Apollodescentguidnce.pdf

GUIDANCE, NAVIGATION AND CONTROL

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CHARLES STARK DRAPER LABORATORY

8'--

71

R-695

APOLLO LUNAR-DESCENT GUIDANCE by

Allan R. Klumpp JUNE 1971

CAMBRIDGE,

CHARLES STARK DRAPER LABORATORY

MASSACHUSETTS,

02139

AC KNOW L E DG ME N TS

This report w as prep ared under DSR P roj ec t 5 5 - 2 38 9 0 , sponsored by the Manned Spacecr aft C enter of the N ation al Aeronautics and Spac e Administr ation th rough C ontrac t N AS 9 - 4 0 6 5 . The P 66 vertic al channel was developed b y C r aig W . Schulenberg.

The

an alytic al design and gain setting of the P 6 6 horizontal ch annels w as done by N icholas J.

Pippenger using concep ts suggested by Jerrold

H.

Suddath.

The concep t of

analytically extr apolating to yield the p r edictive guidance equation for P 63 and P 6 4 w as con c eived by W illiam S . W idnall.

The existence of an an alytic al solution for

the guidance frame orientation to yield zero c rossrange target j erk w as recognized by Thom as

E.

Moore.

The thrust direction filter configur ation for eliminating

thrust-pointing errors due to attitude bias within the digital autopilot deadband was conceived by W illiam S . \Vidn all and Donald W . Keene. The public ation of this report does not constitute app r ov al by the N ational Aer,onautcs and Space Administr ation of the findings or the conc lusions contained therein.

It is published only for the exchange and stimulation of ideas.

(D C opyright

by the M assachusett s Inst it ut e of Technology Published by the C ha rles Stark Draper Labo ratory of the Massac husetts Institute of Tec hnology. Printed in C amb rid ge, Massachusetts, U. S. A. , 1971

ii

R-695

APOLLO LUNAR -DE SC ENT GUIDANC E

ABS T R AC T This report records the technology associated with Apollo lunar- descen t guidance. I t c ontains an introduction plus five m aj or sec tions: 1.

Braking-ph ase and app roach - ph ase guidan c e.

Braking- phase guidance begins

in lun ar orbit pr ior to engine ignition and transfers the lun ar module ( LM ) to a terminus typically 7 8 0 0 - m slant- r ange before the landing site.

Approach-phase

guidance begins at braking-ph ase terminus and transfers the LM to a terminus typ ic ally 30 -m above the landing site.

The b r ak ing-ph ase transfer is near - optim al.

when�as th e app ro ach - ph ase tr ansfer sac rifices propellant-utili zation efficien cy to p rovide landing-site visibility and landing- site redesign ation c ap ability. 2.

Term inal- descent- phase guidanc e.

Initi ated autom atic ally at approach - ph ase

term inus, or m anually any time during the app r o ach ph ase. termin al- descent-phase guidan c e automatic ally nulls hori zontal velo c ity and controls altitude r ate to a reference value.

The referen c e v alue is inc remented or decremented by ast ron aut

m anipulation of a rate-of- descent control switch. 3.

Powered - flight Attitude- m aneuver Routine. The routine connects all powered­

flight guidance p rogram s- descent and ascent- to the digital autopilot. A dep artu re from traditional app r o aches, the routine transfers the L M from any cur rent attitude to any comm anded attitude while avoiding gimbal lock by inherent char acteristics of the m aneuver algo rithm. 4.

Throttle R outine.

No gimbal- lo ck - avoidance str ategy is required.

The routine connects sever al guid ance progr ams to the

descent p ropulsion system ( DPS ) . DPS h ardw are limitations r equi re operation either at a m aximum - thrust point or within a sep ar ate p ermitted- thrust region. The routine alters thrust commands from the guidan c e progr am s when nec essary to meet DPS c onstr aints and issues cor rected th rust inc rement commands.

iii

5.

Braking-phase and A pproach- phase Targeting P rogram .

This groun d - b ase d

p ro g r am is use d at the Drap e r Laboratory and at the N ASA M anned Spacecraft C enter . T h e p r og ram supplies descent targets which are lo aded into guidance computer m emory shortly before laun ch.

by Allan R . Klumpp June 1 9 7 1

iv

TABLE OF CONTENTS Section INT RODUCTION

1

Descent Phases

1

4

Navigation, Guidance, and Control Configuration Navigation Guidance

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Powered - flight Attitude-maneuver Routine

6

Throttle Routine

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Braking-phase and App roach-phase Targeting P rogram

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Digital Autopilot

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DE FINITIONS OF LUNAR DESCENT COORDINATE FRAMES, ATTITUDE ANGLES , AND GIMBAL ANGLES

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BRAKING- PHASE AND APPROACH-PHASE GUIDANCE Guidance Equation Derivatio n

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Braking-Phase Targeting Objectives

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Approach -Phase Targeting Objectives

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Landing-Site Redesignation P rocedure

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P 6 3, P64 Guidance Algorithm P 6 3 Ignition Algorithm

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T E RMINAL-DESCENT-PHASE GUIDANCE P 6 6 Horizontal Guidance Algorithm

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P 6 6 Vertical ( ROD) Guidance Algorithm POW ERED - FLIGHT ATTITUDE - MANEUVE R ROUTINE THROTTLE ROUTINE

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TABLE O F CONT ENT S (Cont . ) Page

Section B RAKING-PHASE AND APPROACH-PHASE T ARGETING PROGRAM Constraints

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Approach-phase Targeting Braking-phas e Targeting R E F E RENCES

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NO MENC L AT U R E Symbols a r e norm ally defined whe r e first introduced. The refore it is necessary to define here only those symbols use d in more than one section of this report. Most symbols are self- defining by being constructed of standard i dentifiers as follows: 1.

Type of variable.

Position and its deriv atives veloc ity, accele r ation,

je rk, and snap are denoted R , V, A, J , S .

Thrust is denoted F, thrust­

accele r ation AF , unit vecto rs 1J.N, c lock-times by lo wer c ase t, and target- refe renced times (times with respect to the target point of a particular mission ph ase ) by uppe r c ase T. 2.

Mission ph ase.

The braking ph ase ( P 6 3 ) is denoted B R , the app r o ach

phase ( P 6 4 ) AP. 3.

Applic able point in phase. Inception is denoted I, terminus F , and tar get point T .

4.

C oordinate fram e of referen c e .

P latform coor din ates are denoted P ,

guidance G , an d L M body B . 5.

Achieved ( as opposed t o nomin al) v alues ar e denoted A .

Thus b y construction, R B R FG A i s the p osition vector e xpressed i n guidance coo rdinates achieved at the b r aking ph ase terminus.

Without a ph ase i dentifier,

B_TG to �TG represent the b r aking or appro ach ph ase targets R BR T G t o §.BRTG o r B_AP TG t o .§.AP TG, whichever phase is current.

Vector elements a r e denoted by

subsc ript, e . g. , V

· Vector m agnitudes are implied whenever a symbol reserved z for a vector lacks the unde rscore. Row vectors of 3 x 3 m atrices are denoted

and m atrix e lem ents are identifie d by row and column, e . g. , C y z is the �X'�Y'�Z ' Z component of the row vector �y· Symbol

Definition

_AFC P

Thrust- ac c e le r ation command

AF P

Thrust- accele r ation me asurement ( computed by dividing the inerti ally sensed velocity inc rement by the guidance sam p le inte rval)

AGS

Abort Guidance System

ATT

P owered- flight Attitude - maneuver R outine

C BP

T r ansform ation to body from p latfor m coordinates

CGP

T r ansform ation to guid ance from p latform coordin ates

vii

D.\P

Digital autopilot

DEC 1\

Descent engine c ontrol

DPS

Descent p ropulsion system

interface between the

oF.\

assembly.

A digital

to analog

LGC and the DPS thr ottle

Tht·ust cor rection in c r ement which must be added to the thrust measurement aver aged over the sample interval to obtain the samp le-instant thrust.

G:\1

Lunar g r avity ( ac celeration of p ositive sign val i d at the lun a r s u r face)

GP

Cur rent g r avity valid at the cu r rent position

I:\ I u

.B_P

Inertial measurement unit consisting of a three - gi m b alled

u: :\DTII\IE

stable member and three acceler o meters A

time

interval

2.2

( typically

seconds)

added

to

the

target-referenced time T in P63 and P64 to account fo r the effective t r ansp o r t lag due to c omputation and system response times

Ll\1 guidance computer

LGC

c\

LII\IIT

function of two arguments yielding the fi rst argument

limited in magnitude to the value of the second argu ment Ll\1

Lun ar m odule

LP

L anding site L a n d in g p o int designator c onsisting of two ret i c les, one

LPD I\ L\.'\. I:\IU :\1 :\1I;'>JI!\1UM

N A.. S,\

on the inside w in d o w p anel, and one on the outside A

function

of

n

arguments

yielding

the algeb r ai c ally

highest argument A function of n a r guments yiel ding the algeb r ai c ally lowest a r gument National Aeronautics and Sp ace Administ r ation

Pitch

" L:\1 attitude angles. See Definitions o f Lunar Descent " Coordinate F r ames, Attitu de Angles, and Gimbal An gles

Yaw Boll

P63

B r aking ph ase o r b r aking-phase g u i dance p rogr am

P64

Ap p r oach phase o r app r o ach-ph ase guidan c e p r o g r am

P66

Term i n al - descent gui d an c e p r o g r am

R .�PTG, Y:M>TG, .:::ir\PTG, JAPTG, �A.PTG,

RBRTG, ll_TG, YBRTG, YTG, ABRTG, ATG, JBRTG, JTG, �BRTG, .§.TG

or

te r m i n al - descen t - p h ase

T ar get position, veloc ity, acceleration, jerk, an d snap of the app r o ach phase, b r aking phase, or c u r rent phase

.B.G, .B_P

Cu r rent position

ROD

R ate of descent

t

c lo c k - t i m e

T

phase

T arget- referenced time ( t i m e w i t h respect t o the t a r get p o int of the c u r rent m ission phase )

viii

THROT

Throttle Routine

!JN F C P

Unit thrust command

!J.N W CP

Unit window comm and

.YG. Y.P

C u r rent veloc ity

ix

IN TRODUC TION This report d e s c r ibes how th e Apollo lunar-desc ent guidance works, why it w as designe d th is w ay, and, in seve r al c ases, how it might h ave been desi gned d iffe rently. The c onc epts desc ribed c an be applied to l anding on any planetary b ody, w ith or without atmosphere, should m an r e s olve to c ontinue this adve nture. s olutions p resented offe r ample opportunity for checking the theory.

The

Such c hecks

h ave been m ade, and all algorithm s ar e known to work as c onceive d, Lunar -desc ent guid ance begins with the lunar module ( LM ) at about 15 - km altitude in a s lightly elliptic al c oasting lunar orbit, and ends with the L M on the lunar surface.

The guidance is pe rformed by the onboard LM guidance c ompute r

( LGC ), which t akes input d ata and c om m ands directly from the LM c rew and via the uplink from N AS A's R e al Time C omputation C ente r in Houston, Texas, c r ew c onsists of a c ommander and a L M pilot.

( See Figure 1. )

The

Standing on the

left, th e c ommander m onitors and c ontrols the descent using visu al cues and v ar ious h and c ontrolle rs and s witch e s .

St anding on the right, the LM p ilot monitors the

c ompute r display, vocally relays pe rtinent d ata to the c om m ander, and e nters any nece s s ary d ata into the computer via the keyboard. The primary guidance m ode for the lunar de s c ent i s automatic; the LGC c ontrols both attitude and th rust.

The c ommander c an, tempor arily or permanently, select

nonautom atic guidance m odes if he wishes to c ontrol, m anually, attitude or thrust or both .

The nonautom atic m ode s , not d e s c r ibed fur th e r in this report, provide

attitude and th rust refe renc e s for the c om m ande r to follow if he chooses to fly the L:\1 manually along the autom atic guidanc e p r ofile. D e s c e nt P h ases The lunar d e s c ent i s a nom inally-planar trajectory c onsisting of three phases illustrated in F igure 2 and d e s c ribed as follow s : 1.

T h e b r aking ph ase ( P r ogr am 6 3 , o r P 6 3 ) is initiated b y keyboard entry aboat

1 0 minutes befor e nominal ignition tim e .

P 6 3 fir st c omputes the p re c ise time an d

attitude for ignition.

N ext, at typic ally 49 2 -km s l ant-r ange from t h e l anding site,

P 63 ignites the DP S .

F i nally, P 6 3 t r ansfe r s the LM to the terminal state required

as initial c onditions for the succeeding app r oach phase. The tr ansfe r takes typically 5 1 4 second s and is ne ar- optim al.

1

LANDING POINT DESIGNATOR RETICLES SUPERIMPOSED

N

LM PILOT "

/;'/k.__

LM COMMANDER

Figure 1

C o mponents of the Luna r - d e s c e nt Guidance System

IGNITION 1 5 k m. ALTITUDE 492k m. SLANT RANGE 11 min 31. 3 sec. 16.2" CENTRA L ANGLE TO TOUCHDOWN

t

START P63 22 MINUTES 45" CENTRAL ANGE TO TOUCHDOWN

w

U LLAGE 15km ALTIT UDE 50 5 k m SL ANT ANGLE 11 m;n. 38.8 sec. 16.6" CENT RAL AN GlE TO TOUCHDOWN

. -·

..--.:;�--.

---�.:.$ .--�-:-.-

t

....;:-4 --..._

START P64 '-... 2231 m AlTITUDE

t--�......;-;".:-.-.-

...

, PpRo401 ' , .... Pf.t45f p64

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AL L NUMBERS ARE TYPIC AL AND DO NOT REPRESENT

""

'

ANY SPECIFIC APOLLO MISSION

)

B RAKING-PHASE TARGET POINT -541 m ALTITUDE 4416 m GROUND R ANGE TO L ANDING SITE

Figure 2

START P66 30m ALTITUDE 11 m GR OUND RANGE 31 sec TO T O UCHDOWN

Luna r - descent -guidance Phases and Targets

PPROACH-PHASt · T ARGET POI NT 29m ALTITUDE 4.8m GROUND R ANGE TO LANDING SITE

2.

Approac h - phase ( P 6 4 ) guidanc e begins w ith initial conditions c ons isting of,

typically a) 2. 2- km altitude and 7. 5-km ground r ange and b) - 44 - m / s ec vertical velocity and 129 - m / s ec fo rward velo c ity.

In typically 1 46 s econd s , P 6 4 t r an sfer s

the Ll\I to a point almost directly above the landing s ite.

P 6 4 provides continuous

visibility of the lunar surface and, spec ific ally, of the landing s ite until around 5 s e c onds befo re terminus .

Du ring P 6 4 the commander c an direct the LGC to land

at any visually chosen point on the lunar s urfac e b y a landing- s ite r edes ignation p r o c edure which c an be continued until initiation of the terminal- des cent phase, 3.

Th e term inal- des c ent phase ( P 6 6 ) b egins autom atic ally at typic ally 3 0- m

altitude and 11- m ground r ange from the landing s ite, o r it m ay b e initiated b y the c om m ander any time during P 6 4 . o nly; there is no position control.

The P 6 6 guid an c e algorithm c ontrols velo c ity P 6 6 nulls the forward and lateral velo c ity

c omponents to p roduce a vertical app ro ach to the lunar surface, an obj ective which c annot be achieved from visual cues when the s urface is obscured by a sheet of radially moving dust.

P 6 6 controls altitude r ate to a r eference v alue that is

inc remented or decrem ented by 0. 3 - m / sec each time the commander r ai s e s or lowe r s a three - p o s ition r ate - of - d e s c ent ( RO D ) c ontrol s witch lo c ated ne ar his left h and. I'\

avig aticn, Guid anc e , and C ontrol C onfigur ation Th e N avigation, Guidanc e, and C ontrol configur ation illu strated in F igure 3

applie s to all LM powered- flight guidance maneuver s . the s olid portions of F igure 3 .

This repo rt des c ribe s only

All routines are p r o c e s s ed once eve ry two s econds,

except the ve rtical ch annel of the P66 guidance algo rithm is p rocessed once per s e c ond, and the digital autopilot is proc e s s ed 10 times per s e cond. Navigation 1\' avigation

1 ( s ee K riegsman ) p rovides an estimate of the current state vector

based on data from an inertial measurement unit (ll\IU ) and a l anding r adar. IMU data are used throughout all thrusting m aneuvers, but, to avoid accumulatiun of inertial errors, I M U d ata are not used dur ing coasting flight except for a m inimum perio d immediately preceding and following e ach thrusting maneuver,

The l anding

r adar p rovides altitude data below typically 1 0-km altitude, and velocity d ata below 6 1 0 - m /s e c . Guidance G uidance transfe rs the LM from the current state to the terminu s of the current phas e . In addition to the current state e stim ate from Navigation, Guidance is based

4

BRAKINCH'HAS£ AND APPROACH-PHASE TARGETING PROGRAM

� J

r4

I GROUND BASED I

,------l

I !STATE VECTOR UPDATE I ROUTINE L ______ _j

�-P64

CURRENT STATE GRAVITY

v

P66 GUIDANCE AlGORITHM

SYSTEM

__ ____

I

r---

:

UNIT THRUST COMMAND UNIT WI NOOW COMMAND

THRUST VECTOR

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I

POWERED -FliGHT

ROUTINE

AT!' I TUDE-MANEUVER

GIMIAL INCREMENT COMMANDS PHASE PlANE DATA

-:!'>I "'

r

�;��l

-D

--

AUTOPILOT ANO CONTROl

- ---,

SPACECRAFT DYNAMICS

L ___

be CJl

L

THRUST -ACCELERATION COMMAND

P63, P64 GUIDA!'�:£ ALGORITHM

�

I

DESCENT

PROPULSION

UIDAI'I:E TARGETS

GUIDAJC:£

NAVIGATION

THROm£ ROUTINE

r------,

THRUST INCREMENT COMMAND

,---if �

__

: I

_j

I TORQU[ VECTOR

L �� � .:__ - _j C

R

NAVIGATION MEASUREMENTS

,-------i

lb===================== I �

INERTIAl SENSORS LANDING RADAR

�I :::!:========:::::= ::: ::::=� = ==== =======::1

I

I L _______ _j

Figure 3

LINEAR ANO ROTATIONAl MOTION

Navigation, Guidance, and C o ntrol C onfiguration

on p r ecomputed targets from th e groun d -based targeting p r ogram .

The outputs

from the Guid ance algorithm are a unit thrust c ommand and a unit window c om m and is sued to the Powered- fli ght Attitude- maneuver R outine, and a thrust- ac celeration c om m and is sued to the Throttle R outine. Through the s e routin e s , Guidance c ontrol s th e thrust vector magnitude and direction with respect to ine rtial spac e . Powered-flight Attitude - maneuver Routine The Powe red- flight Attitude - m aneuver R outine ( AT T ) c onnects all guidan c e p r ogram s , de s c ent and ascent, to the digital autopilot ( D AP).

ATT inputs a r e two

c om m and vector s ; a unit th rust c om m and and a unit window c ommand. AT T e stimates a unit thrust vector from ac celerometer m e asurements, an d i s sues incremental c om m ands to the DAP.

The s e commands c ause the DAP to dr ive the e stimated

unit th rust vector into c oinc idence w ith the unit thrust command and the symmetry plane of the LM into c oinc idence with the unit window c om mand . During P64, as long a s the landing s ite would b e vi sible, the unit w indow c om m and is sued to ATT by Guidance i s the lin e - of- sight vector to the current landing s ite .

By rotating the Ll\1 symmetry plane into coinc idence with the line - of- sight

ve ctor, A.TT sup e r impos e s the landing-point designator reticles of F igu re 1 on th e current landing s ite. Th rattle Routine The Throttle R outine ( T HRO T ) connects several powered- flight guidance p r ogram s to the D PS .

The DPS is used by all d e s c ent guidan c e p rogram s , one of

the two abort p rogram s, and on e guidance program whose purpose is to p r ovide a velocity -vector inc rement c omputed by the R e al Time C omputation C ente r in Hou ston and tran sm itted to the LM. The DPS must be op erated e ither at the m aximum - thrust point ( about 92'7o of th e rated th rust of 46 7 06 newtons ) or within a p erm itted-thrust region ( 1 1 to 6 5o/o of rated th rust ) .

The intervening region ( 6 5 to 93o/o) is forbidden because in this

region oxidi zer flow an d fuel flow m ake independent tran s itions from c avitating to n oncavitating regime s .

The independent tran s ition s c ause gros s deviations from

th e required mixture ratio and produc e exc e s s ive erosion of the DPS noz zle. U s ing a c omputed mas s estimate, a thrust- ac celeration me asurement, an d th e th rust- ac cele ration command from the guidan c e equation s , THROT computes the cur rent and c ommanded th rusts and i s sues th rust inc rement commands to the

6

DPS.

These c omm ands e ithe r 1 ) drive the com puted th rust into c oincidence w ith

the c om m ande d thrust whenever the c omman d e d thrust lies within the permitted­ thrust region , 2) p roduce m aximum thrust whenever the commanded thrust lie s above the p e rmitte d-thrust region, or 3 ) p roduce minimum thrust whenever the com m anded th rust lie s be low the perm itted - thrust region. Braking- ph ase and Approach-phase T ar geting P r ogr am The targeting p r og r am provide s t argets for P 6 3 and P 6 4 . The targets define b r aking-

and

app r oach- phase

reference

t r aj e ctories

as

independent

p olynom ials centered at individu al target p oints as illu strated in Figure 2 .

vector Although

only P 6 3 and P6 4 are t argeted, the targets are des igned to achieve all the guidan c e objectives of P 6 3 , P 6 4, and P 6 6 . Digital Autopilot 2 The DAP ( s ee Widn all ) c ontrol s the attitude of the LM during pow e re d flight by m e ans of c ontr ol effectors c onsisting of a r e action contr ol system and a trim gimbal system.

As the n ame implie s , the trim gimbal s ystem is a slow s ystem

u s e d p rim ar ily for trimm ing the DPS thrust vector through the LM c enter of m as s ,

7

DE F INITIONS O F LUN AR DESC ENT C OOR DINATE F R AMES, A T T I T U DE AN G LES, AND G IMBAL A1�G LES Three coordin ate fram es are required for lunar des c ent guid an c e bec au s e 1) all inertial m easurements are with resp ec t to the s table p latform of the I MU , 2 ) P 6 3 and P64 guidance is with respect to a l anding site which rotates with, and c an b e redesignated along, the lunar surface, and 3 ) thrust- vector determin ation an d landing- site redesignations are with res p ec t to L M body axes .

These coordin ate

s y s tem s are illus trated in F igure 4 and defined as follo w s : 1.

P latfo rm coordinates .

V ariables in p latform coordin ates are tagged P . The

o rigin i s at the center of the moon, the XP- axis pierces the nominal (unred esign ated ) landing s ite at the nom inal landing time, the Z P - axis is p arallel to the orbital p l ane * of the C omm and Module and points forward, and the YP- axis completes the right- hand triad. 2.

Platform coordinate s are nonrotating.

Guidance coordin ates .

Variables in guidance coordin ates are tagged G.

The

o r igin coincides continuously with the current landing site ( the frame rotates with the m oon ); the XG -axi s i s vertic al; the ZG - axis lies in, or near, the plane containing th e I....M and the landing s ite and points forward; and the YG - axi s completes the r ight- hand triad. Thu s , the origin and orientation of the guidance frame are altered each time the landing s ite is redes ignated, Guidance- frame unit vectors exp res s ed in p latform coordinate s are the row vectors �G P , C G Py • C G P of the m atrix Z X CGP. 3.

Body coordin ates .

V ariables in body coordinates are tagged B.

the generally accepted LM coordinates .

Thes e are

The X B- axis is in the direction of the

nominal thru s t vector, the ZB - axis is directed forward, and the Y B- axis completes the right- h and triad.

Body- frame unit vectors expres s ed in platform coordinates

a r e the r ow vecto r s C BP

X

' �BP y • C BPz of the m atrix C BP .

F rom the s e definitions i t i s noted that if the L M lands at the nominal s ite at the nominal tim e in a nomin al erect attitu de, the three frames w ill be p arallel at the ins tant of touchdown. The following conventions are defined for orthogonal m atrices:

*

The LGC transfers state vectors in pl atform coordin ates to an abort guidance sys tem ( AGS). The AGS requires the s tate to be expres s ed in a frame who s e Z axis p arallels the orbital plane of the command module.

9

XB XP

XG

ZB LUNAR ROTATION TO OCCUR DUR ING TIME REMAINING TO NOMI NAL LANDING TIME.

NOMINAL LANDIN G SI T E AT NOMINAL LANDING TIME

Exage rated Scale '------+- zP.

XP, YP, ZP-Piatf o rm ( Inertial Measu rement Unit) Coordinates XG, YG, ZG-Guidance Coo rdinates X B, Y B, Z B -Body ( LM Spacec raft) Coo rdinates

Figure 4

Lunar-des cent Coordinate Frames

10

1)

A m atrix element i s denoted b y two sub s cripts which indic ate the row

2)

and column res pe ctively of the element. Thus C BP denote s the XY Y- component of the row vector �BP . X A m atrix transpose ( inve r s e ) i s denoted b y interchanging t ags.

3)

F rom the definitions , it follows that m atrix p ro ducts are obtaine d b y deleting internal tags .

By c onventions 2 and 3 , a vector V is transformed to body from gui d ance c oo rdin ates by VB

C BP C G P

-1 V G

=

C BP C PG V G

=

C BG V G .

LM attitude angle s are a s et of E uler angle s defined as c lockwise rotation s about the XB- axis ( yaw ), the displac ed YB- axi s (pitch), and the displaced ZB- axi s ( r oll). LM gimb al angles are a s et of E ule r angle s defined as c lockwise rotation s about the YB- axis ( inne r ) , the d i s plac ed ZB-axi s ( m iddle) , and the displac ed XB- axis ( oute r).

11

B R AK I NG -PH A S E AN D APPRO AC H - P H ASE G U IDANC E The guidance p rograms for P6 3 and P6 4 are almost identi c al . The two p h ases use the s am e guidan c e algorithm, the s ame Throttle R outine, and the s ame Powered­ flight Attitude- m an euver R outine. The differences are 1) the guidan c e equ ation s elects different s ets of tar get s , 2 ) the erection of the guidance coordin ate f r ame is slightly different, and 3 ) landing s ite r edes ignation c ap ability is available only in P6 4 . Gu idan c e E quation Derivation To guide a spacec r aft from any initial or current state to a s p ecified target state c an be viewed either as an explic it guidance p roblem or as an implicit guid an c e p r oblem.

Explicitly, we c an r epetitively determine, as the m i s s ion progres s es , a

vector p olynomial function of time that inter s ects the cur rent and target states . G uidance

then commands the cor responding

p rofile of

ac celeration vs time.

Implicitly, we c an define, in advance of the mis s ion, a refer en c e traj ectory as a vector p olynomial function of time that evolves backward from the target state but c annot be expected to inter sect a disper s ed initial ( o r disper s ed current ) state. Onboard guidance then commands an ac celeration vector p rofile c omposed of th r ee term s, n amely the ac c eleration along the reference t r aj ectory m inus two feedb ack term s p rop ortional to the deviation s in veloc ity and position of the actual t r aj ectory with respect to the referen c e traj ectory. In either the explicit o r the implicit c as e, rep etitively solving the guidan c e p roblem p roduces c onvergence upon the spec ified tar get state even though the target point m ay be redes ignated in flight and th e c ommanded ac c eleration is not p reci sely achieved bec au se of cont rol erro r s . The implicit guidance equation derived here i s c ategoric ally superio r t o the explicit guidance equation bec au s e the explicit equation is merely a spec ial c as e, as will be shown. Besides bein g intellectu ally more s atisfying, the implicit equ ation h a s demon strated in s imulations significantly fas ter r eduction of deviations from the reference tr ajec tory. Deviation s come from navigation erro r s an d from disp lac ing the reference traj ectory to inter s ect a redes ignated l anding s ite. of

deviations restores a nom in al

R ap id reduction

app roach to the r edesignated

landing s ite.

U nfortun ately, the implicit equation had not b een developed when the program for Apollo 1 1 was c oded, and the advantages were ins uffic ient to recode the guidance p rogram for later mis sion s .

/3)

(4 5) K lumpp • simplified it (6) for LGC coding and gener ali zed it to nth o rder. Moore et a1 derived an impli c it C her r

derived the exp licit guidance equ ation .

equation which did not generali ze the explicit equ ation. The gener al impli cit guidance equation is now derived and spec i alized to the explicit c ase.

13

It is convenient to think of the reference traj ectory as evolving b ackwards in tim e from the target point, with the time variable T reaching zero at the target point and negative prior to th at point.

Thus target- referenced time ( T ) is to b e

dis tin guished from clock- time ( t ) . Bec au s e guidan c e gains w ould become unbounded, th e target point is never reached . In s tead, a guided phase is terminated at a negative time T and the succeeding phase is started .

Both the terminu s and the target point

lie on the referen c e traj ectory, but the target point lies beyond the portion that is (? ) actually flown, s im ilar to a suggestion of McSwain and Moore. In term s of a vector polynomial func tion of target- referen c ed time, we wish to define a referen c e traj ectory that satisfies a two - point bound ary value p roblem with a total of five degrees of freedom for each of the three components .

This

number of degrees of freedom is required in order to cons train terminal thru st in PG3 and to shape the traj ectory des ign in P 6 4, as is discu s s ed in connection with the targeting program. A quartic polynom ial is the m in imum order with which five cons traints on th e reference traj ectory c an be satisfied.

W ith the reference traj ectory evolving

b ackw ards in tim e from the target point, it c an be defined as (1)

where .B_RG is the position vector on the referen ce traj ec tory in guidance coordinates at the negative tim e T, and R TG, .Y_ TG , ATG , .i_ TG , and � TG are the target pos ition, velocity, acc eleration, j erk, and snap . T h e acceleration to be commanded a t any poin t in s p ac e consists o f three term s :

the acceleration of the reference traj ectory at the p articular time T, minus

tw o feedback term s p roportional to velocity and position deviations from the referen c e traj ectory.

T aking derivatives of Eq. ( 1) a s the veloc ity an d acceleration o n the

referen c e traj ectory yields the three- term guid anc e equ ation AC G

ATG

+

.i_ TG T

+

2 .§. TG T / 2

3 2 - ( .Y_G - V TG - ATG T -.l_ TG T / 2 - .§. TG T / 6 ) K V/ T

(2)

where AC G is the comm anded acceleration, V G and R G are the current velocity and pos ition, and K V and K are the nondimensional feedback gains . R

14

C o mbining like terms in E q. ( 2 ) yields AC G

=

-

RG K

R

/ T 2 - Y.G K / T v

+

2 B_TG K I T R

+

Y:TG ( K V

+

ATG ( 1

+

+

1.T G ( 1

+

+

.§.TG ( 1 / 2

+

K

Kv

R

+

K VI 2 +

)/ T K +

K vl 6

R

(3)

/ 2)

K R I6 ) T +

2 K I2 4 ) T . R

E quation ( 3 ) is the implicit guidan c e e quation. Although the refe r ence tr aj ec tory is quartic, the traj e ctory gene rated by th e implicit guidance equation is obviously not.

The implicit equ ation c an, howeve r , be spe c i alized to the explicit equ ation -

which does gener ate a quartic - b y a spe c ific choice of the feedback gains K v an d K F i rst we note that E q. ( 2 ) m ay be identified with the lin e ar second-order W differ ential equation • •

X

•

+

2 !; w X n

+

2 w n

=

0

by the associations (noting T is negative ) (4) where w i s the undam ped natu r al frequen c y an d !: i s the d amping r atio . Of c ourse n the system is time varying. Howeve r, this assoc i ation does affor d some intuition on gain setting.

Solving Eqs. (4) yields (5)

and where P is the undamped period 2 rrlw n ,

(6) Equ ation ( 5 ) p rovides a m eans of c ontrolling the system response time in terms of th e nondimensional r atio P / T, and E q. ( 6) p rovides a m e ans of setting the d amping ratio .

15

An

interesting set of values to choose for response and damping is

P/ T

=

- rr/f3',

.t

=

{3' / 2 .

This choice yie lds

W h en th ese v alues are substituted into E q. (3 ), the result is the explic it guid ance equation derived in refe ren c e s 3 to 5,

AC G

=

1 2 ( R TG - B_G / ) T

2

+

6 ( Y:TG

+

Y.G / ) T

+

ATG .

( 7)

The discussion of implicit vs explicit guidance is c oncluded by introdu c in g the concept o f a sp ace c ontaining all p e rmissible combinations o f guidance p ar ameters.

I mplicit guidan c e - p arameter sp ace is one qu adr ant of the t, PIT plane or, equivalently, one quadrant of the

KR'

Kv

plane.

E xplicit guid an c e - parameter sp ac e is a single

p o int in eith e r plan e . E quation ( 7 ) presents the expli cit guidance equation assuming negligible transport t ime d elay.

The explicit equation progr amed in Apollo is cor rected to

c ommand an acceleration app ropriate fo r the time at which the ac c el e r ation is p redicted to be achieved.

T

p

T

+

Let this predicted target-refe renced time be

LE ADT I M E

w h e r e LE ADT IME i s the transport delay due to computation and c o mmand execution. An

explicit guidance equation will now be derived that fits a qu artic polynomial

through the target position, veloc ity, and ac cele r ation, and through the current position and ve loc ity.

The ac celeration of the qu artic at the p redicted time T

is P the ac celeration to command at the current time T in order to realize the qu artic

p r ofile. T

=

It w ill be shown that the resulting guid ance e qu ation reduces to E q. ( 7) for

T , and the refo re E q. ( 7 ) gen e r ates a qu artic profile when the tran sport delay

p is ze ro.

C onstr aining the actual traj e ctory to be a quartic fun ction of time allows the cur rent position and velo c ity to be expr essed as

16

l

[:: [:

T T2/2 T3/6 T4/24] RTG 1 T T2/2 T3/6 Y:TG ATG .J.TGA §.TGA

(8)

where !I TG A and .§ TG A are the j erk and snap which would be achieved at the target point, and are not targets loaded into LGC m emory.

l

Solving Eq. (8 ) for the j erk

and snap yields

[�TGA .STGA

[

-24/T3 -18/T2 -6/T 24/T3 -6/T2] RTG 72/T4 48/T3 12/T2 -72/T4 24/T3 Y:TG ATG RG

(9)

VG

The acceleration to be commanded at the current time T and realized at the predic ted tim e T pi s A CG

=

A TG

+

!L TG A T

P

+

§.. T G A T

� / 2.

(1 0)

Subs tituting E q. (9) into E q . ( 1 0 ) and s implifying yields the A pollo lun ar- des c ent gui d ance equation

ACG [3 (T;) - 2 C:) "

+

l

[ {;y - C:)]· VG/T [ {:y - 6 C:) +

+

1] ATG.

N ote that when time T is large in m agnitude comp ared to the transport delay,

(11)

T

p / T approaches unity. all bracketed coefficients in E q. Eq. approaches Eq. ( 7 ) identic ally. The net effect of E q .

(11)

Eq. (7) is a gain reduction as the target point is approached.

(11)

are identical for T

P

=

appro ach unity. and

(11)

not ach i eved by

Bec au s e Eqs. (7) and

T, Eq. (7) generates a quartic profile when the tran s port

delay i s zero.

17

In the derivation of the guidance E q. ( 7) or ( 1 1 ) , nothing c on strained the time T.

At any point in

and

Eq.

a

guided phase, T c ould be s et to any arbitrary negative v alue,

( 7) or (1 1 ) would satisfy the boundary-value problem from that point forward.

Lan ding- site redesignation, which c an arbitrarily stretch or shrink the traj ectory, would produce an unnec essarily s evere guidance response if T were not c or­ respondingly adj u sted.

Becau s e T is arbitrary, it can be computed to s atisfy an

additional boundary c on straint.

In Apollo, this additional c on s traint is imp o s ed on

the downrange (Z) c omponent of j erk . Thus the Z - c o mp onent of the j erk polynomial of E q. ( 9 ) is solved for T by u s ing a target Z- component of j erk JTG 2 . Separating th is scalar cubic p olynomial from E q. ( 9 ) yields JTG

z T

3

+

2

6 ATG z T

+

( 1 8 VTG z

+

6 VG z ) T

+

24 ( R TG z - R G z )

=

0.

(1 2)

One root of thi s cubic i s th e required time T. An

altern ate c riterion for c omputing T reduces the propellant-consumption

p enalty of downrange landing- s ite redesignation s .

Although exten s ively tested, the

altern ate was developed too late for inco rporation in the LGC p rogram. The altern ate criterion sets the downrange pos ition error to zero. quartic STG

Z

4 T /24

+

3 J TG Z T /6

+

2

ATG z T /2

+

VTG z T

+

Thu s T is one root of the

R TG

Z

- RG

z

=

0.

Braking- P h a s e T argeting Obj ectives The near-optimal tran sfer provided by P 6 3 targeting mus t s ati sfy a throttling c on s traint th at the DPS be operated within the permitted - th rust region for, nomin ally, the final two minutes of the phase.

This throttling duration ab s orbs dispersions in

D P S p erformance and errors in lunar terrain modeling. With a total p ropellant (8) shows that the Apollo guidan c e and targeting

c ons umption of over 6 6 0 0 - kg, Y ang

are w ithin 16 - kg of an optimal traj ectory s ati s fying the s ame throttling constraint. To provide thru s t within the 1 1 to 6 5 o/o region for the last two minutes of P 6 3 , th e targets are chosen to p roduc e a constant thrust level o f about 57o/o o f rated thrust at P 6 3 term inu s .

The targeting p rogram accomp li shes thi s by con s training

the magnitude of the terminal thrust- acc eleration vector to be F I M, and con straining the Z- component of terminal j erk to be ( es s entially ) required terminal thrust,

l'vl

K

F

i\1/M2, where F is the

is the es timated terminal mas s , M is the estimated

termin al mas s flow rate (negative), and K is a j erk coefficient to account for the verti c al component of thru s t. The targeting program achieves the two minute duration of constant thru s t by adj u s ting the initial range. P rop erly targeted, the guidance algorithm commands during mo st of P 6 3

a

th ru s t- acc eleration in exc es s of what can be achieved . The throttle routine multiplies thi s thru s t- ac c eleration command by the estimated mas s to yield the guidance thrus t

18

c om m and (GTC ), and p rovide s m aximum thrust until the G TC falls below 5 7%. Figu re 5 illustrates the profiles of GTC and actual thrust for a p roperly targeted P 6 3 phase and shows the effects of adj ustments of the ignition time and of the DPS th rust level. The targeting program computes the P 6 3 targets by p r oj ecting computed termin al conditions forward typic ally 60 seconds . Although the targets are p roj e cted, they are computed to p roduce the traj e ctory.

required terminal condition s on a nomin al

T r aj ectory disp e r s ions c annot be elimin ated prior to the target p oint,

but they c an be reduced suffic iently by terminus to achieve the targeting obj ective s . Both the ignition time and the p rojected targets are c omputed iter atively us ing a d e s c ent s imulation in the ite ration loop. App roach- Phase T argeting Obj ective s The P 6 4 targets are computed to provide lunar - surface vi sibility until about 5 s e c onds before terminus, and c ontinuous throttling. In addition, the P 6 4 t argets are computed to p roduce at te rminus a m atched s et of values for the Z-components of ac cele r ation, veloc ity, and position such that, in the nominal c as e , the P 6 6 algo rithm w ill p ro duce no initial p itch trans ient and will s imultaneou s ly null the Z- compon ents of veloc ity and position. Unlike P 6 3 , the P 6 4 reference traj e ctory c an be determin e d i n c lo s e d form from specified traj e ctory constraints.

Thus t h e proj ected targets

are computed w ithout nume ric al ite r ation. Landing-Site R edes ignation P roc edure To steer th e LlVI via the autom atic P 6 4 guidance to a visually s elected landing s ite, the com m ander u s e s an ite r ative proce dure akin to steering an automobile. The proce dure c on s ists of 1 ) identifying the current landing s ite where the LGC w ould take him in the abs ence of intervention and 2 ) steering the c urren� s ite into c oinc idenc e w ith his visually s elected s ite by c omm anding incremental l anding- s ite displac ements ( rede s ignation s ) . Be c ause the P 64 targets are define d in the guid an c e c oordinate fram e , which i s repetivively e rected through the landing s ite, t h e P 6 4 target p oint is disp lac e d ac cordingly. To identify the currently s elected l anding s it e to the astronauts, the LGC 1 ) o r ients the LlVI about the thrust axis to sup e r impo s e the landing point de sign ator ( L P D ) reticles ( s ee F igu re 1 ) on the current s ite and 2 ) display s a number which is r e ad by the LM p ilot and voc ally relayed to the commander.

By sighting through

th e indicated point of the LP D reticles, the c om m ander identifie s the current s ite. H e registe rs his eye by superimp o s ing the two LPD reticles, one of which i s p ainted on the inside window p anel, and one on the outs ide window p an e l. between reticle s is 2 . 5 em .

19

The s ep ar ation

Guidance Thrust Command ( GTC) Profiles

120

--- NOMINAl

-=--=-

��-----�-

100 IV) ::::> e::: :r: I-

0

I<( e::: I...LJ

"" 0

V) a_ 0 LL. 0 Iz

u e::: I...LJ

a_

I...LJ

.- --�:::.::; ::...:;:::-= --� �, ::=::.:: :.::===-===A:::: ct� ua Thrust Pr ofi , e �

80

-----

ABOVE NOMINAl OP'; TH�U'lT EARliER THAN NOMINAl DPS IGNITION

'�" , : '�\ ii

1L_

\\.

60

40 20

0

GTC and actual thrust coincide

26 seconds to trim thrust through center of mass.

t)" Time /.

50

of first Guidance pass ( GU IDTIME)

100

150

200

250

300

350

TIME FROM IGNITION- Seconds Figure 5

Braking phase terminus approach phase in eption \

400

450

Actual Thrust Profile s and Guidance Thrust C ommand P rofile s During P 6 3

500

550

By m an ipulating his c ontroller left, righ t, forward, or aft, the commander d i r e c ts the LGC to displace the landing site ( and the P64 targets) along the lunar ° surface by a c o rrespondingly dire c te d fixed angular inc rement ( 1 ) with respect to the cur rent line of s ight. The LGC redirects the th rust to guide to the rede signated (now current) s ite, and reorients about the thrust axis to m aintain superp o s ition of the reticles on the current site .

The c om m ander c an c ontinue thi s rede signation p r o c e s s - s te e r ing

th e current l anding site into c oinc idence with his chosen s i te - until 1 0 s econds b efore reaching the P64 target p oint, at which time P 6 6 is initiated.

F igure 6 illustrate s the P 6 3 , P 6 4 Guidance Algorithm.

As shown in F igure 3 ,

the algo rithm receives guidance targets , the current s tate vector, and the current g r avity vector as inputs and i s sues a th rust- ac celeration com m and, a unit thrust c omm and, and a unit window command as outputs. Be c ause the

landing site move s due to lunar rotation and

l anding- site

redes ignation, the L M is guided with respect to the guidan c e coordinate frame, which i s e r e c ted th rough the landing site e ach p a s s .

Guidance targets are fixed in thi s

flo ating frame. O ther inputs and all outputs are exp r e s s e d in platform coordinate s . The landing site ve ctor 1-_ P i s up date d for lun ar rotation (Eq. ( 6 . 1 )) using an approxim ate algorithm that avoids c omputation of tr igonometric func tion s, yet p r e serves the m agnitude of the lunar r adiu s .

The algo rithm accounts for the lunar

rotation rate W :YIOON P and the elap s e d c lock - tim e s ince the p r e ce ding upd ate (t tO LD ). For the landing-s ite redesign ation algorithm ( Eq s . ( 6 . 2 ) - ( 6 . 7 ) ) , whenever th e c omm ander m anipulate s the c ontroller ( F igure 1 ) in the autom atic mode, the LGC is interrupte d and the azim uth command c ount ( N C AZ) or the elevation comm an d c ount

(NC EL)

m anipulation.

i s inc remented

or decremented ac cording

to the dire c tion o f

The rede sign ation algorithm fetche s and r e sets to zero the NC AZ

and N C EL ac cumulators and rotate s LOS P ( the unit line - of- sight vec tor to the cur rent ° landing s ite ) by 1 per c ount (Eq. ( 6 . 3 )). If NC A Z and NCE L are both zero, the G iven that attitude c on trol m aintain s

redes ign ation algo rithm has no effec t.

c oincidence of the ZB,X B p lane and 1-_0S P , the rotations of LOS P are about two axe s normal to 1-_0S P .

E levation redesign ation s rotate LOS P about the Y B- axi s ,

and azimuth redesignations rotate LOS P about an axis normal to LOS P i n the ZB,X B p lane .

The landing- site redes ignation geometry shown in F igure 7 depends upon

the defined LM platform orientation, n amely that the X P - axis i s n e ar verti c al through th e landing site.

The con s tr aint th at LOS P

21

X

be at least as negative as - 0 . 0 2 ( E q .

( 6 . 5 ) ) p revents redes ign ating the landing s ite beyond the horizon.

E qu ation ( 6. 7 )

c o m pute s th e di splac e d point near the surface shown in F igu re 7 and plac e s the r e d e s ignated s ite directly b ene ath this point.

The displaye d LPD angle ( OL P D, Eq. (6. 8 ) ) is the angle between LOS P and the ZB- axi s . The co mputation o f the s tate vector in guidance coordinates ( Eq. ( 6,9 ) ) places th e origin of the guidance frame at the landing site and yields the veloc ity of the LM re lative to the lunar surfac e . Target-refe renced time T i s c ompute d u sing N ewton's m e thod s tarting with a good estim ate ( E q s . ( 6 . 1 2 )- ( 6 . 1 3 )). N o te that the denomin ato r of Eq. ( 6. 1 2 ) i s the de rivative of the num e r ator . T h e guidance equation ( Eq . ( 6 . 1 5 )) is identi c al t o E q. ( 1 1 ) .

The thrust­

acceleration c om m and ( Eq s . ( 6 . 1 6 ) - ( 6 . 1 7 ) ) is merely the total acceleration command m inus current gravity G P , and the unit thrust command ( Eq . ( 6.1 8 ) ) i s the direction of the thrust- ac cele r ation comm and.

The s o - c alled r adial - acceleration guidance

c o rrection des cr ibed in reference 9 is rende red unne c e s s ary by current targeting te chniques and is om itted from thi s report, although presen t in the LGC p rogram . T h e c omputation o f the unit window command presented h e re i s a s implific ation of the LGC c oding which produ c e s the s ame result. The obj ect is to keep the landing s i te in th e c enter of vis ion ( superimp o s e the L P D reticles on the current site ) whenever the geom etry permits and, otherwise, to command a forw ard - fac ing attitude. F igure 8 show s why the landing site c annot always be kept in the c ente r of vision.

F igure

9 shows the geometry p e rtinent to c omputation of the unit window command QN WC P . C om m anding the line - of- s ight vector ( !JN WC P

=

LOS P ) alines the retic les with the

landing s ite; c omm anding the forward vector ( .!J.N W C P fac ing attitude.

=

FO R P ) produces a forward

If the fi r s t alternative is chosen ( _!2N W C P

=

LOS P ) the LM will

rotate about the X B-axis to aline the Y B- axis with the ve c tor LOSP the direction of LOS P

X

C BP

C BP . Thus X indic ates whether a normal forward- facing attitu de

X o r an abnor m al attitude would result from the c ommand _!2N WC P the m agnitude of LO S P

x

x

=

LOS P . In addition,

C BP

measures the degree of indete rmin acy in the X c om m and QNWC P LO S P . The proj e c tion ( P ROJ, Eq. ( 6 . 20 ) ) of LOS P x c;2BP X on the Y G - axis detects both the m agnitude and the direction of LOS P x C BP . X Thus P RO J i s used as the c rite rion for mixing LOS P and £.0R P into UN W C P . If =

the d e s c ent traj ectory i s planar, the m ixing ( Eq. ( 6 . 2 1 )) yields .!J.N W C P LOSP for ° ° nL P D $ 6 5 , .!J.N W C P FORP for OLP D � 7 5 , and U N W C P a mixture line ar w ith ° ° c o s rJL P D for 6 5 < ALP D < 7 5 . R egardle s s of whether the tr ajectory i s p l an ar =

=

o r nonplan ar, it is neve r p o s s ible to com m and a side - facing or a r e ar - facing attitude.

22

------ ----

NCAZ, NCEL �.�

REDESIGNATION COMMAND COUNTS CURRENT STATE CLOCK-TIME TAG OF CURRENT STATE CURRENT GRAVITY BEGIN

T

t

!!.!'

UPDATE LANDING SITI VECTOR FOR LUNAR ROTATION

lP. LP UNIT [.b.P tIt- tOLD I �MOONP X .b.P

<$,?�'

J

Ill

I

I

NCAZ • 0, NCEL • 0 LOSf'x • MINIMUM I LOSf'x .

141 151

+0. 01745 NCEL I �BPy X lOSP I

!,P

L

•

LP UNIT

!lJ' +,lOSP

(

LP - RP X X LOSP X

•

)

LANDING POINT DESIGNATOR ANGLE DISPLAY

9 LPD ·- ARCSIN ll,OSP • £8P I

I

X

;

UPDATE T

•

•

CGP I.!!!'· .IJ' I,

TARGET REFlRENCED

T + It- tOLD I,

t.TCRIT

•

T/128

':J.G

+

T I ME T

tOLD • t

•

1131

?

-(

1141

T

6YGIT •[6

T

( {)

2

·

6

T

(-f-)

+1] �TG

1151

•

COMPUTE UNIT WINDOW COMMAND

• UNIT I �BP X !;_GPy I X PROJ ·l,OSP X £8Px • £GPy yNWCP ·UNIT MAXIMUM I PROJ ·COS 65� 0 I J.OSP

l

-'

181 1

I

COMPUTE CONVERGENCE CRITERION FOR T 1101

1111

r

K

·I

L I

�

1211

P64

J I

K- 0

ERECT GUIDANCE COORDINATE FRAME FOR SUCCEEDING PASS

�

IZ21 �GPx ·UNIT I J.P I 1231 �GPy ·UNIT X {J!P -lP- K I yp · �MOONP X J!P I T /4 �GPz ·kGPX X kGPy 1241 THRUST-ACCELERATION COMMAND AfCP OLJTPUTSo UNIT THRUST COMMAND J.!NFCP UNIT WINDOW COMMAND J.!NWCP

[1r

B

END

Figure 6

1201

+MAXIMUM I COS 75 °- PROJ. 0 I £0RP]

P63

1161 1171 1181

IJq)

£0RP

171

CGP l,l!P- �MOONP X J!.P I 1q11

&

1!21

z

T T 121J!,TG J!GIIT2 •[4(�f) 3(nl6YTGIT

COMPUTE THRUST-ACCELERATION COMMAND AND UNIT THRUST COMMAMJ .lfCP • CPG ACG-liP AFCP • I�Fcpr !J.tf'CP ·UNIT I �FCP I

161

COMPUTE CURRENT STATI VECTOR IN GUIDANCE COOROINATIS .!!,G

��'

·(3(f-)-z {;)] T

121 131

!,OSP ·UNIT llOSP I

T +t.T

--- - -

--- ---- -

-

•(z( n n]

LANDING-SITI REDESIGNATION ALGORITHM J.OSP ·UNIT I J.P- J!PI J.OSP • J.OSP + 0. 01745 NCAZ !;,SP y

0. a? I

•

--

GUIDANCE EQUATION Tp • T +LEADTIME 2 T T �CG

64

•

+

I

E

l'.:l w

-�

COMPUTI T SUCH AS TO SATISFY TIRMINAL JERK CONSTRAINT JTG T3 • 6 ATG Tz +I 18 VTG • 6 VG IT+ 24 I RTG - RG l z Z z Z Z Z --� � �--�· --�t.T. 3 JTG T2 + 12 ATG T +18 VTG • 6 VG Z Z Z Z

-

INPUT VARIABLES:

�

---

P63, P64 Guidance Algorithm

XP

DISPLACED POINT _ R p + LOSP NEAR THE SURFACE--

X

LP.

-

pp X X

LOSP.

PLANE NEW LANDING SITE LP

O LD LANDING SITE LP

CENTER OF MOON

Figure 7

Landing- s ite Rede signation Geometry

24

XG

THRU ST VECTOR THRUST V EC TOR LM ATTITUDE B AC KWARDS

'

N Ul

�

.. -

.. .

.

·�

.

--.

:-

· . · /. · :�. _ ,:, N·.z. ;�. 41':' •

_ :...: ---

I

.

'

-

= C � RRE NT S I T E

.

--

·- -

-

--

' -.1: :. �.-::� "'-"-;'i , ·- _/ · -;c f' -�:.� -- ..: �,· , '.: ·1� 7 �.:� : ::--: · · . :: : -: ::. ..: � ·:,r ;.'" �·�·: ···;y ,. .·. � : .f-.:.--�-.;_��} / /. - - ,. .--e-·:z--·--·�"t.--:-. -· � -.:. .. �>:r. · :.. .· .. .2 :· .. = � : :f,. i " ) :;: a"' � � ,� � , w , ·;�:;=:-� : · -:: ('\ .,., - '·_;;; <:. ...�--"'-' - : :_ , o·• � ' ',...;,. ·;- - . ,. � "-:J:?. ? "" --.' _,;,., , ": ·"<-"" :•V".�/;;-_;-:/"· .._ �· y-.:·.:- ! � 'I

-

.

.

.

-, ..

, :.·

.. ... .

-.-" � Figure g

__

.

.

-

· .. t� :< . •- -

.

,•

· '

.

.

.

.

"

.

.

,,.,.;

'\

-

'

,

____ __

, ,

.

,

- -

-

- -

.

..

.

� --

-.

..

.

-

'

'

-

.

· • • . .. .... - --· ·- · ""' _ --- --. �· . . . .

·,

. • . .

-

-

.

.

·:- .

.

-· ·

�

'�... 2- � ...

- ...

·- ' -

' p

· •• •

W hy K e P p i n g t h e Landing Sitf• i n t lw C c ntl· r o f V i s i o n C a nnot be t h e Sole C r ite r i on fo r C ont roll i n g Attitude . \bout the T h r u st <'\ x i s

XG

X r AXI :l

LI NE- OF-SIG

� BFx

{ A f Pf< OX THf\ UST D / � ) =

HT VECTOR !:_O S P = U N I T ( !:_P - � P )

(

APPROACH PHA SE TE RMI N US

/FO RwARD VEC TOk FO R P 0

----/----f=Jt::�::��:::j=;���

'

CU RRENT S I T E LP

= U NIT (5;8P lC

�� � �

X

£G Py }

RANGE

YG fG Py CR O SS RANGE =

Fi gu re 9

Ge om etry

Pe rti ne nt to C o m pu tin g

26

the Un it Win do w

C o m m and

E r ection of the guid ance c oo rdinate fr ame ( E q s . ( 6 . 2 2 ) - ( 6 . 2 4 ) ) is illustrated in F igure 1 0 .

W ith K

=

1 in P 6 3 , the gui d ance coordinate fram e orientation about

the verti c al X G - axis is such that the Y G - component of j e rk would reach z e ro at the target p o int if the traj e ctory were flown there ( s ee reference 5 ) .

With K

=

P 6 4 , the ZG - axis i s in th e verti c al plane c ontaining the line -of- s ight vecto r.

c ro s s r ange landing- s ite rede signation s , s etting K le s s DPS propellant than s etting K

=

=

0 in For

0 in P 6 4 w as found to c onsume

1 to null the c r o s s r ange j e rk at the tar get

p oint. P 6 3 Ignition Algorithm T r aj e ctory dispe rsions p rece ding P 6 3 require an ac c u r ate ignition time an d attitude to be com puted to 1 ) avoid exce s s ive variation s of the time dur ation of th rottle control in P 6 3 and 2 ) to avoid commanding an exc e s s ive attitude trans ient the fi rst time the P 6 3 , P 6 4 Guidance Algorithm is proce s s e d.

The P 6 3 ignition

p r ocedure consists of: 1)

C omputing onboard the p re c i s e ignition time and attitude about 1 0 minutes in advan c e of ignition

2)

O rienting the LM to the ignition attitude

3)

Initi ating re action control system ullage 7 . 5 se c onds p r ior to ignition

4)

Igniting the DPS at m inimum thrust and holding con stant th rust and c on stant attitude for 2 6 s ec onds, the m aximum time required for the D AP to orient the DP S trim gimbal system to point the th rust vector through the LM c ente r of m as s

5)

Conne cting

the

guidance

algor ithm , which

immedi ately c omm an d s

m aximum thrust and b egins commanding an attitude p rofile ac cor ding to the current state vector and the P 6 3 targets . To dete rmine the required ignition attitude, the ignition algorithm ( F igure 1 1 ) c alls the guid ance algorithm as a subroutine.

The ignition algorithm supplies

inputs c on s i sting of an ac curate extrapolation of the state vector an d the corre sponding gravity vector (both valid at G U IDTIME, the e stim ated c lo ck-time of the fi r s t P 6 3 guidance pas s ) . input s .

In p reparation, Eqs. ( 1 1 . 1 )

-

( 1 1 . 5 ) initi ali ze gui d ance algorithm

On the first ite r ation, the state vector ext r apolation rep resented by Eq.

( 1 1 . 6 ) i s p e rfo rmed by an o rbital integration rouhne and, on sub s equent ite r ation s , h y a K e p l e r routine .

Equation ( 1 1 . 9 ) corr ects the extr ap olated velocity vector by

the veloc ity inc rement imp arted during the 2 6 s econds of minimum th rust p r e c e ding th i s p oint .

( The errors due to not correcting the extrapolated position vector and

not c o r recting for ullage are negligible . )

27

The gui d an c e algorithm p rodu c e s a unit

�

Z G ::= C G P. - z

x a === £

ap x Ve rt ica l Th ro u g h Cu r re n t L an d i n g S ite

- ---

.

I

------ G U i do n c e Coa �at· n ate F YG ==: c ra m e cp XG YG zc Y

Cu rre

n t Lan d i ng S ite .L p

Trajec to ry to Cu rre n !Redes t igna te d 1 La n d in g S ite

_

- K I J(p - WM OONp X BP 1 T/4 LM Ve loc ity Re lat iv e to L u na r S u rtac e K � P63 0 P64 T • Tar g et - Re fere nc ed Tim u rre n t e L M Po s it io n BP

{1

Fi gu re

10

INPUT S

RP_ yP

C U R iif NT STAT£ ON COASTING TRAJECTORY

CLOCK - T I M£ TAG Cl CU RRENT STAT!

t

E STIMATtD Cl OC K - T I M£ FOR f I RST P6J GU I D ANCE PASS

GU I DTIME

THRUSJ.ACCEI F A A T I ON Cl Z o S F C C1 M I NIMUM THRUST

AFTRIM

BEG I�

l

SAVE COPY (I WARfNT STArt J!P

t

YP ANO

ITS C l OCK-TIME TAG I

I N I T I Al \ l£ FOR P6l. P6il GU I D ANCE AlGORITHM

T · · oM � 'fC tOtO • G U I D T I ME LP · L P I G U I D T I ME ' I 0 0: CGP •

I

0I } I

0 I 0 0

IZI

14

U Nf C P • I 0. D. -I I

�: : �: \G�� �DT;;::/1 y• · _y:P "N ·

l

t

.utD ,.,E 1

I I

t

•

GUIDTIMI

SET LOOP COU NTE R 161

171

181

�-

I

1

Ill

EXTRAPCUTf COASTING STATE AND GRAVITY T O GIJ I DT I ME -

1\1 111

I

t

CORRECT EXTRAP(lATtD V I L OC I T Y FOR 16 SECONDS Cl M I N I MUM THRUST

VP

•

CAL

N

•

Af R I M UNIT I

•

VP

Z6 SEC

_191

J

P6l P6il GU I D ANCE ALGORITHM O E C REM£NT LOOP COUNT!R

N - I


NO

I I

I lUI

s

ADJUST GUIDTIME FOR TRAJECTORY D I S PE R S I ON S

� GU I D T I 'IE

•

' KX I RG - R R R I G , K y Rr. X 1 X -

' I RG

7

GUIDT IMF

GIJ I D T I MF

YES

I � K

- RAR IC

r � vc 7

l + K x \J f, x l •

V

I 1/G

/

1111

VRR I G t l

f l;'l

�Gli i D T I M F

� 0

PR[PARf fOR I G N I T ION

I I I G N I T ION

I ·

GU I O T I MI

16 SFC

I I ULLAGE 1 • \ I I G N i fiON I - 7. 1 SEC

l}3f 1 14 1

�

Rf S TOR!. CURRENT RP. VP ANO ITS ClOCK - T I ME TAG I

�

END

OUTPUT� :

t 1 1 c "'J 1 1 1 or� f 1 l lJI I I\r,f I

T H R l r ) l O I RI Cl tON R f () I I I R I

Pfi 3 I gnition Algorithm

29

D Al

t r; N i l l r H � i)Nf r _ f1

th rust comm and !l_f',; FC P , which is the direction to point the XB- axi s .

Bec ause the

d i r e ction of the velo city correction is unknown on the fir st iter ation, the above p r o c e dure is iterated th ric e . An

outer ignition - algorithm loop accounts for dispersions with re spect t o the

nom inal t r aj e ctory. HG

Z

E quations ( 1 1 . 1 1 ) - ( 1 1 . 1 2 ) adjust G U IDTIME to correct the

component of pos ition at G U IDTIME as 1 ) a line ar function of the dispersion

in o rbital speed VG and of the dispe r s ion in the R G X component of position ( e s s entially altitud e ) and 2) as a quadratic function of the out-of-plane position R G y · R B R IG X and R B R IG are nomin al initial altitude and r ange components of p o s ition in guidan c e Z

c oordin ate s ; V B R IG is th e nom inal initial speed; and K , K , an d K x y

c oefficient s .

The nominal initial altitude ,

the targeting p rogram.

range ,

v

a r e correction

and speed a re computed by

The c o r rection coeffi cient s are c omput e d u s ing a m anual

p rocedure based on descent simulations. \V hen c onve rged, this proc e s s yield s a precise time and attitude for igniting the D P S .

T r aj ectory dispe r s ions r e s ult in typ ical variation s of 2 seconds in the

time duration of throttle control and typ i c al attitude trans ients of 2 millir adians c o m m anded by the guidance algorithm on the first P 6 3 p as s .

30

T E R l\I IN AL- DESC E N T - P H ASE G U IDAN C E Hori zontal and ve rtical velocity are controlled in P 6 6 by completely independent algorithm s . P 6 6 p rovide s a non autom atic attitude -hold m ode in which the commander c an c ontrol the L M att itude to translate or not, as he wishes, hori zontally over the luna r surface . P 6 6 i s sue s no unit w indow command; yaw is controlled manually . ( 1 0) d e s c ription of P 6 6 including the nonautomatic mo de s is p rovided by Eyle s .

A

P 6 6 Hori zontal Guidance Algor ithm The P 6 6 hori zontal guidance algorithm ( F igure 1 2 ), p roc e s s e d once every two s e c onds, nulls the hori zontal components of velocity relative to the lunar surfac e by directing the th rust vector a s m all angle aw ay from verti c al in oppo s ition to hori zontal veloc ity.

The hori zontal algorithm neither m e asures nor commands

th rust- accele ration m agnitude; the algorithm is derived on the as sumption that the ve rtical comp onent of thru st- accele r ation equals lun ar g r avity. Just as veloc ity feedb ack damp s a p o s ition control loop, ac celeration feedbac k dam p s a velocity control loop . Because of the s ampled - d ata char acter o f the system, a good m easure of current ac celeration is the ac cele r ation commanded the p re c e ding p as s .

The P 6 6 hori zontal algorithm feeds back the veloc ity error ( current velo city

Z minus lunar surface veloc ity V MOON P y , V MOON P z > an d, to p rovide the r equired dam p ing, feeds back a fr action of the thrust- acceleration command from

VP

Y

'VP

the preceding pass ( Eq s .

( 1 2 . 2 ) - ( 1 2 . 3 ) ).

On the first P 6 6 p a s s , the thrust­

acceleration fed back i s th at com m anded the final P 64 p a s s . The direction o f the thrust - acceleration comm an d i s limited t o 2 0 verti c al ( E q s . ( 1 2 . 4 ) and ( 1 2 . 5 ) ) to m aintain a ne arly e rect L M attitude.

°

from

fhe LIMIT

fun ction of two arguments limits the m agnitude of the first argument to the v alue o f the s e c ond argument. The unit th rust com m and ( E q . ( 1 2 . 6 ) ) i s the direction of the limited th rust­ accele ration com m and. The as sumption in gene r ating hori zontal comm ands that the verti c al c omponent of thrust- accele r ation equals lun ar gravity ( E q . ( 1 2 . 1 )) is re ali zed only if the L M i s not ac celerating verti c ally.

The purp o s e o f ignoring ve rtic al acceler ation i s to

elim inate coupling from ROD inputs to LM attitude . The effect of verti c al acceler ation, which occurs wheneve r the commander m anipulates the R O D switch, is to modulate the gains of the hori zontal chann e l s .

T h i s g ain modulation i s negligible b e c au s e

31

only limited changes in the d e s c ent rate will eve r be commanded; the vertical accele ration c an be s ignific antly non ze r o only for short p e riods of time. P66

V e rtical ( R O D ) Guidance Algorithm The ROD guid ance algorithm, proce s s e d onc e p e r s e c ond, c ontrols altitude

r ate to the refer en c e value by throttling the D P S. The RO D algorithm has no control ove r the L :\1 attitude; the thrust- ac celeration command it i s su e s accounts for any non- vertical ori entation of the thrust vector. The obj ect of the ROD guid ance is to re spond r apidly without ove r shoot to R O D in c r e m ent com m an d s . The algorithm p rovides a time constant o f 1 . 5 seconds, even though the s ample interval i s 1 . 0 s e c ond, by c ap itali zing on the s ampled- d at a characte r o f the system.

U s ing a computed e stim ate o f the total acceleration at

the R O D s ample instant, the R O D algorithm extrapolates s ampl e - instant me asured veloc ity by the effective transport lag of 0 , 3 5 second and thus comm ands an acc eleration app ropriate for the velocity error at the time the ac celeration comman d will be realized. A s ampled data analysis ( referen c e 1 1 ) shows that the compen s ation fo r effective tran s port lag is highly effective in stabilizing the verti c al channel. The signific ant s ystem dynamics reduce to a s ingle zero and two poles in the Z plane .

The zero i s ZZ

= - L AG / ( s ample interval - L AG ) = - 0 . 3 5 / ( 1 - 0 . 3 5 )

- 0 , 53 8 .

One pole i s at the or igin, and the s ec ond pole is Zp

= (time c onstant - s ample inte rval ) / time c on stant = ( 1 . 5 - 1 ) / 1 . 5 = 1 / 3 .

T h e poles are the s ame as f o r an ideal system containing neith e r a t r ansport l ag n o r· an extr apolation. The HOD algorithm has b een s implified for this report as follo w s : 1.

I n the LG C

c oding, the ROD algorithm begin s e ach p a s s by reading the

acc elerometers and recording the time at which they are r e ad. the H O D s ample in stant.

This time is c alled

ROD sample instants oc cur irregularly, but the interval

b etween them, c alled the ROD s ample interval, ave r ages 1 s ec ond. The accelerometer r e adings are used to c ompute a) the th ree - component current velocity vector valid at the R O D sample instant, based on updating the velocity vector supplied by the state vector update routine

(SVUR,

F igure

32

3 ),

and

b)

a thrust- ac cele r ation

m e a s u r e m ent w h i c h is the ave r age ove r the

SVUR

veloc ity vecto r it s upp l i e s , the the S V U R

ROD

s ample inte r v al .

To c ompute the

al s o reads the ac c e l e r o m e te r s . e ac h p as s . at

s am p l e instant o c c u r ring at r e gular

2 - s e c ond i nt e r v al s .

H O D s a m p l e i n s t ants are e s s enti ally asyn c h ronou s with the r e gular instan t s .

C on s equently t h e int e r ac t i o n s b e t w e e n t h e R O D

in up d ating the the

S V U H - s up p l i e d

d ata p r o c e s s e d by the

LG C

SVUR

s am p l e

algo r i th m an d the

velo c i ty v e c t o r are ext r e m ely int r i c ate .

R O D algo r i thm are s h own as input s .

SVUR

In th i s r ep o rt.

H o w the s e inputs

9.

a r e o btained i s d e s c r i b e d in r e fe r e n c e

2.

The i r regular

Although the ve rt i c al o r i entation of the X P - ax i s is c ap itali z e d upon by s eve r al routin e s ,

including

the

P66

h o r i zontal

algo r ithm

an d

the

l anding- s it e

r e d e s ignation algo r it h m . t h e LG C R O D algo r i th m l ab o r io u s ly m an ipul ate s c om p l e t e ­ v e c t o t · s t ate

d at a to m aintain validity for any p l atfo r m align m ent.

Presented here

i s t h e s c alar equivalent val i d f o r the lun ar - l an d ing platfo rm ali gn m ent.

F i gure s am p l e

13

shows

ins tant.

E qu ation

c e l e r ation by adding, ROD

the

HOD algo r ithm .

( 1 3 . 1 ) c o m p u t e s the

input s are all v al i d at the

ROD

s amp l e - i n s t an t total ve rti c al ac ­

to the t h r u s t - ac c e l e r ation m e a s u r ement ( ave r aged ove r the

s am p le interval ) ,

c u r rent

g r avity and

c oncluding the p r e c e ding H O D p as s . by t h e Th rottle R o utin e . v e lo c ity .

The

a c o r r e ct ion fo r the throttle

The th r u s t c o r r e c t i on i n c r e m ent oF

A is

c h an ge

supplied

l.c:quation ( 1 3 . 2 ) ext r ap o l at e s t h e s am p l e - in s t an t m e as u r e d

The c o mm an d e d ve rti c al veloc ity ( r e f e r e n c e alt itude r at e ) i s initi ali z e d

a s the ve rt i c al veloc ity exi sting a t t h e time d e c r e m e nted b y E q . ( 1 3 . 3 ) e ach

P66

i s initiated. and i s in c r e m en t e d o r

R O D p a s s ac c o r ding t o the R O D c o m m an d s i s s u e d

by t h e c om m an d e r s i n c e t h e p r e c e d ing

R O D p as s .

E q u ation ( 1 3 . 5 ) f i r s t c ompute s

th e total v e r t i c al a c c e l e r ation requ i r e d as the n e g ative of the ext r ap o lat e d velo c i ty e r r o r divided by the

R O D time c o n s tant ( 1 . 5

s e c on d s ) .

The equation then obt ain s

the r equi r e d v e r t i c al t h r u s t - ac c e le r ation by s ub t r acting c u r r ent g r avity. dividing by C B F'xx • w h i c h

v e r t i c al X P - axi s ,

Eq.

F in ally.

is the c o s in e of the angle between the X B - ax i s and the

( 1 3 . 5 ) yield s the t h r u s t - ac c e l e r ation c o m m an d A F C P .

To

av o i d an e m p i r i c ally d i s c ov e r e d i n s t ab ility wh i ch o c c u r s when the th rottle routine o r the (13.

6)

DPS

c annot c o mp ly with the th r u s t - ac c e le r ation c o m m an d f r om

an d ( 1 3 . 7 ) r e s t r i c t AF C

P66,

Eqs.

P t o p r o d u c e th r u s t within the p e r m it t e d - th r u s t r e gion .

33

I NPUT,

CU RRENT VHOC I T Y V P

BE G I N

j

C CWI PUT£ UNL I M ill O THRUST-ACCELERA T I ON COMMAND ----- - - -

GM

AFCP X

•

AFCP y

· -

AFCP Z

· -

I V P - VMOONP I I 5 SEC y Y I V P - VMOONP I I 5 SEC Z Z

0. 4

0. 4

Ill

(21

AFCP Y

131

AFCP z

L I M I T COMMANDED THRUST D I RE C T I ON T0 2D° FRCWI VERTICAL AFCP y

· L I M I T I AFCP Y

, AFCP tan 20 ° I X

AFCP z

· L I M I T I AFCP z

, AFC P tan 2D0 x

I S SUE U N I T THRUST COMMAND VNFCP · U N I T I AFC P

I

I

141

151

161

OUTPUT:

U N I T THRUST COMMAND L!NFCP

£NO

Figure 1 2

P 6 6 H o riz ontal Guidance Algorithm

34

I NPUTS : COUNT Of ROD I NP I I T <; S MIPLE l �i S T A NT M E A S U R E D VELOC I TY C U R R E NT GRAV I TY T Y R l! S T � >IC C : : E R AT I n c ; \lE A S U R f l.lf'H

I Averaqed over t h e ROD sample l nl n 'l a l l

T H R U S T CORRECT I ON I N C REMENT I f rom the Throttle Rouline I

M

C U R RF NT M A S S E ST I MATT

BE 1. 1 �

-�-

� -�-��

...______________,

___ _ _ _ _

C OMPUll EX I ST I N G V E R T I CAL A C C E L E RAT I ON AT ROD S AM P L E I N STANT

l

E X T R A POL ATE S AM PLE I N'iTANT MI A S I I R F D VELOC I T Y BY f F F FC T I VE T R A N S PORT l A

U P D ATI COMMANDED VE RT I C Al VFl OC I T Y I N C O RPORAT I NG ROD I N P U T S VCP - VC P X X NROD

-

•

ill

NROD D. l mlsec

141

D

COMPUll THRtJ S T � A ( Cf lE RAT I ON COM M AND F O R T HR OIT lf ROI I T I NI

AF CP

•

- I VP X

- v r rx C BP

1 / 1. 5 SEC - GP X XX

�J I

I

R E S T R I C T T H R U S T ACCELFRATION COMMAND TO PRODUCE

T H R U ST W I T H I N Pf R M I TIED � T H R l ! S T R E G I O N q I/ 3

M I N I MU M I AF C P . (:{) ··�

A F C P · MA X I M U M I A F C P .

AfCP

"',

46706 NFWTONI M I

467DIJ NEWTON/ M I

161

171

O U T PU T : T HR I J '> I AC C f l FRAT I O N C O MMAND A f C P E NO

Figu re 1 3

P fi 6 V e rt i c al ( ROD) Guidance Algo rithm

35

l,O W E H E D - F LIG H T ATTITUDE - M AN E U VE H ROU TIN E A link in the attitude control ch ain of command, the Powered-flight Attitud e ­ m aneuver r outine ( AT T ) connects t h e various powe r e d - flight guidance p r og r am s t o th e D AP . T h e functions o f A T T are : 1.

F o r the sm all attitude changes norm ally r equired e ach guid ance cycle, ATT c omm ands a maneuve r of con stant r ate such as to achi eve the r equired attitude 2 s e c onds late r .

2.

F o r gross attitude m aneuve r s which m ay b e required at phasic inte rfac e s o r upon abort, ATT com m ands a r ate- limited m aneuver which m ay extend ove r s eve r al guidan c e cycle s .

3.

F o r all attitude m aneuve r s AT T avoids th e gimbal - lock region ( middle

gimbal angle > 70° m agnitude ) .

ATT i s sues a gimb al - lock alarm c o de

if and only if the comm anded attitude c o mputed from guid ance inputs lies within the gimbal - lock re gion.

AT T com m an d s a m aneuver which

c i rcumvents the gimb al - lo c k region and i s sue s no gimbal - lo c k alarm c ode when the most direct p ath to the comm anded attitude p a s s e s through the gimbal- lock region. Switching from a d e s c ent program to an abort p ro g r am m ay p ro duce up to ° 1 8 0 change in comm anded thrust direction. A b r e ak with tr aditional app roaches, ATT m akes gimbal lock during any m aneuve r inhe rently impos sible by 1 ) c omputing c o m m anded gimbal angle s , 2 ) lim iting the m agnitude of the middle comm an de d gimb al angle , and 3 ) is suing t o the D A P a s e ries o f inc r emental attitude - m aneuve r * commands that monotonic ally drive the gimbal angles from their cur r ent values to their c omm anded value s .

P rovided the attitude i s not currently in gimb al lock,

and given that the m iddle c ommanded gimb al angle i s m agnitude limited at the gimbal- lock boundary, it is inherently impo s s ible to m aneuver through gim b al lock; the m iddle gimbal angle is c onfined to the r ange between its cur r ent and c omm anded v alue s .

Other attitude- m aneuver s chemes with appended gimbal- lock avoidance

r equire more computation to p roduce s imilar m aneuve r s . F igure 1 4 p r e sents an overview o f the L M powered- flight attitude c ontrol p r o c e s s , including some inform ation on the p ro c e du r e s on the D AP side of the

*

Exc ept the outer gimb al angle p rofile m ay not be m onotonic in the geometric ally c omplex case of a large m aneuve r about multiple axes at substantial middle gimb al angle and with m agnitude limiting of the X - axis attitude angle change on at l e ast one p a s s through ATT.

37

I

I I

m ��: l

R

RATES

PERM !TIED lAG ANGLE I

U N l T TI< RIJ S T COMMAND

tiNlT W l NOOW COMMA�D

w co

THRUSTACCElERA TION III'A SUREIII'NT

OMPUTt COMMA�£ G l 'oi B AL ANGLE S, LIMlT MAGNITUDE CX COMM ANDED MIDDU

I

IMBAl ANGU TO 700

I I

1

COMMANDE D G I MBAL ANGUS

I I I I I I I I

I �:� I !

REF E RE NCE G I MBAL ANGLES

J>1 1 :0,:,

�

DAP COMMANDS

REFERENCE L G L INCREMENTS

I I I I ! I� � 19 �� 1 5�

MEASURED G I MBAL ANGLES

��� �§ �I:�

E�

�ISs ���= �!!I I � \l!�

� �

�lg

�

���

�li� G l�

���

Figure 1 4

LM P owe red - flight Attitude C ontrol

SPACECRAFT ATTITUDE

inte rfac e .

Two c omputational c oordinate frame s are introduced.

F rom gui d an c e

and n avigation inputs , A T T c ompute s a c om m anded- body frame ( tag C B ) t o repres ent the c om m anded attitude inherent in the input vecto r s .

F rom ATT inputs, the DAP

com putes refe rence gimbal angles to compare with m e asured gimbal angle s for c om puting the attitude e r r o r s . The reference gimbal angles define a refe rence- body fram e (tag R B).

ATT c omputes that the attitude errors are zero when the s e two

c om putational coordin ate fram e s coincide.

Of cou r s e , the re m ay be DAP c ontrol

e r r o r s undete cted by ATT, but any thrust pointing error is detected

in

the ste ady

s tate by a thru st- direction filte r, and c o r r ected. The guidance and navigation inputs to ATT, shown in F igure 1 4, consist of a unit th rust comm and, a unit window command, and a thru st - ac c ele r ation measur ement. ,\TT proce s s e s the thrust- ac celeration m e asurement in a thrust- direction filter to

dete rmine an e stimated unit th rust vector with respect to the referenc e - body fr ame. C o rrecting for the offset of th e e stimated unit th rust vector w ith respect to the X R B- axi s , ATT u s e s the unit thrust command and the unit window comm and to e r ect the c om m anded -bo dy frame. F rom the commanded- body fram e m atrix, ATT extracts commanded gimbal angle s which it compares with the refe ren c e gimb al angle s to generate inputs to the D A P .

T en tim e s per se cond, the D1\P update s the refe rence

attitude and gene rate s the corre sponding control command s . The dyn am ic r e sp on s e i s suffi ci ently fast and tight that the refe renc e attitude i s a good me asu re o f instantaneous space c r aft attitude. A feature of thi s c onfiguration is that, although ATT runs at a s ample r ate of

2 s e c onds, clo s e to the fue l- slosh resonant frequency at cert ain points in the mis s ian, it avoids exc iting fuel slosh by avoiding all c oupling with the actual spacecraft attitude except th rough the s lo w thrust- dire ction filter. F igure 1 5 details the Powe r e d - flight Attitude - m aneuve r routine.

The th rust­

d i rection filter compute s the thrust - ac celeration m e asurement in referenc e - body coordinates by constructing th e required t r an sfor m ation from the referen c e gimb al angles ( Eq . ( 1 5 . 1 ) ) . The change in th rust direc tion i s limited on e ach cycle to 7 - mr ( E q s . ( 1 5 . 3 ) and ( 1 5 . 4 ) ), the m aximum travel of the trim gimbal in 2 secon d s .

The

total excurs ion of the e stimated unit thrust vector i s limited to 1 2 9 - mr ( E q s . ( 1 5 . 5 ) and ( 1 5 . 6 )), the mechanical exc u r s ion limit o f the trim gimb al plus mechanic al deflection and thrust offset with respect to the nozzle.

The X - component of the

e stimated unit thrust vector is not needed and not computed. If e ithe r 1 ) guidance p rovid e s a unit w indow c ommand too c lo s ely alined with the unit thrust command to ade qu ately dete rmine the att itude o rientation about the

39

I 'Jf't· l '.

: 1 "' 11 TH�t·)J

!'NIT

CtlM��"-jO W INlHl'll (l�Y.ANO

U�CP

I

!?Hi R! "''rl AllflY

1

RGA x

_g_ R8

A.l'\ \ t f ll ii, I ION,Ri r i � ! �'i nnR�l t NAH '.

1\�'A II Afll!

.

�GA y

,

llGA z

l i M I T MIDDLE COMMA NOLO G IMBAL AM;lE SAVING O R I G I NAl FOR POSSIBL� AlARM Q • CGAz CGA

B EG I N

L

I

· II� II I �fR8

l�fP

· l l �tl

•

I I Y. l l

--�-- --

CGA

()91

ro• 1

z

1101

�

m·

L

II'

6URGA • CGA

r---- ·-

161

M

RGA

RES£1 W I NDOW HAG

If I 1f I

•

t l r'ol tt W'f t r-.oow (OMMANo tr w i T H I N ts0 or uNn THRUST coMMAtfD o\N[) � : \ I f VV I�Oll'W H A,f; ' • <' 0 r - UNH P f.
�URGAy

If COMMAND lNG

f Rf 1-

:

����MA.�Of 0 --

\JBD;( · \.!_M I- r r !-

�CRP,/

llr-.11 l � f\IW( p

I '::: < 'RPX )( ��l

�,- R P ,

l

L...__________ ..._

�-���;ti;-��MM-ANOI [) I

�C B P X

II

��

l

B P 'I"

i,._C R P /

t

•

U N II

AURGA .z

I >41� l >4��

FLAG .

FLAG .

�

BP

BODY

A�\TRIC !

U Nf R B

fj,RGAX

-------

Cf B P)( l

y

i

I

l

0

0

CC B PX I

" RB

4>RB

X

I

r C BP )(

�

flO)

)

$RB

X

Y

�•ez

UTI UBI

OR

Z

I

G I MBAL ANGLE CHANCES

1221

i2}t

Y

;.; '

ll.uRGAX

l'l ATT X

·

•

1241

COS RGA

0

l'lRGA

y

20

Z

°

1 I COS RCA l

t.. URGA y S I N RGA

S I N �GA

+

. [�

z

· L IMIT ( o.� R B X J �o� R B · l i M I T I �J RB · LIMIT

X

OI R B Jl y yi

I • RBz

I • R Bz

�

!1 1

o RB o RB

;

X y 1

10

°

10° . 10

°

j

GA

COS RGA

X

I I 2 a RB

'lff

�'j

J

Q1\

i281

-

liN

COS RGA z COS RGA X COS RGA z S IN R:GA

I,

l

• l:.ftGA I 10 S I N RGA

z

ll�\

1291

6B_GA

I

2 SlC

)(

1 I

1

OUTPUTS :

N

T

l

llOI

011

REFfRENC[ G IMBAL ANGI( INfRI MFNTS

I

Figure 1 5

•

6 ftGA

1131 11•1

I

Y

1211

+

II SUE OAP COMMANDS

I III

CC B Pn

[

If HAG • 0 . 6 ATT X

I Ill

U>HB z I,C BP

�

CCBP y z

)(Y

-- --

I C.CBP Y

y

G I MBAL "NGLES

C C BP

l'lATT X · L IMO

F R A .W: f RfCTION FOR THRUST OffSfT

· �i'!H') X �\B PX:

J (,tl \ · ,!I, P [ T P ) f, l

l'lRGA y · L I MIT 1/j.URGA

Ill\

. J. -

J

10° 1

l'lRGA z • l iMIT (ll,URGA z

--·

I

G I M BAL-l OC K

ALARM COOl

COMPUTE RHER£NCE G IMBAl ANGLE CHANGES l i M I T I NG ATIITUO£ ArK;tl RAH5 TO ° 10 1 5lC i l0°1 l SEC I

1101

I !.;. C B PX

· �rRP)( )( �f.BPy

W�,,( T Cll�MAND£D

j i ,A ,

x

I Ct:RP�

1 (.!> ( · ARCH� I(, I

L

-- - - - I -RAMI - -�l•OY - -

I SSut

I

[)(CESS lYE

� � ,!!,t

:

VM U[

COMPUTE Ulll IMITEO �EFE�EM:E G I MBAL ANGlE CHANGES US I NG MODULAR SUBT�ACTION

141

•

-·-

I

t

111

!

--

• l iMIT

1

'II

�

-

I

RGAf

I

[ D l l lJ�f R B y u ,.,- R B y I 0 001] [ 0 � liNAf RB l Uti-"RBz I 0 001] 11"11- R By · 1 1'-IIT [1 Ur,ifRBy t..U �RBy I. 0 12'] t1NF RB 1 · : I M I T [t L'Nfli'Sz AU�R: 8z I , 0 12'9 ]

t. l l 'f R B y

6 U"' R s7

l 1R; r�- Ac-f 1 ! If I :J\'Wt l �

I

1

THRliS 140 I R�Cl iON f 1. Tf R �fR8 · C I R t; A ); R f. A. y �·�Af R �

--

UM'fCP

C I Mfi:A.l .l.N�;[ f ')

TIIRI • 'll�A.U l l t R II, TiON W A <., U Rf W NT MP

REn

Rf

CE All ITUOf RA I S

PfRM InfO I AG ANGI • :,

END

P ow e re d - flight Attitu de - maneuve r Routine

40

I

� RGA iie

� RB

XCB -ax i s , or 2 ) the guidance p rogram i s mand ) ,

then ATT

P66

(which p rovid e s no unit w indow com ­

p rovid e s a unit window command suitable for e re ction of the

c ommand e d - body frame and resets a flag t o indicate that no attitude rotation i s allowed about the XC B - ax i s .

A T T first p rovid e s the c u r rent Z B -axis ( E q. ( 1 5 . 8 ) ) .

But th i s choice may also b e nea rly c olline a r w ith the unit thru st c ommand, s e c ond poss ibility,

the c u r rent negative

X B - axis ,

is also o ffe re d (Eq.

so a

( 1 5. 9)) .

Because the Z B - and X B -axes ca nnot both parallel the unit t h ru st command ,

no

fu rthe r checks need be made .

The m atrix C C B P , whose row vecto r s are the comm anded- body fr am e unit ve cto r s exp r e s s ed in platform coor dinate s , is computed to s atisfy the unit thrust c o m m and, the unit window comm and, and the thrust offset (th e angu l ar displac e m ent between the e stim ated unit thrust vector and the X R B - axis ) . two step s as illustr ated in F igur e 1 6 .

C C B P is c o mputed in

The first step ( E q s . ( 1 5 . 1 0 ) - ( 1 5 . 1 2 ) ) u s e s

t h e unit th ru st c om m and and t h e unit window command b u t fails t o account f o r thrust offset. The s e c on d step ( E q s . ( 1 5 . 1 3 )-( 1 5 . 1 5 ) ) corrects for thrust- o ffset components U N F RB

y

. Since the s e c o r re ctions are s m all, no unit need be t aken Z A s m all window pointing e r ro r , shown in F igur e 1 6 , i s introduced

and UNF R B

in J:: q . ( 1 5 . 1 4 ).

by the th rust- offset c o r r e ction.

Define d a s the angle between the ZC B,XC B plane

and the unit window command, the w indow pointing error i s the product of the sine of the LPD angle and the thrust- offset angle about the Z C B - ax i s .

Although the

°

t r im gimbal h as a m aximum displac ement of 6 , the m aximum thrust offset during d e s c ent i s about 1 ° , which yields a m aximum window pointing e r ror of 0 ° at 0 ° L P D angle and 0 . 9 ° at 6 5

° L P D angle, the lowe r e dge of the LM window.

B e c au s e the m at r ix e e B P i s the t r an s formation from p l atfo r m to commanded­ body c oordinat e s , it c an be exp r e s s e d in terms o f the I M U gimbal angle s which would place th e body axe s in the com m anded dire ction s .

The r efore, c o m m anded

gimbal angle s c an be e xt r acted from the commande d - body m atrix. E xpre s s in g e e BP as the p roduct of the three m atrices that c o r r e spond to rotations about the thr e e

[

gimbal axes yields

e e BP

=

+C Z

-ex +SX

where S and

e

ey sz sz

ey ey

+ +

: +sz sx ex

I I

SY SY

: -e z I I

SY

: +e x e z : +e x sz I

I

I

I

: - sx c z : - sx sz

SY

+ sx

SY

+ ex ey

CY

l

(13)

indi c ate s in e and c o s in e , and X , Y , and Z ind i c ate the commanded X ,

Y , and Z gimb al angl e s .

F rom E q . ( 1 3 ), i t i s apparent th at the comman d e d gimb al

an gle s are extracted from the elements of AR e TR IG defined a s follow s .

41

e e BP

by E q s . ( 1 5 . 1 6 ) - ( 1 5 , 1 8 ), with

1 c -c s P

1

=

X

UN FCP ( U N I T T H R UST COMMAND) - U N FR B y

---

J

- U N FR B z

UNWC P ( U N I T W I NDOW COMMAND )

- �N F R B y YC B �C B P y - U N F R Bz ZCB fC B Pz

Figure 1 6

Ge ometry of E re ction of C ommand e d - body Frame Viewed on a LM - c e nt e r e d Unit Sphe re

42

The AR C T H IG function of two arguments yields the angle whose tangent i s t h e r atio of th e f i r s t and s e c ond argument s .

ARC TR IG ext r acts t h e angle anywhe r e

in the c i rcle b y u s ing the r atio of t h e s m alle r - m agnitude argument t o the large r ­ m agnitude argum ent a s the tangent of the angle o r its c omplement, and b y u sing the s i gns of the argum ents to dete rmin e the qu adr ant of the angle .

Equations ( 1 5 , 1 6 )

an d ( 1 5 . 1 7) yield the outer and inn e r c o m m anded gimbal angle s anywh e r e in the c i r c l e . B e cau s e the s e c ond argument i s always pos itive, imp lying a p o s itive c o s ine, l·: q . 0 5 . 1 8 ) yields the m i ddle commanded gimbal angle between ±9 0 ° . To p re c lude c o m m anding gimbal lock, E q . ( 1 5 , 2 0 ) limits the m iddle c om m anded g i m b al angle to 70 ° m agn itude. B e c au s e the unlim ite d v alue lay between ±9 0 ° and the oute r and inn e r c o m m anded gimbal angle s were c omputed consi stent w ith the mi ddle c om m anded gimbal angle range, no quadrant s w itchin g of the oute r o r inn e r c omm ands is r equi r e d by g im b al- lock limiting.

If limiting c h anges t h e m i d dle

c o m m and, the gui d ance is com m an ding gim bal loc k, and the gimbal- lock alarm code i s i s s ue d . U nlimited reference gimb al angle c h anges are t h e changes w h i c h would b e req 11 i r e d t o b ring c o m m anded

the D ,; p

gimbal angle s .

's

r e f e r e n c e gimbal angle s into coincidence w ith th e

The se are computed by subtr acting, modularly, the

c u n ent reference gimbal angles from the comm anded gimb al angles ( E q. ( 1 5 . 2 1 ) )

.

The m odular subtractions yield the s m alle r angular diffe ren c e s , i . e . ,

If a Y o r Z gimbal angle change g r e ater th an 4 5

°

i s requi red, t h e flag i s

r e s e t i ndicating n o attitude rotation i s allowed about the XC B - axi s .

T h i s i s n e c e s s ary

to p r event fals e starts about the XC B - axi s as de rived in the appendix of refe rence 1 2. Equations ( 1 5. 2 4 ) - ( 1 5. 2 8 ) yield the reference gimbal angle c h anges oy lim iting ° ° th e m agnitud e of the attitude c h anges to 20 in 2 s e conds (1 0 / s e c ) about e ach of th r e e o rthogonal axe s ; one axi s i s coinc ident with the XC B- axi s and the oth e r axe s l i e i n t h e YC B , ZC B plan e . deg ; s e c .

Th i s p e r m its an angular - r ate vector of length 1 0

{3

N ote that i f the flag i s r e s et, the attitude rotation about the XC B - ax i s i s

m ade z e r o , re sulting i n an outer gim b al angle change t o offset the inne r gimbal an gle c h ange ( E q . ( 1 5 . 2 8 ) ) . The DAP c omm an d s c on s ist of th e refe rence gimbal angle inc rements to be app l i ed by the D .\P e ach 1 I 10 se cond, the referen c e attitude r at e s , and the p e rmitted

43

l a g angle s .

T h e refe r e n c e g i m b al

angle incr em ents are the r eferen c e gimbal angle

c h an g e s m u ltip l i e d by the r at i o of the I L\ P and ATT s am p l e inte rval s ( Eq . ( 1 5 . 2 9 ) ) . The

r·efe r e n c e

attitude r at e s a r e c omputed b y the nonorthogonal transfo r m ation o f

t h e r · cfe rence g i m b al angle c h an g e s shown in E q . ( 1 5 . 3 0 ) , which

The p erm itted l ag angle s ,

a c c ount fo r t h e angle s b y which t h e attitude will l a g behind a r amp angular

c o m m an d

due to the finite ac c e le r ation s available, are computed u s in g the available

a c c e l e r · at i o n gR B , and then individu ally

m agnitude limited ( E q . ( 1 5 , 3 1 ) ) .

The DAP

avoi ci s attitude - r ate ove r s hoot by permitting lagging attitud e e r r o r s equal to the p e r m i tt e d lag angle s .

44

T H HO T T LE H O U T IN E The throttle routine connects the currently ope r ating guidance algorithm to th e D P S , as illustrated in F igure 1 7 .

F o r e as e of und e r s t anding, all thrust levels

are represented as p e r centages of the DPS r ated thrust of 4 6 706 newton s . g enerates

th rust

inc rement com m ands to drive the input

T HROT

thru s t - ac celer ation

m e asurement into c oinc idence with the input th rust - acceleration c omm and wheneve r the resulting thrust would lie within the illu strated p e rmitted - th rust region. W hen the r e sulting th rust would lie below or above the p e rmitte d-thrust r egion, THROT c au s e s m inimum o r m aximum thrust.

The hyst e r e s i s - like region from 5 7 to 6 5 o/o

th rust a\'oids frequent alternation between the m aximum - thrust point an d the p e r m itted-thrust region when the thrust command dwells at the boundary between th e permitted-thrust and forb idden -thrust r e gion s . :\

digital- to- an alog interface between the LGC and the D P S is p rovided by the

d e s c ent engine c ontrol assemb ly ( D E C A). E ach guidance cycle (once p e r two seconds, except once per s e c ond for P 6 6 ) T HRO T gen e r ates the th rust inc r ement c omm and � F C % which i s c onve rted to a pul s e t r ain and is sued to the D E C A.

E ach pul s e

c au s e s about 1 2 . 5 newtons thrust change, an d the pulse r ate i s 3 20 0 I s e c ond. F ollowing i s suance of a thrust inc rement c om m and, th e thrust therefore changes at the r ate of 4 0 , 0 0 0 newton s I s ec ond ( 8 5% of r ated thrust per second ) until the thrust inc rement is achieved.

\V ith a guidance cycle as short as one s e cond and an engine r e spon s e

time which m ay be a substantial fr action of o n e s e cond, i t i s nec e s s ary f o r P 6 6 and T H R O T to ac c ount for this transport delay. c\s illustrated in the rightmost box of F igu re 1 7, in the region from 1 1 to 9 3 % the DPS th rust i s a ne arly line ar function of the pulse c ount ac cumulated by the D E C :\.

:\i ot shown in F i gu re 1 7 i s the m anual throttle command, which is summed

w ith th e DEC A output command by the DPS and p rovides the m inimum 1 1 o/o thrust w hen the D I,:C A c om m and i s zero.

The DPS c ontains a m e c h ani c al stop at typ ic ally

9 3 o/o rated th rust. This thrust level m inimizes p ropellant c on sumption on a nominal

descent, con side ring the lo s s of specific impul s e at higher thrust. th e

DPS

To en su r e that

is driven to the m e c h ani c al stop, the DEC A s atur ates at a sub stantially

high e r thrust level ( about 9 9 o/o ) and the throttle routine drive s the DEC A into s atu ration whenever m aximum thrust is required. � onlinear iti e s

in r e sponse and unc e rtainties in DEC A and DPS scal e factor s

a r e ove rcome b y the th rust in c rement command c on cept. N omin ally, THROT p rovide s dead-beat respon s e to step input s , but w ith down stream nonlineariti e s and s c al e - factor e r rors T H R O T d rive s the thru s t - ac celeration error to zero in the ste ady state .

45

THRUST

1HRUST-CONTRO. LOGIC TO ?J;?OOUCE T Hl REQL I RED RlSPO�;Sf

COMPUH THJlU� T

Af(P ACCEURATIQN

COMMA'NO

CI""''«l

C

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O�HtALL SYSTE\1

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;:: I � �

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�

1�

fCC� � qq,.

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>

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M

RfSPONSl

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C U M U l A T I VE THRUST COMMA"..O

1 ,. I�

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ll<J

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DECA AND D P S

DPS Thrust C o ntrol

,

-

OEI IVE RI D THRUST

m

F i gure 1 8 illu strates the Throttle Routine computation s . The thrust comm an d and

th rust m e asurement are c omputed

( 1 8 . 1 )- ( 1 8 . 2 ) ).

usin g

the input m as s e stimate ( E q s .

T h e input thrust- accele ration m e as urement i s the aver age o v e r the

p re c e ding s ample interval, du ring which a th rust inc rement com m and was i s sued p roducing an instantaneous th rust p rofile as illustrated in F igur e 1 9 . to obtain

the current

s ample- instant

thrust,

Eq .

( 1 8. 3 )

Therefore,

corrects the th rust

m e asurement by addin g the th rust c orre ction inc rement c omputed the previo u s cycle. The th rust- c ontrol logic for p roviding the r equired ove r all system re spon s e illustrated in F i gure 1 7 is t o pick one o f four po s s ible thru sting policies acc or ding to the regions of the p re c e ding and p r e s ent th rust commands ( FC O L D% and F C %). and to r e s et the th rust command if nec e s s ary to s atisfy DPS constr aint s .

Equ ation

( 1 8 . 4 ) or ( 1 8 . 8 ) re sets the thrust command to the thrust actually anti c ip ated .

A

th rust c ommand augment ( F C AUG%) i s c o mputed that e ithe r drive s the DEC A into s atu r ation if the policy is to initiate or retain m aximum thrust ( Eq. ( 1 8 . 5 ) or ( 1 8 . 9 )) , o r c o r re cts for the region between the D P S mechan i c al stop an d the DEC A s aturation valu e if th e policy i s to initi ate th ru sting w ithin the permitted - thrust region ( Eq. ( 1 8 . 7 )) .

No thrust command augment is required when the policy i s to continue

th rusting w ithin the perm itted-thrust region.

No equivalent thrust- control logic i s

needed a t the m inimum - thrust point b e c ause minimum thrust would occur only i f t h e c om m ander c ould i s sue five or more downward ROD c omm ands within a s ingle P 6 6 guidance s ample interval, p ractic ally imp o s s ible.

The th rust increment command ( Eq. ( 1 8 . 1 2 )) i s comp o s e d of the actual thrust in c r em ent �F Ao/o, plus the thrust command augment FC AUG% to drive the DEC A in or out of s aturation, when requi red. P re paratory to com puting the thrust corre ction increment for the s u c c e e ding p as s , Eq. ( 1 8 . 1 3 ) compute s the total effective t r an s port lag. The terms in the effective tr an sport lag are 1 ) the c omputation duration t - tSI, 2 ) the e stim ated DPS time c on st ant of 0 . 0 8 s e cond, and 3) the effective DEC A d elay equ al to hal1 the time r equi red to output the thrust inc r e m ent command pulse tr ain at 8 5 % thrust change per s e c ond.

As long as the actual thrust inc rement AF A ( F igure 1 9 ) i s c ontaine d

enti rely w i thin the s ample interval At, i t is c l e ar that. a s L AG app r o aches ze ro, the th rust measurement ( obtained by diffe rencing accelerometer r e adings at the s ample instants ) app roaches the s ample - instant thrust F .

Similarly, as L AG

approaches the s ample interval At, the thrust m e asurement must be augmented by an amount app roachin g the actual thrust inc r ement AF A to obtain the s ample - in stant th rust F .

F rom this heuristic argument, i t i s app arent that the thrust c o r r ection

inc r ement whi ch must be added to the thrust m e asurement to yield the s ample - in s t ant

47

I NPUTS THRUST ACCH I R A T i fl N COM\IAND THRlJST ACI H I RAT I O N Mf A SU RI '.II NT C l OC K - T I MI

SAMPll I NSTANT Cl CURRENT

Af C P

Af P lSI

GU I D ANCf C YC l E

M

C U R RI N! M A S S f S T I MAIT BI G I N

C OM P U IT TH RU S T COMMAND A N D THRUST Ml A S U REMENT

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COMPUH SAMPlf I NS TMIT T HR \ I S T BY A D D INC TH> ACTUAl THRUST THF PREC f D INC CYCI I

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S £ 1 TH RU S T COMMAND AUGMENT

}

COMPUTE ACTUAl T H R U S T I �C RIMFNT AND S A V l THRUST COMMAND FOR S U C CH D lNG P A S S I'. I A'' I C' ' F �

r ca o '

r c,,

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ALGORI THM

OUTPUTS, THRUST I NC RFMENT COMMAND THRUST C O R R fC T I O N I NC R f MoNT

I N il

Figure 1 8

Throttle Routine

48

M C '\

bfA

�---

Guidance Sampl e Interval �t

Computation Del ay P l us D P S Time

C onstant

Effective

r-; DECA Del ay .1�J

•

.

1 nstantaneous Thrust P rof i l e _ Sampl e

-

Thrust F ..,. co

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Actual Thrust Inc rement 6FA

Th rust C o rrection Increment P revious

o FA

=

NA

LAt 6t C ur rent

Sampl e

Sampl e

Instan t

Instan t Figure 1 9

Thrust Dynamic s Within a Single Guidance Sample Interval

1 nstant

th rust is proportional to L AG as c omputed by E q . ( 1 8 . 1 4 ).

A r igorous derivation of

th is re sult is p r e s ented in Appendix A of reference 1 1 .

The s ole purpo s e of E q.

( 1 8 . 1 5 ) is to inte rfac e the P 6 6 V e rtic al ( RO D ) Guidance Algorith m . \V ith the thrust command FC % either within t h e permitted- thrust region o r r e s et to the value which will actually b e achieved, A F A o/o is an ac c u r ate p rediction o f the actual thrust increm ent, and 8 F Ao/o o r c5F A is an ac c u r ate thrust c o r r e ction inc rement .

a F A o/o or aFA is s lightly in e rr o r w hen initiating thrusting w ithin the

permitted -thrust region.

The slight e rro r is due to ne glecting FCAUG% in the

c o m putation of LAG.

50

B R AK IN G - P H ASE AN D A P P RO AC H - P H ASE T AR G E TING P ROG R AM The targeting program gene r ates m i s s ion - dependent d at a for the P 6 3 Ignition Algorithm an d for the P 6 3 , P 6 4 Guidan c e Algo rithm .

All d at a are exp r e s s e d in

gui d an c e coor dinate s . The ignition alg orithm requ i r e s nominal initial altitude, r ange, and speed d at a that dete r m ine ignition time and indi rectly determine the throttle­ c ontrol du ration.

The guidance algorithm requ i r e s t ar gets for P 6 3 that p rovide an

efficient t r ansfer and targets for P 6 4 th at p r ovide a t r aj ectory m eeting s ev e r al c on s t r aints on geometry, vis ibility, and thrust.

De s c r ibed in detail in referenc e 5,

the P 6 4 c onstraints p r ovide a fast s h allow app roach phase m o r e akin to an ai rplane app r o ac h th an

a h elicopter

app roach,

although the

termin al - des cent

phase

is

e s s entially vertical i n helicopter fashion.

T h e landing s ite m u s t b e appr o ached ° along a ne arly s t r aight- line p ath depre s s e d typic ally 1 6 from hori zontal, terminating

typic ally at 30 m altitude 1 1 m ground r ange.

The l anding s ite must be vis ible

c ontinuously until a few s e c on d s before app r o ac h - ph ase terminus , and the DPS thrust must begin at around 5 7% and must lie c ontinuou s ly in the 1 1 to 65% region.

G eometry, visibility, and thrust during app r o ac h c annot b e specified explicitly. V i s i b ility depends upon the p o s ition and attitude p rofile s , and the s e p rofiles (with the th rust and m as s p rofile s ) are con s t r ained to s atisfy the l aw s of phy s i c s .

The

guidan c e algo rithm will p rovide the t r an sfer from any arbitr ary initial s t ate ( w ithin b ound s ) without regard to any vis ibility o r thrust constraint s . The task of the t ar geting p r og r am is to set up the P 6 4 initial s t ate and guidan c e t argets such th at suit able v i s ibility an d thrust p rofiles are r e al i ze d implicitly. During the final p ortion of P 6 3 , and th roughout P 6 4 , the guidance algo r ithm will gene rate a t r aj e ctory who s e p o s ition vector is a quartic polynomial function of time, as shown in F igure 20.

T argeting consists of 1 ) defining e ach of the two

p olynomials and 2) ext ractin g the guidan c e t argets as the p o s ition vector and its d e r ivative s at a t arget point, lying on the polynomial, sub s tantially beyond p h a s e t e r m inu s . The

P 6 4 targeting

c on c ept i s to c on st ruct the app r o ac h - ph ase quartic b y

i m p o s in g nec e s s ary and suffi cient constr aints . W ith quartic degree, five independent c on st r aint s m ay be im posed in e ac h of three axe s .

The nomin al t r aj ectory i s

arbitr arily m ade planar, requi r in g the Y - components i n guidan c e coordinates o f p o s ition and all its derivative s t o be z e r o and leaving two axes t o specify .

Bec au s e

t h e initial state c an be c ontrolled by t h e p r e c e ding b r aking- phase guidan c e , all five c o n s t r aints in e ac h of the two rem aining axes m ay be specified arbitr arily.

Sinc e

these ten const r aints - c alled a P 6 4 c on s t r aint s e t - completely deter m in e the

51

T o - 1 8 0 SEC T H ROTTLE CONTROL RECOVERY

IGNI T I O N

N O N OU

T o -6 0 S E C BRAKINGPHASE T E R M I NUS

T o - 1 56 S E C APPROAC H PHASE INCEPTION

'\.. /

AR l iC

.

-·

-

(J1 N

T o 0 SEC BRAKING ­ PHASE TARGET POIN T

Figure 2 0

Composition of the Luna r - d e scent T raj e ctory

P 6 4 t r aj ectory, the geometry and vis ibility p rofiles c an be dete rmined in c l o s e d fo r m , and the th rust p r ofile c an be dete rmined from a p r i o r knowledge of m as s . Thus P 6 4 targeting consists of gen e r ating c l o s e d - form s o lutions for a number of P 6 4 constraint s et s and pickin g one which provide s adequate vis ibility and th rust. S p e c ification of P 6 4 con straint sets i s reduced to a two - dimen s ion al s e arch, as w ill be d e s c ribed in the following s e ction . The P 6 3 targeting p roc e s s is not so c le an .

B e c au s e the engine must be run

at fixed m aximum th rust fo r most of the phase, the guidan c e commands are not s ati sfied, and the refo r e , as shown in F igure 2 0 , the m aximum- th rust po rtion of P 6 3 i s not quartic . b e gun.

W hen th rottle c ontrol i s recovered, gene r ation of a qu artic is

But the th rottle recovery point i s not c lo s e to any target point.

Therefo re

the state vector at this point c annot b e controlled, and we h ave no c lo s e d - fo r m solution f o r it .

S i n c e t h e initial p o s ition and velo c ity on the braking-phase quarti c

must be free, there remain only th ree constraints , in e ach of two axes , which c an be imposed arbitr ar ily.

The guidance algorithm p ermits a fourth constraint i n one

axis by s olving for the current target- refe ren c e d time such as to s atisfy a c on st r aint on the ZG - component of j e rk . Thus a P 6 3 c on straint set comp o s e d of seven constraints is spec ified arbitrarily, and the rem aining three condition s required to define th e b r aking- phase quartic are det e r m ined

ite r atively by simulation.

Three or four

ite rations are gen e r ally requi r e d bec au s e there i s b ilate r al inter action b etween the targets and the simulation. C onstraints The P64 constraint s et i s c onstructed as follow s : 1.

F ou r constr aints at a spe c ified target- refe renced term in al time T AP F :

Two

te r m in al verti c al const r aint s , spec ified by the m i s s ion comm ander, are the termin al altitu de

( R AP F G

typ i c ally ) . P66

3 0 - m typ ic ally ) and altitude r ate ( V AP FG - 1 -m / sec X X T w o termin al hori zontal con s t r aints , imp o s e d b y the choice d effective =

hori zontal time

=

constant

T,

are that the

terminal pos ition,

2

veloc ity, and

A A P F G r , V AP F G - AAPFG r. z z z z T h e s e P 6 6 compatibility constraints c au s e the p itch comman d s at P 6 4 terminus ac celeration shall be re lated by R AP FG

=

=

and P 6 6 inception to be identical ( avoiding a p itch tran sient at the p h as i c interfac e ) and c au s e the P 6 6 algorithm t o null the hori zon t al p o s ition e r r o r a s well a s the hori zontal velocity e r ror, without p o s ition feedback.

B e c au s e the P 6 6 hori zontal

algorithm fee d s b ack the prior accele r ation c o m m and, and b e c au s e of the tran sp ort de lay, an effective

r

of 8 s e conds h as been found s atisfactory r ather th an the 5

s e c onds u sed by the P 6 6 algo rith m .

53

2.

Four c on st raints at an unspec ified t arget - referenced m idpoint time T AP M :

T h e m i dpoint constraints are specified by the commander according t o h i s sense o f s afety a n d compatability with a pos s ible m anual t r an sition to P 6 6 . Typic ally, h e ° s lope, completely

m ay spec ify - 5 - m / s e c altitude r ate at 1 50 - m altitude, a n d a 1 6 determ ining the m idpoint state R AP MG ,..Y_APMG given R AP MG

y

=

V APMG

y

=

0.

3.

T w o con st raints a t a n unspecified t ar get- referenced initial time T AP I : The ° initial pos ition is arbitrarily spec ified to lie on the 1 6 p ath and to p rovide an app r o ac h p h a s e o f typic ally 7 . 5 -km length, dete rmin ing the initial p o s ition vector R AP IG given R AP IG

y

=

0.

This c ompletes the P 6 4 constraint set except for spec ifying the time s T AP M and T AP I at which the midpoint and initial constr aints apply. determined

by

running

the

App roach -phase

Targeting

These times are

R outine over the two­

dimens ional s weep of value s of T AP M and T AP I . F rom the c as e s run, one i s p icked that exh ibits suitable attitude and th rust behavior ( b as e d on m as s e stimate ) .

an

a - p r iori P 64 initi al

If sub s equent s imulation p rove s the m a s s estimate exc e s s ively in

e r ror, the initial thrust w il l be un s atisfactory, and an alt e rn ate c as e must be p i cked. The seven P 6 3 constr aints are specified as follow s :

Four constr aints are

spec ified by c omp atibility of the terminal state on the b r ak ing-phase quartic with the initial state on the app roach- phase quartic .

Two constraints are impos e d on

termin al acceleration by requirin g the termin al thrust to be 57% and by specifying the term inal pitch angle . The final constraint is imp o s e d on the hori zontal componen t of termin al j e rk b y requiring zero rate of c h ange of thrust at terminu s , The term in al ° pitch angle, typic ally around 60 , is chosen by t r i al and e r r o r to m inimize p ropellant c on s umption. App roach - phase T argeting F igure 2 1 illustrate s the App roach - phase T argeting Routine.

N o r mally, this

routine is first run separately in search of t argets for the approach phase, and then run j o intly with th e B r aking- phase T argeting R outine ( F igure 2 2 ) to deter m in e t a rgets fo r t h e entire lunar d e s c ent. In the XG - axi s ( altitude), the te rminal ac celeration, j e rk, and snap are computed by Eq. ( 2 1 . 2 ), which is obtained immediately from

54

1 TMF TMF2 / 2 TMF3/6 TMF4:3 / 2 4 RAPFGX TMF TMF2 / 2 TMF /6 VAPFGX 1 T TIF2 / 2 TIF3 /6 TIF4/24 AAPFGX (14) JAPFGX SAPFGX

RAPMGX VAPMGX RAPIGX

0

1

IF

where T l\TF and TIF are the m i dpo int and initial te rm inu s - referenced time s c omputed by E q s .

(2 1. 1).

I n the YG - axi s , pos ition and all its deriv ative s are z e ro t o p rodu c e a planar t i· aj ectory.

P66

In the ZG - ax i s ,

Eq. ( 2 1 . 4 ) i s obtained by substituting the comp atibility 2 - A APFG r into th e ZG-axi s ve r s ion const raints H AP FG A. A P F G r , V A P F G z z Z Z of Eq. and inv e rting. E quation s ( 2 1 . 5 ) and ( 2 1 . 6 ) c om p lete the definition of the =

=

(14)

app r oac h-phase quartic .

It r e m ains to c ompute the app r o ac h - p h as e t argets as the

p o s ition vector and its d e r ivativ e s at th e target po int on the quartic . F o r a quartic polynom i al, a 5 x

5 state t r an s ition m atrix

d efined by

X1 =

(T 1 -T 0 ) 3 / 6

2 (T 1 -T 0 ) / 2

(T 1 - T 0 ) 4 / 2 4

ID

(T

R1

1

(T 1 -T 0 )

Y.1

0

1

Al

0

0

I. 1

0

0

0

1

(T 1 -T 0 )

I.o

�1

0

0

0

0

1

�

theory,

-1

R X A .

v

j

�

r� li

(T 1 -T 0 ) 2 / 2

(T 1 -T 0 )

whe r e H . to S . are row vecto r s . -1

(T 1 -T 0 ) 3 / 6

(T 1 -T o > 2 / 2

( T 1 -T 0)

0

1,

T

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

55

0

) c an b e

Ro .Yo Ao = m (T l , T O )X O,

) c an b e de rived u s ing linear syst e m-s

0

0

0

(T

1

0

0

_m

1, T

0

1

0

R y

A

J

.s._

0:

X

w ith the s olution

where I is the 5 x 5 identity m atrix. The exponential s e r i e s is zero after the fifth i term b e c ause o 0 for i > 5 . All the p roperti e s of s t at e t r an s ition m atr i c e s c an =

be applied t o s c alar and vector polyn o m i al s . Equations ( 2 1 . 7 ) yie ld

the comp lete target and initial states by u s in g state

t r an s ition m at r i c e s and the definition of target- referenced t arget time as ze r o .

B r aking- phase T argetin g To target P 6 3 , w e m u s t c o mpletely dete rmine the b r aking- p h as e quartic shown in F igu re 2 0 . Seven of the ten n e c e s s ary c onditions are dete rmined in c l o s e d form, although th ree are based on a P 6 3 termin al m as s e stimate which must be upd ated by simulation .

The rem aining three c on d itions n ec e s s ary to define the quartic are

determined ite r atively by s imulation.

The terminal p itch angle IJ P BR F is a fixed

input to the B r ak ing - phase T argeting R outine. F i gure 22 illu s t r ates the routine .

*

Four conditions are s p e c ified by s etting

the P 6 3 termin al pos ition and veloc ity equ al to the P 6 3 initial state

(Eqs. ( 2 2 . 1 ))

.

A unit vector in the term inal- th rust direction is c omputed from the terminal pitch angle O P B R F ( Eq . ( 2 2 . 2 ) ), and the terminal ac c e l e r ation i s c alculated by Eq. ( 2 2 . 3 ) u s ing th e term inal thrust F B R F , the P 6 3 term in al m as s e stim ate M B R F , and allowing fo r lunar gravity GM.

The XG - component of terminal j erk must be dete rmin e d by

s imulation an d is therefo r e set to zero for the first ite r ation (Eq. ( 2 2 . 5 ) ) .

The

Z G - c omponent of termin al j e rk is c omputed by E q . ( 2 2 . 5 ) to p ro duce z e r o r ate of c h ange of th rust at terminu s , accounting for the e stim ated terminal m a s s flow r ate c omputed by

Eq.

( 2 2 . 4 ) ; the j e rk c oeffic ient

XG - component of thrust.

KJ, typ i c ally 1 . 2 , accounts for the

The te rmin al snap must b e dete rmined by s imulation and

is the refore set to zero ( E q. ( 2 2 . 6 ) ) for the first ite r ation.

This c ompletes the

fi r s t- ite ration definition of the b r aking- phase quart i c .

*

N ot s hown i s the c ap ab ility of the targeting program to set the P 6 3 terminal s t ate to a b ackwards extr apolation of the P 6 4 initial s t ate to allow for a short tran s ition du ring which the ac c e l e r ation is as sumed to c h ange line arly with time. Thi s c ap ab i lity is n ot alw ay s used, and to show it would unn e c e s s arily c o mp l i c ate the p re s entation of F igu re 2 2 .

56

B r aking-phase targets are c omputed by E q . ( 2 2 . 7 ), u s in g the state t r an s ition m at r ix and the definition of target - referenced

target time a s zero.

U s ing the

c om puted t arget s , a s imulation i s run to produce c o r r e cted d ata.

The nom inal initial r ange used by the ignition algorithm is c o r r e cted by E q . ( 2 2 . 8 ) to c o r r e ct t h e error i n t h e target- refe ren c e d t i m e o f throttle control recovery. The s imulation p roduces a b r ak in g - ph as e quartic s atisfying the t arget v alues of pos ition, veloc ity, acceleration, and Z G - component of j erk.

The rem aining

conditions nec e s s ary to define the quartic c an be obtained from the cur rent s t ate on the l ast pass of the b r akin g - p h a s e s imulation.

The equation for the curren t

state,

[::l [:

RBRTG

T 1

Y:BRTG

T

ABRTG .J.B RTG §B RT G

is re adily s olved to yield the achieved t arget j e rk and s n ap accor ding to Eq. ( 2 2 . 9 ). Solving

for the

ZG - c omponent of achieved target j erk provides a check on the

com put ation of T by the guidan c e algorithm; agreement between achieved and input values is typic ally to s even plac e s . I n p rep aration for c o r r e cting e stimates at the terminus, the complete s t ate at t e r m inus is computed by Eq. ( 2 2 . 1 0 ).

E quation ( 2 2 . 1 0 ) yields a term in al state at

the specified termin al time T B R F p re c i s ely, whe r e as the state R G , Y:G appli e s at the time T which m ay differ from T BR F by up to the 2 - s e cond g r anularity. E quation ( 2 2 . 1 1 ) corrects the P 6 3 t e rminal mas s estimate u s in g the rocket equation.

Equation s ( 2 2 . 1 2 ) - ( 2 2 . 1 4 ) c o r r ect the termin al ac celeration, j erk, and

s n ap u s ing the corrected P 6 3 termin al m a s s e stimate, the achieved X G - component of terminal j e rk, and the achieved X G - and Z G - c omponents of t e rminal s n ap . F inally, state convergen c e test quantiti e s are computed b y E q s . ( 2 2 . 1 5 ) ( 2 2 . 1 7 ) . Since only three conditions ( J B R F G

A:x_ ,

S B R FG AX ' and S B R FG A z ) defining

th e b r aking - ph as e quartic are sought ite r atively, only three convergen c e c rite r i a

57

I NPUTS:

TAPF RAPFGx

T E R M I NAL TARGH-REFI:RENCEO T I ME TER M I NAL ALTI TUDE TERM I NAL ALT ITUOE RATE EFFECTIVE P66 HOR I ZONTAL T I ME CONSTANT

VAPFG X T

TAP M RAPMG, \IAPMG TAP I BAPIG

M I D PO I NT TARGET -REFERENCED T I ME M I D PO I NT STATE

I N IT I AL TARGE'HIEFERENCED T I ME I N IT I AL POS I T ION

BEG I N

f-----

COMPUTE TERM I NUS-REFERENCED T I ME'>

"IMF

•

T I F · TAP I

TAPM - TAPF,

-

(\)

TAPF

O] ;:::�: '� I 1

TERM I NAL X COMPONENTS

TMF TMF

TIF

3 2

3

- TMF

16

-1

12

o

I

-TIF

16

RAPMG X

VAPMGX

RAPIG

TERMINAL

Y COMPON NTS

RAPFG y •

0,

VAPFGy

["'"1 [ 'L,

.-----

TERM I f.IAI

JAPFGz

SAPFG z

�

l

-T

t 2- t

['""]

0,

AAPFGy • 0, J A PFGy • 0, S A PFGy •

7 �" '"""c"'

+ TMF

R A PFG l • AAPFGz

•

'" '

"" l/ 2

TIF + T I F 2 1 2

T

TMF 3 1 6

• w '' TMF 31 6

3

T I F 4 1 24

TMF 2 1 2 TIF

16

t2

VAPFG z •-.\APFG z T

IZI

X

0

[ l ""ffi z VAPMGz R A P I Gz

141 151

161

COMPUTE COMPLHE TARGET AND I N I T IAL STATES VAPTG APTG

�

S APTG JA PTG

·¢ 10, TAPFI

r'"'"

l

VAPFG

�APFG

S -APFG . JAPFG

l'""l r·l "[ VAP I G

�APIG

JAP IG �APIG

· ¢> 1TA P I , O I

VAPTG

fAPTG

JAPTG -APTG

OUTPUT S :

171

COMPLHE TARGET AND I N ITIAL STATES

['""] "j JAPTG

SAPTG

-

f NO

Figure 2 1

A 'A P I G

VAPIG

V A PTG

AAPTG

,

JAP I G SAPIG

-

Appro a c h - phase T argeting Routine

58

I N PUT S ft�PIG. ��PIG Pb4 IN l�l Sl�TE TBRf TERM I l T�RGET-REHREN:ED T I ME 6 PBRF TfRMI I PilCH A�GlE CotiSTRAINT FBRF TERM I .L lHRU S.l C ON STR� I NT MBRF P63 TE I I N�l MASS ESTIMATE VEXBRF DPS EX �UST Vl'LOCITY ESTIMATE fOR P6l TERMINUS KJ J£RK Cl JF ICIENT T O �CCOUNT fOR THE Vl'RTIC�l COMPONENT Of THRUST RBRIG , R B R I Gz MJMINA. INITI�l AniTUDE, RANGE, ANO SPEED X FO THE IGNITION ALGOR ITHM KlG COEFFI ENT fOR CORRECTING MJMINAl iNITIAl RANGE R B RIGz ITHROT REOU I I D TARGET REHRFN:ED T I ME Of THROTILE RECOVl'RY CONVl' :ENGE CRITERION fOR TAR<.ET ·REFERENCED T I M£ OF lHROTILE R EC OVER Y f TIHROT f CONVl' :EN:E C R I TERION FOR ITRMINAl STAT£ ERRORS

CDMPUH COMPtm SI Alf Al llRMINUS ACHIEVED B Y 5 1 MlllATED VBRIG

r'·l ['" ' ff

�B RfG A ·+ (18RF,OJ XBRTG

CORRICT Al l ESTIMATES Al TfRMINUS

[

c.n CD

�BRTG VBRTG ABRTG JIRTG

SIRTG

111 Ill I(I 151 161

[]

SBRfGA X . 0. S B R F G Az l

02

OJ

·

4> 1 0. TBRf I

•

)1

VB R

im G

GA X

ABRfC.

VBRIG

YES

X

X

I I

YES

S I MULATE DESCENT US I NG TARGETS Of CURRENT ITERATION

YES

ITHROTA BBRIGA. YBRIGA

[!�"W''

01 · E

01 , r

OJ · f

i nH ROTA TIHROT I , E TIHROl

YES

TNo

AND SPEED CORRECT FOR IGNITION AlGOR ITHM MJMINAl INITIAL �LTITUOE. RANGE

Ill

RBRIG

t

X

· R BRIGA

X

. R8R1G z · RB R I GA , VBRIG · I YB R I G A I Z

nt T 4

411 Tl

"''

111 r "

w •'

·nt T4

•l�l J

141 T

�IRTG

�BRTG

OUTP UTS :

191

!!.G yc

I

Figure 2 2

1181

COMPl£TE TA RGn STATE.

COMPUTE JERK AND SNAP AT TARGET POINT ACHIEVl'D 8Y S llllJLATED TRAJECTORY }.IRTGA

!171

NO

T II_G . yG M

•

1161

I YB Rf G I I

S IMULAT ION OUTPUT DATA USED FOR CORRECTING GUIDAM:E �lGORITHM TARGETING;

RESET NOMINAl I NI TI AL RANGE TO CORRECT THE Tfft!OTILE CONTROL DURATION KlG TIHROTA · TIHROT I '" """ ''z " """ z

1111 1111
NO

S I MULATION OUTPUT DATA USED FOR CORRECTING IGNIT ION ALGORITHM PARAMEHRS!

EXACT TARGEHIEHREN:ED l i ME Of fiNAl P6l PASS STATE AT TARGET REffREN:EO TIME T MASS AT T�RGET-REHR£NC[O Tlr.t: T

i

NO

+

ACTUAl TARGET ·REFEREM:ED liME Of THROTILE CONTROL RECOVl'RY ACTUAL STATE ON FIRST GUIDAM:E PASS

!lOI

bl

I I I A BR f G I I

Ill

ABRFG JBRFG

L

f

· I I ABRfGA x

RBRFG

VBRFG

t

COMPUTE I l ATE CONVERC.fNCE T E S T QUANT IT IES U l " I ' AOKIGA z . A B R f G 7 I I I �BRfG\1

Ill

t

t

MBRF • M rxPI t lyBRFG I 1 �G p/Vf x BRF l �B RI G · l B R l MBRf I �M' B R fG f GM , Q. O l lBRfG · I J BRF GA x , 0, KJ A B RF G z MBRFI MBRF I

COMPUTE FIRST ITERATION Of P6l COMPLETE TERMINAl STATE

COMPUTE COMPl£1£ TARGET STATE

�BRTC.A

BRfC.A

2B R Fr; - j

J

I B R T <. Ax JBRTGAy JBRTGz

JBRIGA

SEG I N

RBRFG · RAP I G. YBRFG • YAPIG YNFBRFG • I COS 8PBRF , O, - S I N B P BRF I ABRFG · I FBRFI MBRF I UM'BRFG · I GM. O. 0 I �BRf • - FBRFI VEXBRF lBRFG • I 0. 0, - KJ �BRFGz MBRFI MBRF I �BRFG · I D,O.O I

VBRlG

VBRFGA

TRAJECTORY

---

Braking - phase T argeting Routine

END

MJMINAl INITIAl AlT ITUDE AND RANG COMPONENTS Of THE STATE VECTOR I AT HIE f I RS! P6J GUIDAM:E PASS MJMINAl INITIAL SPUD

RBRTG VBRlG �8RTG JBRTG � 8RTG RBRIG VBRIG

X

.

RBRlG

Z

a r e needed.

The three c r iteria chosen are important for guidan c e per forman c e

a n d are related nons ingularly t o three conditions s ought.

If any one of t h e state

c onvergence tests fails, or if the throttle control r ecovery time conve r gence t e s t fails, the b r aking- phase t argets a r e c o r r e cted an d anothe r s im ulation i s run; otherwise the targeting i s concluded by corre cting the ignition algorithm inputs p e r E q . ( 2 2 . 1 8 ).

60

REFERENCES

1.

" R adar- Updated Inertial N avigation of a C ontinuou s ly ­ S p a c e Vehicle ' ' , I E E E Ae rospac e S y s t e m s C onfe rence, S e attle,

K ri e g s m an , Powered

B . A. ,

W ashington, July 1 1 - 1 5, 1 9 6 6 . 2.

W i dn all, W . S . , " Lunar Module Digital Autopilot " , Journ al o f Space c r aft an d Rocket s , Vol. 8 , N o . 1 , J anuary 1 9 7 1 .

3.

C h e r ry , G . W . , 1 1 E G uidance - A G en e r al E xpli c it, Optimizing Guidan c e Law for Rocket- P ropelled Spac e c r aft " , M I T Instrumentation Lab o r atory R eport R - 4 5 6 , Augus t 1 9 6 4 .

4.

Klumpp, A. R . , 1 1 A M anually Retargeted Automatic De s c ent and L anding System for LE M 1 1 , M I T Instrumentation L ab o r atory R eport R - 5 3 9 , M ar c h 1 9 6 6 ,

5.

K lumpp, A. R . , ' ' A M anu ally R etargeted Automatic L anding System for the Lun ar M o dule ( L M ) 11 , Journal of Spacec r aft and R ocket s, F eb ruary 1 9 6 8 ,

6.

?vl oore,

T. E . ,

G.G.

M cS w ain, and J . D .

Montgomery, '' Guidan c e

Laws

for

C ontrolling Off- N om inal LM Powered De s c ent T r aj ectorie s B ac k to the I\ omin al 1 1 , Inte rnal N ote MSC - E G - 69 - 9 , P roj ect Apollo, N AS A, M anned Sp ac e c raft C enter, Houston, Texas, F ebru ary 2 8 , 1 9 6 9 , 7.

l\l c S w ain, G . G . and T . E . Moore, 1 1 F al s e High G ate T argeting fo r L M Powe red Des cent" , Internal N ote M SC - EG - 6 8 - 0 7 , N AS A, M anned Space c r aft C enter, Houston, Texas, M ay 2 7 , 1 9 6 8 .

8.

Y ang, T . L . ,

11

A T ar geting Scheme for F u e l Optim al R o c ket T r aj e ctories -

W ith App lic ations to the LM De s c ent B r aking P h a s e " , Technic al M e mo r an du m T M - 7 1 - 20 1 4- 1 , Bellcomm J anuary 2 2 , 1 9 7 1 .

9.

MIT

C h arle s

Stark

D r ap e r

Laboratory R ep o rt

R - 5 6 7, " Guidance

System

Operation s P l an for M ann e d L M E arth Orbital and Lun ar M i s sions U s in g '. P rogram Luminary 1 D . S e ct ion 5, Guidance Equation s , R ev . 9 , De c e m b e r 1 9 70 .

61

1 0.

Eyle s , D. E . ,

11

Apollo LM Guidan c e and P ilot- A s s i stan c e During the F in al Stage

of Lunar De s cent - Soft w are C onsideration s " , F ourth IF AC Symp o s ium on Automatic C ontrol in Space, Dubrovnik, Yugos l avia, September 6 - 1 0 , 1 9 7 1 . 11.

K lumpp, A. R . and G . R . K alan, " E limin ation of N oi s e and E nhanc ement of Stab ility and Dyn amic R e sponse of the Apollo L M R at e - of - De s c ent P ro g r am " , MIT C h arle s Stark Dr ap e r L ab o r atory R eport E - 2 54 3 , October 1 9 7 0 .

1 2.

K lumpp, A. R . , " F IN DC DU W - Guidan c e Autopilot Interfac e Routine", M IT Instrumentation Labor atory L U M IN ARY Memo N o . 2 7, R ev . 1 , September 2 6 , 1 968.

1 3.

Bennett,

F. V.,

" Lunar

De s cent

and

As cent

T r aj e ctorie s " ,

AIAA

E i ghth

Aerospace Sciences M eeting, N ew York, J anuary 1 9 - 2 1 , 1 9 70 . 1 4.

Bennett, F . V . , " M i s s ion P l anning for Apollo Lunar Module D e s c ent and Ascent", to be publi shed as a N AS A Techni c al Note.

62

R-695 DIST RIBU T ION LIST

Internal:

P. Adle r

G . K alan

S . Albe rt

D . Keene

R . Battin

J. Kernan

L . Berman

A. Klumpp ( 1 5 0 )

H. Blair -Smith

R . Lar son

F . Brun swick

J . L aning

S . Copps

B . M c C oy

L. Drane

R . Metzinger

T . E delbaum

D. Millard

A. E ngel

P. Mimno

D. E yle s

D. Moore

D. Farrar

J. N evins

P. F e lle man

N. Pippenger

S. F e mino

J. R e e d

T. Fit zgibbon

P . Rye

D. F r a s e r

J . Scanlon

R . Gilbert MIT / KSC

R. S chlundt

K . Glick

c.

E . C. Hall

N . Sears

M. Hamilton

R . Stengel

P. He ine mann

P. V ol ante

D. Hoag

R . W e athe rbee

A. Hopkins

P. Wei s s man

I. Johnson

CSDL / T D C ( 1 0 )

M. Johnston

Apollo Library ( 2 )

S chulenberg

Group 2 3A ( 1 8 )

D-1

Ext e rnal: N A SA / RA SPO

(1 )

D E LC O

( 3)

Kolls man

(2)

Raytheon

(2)

M SC :

( 3 7 & 1 R) National Ae ronauti c s and Sp ace A dmini stration M anne d Spacec raft C ente r Houston, T exas 7 7 0 5 8 ATTN : A pollo Document C ontrol Group (BM 8 6 ) M . Holley (EG 1 4 ) T . Gib son ( FS 5 ) B . Kirkland ( FM 2 ) B. Taylor ( FM 2 ) J . Alphin ( FM 2 ) W . Bolt ( FM 2 ) D . J e z e w s ki ( FM 8 ) F . Be nnett ( FM 2 ) J . Garman ( F S 5 ) D . Scott (C B ) A . W o rden (C B ) P . C onrad (C B ) J . Young (C B ) K . Mattingly (C B ) F . Raise (C B ) E . Mitchell (C B ) R . Gordon (C B ) V . Brand (C B ) D. Cheatham (EG) R . Chilton (EG)

( 1 8 & 1 R) (2) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)

KSC :

( 1 & 1 R) National Ae ronautics and Space Administration J . F. Kennedy Space C ente r J . F. Kennedy Space C ente r, Florida 3 2 8 9 9 ATTN : Te chnical Document C ontrol Office R . Pearson (C FK)

LRC :

(1) (2)

National Ae ronautic s and Space A dministration Langley Re search C ente r Hampton, Virginia ATTN : M r . A. T . Matt son GA :

(3& 1R) Grumman A e rospace C o rporation Data Ope ration s and Se rvice s , Plant 2 5 Bethpage , Long I sland , New York ATTN : Mr. E . Stern

D-2

NAR:

(4& 1 R) N o rth Ame rican Rockwell, Inc . Space Division 1 2 2 1 4 Lakewood Boulevard Downey, C alifornia 9 0 2 4 1 ATTN : T . R . W atson; C SM Data Management D / 0 9 6 - 40 2 AE 9 9

N A R RASPO :

(1)

NASA Re sident Apollo Spacec raft P rogram Offi c e N o rth A me rican Rockwell, Inc . Space Divi sion 1 2 2 1 4 Lakewood Boulevard Downey, C alifornia 9 0 2 4 1 ( 1)

GE: General Electric Company Apollo Syste m s P . 0. Box 2 5 00 Daytona Beach, Florida 3 2 0 1 5 E . P . Padgett, J r . / Unit 5 0 9 ATTN :

(1)

H DQ : National A e ronaut i c s and Space A dministration Washington DC 2 0 5 4 6 ATTN : G . C he rry

( 2)

J PL: J et P ropulsion Laboratory 4 8 0 0 Oak Grove Drive P a s ad e na, C alifornia 9 1 1 0 3 ATTN : R . Mo rris W . B re cke nridge

D-3