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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16. DISTRIBUTION STATEMENT (of this Report)
Approved for public release; distribution unlimited.
'7. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If different from Report)
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
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:
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
BSHIIT
CO!MWRS
CURVE
DASH
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
a.41
-4-118
FigMre A- 3
ALtitude FL'LlLbtProfL~e
'-to
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a)f
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11
Fixure A1
PoLL FL'~ht Profi~e
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Yaw FLUqht ProfiLe
116
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0.0 77'0 154-0 231 0 308"0 3850 4 62'0,
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.
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