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Produced by the NASA Center for Aerospace Information (CASI)

NASA TECHNICALMEMORANDUM

Repo rt No. 53900Y

iX

TRAJECTORY APPLICATION METHOD (TAM)

By John P. SheatsAero-Astrodynamics Laboratory

Y

September 12, 1969

b"^rd P 6 A to K#,

a IJM1

NASA

George C..Marsball Space F CenterMaffhallSpace Flight torf AlabdmaCen

N N7 . 0 m 5.2

4 (ACCESSION NUMBER) (THRU)X

(PAGES) (CODNSFC • Farm 3190 (September 1968) )0-— 4 O j^

.r t/ V

(NASA Ck OR TMX OR AD NUMBER) (CATEGORY) _'

NASA-GEORGE C. MARSHALL SPACE FLIGHT CENTER

AFRO-ASTRODYNAMICS INTERNAL NOTE # I-r)7

r • t c

TRAJECTORY APPLICATION MFTHOn (TAM)

Hy

Join P.Sheat

ABSTRAC

A simulation technique (TAM) for the p(st-flight evaluation of the propulsionSystem performance has been developed which incorporates the time history trajectoryParameters from the Post-flight observed trajectory as input. This technique representsa significant reduction in time required to perform staqe propuisioii system evaluation%.The development and some advantages and disadvantages of this technique are given.The propulsion system evaluation was performed on the S-113 stage, of AS-701, AS-203, andAS-20? utilizing both the proposed technique and a conventional simulation technique;the results and comparison of both method3 are presented. Additional detailed specifica-tions of the TAM program are given in the Appendices.

1..

14

1C/

-` v '

GEORGE C. MARSHALL SPACL FLIGHT CENTERAFRO-ASTRODYNAMICS LABORATORY

FLIGHT TEST A14ALYSIS DIVISION

Aero-Astrodynamics Internal Note No. 1-07

March ?I, 1967

TRAJECTORY APPLICP ION METHOD (TAM)

By

John P. Sheats

This note has been prepared for inte-nal use within the Aero-Astroaynarnics

Laboratory , no is transmitted to other agencies only for expeuiency in special

cases. Transfer to parties other than those apLaboratory Director is not authorized.

TABLE OF CONTENTS

Page

1.0 INTROOUCT!ON.................................................... 1

2.0 SIMULATION PROBLEMS ............................................. 1-2

5 ,0 TAM DEVELOrWENT ............ . .................................... 2

3.1 Altitude: and Attitude Lffec`s .............................. ?-3

3.2 Gravity Considerations ..................................... 3

3.3 Estimated Instantaneous Adjustments ........................ 3-4

3,4 Overall Adjustments ........................................ 5

5,5 Advantages and Disadvantages ............................... 5-6

5.6 Flight Results ............................................. 6-7

3 .7 Program Description ........ . ................1400........... 7

4.0 CONCLUSIONS ..................................................... 7

APPENDIXA........................................................... 8-12

APPENDIX13 ................................................ .......... 13-15

APPINDIXC ........................................................... 16

APPENDIX D ........................................................... 17-77

'T

DEFINITION OF SY ►IBOLS

Symbol Definition

*AA Average *Am between two time intervals printed out at thelatter time point.

AENozzle exit area (Fngines I through n).

ALISPO Average instantan6ous c',ea level specific impulse.

*ALT Altitude (from OMPT).

AmTotal calculated platform (inertial) acceleration.

*AmTotal platform (inertial) acceleration from Observed Mas,,Point Trajectory (OMPT).

"AM1CN Average Mach number between t and t -I as observed.

AREA Cross sectional area of vehicle.

A T Throat area (Engines I through n).

AVDFLOX Average propellant mass loss rate.

AVDMM Average total weight loss rate of the vehicle.

AVF IO Average total sea l evel longitudinal thrust.

BDRAG Base drag (input as table or equation).

CD Drag coetficient (table look-up vs. Mach).

CCDD Average CD between two time points.

CF Sea level thrust coefficient.

CFV Vacuum thrust coefficient (pulled from data tape).

(-XCA Required CD as based on acceleration difference (DA).

CXCVE Required CD as based on earth-fixed velocity difference WDVE).

CXCVI Required CD as based on integrated acceleration difference,,(DDIAT).

DA Difference ;n total acceleration (calculated minus measured).

DCXA Required drag coefficien t chance at any time as based on

acceleration difference (DA).

DCXVE Required CD change based on DDVE.

DCXVI Required CD change based on DDIAT.

1:'

iv

F _! I

FJ 10

FLT TT

FNII(F)

FOCI —n)

Symbol

DO

DDVE

DDIAT

DFA

DFLOX

DFVF

DFV I

DIAT

DMA

DEFINITION OF SYMBOLS (CONT'D)

Definition

Average vehicle aerodynamic longitudinal drag bc'ween 1 and t-I.

Difference of OVE between two time points.

Difference between DIAT from one time point to another dividedby time difference.

Required thrust change at any time as based on accelerationdifference (DA).

Propellant mass loss rate (from data tape).

Required thrust change based on DOVE..

Required thrust change based on DDIAT.

Integrated difference betwnen A and *Am m

Required mass change at any time as based on accelerationdifference (DA).

Total vehicle mass loss rate (from data tape).

Required mass change based on DDVE.

Required mass change based on DDIAT.

Total vehicle aerodynamic longitudinal drat).

Difference between calculated and measured earth-fixed velocity.

Total longitudinal drag force.

Vertical buoyancy force.

Local individual engine turbine exhaust thrust (from data tape_).

Sea level turbine exhaust for engines (I-n).

Local engine thrust.

Average local thrust (FJ1) between t and t_ I'

Total local longitudinal thrust.

Total sea level longitudinal thrust.

Flight time as measured from first motion.

Total local lonqitudinal effective force.

Individual engine sea level thrust.

DMASS (w)

DMV[

DMVI

DRAG

DVE

I A I

fB

FE(I-n)

FFl(I_n)

F-ENG(I-n)

FF

too--ft

DI_ F I N I T I ON OF SYM13OL S (CONT' D)

Symbo l De f I n I t Ion

GC177' (T) Time from guidance reference release.

ORR Guidance reference release tim".

IAT Integral of Am.

*IAT Integral of *Am.

K Vehicle firing direction East of North.

KVAL Thrust multiplier (fecal thrust correction constant).

LISP`O Instantaneous sea level specific impulse.

•MACH Mach number as pulled from OMPT.

MASS (M) Instantaneous mess.

M I Initial vehicle mass.

MM Average mass between t and t-I.

()MP( Observed Mass Point Trajectory or Measured Trajectory.

PAF Partial derivative of thrust with respect to accelerationdifference.

PAM Partial derivative of mass with respect to accelerationdifference.

PAW Partial derivative of flow rate with respect to accelerationdifference.

P Individual engine chamber pressure (from data tape).r (I - n)

PO (P ) Sea level pressure.0

Local pressure difference.

*PRESS ("P) Ambient pressure from OMPT.

PVF Partial derivative of thrust with respect to velocity difference.'

PVM Partial derivative of mass with respect to ve!ocity difference.

PVW fart i a I der i va l. i ve of f I ow rate vi i th respect to ve I or i ty

difference.

*Q Dynamic pressure from OMPT.

*QQ Average dynamic pressure between tirne points.

vi 1

i

D EFINITION OF SYMBOLS (CONT'D)

Definition

Radial distance from pad.

Radial distance -rom pad from OMPT.

Local atmospheric density from 0mrT.

Radial distance from geocentric center of the earth to launchpad.

Initial components of the vehicle po%ition vector roferencedto the geocentric center of tho earth.

Range time.

Midpoint time between two data points.

Calculated earth--fixed velocity.

Measured earth-fixed velocity.

Symbol

RAW

*RADD

* RI IO

R0

r0

RTIME 't)

TT

VE (V )

*Vc ( *V )c'

VOLUME

4o

0

m m m

*xm *;gym' *Im

Total vehicle volume.

Geodetic latitude of the launch site.

Geocentric latitude of launch i tr! .

Angular rotational velocity of earth.

C;i l r_u l ated platform (inertial) acceleration components .

Measured platform (i nort i a I) acre' ^ration com r)onents .

` /..,.

1'

vii

^^^^

whom:

Ho - (^o - ^o )

DEFINITION (%F SYMBOLS (CONT'D)

MATRIX IDENTIFICATION

,.In K 0 cos K

r K] 0

-cos K 0 sin K

1 .0 1 NTRO01)CT ION

The ;post -flight propulsior, system analysis on Aach Saturn stage Is usuallyperformed by two methods. The first method of determining the stage propulsionsystem flight ,)erforman;e !s a reconstruction of the telemetered flight dataIncluding enlculated propellant residuals. This flight reconstruction method

Is a mathematical model of the stage propulsion system u t ilizing a table ofinfluence coefficients to determine engine performan-;e. The second methodut i l i . , es a tr a jector i s im:, I at i c a n to generate adjustments that are on forcedon tho results from the flight reconstruction method so that the resulting

calculated trajectory will match the observed trajectory. This second method,trajec.4or- y simula tion, will be the point of discussion in this report.

The post-fl ight propulsion system performance simulation on oa0i S.ittirnstage has, been accomplished using either three-degree or sim-degree-of-freodomtrajectory computer program. These, combined with a weinhted least squaresprogr,im, provide the linear adjustment of the pr,-,t -f I i t propulsion systemparameter ,-. (thrust, mess loss rate and/or initial mass) anu the predictedaerodynamic drag coefficient required for the trajectory parameters of thecomp uted trajectory to match those derived from the tracking data observeddurinq flight. (See Reference I).

All of the tracking data observed durinq flight are converted from thevarious tracker measurements with the origin at the tracking Site to thet rajectory paramete rs in a coordinate system with the origin at the l aunchsite. All of these track;rg (1 0ta are used in conjunction with the guidanceSystem out puts to obtain a _best estimate of the trajectory. The meteoro I og i c^a Idata observed at the launch time are combined with the best estimate oftrajectory to yield whet is c%i I I ed the Observed Mass Point Trajectory (01'APT )or measured trajectory. In all the p ropulsion system simulations performedon Saturn stages f lown, an attempt has L)e . -n made to compute a trajectory whichmatches the measured trajectory. T,iere are -everal difficult problem-, (whichwill he discu s sed I ator) associ ated with + his procedure; however, the ,e nrob I emscould po5-.it;Iy be circumvented to a large degree if some of the trajectoryDarameters representing the a i t i tude--time history of the measured tra ipctoryare used as input to the simulation program.

A. simulation program called the Trajectory Application Method (TAM) wasdeveloped to investigate the advantages and disadvantages of this approach.

7.0 S I MU; AT I ON PPOBLEMS

Two basic problems are associated with the usua! simulation programs thatdo not use the observed trajectory parameters representing altitude-time historyfrom the measured trajectory or 01'Al"T.

ALTITUDE EFFECTS - The meteorological data (atmosnh3ric density, nre'-'C;ure

'e. T Deratu re, and wind data) are independently observed functions of altitude.These data are combined with trajectory parameters from the best estimate of

trajectory to com p ute several altitude-dependent parameter;, such as dynamirpressure, Mach number, and the thrust gain from increased altitude. The relaiion-

ship be t ween altitude and the meteorological data i; r)bsp rved independently andis assumed correct. However, the altitude from convpntion,-1 simulation pro(Iram;will initially be incorrect since the initial value; of the propulsion systemparar^.eter, have not been adjusted. Thus, meteoro I og i ca I d,)ta, which ;ire refnrr. ► i,^edto the correct altitude,are introduced into the trajectory computation at either

- ^_-zA

an earlier or later timo (depending, on the propuls i on parametor adjustments• required) than needed to sntl5 4 y the measured trajectory parameters. This

scheme will eventually converge to the correct r :,'ntionship as the adjustments,to the propulsion sys'em parameter, and the aerodynamic drag coefficient convergeto thu appropriate solutlon.

•

ATTITUDE EFFECTS - The simulation proyrams often compute the -ittitude of -she

vehicle in the s one fashion used in the precaIcuIated or operntionnI trajector,programs, except that the flight sequence and attitude program are consideredfixed. The vehicle attitude as computed in this way is a function of the pro-nulsion and trajectory parameters. (hus, if t he propulsion parameters areincorrec`, the re,ultint -1Ititu,o- l ime history and attitudes will be incorrect.These, too, wi I 1 eventually converge to the r.)rrect relationships as the propulsionp,jrarvx ter- and aerodynamic drag coefficient adjustments convercio to the appropriatesolution. Also, the telemeterod attiti,des are :omf-times w-,on ar, inputs to thesimulation urograms. The problems associated with this app roach are the timein bias snifts which m;,y be inhr,rent in IeIemetured data.

The effect of These t,,o problem,, cannot be suparntod. After severali terat ions these: problems can be r :solved, but a large numhe.r of both man-hours and computer hours, are required. Since this type of i mu I at i o n roqu i re-,i sut ,;tantlal amount of computer time for a sin(tle run, the turn-around time onthe computer is longer than would t•e necessary with a more simplified simulation.

3.0 TAM DEVELOPMEivT

The Trajectory Application Method was developed to circumvent some of theproblems assoc,ated with the usual propulsion System simulations. In addition,a capability to estimate the instantaneous adjustments required was built intothe program. The advantages and disadvantages of the TAM approach and Someflight results are presented in subsequent paragraphs.

i. 1 At i I TUDE AND ATT I TUDE EFFECTS

The altitude and attitude effect problems are handled in the foilowinq way:

ALTITUDE EFFECTS - The following parameter%, representing the altitude-timenistory,are input dire--tly as a function of time: ambient pressure, density,Mach number, dynamic ressure and altitude. These parameters are used whereverthey are required in the various computations ins,u.ing an altitude-time historycompatible with the measured trajectory. Observed ambient temperature and windeffects are included in the calculation of dynamic pressure and Mach number.This eliminates the convergence problems associate6 with an incorrect altitude-time ni;tory, thereby placing the vehicle in its proper nnvironmont for drag

• considerations although adjustments to propulsion parameters may yet be required.

AT TITUDE [FrF('TS - The moasured p l c+t form (inertia!) icce l era t i or. component-,from tht-, or measured trajectory are input as a function of time. The unit

•/ector ^ f this accc I ernt i on Is established.

*A *X 2 ♦ * Y + *7 ;? 1 /2r+ m m m

*X *Y *T*Am ►A i + rAm j + *Am k

m m m

rho ca I cu I cited l•' I gat form accc I :rat ion cornpononts u-. i nq the cornponunts of *AR'

F• P' 1irc computed from th , , total accelerationMAS

S i,ropo^rtional to those obsery

in the mea-;urud t r , juctnry. This mothod rup I aces 1.w I ransformat i can from the body-

f i xe:: to p I i t form cocrd i nato •system.

Xm.. p t^1 1'^A t^IASrn

:n (f^1 IY rr. (* A ttASSn:

*7m_I

h1II I` m *A I ^ 4ASJ

m

This :c`eme el iminato c, tre necess i ty for either comput inq the atti Ludo or directly-muting tic? tnlemeiered att itude information and also insures compatibi l i ty wi ththe measured trajectory parameter components.

?d rPAV I TY CONSI DERATIONS

Tore trajectory porumef ers, of ter a corroct -;ol ut ion for propu I, ion sy-.temociramr--trr:, and aerod •rnnmic draq cooff icient adjustments have been obtr,ined, -,hou1(1t;e i der y t i ca I t(-) tho,,e i ri the measured trajectory or OMPT. Thorefore, the (Irav i ty•on t r i but i on s to t hH t ra_j ectory parameters needed to convert from i nert i ,a l space-f i xed word i riates care identical. The romponents of F;cco I erat i on , ve I oc i t y, andpo5 i t ion due to gravity are input directly from the me,^ :ured tra . jectory a-; functionf t i m-, thu ,., fora i nr1 the gravity contributions to the trajectory parameters in TMA

tc _ -u i va (en ` to rhose in the measured t rya Th is approach ro-ducc . the- voter• f e con , j ,-)r aer _difforential equation:, to be sr)lved i n t h e u ,,u.il ty L. C;f

mu I i' i en s to ^i mn I f.!I i near e^uat i c;ns

3.3 ESTI11ATED INSTANTANEOUS ADJUSTMCNTS

?ejundart i nstnntanecu i adjustment :-1 to the prr)r)"I lion •,y-,tern o.ar-,meter -, andierodynar, i c coof f i c i ents - -e made by compar ing thf ( I ) i nert i -i I aecA I er,at i %n

3

(^) ^Prti•il velocit y , and .1 31 :arch-fixed velocity computed in TA"' wit ►, t l ,ose inputfrc- the measured trajecto,, at each time point. There is a requirement for partial:erivative r. of the trajectory parameters with respect to the parameters for whichrequired adjustments are :ought. These partial derivativ,3s can be obtained by

conventional perturbation methods. however, a highly simplified approach todetermining these partial derivatives can bo developed, when the appropriatesimplifying assumption is made. Thn assumption to be made is tnat the deviationsproduced in inertial and earth-fixed acceleration resultinq from a devi,ition ineither the propulsion parameter: or aerodynamic drag coefficient are approximatelyequivalent. The partial derivatives shown below can be used for both inertial andcarth- fixed accelerations.

aAm

F— MA )S

t t

DAM Am

M. "SS•lA

^ t t

`t A m "U • AR L±

C x t MASS It

These partial der i vat ivos which are used for both inertial jnd cart h-fixed.icce I erat ions havo proven quite adequate in several te ,;t cases. Also, thu separtial derivatives may he integrated with respect to time to yield parti;11derivatives that may be used with velocity and position diff r-rence%. Sinee^.o •; i t ion and vo I oc, i t y data arcs general I y qu i to smooth , i t may be des i rah I e to

• u;e these in Iiou of (or in addition to) acceleration Oata.

The instant^rneous adjustments to tho p ropulsion system parameters and theierod /nami c dr-!g coe f f i c. i ont are determined by dividing the di f ferenc.r^ hetweenthe trajectory parameters computed in TAM and input from the mea•ured trajoctoryor OMPT by thene partial derivatives at each time point. It must Ce ­.sumed tha`the entire difference in the trajectory parametAr-, is a result of any one of the1d juStM ;ts.

Fn9 ineerin,3 judgement combined with a priori knowledge of the accuracy of thennrameters being adjusted can be used to give n estimate of how much each of theparameters to be adjusted contr i bu're to the difference t,e;tween the computed andc-)easured trajectory parameter,;. The estimated instantaneous adjustments c-in beuses' to determine if any sioni f icant tr(;nds or discontinuitie-, could exist in the

to be adj usted. (If any d i ,;c.ont i nu i t i es ox i ­;t in the tr ,i iir(J-)ry p,rr,i-;ter i i.;;ut f rU r^ 0!#'T, th r;ses would also be re f l ectcd i r, the e y,t im.atnd i n-A,int-dneou ,

'^;uaIIy initial r:r,rrr:ctiuns are , appl ic;d tr; the; l,ropuI ,ior1 1),1rrime;t-:r • , t,c:f()r(: ,iny.i'tempt i made to esldt)I ish the adjus tment to the; r,err,rfyn.imi drari rr)i:f f i I orit.

'forma I I y, only a crn stc,nt sh i f t to the propu I s ion p.,r^imeters • 1r rt :on->i evon_ though estimated instantaneous adj u! r tinen is are evil i I at;1 c:. Th(.. i nst.antane ,)us ad j ust-

1nc;ntt, for the aerodynamic draq coefficient are con,,idernd aprjl icablo.

5.4 OVERALL ADJUSTMENTS

The difference between the trajectory parameters computed in TAB" and inputfrom UMPT can also be used w'th n conventional weighted least s q uares program tosalve for an overall constant shift or bias in the propulsion system parameters.Th(% partial derlvatives required can either be those established using pertu-ba-tion techniques or those partial derivatives determined in the simplified approachdiscussed under Paragrapt 3.3. Usually, the a priori knowledge of the accuracyof the parameters to be adjusted is included in the least squares solution. Anyconventional least squares computer program can be used with TAM to obtain pro-pulsion adjustments such as the one shown below.

P I^lo-I ^T WiP-

I (, -1 CT

W IP - I R

wheret

f — (nxl) inafrix of the propulsion adjustments.

W - (nxn) diagonal matrix consisting of the squares of the

accuracies associated with measured propulsion parameters.

(mxn) matrix of partial derivatives of the tr,ijectory datawith respect to the parameters P.

W (mxm) diagonal matrix consisting of the s q uares of theaccuracies associated with trajectory data.

R = (mxl) matrix of the difference between the calculated andobserved trajectory usually referred to as the residual matrix.

The diagonal elements of the covariance matrix W -1 + CT ^^ - 1 r -I

o I F'.ire the statisticaI variances of the parameter adju-;tments, arid the off-iawnaI elements are the covariance ,, of the parameter adjustments. The

I.nuare root: of the diagonal elements are the standard deviations of theadjustments and the off-diagonal elements are an indication of the correlationbetween the adjustments.

The matrix Z(r T W IP -I P) represents the sum of the weighted squares of

the residuals that are to be minimized subject to the constraints imposed by W0

3.' ADVA t ITAGES AND U I SADVANTAGES

There are several advantages and at least one d i sadviintax)e of the TAMap p roach over conventional Jmulation techn i raun- .

ADVANTAGES _- The advantages, other than those for which the scheme was originallydevised, are discussed as follows:

Altitude Effects

a). The altitude-dependent functions are given versus time therebyeliminating tape interpolation.

b). Fewer equations are required, thus eliminating unnecessary computations.

• Attitude Effects

a). Control equations are eliminated.

b). Moment find angular motion equations are eliminated.

c). Computation or input of attitude angles is eliminated.

Gravity Cor,,iderations

a). The system of second order differential equations, usually requiredin most sioulations, is reduced to simple linear equations.

b). Complex integration schemes are not required.

Partial derivatives

The simplified approach for computing the partial derivatives eliminates thenecessity of consecu'ive computer runs usually required for the conventionalperturbation schemes.

All of these effPr.ts aid in separating the propulsion parameter and aerodynamicdrag coefficient adjustments and also reduce the number of iterations renuired toobtain a valid solution. The TAM simulation technique is far less complex andmore economical with respect to machine time than either the six--degree-of-freedomor three--degree-of-freedom simulation programs. Both the man-hours and machine-hours req uired for an evaluation are significantly reduced through the use ofth i.s progr.3m.

DISADVANTAGES - The TAM program was devised for use in post-flight evaluation ofpropulsion system performance with a high degree of dependence on input dataobtained t mm the measured trajectory. This is the principal limitation anddisadvantage of this approach. since the altitijdn-timA hi,tr,r y inH vAhirloattitudes are used as input-., TNI cannot be used to show the effect: of propulsionparameter and aerodynamic drag perturuations on Irajectory pararneter5.

3 . F FLI;NT RES^!LTS

fhe flight results usinq this simplified technique are compared with the

flight results using a conventional three-degree-of-freedom (3[)) simulation

p rogram for three S-113 stage Saturn IB flights in the table belo4. Thistable shows the exceIIent result obtainable with the 1AM simulation technique.

This ap p roach is as efficient and reliable for use with the latter or upper

stage-, as with these stages for which drag effects are of more concern.

6

W+^-• -'

AVERAGE SEA LEVEL AVERAGE TOTAL AVERAGE SEA LEVEL

LONG. THRUST (LEA) PROPELLANT FLOW- LONG. ISP (SEC)

RATE (LUSEC)

AS -201 Sit 1 , (,1 i , 5b0 6155.98 262.20TAM 1,612,754 6151.80 262.16;, UEV. -.05% -.034% -.G15%

A`)-103 iD I , o60, 4 71 6285.18 264.19TAM 1,')59,928 6283.40 264.18% UEV. -.03% -.03% -.0O5%

5-202 31) 1,031,558 6234.70 20I . W)TAM 1,631,374 6234.86 261.65% DFV. -.UI% -.003% -.001%

% I)EV.TAM- 3D

X 100

5.7 PROGRAM DESCRIPTION

in A ppendix A. T:le integrationsaccomplished using eithercomplex equations of motion areequations are defined inppend'x A are defined on pages

to satisfy the equations aregiven in Appendix C.

Output formats are generally considered arbitrary; however, in order toillustrate the Trajectory Application Method, a sample print format is givenin Appendix U. This sample print is extracted from a typical S-II; stagecalculated trajectory. This table shows the re-,iduals between the measuredand calculated trajectory parameters along with the instantaneous correction,to the propulsion parameters. Any one of these correction., will explain thedifference between the two trajectories.

4.0 CONCLUSIONS

The TAM simulation technique, described under Paragraph 3.0, yields resultswhich are well within the accuracy tolerances of the more conventional simulationschemes. The use of this program for the trajectory simulation represents asignificant reduction in both man-hours and machine-hours required for anevaluation of the propulsion system performance. The TAM Jrrulation techniqueis not a tool for studying the effects of propu i s ion y -,tom parameters and :, p ro-dynamic drag coefficient perturbations upon the trajectory parameters representingthe altitud e -time history or vehicle-attitude, but r#-prA;e.nt, a most efficientmeans of ob to i n i nq the post-f I i qht evaluation of the propu I s i on system pc-r formancP .

7

The comp lete set of TAM equations is givencalled for in these equations can be accuratelySimpion's Rule or the Trapezoidal Rule since nopresent. The coordinate systems utilized in the.Appendix i?. The symbols and matrices used in Aiv through viii. The input parameters required

sip,a_ -

I ar

APPENDIX A

EQUATIONS:

NIF = FV -AEI-n Po

TC

(I-n)C

( I -n) A PCI-n 1-n

#2 F0 1_n (CF) 1-n (A T ) 1-n (PC)1-n

#3 F-ENG 1-n = FO n + A L (Po - *PRESS)

I-n

#4 **FJ I = [-cos 6 * E ( F-ENG 1-4 I - 4

+ FE ) + cos 3* 1v ( F-ENG 5-8 ) + E. FE 5-8 I KVAL

115 FM I = FJ I ; FA I + F©

#(, FB = (*RHO) (VOLUME)

117 FAI - - (BDRAG) - (DRAG)

03 [)RAG = (CD) ( *(J ) (AREA)

#9 **FJIO = cos 6* F.(F0 1-4 + FE1 1-4 ] + cos 3 0 E EF05-8 ] + E FE15-h

#10 LISPO - i,j,oD` LOX

#11 ALISPO = AVFJIOAVDFL0X

t n

12 AVEJ I^) FLTT 1 (FJ 10) dt

t.i

* Input data from tape

** The equations i I I ustrate the S-113 stage, Saturn I B, where the 4 inboardengines are canted 3° and the 4 outboard engines are canted r". However,

on stages where engines are not canted these considerations c-,n be dropped.

8

_.^Y{W .._^^.. ^.. .. a ^^—.^.^__ ^^.^+^►wa.^A^^ rya ^^ ^ 1'_

APPENDIX A (CONT'D)

tI n

#13 AVDFLOX - MFLOX) dtF LTTT ft.

i

tn

# 14 MASS - M . + ( [?MASS) dtit

i

#15 AFMI

m MASS

X16 X -*Xm

A Y =*Ym

A *7mm *Am m m *A M *A Am

I m m

_a t n _ t nN17 X =

M ( m) dt,

^(Xm) dtJt ; t.

t

nt

n7118 IAT (A ) (it *IAT (*A ) dtm m

t^ ti

#19 FIAT = (,AT - *IAT)

-.a v _ay2^ y = X + *Xs m G

x = Y, + X + * X-S 0 m G

*XSc

r0 + XSO

I + Xm + XO

--^ _. _.sY = X - rs sc 0

!#21 e = {C KITFso ^ TCw^ TF9 -1[K] X } + {[K] jFo T Cw^ Tf _)FK..

) X - }

01o sc

e = {L KIT C^o JT[w]T E^oI[KI x c } - r 9

PA,

IP

APPENDIX A (CONT' D )

#22 CA - j A - "Am m

X23 V X 7 ♦ y 2 1 12

e e e e

#74 DVE V - "Ve

m R. m m

(;)VE t - CVEt -I )#1`6 DOVE -

(t - t- I)

APP N() I X A (CON T ' [) )

b15 OFA = -(DA) (MASS)

036 DFVE = -(DDVE) (MM)

Nil DFVI (DDIAT) (MM)

k3P, OMA CAI -

FF DD

MASSm

- ^iq DMVE _ *AA - DOVE

AIM_

# .10 UMV II= *AA - DDIAT - MM

Nnl *AMACH = ( *MACH t ♦ *MACH t-I)

N42 OCxA - [(DA) (MASS)

*0 (AREA)

#43 DCXVE = (ODVE) (MM)[*QQ (AREA)

N44 DCXVI - DDIAT) (MIA)

*QO (AREA)

YY45 CXCA = CO + DCXA

#46 CXCVF = CCDn + DCXVE

#47 CXCVI = CCDD f DCXVI

APPENDIX A (CONY D)

tae PvM -am

aDvE

#4Q NAM - _am MASS

aA Am m

#50 PV - a3F - MM

#51 PAF = aA - :-- MASSM

X52 PVW = aw _ _ (MM)

aDVE (*AA) (TT)

X53 PAW = aw _ MASS

aA A tm m

#54 PAOO `X P ` + Y 1 + 7^`

J _so o#`)5 X = E K _1 FE 1 T E4 ][K]jf m K^

o

MM_

"

12

APPENDIX 8

r't.UMI)L I NE COORDINATE SYSTEM DEFINITIONS

1. Earth-Fixed Coordindte System. The earth-fixed coordi nate system i%defined as a right-handed Cartesian system with the p rojection of the centerof gravity of the complete vehicle at or prior to First Motion (FLTTT • 0) on thereference ellipsoid as the origin.

The X-Z plane is tangent to the reference ellipsoid at the origin of thecoordinate system. Tho positive X-axis is oriented in the flight azimuth direciion;the positive Y-dxis is above and normal to the X-7 plane; the positive 7-axisis in a right-handed relation to the X-Y axes. The orit,in of this earth-fixedsystem rotates with (-In angular velocity equal to that of the earth.

Launch pad coordinates are defined with respect to the reference ellipsoidchosen to represent the earth and its gravitational field. The elevation of thelaunch site above mean •-;e,) level and the position of the center of gravity of thecomplete vehicle are treated as an elevation above the reference ellipsoid.

._.aX E X X X E_

X YE Earth-Fixed Cartesian displacement - YYYE

ZE

componentsZ77E

X E DXXE

X YE Earth -Fixed Cartesian velocity = DYYE

Z

E

componentsDZ.'7E

^

X E DDXF

Y E= Earth-Fixed Cartesi an acceleration - DDY1

lcomponen t s

E F)DLE

7. Space. -Fixed Coordinate System. The orientation of the space-fixedcoordinate system is identical to the earth-fixed system at and prior to guidancereference release (GCTTT - 0). The origin is a point fixes; in space, and thecoordinate system remains fixed in space as oriented at guidance reference release(GRR).

13

APPENDIX D (CONI'D)

X S XXXS

X s = YS Space-Fixed Cartesian displacement = YYYS7S

componentsI7ZS

-h xS I DXX ;

• XS - YS - .Space-Fixed Cartesian velocity DYYSjS

componentsDZIS

X S DDXS

X s YS _._Space-Fixed Cartesian acceleration DDY' -

Z5r'.xnponents

DDZS

S. Inertial P latform _Coordinate System. The inertial platform is a gyrostat' l ized reference elemont oriented at guidance reference relea;e ((RR) timeidentical to the earth-fixed and space-fixed coordinate systems. The coordinatesystem remains fixed in inertial space as oriented at GRR. Coordinates in theinertial system do not irclude tho effects of gravity and the initial rotationalvelocity of the earth,

X XXXM

X - Y Platform displacement components - YYYMm

IM 7.ZZM

AM = DXXM

Y Platform velocity components - DYYM

IM DZZM

XM DDXM

Y Platform acceleration components - DDYM

2 DDZM

4. Gravitational Components

X^. I XXX("

Y F Grav i tat i ona I d i sp I ae emen t component ,, - YYY(;

IZ

!Initiated at (;RR Time)177("G

1 1

V. wit

,.y.

m

Xm

Xq

JX

n

Imo. djeft—

APPENDIX I3 ( CONT' D )

4, Gravitational Components (Cont'd)

XG DXXGJX Y, Gravitational velocity components = DGYYg

I

Z `' (Initiated at GRR Time)DZZGr

A^ DDXG

X9 = 1'GI - Gravitational acceleration components DDYC

IDWOI GI

APOLLO CooRDIHATE SYSTEM (SEE REFERENCE 2).

The p revious coordinate systems as described are similar to Project ApolloCoordinate System Standards No. 10, 13, and 12. A simple matrix rotation and,in case ct the space-fixed system, a shift of origin will convert the data toApollo Standard Systems.

0 1 0X (Apollo) = 0 0 1 Xe

1 0 0e

0 I J

X (Apollo) - 0 0 1 (X + i)' c 1 0 0 s o

i the data tape.

PC(1-n)

irameters.

.0

APPENDIX C

INPUTS:

A. The following parameters are preloaded as constants.

A EK mo Ro0-n)

AREA Mi PO VOLUME

AT(.-n)w 4o FEI(1

-n)

Mass Point Trajectory (OMPT).

*X9

*Y9

*Z9

B. Parameters input from the Observed

"Q.LT *RHO *X *X9 9

*A *X *Y *Ym m 9 9

*MAC'H *Y *Z *Z9 S

RADD

C, O O O O O O n C1 (` 0 r) O C' C G

Z

K

O

V

WrO

LO

x 7._ ZQ O

7-wa_a_Q

W

O

O

>C

K

pppppp Opi

pp pppp eQ o=0 0 C1 O O O O O 00 O O O G O O O O O O O O O O O O O O O O O O O O O O O O O O O O e 0 •i O O O• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •-^ N n ♦ M ♦ 0- •P p r N n! •P ! T P O r ry n ♦ •/^ d r` ! P o --. N n J .N d e P r7 .; 1V n J •P1 F. m PAArr ••+r.^ rrrN NN NN NNNN Nn M nn n n n MMM r/ ♦ ♦ ♦ ♦ •/ •I

•. O J Z c :v ! '4 ^ .^ •O J m n n A ^. a -+ J f► N r - . T 7 f -• J -+ '^ A •+ r •w N O ► ♦ .DJ dJ d.HnJ JA d O O-.mr f1NP J•d^Or •VPO JO ON ♦ ^ PrJ•rr^ J^IP1J nOmmM10 P► ON V1• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •OrM•^ OmNNC d00 dl t ►-r<p OnM► - >D P'• J1 ! •r TP^ O O^ n n d n r J N n N o m 01 f a C f J P` P- J\ -+ 0 0 J J1 f O m P- f O f r N d J♦ O C O 0 0 N /+ 1` O O r a T-• '^ IN m r •^ n ^1 v C D m D O n •^ .7 J •^ n V v m O s ! M v -+ N P '^ mr r .r r .^ .^ r -r r .r ..• .r .+ N N N V •^ ••1 n n n N \I N N ry 'V N N .••• r .... r r 1 1 1 I 1 1 1 '^ "'•

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 1

iO N a J 0 'v 'v d no .r P J V P d 7 N J n J1 J N f w r O m m Q V% J n J^ + -+ P d N A ♦ -. N N -+ .^ J1 m m mO •'1 0NNNONmON". • % r n t 1 mONR P P % P J ♦ ♦ NOV^w om/^ lr^rnr^

1

J• N N N O r d •f1 -+ N A ...r P n O r n r_ m O r f! J O m r- d N N O O N .!1r .^ f^ J r O f d r•1 n r m P- J ry N m F, ^'+ r f P; J V J1 N f r m r M m n rV V1 f P m N 0 P- M M N M J ON O4. d 3` f 3• a /^ N m O 'Ni J d O N n n N O P OW 0 J N ID U1 n ry •y •+ O F. { f M r Ir I^1 /► : d

rr rr r..r rr .^ NN V N Nnn n n ••1 N NNN N NN r.^r.^ r( 1 I I 1 I 1 ( r r1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Q P 1^ 0P P^ 4 lQ dJ1A d mP 'NO NOOP^ v1 0% PNN J•NMm PN0^+ N J n M f Nm 1 •fi Om rO Oramen dMNnr+P^ ? OJ n O y •n P r^v^J rO..N m++v+NmP •n P^ 1 ^ J OP tOP P a 1l^m OP• NO d• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •I P? J• O M J -+ A T P J1 J J ONTO V1 •D +^ O n ! N m rJ d •^ m D TNT d o •n N-^ P r m d d ^• X 0 0 JPm HOOP m dm nom J J rr•. Oddn Nm J1 NONPO4 NP40z 4 dmNn Mfr Pro Ord dM d d d d 1 ^^ •n .P1 d A h m O N J d T N n n N ''1 m m m 0- V. J r m M N - a (• A M1 n fy r r r ♦ .n 0 .•1 mN .-• r .r .r r.rr r r.... .r .v V NN N NnM M M MN tv N NN - - - - .v 1 1 1 1 I 1 1 ( •+^'• •"•M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1+ 1 1 1 1 1 1 ^

i INJ1 j/rA^ O ION U1J 0 0 d m PmNV11-0% JP mrPd d? d 0 d M f- O O n r n N L1 IN 3• r J• •^ J, h T r ry m d J 0 N m A4 M r O /^ P J d d P^ P- D n J^ P ••+ O^r1d0 J O^ r dPrf rP._.r ♦00 dd OrNPNNOrOMA U1rm PNP- J O.•10dOmOr Pr^ •'^• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •d 1.0- .P N c /- 40 1 n m J J O P d J1 m N n m m P P fn n J N •O O p- ! O a N 0 P- M M J d '0 n M1.O! J MnMn Nfy N n 11% -# VN d/- Pr n H1 J1t Tr000P Q.0 r PP40m P. 1P1 J Mf^1 Nr 1 r Nmrr r - - r.•+r-- -.ter r - - r N N N NN N N N NNr rr.+ 1 1 1 ^+^ i i I I I 1

QM^/+H1 JO M n...a

J

m OP J '4Od^n d ONOnNr^ M f m d.^ J O aNP ON, O •O n low/^PM.Mh dm

O d ^+ J N P^ P d J\ M 1/1 0 0- d J P n J1 d- p- J\41 o n o d M d T m J P -+ N m n N d f N O m r O o m n J CTOmnOf Q 1O 00' ^ 1. n om dPnm aOr.-•/► PNPm vN P.+N 01 P Mt 00^ VI II 0V: pv• • • • • • • • • • • • • • • • • • • • • • • • • .- • • • • • • • • • • • • • • • • • • • • • •y1 rA ^/•rrm 4) 0'% •'P- PNA J1 d ti f rNm ♦ Pf m AJ dN dN fO J d J OP CID 'W cc

OJ J n J n NNNMnJ J Jl r-m 7N.f.n J'• J NraOP q P.0 MOP •1D m^ D.hn NNr 1 Jr Or .r r .•. r r r r r r r r r r .r r fV 'v N N N 'M N N NNr M r r .•r r 1 1 ;

IIo-•' dit rf Im

dON D ff rRnN^NQP#,I N ^AP ri 'IN 0P.-. m J rry 4I J rrd OM

O etm ^ -1 J

^n J j n N •^ M •. d O P r P Q' J •y J P d .•1 N J ^ r1 0' .-. d O J• O N ti d N d P N P P J• ^ J N rQ .^ M J P d M d D r ^+ P. 'N' O 6P m h o M P J1 P rV 'v P- P 3\ m lt PI J n n n P d A d 611 m It ! ►^ fv r A P m m"•NW Nrf1m dT rd^ C• ff.+ d 'n d r- n P N a P 'O n D ,/1..n m P- d ••. n.P%r0 ONOPT •O 0m ^^ dd J 0' n n M IV N fV T M J J .P1 m C N d' 0 J J N U C C P P^ .f1 N P m r •O .P1 0" fv r r 1- N n •O m •+N r r i r r r r r i r r r r r N N N N N M N N N N r r r- 1 1 1 1;

1ON m J ON n J nP 4 W% MJ m M ra P M.O NP ..F / CD r OP dN T m J P •O - r f• I •^ /. O J1 O -•. ^+ N .. ^ h J ^' P P O J• J m M r r r-Q `3 J•• 0 C U v O 0V' ( 0IIN Jr n TrU-. y,O n d ^nN V. •^ d P m J 4)0 & J W I WI O!• y n.. I.- p- f- F► a d 0ON.•' •te rJ, Or rOd NO LL J P N '09% On 0 .O NO P.- Oj0J •M PP Pr r yN N • V 'v f v V v N N n n n n J f f.0d d J` ^ f^ J• 0 J "N,) N N N N N r r - O O 000 r N noo0o p 0000001 00000 00000 00000^0000o 00000 00000 000

^ I I 1 1 1( 1

0 R'.O.O NO 11 ON 0P. Tr 1-P. I C ror fv Pn.D Pr O ►-P N 1I•r O/vr dm d p- MhA ♦ W-0 n0;J P- -• kWrn dPN .OPntCNmJO'ON^nmos { mN OPN h Pr Nn .01 0ddd .h 1 00OOO-+rrNti Nfr1r'1n J j Y10CAP-mmP POO•} •^rNNfVM nn J J J J J Jt J { J •I f^

r A r r r r r r r A r r r r ..• ••A r r r

N NON MQK 4-mf r. +•O.r In 0 O jm Y1 f rf P` PO?fl P NOM •- .O N Mr Pry OP► P, 41 of Jn V• m CT 4 4 M Cn.00J 404.1 W +mMO TQ'N 1f 'nr T in ►.- P O•+M.. Nryr00•r^+0J1 - •^1 ^I N N •1 n T J f! ./1 J1 l^ D O A •p a p ? O 1 r r M-4 +1 n .n ! J f J 'n V, f, ^n •% In ^r1 N

0 0 o C o 0 0 0 0 0o 0 0 0 0 0 0 0 0 o c

0 0 0 0 0 o O o 0 o 0 0 0 0 0 0 00000 000o 0r000c00000 p 00000p0000o0oa0 00 00000^c0000 00000 coo• • . • • • • • • • . • • • • . • • • • • • . • • • w.y

• • . • P • • • • • • P. 0. •r n j Y1 d •^ m P 0 -• N n ♦ .P1 p- 0 ; z N M J U% a /- P O -+ N M J1 O A m a r fv M ♦ v1 • P

r .r r r r r r r N N PV N N ry ONO N M M M n A /.1 • P1 M M n ♦ 1 f' 4i

1•

I

1 I i hhhl I IO c O O O O O 0 O U G u O O O O O

^7

O O O O o o o O C O Oi I

O •

oococo^0oo^00000co^000c0000000000^000000^^^0000$000i 000cooc^0000000000000000c000000000000000 00000 000 ,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

orN••+! ono.-mo •^rN ^J,neh sPOrN^ t•no^-mP^rti- f,r o.-oo or•„•^.J.n o.-m o< I rn N N N N J• rn N N n O d d O d d O d d d A A A A A A A h A A m O m m m o m m m 0 P P P 0 0 0 P O P P

. ► ^ ^I r+ ^ c^ m y r I'•1 on .• mNr J N OP df NO ! "^T •n J r l •+MI ••>Dr^ ♦ m r-.O J vN•..r .. ...P r,: D ► w. d J rdt+.+PLV C

• • • • • • • • • • • • • • • • • • • • • • • • • • •O m J m •^ h an O P O VIN O h ow O N A A m P n r p P O m J In A m r In J m m d O N J O III h ♦ N Nm Jh^^OP .0O JNr•ONO OOAOrNOr A nNJ dP J ♦ V 3` Jr ! O N000 OmN?r0•••dPO

> v r 0 1 O N n n 7! /+ n v D 7 0 h •n ? JI T h n I^ P O •^ I+ O D r v J ./^ ^ ^+ P •/^ •- r J •w ^ rN+V I•I ^ ♦ ♦ ♦ NO d OAA r1w ♦ N00, Oa0 rrOPA O ♦ A PI rVr 1 1 rNN•n ! ♦ NNMN%I%1 1 1 1 1 r-.rr 1 1 1 1 1 1 1 1 1 1 1 1

1 I iI

O w n N a 00 •V O 0 N •D d VI 0 O m d m m .^ A •n P O ♦ A N Pd r •. J O P N P wI h N i m J N f O O t O m P P d P J N N N A 0 0 0 Y" 31 P d N N h O r m 0 ♦ N O •n N O• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •C r! A O J N f V' d f N I•• N •^ r I- A •^ J 4 rY O r /^ d 'n d? N M\ S t N J N ,D N r r A In N m N O 1 00 r

v. d •+ n P d N P P O•I J J P O O J .+ N f O J .n m O d N N N N O P .n N h T P P AI A 31 ►- ?< d m 3\J G_ O ^I O N d N P +I J• ►- N J O m 00 m P O O h N m -)Z ! P 3• T 'AO O r J I f N O m A N O P N ••I w1 N rIV rV N & J•nN OddA J 1 In 0/► MI 1 JAP r rPm dNJ O N I•INr 1 rNNPI ♦ ♦ ♦ NNV%1 1 1 1 -• r -+ 1 1 `I 1 1 1 1 1 1 1 1 1

J O-+N M 3`rN ♦ rrdC- rNO0 PF- VA N JO J m Jh +n OInN 3\J O J a1A nA V` ♦ ANNO Cr n P P r N N Q n O n .n ra s r Z d n J d N d O r A J r+ O f n V 0 3 A ac ♦ N N d h N d m A d P 0 T

• > f T h r J a J J .+ rj P •T N N J A 1 M m h A ♦ N T 'D (•) o J h O J ./'I f O! •^ d o O r A A N f h h N d mY. N h n J m O h I.I J O P .+ O rv? .• .n J J '^ O In h N .n n r- N r 0 0 P m VI VI d N MI + VI J r In A P N m • NO PI m OOP nPf rA A OJ hrd h RPLNA h J PNhf r ,O rn Nn wlrNrNO O m Ar d m TT JNrNNIn In In J ! 3` d d d'o lr.n ♦.ON 1 JA P -+-+-•PmAN ♦ 0" AI Nr 1 1 rN NI^IJ J JNNNNN

1 1 ^ 1 1 r r ,.• 1 1 1 1 1 I 1 1' 1 1 1

A A al P V^ R 1! 0 0 0 0 ♦ N In A •r N d d O A IP •H J d P r r J N •D A IV r .•1 J A A r .n m O r O j r A r NP ^+ O A N J O O^ N W V P h 0 j J r T m r A m r O m J r r 0 ••I MIA N N h wI d r r J! P N O aJ O N

d d 0 .n d G VI N P J T h r^ m d N? m P m O A ! f P •^ O r I •` v` J N .n N O d J o Q Y'• C .n m In h r r O n ♦r-. O T ♦ O m •^ h r A r p• f d J J w P w h A In d N P 040.1 f O U'% d J T 0 A O r r rz ID O A h m m T r N dQ J V: PN ♦ v0 •^dmmONa7PJ I •OOP f Pm J'" jAOr MI m P. ♦ 1 ^•JAOJA POr N IV Nr01 0NNN N •nTnTJ J 1 r N +►IJ N I I•IJ f d dN J J TN rrr I I rrr rNNN MrIIN N i

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N 3` m C .•I N w t•. 'D(Y.my P 0 O MN ry j r Jm r r•1f m rA JOHN 149% In 4m0 — qI NrOr N N N ^I T A l T r I I J N N M J • N ♦ f d d 'A W J T N N ••• r .. • 1 r r N N N N N N

Q

1 1 1 1^ 1 1 1 1 1^ 1 1 1 I I 1 1 1 1 1( 1 1 1 1 1 1II

I

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• • • • • • • • • • • • • w • • • • • • • • • • • • • • • • • • • •^1 1 1 1 I 1 1 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I

a• J O d CI A •V wI P J J` P w I. o r rh .• N r •- & I►• V` J J r m Q J m N T N r I^ r In Cr O O .O 0 A N J4 I^ Z N J b•c O J 7d •+NG r PONPIn rf dR V• r •^ZPf► N •^ .+d p-+-^ PYIO ^•r a-00 J Nd O.•IN Nr p PmA d J .hr O -^OC.••NJ J rnrpin dPrw+ f A m POO.a •-• •-a 000m O VI JN PA dO • • • • • • • • • • • • • • • • • • • • n • • • • • r • • • • • . • • • • • • • • • • • • ••^ ^• ••+^• I I I ' 1 1 1 rr r r..•.rNN MNN r,4 NN r rr.r .• 1 1 1

1 1 1 ^ 1 ^ 1 1 1 1 1 1 1 1 1 ^1

f

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REFERENCES

I. Jackson, J.w., "Propulsion System Evaluation throuqh Flight Simulation",MTP-AERO-62-69, dated August 31, 1962, MSFC.

2. Office of Manned Spaceflight: Project Apollo Coordinate System Standards,SE-M8-001-I, June 1965.