c AFAL-TI-76't47CID

PROMEN - A COMPUTER PROGRAM FOR GENERATINGMOWT PROFILES

REFWRNCE SYSTFMS BRANCHRECONNAISSANCE AND WEAPON DELIVERY DIVISION

NOVEMBER 19713

- TECHNICAL REPORT AFAL-TR-76-247FINAL REPORT FOR PERIOD JUNE 1975 - FEBRUARY 1976

- f u" pub&ii IrWl; ..... L II UL

AIR FORCE AVIONICS LABORATORYAIR FORCE WRIGHT AERONAUTICAL LABORATORIES

SAM FORCE SYSIPsM COMMAND-JWRGH.PATEiR5SON AIR FORSCE BASE, 0OHI 45m

Whe~n Goversont dxwijvga 1 .iecifiooti~ona, or ather data ar* aimed for ma pother than ±-- ~opaamctioza with a defmtizihy~ related Govrimwnt provarammut opuzustiesthe United Statim Govmmmet Varf nomr .smponalhity Aou enV £h*gqtimwhatmoeveri and the fact tta~t the gyzvumst way bae" formulatmod fugiuiahg, or Iam~ way suppliled the said lxawings, .pcfoti~s or otho data, Is not to Jwregarded by impl-ication cc otherwis, as in Any mineur licenwiug the holder or =Vyother ~parmon or corporation, or onow"yla any rights or psrudazsion to mzmfactum,use, or &..%Z any patented iavertion theý iny iA any imp Jo related thereto.

Thin report baa beew rewimmWe by the Infotuation Officoe (01) and is releasable

STANTON 1. M~SICK, Engineer

FOR THE COMNANDM

RONLD L. RIM.O Acting ChiefReference System AranchReconnaissance & Weapon Delivery Division

'A

Copies Of this zoport shotad not he retwmned wa~w rmtin ft reqUird j* pamwityconsideratIons, riltractUal ob,71gatAoMm., CC AptiM o ag epmaiflC &XMMt.AI FRC - 22 O9EC67R 76 - lt00

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, ,PEPORT DOCUMENTATION PAGE READ INSTRUCTIONSDBEFORE COMPLETING FORM

2. GOVT ACCESSION NO. S. RECIPIENT'S CATALOG NUMBERAFA pfR-76-24

Final -f-)PROFGEN - A Computer Program for_•Fnl •1O•

Generating Flight Profiless Jun 4751 - Feb 076 &

7. AUT NORSt 8. CONTRACT OR GRANT NUMBER(*)

JO Stanton H. sick ,

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

Reference Systems Branch AREA 8 WORK UNIT NUMBERS

Reconnaissance and Weapon Deli ry Division 6095 0501Air Force Avionics Laboratory, WPAFB, OH 45433

11. CONTROLLING OFFICE NAME AND ADDRESS

Same as 9 , .. . '200

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Unclassified019.. DECLASSIFICATION/DOWNGRADING

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Approved for public release; distribution unlimited.

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on revers. side if necessery and identify by block number)

Trajectory driver Numerical integration ............ -

Profile generator NavigationComputer simulation Flight path , ,/-/,, / t/

. . "Six degree-of-freedom Wander azimuth \20. ABSTRACT (Continue on reverse side If neceessry end Identify by block numbe*j/

... - ./ c.,This report describes- Acomputer program/that calculates flight pathdata for an aircraft moving over the earth. X'he program ip_!alled PROFGEN,is written in FORTRAN, and isrii-Tn-ded t-o-su-pp6r-simu-l--atdons that requirea six degree-of-freedom trajectory driver. -

(Cont'd on reversi "

DD I J AN 11 1473 TNOF INOV 65 IS OBSOLETE Unclassified- SECUR. rY CLASSIFICATION OF THIS PAGE (When .ate Entered)

C4'-- . .. LI I .. . . . . ,i i i~ i -.... .

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i...PROFGEN computes the position, velocity, acceleration, attitudeand attitude rate of an aircraft flying over an ellipsoidal earth andresponding to maneuver commands specified by the program user. Fourtypes of maneuver commands are available: vertical turn, horizontal

turn, sinusoidal heading change and straight flight. In addition, aspeed change may be superimposed on any maneuver. Extended flightpaths are created by stringing together a sequence of maneuvers.

PROFGEN uses a fifth-order numerical integrator to solve thekinematic equations of motion. , This high-order integrator can operatein a self-analysis mode t i produce a highly consistent set of valuesfor position, velocity, acceleration, etc. In addition to using suchan integre'.or, PROFGEN insures self-consistent and accurate results by(1) adjusting the step size to suit the problem's dynamics, (2) usingthe exact non-linear differential equations of motion, (3) avoiding

integrations that span abrupt rate changes and (4) stopping theintegration process to make output only when required by the user.

PROFGEN was developed on a CDC CYBER-T4 computer where it compilesin about six seconds and uses less than 60,000Awords of memory. Theprogram includes a plotting capability that increases the memoryrequirement to 137,000e when installed.

f t t

I

UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE(Whien Date Entered)IIP

FOREWORD

This technical report was prepared by Stanton H. Musick of the

Reference Systems Branch, Reconnaissance and Weapon Delivery Division,

Air Force Avionics Laboratory, Wright-Patterson AFB, Ohic.

This work was initiated under Project Work Unit Number 60930501

and spanned the period from June 1975 through February 1976. The final

manuscript was typed by Mrs. Shirley Suttman and was originally released

in March 1976 as AFAL-TM-76-3.

Since the initial release in March 1976, one minor sign correction

has been made in the PROFGEN program (see Subroutine GRAVITY in the

listing) while numerous revisions have been made in this manuscript to

correct mistakes and improve its readability.

ii

tWOO

ACKNOWLEDGEMENTS

The author would like to recognize two people for

their substantial contributions to the development of

PROFGEN: Jay Burns for developing and documenting the

equations necessary to maintain flight in a great circle

plane, and for doing the analysis that lead to a companion

program named HEADING (see page 35); and Dave Kaiser for

the writing, debugging and testing of HEADING and of all

the code that produces plotted output in PROFGEN.

The author would also like to thank Shirley Suttman

for her patience and skill in typing this report.

iv

13!

CONTENTS

SECTION Page

I INTRODUCTION 1

II GENERAL CHARACTERIZATION 2

III USERS GUIDE 7

3.1 PRDATA Input 7'3.2 PASDATA Input 123.3 Program Limitations (What Happens If ... ) 213.4 What to Expect from Each Maneuver 23

3.4.1 Vertical Turn 233.4.2 Horizontal Turn 263.4.3 Sine Maneuver 313.4.4 Straight Flight 33

IV ANALYTICAL DEVELOPMENT 36

4.1 Coordinate System Descriptions and Relationships 374.1.1 Frame Descriptions 374.1.2 Frame Relationships: Direction Cosines and

Euler Angles 424.2 Trajectory Equations 48

4.2.1 Direction Cosine Rates: Location andAttitude 48

4.2.2 Angular Rate - Nay Frame w.r.t. Earth Frame 534.2.3 Velocity w.r.t. Earth 58

4.2.4 State Vector 604.2.5 Other Trajectory Relationships 61

a. Specific Force E2b. Attitude Rates 65c. Gravity Model 67

4.3 Path to Nay Rotation Rates and Control Equations 754.3.1 A General Expression for w 764.3.2 Vertical Turn 78

a. w Equation 78b. Control Derivation 79

4.3.3 Horizontal Turn 81a. w Equation 81b. Control Derivation 83

vrIi- l

CONTENTS (Continued)

SECTION

4.3.4 Sine Maneuver 90

a. W Equation 90

b. ip Equation 90c. rýx Equation 924.3.5 Straight Flight 93a. w Equation 934.3.6 Heading Angle Turning Rate for a Great CirclePath 94

V PROGRAM ORGANIZATION 100

APPENDIX A: SAMPLE RUN OF PROFGEN 107APPENDIX B: PROFGEN LISTING 123

REFERENCES 188188 i

i j~ivi

ILLUSTRATIONS

Figure Za

1 Coordinate Frame Geometry 3

2 Sample of PRDATA Input 13,

3 Sample of PASDATA Input 19

4 Two Examples of Constant-Speed Veritcal Turns 24

5 Roll Angle Behavior in a Horizontal Turn 28

6 Two Examples of Constant-Speed Horizontal Turns 29

7 Sine Maneuver Ground Tracks 32

8 Example of Sine Maneuver 34

9 Earth, Inertial and Navigation Coordinate Frames 38

10 Navigation and Path Coordinate Frames 41

11 Relationship of n z, a and i 47

12. Geometry for Deriving p 54

13 Geometry for Deriving i 58

14 Geometry for Deriving Level Gravity 72

15 Balancing Accelerations in a Coordinated Turn 82

16 Roll Angle History (Case A) 84

17 Roll Angle History (Case B) 89

18 Great Circle Geometry 95

19 Macro-Level Logic Flow Diagram 102

20 Numerical Integration of x = F(t,x) from t0 to t 0 +h 103

21 Subprogram Dependency Chart 104-106

vii

TABLES

Table

1 Def~inition of Turn Parameters 202 Azimuth Angle Mechanization sc~hemes 57

vii.i

NOTATION

Subscripts, Superscripts, Prefixes

Equals by definitionEquals approximately

() Physical vector

U Math vector with components in . frame

Tj( )T Matrix or vector transpose

( ) Time derivative

A( ) The change over time of the variable ( )

() Average value

Transformation matrix, frame . to frame k

Coordinate Frames

Frame Symbol Components

Inertial i Xi±Yi, Zi

Earth e Xe 'Ye 'Ze

Navigation n x, y, z

Cardinal navigation - N, W, U

Path p pyp,Zp

ix

I. INTRODUCTION

This report describes a computer program that calculates flightpath data for an aircraft moving over the earth. The program is called

PROFGEN and was written in FORTRAN. Its primary intended use is to

support simulations that require a six degree-of-freedom trajectory

driver.

This version of PROFGEN evolved from one written in 1973 that

became obsolete because it lacked a wander-azimuth capability and

employed an unrealistic roll control mechanization. These shortcomings

are corrected in the revised version and several new features are added

including output at user-determined times. the computation of attitude

rates, an improved gravity model and the ability to turn through a

precise angle without overshoot. In addition the revised version is

coded in a modular fashion for ease of understanding and change.

This report will document PROFGEN in full. Section II is a

general description of PROFGEN's capabilities and l3mitations that

should allow the reader to determine the program's applicability to

his problem. Section III is a user's guide that tells how to construct

a flight profile with the available input parameters. Section IV

develops the equations tnat PROFGEN solves. Section V describes the

program itself. Appendix A presents an example problem and Appendix

B gives a listing of the program source deck.

F 1

II. GENERAL CHARACTERIZATION

PROQ±GEN computes position, velocity, acceleration, attitude

and attitude rate for an aircraft moving over the earth. Position

is given as (geographic) I -itude, longitt.de and altitude (see

Figure 1). Velocity with rcipect to earth is componentized and

presented in a local-vertical frame (x-y-z in Figure 1) that will

be called the navigation frame. Acceleration consists of velocity

rates-of-change summed with Coriolis effects and gravity. Attitude

consists of roll, pitch and yaw, the Euler angles between the path

frame and the navigation frame. These quantities will be defined

precisely in Section IV.

Although the descriptions herein always refer to "aircraft"

flight paths, PROFGEN has applicability to path generation for land

and sea craft as well. In general PROFGEN is suited for simulation

of any craft under continuous control. It is not well suited to

describing bodies in free fall or earth orbit where mass attraction

is the primary forcing function.

PROFMEN models a point mass responding to maneuver commands

specified by the user. These maneuvers are available:

"* vertical turns (pitch up or down)

"* horizontal turns (yaw left or right)

* sinusoidal heading changes (oscillates left and right)

• straight flights (great circle or rhumb line path)

2

N-W-U - Geographic Coordinatesx-y-z - Navigation CoordinatesX - Longitude* - Latitude (geographic)h - Altitude

Aircraft position

Earth's reference ellipsiodwith exaggerated flattening

Figure - Coordinate Frame Geometry

1 3

-£

All horizontal-plane maneuvers are executed in a coordinated fashion.

This simply means that the aircraft is rolled to an angle where the

vector sum of the centrifugal turning force and the force of "gravity"

(32.2 ft/sec 2 ) acts perpendicular to the winigs. Only one type of

maneuver may be executing at any given ti1me but it can commence from

any aircraft attitude. For example, the aircraft may go into a left

turn while in a dive.

In addition to-the four basic maneuvers, the user also has

control of path acceleration by which the aircraft can be forced to

change speeds. Path acceleration may be superimposed over any

maneuver. This would allow, for example, an accelerated diving turn.

PROFGEN is used to create an extended flight profile by stringing

together a sequence of maneuvers chosen from the basic four. The user

specifies how long each maneuver shall last and thereby divides the

profile into flight segments. Up to fifty flight segments may be

strung together to produce a varied total profile. The final values

of the variables in each segment are passed along as the initial values

for the start of the next segment thereby creating uninterrupted time

histories for all output variables.

The program allows step changeL to occur in displacement accel-

eration and in rotational velocity. This produces continuous time

histories for displacement velocity and rotational position (roll,

pitch, yaw) but results in infinite jerk (rate-of-change of displace-

ment acceleration) and infinite rotational acceleration. L~'14

Acceleration, velocity and position are related instantaneously

by integration and differentiation to within the accuracy of the

Kutta-Merson numerical integrator. Every effort has beun made to

configure this integrator to produce an accurate result so that the

output variables form a self-consistent set. Thus the integrator is

fifth order and can adjust its step size automatically to control

the growth of errors. To illustrate, a great circle path from Dayton toMoscow accumulated less than 15 feet of error over its 5000 mile distance.

PROFGEN is limited in its capability to simulate intricate

fighter maneuvers. This arises in part because PROFGEN forces the

aircraft body and path frames to be coincident and thereby loses

the ability to simulate slipping or crabbing motion. Thus, for

example, one could not simulate a fighter aircraft doing a barrel

roll or an Immelmann. On the other hand, one .ould simulate a

complete loop of arbitrary radius since severity of maneuver is

not restricted. In general, PROFGEN can simulate any maneuver

possible with a bomber or cargo aircraft.

The earth is modeled as ea perfect ellipsoid naving values for

eccentricity, semimajor axis length, spin velocity and gravitational

constant equal to those of the DOD World Geodetic System 1972

(Ref. 1). Earth's gravity is modeled as a function of latitude

and altitude, having both radial and level components. This model

is not overly precise (probably no better than 25 micro gees)

and may need revision for some applications.

U n

_____ _____ __IT,

PROFGEN compiles and executes in less than 60,000 words of

CDC CYBER-74 memory. It uses only single precision variables and

all source code is FORTRAN. The program takes six seconds of

central processor time to compile. The ratio of simulated time

to execution time improves as problem dynamics become less severe,

reaching 20267 :1 for straight flight segments but falling to

4 1 for a 10 gee horizontal turn.

6

III. USER'S GUIDE

This section defines the inpuat data that the user supplies to

run PROFGEN. The input data specifies

* initial conditions

* maneuver characteristics

* integrator control

* output control

All data is entered under a NANELIST format that permits the

entry of character strings. A character string is a parameter name

followed by its values written in the user's choice of format

specification. The use of NAMELIST on the CDC CYBER-7Th will be

illustrated in Figures 2 and 3.

Two NAMELIST input data lists are used, PRDATA and PASDATA. The

PRDATA (Problem Data) list contains 15 parameters that remain fixed

for the entire run. These parameters specify all initial conditions

and control output.

The PASDATA (Path Segment Data) list contains 13 parameters that

remain fixed only for the length of a segment. These parameters specify

and describe each maneuver, control the numerical integrator, and

control the output frequency.

3.1 FRDATA Input

Fifteen parameters are entered through the PRDATA list. Failure I

to specify any one of these results in program termination. All

, '7

parameters are single precision and all must be entered In units of jfeet, seconds and/or degrees. The following format will be used

to describe input parameters throughout this section and the next.

II

Parameter (Type) Units (If Any)'

IPROB (Integer)

The problem identification number. It is set by the userfor identification purposes only.

NSEGT (Integer)

The total number of path segments required to complete theentire problem. This number may not en:oed 50 as the programis now configured.

LL.TECH (Integer)

The local-level azimuth angle mechanization index.See Section 4 and Table 2.

LLMECH Azimuth Mechanization

1 Alpha Wander2 Constant Alpha3 Unipolar4 Free Azimuth

TSTART (Real) seconds

The initial time. It is used to begin the problem at anydesired point. It may be negative.

E.71

TO (Real) feet per second

The initial magnitude of total velocity with-respect-to the

earth. VTO must be non-negative.

PHEADO (Real) degrees

The initial heading angle of the path coordinate frame.It is specified as positive clockwise from North. Itsrange is the closed interval [-180., +180.].

PPITCHO (Real) degrees

The initial pitch angle of the path coordinate frame. Itis specified as positive in the upward direction. The pathframe is level when the pitch angle is zero. Its range is[-90., +90.].

ALFAO (Real) degrees

The initial alpha angle. Alpha is the navigation frameheading angle and is specified positive counterclockwisefrom North. Its range is [-18o., +180.).

LATO (Real) degrees

The initial geographic latitude. Its range is the openinterval (-90., +90.). Since the program falters whentrying to compute at exactly 90 degrees, these two extremepoints must be avoided.

9

- j ,•"A4.Z fr d

LONO (Real) degrees

The initial longitude. It has no effect on the problem'Idynamics but is necessary to establish a reference pointfor the calculation of current position. Its range is[-180., +180.].

ALTO (Real) feet

The initial altitude above the reference ellipsoid.ALTO may be negative.

IPRNT (Integer)

Print control index having control, in part, over what iswritten on TAPE6. This tape is considered to be printedoutput. All TAPE6 output is formatted.

IPRNT Action

I Output on TAPE6 at time-intervals specifiedby DTO (a PASDATA parameter)

01 Output at DTO intervals is turned off. j

Regardless of the state of IPRNT, the following output alsoappears on TAPE6:

"date and time

" input data from PRDATA and PASDATA lists

" variable values at start of each segment and at t-final

" error messages

" post-run assessment of numerical integrator performance

10

-.. .. "- - , ,

IRITE (Integer)

Write control index. This output is written on TAPE3 andis designed for compact storage of data for subsequent useby sAiother program. All TAPE3 output is unformatted.

IRITE Action

1 Output on TAPE3 consisting of date, time, inputdata and variable values beginning at TSTARTand continuing at DTO intervals.

01 No output on TAPE3.

IPLOT (Integer)

Plot control index. This output is on PLFILE for post-run

graphing using DISSPLA, a CALCOMP plot library.

IPLOT Action

1 Program plots five graphs, latitude vs. longitudeand time histories of altitude, roll, pitch andyaw. Up to 501 points are plotted in each graph,ihe first bcing at TSTART and all thereafterat IWO intervals.

'1 No plotted output.

ROLRATE (Reel) degrees per second

Norinal aircraf' roll rate. When the aircraft must bank toexzetute 9 coordinated horizontal turn, it rolls to the properbank angle at a rate of ROLRATF, In sine-heading-changemaneuvers, ROLRATE serves as the limiting value for thederivative of roll. ROLUATE must be positive.

j

Figure 2 is a sample of a PRDATA card input set. Note that the

data items may be listed in asy order so long as they all appear

between the beginning identifier, $ PRDATA, and the ending identifier,

3.2 ?ASDATA Input

Thirteen parameters having up to 50 values each ate entered

through the PASDATA list. Each parameter is dimensioned in the

program as a 50 element array, the number 50 corresponding to the

maximum zumber of segments allowed. Each parameter value must be 4

assigned to the array element corresponding to its segment number;

for example, if the output spacing in the sixth segment is to be

25 secords, one would input DTo(6) a 25. roach parameter name in I

the list that follows has the argument i appended to it to indicate

its dependence on segment i, 1 <-i i--50.

Six of the PASDATA parameters (TURM, NPATH, PACC, TACC, HEAD, Al

PITCH) describe the maneuver and four (MODE, ERROR, HMAX, HMIN) are

associated with numerical integration. The other three control output

frequency (DTO), set segment length (SEGLNT), and control initial

conditions (RESTART). Each parameter has a default option that is

invoked in lieu of input data. The default saves the user the

trouble of specifying values that often recur. All parameters are

single precision and all must be entered in units of feet, seconds,

gees (I gee 32.2 ft/sec. 2) and/or degrees.

Wý7_1ý__Y17_77:12

SPROATA IPROS=6509

NSEGT=I7,

LLMECH=29

TSTARTzOo,

V7OIOO0000

PHEAOOZI80*,

PPIrcHOUQ0.,

ALFAO=4599

ALTr =30000.o

LArO= 39.,

LONO=-84*9

ROLRATE=25O.,

IPRNT~it

IR!TE=09

I PLOT=I.

Figure 2 Sample of PRJDATA Inpuxt

13F

.771 'ALI,

Parameter Units (If Any)

SEGLNT(i) (Real) seconds

The time interval of the ith segment. SEGLNT(i) can beany non-negative number, including zero. The programremains in segment i until exactly SEGLNT(i) seconds have Abeen simulated. The default value is zero seconds. I

RESTART(i) (Intetter)

The index number for control of the initial conditions at

the beginning of each segment.

RESTRART(i) Action

1 All variables in the state vector are resetto the conditions that existed at TSTART,namely those in PRDATA. RESTART 1 1 isuseful when one wishes to produce a referenceflight, and a variation of that flight, allin one run.

$i The variable values at the beginning ofsegment i equal those at the end ofsegment i-l.

The default value is zero, no reset performed.

TURN(i)

The index number for the type of maneuver to be used.

TURN(i) Action

I vertical turn

2 horizontal turn

3 sinusoidal heading change

4 straight flight

14

I

All maneuvers begin at the start of a segment. Vertical andhorizontal turns are complete when a specified turn angle isreached. If specified angle is reached and time remains inthe segment., PROFGEN reverts to a straight flight mode(TURN a 4) for the remaining seconds of the segment. IfTURN(i) is 3, a "sinusoidal" path (oscillatory yawing motionin the horizontal plane) is flown for SEGLNT(i) seconds.For sine maneuvers, the user must select a segment lengththat is a multiple of Tp/4 where Tp is the period of thesinusoid. If TbRN(i) is 4, a straight-flight segment willbe flown over a nominal path determined by the value ofNPATH(i) for SEGLNT(i) seconds. Section 3.4 discussesthese maneuver characteristics more fully. The defaultvalue is 4, straight flight.

NPATH(i) (Integer)

The index number for the nominal 2a.th.

NPATH(i) Action

1 Great cricle path

2 Rhumb line path

When a rhumb line path is chosen, the aircraft maintains aconstant heading angle during straight flight periods. Whena great circle path is chosen,the aircraft flies in a fixedplane during straight flight periods. The aircraft maintainsthis fixed-plane flight over the ellipsoidal earth, even whenaltitude changes, by correcting heading continuously. Whennot in straight flight (i.e. TURN = 1, 2 or 3), the rhumbline or great circle actions are superimposed on the chosenmaneuver. The default value is 2, rhumb line path.

PACC( i) (Real) gees

The signed value of the constant acceleration along thevelocity vector, i.e. along the path x-axis. The programconverts PACC(i) in gees to path acceleration in feet/second2

by multiplying by 32.2. PACC(i) may be assigned any real

15

it

value; it remains that value for the entire segmentregardless of maneuver specification. Positive (negative)values cause the aircraft to gain (lose) total speed.Since all active maneuvers (TURN = 1, 2 or 3) require adivision by total speed (VT) to compute acceleration, theuser must assign PACC(i) so VT is never zero during theactual turning portion of such maneuvers. PACC(i) mayforce VT to zero anytime during a straight flight segment.The default is zero gees.

TACC(i) (Real) gees

The magnitude of the maximum centrifugal accelerationduring either a vertical or horizontal turn. The progr~mconverts TACC(i) in gees to acceleration in feet/secondby multiplying by 32.2.. TACC(i) must be positive forvertical and horizontal turns, The default value is zerogees.

HEAD(i) (Real) degrees

HEAD(i) has two uses.

For horizontal turns, HEAD(i) is the desired change inheading angle. Other factors permitting (SEGLNT, TACC,ROLRATE, PACC, VT) this turn angle will be executedaccurately. The magnitude of HEAD(i) may be greaterthan 360 degrees. A positive (negative) HEAD(i) forcesa right (left) turn.

For sine maneuvers HEAD(i) is the maximum variation ofthe heading angle and its absolute value must be lessthan 90 degrees. A positive (negative) HEAD(i) fornesthe sine maneuver's ground track to lie right (left)of the initial ground track. The default value iszero degrees.

16

I

PITCH(i) (Real) degrees or deg/sec

PITCH(i) has two uses. *1For vertiLal turns PITCH(i) is the desired change in pitch Iangle in degrees. Other factors permitting (SEGLNT, TACC,PACC, VT) , this value will be achieved precisely. PIM.CH(i)may exceed 90 degrees. A positive (negative) PITCH(i)forces the pitch angle to increase (decrease).

For sine maneuvers, PITCH(i) is the frequency of the sinusoidalrate of change of 'heading in degrees per second. It mustbe non-zero. The sign of PITCH(i) has no eff-ct on thesine maneuver. The default value is zero in degrees ordegrees per second, as the case may be.

DTO( i) (Real) seconds

The time interval between required output times. DTO(i)

is referenced to zero seconds; e.g., if DTO(i) = 6, outputwould be available at T = (..., -12, -6, 0, 6, 12,... ). IDTO(i) must be positive. DTO(i) controls output frequency

for printing, writing and plotting (see IPRNT, IRITE, IPLOT).Careful sizing of DTO(i) is a necessity, especially whentwo or three output modes are used simultaneously. Thedefault value is 100 million seconds corresponding to nooutput at all.

MODE(i) (Integer)

The index for control of step size in the numericalintegration routine.

MODE(i) Action

0 Fixed step-size integration.

1 Variable step-size integration.

The step size is HMIN(i) when fixed step-size integrationis used. A fifth order numerical integration is performed.

17

:<- .... 2

With the variable step-size mode, the program begins theintegration with a step size of HMIN(i). The numericalintegrator adjusts the step size upwards from there whilekeeping the within-step error below the value specifiedin ERROR(i). If problem dynamics are mild, the step sizecan grow very large, limited finally by HMAX(i). If problemdynamics are severe, the minimum step size may not beadequately small to satisfy the error criterion in whichcase an error message is printed.

In summary both integration modes perform fifth ordernumerical integrations but MODE = 1 adjusts step sizeautomatically to conform to an error criterion. Thedefault value is variable step-size integration.

ERROR(i) (Real)

The allowable within-step integration error. It must bepositive. The default value is 10- , a value that hasproven satisfactory during program development.

HMAX(i) (Real) seconds

The maximum step size when variable step-size integrationis used. It must be positive. The default value is10,000 seconds.

HMIN(i) (Real) seconds

The minimum step size when variable step-size integrationis used. With fixed step-size integration, HMIN(i) isthe size of each step. It must be positive. The defaultvalue is one second.

Table 1 shows the relationship of TURN, TACC, HEAD and PITCH.

Figure 3 is a sample of a PASDATA card input set. Note that some

parameters are not specified because the desired values agreed with

the default option. Also note the capability to specify repeated

values using a repetition factor.

18

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3.3 Program Limitations (What Happens If ... )

PROFGEN will not begin profile generation until each parameter

lies within its permitted range as specified in 3.1 and 3.2. Subroutine

VALDATA range-checks NSEGT, LIS'CH, VTO, PHEADO, PPITCHO, ALFAO, LATO,

LONO, ROLRATE, SEGLNT, TURN, NPATH, TACC, HEAD, PITCH, DTO, MODE,

ERROR, HMAX and HMIN. A message is printed for each range-check

that fails and the program is terminated.

Error messages can also occur during profile generation (i.e.

after TSTART). One such mid-run message occurs if and when the

integrator reduces step size to HMIN and is still not able to satisfy

the error criterion (ERROR). In such cases this message is printed:

THE INTEGRATION ERROR EXCEEDS ITS ALLOWED VALUE

When this occurs PROFGEN is designed to continue to run, doing the

beat it can with HMIN. The value of the result is questionable,

however, and thp best advice is to scrap the output, reduce HMIN

by at least a factor of ten, and rerun the program.

Another mid-run error message occurs if and when the product of

computed roll rate and minimum step size would produce a roll bank

angle in excess of 90 degrees. Since the aircraft must bank to

execute either a horizontal turn or a sine maneuver, excessive roll

angles could occur in either type of maneuver. PROFGEN avoids this

problem in a horizontal turn but succumbs to it in a sine maneuver;

prior to each sine maneuver the program checks for the problem and,

if it exists, prints the following warning message and then terminates

execution.

21* II•2: Li . tS! ~a U~M - .k&~l--~-~t,,~.<~ t&~b~4

CHJCSHC MESSAGE -THE PRODUCT OF COMPUTED

ROLL RATE AND MINIMUM STEP SIZE EXCEEDS

90 DEGREES. BANK ANGLES IN EXCESS OF

90 DEGREES ARE NOT ALLOWED. PROGRAM

Again the solution is to reduce HMIN for that segment. J

Another mid-run message occurs if and when the cosine of pitchJ

is exactly zero. This would happen, of course, if pitch magnitude

were exactly H1/2 radians (90 degrees). At 90 degrees, the algorithm

for computing yaw rate arnd roll rate would make both of these

quantities infinite. ?ROFGEN recognizes the situation and prints

the following warning message from subroutine ETADOT.

ROLL AND YAW RATES ARE UNDEFINED

WHEN PITCH IS 90 DEGREES. THUS

ALL RATES HAVE BEEN TEMPORARILY

ZEROED.

No divisions by zero are attempted so the program continues to

execute. In short PROFOEN handles a pitch angle of 90 degrees

by avoiding the fatal rate computations.

If latitude becomes ! 90 degrees, PROFGEN attempts a division by

zero in LANDOT and suffers a fatal error in which the CDC operating

system kicks the program off the machine. Similar zero-division

failures occur when one attempts a horizontal plane maneuver

(horizontal turn or sine maneuver) with horizontal velocity equal

22V;I

zero, or when a vertical turn is attempted with total velocity equal

zero, or when the aircraft is flovn into the earth's center. Other

zero-division situations would be even rarer than these and are not

worth mentio•iing.

3.4 What to Expect from Each Maneuver

This section desuribes each maneuver in depth to see what it

does and how it does it. These descriptions form the basis for

the development of the control equations in Section 4.3.

3.4.1 Vertical Turn

A vertical turn is a pitch-up or pitch-down maneuver that takes

place in a vertical plane. As with all maneuvers, vertical turns

begin executing at the start of a segment (TI). Pitch angle advances,

at a rate controlled by TACC and aircraft speed, until the time in

the segment runs out at TF or until the change-in-pitch reaches PITCH

degrees at TDONE, whichever time comes first. Altitude, pitch and

acceleration curves for two vertical turns are shown in Figure 4.

Let a represent turn acceleration normal to the flight path.n

PROFGEN holds a ( =TACC fp4 2 ) constant while pitch advances. Sincen

VI"-- ve

the turn's radius of curvature, r, and its advancement rate, 9,

are also constant as long as total speed, V, remains fixed.

23

, • , q ... • s.. _ . _ .. ._. .. -

II I -

i0

4-2,

S 44

'0 .

24,

il- JJ' ' I .•-• ' " -..

SI Iq I I

Turning action is enabled by switching 6 on at TI and then off

at min (TF, TDONE). This produces a pitch-rate discontinuity at

min (TF, TDOE) that the numerical integrator, KUTME, cannot

handle. PROFOEN solves the problem by splitting the segment into

two pieces one from TI to TDONE and the other from TDONE to TF.

(If TDONE > TF, only one piece is necessary, viz. TI to TF.)

KUTMER integrates the two disjoint pieces separately and thereby

avoids a time step that would span the pitch-rate discontinuity.

The switching action on 6 may be observed in the program's

pitch-rate output which is a non-zero constant while pitch is

advancing and zero thereafter. Vertical plane maneuvers induce

no rolling or yawing motion.

TDONE is computed in subroutine TSETUP1 before segment

integration begins. The computation for TDONE assumes two things:

"" turn acceleration is constant

" total speed does not drop to zero

The first assumption is guaranteed by the program's construction.

The user must guarantee the second assumption by choosing PACC so

total speed will remain positive. When these assumptions hold

the aircraft's PITCH angle will advance exactly PITCH degrees

in the interval TI to TDONE as illustrated in Example I of Figure 4.

If TDONE exceeds TI, the change-in-pitch will fall short of PITCH as

illustrated in Example 2 of Figure .4.

The minimum time required to complete a vertical turn through

an arbitrary pitch angle AO is as follows: '

25

i!Ca V0

At ,V~

where At - time required to pitch through A8 radians (>0)

V 0 total speed at TI (>o)

A turn angle -IPITCH I(>o)

an W normal turning acceleration * TACC (>o)

V - tangential acceleration a PACC0

A derivation of this result is given in Section 4.3.2. Equation (2)

is useful for computing flight time in a pitch maneuver.

3.14.2 Horizontal Turn

In a horizontal turn the aircraft heading swings left or right

to force the aircraft to follow a pseudo-circular path over the

ground. Such a turn can be performed in any pitch attitude except

± 90 degrees. Horizontal turns are always performed in coordinated

fashion. (Coordinated turns are also termed symmetric.) A coordinated

turn is one in which the aircraft roll (bank) angle if controlled

so that the vector sum of the horizontal turning force and the

2vertical force of "gravity" (defined for this purpo~e as 32.2 ft/sec2)

acts perpendicular to the wings. For example, in a level one-gee

turn to the pilot's right, the aircraft rolls about its long axis

to a bank angle of 45 degrees, right wing down. Because heading and

roll must both be controlled, the software implementation for the

horizontal turn is more complex than that for the vertical turn.

j*. 26-7

rnw!W ,l* • • W' . . •.3,,'r- ,, .- ,-' S r -.••i.,• I . •:

As was true with pitch in the vertical turns heading advances

in the horizontal turn until the time in the segnent runs out at

TF or until the change-in-heading reaches HEAD degrees at TDONE,

whichever time comes first. Another way to say this is that the

aircraft turns in the time interval between TI and min (TF, TDONE).

During this turning interval, Ohile Leading advances continuously,

roll also goes through its own set of gyrations in order to

implement a coordinated turn. Representative roll curves are shown

in Figure 5.

Note that roll always begins and ends at zero and remains in

the interval (-900, ÷190). Also note that when roll changes, it

does so at the constant rate, ROLRATE.

In contrast to the vertical turn where a was constant, a forn n

the horizontal turn follows a curve similar in shape to the roll

curves from Figure 5. a is given byn

where rl is (constant) pitch and nx is roll. Note that, since n

varies with time, an does also thereby producing a path with a

variable radius of curvature. (The radius of curvature is infinite

at the two ends of the turn and reaches a minimum when bank angle

peaks.) Lat-long, yaw, roll and acceleration curves for two

horizontal turns are shown in Figure 6.

27

Case A - Max roll reached and turn completed

LIrI TOFF rosy rcow, TP

Case B - Max roll not reached but turn completed

71 TA'P T•OE TF

Casc C - Max roll reached but turn not completed

TON ifTOON£•

Case D - Max roll not reached and turn not completed

Figure - Roll Anle Behavior in a Horizontal Turn

28

1 .: ,•- --; -{ -..,,i.L,. •-.:.i' ''r•. .••• -

Latitude -xample 1 Example 2

1'F

/ 00ý

Longitude

Yaw

TX W 7w h) 7- 7X" TOPP TPA "" -Roll

•n

p -

Figure 6 - Two ExAmples of Constant-Speed Horizontal Turns

29- i.

It is apparent from Figure 5 that roll rate has from one to

three points of discontinuity within the segment - one in Case D,

two in B and C and three in A. Again, the numerical integration

problem that this presents is handled by piecewise integration as

explained in Section 3.4.1.

Before integration begins, time points TOFF, TON and TDONE

(defined in Fig. 5) are computed in subroutine TSETUP2. The

condition on TDONE is that heading at TDONE should be different

from heading at TI by HEAD degrees. To compute TDONE, TSETUP2 must

account for variations in both acceleration (a n(t)) and speed. The

exrc-t equations for doing this are very non-linear and have been

approximated in PROFGEN as quadratics in TDONE. If TSETUP2 finds

TDONE is larger than TF it makes T0ONE equal to TF to keep the

turn within the time limit of the segment. Once TDONE is known,

TOFF and TON are easily computed based on max roll angle and

ROLRATE. As in the vertical turn, PROFGEN assumes that speed

remains positive throughout the turn segment, a condition that

the user must guarantee.

The following equation is an approximate expression for the time

required to complete a turn through Ai radians.

At +Z (rofrF-r)v

30 I

,.,,.: ,,••,g,.•.i•[... ,,:•.•., ,v~,.,.,,,, ,,..•, • :'• " • • TM•

r7ý

where

At = time required to turn A# radians (>o)

V = total speed at TI (>o)0

AiP turn angle = IHEADI (>o)

a = normal turning acceleration = TACC (>o)n

V = tangential acceleration = PACC

2(TOFF-TI) = time required to roll into and out of turn

= 2ta-1%f a n ,, ROLRATE2tan-l 32.2 coas-r

ThiL. equation is approximately correct for a turn that rolls quickly

to its maximum bank angle, holds that angle for awhile and then rolls

quickly back to zero (Case A in Figure 5 ). The error in this equation

grows large as Ab and ROLRATE grow smaller and as PACC and TACC grow

larger.

3.14.3 Sine Maneuver

In a sine maneuver the aircraft follows a ground path like that of

Figure 7a. This path results when ground heading, 0(t), is controlled

by the qquation

- * lIt , 4 c70' rp

where A Is maximum heading variation (HEAD) and w is oscillation

frequency (PITCH).

31

V77

rr

II

o I4'

E--4

•I.

4'4

324

".4 -,

:s .. ...:, . . .. . ...• - ,. ,..,. . ...,4.,. .... - ,12•.; iK .',,: w " • .... . . • • n

Repeated cyles of 7a are shown in 7b and are produced by simply

iterating the above equation to yield a longer maneuver similar to

Jinking. Note that neither 7a or 7b are properly scaled.

A sine maneuiver may execute in any pitch attitude except 900

degrees and is always performed"in coordinated fashion. Againtheading

and roll must both be controlled but the governing equation is the

one for heading given above. The companion equation for roll that

produces coordinated maneuvers is

where V is total speed. Since rx has no discontinuitiesthe numerical

integration can proceed uninterrupted ;,-" the sine maneuver thereby

avoids complex event-time calculations like those for a horizontal turn.

Figure 8 shows ground track, roll and heading curves (to scale) for a

sine maneuver where A is -20°, T is 10 seconds, V is 1000 fps andp

SEGLNT is 12.5 seconds. Note that roll passes through zero at

multiples of T /4 seconds so that the aircrafts wings are level whenp

the segment is finished at 12.5 seconds.

3.4.4 Straight Flight

Complete straight-flight segments occur when TURN is 4 and partial

segments occur anytime a vertical or horizontal turn has reached its

max turn angle with time remaining in the segment. Neither roll nor

33

I

y(ft)

800

6oo

I4oo

200

5000 10000

yaw (degrees)20

10

0 P t (seconds)

-10

1

-20

roll (degrees)80

0 n t (seconds)

-4o0

-80

Figure 8 - Example of Sine Maneuver

34

J'

Pitch vary in straight flight segments and heading is governed by the

users choice of nominal path (WpATH). Heading is constant over a

rhumb line path whereas, for a great circle path, heading must vary

to keep the aircraft in the great circle plane. Rhumb line flights

that continue long enough spiral in on one of the earth's poles and

end up causing a division-by-zero failure.

Total speed, which had to remain positive during turning maneuvers,

may be zero in straight flight segments. At such times, aircraft

position is fixed and attitude is that which existed just prior to

speed becoming zero.

To aid the user in constructing straight flight segments between

locations over the earth, a program called HEADING has been written.

In response to user inputs of lat, lon and altitude at origin and

destination, HEADING computes the heading angle at origin neet. -1 to

reach destination over a great circle path. HEADING also computes the

great circle distance from origin to destination. HEADING is a

double precision FORTRAN program that can be made available to

interested users.

354

IV. ANALYTICAL DEVELOPMENT

This section develops the equations that govern the trajectory

of an aircraft under continuous control in the earth's gravity field.

These equations can be conveniently divided into two groups, control

equations and trajectory equations, which are related schematically

as follows: H(LocationControl Trajectory Velocity

Input Data -Specific Force

Equations Equations SpAttitude

'Attitude Rate

The control equations are the relationships that specify turn rates

according to the user's input data.

The trajectory equations are a collection of differential and

algebraic equations that produce position, velocity, specific force,

attitude and attitude rate in response to the imposed control. They

are, in short, the equations of motion for a body free to move in

six directions in inertial space.

The trajectory equations are kinematic relationships, i.e. they

deal with motion in the abstract without reference to force or mass.

Since force/mass concepts are immaterial, PROFGEN avoids all aircraft-

" specific considerations such as moment of inertia, aerodynamic force

and thrust force. It follows that the aircraft modeled here is a

weightless body that can be displaced and rotated, without restriction,

to suit the users, demands.

36

I,.I '

*-

In the following development thoue equations that became part

of the actual code in PROFGEN have stare (*) beside their numbers.

14.1 Coordinate System Descriptions and Relationships

The coordinate systems of particular interest in this report are

the inertial, earth, navigation and path systems. These four systems,

or frames$ will be defined shortly as right-handed orthogonal frames.*

The relationship of the earth and navigation frames will determine

aircraft location (longitude, latitude, alpha) while that of the

navigation and path frames will determine attitude (roll, pitch, yaw).

Loca't'ion and attitude data will be carried in two direction cosine

matrices (C n and C n) that describe the rotations between pairs ofe p

coordinate frames. The subsequent portions of this section describe

the four frames, define the two direction cosine matrices and delineate

the extraction of location and attitude angles from each of7 these

matrices.

Frame Descriptions

.Inertial frame (i frame: X,, Y, Zi axs

The inertial frame has its origin at the earth's center

of mass and is non-rotating relative to the stars. This

frame is important mainly as it applies to the computation

pecific force. Its relationship to the earth frame

in portrayed in Figure 9.

I ____3T

x xe

Position

ii

Figure 9 -Earth. Inertial and Navigation Coordinate Frames

38

Earth frame (e frame: X Y Z axes)

The earth frame has its origin at the earth's center of

mass and has axes fixed in tht earth, Figure 9. Axes

Ye1 Yi, Ze and Zi all lie in the earth's equatorial plane

while axes Xe and Xi are coincident, passing through both

poles. The rate of rotation between these two frames is

the earth sidereal rate, designated Q. WGS-72 (Reference 1)

gives this value for S which is denoted WEI in PROFGEN:

2411 a 0.7292115147 x 10" rad/sec

Navigation frame (n frame: x, y, z axes)

This locally-level frame has its origin at the aircr.aft A

center of mass with x and y in a plane tangent to the

reference ellipsoid and z perpendicular to the ellipsoid,

Figure 9. (Center of mass and center of rotation are

coincident in this development). PROFGEN solves the

trajectory equations in the navigation frame. Aircraft

location is specified relative to the earth frame by the

three-tuple (X, 0, a) where X is longitude, 0 is geographic

latitude and a is the navigation frame heading angle,

referred to variously as alpha, wander angle or wander

azimuth angle. Figure 9 shows that 0 is geographic latitude,

not geocentric latitude. Thus z is normal to the elliptical

I 39

surface of the earth rather than in the directicnn of the

earth center. The values for A,, *.etnd ot will be computed

from the direction cosine matrix C.e,

Path frame (p frame: x P, y p z paxes)

The path frame, depicted in Figure 10, has its origin at

the aircraft center of mass. It takes its name from the

fact that the x-axnis follows the aircraft path by staying

aligned with the total velocity vector, V.(,velocity Ji

with respect to the earth, will be defined precisely in

section 4.2.3)

In general V is misaligned from the aircraft's longitudinal

axis by an angle of attack and a crab angle. In this

developmnent we assume these angles are zero. The effect

of this assumption is to weld the path frame to the

aircraft's body thus causing x to pass through thep

aircraft nose and y p to point out the right wing.

Z points down in level flight but rotates aboutxp

during coo~rdinated turns so there is never any maneuver

acceleration along yp

Since path and body are coincident, the familiar body frame

terms of roll, pitch and yaw will be borrowed to describe

the Euler angles between the path and navigation frames.

Roll, pitch and yaw are denoted fli n and nand arex y

4o0

-7 ,

i

plane

nI

y •

II

North

:I

Note: Origin of path frame displaced from that of nay frameonly for clarity of diagram; they are actually coincidentat aircraft center of mass.

Figure 10 - Navigation and Path Coordinate Frames

I: ~i1

measured around x , yp and z respectively. A right turnpp p

produces a positive yaw rotation, a pitch up is a positive

pitch rotation, and a clockwise roll (as viewed from behind

the aircraft) is a positive roll rotation. The values of

•x, ny and nz will be computed from the direction cosine'

matrix Cn.p

4.1.2 Frame Relationships: Direction Cosines and Suler Angles

0 Earth and Navigation Frames

Figure 9 presents the relationship between the earth and navigation

frames. When A, 0 and a are zero, the navigation frame is directionally

aligned with the earth frame. Beginning at the aligned position, the

rotations necessary to go from earth to nav coordinates form the

direction cosine matrix Cn. This matrix is the ordered product ofe

three individual matrices describing these rotations: an x rotation

of X degrees, a y rotation of * degrees and a z rotation of a degrees.

Using an "s" prefix for the trigonometric sine and a "c" prefix for the

cosine, Cn ise

-c 4W X X 4A. (r

40 6 4j 0 -4 A

"42

Now if the elements of Ce are identified as

; I t CEA', C ("¼ ICaN,, CENIS C EN6

LCEA-, CENza CEAl4t

then the individual elements are "

CEN, 1 AN(A 0** CL

cEN,, ~,, ,• ,v . - Ccd• 4o u'4' 4 •6•A. "

CCAI,,

CEM~j A'Cc4~t XA. 01 AWk 040

CEM 4M 40V ALeX74~ ~( ~

K * Coded for implementation in PROFGEN.

43

F .- • III _ A, .. .... ...

tIt4

To extract latitude, longitude and alpha from the elements of Cne

the following calculations are made

I

where the initial values for ¢, A and a are

LATO

S= LONO

a=ALFAO

The FORTRAN functions SIN ( ) and ATAN2 ( , -) were used to implement

(10), (11) and ',12) because their range agrees with that desired for

and a. An important aspect of the computation for A in Equation

(11) is that 0 e [-ff/2, ff/21, which means cos (4) is always positive,

which in turn makes the sign of CEN and CEN depend solely on X,~32 33

which removes any doubt as to the quadrant where A lies. A similar

statement applies to a as computed in (12).

44

* Path and Navigation Frames

Figure 10 presents the relationship between the path and navigation

frames. Beginning at the nonaligned position shown there, the ordered

sequence of rotations necessary to form the C~ matrix is as follows:

a roll about x of n degrees to get the wings level; a pitch aboutp

yof fl degrees to get the nose level; a yaw about zof ii degrees

to align the x and x axes; finally, a flip about x of 1800 to alignp P

zP, which is nominally down, with z which is always up. Thus

/ *0 Ct af o 4;.), A

nNow if the elements of C are identified as

C II 11

CAM2, CP/aj CPk,.('4

CPN31, CR PA CPA',,

+4i

then the individual elements are

CPR co1 = eosry

CR22 =-in siflrlinX COST)z

CR111 on sinn32. y

12z . ity xOT +snz si K

CPRI 23 osin sint) sinnr S - COT)22 z y xos01'

CPR13 ~COST) sny xO~

33 yx 24

nIR0ll, pitch and yaw are extracted from the elements of C~ as follows:

p

I.L

where the initial values are

l rix = 0

ny = PPITCHO

• = ALFAO + PHEADO

Again SIN (- ) and ATAN2 ( - , ° ) were used to implement (16), (17)

and (18). As with X and a, the key to the computations in (16) and

(18) lies in the fact that ny has a restricted range which makes its

cosine always positive. The relationship between a, nz and • (heading)

is illustrated in Figure 11.

North

x

a

l1z a +i xp

West i

y

Figure 11 - Relationship of n a and *

47

4.2 Trajectory Equations

Sections 4.2.1, 4.2.2 and 4.2.3 will develop first order

differential equations to describe the motion of a body in six

degrees of freedom. Section 4.2.4 defines the states of the

state vector, x. The companion algebraic relationships for

specific force, attitude rates and plumb-bob gravity will be

developed in Section 4.2.5.

4.2.1 Direction Cosine Rates: Location and Attitude

At least three methods are available for keeping track of the

rotation angles between frames, including direct integration of the

Euler angle rates, propagation of four quaterion parameters representing

a complete direction cosine matrix (Reference 5), and propagation of

the direction cosine matrix. The last approach was chosen for

PROFGEN because of its simplicity and versatility. This section

derives a general expression for the direction cosine rate and

then displays the result in notation appropriate to Cn and Cn.e p

For any two frames, a and b, the Theorm of Coriolis can be

written for any vector u as

- 6 d/-, i:i

"...1

48_

This equation is in "physical vector" form. It states that the

time rate of change of u, as observed in the a frame (i.e. with

respect to the a frame), equals the time rate of change of u, as

observed in the b frame, plus the angular rate of change of frame

b with respect to frame a crossed ont6 u. The addition and

multiplication in (19) are physical-vector addition and physical-

,rector cross multiplication. When (19) is coordinatized in the a

trra-,e, these "math vector" relationships follow:

4?

or

49

L .:27, ....... .

where %ais a "cross-matrix"' that produces a result on a math

vector identical to that of cross multiplication on a physical

vector. B a is defined below. Continuing

Ai4 4 J

A ~ &

Equating (20) and (21) yields

and, since ui is any vector, it follows thatI

50

where

b. 0(4)

The specific notation chosen to implement (22) and (24) for

C and Cn is shown below:e p

o - "e

0_

where

vie WO

51

e•% = -3.

"Also o

-4' 0

where

For writing convenience, ne and n will be referred to hereafter-ne --p1

as . and w. In (25) and (27) we have expressions for keeping track

of location and attitude provided p and w can be computed. Sections

4.2.2 and 4.2.3 deal with p. The computation for W will be given

in Section 4.3 where turning rates are discussed.

52

Mi =-777,2

4.2.2 Angular Rate - Nay Frame w.r.t. Earth Frame

p is the angular rate of the nav frame with respect to the earth

frame. The fact that the x and y axes of the nav frame remain tangent

to the earth will be used to derive expressions for px and p y. pz will

be determined by the users choice of azimuth-angle mechanization.

Consider the geometry of Figure 12, a section of the earth

ellipsoid, where V and V. denote North and West velocity components.N

The North-West-Up (N-W-U) frame differs from the nay frame only by

the rotation a. If V is earth frame velocity, its navigation frame

components are denoted

~, A

Then from Figure 11, VN, VW, Vp are given by

N Wo UP

SV AX o o o

= Am A 0 (30)

LL

53

Polar Axis

I VN

IU

I ii5h

'"W-

Figure 12 -Geometry for Deriving p ••

r- ,* 1 .. .. .. i . . .i i - .. !

The angular rates required to keep the N-W-U frame level over the

earth ellipsoid are deduced from Figure 12 as

Vw

SVN ~(3:) !

ew

where h is altitude above the ellipsoid, Rm is the radius of curvature

of an ellipsoid meridian line and R is the radius of curvature of thep

ellipsoid in a plane through the normal and at right angles to the

meridian. (It can be shown that R is the distance o'• where c liesp

on the polar axis.) R and R vary with 0 according to the followingm p

equations (Ref. 2, pp 168-170):

R 6- e') c)" !

"a 1-1

SOF

551

where

2 2 Rf-be eccentricity 2 ±e.---• O.o06694317778 (WGS-72 data)

= semimajor earth axis = 2092564o feet (WGS-72)

b = semiminar earth axis a 20855481 feet (WGS-72)

PN and pW lie in the x-y plane of the nay frame and can be resolved

into components along x and y as follows:

W eN C( + 4 (

The general relationship between pz and a can be deduced from the

geometry of Figure 12 as

The value for & depends on the azimuth angle mechanization (LLMECH)

desired by the user. The various choices and the resulting p values

are tabulated in Table 2.

56

LLMECH Name - .1

1 Alpha Wander -X sin 0

2 Constant Alpha 0 A sin *3 Unipolar j t A (sin -J)

4 Free Azimuth -(Q + A) sin cpt -Q sin

t J sign($) 0.7292115147 x 10-4 rad/sec

Table Z - Azimuth Angle Mechanization Schemes

iFigure 13, a section of the earth, is drawn so VW is perpendicular

to the paper at the indicated point. (Note again that R terminatesP

on the polar axis.) Examination of this figure shows that the

equation for A is

Px' p and pz, as derived in this section, depend on a, $, Vx, Vy and

Vz * aand 0 can be obtained from Cn using Equations (10) and (12) whileZ e

expressions for V , Vy and Vz will be derived in the next section.

.•:•;57

Polar Axis

(R +h) cos Vp

hh

R

i-i

Equatorial Axis

Figure 13 Geometry for Deriving X,

4.2.3 Velocity w.r.t. Earth

Referring now to Figure 9, the vector R connects the earth's

center with the airclaft location at all times. By definition of V

and by Coriolis' Law

Alei

I 58

Coordinatize (39) in the nav frame and use (22) and (28) to produce

the following equivalent expressions in math-vector form:

¢<+:',," " + ,-, "'j'\,W \

I) 4) .zV~eeh4d 0 V4o

Since velocity in the path frame lies entirely along the xp axis

0

59

.. ,-,, -.tt.~i a. =..

Substituting (42) in (40) and writing out the individual equations

yields

=CPNU VT -V

' !V6=~ CPNIvT4V ýr

VT will be recognized as PACC, the path acceleration needed to alter

the magnitude of V.

P'ACC

In (43) we have a differential equation for earth frame velocity that

Tdepends only on factors already specified save for w = (Wx w y Z)

To repeat, w will be derived in Section 4.3 xy,

4.2.4 State Vector

PROFGEN carries a state vector, x, Icontaining 23 states in a 23

element, labled-common array named STATE:

x (V V V VT h CPNll CPN .. CPN c c ... CEN ) T

11 y 21 CP33 CE11 2EN1 33 2 3X1

I.

60 • Li

The appropriate differential equations for the elements of x are

Equation (43 ) for the velocity components Vx, Vy, Vz; Equation (44)

for the total velocity VT; Equation (25) for attitude data in Cp;

nEquation (27) for the location data in Ce and this differential

equation for altitude, h;

4.2.5 Other Trajectory Relationships

The following three topics are discussed now to conclude the

derivation of the trajectory equations:

a. Specific Force

b. Attitude Rates

c. Gravity Model

Topic c supports topic a. Topics a and b are important only insofar

as they provide a way to compute specific force and attitude rate for

PROFGEN output. Specific force and attitude rate are algebraic

expressions not required during state vector propagation; therefore,

in some sense, these equations lie outside the mainstream of

PROFGEN' s calculations.

A6

• 61

a. Specific Force

Specific force, F, is the acceleration that a velocity meter

(accelerometer) aboard the aircraft would detect. Specific force

is the total inertial acceleration minus the mass-attraction

gravitational acceleration; i.e. specific force is the second rate

of change of R as viewed by an observer fixed in inertial space,

minus mass-attraction gravity, k.* The physical vector equation

for this (see Reference 3, p. 121), where + and -are physical

vector operations, is

Recall from (38) that

- - (38)

The e frame rotates at rate (,e ( =(Q 0 0 )T) with respect to the

inertial frame so we can write

'2 xR

W7 wa;_ /e

'I62

AI

The navigation frame rotates at rate T (Z _p + Q) with respect to

the inertial frame. Differentiating (147), substituting the result in

(46), and continuing with the expansion gives

dtt

C/ V~C~ A ) 6,

(.5)

4I-J

4*1 7L. itV 7-cA)

I,,,.,

63

where we have used the fact df_/dtIl e-0 and where

- i eq

The vector g is the usual plumb-bob gravity composed of both mass

attraction and earth rotation components. The vector E points

downward. Recalling from Section 4.2.3 that

-1 (V. .

we can componentize (48) in the nav frame (subscripts x, y, z) as

follows

FA,

W5 F Vg 2D.)2 V7 -4 fzn Vx ~L

S64:. /6wi

where

Gravity (gx, gy, gz) will be discussed presently. Ox, y and Qz

are the projections onto the nay frame of the angular Nelocity

between the earth and inertial frames; these quantities are not

related to w x, w and wz except that both share the greek letter

"omega",upper and lower case. All other quantities needed for

computing (50) have already been discussed.

b. Attitude Rates

This section deri-es an expression for each Euler angle rate

(X.' Dy, ;z) as a function of the commanded turning rates, wx wy and

w . Recall these formulas from (15) and (27):z

cPA(,~(521)

65 1

7 7.. .] - .;

CPA, 15 =-o CPN, + 4b. cPA'a, cR:)

Differentiate (52c) to get

CPN31 (cos •y) ry

and equate this to (52f) to get

-wy CPN +w O cPN = (cos Iy) y

l•Y (co s nl CO s n y) + w (-sin n, COs ) = (Cos Iy) y

Assume now that pitch is not i900 and cancel cos ny to yieldy

I 66

.. .... --

which is the desired expression for ry. Similar manipulations of

(52) produced the following expressions for and :

PROFGEN does not attempt to make attitude rate calculations when

cos n 0. It simply prints a warning message and goes on (seey

Section 3.3).

c. Gravity Model

Throughout this report the ellipticity of the earth has been accounted

for while higher order effects and local geoid perturbations have been

neglected. The purpose here is to derive equations for gx, gy and gz

that are consistent with this philosophy for modeling the earth. The

normal component gz will be tackled first following the approach

beginning on page 78 of Reference 4.*

n Derivation for Normal Gravity, g

Define y as gravity normal to the ellipsoid at altitude zero.

Then for an altitude h above the ellipsoid, g at this altitude can be

expanded in a MacLaurin series of terms in h:

67

• ! .5:Z-

iL 2

2-d93). Takinh iscbased aln anelpoda exadninathimodel:eresgie

RIO/

whreuRcatnd ths areathepioncipl radiig ofem craturepin deinherode b te(33)

anodue (3h) Takoing r~eipoult n adn nabioilsre ie

RRIO

Trnctig heeeqaton, din tem addrppnghihr rdr6ers

S• , m'y,,w wr•, r• • ' • m • ,o,• . '"" ,'., .. . . .- ., . . . .. 1A .. .. . •

.i

41,

I

If Ye is the value of y at the equator at hno, the first order2

relationship between ye and Q is e - mye /Re where m is 0.003449783.

Substituting this and (58) in (57), and simplifying, yields

1~J,

The second .,.tive aVah may be taken from the spherical

approximation obtained when earth flattening is neglected entirely.

Then according to Newton's law of mass attraction

2

yk/E

where M is earth's mass and k is the uni •ersal gravitational constant.

ah 8R R3e e

2 22_X D 6kM2 2 4,

e e SIso that

2

ah Re

69

Combining (59) and (60) with (56) produces the desired approximate

equation for normal gravity.

+ e a'

where y is gravity a, the ellipsoid surface which is given in

Reference 1, page 22, as

U ire 0. OOSZ7jI l4 4 000 IL 0060eo346/A14W ) (

do= -32. 08 77057 ('3) *1

Combining (62) and (63) with (61),and evaluating all constants,

produced this final expression for g:

08 77SI 0d/1fJ?c9/'0" 0 0 #40OW i

U Derivation for Level Gravity, gx and gy

At first it is somewhat surprising to realize that plumb-bob

gravity has a level component. Such component arises because level

surfaces at different altitudes (but same latitude) are not parallel.

This fact is evident when one considers these two extremes: at

70,,., I .. ;

hoo the level surface is the ellipsoid and gravity points along

0; at the same latitude but elevated to h-c*, gravity points at

earths center of mass. Between these extremes the difference inA

slope of the two gravity vectors is the difference between

geographic and geocentric latitude.

Another way to view the level gravity phenomenon is through

the curvature of the norm~al. plumb line as illustrated in Figure 14.

Curvature is zero in the east-west direction owing to the rotational

symmetry of the ellipsoid of revolution. Thus level gravity is

entirely a north-south acceleration.

From Figure 14, observe the following relationship

The plumb line's radius of curvature, r, 16 given by (2-22a) in

Reference 4:

with (65) an raranin

1131

1 71

S~.~F.7..-

reference ellipsoid aircraft position

Fitgure 14 -Geometry for Deriving Level Gravity

72

... .. .. ..

The change in plumb line direction, 8, between h=o and h=h is j

An approximate relationship for d is d R R. Then 'e

Thus (63) becomes'

To obtain a closed form expression for (69), simplify g a3 follows:

Re,:

"/ /"--Z A

ni,,

O~r (1-74 4 A.A/eeeM

73

Lakt.; j, , 4k2

Then

Substitute this in '69) and integrate

0 i

h _ __ _ (740)

Re A

B is the tilt angle through which the gravity vector tips over as

altitude increases. Projecting the magnitude of gravity (approxi-

mated here as lye) through a and onto the level surface gives for

gn (g north)

39 (70)

Now rotate gn through a to obtain gx and gy

4 V (73)

74

which may be stated in terms of the elements of C as

W /0~,i a l CEV 3 CEA',N, (74)

=1( C1 -N~A E~ CF/V, I

This derivation of level gravity was based on material in

Section 5-6 of Reference 4 where it is pointed out that the effect

of topographic irregularities on the curvature of the plumb line

often overwhelms zhe value from equation (71). In high mountains

the actual deflection could be 10 times greater so the limitations

of (71) are apparent.

I

4.3 Path to Nay Rotation Rates and Control Equations

The relationships derived here for w will produce turning rates

commensurate with the input data and with the restriction that level-

plane turns be coordinated. In addition, equations for controlling

the application of W will be derived. This control will usually take

the form of a switch to turn W on or off at a critical event time.

The control equation will compute the event time; e.g. the time atwhich ý should be disabled in a vertical turn to make An_ = PITCH

y sThis section evolved from the work in Section 3 of Reference 6.

75IIL

'I

As a preface, we list some basic kinematic equations for the

illustrati'r, below where S is arc length, V is speed tangent to the

path, r is radius of curvature and a is acceleration normal to then

curved path:

flight ath

cis

At

Combining these equations produces this relation for angular rate

a.i

I '4.3.1 A General Expression for w13

Now recall equations (13) and (28) defining Cn and w:p -0. . -f4 ff, : 1 . 4 , . .

oo /0 0 4t t-i 0*,$°'" '3'

0 a

76

iUS

(4 J

where TI 8 0 , Tz, T and T are introduced here for reasons that will be18'z y x

apparent shortly.

Each Euler angle, nxi, ny and nz, has an associated rate, x ny

and zj " For a given Iath to nay orientation, the vector associated

with n is directed along x . If the given path frame is rotated so

that roll is zero (nx=o), the vector associated with ry is directedx y

along the new (, rolled) y axis. When the new path frame is rotatedp

again to remove pitch (n y=o), the vector associated with r1z is

directed along the new (unrolled and unpitched) z axis. Note thatPthese three vectors are not mutually orthogonal. When transformed

into the nav frame and added vectorially, they give the entire path

to nay rotation velocity. Thus

S -

's~x 0

____ ____ ___ ____ ____ ___77

This may be simplified vsing n c= + ip (Figure ii) and the definitions

of TI 80 T and Cn in (78):

0 40 (g)T. .

Equation (80) is the most general relation for angular rate between

the path and nay frames. It will simplify considerably depending

on (1) the type of maneuver (2) the nominal path (great circle or

rhumb line) over which that maneuver is superimposed, and (3) & which

is given in Table 2 as a function of the nav frame mechanization

choice. In the following four subsections it is assumed that the

reader is familiar with Section 3.4.

4.3.2 Vertical Turn

a. w Equation

Since the aircraft's wings remain level in a vertical turn, T = Ix

and n = 0. Thus (80) becomesX

00 0

78

or

where fy is given by (77) as

AlLall, (ale)

41r

and

VT(t) Vr(~ U + (t-tZ) '4 B

a. TACC • , (P/rcI ) (O) *

Note that TACC is positive and in units of ft/sec2 . A vertical turn

will have a slight heading rate if the aircraft is following a great

circle path so J 0 I humb linegrea (PSI)

)great circ~e (Section 4.3.6)

b. Control Derivation

Equation (82) can be integrated to yield change in ny over the interval

(ti, t). In the case where V varies linearly with time (• = PACC # 0),ST ýTA

79I.M

n*,nr' . Vt'",I I I iiii ~ r niii ~ i *'-........2W~ f -, _ i~v~ ~ i ;i•v • • - I

'--

t I

ereez I

In the case where V, is constant

t

LI7~~),/ Wr =~A(t -tC ) Vr'~ (*47)Vr Yr

Equations (86) and (87) may be inverted to compute a time, t TDONE,

when Any (TDONE) = IPITCHI:

rD + -PA V 0

(Sit)

80

4.3.3 Horizontal Turn

a. w Equation

Since the aircraft does not pitch in a horizontal turn, Ty is zero

and (80) becomes

where nbehaves as pictured in Figure 5. isethronoofx x

When on, nx -+ ROLRATE.) is the sum of , the nominal path

contribution from (85), and PM) the maneuver contribution due to

TACC:

, rhumb line

f M + • , great circle

* Coordinated Turn Requirement

During the turn the normal acceleration, a (t), progresses fromn

2zero to a peak - a flat peak has a magnitude of TACC ft/sec - and

back to zero (see Figure 6). This progression occurs because a (t)n -.

I 81'27 MM 7

must "follow" nx(t) to satisfy the i quirement for coordinated turns.

This requirement manifests itself in this way:

am(i)= 32. - o

Equation (92) shows the aircraft will turn only if its wings

are banked. The genesis for (92) is provided in Figure 15, a nose-on

view of the aircraft in a right turn with pitch zero (n 0).

y

horizon

32.2Figure 15 - Balancing Accelerations in a Coordinated Turn j

41

The vector sum of 32.2 and an must act perpendicular to the wings in

order to implement the coordinated turn. Thus

When pitch is nonzero, Figure 15 is altered by making the downward

component of gravity 32.2"cos ny instead of 32.2. Equation (92) then

follows immediately.

82

I •. 'P " " " " ' • -" " ' .. .

II

i

; 'M Equation I

Defining VL as the level-plane component of total speed

(VL - VT cos ny) we may plug (92) in (77) to got the mazouverturning rate:

=!

Vr W(}

b. Control Derivation

Examination of (89) and (94) shows that roll and roll rate must be

known before w can be computed. Their deteirmination rests on

choosing the appropriate roll history from Figure 5 and then on

computing TOFF, TON and TDONE. The logic and calculations for

accomplishing this are contained in PROFGEN subroutines TSETUP2

and YAWCHG and are outlined below.

[I

83

- ,?-

lil•"• r.F i .,• . I _ I jl , __ , , .'Mr.. . ...... . ... ' - - : - . .. . ..-..

A 'M Computation

The most general roll history is pictured in Figure 16. (This

roll history is for a right turn. A left turn would be the negative

or Figure 16).

It '

Ro-11 Into Tuzrn Hold Bank Angle Roll Out of Turn1 t(TF)

t (TI) t (TOP) t 2(.TON) td(TDoNE)

Figure 16 - Roll Angle History (Cse A)s

We wish to compute the change in heading, tip that would occur if

Figure 16 was the roll history. For this purpose we may assume t

and t are time increments measured from a ti of zero. Set up the2 dintegral of (94) as follows: j

J* V-r V?()

4 74ILJ

From (92)

~~tf T14CC (?

Recalling (83) for VT(t), the middle integral in (96) is

tJ,

4(re

rq( (o -t,) ,Vq-

The first and third integrals in (96) are not closed-form integrable unless VT0o.

Satisfactory approximations have been obtained for them by substituting

VT, average speed, for VT(t) as follows:

ti ti

fa V = - " - -IP;.

S •85 :. -

.-.. - -

whereix -- ROLRATE and VTl, average speed in (ti, t), is

-r1 V (t = w (4)V U + tV fo

Similarly

V~~d Vr (te) -0 V7(061) VU)+(+ V 7 ()"Vr• (tj 0 + Vr Ce,)

Inserting (98) - (i01) in (96) and simplifying gives

÷ r r f])tI)] V, (MICC *r(o

4 '44 A[L TA(4,)], xwCC(&t,)

Since nXMeX and x (=ROLRATE) are known,t is1

- "( c/szz ee,) ( .

86

•'T • :' .-• --• - . . . . "• "" .. e'

* Reasoning on A&Max

PROFGEIT determines if the maneuver can be completed (HEAD reached)

by seeing how far the aircraft would turn if turn acceleration was left

on for the entire segment, t. to tf. Equation (103) is used for this1 f*

purpose where t is obtained from (104) and t is placed t seconds1 '2 ispae 1 seod

short of t fd = tf (If 2 t 1exceeds SEGLNT, tI is:set to SEGLNT ÷ 2.)

PROFGEN solves for A•Iýa in subroutine YAWCHG using (103).

If AMmax exceeds IHEADJ, the turn can be completed and either

Case A or B of Figure 5 is appropriate. Having decided A or B (not

C or D), the problem becomes determination of t and t2

(In Case B, tI = t 2 .) The following paragraphs will der•.e equations

for t and t for both Casc A and Case B.

If A'Mmax falls short of IHEAD(, the turn cannot be completed and

either Case C or D of Figure 5 is appropriate. For Cases C and D,

the determination of t and t is trivial.

* Case A Roll History

The roll history shown in Figure 16 is identifiable as Case A

from Figure 5. Setting A&M = IHEADI and obtaining tI from (104),

everything is known in (103) except t 2 , the time at which roll-out

8,7

87J

,, . . -.--..-.- , _ . _. - ... _. . . . . .. . . . .. ,, . , . - -• ..__...... .i; ": -' " ' ' , .

should begin. Unfortunately, (103) cannot be easily inverted for t2

This difficulty was overcome by ridding (103) of its "ln" function

which was accomplished by approximating the middle integral of (96)

just as 'ie first and last integrals in (96) were approximated earlier.

Thus (98) becomes

ta

r(-r) eZ~4 VrX

where

Now replace the first term in (103) with (105) - (106), set A•PM = IHEADI

and simplify to get this quadratic equation in t2

t4 [49 A ' ()A. / 1, - 74C] )

ýjr 7 ,ck 7r4cc t,1]

J A, 2 . 7LA,-cct] + co, ,

88

where b

6, = Jgt,.) ÷ ýt, r)

Note that (107) reduces to a linear equation in t2 if VT is zero.

The coefficients in (107) are computed in TSETUP2 and supplied

to QUADRT where t2 is computed. TOFF, TON and TDONE are given below.

To reference them to true time instead of ti-O, merely add TI to each one.

TDoFF , 4 # t,

0 Case B Roll History

A "Case B" type roll history is illustrated below.

o. tRoll Into Turn I Roll Out of Turni t (TF)

t (TI) t (TON-TOFF) td(TDONE)

Figure 17 Roll Angle History (Case B)

i e 1' _ _"..-

•=" '; 89 ..

The following eqi.~ation in t was obtained using a procedure like that

which lead to (103):

64-4A4C ~tJ)]*1e/ts)Vt V4,h Ct-/o=f)

2If in (cos x) is approximated as -.632x ,(109) becomes

t12 0. ý, Vr N A 0 0 L

The coefficients in (110) are computed in TSETUP2 and supplied to

QUADRT There t 1is computed. TOFF, TON and TDONE follow immediately

(see Figure 1T) when t 1 is known.

~4.3.4 Sine Maneuver

Since the aircraft does not change pitch in a sine maneuver, Tj is zero.

Henceig In (80) reduces to a form identical to that for a horizontal turn:

0

90 ;'

'is the sum of the nominal path contribution from (85), and

the maneuver contribution due to an(t):

6 (90)

-- , rhumb line

+41 , great circle

The next two paragraphs derive expressions for PM and nx. Expressions

ford& and • are given in Table 2 and Section 4.3.6, respectively.

b. 'N Equation

For the aircraft to fly a sine maneuver, its heading must vary per

Equation (5):

(-A az , i•t <r&

where

A = max heading variation

w = heading oscillation frequenrny

T = 2r/w = period of one full oscillation

p

I 917-7,i

Differentiating (5) twice gives

A AV Ai(r(' ,t) ON*

-2A Am (Z&rt)

C. x Equation

In the context of a sine maneuver, (77) becomes

* ) - a_ _ (e__)__

where a (t) acts in the level plane and V is level-plane speed. Nown Lrecall (92), the coordinated turn requirement relating an(t) to fix(t).

a~( 3Z. 2 ad.~ (92)

Plug (92) into (113) and rearrange to get

92

from which it fellows that

- .9Z2 VT

5 2.Z z 4' V

The trouble that was experienced in establishing roll control in the

horizontal turai is entirely avoided in the sine maneuver because there

is no need to compute any special "event times".

4.3.5 Straight Flight

a.,. w Equation

Since aircraft attitude remains fixed in straight flight, (80)

simplifies to

4 ,NO

where

0rhumb line

(p~ ,great circle

"events" occur in straight flight so (116) tells the whole story.

93

4.3.6 Heading Angle Turning Rate for a Great Circle Path

Figure 18 shows the geometry associated with the problem of determining

the rate of change of heading along a great circle route. (The E frame

in Figure 18 is established here to facilitate this analysis). A great

circle route lies in a single plane, Plane I, which passes through the

center of the earth. This plane is described by XAq and eq where Xeq

is the longitude at which the great circle plane, Plane I, intersects

the equatorial plane and *eq is the heading at the aforementioned

intersection.

Consider a vehicle at point P proceeding along a path lying in

Plane I. The coordinates of this point are given by A-Xeq' 4c and R

where c is the geocentric latitude and R is the length of thec

geocentric radius vector.

The geocentric heading, 4c at point P is given as the anglec

between the horizontal velocity vector and a vertical plane, Plane II,

erected at longitude A-Aeq and containing point P. Thus 4c is the

angle between Planes I & II. In rectangular coordinates(XE YE zE)

the equations for Planes I & II are respectively

XE- It Zo (A1?0

94

IA

Y E

446I

yeIQUTA

Figure 18 -Great Circle Geometry

95

The angl,. is therefore given by

The primary concern here is with the geographic heading angle

rather than the geocentric heading c'. These two angles differ due

to the deviation angle D between the local vertical and the geocentric

position vector, see Figure 18. If * denotes geographic latitude.then

The projection of ic onto a local level coordinate system (rotation

through angle D about the local east axis) yields

- 5 # (/1') 'and

Differentiation of (121) with respect to tine yields the desired

quantity:

(PC

96

The remaining steps are conwerned with determining usable expressions

for the right side of (123);,

The time derivative of 4c is obtained from equation (119) asC

From Figure 18 it is seen that

and

Combining equations (117), (118), (119), (125) and (126) yielas

IL2 a

which when substituted into (124) gives the simple expression

Also required is the inverse solution of equations (121) and (122)

vrz;A2' -.4;W9D 40~'

97

Substituting (128), (129) and ( 30) into (123) yields the expression

for the turning rate of the heading angle

Since this is the value of ' required to maintain flight in the great

circle plane, it is the quantity labeled previously as ýG" Thus

ýG <-> Equation (131)

The angle D and its time derivative D are given for the ellip-

soidal earth by the following relationships:

I-o ft I Ii

, = ,,- ____.__ _ (€e

ese

R=~[u~~r e~c. /-(ez 1 lRe,~m

98

RIM ... .

Equations (131) through (135) are the exact relationships for

an el±ipsoidal earth. If the earth had been assumed spherical, its

eccentricity would have been zero and (131) through (135) would

reduce ÷

Ssin € (13)

D =0

R =R +he

D 0

The earth model in PROFGEN may be converted from an ellipsoid to a

sphere by simply setting e2=0 in BLOCK DATA. However, to take full

advantage of the spherical-earth simplification would require replacing

(131) through (135) with (136) and revising the gravity and earth radii

computations.

99

SECTION V

PROGRAM ORGANIZATION

Previous sections have described what PROFGEN does, explained

how to use it, and derived equations for its implementation. This

section assembles these equations in a sequence amenable to solution

in FORTRAN code. Flow of equations and code are both presented.

Two principles will guide us now (Reference 7):

*The most reliable documentation for any program is the code itself.

Therefore our purpose is not to describe the code in minute detail-

such a description would be unreliable, redundant, and probably harder

to read than the code itself - but merely to show how large pieces of

code interact.

* Each subprogram contains comments giving a readable description of

what that subprogram is supposed to do. These comments form the core

of the micro-level documentation and do not need to be repeated here.

Figure 19 is a macro-level flow chart emphasizing overall

computational structure, especially with regard to control of step

size, h. The name(s) beside each block in Figure 19 designates the

subprogram(s) where the action in that block occurs. Each of these

subprograms usually calls one or more other subprograms to complete

100

this action (see Figure 21). The main program and master executive

is named PROFGEN. The subexecutive for controlling numerical

integration during each maneuver is FLTPATH.

Figure 20 is an expansion of the integration block that appears

in heavy outline in Figure 19. Figure 20 was included here to show

how the differential equations in Section IV actually get solved.

Figure 21 is a dependency chart showing what calls what.

Although timing relationships are vague, (Figures 19 and 20 deal

with timing) this chart is nevertheless u'zeful for getting a bigger

picture of how PROFGEN fits together. It was kept during program

development to help assess the impact of proposed changes.

I

101

I,

Start ) PROFGEN

Qbtain valid PROFGENinput data VALDATAin ft,sec,rad NEWUNIT

nit

Initialize t PROFGEN

times t i and t• RFE

CInitialize state vector

,and step size (h) FLTPATH

Vertical Typerzots

ICompute event V " Compute event TS-TTYP2TSETUPI itime t _ Other itimes toltnal

Vertical Horizontal Type Sinusoidal StraightOf

t<turn FLTPATH

Adjust h to Adjust h to Adjust h to Adjust h toprevent integrating present intei'rating prevent integrating prevent integrating

Pasttf or tout past t or out .HLIMIT. past tf or tout Past t or tps orHLIMIT paortttf otout

HLIMIfTIAdjust h to Adjust h to Adjust h to

prevent stradling prevent stradling prevent stradlingt dHCHOP [toff, ton or t d HLIM2 half-period pointE HLIM3

KLTM~ER

KUT.ER - KUTMRT

lOutput as PRNTOL?' lOutput as PNOUT, Output as Output asRrequired at to RITEOUT required at t RITEOUT • required at tout required at toutARAYFIL ou ARAYFIL

Segment N N egment egisent N N egmentcomplete cotlet c mplete ctPle~ atat) nTPATH t FTPATH FLTPATH FLTPATH

StpPost-rnm Y Proble N AStpresults complete A

KMPERF PROFGEN

Figure 19 Macro-Level Logic Flow Diagram

102

Enter KUT1MThR

from FLTPATH

Save 2End

Y,)=X (r')-Y1-yO+ {F'/.!) F TOqyO0),*Y,=YOQ. (1l/:) F(To ,YO) + ( 1 4/6) F (T)F+H/T÷ ,y.,)

Y 4= YO+ (14/ ý)F( to, Yai + (74/9) F {TO+H/ ',, V'2)Y " Y O fF / ? r ( '" 'C"I' " H / 2 ) (T r)÷ H / -;,,Y ,I) ý-( ? .H ) 'ý(T o +11/ 1, Y 7)

Y '.-:= Y 0* (H / cý) 9 ( T '3 Y ;)) + ( L.ý!41 ) .- T H I 7 y 3;) 4- ( W 1 r) Or T ) " U-• Y 4+ )=y (YO+H)

Note: No Variable

All computations -Step - Sizeoexcept F(t, x) Itegrati

occur in KUTMER.-+

Yes

P-maxlY 5-Y IJ

1S iS n

Adjust h basedon ratio of Pand errorcriterion

SYes Repeat Integration

t *t +hrn RetrntoZ(to+h)uy FLTPATH

Figure 20 - Numerical Integration of : F(t, x) from t to t +h A0 -0

103

141

.... " ...

crii

1051Uf

MAXM4INSCL

BANGLE

BASALI'

BGNPL

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CO!MWRS

CURVE

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DONEPL

ENDPL DISSPLA SubprogramsFRAME(theme sources are notFlisted in Appendix B)

GRAPH

HEADIN

HEIGH{T

MAPDATA

MIXALF-

PHYSOR

RESET

RL!'ESS

TITLE

Figure 21 (Continued)

lo1

iio6I-I

APPENDIX A

SAMPLE RUN OF PROFGEN

The sample run described here was constructed in seventeen

segments to exercise most of PROFGEN's code including at least two

segments of each of the four types of maneuvers. Throughout the

sample run the nominal flight peth is a great circle, the output

interval is one second, and the integration step-size is variable.

Figures 2 and 3 are the PRDATA and PASDATA lists that were used as

input.

* Printed Output

Figure A-I is a rortiun of the printed output from the sample

run. The first page of printed output consists of a banner

(automatically printed by the CYBER-74 computer) followed by the

date and clock time of the run. The second and third pages are

listings of the PRDATA and PASDATA lists as read from TAPE9, the

local file o., whinh the input should reside. These listings simply

echo thi !ata, including its mistakes if any.

?age 4 of Figure A-I begins the printed output generated during

the c~mputational portion of the zrn with IPRNTl1. This output

consits o^ a header ,n - list of variable values at the start of

each segment, foYlnwed by output at DTC' intervals (one second in this

run) during the segment. The liat of variables printed does not change

SL___107

and the definition for each such variable, with itn units, is given in

Table A-1. Pages 5, 6 and 7 of Figire A-1 show output up through the

beginning of segment 2.

The last page of the sample printc•lt contains, in addition to

output spaced at DTO intervals, output at t-final (460.5 seconds in

this run) plus a post-run assessment of the numerical integration

burden. In this case 5393 numerical integration steps were used and

F was called 34675 times.

* Plotted Output

Figures A-2 through A-6 are the plotted output for the sample

run. The small numbers appearing along the curve in each figure are

segment numbers designating approximately where each new segment

began. The latitude - longitude plot in Figure A-2 is constructed

with the latitud. and longiti!.de axes at the same s-qle.

• Other Output

TAPE3 output was suppressed in the sample run by setting IRITEuO.

If TAPE3 output had been specified (unformatted binary records), each

record would. havo contained the following list of variables in units

of feet, seconds and/or radians: time, latitude, longitude, alpha,

altitude, roll, pitch, yaw, velocity components along nay x, y, z

and specific force components along nay x, y, z. Subroutine RITEOUT

should be consulted if a more definitive description of TAPE3 output

is needed.

108I

Table A-i - Output Variables

Variable Units Description

TIME sec. time (t)

LAT deg. geographic latitude (•)

LON deg. longitude (X)

ALPHA deg. angle between north and nav X-axis (a)

ALT feet altitude from ellipsoid (h)

ROLL deg. roll (nx)

PITCH deg. pitch (n

YAW deg. yaw (NZ)

PSI deg. ground heading angle measured positivecw from north (i)

DROLL deg/sec derivative of roll (;x)

DPITCH degi'sec derivative of pitch (iy)

DYAW deg/sec derivative of Yaw (AZ)VX ft/sec velocity w.r.t, earth along nay x-axis (V x

VY ft/sec velocity w.r.t. earth along nav y-axis (V

VZ ft/sec velocity w.r.t. earth along nay &-axis (VZ)

VPATH ft/sec magnitude of total velocity (VT)

ft/sec2 specific force along nay X-axisFY ft/sec' specific force along nay y-axis (F

FY f/seca specific force along navy -axis (FFZ ft/sec2 specific force along nay z-axis (F Z)

APATH ft/sec2 acceleration along path X-axis (i.e. alongV)

109

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Lat'tude/Longctude FL'Goht Prof'Le

Q) 7

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ALtitude FL'LlLbtProfL~e

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Fixure A1

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T12e (sec)

122

APPENDIX B

FROFGEN LISTING

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REFERENCES

1. T. 0. Seppelin, "The Department of Defense World Geodetic System1972", Defense Mapping Agency, Washington, D.C., May 1974.

2. G. I. Hosmer, Geodesy, Wiley and Sons, 1930.

3. J. C. Pinson, "Inertial Guidance for Cruise Vehicles," inLeondes, C. T. ed., Guidance arid Control of Aerospace Vehicles,McGraw-Hill, New York, 1963.

4. W. A. Heiskanen and H. Moritz, Physical Geodesy, W. H. Freeman,1967.

5. P. S. Maybeck, "Wander Azimuth Implementation Algorithm for aStrapdown Inertial System", AFFDL-TR-73-80, AD784752, Air ForceFlight Dynamic Laboratory, WPAFB, Ohio, Oct 1973.

6. R. F. Osborn and I. J. Dotterer, "SAMUS,A Program for State SpaceAnalysis of Multisensor Systems", Vol III, T70-428/201, Autonetics,Anaheim, California, May 1971.

7. B. W. Kernighan and P. J. Plauger, The Elements of ProgrammingStyle, McGraw-Hill, 1974.

188S" F', F, I , •,! i~ , iiF[ ' I FF111 [Iii 1F I ( F l-F. , F. - ,.'-F•FIII 1 (IT-