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NATIONAL AERONAUTICS AND

. . /,X__ '__v_/_ OCT2 I. 138@

IJ

SPACE ADMINISTRATION

MSC INTERNAL NOTE NO. 68-FM-256

October 11, 1968

VERIFICATION OF SUNDISK ORBITAL

NAVIGATION PROGRAM

MISSION

".'.'. ..... (NASA°T[_XQ_9716)

!:i:i:!:i:!:!:i:o __I $AL _,¢VI _A$_0°°.°°%°°°°°°°°°

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.'.'.-o.o..-oo=.

°°°°°°°°°%°°°°°.°.'.'.-o...o*o°

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"'5 , ' I _'; "_

Mathematical Physics Branch

PLANNING AND ANALYSIS DIVISION

MANNED SPACECRAFT CENTER

HOUSTON,TEXAS

VERIFICATION OF SUNDISK N7;4-70626

PI_OGHA_ (NASA) 112 p

Unclas

00/99 16149

MSC INTERNAL NOTE NO. 68-FM-256

PROJECT APOLLO

VERIFICATION OF SUNDISK ORBITAL NAVIGATION PROGRAM

By Richard Eo EckelkampMathematical Physics Branch

October ii, 1968

MISSION PLANNING AND ANALYSIS DIVISION

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

MANNED SPACECRAFT CENTER

HOUSTON, TEXAS

'W

llar_es C. Mcl:rSerson, Chief - -1_tthematical Physics Branch

Approved: _"_. 0 (_"1_( _--_.,_,

John P_ayer, Chief -'--=r"

Missio_=:qanning and Analysis Division

_J

.i

CONTENTS

Section Page

SUMMARY ............................. I

INTRODUCTION ........................... 1

The Kalman-Schmidt Filter ........ ........... 2

SUMMATION OF ORBITAL NAVIGATION EQUATIONS ............ 3

Coordinate Systems ....................... 3

Orbital Integration ...................... 4

Use of Landmark Sightings in the Kalman-Schmidt Filter ..... 6

The b-vector ......................... 7

The W matrix and a2 ..................... 9

The observation residual ................... ii

The CMC's Implementation of the Kalman Filter ......... 12

Verification of the Kalman factor .............. 13

Verification of the W updating equation ........... 14

Unknown landmarks ...................... 17

THE CMC CODING .......................... 20

Results of Coding Review .................... 37

Flow chart comparison .................... 37

Comparison of equation and astronaut procedure to coding. • 37

Illegal interfaces ...................... 37

ASTRONAUT PROCEDURES ....................... 37

BIT-BY-BIT TESTING ........................ 37

CONCLUSION ............................ 38

APPENDIX A - MIT FLOWCHARTS FOR SUNDISK REV 212 P22 ....... 39

APPENDIX B - MIT APOLLO COMPUTER LOGIC CHECKLIST INTERFACE

(GSOP IV) ........................... 61

iii

Section Page

APPENDIXC - APOLLO7 CREWCHECKLISTFORP22.......... 79

APPENDIXD - SUNDISKP22VERIFICATIONTEST- TWOKNOWNLAND-MARKSCASE.......................... 85

APPENDIXE - SUNDISKP22VERIFICATIONTEST- ONEKNOWN,ONEUNKNOWNLANDMARKCASE.................... 97

REFERENCES........................... 107

t

iv

TABLES

Table

D-I

D-II

D-Ill

D-IV

D-V

D-VI

D-VII

D-VIII

D-IX

E-I

E-II

E-III

E-IV

E-V

E-VI

E-VII

E-VIII

Page

DIFFERENCE BETWEEN THE BIT-BY-BIT AND THE

ENGINEERING-SIMULATION UNIT b-VECTOR ......... 87

THE BIT-BY-BIT UNIT b-VECTOR ............. 88

THE BIT-BY-BIT VECTOR UPDATES ............. 89

ENGINEERING-SIMULATION VECTOR UPDATES ......... 90

BIT-BY-BIT ESTIMATED LANDMARK POSITION ........ 91

ENGINEERING-SIMULATION ESTIMATED LANDMARK POSITION 92

DIFFERENCE BETWEEN THE BIT-BY-BIT AND THE

ENGINEERING-SIMULATION W-MATRIXAFTER THE SECOND

INCORPORATION .................... 93

DIFFERENCE BETWEEN THE BIT-BY-BIT AND THE

ENGINEERING-SIMULATION W-MATRIX AFTER MARK 5,.SECOND

INCORFORATION .................... 9h

BIT BY BIT W MATRIXAFTER MARK 5, SECOND

INCORPORATION ..................... 95

BIT-BY-BIT VECTOR UPDATES ............. . . 99

ENGINEERING-SIMULATION VECTOR UPDATES ......... 100

BIT-BY-BIT ESTIMATED LANDMARK POSITION ........ 101

ENGINEERING-SIMULATION ESTIMATED LANDMARK POSITION . . 102

DIFFERENCE BETWEEN THE BIT-BY-BIT AND THE ENGINEERING-

SIMULATION ESTIMATED POSITION AND VELOCITY VECTOR . . 103

DIFFERENCE IN THE RESIDUALS (ACTUAL MINUS ESTIMATED)

OF THE BIT-BY-BIT AND THE ENGINEERING-SIMULATED

POSITION AND VELOCITY VECTOR ............. 10h

DIFFERENCE IN THE DIAGONAL OF THE BIT-BY-BIT AND THE

ENGINEERING-SIMULATION W-MATRIX 105

DIAGONAL VALUES AT SPECIFIED POINTS IN THE

BIT-BY-BIT W-MATRIX .................. 106

v

VERIFICATION OF SUNDISK ORBITAL NAVIGATION PROGRAM

By Richard E. Eckelkamp

SUMMARY

This paper presents a detailed explanation and test verification

of SUNDISK 282 orbital navigation program 22 _P22). Included are a

step-by-step analysis of the coding and flow charts, a summary of the

equations utilized, an outline of astronaut procedures, and charted

results of bit-by-bit test cases.

INTRODUCTION

Orbital navigation, as contained in the roped version 282 of

SUNDISK for the command module computer_(CMC), will be exercised for

the first time on Apollo 7. This report fulfills a need of gathering

and interpreting the official documentation concerning the programunder one cover.

In orbital navigation, optical sightings on landmarks are statis-

tically weighed to correct the vehicle position and velocity in the

CMC as well as the position of the landmarks. This process can thus

determine both the orbit and the landing site.

The explanation of P22 will begin with a summation of the formula-

tion found in parts 5.1 to 5.2.8 of Section V of the GSOP (ref. 1).

A line-by-line interpretation of the coding, as presented in reference

2, will then be given. The MIT flow charts are available to aid in the

understanding of the coding. A resume of astronaut procedures and thetest results follows.

2

The Kalman-Schmidt Filter

Accurate estimates of the state vectors of bodies relative toparticular reference systems maybe computedby integration of theequations of state complementedwith processing of navigational obser-vations. Onemethod of processing observations, the Kalman-Schmidtfilter, adapted from filter theory in electrical engineering, statis-tically weighs single observations and uses them to correct statevectors while smoothing observational noise.

In orbital navigation the state vector consists of the commandmodule's position and velocity and the position of a given landmark inan inertial coordinate system. Telescope or sextant observations ofthe landmark relative to an inertial platform constitute the measurement.

Before presenting detailed information of the navigation processes,a brief outline of the Kalman-Schmidtmethod is useful.

To correct the state at time t.i

a) Make an observation Qi at t i

b) Integrate the state vectors as stored in the vehicle computer

to t.1

/Fr-v-c

r-c

v-c

i-i i

where r-c = position vector of command module

V-c = velocity vector of command module

c)

_r£ = position vector of landmark

^

Compute from this state vector the estimated observation Qi

3

^

d) Find the observation residual Qi - Qi = _Qi

e) Compute the weighting factor _. for the residual at t.-1 l

This factor is obtained by propagating the square root of the state

convariance matrix (or matrix of state uncertainty) to t..1

f) Correct the state

. . +r-C

V-C

i

i r-C

V-C

+ __iSQi

g) Also correct, or update, _ to -q_'+'where the plus denotes the

updated quantity.

This statistically weighting of single observations for vector

correctionworks well and is equivalent to the modified weighted least

square (WLS) method used for ground-based navigation. Generally, the

number of observations required to reduce initial errors to within the

system noise level and maintain this level of accuracy varies with

mission phase and navigation requirements.

SUMMATION OF ORBITAL NAVIGATION EQUATIONS

Coordinate Systems

To locate any body at a given time a coordinate system is

necessary. For the Apollo program, the basic reference system is the

nearest Besselian year (NBY) coordinate system.

The NBY system is defined in an earth-centered inertial Cartesian

system. The X-axis is along the line of intersection of the mean

equatorial plane and the mean orbit plane of the earth (equinox); Z is

along the mean north pole; and Y completes the right-handed triad - all

defined at the time of the beginning of the NBY. Vectors dated after

the Julian date of June 30 _re referenced to the following year. Forexample, the Besselian year 1968 begins on Julian day January 1.283,

1968; the Besselian year 1969, January 0.525, 1969. The center of this

system is translated to the moon's center when the vehicle is in lunar

reference.

4

As stated in reference 3, the CMC programs, including orbital navi-

gation, utilize an approximate system to locate earth fixed targets.

The transformation from this system to the NBY system for a state vector

_A in this CMC system is

io n0Z Z ~

_ANBY = 0 i i_ Az cos Az __y -A iX

where _A and _ANBY are column vectors. The onboard state vector _

first rotated about the true Z-axis by an angle -A whereZ

A = A + _(t + t )z zo ephem

is

A = angle between the X-axis and the Greenwich meridian atzo

midnight Just prior to the year preceding the reference Besselian year

tephe m = elapsed time between July i and the time that the

CMC clock was zeroed

t = time indicated in the CMC clock

= sidereal rotation rate of the earth

The transformation is completed by making small angular rotationsabout the X- and Y- axes. The values of A and A are constant for a

x y

given mission. These introduce an approximation. The resulting errors

are proportional to the elapsed time between t and the time associated

with the evaluation of A and A .x y

Orbital Integration

The state vector for the CSM is periodically integrated using

either the conic or the Encke method. In the conic mode, the perturbing

acceleration, ad(t), in the basic equation

' d 2 . "_e-- r(t) + r(t) =dt 2 -- 7-- _d(t)

is omitted. In the precision mode, an oscillating conic is defined atto, again ignoring _ad(to). The perturbing acceleration is then integratedseparately for any t + At using Nystrom's method. Whenthe perturbed

0

portion of the vector reaches a certain magnitude, currently defined as

8 kilometers, a new conic is defined, a procedure called rectification.

The time of the new conic, t ', iso

t ' = t + nat, where nAt is the elapsed time since the0 o

last rectifications at tO"

The square root of the state covariance matrix, W, is also propa-

gated forward. The covariance, or correlation matrix, is defined for

orbital navigation as

E(t) =

T T BTg g g O E

T T BTrl E q q tl

T 8Tea T 8 n a9 x9

where _, _, and _ are partial estimates of the errors in CSM position,

CSM velocity, and landmark positive, respectively. The W matrix isdefined by

E(t) = W(t)W(t) T

(T denotes transpose)and extrapolated by numerical integration of

10 i i]dw(t) = (t) 0 W(t)dt

L.O 0 0

9 x9

where_e

G(t) :r5(t)

[3_(t)[( t)T - _r2(t)I]3 x 3

I = 3 x 3 identity matrix

0 = 3 x 3 zero matrix

Despite the use of rectification, initial errors in the state and

dynamic biases neglected in the equations, for example, drag, eventuallycause the CMC vector to diverge from the actual vector. The W matrix

is subsequently affected the same. Navigation measurements are utilized

to correct both the state and the W matrix. In orbital navigation landmark

sightings are processed through the Kalman-Schmidt filter for thispurpose.

Use of Landmark Sightings in the Kalman-Schmidt Filter

CSM

u r _ landmark

Figure i.

Given that a mark relative to the navigational base coordinate

system has been made on a landmark and transmitted to the CMC,program 22will utilize this data to correct the state vector.

As outlined in the introduction, an observation is computed for the

same time as the observed data. The difference between the actual and

computed observation, _Qi' is weighted with _. and added to the statevector. -i

The b-vector.- In the computation of _., where--l

T i zTwT-_ = 2 ''_[2Z +

z = WWb

the b-vector, defined as

b = 8(COMPUTED OBSERVATION)- 8_COMPUTED STATE)

i)

_Qi (2)_X.

1

indicates how much and in which direction thestate will change for a

given change in the observation. Referring to figure 1 and following

the derivation from reference 3, let u be a unit vector pointing-S

toward an imaginary s,tar.

Assume u s × _rcA # O.

To get the b vector, take the dot product

rcACOS 8 = _c£ " _s

Take the differential:

6rcACOS 8 - rcASin e de =6Ec A . _s (3)

Now

= [r_.... E ]rcA

c£ cA

112

_rcA = ll2[_cA " r--cA]-ll2[6r--cA " r--cA+-_cA " _cA ]

[c£ " c_£

_rc£ = rc£(_)

Substituting ,(4) into (3) and rearranging,

re_ • 6rc£. - cos e - rc£ sin 8 de = 6rc£ u--S

rc£

rczCOS e ]- - us z___A____ . _rc_

68 [ rc £ rc£ sin 8

Now if we choose u perpendicular to r_--S

u-s

6e .... 5_c£rc_

This is the relation which defines b for updating the relative position

_c£" Since the CMC updates -cr and _, note

+ = [,%r_c r_c_

Then (3) becomes

6rc + 6rc_ = 6r__

-U

,se =-.-As • [6r_- 54]rc£

u

s _]=-- • [6r -

rc_ -c

The b - vector can be written compactly asm

b= l-l- I_-- rc£

Each observation point is incorporated twice, since two degrees

of freedom are available perpendicular to _c£" For the first directionof correction,

9

u (i) = unit × ×

In case the computer comoutation is difficult resulting from r c _being

parallel or near-parallel to , use

U--s

(i) = unit

For the second correction use u (2)= unitI_c × u (i_--S _ -'S '

where the plus indicates that _ has been updated by the first correction.

The W matrix and a2._ The 9-by-9 W matrix in equation (1) represents

a numerical estimate of the uncertainties associated with the mathematical

description space. These uncertainties include noises and bias on the

state vector and prediction models (lack of drag, for example) and limita-

tions of their representation for programming. For convenience, the

W matrix may be divided into nine 3-by-3 matrices:

W

Wo

W3

W6

W I W 2

w 4 w 5

w7 w8

(6)

At the beginning of a navigational sequence, this matrix must be

set to an initial value representing, among other things, an estimate

of state uncertainty. Each 3-by-B matrix is taken initially to be a

diagonal, since the initial self-correlations of vehicle position,

vehicle velocity, and landmark position, W O, W 4, W 8 respectively, cannot

be accurately determined a priori.

i0

Further, the initial correlation between W0,W4, and W8 are

simplified. The correlation betweenW0 and W4 represented by WI and

W3, are taken to be zero. For a known landmark, correlations between

W8, WO,andW4 are also taken to be zero.

For an unknownlandmark (un£), however,

W6 = K0 W0

W7 = KI WI

W8 = wun£ I + K2 W0, where K0, KI, K2, and wun£ areconstants.

These artificial correlations attempt to reflect the fact that the initialvalue of the unknownlandmarks position will be heavily affected by errorsin the vehicle state. In practice, K0 and KI are set to zero inSUNDISK. For the program COLOSSUSa more realistic approach exists forthe definition of errors associated with unknownlandmarks.

As a marking sequencing continues, the numerical size and character-istics of the Wmatrix change. Correlations between the various elementsgrow. After each state vector update from mark data, the Wmatrix isalso updated. The Wmatrix is extrapolated in time by integration, asoutlined in the introduction.

_2The term a in equation (i) is a constant representing the uncer-

tainties of the observational space. Noises and biases of the observinginstruments and the observer are included.

It should be emphasizedthat since the filter employed in orbitalnavigation is nonoptimal, i.e., all knowndynamic forces and dataerrors are not modeled, and the numerical values of _2 and the Wmatrixdo not reflect the actual estimates of instrument and state vectorerror. Further, since the filter is linear, the relative value of W

_2and a is the principle factor affecting the operation of the filter.For example, the filter will operate identically if values of W0 and_2a0 or k WQand k2 a2 are utilized, where k is any constant within thenumerical limits of the computer.

ll

The observation zesidual.- Recall from the introduction the equationfor correcting the state vector:

r-c

v-c

r_m

÷m m._

r-cl

= v + _6Q i-cm (7)

One must compute 6Qi:

Let _m be a unit vector in the NBY coordinate system pointing along

the measured line of sight. Since the optical measurement occurs in

thenavigationbase coordinate system, the transformation to NBY

coordinates is

u = [REFSMMAT] T [GIMBLE] [NAVGMBL] u

-mNB Y -m NBS

where the matrix [REFSMMAT] transforms from NBY to IMU coordinates,

[GIMBLE], from body to IMU coordinates, and [NAVGMBLE], from navigation

base to body coordinates. Superscript T denotes a transpose.

For the actual observation,

U U = COS 6)-S - m

cos • u) = eACTUAL"

For the computed observation

cos-I (us . _Um) =OCOMPUTED,

unit vector.

where the prime denotes a computed

12

Now

_Q= 8ACTUAL 8CO_UTED cos-l(_-- _ • u ) - cos-l(u • -=u')

6Q = cos-l(_ . _m ) _ _H , since -sU is perpendicular to the

estimated line of sight, _m"

The CMC's Implementation of the Kalman Filter

To utilize equation (7), up to five marks are made with the

telescope or sextant on a particular landmark. A CMC routine, auto-

optics, can aid the astronaut by aiming the observing instrument at the

estimated landmark position. (The accuracy of the aim is proportional

to the accuracy of the CMC state vector.) A residual, 6Q, is computed

with equation (8) and the b vectors are computed.

For computational purposes the W matrix is divided into 27 column

vectors within the CMC. For example, W0 in equation (6) may be written

_23

The nine dimensional quantities b, _, and _ are also divided as

zI

The Kalman computations are then performed as

2

zi:_ wi÷3j_-ji:0 l,2j=O , , (8)

13

2

j=o=-j 3i+J(9)

where 8 =1

8x. = 6Qm., where--1 ---i

(10)

Verification of the Kalman factor.- To verify that -_i' defined

in equation (9), is the Kalman weighting factor, which is normally

defined as

EM_T(MEM_T _-21-i (ii)_m= +

begin with the co.act definition given in reference i, as

i zT__= 2 __2--

z +_

which implies

_ 1 Wz (12)2 2 --

Z + I_[

To get (12) expand the Kalman equation (ii) using MIT terminology

_ = +_

(Z3)

14

From reference (i),

z = wTb

Since the basic Kalman filter in (ii) considers b as a row vector,

z = WTbT

(l_)

Substituting (14) in (13)

= W_(zz__T+ _2)-1

which is MIT's fo_ation (12)

_=

IWz

2 _2 --Z +

Verification of the W updating equation.- As noted in the

introduction, the W matrix is updated after each incorporation of

mark data. In CMC notation the update is

i = 0, i, ..., 8

_+gj = w - yzi_._i+9j --O

j = 0, i, 2

1¥ =

1+_

(15)

To verify that equation (15) follows from Kalman theory, proceed asfollows :

First equation (15) may be written (ref. i) as

T_Z

W+ =W --

i+ _+_

(16)

The Kalman update is defined as

_,+= (I - _)_,

Expanding in MIT notation,

(wwT)+ = wwT _ _bWWT

= wwT _ WWTb_T(b_WWTUT + _2)-IbWWT

15

Now following the derivation in reference 4,

(wwT)+ = w[z - _(z2 + _2)-ZzT]wT (17)

In order for equation (17) to be valid, the right side of the equation

must consist of some quantity times its transpose to update W.

Otherwise, only E could be updated and the CMC W scheme would be invalid.

Assume this quantity to be of the form

W[I- 8zz T ]-- (18)

where 6 must be found.

Equating equation (17) to equation (18) times the transpose of

equation (18):

W[I - _(z 2 + _2)-Iz_T]wT = W[I - 8_T][I - 6_T]w T

I - zzT(z 2 + N2)-I = I - 26zz T + 82zzTzz Tn--

wm n-- --w

82z 2 - 26 + (z2 + _2)-i = 0

Using the quadratic formula

2 ±_ - _z2(z2 + _2)-z6 =

2z 2

1 ±_1 _ z2(z 2 + _2)-1

2z

In order to limit the amount by which W is decreased during each

update, take the smaller 6.

Multiply by _rzz2 + ct

16

#z _22+_ _

z2 _zz2 +

Vz __2 . zII i

_2(%

_z 2 + j - _.2

2 Vz 2 _2z + c_

Since W+ = A + Bzz Tw = w+ _w=w[z- ]

AW = ,WSzz T.

Using (19)z2 +-d- - T

AW = -w _z_z

[z2 _z2 + "_2

_/iz2÷_ _ _,)(42+._2÷ _'t= -- "- P 2 + c_)

T i= -Wzz

-- 2 ._2 c_z 2 _2z + + + ol

T= -Wzz

i

-d2

(z2 + _.2) + z2 +(21)

17

Regrouping,

-Wzu

_W =2 _2

Z +

_Wz T

Tz

_z -2i+ 2+_ _2

i + z2 +7 2

Therefore, from (20) and (21)

W+= W-

which is equation (16).

Wz T

i+ 2+_ _2

UnknoSn landmarks.- The orbital navigation program can also use

optical marks made on unknown landmarks. In this mode of operation

the program uses the first mark to define the location of the landmark.

Following the derivation in reference 3, consider the plane deter-

mined by the planet center, landmark, and vehicle shown in figure 2below.

Landmark

,--e

I _u

I

ii-- CSM

Planet r--C

center Figure 2.

18

By vector addition

_£=r +z--c

To find _ and thus, [£, notice

(22)

z COS _ + r Z cos _ = r C

Z

r c - r£ cos _ rc

COS _) COS _)- -- COS

r C

Using the law of sines,

sin I sin-- = and

r r£C

rc

sin I = --sinr£

2and using the identity cos I + sin 2 i = i,

-IrqCOS I = _i- sin2 I = 1 k_/ sin 2

Now, from the figure

cos _ = cos[_ - (x + _)] = -cos(X + ,_)

cos _ = sin I sin _ - cos I cos

Combining (24), (25), and (26)

r__c 2 ,_/: _rr_£.c£')2 2cos _ = r£ sin _ + cos _ - sin

r£ 2 r£ rc 2--cos u = sin _ + cos _ -- L - l--J sinr rc C

Recall equation (23)

Z ___ w

rc

COS -- -- COSr c

(23)

(2h)

(25)

(26)

19

Z _ i - sin 2COS %)

%) -- COS _)-

r C

rc I 2COS %) COS%) cos _ L sin 2-- _ -- %)

r c

= rc[COS _ -_r_>2 - sin2 _]

So equation (22) becomes

r£ = r + r os %) - sin-c c _m

where u is the measured line of sight

(27)

o

Looking at the result for the landmark position, equation (27), onenotices the r£, the magnitude of the vector, is required. Since this is

unavailable for an unknown landmark, r£, is taken to be the mean radius of

the planet.

This approximation introduces considerable inaccuracies into the land-

mark position estimate. For the remainder of the marks, the landmark

is considered known. The resulting updates decrease the landmark error.

Having reviewed and approved the equations used in P22, this paper

will next provide an explanation of the coding of these equations.

2O

THECMCCODING

Review of the coding for the orbital navigation program has beenaccomplished chiefly through analysis of the MIT flow charts and studyof the decoding in reference 2. The Program 22 flow charts arepresented in Appendix A.

The coding analysis will follow through reference 2 line by line.

Explanation of each line will be given to the right of that line.

1. Start program

PROG 22

CODE

2. Perform R02

.

,

.

Set TARG2FLG = i

(bit 9 of FLAGWRD i)

Set TARGIFLG = 0

(bit l0 of _LAGWRD i)

Set RNDVZFLG = 0

(bit 7 of FLAGWRD 0)

6. Tdecl = Tno w

EXPLANATION

2. IMU STATUS CHECK;

if REFSM = i, IMU orientation

is known by computer and proceed.

if REFSM = 0 and IMU is on, per-

form P51 to determine alignment.

if REFSM = 0 and IMU is off, go

to POS to start up GNCS.

3. Target is a landmark.

4. Target is not L]94.

5. P20 is not running.

. Time to which integration is to

be performed is present time.

7. Perform CSM Conic 7. CSM R & V are integrated conically. •

21

8. PMGA=-i

cos [lunit (Vatt x R_att)

•

a Set RENDWFLG

(bit 1 of FLAGWRD 5)

10. TS = 0645vn

ll. Perform GOFLASHR

12. TS = 0112 Perform BLANKET

PROG. 22A

13. MARKINDX = 5

14. TS = 000118

15. Proceed to GOPERF1

16. If No R52, proceed.

If Yes R52, perform R52 and

R53 within and proceed to 18.

.

10.

ll.

16.

Assuming X of spacecraft is in

orbit plane, this checks to see

if IMU orientation is satisfactory

for P22. PMGA (displayed) is

maximum middle gimbal angle• > O .posslble. If 60 , and tlme

permits, IMU should be realigned

(P52), then key in V37E22E and

proceed. If not, proceed.

Maneuver to acquire landmark.

9. W matrix is invalid.

Machine channeling for 8. (display)

(specifies VN PATTERN for display

routine).

Display of 8; exit to realign

or continue as decided.

12. Display interface routine.

13.

14.

15.

Communication cell with R53.

Channeling to display.

Display interfacing for choice of

auto-optics (R52).

Monitor and respond to landmark

parameters. R53 is called within

R52 automatically or by selection

of manual optics.SCT trunnion or shaft and

trunnion may be driven toward

CMC's estimate of landmark posi-

tion.

22

17. Perform R53

18. NUM8NN= QPRETmk

19. NUMBKK= i

20. $22LOC= SVMRKDAT

21. SVMRKDAT+ i = EMARKSTAT+ i

i = 0, i, ...35

22. Perform 22 LMKID.

23. 22LMKIDa. LANDMARK= bits 9-4 of

22LMBD0

17.

18.

21.

Marks are made; then accepted

or rejected; up to 5 unrejected

marks may be stored.

Total number of marks taken =

total number of marks taken.

Serial no. of mark being processed.

Buffer cell location of 1st mark

data to be processed.

35 cells divided into 5 sets,

i for each mark, containing

(TIME 2:1) (TIME 1:2) (CD_),

(CDUS), (CDU z) (CDUT), (CDU x)

composing a table to avoid lossif restart occurs.

a. ID for first mark is placedin 22 LMBDO and coded as

A, B, C, D, and E where

A = i known

= 2 unknown

B = i coordinates stored

= 2 coordinates not stored

CD = landmark serial number

E = i (earth landmark)

and initialized as A = l,

B = l, CD = ID, E = 1

23

be

C.

d,

ee

f.

g.

h.

if LNDMKSTR = O,

(bit 7 of FLAGWRD 2)LANDMARK -- LANDMARK + 210

if LNDMKSTR = I,LANDMARK = LANDMARK + 29

if LNDKNOWN = 0,

(bit 8 of FLAGWRD 6)LANDMARK = LANDMARK + 21B

if LNDKNOWN = i

LANDMARK = LANDMARK + 212

_DMARK = _DMARK + I

TS = o57o w

Perform GOFLASHR

i. 22LMBDO = bit 9-4 of LANDMARK

J. Set bit 7 (LNDMKSTR) of

FLAGWRD 2 = bit l0 of LANDMARK

k. Set bit 8 (LNDKNOWN) of

FLAGWRD 6 = bit 13 of LANDMARK

Perform MKRELEAS

MKRELEAS

a. Release VAC areas assignedto marks.

b. MARKSTAT = 0

c. Inhibit interrupts

b, If landmark coordinates are

not stored, B = 2.

c. If landmark is stored, B = 1.

d. If landmark is unknown, A = 2.

e. If landmark is known, A = i.

f. Landmark is an earth landmark.

g. Machine channeling for display.

h. Displays LANDMARK, proceedor terminate and load new data.

i. Corrected ID is placed back in22LMBD0.

J. Landmark stored or not stored.

k. Landmark known or unknown.

a•

be

C.

VAC area is now available

for other use.

Mark storage cells are now

available for a future set

of marks.

Following sequence will not

be interrupted.

24

26.

27.

28.

d. Set bit 9 of OPTMODES = 0

e. OPTIND = -i

f. Set bit 2 of channel 12 = 0

g. Return

If ORBWOK (bit 6 of FLAGWORD 3)

= i, set WDIM91NC (bit 9 of

FLAGWORD 5) = i

If 0RBWOK = 0

w. = 0 (i = o, i, ..., 53)i

a. W0 = CWORBPS

W 4 = ,,

W 8 = ,,

W36 = CWORBVL

W40 = ,,

W44 = ,,

b, Set 0RBWOK = i

c. Set WDIM91NC = 0

Perform S22FLAGS

a. Perform INSTALL

b. Tdecl = ES22LOC dP

d. Optics switched from computercontrol mode.

e. Driving of optics is bypassed.

f. Disable optics CDU error

counter.

g. --_

26. If W matrix is valid for

orbital navigation, it is set

as 9 × 9 for incorporation

purposes.

27. If W matrix is invalid for

orbital navigation, ...

a. Reinitialize upper 6 × 9

to a diagonal in the

upper left 6 × 6.

b. W matrix is valid for

orbital navigation.

c. W matrix is not a 9 × 9

for incorporation purposes.

28. Preparation for integration.

a. Grabs integration packageand secures it from other

users.

b. integration will be up to

time of i mark, stored in E.

25

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

c. Set bits 5(STATEUP), c.

3(CSMINT), 2(WDIMEN 9), and

l(WMATINT) of FLAGWRD 3 = 1

CMpermanent state vector and

9 _ 9 W matrix are to be

integrated.

d. Let CONINT (bit 4 of

FLAGWRD 3) = 0

d. Encke integration.

e. Return e. --

Set WDIMEN9 (bit 2 of

FLAGWRD 3) = 0

29. W matrix is 6 × 6.

If WDIM9INC = 0, set WMATINT 30. W matrix is to be integrated.

(bit 1 of FLAGWRD 3) = 0

Perform INTEGRV 31. Integration.

CSMPOS= _Ratt

If LNDKN0WN = i, proceed to

22LMKI.

if LNDKNOWN = 0, Proceed

32. Answer.

33. If landmark is known, proceed

to 22LMKI; otherwise, proceed

[We are now treating an unknown landmark]

MARKDATA = $22LOC 34. Address of next mark data.

w. = o (i = 5h - 8o)1

W72 = Kwun

W76 = Kwun

W80 = Kwun

35. - 39. Lower right 3 x 3 ofW matrix is initialized for

unknown landmark tracking.

K2=I

[w8] = [w8] +K 2 [Wo]

X i = - MABKDATA

MARKDOWN = i = EMARKDATA + i,

(i = O-6)

h0. - 42. Mark data is shoved into

buffers for computations.

SI = MARKDATA + 2

Perform SXTNB 43. TS converted to double precision

for pointing vector computation.

25

44. Perform NBSM

45. uM = TS [_FS_T]

46. TS1 = unit CSMPOS

47. ALPHAV z = TSI, z

48. Perform GETERAD

49. oRBCOSA= -gM" TS.

50. ALPHAV = CSMPO_ = t COMPO_I

_R3C0SA '_/(i EP'ADM _ 2- co_o_l]

-(1 - ORBCOSA)J_UM

51. Delay 2 secs (via DELAYJOB)

52. Proceed to S22BIG2

S22BIG2

a. Set ERADFISC

(bit 13 of FLGWRD1) = 1

b. TS = ES22LOCdp

c. Perform LAT - LONG

d. If NUM8NN - NUM8KK _ 0,'

go to 9DWT06DW

e. NUMBKK = NUMBKK + i

f. $22LOC = $22LOC + 7

44. Computes pointing vector in IMU

coordinates.

45. Pointing vector in NBY coordinates.

46. For 50.

47. For &8.

48. Compute Fischer radius

at latitude of landmark.

49. For 50.

50. First estimate of [1.

a. Use Fischer ellipsoid radius.

b. Time of mark.

Co

do

eo

fo

rI converted to latitude,

longitude, and attitude.

If this is last mark, go

to ......

Serial nt_nber of mark date

is increased by one.

Starting address of next

mark data.

g. Proceed to S22BIGI g. --

27

[Now if we had marked on a known landmark, we would be back at 33 and

starting 22LMK1 ]

22 LMKI

a.

b.

If LNDMKSR (bit 7 of

FLAGWRD2) = 0, proceed to

22LMKDAT; otherwise, proceed.

TS = bits 14 - 4 of

22LMBO0, shifted right 3

places

c. Xl = LLATAB + 6TS

a. If landmark coordinates

are not stored, ....

c. Address of landmark

table entry.

d. LA_T-- -Xl d. Latitude, longitude, and

altitude stored.

e. Proceed to $22.16A e. mm

[If coordinates had not been stored in a above]

22LMKDAT

a. LANDALT = ALT/K2dcptl a. Scale altitude for display;store in LA_TDALT.

b. LANDLONG = 1/2 LONG b. Longitude/2 displayed for

better accuracy.

C.

d.

TS = 0689vn

Proceed to GOFLASH;

(1) If terminate proceed

to GOTOPOOH

c. Channeling for display.

d. Display of landmarkcoordinates.

(2) If proceed, proceedto $22.16

(2) if accept data, ...

(3) 0therwise, proceed. (3) if reject data, ...

e. ALT = K2dcptl LANDALT

f. LONG = 2LONDLONG

e., f. Data has been modified.

This is rescaling asabove.

28

a.

Proceed to c

w. = o (i = 5_-8o)1

b. W72 = CWmk

c. W76 = Cwm k

d. W80 = CWmk

e. If LNDMKSTR = i,

f.

S22BIGI

53.

54.

55.

56.

57.

[W8] = K3d 4 [W8]

Proceed to $22B161

Set ERADFISC

(Bit 13 of FLAGWRD i) = i

TS = ES22LOCdp

Perform LALOTORV

X789 = Krfactl ALPHAV

Perform S22FLAGS

58. Perform INTEGRV

59. CSMPOS = Rat t

6o. M_mDATA : S22LOC

61. X1 = - MARKDATA

62. MARKDOWN + i = EMARKDATA + i

_: (0-6)

63. SI = MARKDATA + 2

53.

g. Recycle from c till satisfied.

Initialize lower right

3 × 3 for known landmark

incorporation.

f.

If landmark is stored,

reduce W 8.

Use Fischer ellipsoid radius

for coordinate computations.

54. Time of mark.

55. Gets Ax, A, Az, and _.

56. Rescales r to kilometers.

57. Sets flags for and time of next

mark for integration.

58. W matrix and CSM vector are inte-

grated to specific time.

59. Answer.

60. Address of next mark data.

61-63. Mark data is shoved into

buffers for computations.

29

64. Perform SXTNB

65. Perform NBSM

66. UM = TS [REFSMMAT]

67. RCLP = X789 - Krfactl CSMPOS

64. TS converted to double precision

for pointing vector computation.

65. Computes pointing vector in IMU

coordinates.

Pointing vector in NBY coordinates.

Relative landmark vector.

68. Set FSTINCRP = i

(bit ll of FLAGWRD5)

69. TS = RCLP * UNIT Z

70. IfITsl<2-6km,USTAR=UNIT Y

68.

71. IfITsl__2-6_, usT_=UNITTS

72. Proceed to $22B164

$22B164

73. VARIANCE (RCLP) 2= _ (KscTVAR +

K )IMUVAR

First incorporation of measure-

ment data is being made.

69. TS now contains star unit vector

u- (o1.

-s [i]70. If 69 overflows, U (°) = 0--S

NBY

Perform BVECTORS

BVECTORS

a,

71. If 69 doesn't overflow, .....

72.

74. --

75. BVECTOR and other intermediate

Kalman computations.

BVECTOR 0 = unit(UNIT RCLP *- USTAR) -

b. UST__= B_CTO_Ro

DELTAQ = K2p i 1_c<PI

[cos -1 (BVECTOR_o . _UM) - 1/4],

(i/4 = 90 O)

C.

3O

d. BVECTOR, = 0

e. BVECTOR 2 = 0

f. RETURN

76. BVECTOR 2 =-USTAR_

77. Set WDIM91NC (bit 9 of

FLAGWRD5 ) = ]

78. Set CSMUPDT (bit 8 of

FLAG_DZ) = Z

79. Perform ]NCORPI

80. INCORPI

76. BVECTOR for update of landmark.

77. W matrix is 9 × 9 for incorpor-

ation purposes.

78. CSM state vector is to be

updated.

79. --

80. --

b.

= [wo] B_CTOR_0+ a-c.

[W3] B_CTO_RI + [W6] B_CTO_R2

zI = [WI] B_CTOR_o +

[W4] BVECTO_R 1 + [W7] BVECTO_R 2

c. z 2 : [W2] BVECTOR +

[W5] BVECTO_R 1 + [W8] BVECTO_R 2

d. If WDIM91NC = 0, d.

z2= 0

2 2 2e. LITLA = z + z 1 + z 2 e.-0

+ VARIANCE

TS =QLITLA VARIANCE f-h.f.

g.

h.

GAMMA = 1/(TS + LITLA)

DELQDA = DELTAQ/LITLA

Computation of z, where

z=wT b

If W matrix is not a 9 × 9

for incorporation purposes.

2 _2z +

Intermediate steps

in computing the Kalman.

31

81.

82.

i.

J ,

k,

i.

m.

n.

OMEGA 0 = z0 [W0] + zI [WI] i-n.

+ z2 [w2]

0MEG_A1 = zo [W3]

+ zI [W4] + z2 [W5]

OMEGA 2 = z0 [W6] + zI [W7]

+ z2 [w8]

DELTA X = DELQDA OMEGA-0 -0

DELTA X1 = DELQ,DA OMEGA 1

DELTA X2 = DELQDA OMEGA 2

o. RETURN

DSTEMldp = Kkmmtr 2 81,82.

IDELTA X01

DSTEM1 + 2dp = Kkmmc s 2

IDELTA Xll

TS = 0649vn 83-85.

Perform GOFLASHR

If terminate, proceed to

fourth line of S22BIG2 ;

if proceed, go to 87;

otherwise, proceed

TS = 002 and perform BLANKET

END of JOB 86. --

More intermediate steps and

state correction computa-tions.

Scaling for display.

Display routines and

incorporation decision.

87. Perform 1NCORP2 87. --

32

88. 1NCORP2

a. EGRESS = return address

b. Perform INTSTALL

88_

C, OMEGAM = GAMMA OMEGA-0 -0

_ o_o_i-- o_ O_GAI

e. OMEGa2 = OAn_A OMEG_A2

f. Set INTPHS (bit i3 of

FLAGWRD5) = i

g.

h.

i.

j ,

k,

[wo] --[wo] -zo OMEGa_o

[WI] --[Wi] - ZlO_G__0

[W3] = [W3] - Z_OO_G__I

[w_]= [w_]- zI OM_O_I

If WDIM91NC = i,

[W 2] = [W2] - Z20N[EGAM_o

[W5] = [W5] - Z 20MEGAM_ I

[w6]= [w6] - Z_oO_G__2

[W7] = [W7] - ZI OMEGA/J_2

[W8] = [W8] - Z 2 OMEGA/J_2

b. Grabs and secures

orbital package for matrix

updating procedures.

c-e. --

f. To preclude loss of

integrating package to

priority if a restart occurs.

g-j. W matrix is updated.

k. Additional W matrix updating

for a 9 x 9

i. If CSMUPDT = i

(bit 8 of FLGWRD1)

i. If the CSM state vector is

to be updated, do so.

33

(1) TSl = D_TA[cm

DELTAX-O

(2) TS2 = NUV_cm + DELTAX_ 1

(3) If there is no over-

flowing in computing

TS1 and/or TS2:

DELTAIc m = TSl_

NU__cm = TS2_

(4) If overflow has occurred

in computing TS 1 and/or

TS2:

_rectcm = RVCcm +

DELTAV + DELTAXcm -O

RCV =-cm _rectcm

V = VCV +-rectcm -cm

NUVcm + DELTAX1

VCV = V-cm -rectcm

DELTAV = 0-cm

NUV = 0cm

T --0ccm

XKEP = 0cm

(5) If WDIM91NC = 1

x 789_= x 789_+ D_._T__2

(1)-(3). For no overflow

condition add weighted

residual to perturbation

portion of vector.

(4 For overflow condition

add weighted residual to

the whole state vector,

i.e., the conic portion

plus the perturbed por-

tion, and consider this

update as a new rectifi-cation.

(5) If matrix is 9 × 9,

update the landmark

coordinates.

34

(6) Perform PTOACSM

(7) R = K- rfctr (RCV + TDELTAV)

(8) Z = Kvfct r (VCV + TNUV)

(9) Tpptm = Tet

(6) New vector is labeled

as the permanent state

vector.

(7)-(9) --

89.

90.

m. If CSMUPOT = 0 m.

n. QPRET = EGRESS

_. PROCEED TO INTWAKE

CSMPOS = Klk b 15 (RCV-cm

+ DEL AVm)

If FSTINCRP (bit ii of

FLAGWRD 5 ) = 1

RCLP = X 789 -

- Krfactl CSMPOS

n.

(9.

If LM state vector is to be

updated, perform i, but with

LM vectors.

Which returns

89. CSM position vector rescaled

90. If first incorporation of a

measurement, compute relative

landmark position and recycleto $22B164

Proceed to S22B16h

91. Proceed to $22B162.

92.

91.

ithin S22BI62. 12.d. If NUMSNN - NUMSKK < 0go to 9DWT06DW

9DWT06DW 92.

Recycle to S22BI62 to process

next mark

I_f all marks have been]

rocessed go to 9DWTO6D_

Converts upper 6 × 9 of 9 × 9

matrix to an equivalent 6 × 6

35

93. $22.1P5

a. I f LNDMKSTR _= 0

(bit 7 of FLAGWRD 2)

(1) If 22LMBO = K22mdmx '

22LMBDO = 0

93.

a, If landmark is not stored,

serial number of cell where

landmark data is stored is

kept as one, since only onelandmark can be stored in

Sundisk

(2) If 22LMBD _ K22mbmx '

22LMBD0 = 22LMBD

b. Proceed to $22EX3 b, mm

94. $22EX3 94.

a. LANDMARK = 22LMBDO a. Landmark code = number in

22LMBD0

b. Proceed to'S22EX33 b, mN

95. $22EX33 .....

a. TS = 6570vn

b. Perform GOFLASHR

95.

a. For display of landmarks ID.

(i) If terminate, proceed (i)

to 22TERM,

If proceed, go to l;(2)

(3)

If one does not wish

to see display of updated

landmark, ...

(2) If one wishes to see

them, ...

otherwise, proceed to S22EX3 (3) Otherwise, recycle ....

c. TS = 1012 and perform c. Kills displayBLANKET

d. End of Job d. --

e. LANDALT = ALT/K2dcptl e

f. LANDLONG = 1/2 LONG

e. Scaling for display

f. For display

36

96.

g. TS = 0689vn

h, Proceed to GOFLASH;

(1) if terminate, proceed to

22TERM ;

(2) if proceed, proceed;

(3) otherwise, go to i.

i, TS = bits 14-4 of 22LMBDO,

right shifted 3 places

j. XI = LLATAB + 6 TS

k. EXl--_AT

1. If LNDMKSTR (bit 7 of

flagword 2) = 0

22LMBDO = 22LMBDO + Klbll

m. Proceed to 22TERM

22TERM

a. TS = OOO178

b. Proceed to 60PERFI;

(l)

(2)

96.

If terminate, proceed

to GOTOPOOH ;

If proceed, proceed to

GOTOPOOH;

(3) Otherwise, proceed to

PROG22A

g.

h.

Verb-noun flash for display

of updated landmark coordi-

nates

(z) If one doesn't want

updated landmark coordi-

nates to be stored as the

landing site, .._.

(2), (3) --

i-k. Landmark data is stored.

i, If landmark is not tagged as

stored, indicate that one

has now been stored.

m. --

ao

b.

(i) and (2) P22 is finished.

(3) Recycle to perform

another sighting

37

Results of Coding Review

Flow chart comparison.- A comparison of the above coding and the MIT

flow charts in appendix A reveals no differences except occasional

insignificant order changing of steps and the omission of the W8 down-grade (page 9 of appendix A) for a stored landmark in the coding. In

this latter instance, as at other times during the review, reference to

the actual SUNDISK revision 282 listing was used to resolve difficulties.

The W 8 downgrade is not contained in the 282 listing.

Comparison of equation and astronaut procedure to codin6.- Besides

the comparison between coding and flow charts, checks have been made

between coding and equations, and coding and astronaut procedural inter -

faces, as outlined in reference 5 and reproduced in appendix B. In all

instances, the comparisons were satisfactory. P22's interfaces with R52,

R53, R02, CSMCONIC, P52, and P00 also appear satisfactory.

Illegal interfaces.- Not all interfaces with P22 are valid, however.

Generally, none of the P30's or P40's (P3X or P4X) should be called while

P22 is running, or vice versa, due to shared erasable cells (ref. 6).

Verb 82 cannot be used during P22 for the same reason (ref. 7).

Programs 22 and 20 cannot be operated simultaneously since P22 initializes

the W matrix to a 9-by-9 (ref. 8). Program 22 or any program cannot be

called by verb 37 if integration is in process, or in P05 until the time

specified in reference 8. Before running any program, reference 8 andits updates should be read.

ASTRONAUT PROCEDURES

Crew procedures for exercising the orbital navigation program are

presented in appendices B and C. Appendix B contains the procedures and

interfaces among the crew, CMC, and the ground, as written by MIT in

reference 5. Appendix C gives the concentrated crew procedures for P22

which are to be carried onboard the CM during Apollo 7 (ref. 9).

Comparison of appendices B and C reveals no differences save the

omission of the IMU status check (R02) in appendix C.

BIT-BY-BIT TESTING

The final "roped" version of SUNDISK 282 has been tested on the

MSC bit-by-bit simulator. This simulator contains an exact representation

of the CMC, i.e., the machine language coding is the same.

38

Results of the bit-by-bit tests for orbital navigation are contained

in appendices D and E. The tables (ref. i0) contain comparisons between

the W-matrix, b-vector, and state vectors as computed by SUNDISK 282 P22

and by a reliable onboard bench program outlined in reference 3.

Two bit-by-bit cases were run. The first case (appendix D) involved

sightings on two known landmarks. The second case (appendix E), is a rerun

of the first case with the first landmark considered unknown. In both

cases, only three marks could be taken on the second landmark due to

constraints within the environmental program associated with the bit-by-bit simulator.

For both cases the results are excellent. Differences between the

SUNDISK program and the engineering simulations are small and can be

attributed to differences between fixed and floating point machines. These

results, added to other studies of P22 with the engineering simulator,

indicate the program operates correctly and performs well in flight

configuration.

CONCLUSION

From the study of equations, coding, astronaut procedures, and

testing which has been presented, SUNDISK 282 orbital navigation program.

P22 has been verified as valid for earth orbital flight.

"39

APPENDIX A

MIT FLOWCHARTS FOR SUNDISK REV 212 P22

41

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KNOWH LAT XX .XXX

K2-12KXI LONG/2 XX. XXX

. . _ _T , ..xxx.xx_I [[_0 LDD

1

[oKk_N V06 N49 ORB PARAMETER

$TRD FL

r UNKWN/ AR KXXX.X NM_'NSTRD AV XXXX .X FPS

PRO---_UPDATE STATE V34E---_REJECT .MARK DATA

I VECTOR (Tillall marks used)

ALTER LAST MARK US_)

I FL V05 NT0 IL_K ID

"I R2 x,xc_x....

_gQ V_d_-pITEP 15!

L

s_

P22,23

q

0

4-4

RECORD

PRO (stores in CMC)

14 FL V06 N89

LAT XX. X_X.X

LONG/2 XX, XXX

ALT XXX. XX

V34E (Don't store)

FL V5G N25 | PERFORM

R1 00017 ] ADDITIONAL SiG_TINGS

IENTR--_RET STEP 6

l

_ iIPRO

FL V50 N07 ]

,_,K IDENTITY

DEG (+NORTH)

DEC; (+EAST)

N.N

17 G/N PP_ OPTICS - OFF

_xiP23-CSM-CISLUNAR_MIDCOURSE NAV MEASURE_._ENT PROC_IM

CMC - ON(Req) pg 0/2-1

G/N PWR OPTICS - ON (up)CMC ATT - IMTJ

.05G sw - OFF

SCS LOGIC - BUS 3

OPTICS MODE - MAN

OPTICS MODE - ZERO (15 secs)

Move RHC to LEB

Select SC control

PERF STAR - L_[K ACQ

1 K_y V37E23E

_ev59 J rERFORMOPT:CSCA_IE2 !1 iCtr moon in SCT PRO----_STEP 4

ACQ LUN_ I./,IK IN SXTSUPERDIPOSE SI,OS & LLOS

MARK

l

I FL V06 N87 _ BIAS ANGLE(TRUN)R2 XX.XXX J DI';G

PRO V32E--_RI,;T STEP 2

I

I-

O"

oo r

f|

._ e- L

_3

o

85

APPEN IX D

SLTNDISK F22 VERIFICATION TEST - TWO KNOWN LANDMABES CASE

87 ¸

TABLE D-I.- DIFFERENCE BETWEEN THE BIT-BY-BIT

AND THE ENGINEERING-SIMULATION UNIT b-VECTOR

Mark no.

1

2

Incorporation

no.

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

bl

.00010925

-.00002955

.00625410

.00000434

.oooo187o

-.oooo1613

.oooo54oo

.00004329

.00003496

-.00021332

.00006993

.000003

-.00000597

-.00001594

.00004537

.01065872

b2

.00039536

.00012107

.O0OOO885O

-.00002697

.00008554

-.00000897

.00006314

.00035168

.00006639

.00015567

.00029096

-.00002593

-.00016595

.00004936

-.0000474

.00019926

b 3

.00002127

.000000174

.00000745

.00000037

.00001657

.00000034

.00001012

.OOOOOO38

.00000435

.00000072

.OOO129O3

.000012

.00026864

.OOOOOO18

.00001425

.0OO0523O

88

TABLE D-II.- THE BIT-BY-BIT UNIT b-VECTOR

Mark no. IncorporatiOnno. bl b2 b3

i

2

3

4

i

2

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

First

Second

.01722365

.96760445

.01692410

.99287134

-.01325370

.99284787

-.06199400

.87520529

-.09830496

.59285168

-.02082393

.99875302

-.10489403

.97362556

-.39230337

.23289772

.06601096

-.25247107

.08906350

-.11919103

.11024054

.11938646

.11216514

.48375168

.07236639

.80531167

.41681896

.04992407

.44763405

.22815184

.09168866

.97249826

-.99767023

-.OOOOO018

-.99596855

.000OO063

-.99381657

.00000234

-.99175388.00000262

-.99252165

-.00000277

-.90875097

.00001247

-.88804336

.O0000115

-.91525475

-.00240317

89

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HL0 I O l _0 • ,--I I I 0J0"_I L0 ,'-I I b-- I"

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CO O0 CO O0 O0 C_ O_ O_

° 91

TABLE D-V.- BIT-BY-BIT ESTIMATED

LANDMARK POSITION

Mark no.

i Before inc. i

After inc. 2

2 After inc. 2

3 After inc. 2

4 After inc. 2

5 After inc. 2

1 Before inc. 1

After inc. 2

2 After inc. 2

3 After inc. 2

X

-ii 915 844.

-ii 920 412.

-ii 932 272.

-ii 946 187.

-ll 960 939.

-ll 974 579.

-16 71_ 688.

-16 720 517.

-16 727 617.

-16 746 h19.

r£, ft

12 998 932.

13 000 596.

12 989 420.

12 974 770.

12 959 903.

12 946 970.

7 377 763.3

7 379 7o2.1

ii 228 431.

ii 221 747.

ii 221 465.

ii 221 155.

ii 220 926.

ii 220 790.

io 167 490.

i0 163 640.

7 362 582.6 lO 161 587.

7 313 567.1 lO 158 936.

92

TABLED-VI.- ENGINEERING-SIMULATIONESTIMATED

LANDMARKPOSITION

r£, ftMark no.

X Y "Z

i

2

3

4

5

i

-11 915 826.

-ii 920 460.

-ii 932 336.

-11 946 184.

-ii 960 923.

-ii 974 566.

-16 714 671.

-16 720 415.

-16 727 508.

-16 746 321.

12 998 456.

13 000 166.000

12 988 976.

12 974 563.

12 959 814.

12 946 910.

7 377 346.9

7 379 474.1

7 362 355.6

7 313 570.4

ii 228 979.

ii 221 841.33681

ii 221 542.

ii 221 259.

ii 221 037.

ii 220 907.

i0 167 805.

i0 162 551.

lO 161 539.

i0 159 026.

95

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.97

APPENDIX E

SUNDISK P22 VERIFICATION TEST - ONE KNOWN, ONE UNKNOWN LANDMARK CASE

E_<

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i01

TABLE E-III.- BIT-BY-BIT ESTIMATED LANDMARK POSITION

Mark no.X Y Z

Unknown

2 Before Inc. i

After Inc. 2

3 After Inc. 2

4 After Inc. 2

5 After Inc. 2

-ii 931 967.

-ii 932 667.

-ii 946 634.

-ii 961 449.

[-ii 974 998.

i12 992 613.

12 992 693.

12 979 028.

12 962 645.

12 949 ll9.

ii 220 524.

ii 220 577.

ii 220 454.

ii 220 120.

ii 219 936.

Known

i Before Inc. i

After Inc. 2

2 After Inc. 2

3 After Inc. 2

-16 714 688._

-16 720 707.

-16 727 834.

-16 746 721.

7 377 776.1

7 380 254.o

7 363 162.1

7 315 078.5

10167 490.i0 161 428.

i0 160 369.

lo 158 148.

102

TABLE E-IV.- ENGINEERING-SIMULATION

ESTIMATED LANDMARK POSITION

r£, ft

Mark no.

X Y Z

Unknown

2

3

4

5

-ii 931 917.

-Ii 932 656.

-ii 946 618.

-ii 961 434.

-ii 974 969.

12 992 613.

12 992 689.

12 979 045.

12 962 673.

12 949 141.

Known

i

ii 220 566.

ii 220 609.

ii 220 501.

ll 220 168.

ii 219 999.

-16 714 671.

-16 720 716.

-16 727 763.

-16 746 629.

7 377 346.9

7 380 018.0

7 363 222.7

7 314 991.9

i0 167 805.

i0 161 313.

i0 160 388.

i0 158 136.

103

TABLEE-V.- DIFFERENCEBETWEENTHEBIT-BY-BIT ANDTHEENGINEERING-

SIMULATIONESTIMATEDPOSITIONANDVELOCITYVECTOR

Mark no.

[

2

3

5

|

Time of state vector

comput at ion

Initialization

Before incorp.

After second incorp.

After second incorp.

Before incorp.

After second incorp.

Ar , ft-C

....AX AY

0.0 0.0

Unknown

AZ

0.0

2.0 2.0 -6.O

6;o 24.0 -2.0

-44.0 26.0 -3.0

Known

ii.6 2.0 2.0

17.o 78.7 -30.

Av , fps-C

0.0 0.0

-.002 .005

-.o24 .029

.046 -.042

-.016 -.004

-.022 .005

.OO8

.024

-. 028

.0O2

.040

104

TABLE E-VI.- DIFFERENCE IN THE RESIDUALS (ACTUAL

MINUS ESTIMATED) OF THE BIT-BY-BIT AND THE

ENGINEERING-SIMULATED POSITION AND VELOCITY VECTOR

Mark no.Time of state vector

computation

Initialization

2 Before incorp.

After second incorp.

After second incorp.

Before incorp.

After second incorp.

After second incorp.

6rc, ft _ _v c, fps

-AY _ _,

O.OlO.O o.o o.oUnknown

-2.0

-4.7

-31.4

Known

-1.7

20.0

-76.

_ -.o05

-.oo5

.010

.004

.011

-.OO7

.ooo6

-.0006

105

0L'q<_ 0

4-_

<30

.,::3

0

<3 0

0_I

_£_ oI

Pq _

o_ _ >'_

,M

H_ 4._

e_

' J_ _ o

O

._ o

O o ,..-I

_ ,_--t

0

o _ _,o cO 0"_

o I I" I

,.H0", 0"_ ,-I

0 00 u x o

0 o.I t" OJ

o _0 _ _ _ r

_0 I I

OJ

I I l

0"_o oJ kO

o

I I I

,-i

o _ o

I I I

0,-I ,-4 ,-0

0 I" I

0 0 oJ

0 I I

oJ oJ aO,-I ,M o_

0 o

o I" I

OJ

_ o

_D 0OJ I._

I I I

CO OJ COOJ _0

OJ I r_I

',D

,--4 ,_

r_

I OJ

O

I I I

0e3 m ,--I

I° I

I I I

106

H

H

D_0_

CO

_q

OO

H

I

HHI--I

I

F_

I

E-4H

I

I

H

H

HO

_ 4O_ o.I OJ

0 C_ 04oJ _- _ ,-I

_ 0

.t.q

o>l

.X

o

X

O

o 0

-_ 4a

O@ o

cO

CtlO

t"q

0d

0", CO ed0", _ M3

C_ _ oOoJ .--I

8

O4

Oq Lf'X _04 .._ Lr_0 O_ O0 0

Dq I" II

Lf'X L_ Oq

0 0 r--I r--I

_ l" I I"

O_l0 0

d _ 4o_ _ co

o4 c_l

_ 4

(xl ctl

_ 40", c_l0,1 oJ

0.M4_

b_

,-t.rq4_.et

.-_ oqo C_

ox c_ 0

I I°

_ ,O

J _; 4_-- Lr, '.__q _ ,_I

I I I

4 _ 4O_ Oh o',

8 _ d_-I _-_ ,-M

I I I

_._

107

REFERENCES

i. MIT: Guidance System Operations Plan for Manned CM Earth Orbital

Missions Using Program Sundisk, Section 5, Guidance Equations

(Rev. 2). MIT Instrumentation Laboratory, March 1968.

o TRW: Programmed Guidance Equations for Sundisk Command Module Earth

Orbital Program, Revision l, based on DISK 282 Program.

TRW/Houston, NAS 9-4816, February 25, 1968.

. Clifford, J. B., Jr.: Apollo Coasting Flight Navigation Simulation -

OBSIM/NAVSIM Program Formulation. TRW note no. 67-FMT-5B1, July 25,1967.

_. Phillips, Laurel A.: Apollo Onboard Orbit Determination Equations.

MSC memorandum no. 66-FM42-228, August 4, 1966.

. Guidance System Operations Plan for Manned CM Earth Orbital Missions

Using Program SUNDISK, Section 4, GNCS Operational Modes (Rev. 2).

June 1968.

6. Kimball, Garner R.: Additional Sundisk Program Notes. MSC Draft

for Corrections to Sundisk Program Notes, July 8, 1968.

7. Fox, M.: Discrepancy Report Status (June 28, 1968).

TRW memorandum no. 68:7252.1-99, July 9, 1968.

8. Kimball, Garner R.: Updated Sundisk Program Notes and Discrepancies.

MSC memorandum, May 18, 1968.

9. Flight Crew Support Division, Spacecraft Systems Branch: Crew Check

List, Apollo 7. May l, 1968.

10. Olah, G. T.: Sundisk P22 (Orbital Navigation) Verification.

TRW memorandum draft, September 1968.