are frequently cumbersome and very susceptible to user error

in their interpretation and manipulation. The charts present

performance information for all aspects of flight operations,

from take-off to final landing. Therefore, it is imperative

that flight crews be able to interpret and utilize these

charts to obtain optimum and efficient performance from their

respective aircraft in all phases of the specified mission.

Mission planning is an essential and key element of flight

operations and must insure a high degree of safety and maximum

utilization of the aircraft performance in order to success-

fully accomplish the mission. This planning requires use of

the NATOPS charts to determine take-off distances, climb

profiles, cruise distances, low-level distances and fuel used

in all mission phases. Such mission planning is very time

consuming and very susceptible to error since it requires

8

I

eye-ball interpolation between curves which are as wide as

the user's pencil lead, and various information is transferred

from chart to chart. For example, even with extremely precise

and careful interpretation of take-off distance, an error of

+_ 200 feet could be made due to pencil width alone, exclusive

of interpolative errors. The combination of these human and

mechanical errors could build to as much as 500 feet, which

would be very critical when taking off at maximum gross weight

on a hot day. Errors in fuel management would be equally as

critical, if the mission profile dictated return to base or

ship with minimal fuel. Also, it is noted that utilization

of the NATOPS performance charts is generally limited to pre-

flight planning and not feasible for use in high performance

aircraft, and in several of the helicopters, for" in- flight

changes in mission profile. A NATOPS pocket check list is

used for in-flight planning, which is small and therefore

subject to even higher human and mechanical interpolation

errors

.

Therefore, a much improved and more accurate method of

mission planning, both before flight and during flight, is

needed. A desk top calculator or hand-held computer with

pre-programmed chips could eliminate these errors during

pre-flight planning, while in-flight errors could be reduced

substantially with performance equations programmed into an

on-board systems computer of the aircraft, or a hand-held

computer with a dedicated or read only memory (ROM) , a pro-

gram chip, capable of storing performance equations for all

mission phases. Not only would computerization reduce the

human errors of flight crews who have the NATOPS charts avail-

able, it would reduce the accident potential of flight crews

who are airborne without NATOPS and make performance decisions

from past experience. Such decisions, without accounting for

all variables which affect aircraft performance, could lead

to large errors in predicting the capability of the aircraft

to safely complete a mission phase; and have been cited as

the cause of a large percentage of Navy/Marine Corps accidents

in both fixed and rotary wing aircraft. Computerization would

also greatly enhance shipboard operations by providing in-

stance Bingo (minimum fuel) profiles to both flight crews and

the air base, reducing fuel management errors based on exper-

ience and interpolation of charts.

This thesis computerizes A-6E aircraft NATOPS take-off and

cruise performance charts [Ref. 1]. However, the feasibility,

procedures and techniques are applicable to any aircraft,

both fixed and rotary wing. The feasibility aspect of the

thesis is a continuing application of thesis research con-

ducted for computerization of A- 7E performance [Ref. 2]. The

programs are written in BASIC language for the Hewlett-Packard

9830A computer, but knowledge of the language is not required

to demonstrate the feasibility of accurate performance com-

puterization. The performance equations within the programs

can be directly utilized in most scientific computer languages,

with easy adaptation to any computer such as hand-held, desk

top, or aircraft systems computers.

10

II. EXPERIMENTAL PROCEDURE

A. GENERAL ORGANIZATION

Two separate, but equally important, A-6E aircraft per-

formance charts were evaluated for computerization feasibility.

The first chart, "Take-off Performance" [Ref. 1, Fig. 11-11],

was utilized since its family of relatively simple, or low

order polynomial equations, provided use of basic curves in

studying and developing the programming techniques. The

second chart, "Cruise Performance," [Ref. 1, Fig. 11-110],

provided a more complex family of curves which required more

extensive analysis in program development. Empirical data

were taken from the specific NATOPS chart and the Least-Squares

method of polynomial approximation was used to determine the

coefficients of each curve in the respective family of curves.

A polynomial expression was then approximated using the same

method to determine the family of coefficients. In other

words, polynomial expressions were determined which approxi-

mated the coefficients of any curve which would fall within

the bounds of the respective family of curves. This technique

precluded the need for interpolation between curves. A

detailed example of the Least Squares method is presented in

Appendix A, while Figure 1 presents a simplified example of

approximating a family of curves.

11

YCl)

XCl)

Figure 1. Family of curves.

Least Squares approximations yield polynomial expressions

for each curve in the form of

Y(l)

Y(2)

Y(3)

Y(4)

^0 " ^11

^20 " ^21

^30 " ^31

^40 " ^41

^1 ^ ^12

^2 -^ ^22

^3 ^ ^32

^4 ^ ^42

where for a... • x., i is the specific curve and j is the

coefficient subscript.

A polynomial expression for each coefficient family is

determined in the form of:

ACO) = b^ + bj_ . C + b2 • C^

12

ACl) = b^ + b, • C + b^ • C^^ -^ 1 2

A(2) = b + b. • C + b^ • C^

where "C" is the curve variable.

Then, the expression for the curve passing through point

A in Figure 1 for X. = C2 r is:

YC2.5) = A(o) + A(l) • X2^5 + A(2) • ^2,^'

The Least Squares method was programmed for the Hewlett-

Packard 9830A computer complemented with the Hewlett-Packard

9266A plotter which was used to verify that the plot of the

polynomial expressions did, in fact, duplicate the specific

NATOPS curve. The accuracy of the polynomial approximations

was determined by evaluating the percentage of deviation of

the approximated curve at specific points from the same points

on the actual NATOPS curve. It was possible to achieve greater

than ninety-seven percent (97%) overall accuracy.

Accuracy of the polynomial expressions for each coefficient

family was determined by comparing the computed value of the

coefficient evaluated at each curve of the family with the

original coefficient approximated for each curve. A more

detailed description of the computer techniques is contained

in Ref . [2]

.

B. TAKE-OFF PERFORMANCE

The A-6E aircraft NATOPS normal take-off performance chart

(Fig. 3] was photographed and enlarged five times its original

size to allow more accurate measurement of empirical data.

13

The chart was then separated into sub-charts composed of

families of curves for individual evaluation of computer-

ization feasibility. The dependent variable or output of

the first sub-chart became the independent variable or input

of the succeeding sub-chart. Table I shows the relationship

of sub-charts to the independent and dependent variables.

TABLE I

TAKE-OFF PERFORMAiNCE SUB -CHARTS

Sub-Chart Family of Curves Parameter Input Output

I Pressure Altitude Temperature ReferenceIndex

II Gross Weight ReferenceIndex

No WindTake-offDistance

III Take-off Distance Wind Take-offDistance

Empirical data were taken from the respective sub-chart,

the coefficient of each curve was determined, and the family

of curves was plotted to verify accuracy with the original

sub-chart. Once the points verified for each curve were

within one percent of the original empirical data, the coef-

ficients of the family of curves were determined. A computer

program was written and the coefficients were verified for

accuracy within two and one-half (2-1/2) to three (3) percent

of the coefficient previously obtained for each curve. The

independent variable for the coefficients of the family of

curves is the specific "family of curves parameter," such as

14

pressure altitude in Sub-Chart I. The dependent variables

are the coefficients of the curve passing through the specific

family of curves parameter. These coefficients are then used

in the output equation of the specific sub-chart.

In Sub-Chart I, "Pressure Altitude," it was relatively

simple to determine the coefficients of the family of curves,

since there were the same linear relationships between each

curve. But Sub-Charts II and III, "Gross Weight" and "Take-

off Distance with Wind," had to be broken down into three and

two sub-families of curves, respectively. This was required

since the non-linear relationship between curves would pro-

duce inaccurate coefficients for the family of curves. By

overlapping smaller sub-families of curves whose relationships

were more linear, the desired accuracy could be achieved. The

range covered by each sub-family of curves was determined by

trial and error iteration until the desired accuracy was

achieved. Coefficients for the sub-family of curves would

then be used in the output equation for the specific sub-chart.

The sub-family coefficients used depend upon the range in

which the desired family of curve parameter would lie. The

computer program was written to check for the limits of each

sub-family range.

Once satisfactory programs were written for each sub-chart,

they were integrated into one computer program which would

yield a final output of take-off distance with wind. The

final program also includes equations which calculate take-off

velocity, and unsafe and not-recommended take-off distances.

15

The coefficients for these equations were determined in the

same manner as individual curves for the sub-charts. A wind

component computation using basic trigonometry identies was

used to input winds into Sub-Chart III, and a round-off

routine was used prior to the final output printout. The

round-off criterion established for take-off velocity was

one-half (0.5) knots, while that for take-off distance was

twenty-five (25) feet. Twenty-five feet was used for take-

off distance round-off, since at high take-off velocities

this value would be negligible in rounding down, but would

still provide a factor of safety for round-up to the nearest

one hundred (100) feet.

C. CRUISE PERFORMANCE

The A-6E NATOPS Cruise Performance Charts are a series of

separate charts or four phases which require that the output

of each chart be transferred as the input to the succeeding

chart until arriving at the final output of "fuel flow."

These four phases are:

Phase I Clean Aircraft Transfer Scale

Phase II Aircraft Reference Number

Phase III Pounds of Fuel per Nautical Mile

Phase IV Fuel Flow

Since Phase I chart consisted of the most non-linear

curves, in contrast to the linear curves of Phases II, III

and IV, and in view of project time constraints, only Phase I

was evaluated for computerization feasibility. This chart

16

was also photographed and enlarged to facilitate extraction

of accurate empirical data. As in take-off performance, the

chart was separated into sub-charts for individual evaluation

of families of curves. This technique varied from that of

take-off performance in that Sub-Chart II had an overlying

family of curves Vhich had to be treated as separate entities

The dependent variable or output of one curve family became

an independent variable or input to the second curve family.

Table II presents the sub-charts with respective curve

families and variables.

TABLE II

PHASE I CRUISE PERFORMANCE SUB- CHARTS

Sub-Chart Family of Curves Parameter Input Output

I

II

III

Pressure Altitude

Guideline

Drag Count

Gross Weight

Mach

TransferScale

Mach

TransferScale

Mach

Determination of the coefficients of each curve, accuracy

verification, and determination of curve family coefficients

were accomplished using the same method described in Take-off

Performance. Sub-Chart I, "Pressure Altitude," and Sub-Chart

II, "Guideline," had to be broken down into two overlapping

sub-families of curves to achieve the desired accuracy. The

range of each sub-family was determined by trial and error

until the desired accuracy was achieved. Coefficients for

17

each family of curves were used in the output equation for

the computer program written for each family of curves. Each

program with sub-families was checked for the range in which

the family of curves parameter would lie to ensure that the

proper coefficients were utilized.

Since Sub-Chart II had overlying curve families, which

required that the input variable, Mach number, would intercept

a guideline at the zero drag count, and proceed down that line

until intersecting the desired drag count line, a comparing

and decrementing routine had to be developed. A graphical

example of this routine is shown in Figure 2.

J

LIHE

HflCH

SEHG

CODHT LINES

Tl II

TRflNSFO? SCRLE

Figure 2. Mach comparison and decremation-

18

In this routine, initial Mach (M0) from Sub-Chart I is

input, as the independent variable, into the zero drag count

equation which outputs the dependent variable, initial transfer

scale (Tj3) . T0 then becomes the independent variable which

determines the specific guideline to be used by establishing

its coefficients. The independent variable in the guideline

equation is Mach, which calculates the dependent variable,

Transfer Scale (Tl) . Tl is then input as the independent var-

iable into the selected drag count equation which calculates

the dependent variable, Mach (Ml) . Ml is then compared to

the previous Mach which was input to the guideline equation.

For the first iteration. Ml is compared with M0 . If the

absolute value of the difference between the two Machs is not

_J.ess than or equal to 0.01, then M0 is decremented, A = .01,

and a new transfer scale, Tl'

, is calculated and input back

into the drag equation to output's new Mach, Ml'. This iter-

ative and comparison routine is continued until the comparison

parameter is satisfied. The compared Machs are then averaged

and output as Best Cruise Mach, M2, while the last computed

Tl is output as Transfer Scale, T. M2 and T are the final.

outputs of Sub-Chart II.

Once satisfactory programs were written for each sub-chart

and verified for accuracy, they were integrated into one

computer program to yield the final output of Best Cruise Mach

and Transfer Scale. The numerical values of these outputs

were not rounded off for aircrew use, since they would be

19

transferred as inputs to subsequent phase charts of the cruise

performance computer program in follow-on thesis work.

20

III. RESULTS AND DISCUSSION

A. TAKE-OFF PERFORMANCE

The A-6E NATOPS normal take-off performance chart. Figure

3, was used to study the feasibility and to expand the basic

techniques of computerizing NATOPS charts. As previously

discussed, each sub-chart was evaluated separately and then

integrated into one computer program. However, Sub-Chart IV,

Obstacle Clearance, was not utilized in the program since

this aspect of take-off is seldom critical for the A-6E air-

craft. Each sub-chart family of curves was relatively simple

to reproduce using the Least Squares method since the highest

order of polynomial equation approximated was third order.

The accuracy of each reproduced curve was verified to be

greater than ninety-nine percent (99%) . A comparison of Sub-

Chart II, Gross Weight, of Figure 3 with that plotted from

the approximated equations. Figure 4, will show that an almost

exact reproduction of the NATOPS charts can be made by Least

Squares approximations. The ordinate system of Figure 4 is

transposed relative to that in Figure 3, since the output

variable of Sub-Chart I is the input variable to Sub-Chart II.

While the individual curves of each respective sub-chart

could be reproduced with a high degree of accuracy, high

IIerrors were generated when approximating coefficients for the

family of curves in Sub-Charts II and III. These errors were

a direct result of the curves in Sub-Chart II becoming more

21

NAVAIR 01-85ADA-1 PERFORMANCE DATATakeoff and Climb

TAKE'OFf DISJAHCE (normal)

AIRCRAFT CONFIGURATION:TAKEOFF FLAPS; GEAR DOWNALL EXTERNAL STORE CONFIGURATIONS

MILITARY POWERHARD DRY RUNWAY

OAH 15 FEBRUARY 1971

DATA BASIS: ESTIMATEDFUEL GRADE: JP-5

FUEL DENSITY: 6.8 LB/GAL

ll|ill|iil|i|ljllijlll

30 40 60 80 100 120

AMBIENT TEMPERATURE - *F

rr30-10 10 30 30 40 SO

AMBIENT TEMPERATURE - 'C

Sub-Chart I

10 12

GROUND RUN- 1000 FT.

I

I Sub-Chart IIII

I

r

Sub-Chart IV

A-ADA1-460

Figure 3. A- 6E Normal Take-off Distance

22

V)

<D

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CJ

t3<D

•MCj

S•HXX O

Ldr-l

lili Ph<

~~*1—

1

1—

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LU U1m _Q

liA :3

U- CO

LJ <-H

rr. M-l

oI

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(30

•H

F9 5 S a S ^ S ? S S 2

133 J - 33NHi5I<I JJD-3:iHi

23

non-linear as the gross weight increased, and of the large

slope changes in Sub-Chart III curves as the take-off dis-

tances increased. The errors in these sub-charts were

reduced to less than two percent (2%) by overlapping smaller

sub-families of curves of more nearly the same linearity and

slope. The gross weights were reduced to an overlapping

family of curves reflecting gross weight changes of ten

thousand (10,000) pounds, while take-off distance was over-

lapped at the mid-curve of the family. The specific overlap

points were determined by trial and error until an acceptable

error tolerance was achieved.

The final computer program which culminated the respective

sub-chart programs into one A-6E NATOPS take-off performance

computer program is included in Appendix B. Sample program

results are in Appendix E, while Table III shows the relative

accuracy between computer and manual chart determinations of

take-off distances for representative input parameters.

24

TABLE III

TAKE-OFF DISTANCE ACCURACY

1 2 3 4

Gross Weight (lbs.)

Pressure Altitude (ft.)

Temperature, Deg . (F)

Runway Heading, Deg.

Wind Direction, Deg.

Wind Velocity

Headwind Velocity

Take-off Distance:

Manual Chart

\ Computer

34,000 •

60

360

360

10

10

1,200

1,200

40,000 •

1,000

100

360

318

20

15

2,550

2,500

48,000 -

5,000

60

360

042

20

15

4,400

4,300

53,000

80

360

060

20

10

4,400

4,400

Accuracy (%) 100 98 97 100

B. CRUISE PERFORMANCE

The A-6E NATOPS Cruise Performance Chart - Phase I is

shown in Figure 5, while all four phase charts are included

in Appendix C. The same basic tachniques used in take-off

performance were applied to cruise performance with exception

of the iterative routine required for the overlying drag count

and guideline curves of Sub-Chart II. As in take-off perfor-

mance, Sub-Chart I, pressure altitude curve family, was rela-

tively easy to reproduce with individual curve accuracies

verified as greater than ninety-nine percent (99^)

.

25

NAVAIR 01-85ADF-1 PERFORMANCEMission Planning

f BliB

CRUISE PERFORtAAHCh

tgmm'su^'^i^'i*^:^

dA<atSti:i^>5^I^-.>^-:

PHASE I - CLEAN AIRCRAFT TRANSFER SCALE

DATE 15 MARCH 1971

DATA BASIS: ESTIMATEDFUEL GRADE: JP-S

FUEl DENSITY: 6.8 IB./GAL

RECOMMENDEDMACH NUMBER FORMAXIMUM RANGEAT DRAG COUNT ^200

.5 .6 .7

MACH NUMBER - M

Sub-Chart II

70"

60<

I 50

Xo

5 30"fie

O20 •

BEST CRUISEALTITUDE ATDRAG COUNT

Sub-Chart I

Figure 5. Cruise Performance, Phase I.

AOf 1-311

26

However, high errors were generated when approximating coeffi-

cients for the curve family due to the increased non-linearity

as pressure altitude increased. These high errors were

reduced to less than two percent (2^) by overlapping sub-

families of curves from zero (0) to thirty thousand (30,000)

feet and thirty (30) to forty thousand (40,000) feet.

In order to achieve accuracies of greater than ninety-nine

percent (99%) for the individual curves of the drag count

family in Sub-Chart II, a high number of empirical data points

were used and the approximated coefficients were limited to

fourth order. This was required to smooth the curves out and

prevent oscillations between points. In addition, some points

in the zero (j3) drag count curve were adjusted slightly to

add additional curve smoothing and better curve reproduction.

While adjustments were necessary to achieve individual curve

accuracy, none was required to approximate coefficients for

the entire family of drag curves, which resulted in greater

than ninety-eight percent (98%) accuracy.

Due to the extreme non-linearities of the guideline family

of Sub-Chart II, the individual curve approximations were

limited to the range that overlaid only the drag count curves.

This restriction was required to achieve greater than ninety-

nine percent (99%) individual curve accuracy, and to limit

the overlapping sub-family of curves to two. The sub-families

overlapped at the fourth guideline, as counted from bottom to

top in Figure 4.

27

Since the Hewlett-Packard 9830 Computer Plotting Routine

requires that the X and Y axis increase from the ordinate,

the transfer scale of Sub-Chart II, Figure 4, had to be in-

verted to reflect values from zero to thirty, vice thirty to

zero. This inversion of the Y-axis caused the approximated

guideline curves to be plotted inverted. Figure 2. However,

this did not affect the output, transfer scale, since the

initial point of entry into the guideline curve family is

only dependent upon the initial transfer scale, i\;hich is de-

rived from the zero drag count equation regardless of curve

orientation. Also, as discussed previously, the iterative

routine required that the approximated drag count family of

curves be transposed. Figure 2, so that Best Cruise Mach could

be determined by comparing input Mach to the guidelines and

output Mach from the drag counts. An attempt was made to

keep the drag count curves oriented the same as the guideline

curves, but the program would not step out of its iterative

routine when the input Mach would not pass through the desired

drag count curve when intercepting the zero drag count curve.

For example, as depicted in Figure 4, an input Mach of 0.73

does not pass through a desired drag count of fifty (50).

The final computer program which culminated the respective

'Sub-chart programs into oneA-6E NATOPS Cruise Performance Phase

I computer program is included in Appendix D. Sample Program

results are in Appendix E, while Table IV shows the relative

accuracy between computer and manual chart determinations of

transfer scale and best cruise Mach for representative input

parameters

.

28

TABLE IV

PHASE I CRUISE PERFORMANCE ACCURACY

' 1 2 3 4'

'i

Pressure Altitude (ft.)

Gross Weight (lbs.)

Drag Count

33,000 .

i

48,000 !

20

25,000 i

50,000

100

15,000 i

58,000

150

37,000

42,000

20

Best Cruise Mach

Manual

Computer

Accuracy (%)

.710

.714

99.4

.620

.623

99.5

.559

.557

99.6

.718

.722

99.4

Transfer Scale

Manual

Computer

Accuracy (%)

13.75

13.65

99.2

9.8

9.9

99.

7.8

7.9

98.7

14.4

14.5 '

I

99.3 i

29

IV. CONCLUSIONS AND RECOMMENDATIONS

Within the scope of this thesis project, computerization

of aircraft Naval Air Training and Operating Procedures

Standardization Program (NATOPS) flight performance charts

has been shown to be feasible with a high degree of accuracy.

The computer programs developed for the A-6E aircraft are

adaptable to hand-held calculators, desk calculators, or the

existing aircraft systems computer. The feasibility, pro-

cedures, and techniques discussed are applicable to any air-

craft, fixed or rotary wing. Utilization of computerized

NATOPS performance charts for before- f 1 ight and during- flight

mission planning will significantly reduce the human and

mechanical errors inherent to visual interpretation and inter-

polation of the charts.

This reduction in chart errors, when coupled with aircrews

having computer access to make airborne aircraft performance

decisions based on all performance variables, rather than on

past experience only, will greatly enhance safety of flight

in completing any or all mission phases. By enhancing safety

of flight through NATOPS performance chart computerization

which is accessible to all flight crews, the operational readi-

ness of all Navy/Marine Corps fleet units will be increased.

Development of NATOPS performance charts computer programs

should be expanded to encompass all phases of mission planning

to include shipboard Bingo (minimum fuel) profiles and tactical

30

weapons delivery. These computer programs could be incor-

porated into hand-held calculators with dedicated read only

memory (ROM) for use in ready rooms and in aircraft without

on-board systems computers, such as most helicopters and light

attack fixed wing aircraft; into desk top computers, capable

of utilizing cassette tapes to program performance equations

for all operational Navy/Marine Corps aircraft for use in land

and ship base operations; and into existing on-board system

computers, for such aircraft which have the capacity for

computer software expansion.

Manual inputs to performance equations incorporated within

on-board systems computers could be significantly reduced by

utilizing signals from air data computers for temperature,

pressure altitude, and velocity inputs; fuel quantity, sensors

for gross weight inputs; inertial platforms for angle inputs

such as angle of attack, climb angles and glide paths; and

weapons selection tapes for drag count inputs.

Future thesis research should address the interface of

existing aircraft sub-system output signals as input signals

to NATOPS performance chart equations incorporated into on-

board aircraft systems computers.

31

APPENDIX A

LEAST SQUARES METHOD OF POLYNOMIAL APPROXIMATION

The Least Squares method of polynomial approximation is

described in detail in reference [5]. Since data points are

subject to experimental errors which may be relatively large

in magnitude, the best curve fit through these points is one

which minimizes the errors. The Least Squares method of

approximation minimizes the maximum errors between the data

points and the curve by making the sum of the squares of the

errors a minimum. The errors are measured by the vertical

and horizontal distances from the points to the line as

depicted in Figure 6.

Figure 6. Error Measurement.

The sum of the squares of the errors is used since the

function has no derivative at the origin and this could be a

single large error. The Ler.st Squares method is also in accord

32

with the maximum likelihood principle of statistics. If the

measurement errors have a normal distribution and if the

standard deviation is constant for all the data, the sum of

the squares can be shown to have values of slope and intercept

which have a maximum likelihood of occurrence.

In the development of an n degree polynomial which will

represent the best curve fit through empirical data, n+1 or

greater pairs of data points are required. Let N represent

the number of (x,y) data pairs. The functional relationship

7 ny=a +aiX + a^x"'+....+a x (1)' o 1 z n ^ -^

is assumed with errors defined by:

^^ A/ 2 ne. =Y. - y. =Y. - a - a,x. - a^x. - ....-ax. .

1 I'^i 1 o li 2i ni(2)

Th e Y. is used to represent the empirical value correspond-

ing to X. of the data pair. The Least Squares criterion

requires that

2 2 2 5! 2S = e, + e^ + . . . . e^ = E e -

1 z^^ i = l ^

2

^ r^ 2 - a x.^ ) (3)= Z (Y .

- a - a^x. - a^x. -.... n i ^ ^ '.,^1 o li 2i1 = 1

be a minimum.

1

The minimum is reached by proper choice of the coefficients,

a through a . Therefore, the coefficients are the variableso =' n

of the problem. At the minimum for S, the partial derivatives

of S with respect to the coefficients are zero. Writing the

equations for the n+1 derivative equation give:

33

N3S/3a^ = = _E^2(Y. - a^ - a^x. -

. . . . -a^x .^) ( - 1) ,

N3S/3a^ = = Z 2(Y.

i-1- a^ - a-,x

.

o 1 1

NdS/dSL = = Z 2(Y.

i=l ^a - a , X .

- .

o 1 1

n. .- a^x, )

(- x.),n 1

(4)

n 1 ^ '^ 1 -^

Dividing each of these equations by -2 and rearranging gives

the n+1 normal equations to be solved simultaneously:

a^N + a, Zx. + a^Zx/ + + a Zx.^ =ZY.,oli2i ni i'

a^Zx, + a-,Zx.^ + a^Zx.^ + + a Zx.^"""'- = Zx.Y.,n 1 1 1

'

1 1 12i-t J\ •

1

2 3 4a Z X . + a, Zx- + a-^Zx- +....+ a Zx.01 li 2i ni n+2 Zx.Y.,

1 i'

(5)

o V ^ n ^ ^ n+1a^ Z X . + a. Z X

.

1 1 1

n+ 2 ^n n+ a^Zx. +...,+ a Zx."" =Zx Y.

2 1 n 1 11All the summations run from 1 to N.

Numerical methods are used to solve these equations simul-

taneously and are discussed in detail in reference [3]. These

particular equations have an added difficulty in that they have

the undesirable property known as "ill-conditioning." The

result is that round-off errors in solving them cause unusually

large errors in the solutions, which are the desired values of

the coefficients, a.'s. Up to n=4 or 5 the problem is not too

great, but beyond this point special orthogonal polynomial

34

methods are needed, which is a separate topic in reference [3]

However, from the experimental point of view, functions more

complex than fourth-degree are rarely needed, and when they

are, the problem can often be handled by fitting a series of

polynomials to subsets of data.

To illustrate the use of equation (^S) , a quadratic is fit

to the data of Table V.

TABLE V

CURVE FIT DATA

Data Pairs

X .

10.0 0.2 0.4 0.7 0.9 1.0

Y.1

1.016 0.768 0.648 0.401 0.272 0.193

Summations for N=6

X .

1

2X .

1X .

1X.41

Y.1

x.Y.1 1

x.^Y.1 1

3.2 2.5 2.144 1.9234 3.298 1.1313 0.74421

The data are actually a perturbation of the relationship

y=l-x+0.2x^ (6)

The curve of equation (6) and the perturbated data are

shown in Figure 7.

35

0.2 0.4 06 0.8 1.0 X

Figure 7. Curve fit data plot.

The summations tabulated in Table V are substituted into

equations (5) to give the following set of equations:

6a + 3. 2 a,o 1

+ 2,5 a^ =3.298,

3.2 a^ + 2.5 a-^ + 2,144 a^ = 1.1313,

2.5 a^ + 2.144 a^ + 1.9234 a^ = 0.74421.

(7)

Using numerical methods of solution to simultaneous

equations yield:

a = 0.9986, a, = -1.0060, a^ = 0.2103.o * 1 '2

Therefore, the equation of the Least Squares approximated

polynomial through the data of Table V is:

y = 0.999 - 1.006 x + 0. 210 X^

36

(8)

The coefficient of equation (8) is slightly different from

that of equation (6) because of the errors in the empirical

data.

37

APPENDIX B

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APPENDIX C

CRUISE PERFORMANCE PHASE CHARTS

NAVAIR 01-85ADF-1

CRUISE PERFORMAm

PERFORMANCEMission Planning

PHASE I - CLEAN AIRCRAFT TRANSFER SCALE

DATE 15 MARCH 197TDATA BASIS: ESTIMATED

FUEL GRADE: JP-S

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Sub-Chart II

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Figure U-U'47

PERFORMANCEMission Planning

NAVAIR 01-85ADF-1

CRUISE PERFORMAm

PHASE II ~ AIRCRAFT REFERENCE NUMBER

DATE 15 MARCH 1971

DATA BASIS: ESTIMATEDFUEL GRADE: JP-5

FUEL DENSITY: 6.8 IB./GAL

Figure 11-12

A0f1-J12

11-18

48

NAVAIR 01-85ADF-1 PERFORMANCEMission Planning

CRUISE PERFORMANCE

PHASE III - POUNDS OF FUEL PER NAUTICAL MILE

DATE 15 MARCH 1971

DATA BASIS: ESTIMATEDFUEl GRADE: JP-5

FUEl DENSITY: 6,8 LB./GAL.

AOfl-313

Figure 11-13

11-19

49

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NAVAIR 01-85ADF-1

t " isas

CRUISE PERfORMAHCE

PHASE 3V - FUEL FLOW

DATE 15 MARCH 1971

DATA BASIS: ESTIMATEDFUEl GRADE: JPSFUEL DENSITY: 6.8 IB./GAL.

as"

11-20

Figure 11-14

AOfl-jU

50

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BIBLIOGRAPHY

1. NAVAIR 01-85ADA-1, NATOPS Fl ight Manual , Navy Model,A-6A/KA-6D/A-6E Aircraft, Commander, Naval Air SystemsCommand, 1 August 1976.

2. Siegel, W. M. , Computerization of Tactical AircraftPerformance Data for Fleet Application , MSAE Thesis,Naval Postgraduate School, Monterey, June 1978.

3. Gerald, Curtis F., Applied Numerical Analysis, p. 284,2nd Edition, Addison-Wesley Publishing Company, 1970.

61

INITIAL DISTRIBUTION LIST

No. Copies

Defense Documentation Center 2

Cameron StationAlexandria, Virginia 22314

Library, Code 0142 2

Naval Postgraduate SchoolMonterey, California 93940

Department Chairman, Code 67 1Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

Associate Professor Donald M. Layton, Code 67Ln 5

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

Major Johnny D. Restivo, USMC 2

Headquarters and Maintenance Squadron-12Marine Air Group-12, First Marine Air WingFleet Marine Force, PacificFPO Seattle, Washington 98764

LCDR W. M. Siegel, USN 1

578 Tamarack Ln.Lemoore, CA 93245

Deputy Chief of Staff for Aviation 1

Commandant of the Marine Corps (Code MEA)Headquarters Marine CorpsWashington, D.C. 20380

Deputy Chief of Staff for Research 5 Development 1

Commandant of the Marine Corps (Code MC-RD)Headquarters Marine CorpsWashington, D.C, 20380

Commanding General 1

Marine Corps Development ^ Education CommandQuantico, VA 22134

Commanding General 1

First Marine Air WingFMFPACFPO San Francisco, CA 96602

62

11. Commanding GeneralSecond Marine Air WingFMFLANTMCAS Cherry Pt., N.C. 28533

12. Commanding GeneralThird Marine Air WingFMFPACMCAS El Toro, CA 92709

13. CommanderNaval Safety CenterNAS Norfolk, VA 23571

14. Director of Aviation Safety ProgramsNaval Postgraduate SchoolMonterey, CA 93940

15. Assistant Commander for Research ^ TechnologyNaval Air Systems Command (NAVAIR-03)Washington, B.C. 20361

16. Assistant Commander for Research 5 TechnologyEngineering DivisionNaval Air Systems Command (NAVAIR-520)Washington, D.C. , 20361

17. Project Manager for A-6/EA-6Naval Air Systems Command (PMA-234)Washington, D.C. 20361

18. Project Manager for Assault HelicoptersNaval Air Systems Command (PMA-255)Washington, D.C. 20361

19. Project Manager for V/STOLNaval Air Systems Command (PMA-257)Washington, D.C. 20361

20. Project Manager for H-53/H-46Naval Air System Command (PMA-261)Washington, D.C. 20361

63

Thesis

c.l

177529Kesti vo

Computerization ofaircraft naval air

ipnrr'^o"'"^ ^"^ Operating^Profcedures standdi^d?^-

tion (NATOPS) flightperformance charts.

•2DEC7813 JUN79

S2AtfG882 70232 6 60 7

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7529Restivo

^

Computerization ot

aircraft naval air

training and operating

procedures standardiza-

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performance charts.

thesR3634

Computerization of aircraft naval air tr

3 2768 002 01315 3DUDLEY KNOX LIBRARY

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