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MSC INTERNAL NOTE NO.LptfnOiE

PROJECT GEMINI

LOGIC AND EQUATIONS FOR REAL-TIME COMPUTATION OF ARRIVAL TIME

AT AN EQUATORIAL CROSSING, A SPECIFIED LONGITUDE,

A SPECIFIED RADIUS, AND APSIS POINTS

7

Prepared by: Loma Lue rag?Thomas Buckman

Prepared by: aaaStan Mafhn

Approved:areeaLe

Chief, Rendezvous Analysis Branch

Approved:

John P. MayerChief, Migbion Planning and Analysis

Division

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

MANNED SPACECRAFT CENTER

HOUSTON, TEXASDecember 14, 1964

TABLE

Section

1. Introduction

2. Discussion of Subroutine

3. Discussion of Subroutine

4. Discussion of Subroutine

5. Discussion of Subroutine

6. Required Inputs

7. Subroutine Outputs

APPENDIX

OF CONTENTS

STLO

STEQ

STCIR

STAP

Le Detailed Flow Charts

II. Definition of Flow Chart Symbols

rhco

fF

WW

10

-l-

1. Introduction

The purpose of this internal note is to discuss the development of four

real-time computer program subroutines which are used in the General Pur-

pose Maneuver Processor (GPMP) and the Agena Corrective Maneuver Processor

(ACMP). The subroutines discussed herein are:

o STLO - Determinesthe next time of arrival of an orbiting vehicle at some

specific input longitude

o STEQ - Determines the time of arrival of an orbiting vehicle at the next

equatorial crossing point

o STCIR - Determines the next time of arrival of an orbiting vehicle at

some specific input radius

o STAP - Determines the time of arrival of an orbiting vehicle at the next

true apogee or perigee

These subroutines form a part of the ACM and GPM Processors and are utilized

to establish the time at which certain maneuvers should be scheduled.

ie Discussion of Subroutine STLO

Subroutine SILO requires the input of a set of vehicle orbital elements for

a given time and the desired longitude (A). Since the longitude of ascending

node (hy) of the input vector will be based on an inertial reference, it

mst be changed to an earth fixed reference. This is accomplished by sub-

tracting the earth rotation (wt). The distance along the equator (VY) be-

tween the present earth fixed ascending node (nh) and the desired longitude (0)

is found by subtracting (h,) from (0). A negative distance along the equator

would be meaningless here, so 2x mst be added to the above result if Qa = h,)

is negative. The argument of latitude of the desired longitude crossing

(U,) is computed with the use of a spherical triangle that includes the incli-

nation (I;) and the proper equatorial distance (v). An initial time of arrival

estimate (At) is made by applying the vehicle mean motion (n) to the difference

between the argument of latitude of thedesired longitude (U,) and the present

argument of latitude of the vehicle (U;). However, this time estimate will

not be entirely correct because the earth rotational rate will affect the

~-2.

actual time of travel between the present position and the desired longi-

tude. This discrepancy is initially corrected by adding the earth rotation

(war) to the desired longitude (XY. A more accurate time estimate is then

computed using the "adjusted" longitude (,,). When the longitude has been properly

adjusted to account for earth rotation, the AEG Subroutine is called and the

initial vector is updated to a time equal to the initial time plus the latest

At estimate. The argument of latitude of the vehicle at this time is checked .

against the argument of latitude of the adjusted input longitude. If the

difference between these two parameters indicates a time difference greater

than an input tolerance, a new At is computed and the AEG is called again.

This iteration continues until a convergence on UL is attained. The time

that permits this convergence will then be the desired time of arrival at the

specific longitude (\) and is denoted as tron:

3. Discussion of Subroutine STEQ

Subroutine STEQ begins the search for the next equatorial crossing with a

set of vehicle orbital elements for a given time. The value of the vehicle

argunent of latitude (U,) at this time is checked against x to determine if

the next equatorial crossing will be the ascending or descending mode. If the

upcoming node is ascending, the argument of latitude of the next crossing point

will equal 2x. If the upcoming node is descending, the desired argument of lati-

tude will equal x. A first estimate for the time of arrival is made by applying

the vehicle mean motion (ny) to the difference between the argument of latitude

of the vehicle and the argument of latitude of the next crossing point. The

110 in the flow charts. This time

estimate is used as the initial estimate in an iteration loop using the AEG

Subroutine. The AEG is called and the vector is updated to the time estimate.

If the vehicle argument of latitude at this time is not within a pre-set tole- '

rance of the value Ki40 a new time estimate is computed and the iteration loop

is entered again. The time corresponding to the argument of latitude that per-

mits a convergence in the iteration loop will then be the desired time of equa-

torial crossing (tag)

4. Discussion of Subroutine STCIR

Subroutine STCIR requires the input of a set of vehicle orbital elements for

a given time and the desired radius of circularization (Rotp)- This subroutine

later parameter is denoted by the value K

« 3 «

begins its search for Ror at the time of the input elements. An initial

calculation of the true anomaly of the vehicle at the desired radius is

made using the following Keplerian relationship:

a(1 - e°) -RrR CIR) (1)fie cos”CIR |

This true anomaly (f,) is compared to the true anomaly of the present posi-

tion (fz) and an initial time of arrival estimate is computed by applying

the mean motion to the difference between these parameters. Since the

computed true anomaly is not a unique solution due to the existence of

two Ror positions in the orbit, a further test mst be performed to insure

the use of the next desired radius point. The AEG is updated to the initial

time estimate, and the true anomaly of the resulting vector is noted. A

more accurate computation of the desired true anomaly is made using equation

(1) with the updated elements. This computed true anomaly is again compared

with the actual true anomaly at the updated position and if the difference

between them is greater than a pre-set tolerance, a new time estimate is :

calculated and entered in the iteration loop. In the course of iterating

on the proper true anomaly, it is possible for a radius (Rorp) that initially |

exists in the orbit to become non-existent in the present orbit due to drag.

The presence of this situation is revealed when the value of cos (f,) becomes

greater than one. Should this situation occur the subroutine sets an error

flag (Ki9,) and returns to the point where it was called. The time estimate

that produces convergence on the computed true anomaly will be the proper time

of arrival at the input radius and is denoted as torr:

5. Discussion of Subroutine STAP

The STAP Subroutine requires the input of a set of vehicle orbital elements

for some time and a Ki00 flag which determines which Apsis point is desired

(Ki09 = 0, Apogee —- Kyo, = 15 Perigee). A first estimate of the time of

arrival at the next desired apsis point is made by comparing the present

mean anomaly to either x or 2x, depending on the Koo flag setting. If -

the apsis point desired is apogee, a check is made to determine if the apogee

~4&.

point has already been passed in the present orbit. If this is the case,

then the present mean anomaly is compared to 3 rather than x to obtain

the proper time estimate. The AEG is called and the present vector is

updated to the initial time estimate. The mean anomaly at the updated

position is then compared to the proper multiple of x (Apogee -- x, Peri-

gee --2x). If the difference between these values is greater than a pre-

set tolerance, than a new time estimate is calculated. This updated time

estimate is then re-entered in the iteration loop. The time that permits

convergence at the proper mean anomaly will be the correct time of arrival

at the apsis and will be denoted as taps

6. Required Inputs

The input that is required for each subroutine is divided into two major

categories: constant input and variable input. The constant input is also

subdivided further into three groups: 1) input that will not change after

three weeks prior to launch, 2) input from three weeks prior until the

program is loaded, and 3) input required after the launch.

This input is listed in the following sections:

3.1 STLO

3.2 STEQ

3.3 STCIR

3.4 STAP

3.5 AEG

3.1 Input for STLO

I. Constant Input

a. Category

1. 15 O52 Kigy> Kyoo? Bios

2. 66, St

3. None

II. Variable Input

h, t, h, U, gn

TIT. Output

t e, I, g, h, £,; R, U, &, nLon? %

IV. Constant and Variable Input Required for:

AEG/DRAG

3.2 Input for STEQ

I. Constant Input

a. Category

lL.

2. 6t

3. None

II. Variable Input

t, & u,n

III. Output

te? a, e, I, g, h, &, R, I"

IV. Constant and Variable Input Required for:

AEG/DRAG virsnitees

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ltinaapy

apitinsoneneniteen

hitibi

tbiime

annsn

ibitib

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t

3.3 Input for STCIR

I;

IT.

IIL.

Iv.

Input

Le

it.

Tir.

Constant Input

a. Category

1. tT, Koy

2. &t

3. None

Variable Input

Rotp? ts 8 Us &, on

Output

torr? a, e, I, g, h, 2, R, u

Constant and Variable Input Required for:

AEG/DRAG

for STAP

Constant Input

a. Category

1. 7

2. 6b

3. None

Variable Input

Ki00? t, 4, g, n

Output ‘

tap? a, e, I, g, h, £2, R, u, & n

Constant and Variable Input Required for:

AEG/DRAG

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-7-

3.5 INPUT FOR AEG

I. CONSTANT INPUT

a. Category

1. my Uy Oy Jy, HL Ky Ry, Rooper Ssgs At, pj(j =1, 12),eas Ky

2. A, C,

3. None

II. VARIABLE INPUT

Ky» Ko, Kz, Ly, Ly, W, t, a, 0, I, h, hy 1

III. OUTPUT

Bes Ops Ios Bes Ho» les Des g; h, Ly, Uy, By r, h", I"

IV. CONSTANT AND VARIABLE INPUT REQUIRED FOR:

DRAG - x, u, Os Ja Hy Kos Ro? Roorp? Caos Ato» ej(j =1, 12), 1»

Ko» Ap» Cp» a» ©, g, (easing), (ecosg), L, u, Wye Bs Bs Y's Cys Co,

Crs C309 By tp» e", L", at

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main

ienn

amma

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7. Subroutine Outputs

The outputs for the subject subroutines are as follows:

STLO - troy» a» > I, gh, 2, R, U, Bn

STEQ - tag? a, e, I, g, h, £, R, I"

STCIR - torr? a, e, I, g, h, £, R, U

STAP - tap» @ e, I, g, h, 4, R, U, gn

APPENDIX I

Detailed Flow Charts

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“SUBROUTINE STLO

V

CONSTANT ANPUTS

We = O25CS6245 vES/MIN

TT= B.1415927

Kia - C.Kie2z =O,

Ries 7O.

&EQSr> C]OOR RAD,

St = aol MIN.

|V

VARIABLE INPUTS

= LONG(TUISE INRQLESTICN

\NITARW VECTOR

Gx. Tr 495 hy Ax, %)

Urs 9po Me

SET,

N= X+2TT

he NWS CES INED

AS THE DISTANCE

BLONYG THE BQLUATOR

IRBETWEENY THE Peewe’T

ACTH FIXED ASTEDG NS

Nove Che) AND THE

|||

DESIREDLonaiTunE(Ay)

wv

Ps

wee

we

irais LOGIi2n PTHS10 WES

THE PEPER RLAPRALUT

OF TNE LONGITUDE (AY)tn ORDER To Sue“Tet leetRIC AL

Te LAWS FCR APG SM ENT“Gh LATITUBE CULL) OF

THE BEES .PED LonaiTupe,, THe Kio ies FLAGS'APE CHECKED LATERUS THE SUBROUTINE"Te PeePteer ADJUSTj THIS Com poTenp uU,

ser |Vartt-v7

Kiezzt

cose == SIN(I)* coe @~

sINC = [{$-(cosc)*

SINCU)= SIN(Y) JSINC

cos(u)= Ji- ant!

U_ = TAN Gu Wi) /osws) 4.4

Page Jos feaetd

ists!

[Ser | Se

fe egi

Yo Ut 1 i a

Le

‘fe AN AEG AS USED [rear eweeaeeytN WHICH THE ARGUMENT

OF LATITUDE \S

TRUNCATED AT 3o0,THERE IG A POeseiLitr !CR LOSING THE CcomemaT |

RELATIONSHIP BETWEENU, € Ur AFTER THE

EARTH ROTATION TERMWe At) 1S ADE To

aX THIS COMODITION

Witt BE CRITICAL ONLY |

WHEN “THE UL AND THE

ARGUMENT OF LATITWO S|

OF THE PRESENT | POZE (Tion (Uz) 1S VERY

CLese€ TO 320°

THE Kiog FLAG WILLPREVENT THE ABove

CONDITION WITH A SHUNTOF THE TEsT ON THE

DIFFERENCE BETWEEN

L£ Up ONCE THEPELATIONSHIP OF THIS

ESTABLISHED . 7 ($4——— -PAPAMETERS HAS BEEN

THE POUTIAL

APPL:TATIONS Se THE

TIME To Agee ATTHE A, (IS COM SITES

BY USING THE

THE PRESENT ARS CLIENT

OF LATITUDE (Jy) <F

THE VEHICLE AND Tre

| NReSIMENT OF

| LATITUDE. (U,) oFdy

DiFEERENCEL BSTWStH |

baw

D4

ee

SINCS THe GAeCTHWILL ROTATE DURINGTHE PERIOD WHEN THEVEHICLE IS ADVANCINGTOWARD THE UL, THEINPUT LONGITUDE CX)MUST BE ADJUSTEDWITH AN EARTHROTATION TREK(ae"AL

VTS GORRSC

THE UL COMPUTED

(HN THe AGove

ITERATION (5

ITERATED YUPon)

Te DETER MiIt>bS

T TIMeOF ARRIVAL AT THE

ARS SOMENT OF LATTTUOGOF THe DESIRED

LoenmGiTweo&

eeoare eee

THRE WuITIAL TIME °

ESTIMATE Gt) MUSTBE AD)LSTED To

REFLECT THEEARTH ROTATION

TERM. THIS \S '

Done BY PEPLACING

THE INITIAU Lone iTuDeOQ) with THEADSVUSTED (-ONGITOOE(Ag) ANOS RE-EUTERING

THE Loaic TO CONPUTE

ate

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SIC IESASS OTINE

CONSTANT IN Pot!

St= 0-01 MIN,

W-+ 31415927Keg. = Ory

VARIABLE INPUTS

Reve — RADIOS, DESIMWEL

VECTOR AT TIME te

oO92, hz, f+ yey

ty Geo We

Piecr ESTIMATE OF VTe OS AOYMATSY OF

Le

(“CF)-Rere\“ je. Reve Z

= =I (OQJLwee ernrion 1°? Ter cos (Ponsr int ORRIT

= U.-3,| '4

j]

»

YO s INDORSED THAT TROE 49.9116

\ OF MEAT DESImeD Kh ft,

| Ponor IS DUSEWU

- oY

fee"‘z) Z- + ze Ae

an’=) TOT Le

> “ s|

- i .

pee | .

Gen.>

CALL AEG2h V \catVEcToR/S}

>»6

(fase2mfy)4t= (9, + nz) } :

-e*jy-foe) (Cagees) ERPS. FAS?FE cue VDESIPED Khon.

f= Ve- elo .°*] DOES 1ST EGET° IN O@EIT ‘

Hf= COS(cosff,.)

ITESZATIOD LOOP THO DSTEteHVe

TIME OF ARRIVAL AT DESIRED

RADIUS WITHIO & TIME AGEJoe PCS

Tovewance Sr 2 2

“~

©STE SISROa

DETERMINES THE

ARGUMENT OF

LATITUDE OF THENEXT EQUATORIAL

CROSSING POT

FIeSr ESTIMATE. oF

TIME TO NExTEQUATORIAL CROSSING

+CONSTANT INPUTS

St= 0-01 MIN.

. TF B14iso27

“vesehiceVARIABLE(NDUTE

| VECTOR AT Time t-

(25,25, Ti, Qryhr Ke a,|

|

Vir gon

SCS dee eR < Sarr

anKune = 27

ahKure = T_

tjAt= CKito —Up/ (4,+ ny) '

im 4

2)1.2 Sstyi"4]aes)

:tt

|}

|

é

it

rhMANUISSOUD

“SeOP3OBABH

2MOMaesOOBWI

at)3UVaALaAg

C5awe]Nolveaat

NYSrTttioesSIRL

7Sererew

100

= 106 =

APPENDIX II

Definition of Flow Chart Symbols

- semimajor axis

- eccentricity

- inclination

- argument of perigee

- longitude of ascending node

- mean anomaly

- time of vector

- argument of latitude

- secular rate of change of argument of perigee

- mean motion constant

- true anomaly

- input longitude

- distance along the equator between the present earth-fixed as-

cending node and the desired longitude

- earth-fixed longitude of ascending node

- initial setting of A in iteration loop

- computed argument of latitude of the desired longitudinal crossing

- cosine of wedge angle between longitude meridian and vehicle path

102 ~ flags to establish correct quadrant of longitude A)

- time interval used in iteration loop

- flag to set proper relationship between the present argument of

latitude and argument of latitude of desired longitude

- change in input longitude due to earth rotation factor

- angular tolerance in iteration loop

- "adjusted" desired longitude dve to earth rotation factor

- option to search for either apogee or perigee

f - true anomaly of point of desired radiusCIR

Kino - flag to determine next node (ascending or descending)

r" - mean inclination

Koh - error flag in STCIR - desired point does not still exist inorbit

tron - time of arrival at specific longitude

tag - time of arrival at next equatorial crossing

torr - next time of arrival at specific radius

tap - time of arrival at next apogee or perigee

Subscripts

I - initial

F - vector at desired time