GUIDANCE, NAVIGATION AND CONTROL

Approved,JLd. � Datd� 7/ G. M. LEVINE, DIREC OR, GUIDANCE ANAL S APOLLO GUIDANCE AND NAVIGATION PROGRAM

Approved: J!. . l � \"(, ;:;J5;;, Date: 1'1 0, . ._. ·r 1 R. H. BATTIN, DIRECTOR, MISSION DEVELOPMENT APOLLO GUI NCE AND NAVIGATION PROGRAM

..u..i[£-'-'<4PJ...L..(,.<j;;__,...:::..;r+=::J..loo�b-- Date: /�t..IH1l( ATION PROGRAM

Approved: )ft�ji( Jl� Date:/88'-- 71 R. R. RAGAN, PUTY DIR CTOR CHARLES STARK DRAPER LABORATORY

R-695

APOLLO LUNAR-DESCENT GUIDANCE

by Allan R. Klumpp

JUNE 1971

CHARLES STARK DRAPER CAMBRIDGE, MASSACHUSETTS, 02139 LABORATORY

AC KNOW LEDG MEN TS

This report was prep ared under DSR P roj ect 5 5 - 2 3890, sponsored by the

Manned Spacecr aft C enter of the N ational Aeronautics and Space Administration

th rough C ontract N AS9 - 4 0 6 5 .

The P 66 vertic al channel was developed b y C r aig W . Schulenberg. The

analytic al design and gain setting of the P 6 6 horizontal channels w as done by N icholas

J . Pippenger using concepts suggested by Jerrold H. Suddath. The concept of

analytically extrapolating to yield the predictive guidance equation for P63 and P 6 4

w as conceived by W illiam S . W idnall. The existence of an analytic al solution for

the guidance frame orientation to yield zero crossrange target j erk was 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 William S. \Vidnall and Donald W . Keene.

The public ation of this report does not constitute approval by the N ational

Aer,onautcs and Space Administration of the findings or the conclusions contained

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

(D C opyright by the Massachusett s Inst itut e of Technology Published by the Charles Stark Draper Laboratory of the Massachusetts Institute of Technology. Printed in C ambridge, Massachusetts, U. S. A. , 1971

ii

R - 6 9 5

APOLLO LUNAR -DE SC ENT GUIDANCE

ABS T R AC T

This report records the technology associated with Apollo lunar- descent

guidance. It contains an introduction plus five maj or sections:

1 . Braking-phase and approach - phase guidance. Braking- phase guidance begins

in lunar orbit prior to engine ignition and transfers the lunar module ( LM ) to a

terminus typically 7 8 0 0 - m slant- r ange before the landing site. Approach-phase

guidance begins at braking-phase terminus and transfers the LM to a terminus

typic ally 30 -m above the landing site. The braking-ph ase transfer is near - optimal.

when�as the app roach -phase transfer sac rifices propellant-utili zation efficiency to

p rovide landing-site visibility and landing- site redesignation c apability.

2. Term inal- descent- phase guidance. Initiated automatic ally at approach-ph ase

term inus, or m anually any time during the approach phase. termin al- descent-phase

guidance automatic ally nulls horizontal velocity and controls altitude r ate to a

reference value. The reference value is inc remented or decremented by astron 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 arture

from traditional approaches, the routine transfers the LM from any cur rent attitude

to any comm anded attitude while avoiding gimbal lock by inherent characteristics

of the m aneuver algorithm. No gimbal- lock- avoidance strategy is required.

4. Throttle R outine. The routine connects sever al guidance programs to the

descent p ropulsion system ( DPS) . DPS h ardware limitations require operation either

at a m aximum -thrust point or within a separate permitted-thrust region. The routine

alters thrust commands from the guidance progr ams when necessary to meet DPS

constr aints and issues corrected thrust inc rement commands.

iii

5 . Braking-phase and A pproach-phase Targeting P rogram . This ground-based

p rogram is used at the Draper Laboratory and at the N ASA Manned Spacecraft C enter .

T h e prog ram supplies descent targets which are loaded into guidance computer

m emory shortly before launch.

iv

by

Allan R . Klumpp

June 1 9 71

TABLE OF CONTENTS

Section

INT RODUCTION

Descent Phases

Navigation, Guidance, and Control Configuration

Navigation . . . . . . . . . . . . . . • . . . . . . .

Guidance • . . . . . . . . . . . • . . . . . . • . .

Powered - flight Attitude-maneuver Routine

Throttle Routine . . . . . . . . . . . . . . . . . .

Braking-phase and Approach-phase Targeting P rogram

Digital Autopilot . . . • . . . . . • • • . . • . . • • . • .

DE FINITIONS OF LUNAR DESCENT COORDINATE FRAMES,

ATTITUDE ANGLES , AND GIMBAL ANGLES . . . . . . . • .

BRAKING-PHASE AND APPROACH-PHASE GUIDANCE

Guidance Equation Derivation • . • . . .

Braking-Phase Targeting Objectives

Approach -Phase Targeting Objectives

1

1

4

4

4

6

6

7

7

9

1 3

1 3

1 8

1 9

Landing-Site Redesignation Procedure 1 9

P 6 3, P64 Guidance Algorithm . . . . . . 2 1

P 6 3 Ignition Algorithm . . . . . . . . . • . • . . . . . . . • . • • . . . . . . . 27

T ERMINAL-DESCENT-PHASE GUIDANCE 31

P 6 6 Horizontal Guidance Algorithm . 31

P 6 6 Vertical (ROD) Guidance Algorithm 3 2

POWERED - FLIGHT ATTITUDE -MANEUVER ROUTINE 37

THROTTLE ROUTINE . . • . • . • . . . . . . . . . . . . • . . . . • . • . . . . . . . 45

v

TABLE O F CONTENTS (Cont . )

Section Page

B RAKING-PHASE AND APPROACH-PHASE T ARGETING PROGRAM 51

Constraints . • . . . • . • . . . . • • • . . , . • . . . . . . . . • . . • . . • . . 5 3

Approach-phase Targeting . . . . • . • . • • . . . . . . . . . . . • . . . . . 54

Braking-phase Targeting . . . . . . . . . . . . . . . . . • . . . • . . . . . . 5 6

REFERENCES . . . • . • . . . . . . • . . . . • . • • . • . • • . . • . • . • . . . . . . 61

vi

NO MENC L ATURE

Symbols are norm ally defined where first introduced. Therefore it is necessary

to define here only those symbols used in more than one section of this report.

Most symbols are self- defining by being constructed of standard identifiers

as follows:

1 . Type of variable. Position and its derivatives velocity, accele r ation,

je rk, and snap are denoted R , V, A,J,S . Thrust is denoted F, thrust­

acceleration AF , unit vectors 1J.N, clock-times by lower c ase t, and

target- refe renced times (times with respect to the target point of a

particular mission phase ) by upper c ase T.

2 . Mission phase. The braking phase ( P 6 3 ) is denoted BR, the approach

phase ( P 6 4 ) AP.

3. Applic able point in phase. Inception is denoted I, terminus F , and target

point T .

4. C oordinate frame of reference. P latform coordinates are denoted P ,

guidance G , and L M body B.

5. Achieved ( as opposed to nominal) values are denoted A.

Thus by construction, R B R FG A is the position vector e xpressed in guidance

coo rdinates achieved at the braking ph ase terminus. Without a phase i dentifier,

B_TG to �TG represent the braking or appro ach ph ase targets R BRTG t o §.BRTG or

B_AP TG to .§.AP TG, whichever phase is current. Vector elements are denoted by

subsc ript, e . g. , V z· Vector m agnitudes are implied whenever a symbol reserved

for a vector lacks the underscore. Row vectors of 3 x 3 m atrices are denoted

�X'�Y'�Z' and m atrix elements are identified by row and column, e . g. , C yz is the

Z component of the row vector �y·

Symbol

_AFC P

AFP

AGS

ATT

C BP

C G P

Definition

Thrust- acceleration command

Thrust- acceleration me asurement ( computed by dividing the inertially sensed velocity inc rement by the guidance sample interval)

Abort Guidance System

P owered- flight Attitude - maneuver R outine

Transformation to body from platform coordinates

T r ansform ation to guidance from platform coordin ates

vii

D.\P DEC 1\

DPS oF.\

G:\1

GP

I:\ I u

u: :\DTII\IE

LGC

LII\IIT

Ll\1

LP

LPD

I\ L\. '\. I:\ IU :\1

:\1I;'>JI!\1UM

N A .. S,\

Pitch Yaw Boll

P63

P64

P66

R .�PTG, RBRTG, ll_TG, Y:M>TG, YBRTG, YTG, .:::ir\PTG, ABRTG, ATG, JAPTG, JBRTG, JTG, �A.PTG, �BRTG, .§.TG

.B.G, .B_P

R O D

t

T

Digital autopilot

Descent engine c ontrol assembly. A digital to analog interface between the LGC and the DPS throttle

Descent p ropulsion system

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

Lunar g r avity ( ac celeration of p ositive sign val i d at the lun ar surface)

Cur rent g r avity valid at the cur rent position .B_P

Inertial measurement unit consisting of a three - gi m b alled stable member and three acceler o meters

A time interval ( typically 2 . 2 seconds) added to the target-referenced time T in P63 and P64 to account fo r the effective t r ansport lag due to c omputation and system response times

Ll\1 guidance computer

c\ function of two arguments yielding the fi rst argument limited in magnitude to the value of the second argu ment

Lun ar m odule

L anding site

Landin g p o int designator c onsisting of two ret i c les, one on the inside w indow 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 arguments yielding the algeb r ai c ally lowest argument

National Aeronautics and Sp ace Administ r ation

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

B r aking ph ase o r b r aking-phase guidance p rogr am

App r oach phase o r app r o ach-ph ase guidance p rogr am

Term inal - descent phase o r terminal - descent-phase gui d ance p ro g r am

T arget position, veloc ity, acceleration, jerk, and snap of the app r oach phase, b r aking phase, or c u r rent phase

Cu r rent position

R ate of descent

c lock-time

T arget- referenced time (time with respect to the target point of the c u rrent m ission phase )

viii

THROT !JN FC P

!J.N W CP

.YG. Y.P

Throttle Routine

Unit thrust command

Unit window comm and

C urrent veloc ity

ix

IN TRODUC TION

This report describes how the Apollo lunar-descent guidance works, why it

w as designed this w ay, and, in seve r al c ases, how it might h ave been designed

diffe rently. The c oncepts desc ribed c an be applied to landing on any planetary body,

w ith or without atmosphere, should m an resolve to continue this adventure. The

s olutions p resented offe r ample opportunity for checking the theory. Such checks

h ave been m ade, and all algorithm s are known to work as conceived,

Lunar-descent guidance begins with the lunar module ( LM ) at about 15- km

altitude in a slightly elliptic al coasting lunar orbit, and ends with the LM on the

lunar surface. The guidance is pe rformed by the onboard LM guidance c omputer

( LGC ), which takes input data and commands directly from the LM c rew and via

the uplink from N ASA's R e al Time C omputation C ente r in Houston, Texas, The

crew consists of a commander and a LM pilot. ( See Figure 1. ) Standing on the

left, the commander monitors and c ontrols the descent using visu al cues and v arious

h and controlle rs and switches . Standing on the right, the LM pilot monitors the

computer display, vocally relays pe rtinent data to the commander, and enters any

nece s s ary data into the computer via the keyboard.

The primary guidance mode for the lunar descent is automatic; the LGC controls

both attitude and th rust. The commander c an, temporarily or permanently, select

nonautom atic guidance modes if he wishes to control, manually, attitude or thrust

or both . The nonautom atic modes, not described further in this report, provide

attitude and thrust references for the com m ander to follow if he chooses to fly the

L:\1 manually along the autom atic guidance profile.

Descent Phases

The lunar descent is a nominally-planar trajectory c onsisting of three phases

illustrated in F igure 2 and desc ribed as follows :

1 . The braking phase ( P r ogram 63 , or P 6 3 ) is initiated by keyboard entry aboat

1 0 minutes before nominal ignition time. P 63 first c omputes the p recise time and

attitude for ignition. N ext, at typic ally 49 2 -km slant-range from the landing site,

P 63 ignites the DPS . F inally, P 6 3 transfe rs the LM to the terminal state required

as initial c onditions for the succeeding approach phase. The transfer takes typically

5 1 4 seconds and is near- optim al.

1

N

LANDING POINT DESIGNATOR RETICLES

SUPER IMPOSED

LM COMMANDER

" /;'/k.__

Figure 1 Components of the Lunar-descent Guidance System

LM PILOT

w

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

t START P63 22 MINUTES 45" CENTRAL ANGE TO TOUCHDOWN

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--� ...... ;-;".:-.-.-

AL L NUMBERS ARE TYPIC AL AND DO NOT REPRESENT ANY SPECIFIC APOLLO MISSION

START P66 30m ALTITUDE

t ....;:---..._4 11 m GR OUND RANGE

START P64 '-..., PpRo401 31 sec TO TOUCHDOWN

2231 m AlTITUDE ' , Pf.t45f p

... ·.•., �-:: �---�·;;,:-:--;-� .

.... 64

"" '

B RAKING-PHASE ) TARGET POINT

-541 m ALTITUDE 4416 m GROUND R ANGE TO L ANDING SITE

PPROACH-PHASt · -T ARGET POI NT 29m ALTITUDE

4.8m GROUND R ANGE TO LANDING SITE

Figure 2 Lunar- descent-guidance Phases and Targets

2. Approach-phase ( P 6 4 ) guidance begins with initial conditions c onsisting of,

typically a) 2. 2- km altitude and 7. 5-km ground range and b) - 44 - m / s ec vertical

velocity and 129 - m / sec forward velocity. In typically 1 46 s econds, P 6 4 transfer s

the Ll\I to a point almost directly above the landing site. P 6 4 provides continuous

visibility of the lunar surface and, specifically, of the landing site until around 5

sec onds before terminus . During P 6 4 the commander c an direct the LGC to land

at any visually chosen point on the lunar surfac e by a landing- s ite redesignation

procedure which c an be continued until initiation of the terminal- des cent phase,

3. The term inal- descent phase ( P 6 6 ) begins autom atically at typic ally 3 0- m

altitude and 11- m ground range from the landing site, or it m ay b e initiated b y the

commander any time during P64 . The P 66 guidance algorithm controls velocity

only; there is no position control. P 6 6 nulls the forward and lateral velocity

components to p roduce a vertical approach to the lunar surface, an obj ective which

c annot be achieved from visual cues when the surface is obscured by a sheet of

radially moving dust. P 6 6 controls altitude r ate to a r eference value that is

inc remented or decrem ented by 0. 3 - m / sec each time the commander r aises or lowe rs

a three -position r ate - of-descent ( ROD) control switch loc ated near his left hand.

I'\ avigaticn, Guidance, and C ontrol C onfiguration

The N avigation, Guidance, and C ontrol configur ation illustrated in F igure 3

applie s to all LM powered- flight guidance maneuvers . This report desc ribes only

the solid portions of Figure 3 . All routines are processed once every two seconds,

except the ve rtical channel of the P66 guidance algorithm is p rocessed once per

s econd, and the digital autopilot is processed 10 times per s econd.

Navigation

1\' avigation ( see Kriegsman 1) p rovides an estimate of the current state vector

based on data from an inertial measurement unit (ll\IU ) and a landing radar. IMU

data are used throughout all thrusting m aneuvers, but, to avoid accumulatiun of

inertial errors, IMU data are not used during coasting flight except for a minimum

period immediately preceding and following each thrusting maneuver, The l anding

radar p rovides altitude data below typically 1 0-km altitude, and velocity data below

6 1 0 - m/sec .

Guidance

Guidance transfers the LM from the current state to the terminus of the current

phase . In addition to the current state estim ate from Navigation, Guidance is based

4

CJl

,------l I NAVIGATION

!STATE VECTOR UPDATE

I ROUTINE L ______ _j CURRENT v STATE GRAVITY

NAVIGATION

MEASUREMENTS

BRAKINCH'HAS£ AND APPROACH-PHASE

TARGETING PROGRAM

I GROUND BASED I r4 THROm£ ROUTINE

J��-P64 UIDAI'I:E TARGETS

THRUST -ACCELERATION GUIDAJC:£ COMMAND

P63, P64 GUIDA!'�:£ ALGORITHM

P66 GUIDANCE AlGORITHM UNIT THRUST COMMAND UNIT WI NOOW COMMAND

be POWERED -FliGHT

AT!' I TUDE-MANEUVER

ROUTINE

,-------i

THRUST INCREMENT COMMAND

r------, DESCENT I THRUST VECTOR

PROPULSION � SYSTEM I L __ ____ _j

r--- ----, I SPACECRAFT : : DYNAMICS I L ___ __ _j

r --D�;��l-- ,---if -:!'>I

GIMIAL INCREMENT "'

AUTOPILOT � COMMANDS

PHASE PlANE DATA

ANO CONTROl I TORQU[ VECTOR

L ��C�R.:__-

_j

I INERTIAl SENSORS �I :::!:========::::::::=::::==�===========::1 lb=====================� LANDING RADAR I LINEAR ANO ROTATIONAl MOTION I L _______ _j

Figure 3 Navigation, Guidance, and Control C onfiguration

on precomputed targets from the ground-based targeting program . The outputs

from the Guidance algorithm are a unit thrust c ommand and a unit window com m and

is sued to the Powered-flight Attitude- maneuver R outine, and a thrust- acceleration

c ommand is sued to the Throttle R outine. Through these routines, Guidance controls

th e thrust vector magnitude and direction with respect to inertial space.

Powered-flight Attitude - maneuver Routine

The Powe red- flight Attitude- m aneuver R outine ( ATT) connects all guidance

programs, de scent and ascent, to the digital autopilot ( D AP). ATT inputs are two

c om m and vectors; a unit thrust command and a unit window command. ATT e stimates

a unit thrust vector from accelerometer measurements, and is sues incremental

c ommands to the DAP. These commands cause the DAP to drive the e stimated

unit th rust vector into c oincidence with the unit thrust command and the symmetry

plane of the LM into c oinc idence with the unit window command .

During P64, as long a s the landing s ite would b e visible, the unit window

c ommand is sued to ATT by Guidance is the line-of- sight vector to the current landing

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

vector, A.TT superimposes the landing-point designator reticles of Figure 1 on the

current landing site.

Th rattle Routine

The Throttle R outine (THROT) connects several powered-flight guidance

program s to the DPS. The DPS is used by all descent guidance program s , one of

the two abort p rogram s, and one guidance program whose purpose is to provide a

velocity-vector increment computed by the Real Time C omputation C enter in Houston

and transmitted to the LM.

The DPS must be operated either at the maximum- thrust point ( about 92'7o of

the rated thrust of 46 706 newtons ) or within a permitted-thrust region ( 1 1 to 6 5o/o

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

region oxidi zer flow and fuel flow make independent transitions from c avitating to

noncavitating regime s . The independent transitions c ause gros s deviations from

the required mixture ratio and produce excessive erosion of the DPS noz zle.

U s ing a computed mas s estimate, a thrust- ac celeration measurement, and

the thrust- acceleration command from the guidance equations, THROT computes

the current and commanded thrusts and i s sues thrust increment commands to the

6

DPS. These c omm ands either 1 ) drive the computed thrust into c oincidence with

the com m anded thrust whenever the commanded thrust lies within the permitted­

thrust region , 2) produce m aximum thrust whenever the commanded thrust lie s above

the pe rmitted-thrust region, or 3 ) produce minimum thrust whenever the com m anded

th rust lie s below the perm itted-thrust region.

Braking- phase and Approach-phase T ar geting Progr am

The targeting program provide s t argets for P 6 3 and P 6 4 . The targets define

braking- and approach- phase reference traj e ctories as independent vector

polynomials centered at individual target points as illustrated in Figure 2 . Although

only P 6 3 and P64 are targeted, the targets are designed to achieve all the guidan c e

objectives of P 6 3 , P 6 4, and P 6 6 .

Digital Autopilot

The DAP ( s ee Widn all2 ) controls the attitude of the LM during powered flight

by m e ans of control effectors consisting of a r e action control system and a trim

gimbal system. As the n ame implies , the trim gimbal s ystem is a slow system

used prim arily for trimming the DPS thrust vector through the LM center of m as s,

7

DE F INITIONS OF LUN AR DESC ENT C OOR DINATE F R AMES,

ATTITUDE AN G LES, AND G IMBAL A1�G LES

Three coordinate frames are required for lunar descent guidance bec ause 1)

all inertial measurements are with respect to the stable platform of the IMU , 2 )

P 6 3 and P64 guidance is with respect to a landing site which rotates with, and c an

be redesignated along, the lunar surface, and 3 ) thrust- vector determination an d

landing- site redesignations are with respec t to L M body axes . These coordinate

system s are illustrated in F igure 4 and defined as follows:

1 . P latform coordinates . V ariables in platform coordinates are tagged P. The

origin is at the center of the moon, the XP- axis pierces the nominal (unredesign ated)

landing s ite at the nom inal landing time, the ZP - axis is parallel to the orbital plane

of the C omm and Module*

and points forward, and the YP- axis completes the

right- hand triad. Platform coordinate s are nonrotating.

2 . Guidance coordinates . Variables in guidance coordinates are tagged G. The

o r igin coincides continuously with the current landing site ( the frame rotates with

the moon ); the XG-axis is vertic al; the ZG - axis lies in, or near, the plane containing

the I....M and the landing site and points forward; and the YG - axis completes the

right- hand triad. Thus , the origin and orientation of the guidance frame are altered

each time the landing site is redesignated, Guidance-frame unit vectors expres s ed

in platform coordinate s are the row vectors �G PX, C G Py• C G P Z of the matrix

C G P .

3 . Body coordinates . V ariables in body coordinates are tagged B. These are

the generally accepted LM coordinates . The X B- axis is in the direction of the

nominal thrust vector, the ZB- axis is directed forward, and the Y B- axis completes

the right- hand triad. Body- frame unit vectors expres sed in platform coordinates

are the row vector s C BPX' �BPy• C BPz of the m atrix C BP .

F rom the s e definitions i t is noted that if the L M lands at the nominal site at

the nominal time in a nominal erect attitude, the three frames will be p arallel at

the instant of touchdown.

The following conventions are defined for orthogonal m atrices:

* The LGC transfers state vectors in platform coordinates to an abort guidance system

( AGS). The AGS requires the state to be expres s ed in a frame who se Z axis p arallels the orbital plane of the command module.

9

XB

ZB

NOMINAL LANDING SIT E AT NOMINAL LANDING

TIME

Exage rated Scale

XP XG

LUNAR ROTATION TO OCCUR DUR ING TIME REMAINING TO

NOMI NAL LANDING TIME.

'------------------+- zP.

XP, YP, ZP-Piatform ( Inertial Measurement Unit) Coordinates

XG, YG, ZG-Guidance Coo rdinates

XB, Y B, ZB -Body ( LM Spacec raft ) Coo rdinates

Figure 4 Lunar-descent Coordinate Frames

1 0

1 ) A m atrix element is denoted by two subscripts which indicate the row

and column respectively of the element. Thus C BP XY denotes the

Y- component of the row vector �BP X.

2 ) A m atrix transpose ( inverse ) i s denoted b y interchanging t ags.

3 ) F rom the definitions , it follows that m atrix products are obtained by

deleting internal tags .

By conventions 2 and 3 , a vector V is transformed to body from guidance coordin ates

by

V B C BP C G P - 1 V G = C BP C PG V G = C BG VG .

LM attitude angle s are a s et of Euler angle s defined as clockwise rotations

about the XB- axis ( yaw ), the displaced YB- axis (pitch), and the displaced ZB- axis

( roll).

LM gimb al angles are a set of Euler angles defined as clockwise rotations

about the YB- axis ( inner) , the displaced ZB-axis (middle) , and the displaced XB- axis

(oute r).

1 1

BRAKING -PHA SE AND APPRO AC H- P H ASE G UIDANC E

The guidance p rograms for P6 3 and P6 4 are almost identic al . The two phases

use the s ame guidance algorithm, the same Throttle R outine, and the s ame Powered­

flight Attitude-maneuver Routine. The differences are 1) the guidance equation s elects

different sets of targets, 2 ) the erection of the guidance coordinate frame is slightly

different, and 3 ) landing site redes ignation c apability is available only in P64 .

Guidance E quation Derivation

To guide a spacec r aft from any initial or current state to a specified target

state c an be viewed either as an explic it guidance p roblem or as an implicit guidance

problem. Explicitly, we c an repetitively determine, as the mission progres s es , a

vector polynomial function of time that intersects the current and target states .

Guidance then commands the corresponding p rofile of ac celeration vs time.

Implicitly, we c an define, in advance of the mis sion, a reference traj ectory as a

vector polynomial function of time that evolves backward from the target state but

c annot be expected to inter sect a disper s ed initial (or dispersed current ) state.

Onboard guidance then commands an ac celeration vector p rofile composed of three

term s, namely the ac celeration along the reference traj ectory minus two feedb ack

term s p roportional to the deviations in velocity and position of the actual traj ectory

with respect to the reference traj ectory. In either the explicit or the implicit c ase,

repetitively solving the guidance problem produces convergence upon the specified

tar get state even though the target point may be redesignated in flight and the

c ommanded acceleration is not p recisely achieved bec ause of control error s .

The implicit guidance equation derived here i s c ategoric ally superior t o the

explicit guidance equation bec ause the explicit equation is merely a special c ase,

as will be shown. Besides being intellectually more satisfying, the implicit equation

has demonstrated in simulations significantly faster reduction of deviations from

the reference tr ajectory. Deviation s come from navigation erro rs and from displacing

the reference traj ectory to intersect a redesignated landing site. R apid reduction

of deviations restores a nom inal approach to the redesignated landing site.

Unfortunately, the implicit equation had not been developed when the program for

Apollo 1 1 was coded, and the advantages were insufficient to recode the guidance

p rogram for later mis sion s .

C her r/ 3 ) derived the explicit guidance equation. K lumpp( 4•5 ) simplified it

for LGC coding and generali zed it to nth order. Moore et a1( 6 ) derived an implicit

equation which did not generalize the explicit equation. The general implicit guidance

equation is now derived and specialized to the explicit c ase.

1 3

It is convenient to think of the reference traj ectory as evolving backwards in

time from the target point, with the time variable T reaching zero at the target

point and negative prior to that point. Thus target- referenced time ( T ) is to be

dis tinguished from clock-time (t ) . Bec au s e guidance gains would become unbounded,

the target point is never reached. Instead, 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 reference traj ectory, but the target point lies beyond the portion that is

actually flown, s imilar to a suggestion of McSwain and Moore. (?)

In term s of a vector polynomial function of target- referenced time, we wish

to define a reference traj ectory that satisfies a two-point boundary 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 constrain terminal thrust in

PG3 and to shape the traj ectory des ign in P 6 4, as is discussed in connection with

the targeting program.

A quartic polynom ial is the minimum order with which five constraints on

the reference traj ectory c an be satisfied. With the reference traj ectory evolving

backwards in time from the target point, it c an be defined as

(1)

where .B_RG is the position vector on the reference traj ectory in guidance coordinates

at the negative time T, and R TG, .Y_ TG, ATG, .i_ TG, and � TG are the target position,

velocity, acceleration, j erk, and snap.

The acceleration to be commanded at any point in space consists of three

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

two feedback terms proportional to velocity and position deviations from the reference

traj ectory. Taking derivatives of Eq. ( 1) as the velocity and acceleration on the

reference traj ectory yields the three- term guidance equation

AC G ATG + .i_ TG T + .§. TG T 2 / 2

- (.Y_G - V TG - ATG T -.l_ TG T 2 / 2 - .§. TG T3/ 6 ) K V/ T ( 2 )

where AC G is the comm anded acceleration, VG and R G are the current velocity

and pos ition, and K V and K R are the nondimensional feedback gains .

14

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

AC G = - RG K R/ T 2 - Y.G K v

/ T

+ B_TG K RIT2

+ Y:TG (KV + K R ) / T

+ ATG ( 1 + Kv + K R/ 2 )

+ 1. T G ( 1 + K VI 2 + K R I 6 ) T

+ .§.TG ( 1/ 2 + Kvl6 + KR

I2 4 ) T2 .

( 3 )

E quation ( 3 ) is the implicit guidance equation. Although the reference traj ec tory

is quartic, the traj ectory gene rated by the implicit guidance equation is obviously

not. The implicit equation c an, however, be specialized to the explicit equation -

which does generate a quartic - b y a specific choice of the feedback gains K v and

KW Fi rst we note that E q. ( 2 ) m ay be identified with the linear second-order

differential equation

• • • 2 X + 2 !; wn X + wn = 0

by the associations (noting T is negative )

( 4 )

where wn i s the undamped natur al frequency and !: i s the damping r atio. Of c ourse

the system is time varying. However, this association does afford some intuition

on gain setting. Solving Eqs. (4) yields

where P is the undamped period 2 rr lw , and n

( 5 )

( 6 )

Equation ( 5 ) provides a m eans of controlling the system response time in terms of

the nondimensional ratio P / T, and E q. (6) p rovides a means of setting the damping

ratio .

15

An interesting set of values to choose for response and damping is

P / T = - rr/f3', .t = {3' / 2 .

This choice yields

When these values are substituted into E q. (3 ), the result is the explicit guidance

equation derived in references 3 to 5,

AC G = 1 2 ( R TG - B_G )/ T2 + 6 (Y:TG + Y.G )/ T + ATG. (7)

The discussion of implicit vs explicit guidance is c oncluded by introducing

the concept of a space c ontaining all permissible combinations of guidance p arameters.

Implicit guidance-p arameter space is one quadrant of the t, PIT plane or, equivalently,

one quadrant of the KR' Kv plane. Explicit guidance- parameter space is a single

point in either plane.

E quation ( 7 ) presents the explicit guidance equation assuming negligible

transport time delay. The explicit equation programed in Apollo is cor rected to

c ommand an acceleration appropriate for the time at which the acc eleration is

p redicted to be achieved. Let this predicted target-referenced time be

Tp T + LE ADTIME

where LE ADTIME is the transport delay due to computation and command execution.

An explicit guidance equation will now be derived that fits a quartic polynomial

through the target position, velocity, and acceleration, and through the current

position and velocity. The ac celeration of the quartic at the p redicted time T P is

the acceleration to command at the current time T in order to realize the quartic

profile. It will be shown that the resulting guidance equation reduces to E q. ( 7) for

T p = T , and therefore E q. ( 7 ) generates a quartic profile when the transport delay

is zero.

C onstraining the actual traj e ctory to be a quartic function of time allows the

cur rent position and velocity to be expressed as

1 6

[ :: l [: T T2 /2 T3 /6 T4/24]

RTG

1 T T2/2 T3 /6 Y:TG (8)

ATG

.J.TGA

§.TGA

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 memory. Solving Eq. (8 ) for the j erk

and snap yields

[�TGA l [ -24/T3 -18/T2 -6/T 24/T3 -6/T2

] RTG

.STGA 72/T4 48/T3 12/T2 -72/T4 24/T3 Y:TG

ATG (9)

RG

VG

The acceleration to be commanded at the current time T and realized at the predicted

time Tpis

A CG = A TG + !L TG A TP + §.. TGA T� / 2 . ( 1 0 )

Subs tituting E q. (9) into Eq. ( 1 0 ) and simplifying yields the A pollo lun ar- descent

guidance 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 compared to the transport delay,

T p/ T approaches unity. all bracketed coefficients in E q. (11) approach unity. and

Eq. (11) approaches Eq. ( 7 ) identic ally. The net effect of Eq. (11) not achieved by

Eq. (7) is a gain reduction as the target point is approached. Bec ause Eqs. (7) and

(11) are identical for T P = T, Eq. (7) generates a quartic profile when the transport

delay is zero.

17

In the derivation of the guidance E q. ( 7) or (11 ) , nothing con strained the time

T . At any point in a guided phase, T could be s et to any arbitrary negative value,

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

Landing- site redesignation, which c an arbitrarily stretch or shrink the traj ectory,

would produce an unnecessarily severe guidance response if T were not cor­

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

additional boundary con straint. In Apollo, this additional con s traint is imposed on

the downrange (Z) c omponent of j erk. Thus the Z-component of the j erk polynomial

of E q. ( 9 ) is solved for T by using a target Z- component of j erk JTG2. Separating

this scalar cubic polynomial from E q. (9 ) yields

3 2 JTG z T + 6 ATG z T + ( 1 8 VTG z + 6 VG z ) T + 24 ( R TGz - RGz ) = 0 . ( 1 2 )

One root of this cubic i s the required time T.

An alternate criterion for computing T reduces the propellant-consumption

p enalty of downrange landing- site redesignations . Although extensively tested, the

alternate was developed too late for incorporation in the LGC p rogram. The altern ate

criterion sets the downrange pos ition error to zero. Thus T is one root of the

quartic

4 3 2 STG Z T /24 + JTG Z T /6 + ATGz T /2 + VTGz T + R TGZ - RGz = 0.

Braking- Phase Targeting Obj ectives

The near-optimal transfer provided by P 6 3 targeting mus t satisfy a throttling

constraint th at the DPS be operated within the permitted- thrust region for, nominally,

the final two minutes of the phase. This throttling duration absorbs dispersions in

D P S performance and errors in lunar terrain modeling. With a total p ropellant

c onsumption of over 6 600- kg, Y ang( 8 ) shows that the Apollo guidance and targeting

are within 16 - kg of an optimal traj ectory satisfying the same throttling constraint.

To provide thrust within the 1 1 to 6 5 o/o region for the last two minutes of P 6 3 ,

the targets are chosen to produc e a constant thrust level o f about 57o/o o f rated

thrust at P 6 3 term inus . The targeting program accomp lishes this by con s training

the magnitude of the terminal thrust- acceleration vector to be F I M, and constraining

the Z- component of terminal j erk to be (es s entially) K F i\1/M2, where F is the

required terminal thrust, l'vl is the estimated terminal mas s , M is the estimated

terminal mas s flow rate (negative), and K is a j erk coefficient to account for the

vertical component of thrust. The targeting program achieves the two minute duration

of constant thrust by adjusting the initial range.

Properly targeted, the guidance algorithm commands during most of P 6 3 a

th rust- acceleration in exces s of what can be achieved. The throttle routine multiplies

thi s thrust- acceleration command by the estimated mas s to yield the guidance thrus t

1 8

com m and (GTC ), and provide s maximum thrust until the GTC 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 adjustments of the ignition time and of the DPS

th rust level.

The targeting program computes the P 6 3 targets by proj ecting computed

terminal conditions forward typic ally 60 seconds . Although the targets are p roj e cted,

they are computed to produce the required terminal conditions on a nomin al

traj e ctory. Traj ectory dispersions c annot be elimin ated prior to the target point,

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 iteratively using a

descent simulation in the ite ration loop.

Approach- Phase T argeting Obj ective s

The P64 targets are computed to provide lunar - surface visibility until about

5 seconds before terminus, and continuous throttling. In addition, the P 6 4 targets

are computed to p roduce at te rminus a m atched s et of values for the Z-components

of acceleration, velocity, and position such that, in the nominal c ase, the P 6 6 algorithm

will produce no initial pitch transient and will s imultaneously null the Z- components

of velocity and position. Unlike P 6 3 , the P 6 4 reference traj ectory c an be determined

in c losed form from specified traj e ctory constraints. Thus the proj ected targets

are computed without nume ric al ite ration.

Landing-Site R edesignation P rocedure

To steer the LlVI via the automatic P 6 4 guidance to a visually selected landing

site, the com m ander uses an iterative procedure akin to steering an automobile.

The procedure consists of 1 ) identifying the current landing site where the LGC

w ould take him in the absence of intervention and 2 ) steering the curren� s ite into

coinc idenc e with his visually s elected site by comm anding incremental landing- s ite

displacements ( rede signations ) . Because the P 64 targets are define d in the guidance

coordinate frame, which is repetivively e rected through the landing site, the P64

target point is displace d accordingly.

To identify the currently selected landing site to the astronauts, the LGC 1 )

orients the LlVI about the thrust axis to superimpos e the landing point de sign ator

( L P D ) reticles ( see F igu re 1 ) on the current site and 2 ) displays a number which is

read by the LM pilot and vocally relayed to the commander. By sighting through

the indicated point of the LP D reticles, the commander identifie s the current site.

He registers his eye by superimposing the two LPD reticles, one of which i s p ainted

on the inside window panel, and one on the outside window panel. The s ep ar ation

between reticle s is 2 . 5 em.

1 9

120

100

I-V) ::::> 80 e::: :r: I-0 I...LJ I-

"" <( 0 e:::

60 V) a_ 0 LL. 40 0 I-z I...LJ u e::: 20 I...LJ a_

0

Guidance Thrust Command ( GTC) Profiles -=--=-

===-===A:::: ct�u�a �, T::=:h:.:: r:.::u-:::.::; st::...:;P:::-= r--o-�f i -.-, e�--------'�" , : '�\ ii

--- NOMINAl

��----- ABOVE NOMINAl OP'; TH�U'lT

-�- EARliER THAN NOMINAl DPS IGNITION

\\. Braking phase terminus 1L_ approach phase in\eption

GTC and actual thrust coincide 26 seconds to trim thrust through center of mass.

t)" Time of first Guidance pass

/. ( GU IDTIME)

50 100 150 200 250 300 350 TIME FROM IGNITION- Seconds

400 450

Figure 5 Actual Thrust Profile s and Guidance Thrust C ommand P rofile s During P 6 3

500 550

By manipulating his controller left, right, forward, or aft, the commander

directs the LGC to displace the landing site ( and the P64 targets) along the lunar

surface by a correspondingly directed fixed angular inc rement ( 1 ° ) with respect to

the current line of sight.

The LGC redirects the thrust to guide to the rede signated (now current) site,

and reorients about the thrust axis to m aintain superp o sition of the reticles on the

current site . The commander can continue this rede signation process - steering

the current landing site into coinc idence with his chosen site - until 1 0 s econds

before reaching the P64 target point, 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 algorithm receives guidance targets , the current state vector, and the current

gravity vector as inputs and i s sues a thrust- acceleration command, a unit thrust

c omm and, and a unit window command as outputs.

Because the landing site move s due to lunar rotation and landing- site

redesignation, the LM is guided with respect to the guidance coordinate frame, which

is erected through the landing site each pass . Guidance targets are fixed in thi s

floating frame. Other inputs and all outputs are expressed in platform coordinates .

The landing site vector 1-_ P i s updated for lunar rotation (Eq. ( 6 . 1 )) using an

approxim ate algorithm that avoids computation of trigonometric function s, yet

pre serves the m agnitude of the lunar r adius . The algorithm accounts for the lunar

rotation rate W:YIOON P and the elapsed clock - time since the preceding update (t -

tOLD ).

For the landing-site redesignation algorithm (Eq s . ( 6 . 2 ) - (6 . 7 ) ) , whenever

the c omm ander m anipulate s the controller ( F igure 1 ) in the autom atic mode, the

LGC is interrupted and the azimuth command count (NC AZ) or the elevation command

c ount (NC E L ) is incremented or decremented ac cording to the direction of

m anipulation. The rede signation algorithm fetche s and resets to zero the NC AZ

and N C EL accumulators and rotate s LOS P ( the unit line - of- sight vector to the current

landing site ) by 1 ° per count (Eq. ( 6 . 3 )). If NC AZ and NCE L are both zero, the

redesignation algorithm has no effect. G iven that attitude control m aintains

c oincidence of the ZB,X B p lane and 1-_0SP, the rotations of LOS P are about two

axe s normal to 1-_0S P. E levation redesign ations rotate LOSP about the Y B- axis,

and azimuth redesignations rotate LOS P about an axis normal to LOSP in the ZB,X B

p lane. The landing- site redesignation geometry shown in Figure 7 depends upon

the defined LM platform orientation, n amely that the X P - axis is near vertical through

the landing site. The constraint that LOSPX be at least as negative as - 0 . 0 2 (Eq.

21

(6 . 5 ) ) p revents redesignating the landing site beyond the horizon. Equation ( 6. 7 )

computes the displaced point near the surface shown in F igure 7 and places the

redesignated site directly beneath this point.

The displaye d LPD angle (OLPD, Eq. (6. 8 ) ) is the angle between LOSP and

the ZB- axi s .

The computation o f the state vector in guidance coordinates ( Eq. ( 6,9 ) ) places

the origin of the guidance frame at the landing site and yields the velocity of the

LM relative to the lunar surface.

Target-referenced time T is c omputed using N ewton's method starting with a

good estim ate (Eqs. ( 6 . 1 2 )- ( 6 . 1 3)) . Note that the denominator of Eq. ( 6.1 2 ) i s the

de rivative of the numerator.

The guidance equation ( Eq . ( 6 . 1 5 )) is identical to Eq. ( 1 1 ) . The thrust­

acceleration command ( Eqs . ( 6 . 1 6 ) - ( 6 . 1 7 ) ) is merely the total acceleration command

minus current gravity G P , and the unit thrust command ( Eq. ( 6.1 8 ) ) is the direction

of the thrust- acceleration comm and. The s o - c alled radial - acceleration guidance

c o rrection described in reference 9 is rendered unnecessary by current targeting

te chniques and is om itted from this report, although present in the LGC program .

T h e computation o f the unit window command presented here is a s implific ation

of the LGC coding which produces the same result. The obj ect is to keep the landing

site in the center of vision ( superimpose the LPD reticles on the current site ) whenever

the geom etry permits and, otherwise, to command a forward- fac ing attitude. F igure

8 shows why the landing site c annot always be kept in the c enter of vision. F igure

9 shows the geometry pertinent 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 reticles with the

landing site; comm anding the forward vector ( .!J.N W C P = FOR P ) produces a forward

facing attitude. If the first alternative is chosen (_!2N W C P = LOSP ) the LM will

rotate about the X B-axis to aline the Y B- axis with the vector LOSP x C BPX. Thus

the direction of LOSP X C BP X indic ates whether a normal forward-facing attitude

o r an abnormal attitude would result from the c ommand _!2N WC P = LOSP. In addition,

the m agnitude of LOS P x C BP X measures the degree of indetermin acy in the

c ommand QNWC P = LOS P . The proj ection ( PROJ, Eq. ( 6 . 20 ) ) of LOSP x c;2BPX on the YG- 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 riterion for mixing LOSP and £.0R P into UN WCP. If

the descent traj ectory is planar, the m ixing ( Eq. ( 6 . 2 1 )) yields .!J.NWC P = LOSP for

nLPD $ 6 5°, .!J.NWC P = FORP for OLPD � 7 5°, and UNWC P a mixture line ar with

cos rJLP D for 6 5° < ALPD < 75° . R egardles s of whether the trajectory is planar

or nonplanar, it is neve r possible to command a side -facing or a rear -facing attitude.

22

l'.:l w

-------- ---- --- -� --- - -

INPUT VARIABLES:

REDESIGNATION COMMAND COUNTS NCAZ, NCEL CURRENT STATE �.� CLOCK-TIME TAG OF CURRENT STATE t CURRENT GRAVITY !!.!'

BEGIN

� UPDATE LANDING SITI VECTOR FOR LUNAR ROTATION I lP. LP UNIT [.b.P tIt- tOLD I �MOONP X .b.P J Ill I

L I

<$,?�' E

64

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

+ 0. 01745 NCEL I �BPy X lOSP I NCAZ • 0, NCEL • 0 LOSf'x • MINIMUM I LOSf'x . • 0. a? I !,OSP ·UNIT llOSP I

!,P • LP UNIT !lJ' + ,lOSP X ( LP - RPX ) LOSPX

• LANDING POINT DESIGNATOR ANGLE DISPLAY 9 LPD ·- ARCSIN ll,OSP • £8PX I

;

121 131

141 151 161

171

-' 181 1

COMPUTE CURRENT STATI VECTOR IN GUIDANCE COOROINATIS I .!!,G • CGP I.!!!'· .IJ' I, ':J.G • CGP l,l!P- �MOONP X J!.P I 1q11

+ UPDATE TARGET REFlRENCED T I ME T & COMPUTE CONVERGENCE CRITERION FOR T

T • T + It- tOLD I, tOLD • t 1101 t.TCRIT • T/128 1111

+ COMPUTI T SUCH AS TO SATISFY TIRMINAL JERK CONSTRAINT

JTGZ T 3 • 6 ATGZ Tz +I 18 VTGZ • 6 VGZ IT+ 24 I RTGz- RGz l t.T. - --� -- � �--�· --- -�- --- -----

3 JTGZ T2 + 12 ATGZ T + 18 VTGZ • 6 VGZ T • T + t.T

I ��' GUIDANCE EQUATION Tp • T +LEAD TIME

T 2 T T z T �CG ·(3(f-)-z{;)] 121J!,TG J!GIIT2 •[4(�f) 3(nl6YTGIT

T ? T T 2 T •(z( n -( n] 6YGIT •[6 ( {) ·6 (-f-) + 1] �TG

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

• COMPUTE UNIT WINDOW COMMAND

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

+MAX I MUM I COS 75 °- PROJ. 0 I £0RP]

� r L P63 P64 J � K ·I I I K- 0

ERECT GUIDANCE COORDINATE FRAME FOR SUCCEEDING PASS �GPx ·UNIT I J.P I �GPy ·UNIT [1r X {J!P -lP- K I yp · �MOONP X J!P I T /4B �GPz ·kGPX X kGPy

OLJTPUTSo THRUST-ACCELERATION COMMAND AfCP UNIT THRUST COMMAND J.!NFCP UNIT WINDOW COMMAND J.!NWCP

END

Figure 6 P63, P64 Guidance Algorithm

1!21

1131

1141

1151

1161 1171 1181

IJq) 1201 1211

IZ21 1231 1241

PLANE

O LD LANDING SITE LP

XP

LP. - p p DISPLACED POINT _

R p + LOSP X X

NEAR THE SURFACE-- - LOSP. X

NEW LANDING SITE LP

CENTER OF MOON

Figure 7 Landing- s ite Rede signation Geometry

2 4

N Ul

Figure g

THRU ST VECTOR XG

THRUST V ECTOR

�.-::� ·1� 7 I

.;_��} / ,.

LM ATTITUDE B ACKWARDS

--. . :� _ ,:,N·. z.· . · /.· _ :...:--- • '

I

.. - - : - . ,,.,.; . - .

" '-"-;'i, ·- _/ ·-;cf'' -.1: = C � RRE NT S IT E-- -- - ·- -

' � ... .

. . ·� ;�.

41':' . . �--

'\

-,.. ;.'" ·:,r::.-:::: . . . ·..:· ::--::�.:�-�.�-- ·:- . . .

/ ,:.· .·. � :.f-.:.�-�·�·::···;y..:�,· , '.: :. . ,a"' " .- . - - , .. --e-: -� :- - - - ·

- � --

<:.('\�� .. � ... ... . ·, z t. ""'_ . .. .... • • . .

... �-- " ,.� · ·-- · " �· - --- - - --··- -·

o·• >:r. =- ·- --- . . -- -

---"f>.c-<-� . -::· :·;�:;=:-� w�:;:)-i:f-: .. :· . . · : ... ... . . -· �-.:. ..

•V"-v' - __

. . , • • . . . - , - , , , -�- - - - - . 2

�· y--

-- - ,. �- ' ____ __ .,., - ' _;;; ,, .

-.-"

: - _,; ,.,, '· .": - -� · "-:J:?. '?' ,...;,. ·;-- . : :_ ., -· ·

� . •- -:<'-

-·"<-"" � < "" ' '--"'-'·-- - .. t� . .. · .:·.:-! � .�/;;-_;-:/"·.._ -... - ... · - - ' ' p '� ... 2- � - · •• •

W hy KePping the Landing Sitf• i n t lw C cntl· r of V i s ion C annot be t h e Sole C rite r i on fo r C ont roll i n g Attitude . \bout the T h ru st <'\xis

X r AXI:l = �BFx { Af Pf<OX THf\UST D / � )

LINE-OF-SIGHT VECTOR ( !:_O S P = UN I T ( !:_P-�P )

X G

APPROACH PHASE TERMINUS / ' FORwARD VECTOk 0 FO RP = UNIT (5;8P XlC £GPy } ----------------f=Jt::�::��:::j=;�������/ �� RANGE

CURRENT � S I TE LP � YG = fGPy CROSS RANGE

Figu re 9 Geometry Pe rtinent to Co mputing the Unit Window Command

2 6

E rection of the guidance c oordinate fr ame (Eqs . ( 6 . 2 2 ) - ( 6 . 2 4 ) ) is illustrated

in F igure 1 0 . With K = 1 in P 6 3 , the guidance coordinate frame orientation about

the vertical XG- axis is such that the YG- component of j erk would reach zero at the

target point if the traj ectory were flown there ( see reference 5 ) . With K = 0 in

P 6 4, the ZG - axis is in the vertical plane containing the line -of- sight vecto r. For

c ro s s r ange landing- site rede signations , setting K = 0 in P 6 4 w as found to c onsume

le s s DPS propellant than setting K = 1 to null the crossrange j erk at the target

point.

P 6 3 Ignition Algorithm

Trajectory dispe rsions p receding P 6 3 require an accurate ignition time and

attitude to be computed to 1 ) avoid exce s s ive variations of the time dur ation of

th rottle control in P 6 3 and 2 ) to avoid commanding an excessive attitude trans ient

the first time the P 6 3 , P 6 4 Guidance Algorithm is proce ssed. The P 6 3 ignition

procedure consists of:

1 ) C omputing on board the precise ignition time and attitude about 1 0 minutes

in advance of ignition

2 ) O rienting the LM to the ignition attitude

3 ) Initiating reaction control system ullage 7 . 5 seconds prior to ignition

4 ) Igniting the DPS at minimum thrust and holding con stant th rust and

con stant attitude for 2 6 sec onds, the m aximum time required for the

D AP to orient the DPS trim gimbal system to point the thrust vector

through the LM cente r of m as s

5 ) Connecting the guidance algorithm , which immediately comm ands

m aximum thrust and begins commanding an attitude p rofile according

to the current state vector and the P 6 3 targets .

To dete rmine the required ignition attitude, the ignition algorithm ( Figure

1 1 ) c alls the guidance algorithm as a subroutine. The ignition algorithm supplies

inputs consi sting of an accurate extrapolation of the state vector an d the corre sponding

gravity vector (both valid at G UIDTIME, the e stimated clock-time of the first P 6 3

guidance pas s ) . In preparation, Eqs. ( 1 1 . 1 ) - ( 1 1 . 5 ) initi ali ze guidance algorithm

inputs . On the first iteration, the state vector extrapolation rep resented by Eq.

( 1 1 . 6 ) i s performed by an orbital integration rouhne and, on subsequent ite rations,

hy a Kepler routine. Equation ( 1 1 . 9) corrects the extr ap olated velocity vector by

the velocity inc rement imp arted during the 26 seconds of minimum thrust preceding

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

not correcting for ullage are negligible. ) The guidance algorithm p roduces a unit

27

ZG ::= CGP.�------GUid C a ·

- z . once oa� tnate Frame XG YG zc

I YG ==: ccp

xa ===£apx - ---

- Y Vertical Th rough

Cu rrent Landing Site

Cu rrent Landing Site .Lp

Figure 1 0

_ Trajectory to Cu rrent !Redesignated 1 Landing Site

-K I J(p - WMOONp X BP 1 T/4 LM Velocity Relative to Lu na r Su rtace K � { 1 P63 0 P64 T • Target-Referenced Time u rrent LM Position BP

INPUTS C U R iif NT STAT£ ON COASTING TRAJECTORY CLOCK - T I M£ TAG Cl CU RRENT STAT! E STIMATtD Cl OCK-TIM£ FOR f I RST P6J GU ID ANCE PASS THRUSJ.ACCEI FAATION Cl ZoSFC C1 M INIMUM THRUST

BEG I�

l SAVE COPY (I WARfNT STArt J!P YP ANO ITS Cl OCK-TIME TAG I

t I N I T I Al \ l£ FOR P6l. P6il GU I D ANCE AlGORITHM T · · oM � 'fC 1 \ 1 tOtO • G U I D T I ME IZI LP · L P I GUIDTIME I 111 ' I 0 0 : CGP • I 0 I 0 I 14 1

0 0 } I

UNfCP • I 0. D. -I I Il l

t

RP_ yP t GU IDTIME AFTRIM

EXTRAPCUTf COASTING STATE AND GRAVITY TO GIJ I DT I ME SET LOOP COU NTE R �: : �: \G���DT;;::/1 y• · _y:P .utD ,.,E 1 t • GUIDTIMI 161 171 "N · l-181

� -

I

I I

t CORRECT EXTRAP(lATtD VILOCITY FOR 16 SECONDS Cl M I N I MUM THRUST VP • V P • Af R IM UNIT I Z6 SEC

J CAL P6l P6il GU I D ANCE ALGORITHM OEC REM£NT LOOP COUNT!R N • N - I

NO <? s

ADJUST GUIDTIME FOR TRAJECTORY D I SPE R S I ONS � GU I D T I 'IE • '

-KX I RGX - R RR I G X 1 , K y Rr./

' I RG7 - RAR I C l I � K V I 1/G VRR I G t l r � vc7 + K x \J f, x l

GUIDT IMF GIJ I D T I MF • �Gli iDTIMF

YES

� 0

PR[PARf fOR I G N I T ION I I I G N I T ION I · GU I O TIMI 16 SFC I I ULLAGE 1 • \ I IGNi fiON I - 7. 1 SEC

� 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

_191

I lUI

1 1 1 1

f l;' l

l } 3 f 1 14 1

I I

THRlr)l O I RI Cl tON R f ()I I I RI D Al t r;N i l l r H � i)Nf r _ f1

Pfi 3 Ignition Algorithm

29

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

direction of the velocity correction is unknown on the fir st iter ation, the above

procedure is iterated th rice .

An outer ignition- algorithm loop accounts for dispersions with respect to the

nom inal trajectory. Equations ( 1 1 . 1 1 ) - ( 1 1 . 1 2 ) adjust GU IDTIME to correct the

HG Z component of position at G UIDTIME as 1 ) a linear function of the dispersion

in o rbital speed VG and of the dispe rsion in the R GX component of position ( e s s entially

altitude) and 2) as a quadratic function of the out-of-plane position RGy· RBRIGX and R B R IG Z are nomin al initial altitude and r ange components of position in guidance

coordinate s ; V B R IG is the nom inal initial speed; and Kx, Ky , and K v are correction

coefficient s . The nominal initial altitude , range , and speed are computed by

the targeting program. The correction coefficient s are computed u s ing a manual

p rocedure based on descent simulations.

\V hen conve rged, this process yields a precise time and attitude for igniting

the DPS. Traj ectory dispersions result in typical variations of 2 seconds in the

time duration of throttle control and typical attitude transients of 2 millir adians

commanded by the guidance algorithm on the first P 6 3 p as s .

3 0

T E R l\IIN AL- DESC E N T - PHASE GUIDAN C E

Horizontal and ve rtical velocity are controlled in P 6 6 by completely independent

algorithms . P 6 6 p rovide s a nonautom atic attitude-hold mode in which the commander

c an control the LM attitude to translate or not, as he wishes, hori zontally over the

lunar surface . P 6 6 i s sue s no unit window command; yaw is controlled manually. A

desc ription of P 6 6 including the nonautomatic mode s is provided by Eyle s . ( 1 0)

P 6 6 Horizontal Guidance Algorithm

The P 6 6 hori zontal guidance algorithm ( Figure 1 2 ), processed once every two

seconds, nulls the horizontal components of velocity relative to the lunar surfac e

by directing the thrust vector a small angle away from vertical in oppo sition to

horizontal velocity. 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

vertical component of thrust- acceleration equals lun ar gravity.

Just as velocity feedback damp s a position control loop, ac celeration feedback

damps a velocity control loop . Because of the sampled- data char acter of the system,

a good measure of current acceleration is the acceler ation commanded the p receding

pas s . The P 6 6 hori zontal algorithm feeds back the velocity error ( current velocity

V PY ' V P Z minus lunar surface velocity V MOON Py , V MOON P z > and, to p rovide the

required damping, feeds back a fraction of the thrust- acceleration command from

the preceding pass ( Eqs. ( 1 2 . 2 ) - ( 1 2 . 3 ) ). On the first P 6 6 pass, the thrust­

acceleration fed back is that commanded the final P 64 pass .

The direction of the thrust- acceleration command is limited to 20° from

vertical (Eqs. ( 1 2 . 4 ) and ( 1 2 . 5 ) ) to m aintain a nearly erect LM attitude. fhe LIMIT

function of two arguments limits the m agnitude of the first argument to the v alue of

the second argument.

The unit th rust command (Eq. ( 1 2 . 6 ) ) is the direction of the limited thrust­

accele ration command.

The as sumption in gener ating hori zontal comm ands that the vertical component

of thrust- acceleration equals lunar gravity (Eq. ( 1 2 . 1 )) is realized only if the L M

is not ac celerating vertically. The purpose o f ignoring vertic 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 whenever the commander m anipulates the ROD switch, is to modulate

the gains of the hori zontal channels. This gain modulation i s negligible because

3 1

only limited changes in the descent rate will ever be commanded; the vertical

accele ration can be significantly nonzero only for short periods of time.

P66 V e rtical (ROD) Guidance Algorithm

The ROD guidance algorithm, proce ssed once per second, controls altitude

rate to the reference value by throttling the DPS. The RO D algorithm has no control

ove r the L :\1 attitude; the thrust- acceleration command it i s sues accounts for any

non- vertical orientation of the thrust vector.

The obj ect of the ROD guidance is to re spond r apidly without over shoot to

ROD increment commands. The algorithm provides a time constant of 1 . 5 seconds,

even though the sample interval is 1 . 0 second, by c apitalizing on the s ampled- dat a

characte r o f the system. U s ing a computed e stim ate o f the total acceleration at

the ROD s ample instant, the ROD algorithm extrapolates s ample-instant measured

velocity by the effective transport lag of 0 , 3 5 second and thus commands an

acc eleration appropriate for the velocity error at the time the acceleration command

will be realized. A s ampled data analysis (reference 1 1 ) shows that the compen s ation

fo r effective transport lag is highly effective in stabilizing the vertical channel.

The significant system dynamics reduce to a s ingle zero and two poles in the Z

plane . The zero is

Z Z = - LAG / ( s ample interval - LAG ) = - 0 . 3 5 / ( 1 - 0 . 3 5 ) - 0 , 53 8 .

One pole i s at the origin, and the second pole is

Z p = (time constant - sample interval ) / time con stant = ( 1 . 5 - 1 ) / 1 . 5 = 1 / 3 .

The poles are the same as for an ideal system containing neither a transport lag

n o r· an extrapolation.

The HOD algorithm has been s implified for this report as follows:

1 . In the LGC coding, the ROD algorithm begins e ach pass by reading the

accelerometers and recording the time at which they are read. This time is c alled

the H O D sample instant. ROD sample instants occur irregularly, but the interval

between them, c alled the ROD sample interval, averages 1 second. The accelerometer

readings are used to compute a) the three- component current velocity vector valid

at the ROD sample instant, based on updating the velocity vector supplied by the

state vector update routine (SVUR, F igure 3 ), and b) a thrust- accele r ation

32

m e a s u r e ment which is the ave r age ove r the R O D s ample interval . To c ompute the

veloc ity vecto r it s uppl i e s , the S V U R als o reads the ac cele romete r s . e ac h p as s . at

the SVU R s ample instant o c c u r ring at r egular 2 - s e c ond interval s . The i r regular

H O D s ample inst ants are e s s enti ally asyn c hronou s with the r egular S V U R s ample

instan t s . C on s equently the int e r actions between the ROD algo rith m an d the S V U R

in up dating the S V U H - suppli e d velocity vector are ext r e mely int r i c ate . In th i s r ep o rt.

the data p r oc e s s e d by the R O D algo r ithm are shown as input s . How the s e inputs

are o btained i s d e s c r ibed in refe rence 9.

2. Although the ve rti c al o r ientation of the X P - axi s is c ap itali zed upon by s eve r al

LG C routin e s , including the P 6 6 h o r i zontal algo r ithm and the l anding- s it e

r e d e s ignation algo r ithm. t h e LG C R O D algo rithm l ab o r io u s ly m an ipulate s c omplete­

vecto t · state data to m aintain validity for any platfor m align m ent. P r e sented h e r e

i s t h e s c alar equivalent valid f o r the lun ar - lan d ing platfo rm ali gn ment.

F igure 1 3 shows the HOD algo r ithm . The input s are all val i d at the R O D

s ample instant. E qu ation ( 1 3 . 1 ) computes the s ample- instan t total ve rtic al ac ­

c el e r ation by adding, to the thrust- ac c e l e r ation m e asur ement ( ave r aged ove r the

R O D s ample interval ) , c u r rent g r avity and a c o r r e ct ion fo r the throttle c h an ge

c oncluding the p r e c e ding H O D p as s . The th rust c o r r ection increment oF A i s supplied

by the Th rottle R outin e . l.c:quation ( 1 3 . 2 ) extr apolat e s the s ample- in stan t m e as u r e d

velo c ity . The c omm anded ve rti c al veloc ity ( reference altitude r at e ) i s initi ali z e d

as the ve rtical veloc ity exi sting a t t h e time P66 i s initiated. and i s inc remented o r

d e c r em ented b y E q . ( 1 3 . 3 ) e ach R O D p a s s ac c o r ding t o the R O D comm 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 ce ding R O D p as s . Equation ( 1 3 . 5 ) f i r s t c ompute s

th e total vertical accele r ation requ i r e d as the negative of the ext r apolat e d velo city

e r r o r divided by the R O D time cons tant ( 1 . 5 s e con d s ) . The equation then obt ain s

the r equi r e d verti c al thrust- ac cele r ation by s ubtr acting c u r r ent g r avity. F in ally.

dividing by C B F'xx • which is the c o s ine of the angle between the X B - ax i s and the

vertic al X P - axi s , E q . ( 1 3 . 5 ) yield s the thrust- ac celeration c o m m an d AFC P . To

avoid an e m p i r i c ally d i s c ov e r e d instab ility whi ch o c c u r s when the th rottle routine

o r the DPS c annot c o mply with the th rust- ac c ele r ation c o m m an d from P 6 6 , E q s .

( 1 3 . 6 ) an d ( 1 3 . 7 ) r e s trict AF C P t o produce th r u s t within the p e r m itted-th r u s t r egion .

3 3

I NPUT, CU RRENT VHOC I TY V P

BE G I N

j C CWI PUT£ UNL I M illO THRUST-ACCELERA T I ON COMMAND

----- - --AFCPX • G M I l l

AFCPy · - I V Py - VMOONPY

I I 5 SEC - 0. 4 AFCPY (21 AFCPZ · - I V PZ - VMOONPZ I I 5 SEC - 0. 4 AFCPz 131

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

AFCPy · L I M I T I AFCPY , AFCPX tan 20 ° I 141 AFCPz · L I M I T I AFCPz , AFC Px tan 2D0 I 15 1

I S SUE U N I T THRUST COMMAND

VNFCP · UNIT I AFC P I 161

OUTPUT: U N I T THRUST COMMAND L!NFCP

£NO

Figure 1 2 P 6 6 Horizontal Guidance Algorithm

3 4

I NPUTS :

COUNT Of ROD I NP I I T <;

S MIPLE l �i S T A NT M E A S URED VELOC I TY

CURRENT GRAV I TY

TYR l! S T � >IC C : : E RAT I n c ; \lE A S U R f l.lf'H I Averaqed over t h e ROD sample l nln'lal l

THRUST CORRECT I ON I NC REMENT I f rom the Throttle Rouline I

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

BE 1. 1 �

M

-�- � -�-�� ___ ____ ..._ ___________ __, C OMPUll EX I ST I NG V E R T I CAL A C C E L E RAT I ON AT ROD SAMPLE I N STANT

l EXTRAPOL 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 TRANS PORT l A

U P D ATI COMMANDED VE RT I CAl VFl OC I T Y I NCORPORAT I NG ROD I N PUTS

VCPX

- VC PX

• NROD D. l mlsec

NROD - D

ill

141

COMPUll THRtJ S T � A ( CflE RAT ION COMMAND F O R THROITlf ROI IT I NI

AFCP • - I V P

X - v r rx 1 / 1 . 5 SEC - GP

X

C B PXX

�J I

I R E S T R I C T THRUST ACCELFRATION COMMAND TO PRODUCE THRUST W I T H I N PfR M ITIED �THRl!ST REG ION

A F C P · MA X I MUM I AFCP. q I/ 3 "', 46706 NFWTONI M I 161 AfCP M I N I MU M I AF C P . (:{) ··� 467DIJ NEWTON/ M I 171

OUTPU T :

THRIJ '> I ACCfl FRAT I O N C O MMAND A f C P

ENO

Figure 1 3 P fi 6 Vertical ( ROD) Guidance Algorithm

3 5

l,O W E H E D- F LIGHT ATTITUDE - MAN E U VE H ROU TIN E

A link in the attitude control chain of command, the Powered-flight Attitude­

m aneuver routine ( AT T ) connects the various powered-flight guidance prog r am s to

the D AP . The functions of ATT are :

1 . For the small attitude changes normally required each guidance cycle,

ATT comm ands a maneuve r of con stant rate such as to achieve the

required attitude 2 seconds later .

2 . F o r gross attitude m aneuver s which m ay be required at phasic inte rfac e s

o r upon abort, ATT commands a rate- limited m aneuver which m ay extend

ove r sever al guidance cycles .

3 . F o r all attitude m aneuve rs ATT avoids the gimbal- lock region ( middle

gimbal angle > 70° magnitude ) . ATT issues a gimbal- lock alarm code

if and only if the comm anded attitude computed from guidance inputs

lies within the gimbal - lock region. ATT commands a m aneuver which

circumvents the gimbal - lock region and is sue s no gimbal-lock alarm

c ode when the most direct p ath to the comm anded attitude passes through

the gimbal- lock region.

Switching from a descent program to an abort p rogram m ay p roduce up to

1 8 0° change in comm anded thrust direction. A break with traditional app roaches,

ATT m akes gimbal lock during any m aneuve r inhe rently impos sible by 1 ) computing

commanded gimbal angle s , 2 ) limiting the m agnitude of the middle comm ande d

gimbal angle, and 3 ) is suing t o the D A P a s eries o f inc remental attitude - m aneuver

commands that monotonic ally* drive the gimbal angles from their current values

to their comm anded value s . P rovided the attitude is not currently in gimbal lock,

and given that the middle commanded gimb al angle is m agnitude limited at the

gimbal- lock boundary, it is inherently impos s ible to maneuver through gimbal lock;

the middle gimbal angle is confined to the r ange between its current and comm anded

value s . Other attitude- maneuver s chemes with appended gimbal- lock avoidance

require more computation to p roduce similar maneuve r s .

F igure 1 4 pre sents an overview o f the L M powered-flight attitude control

process , including some information on the p rocedures on the D AP side of the

* Except the outer gimb al angle p rofile m ay not be monotonic in the geometric ally

complex case of a large m aneuver about multiple axes at substantial middle gimbal angle and with m agnitude limiting of the X - axis attitude angle change on at least one pass through ATT.

3 7

w co

UNlT TI<RIJST COMMAND I

tiNlT WlNOOW I COMMA�D

I I I I I I I

THRUST- I ACCElERATION

III'ASUREIII'NT I J , >1 1 :0,: � E � � I S s � � � = � I � � !!I � \l!

� l i � G l �

OMPUTt COMMA�£

Gl'oiBAL ANGLE S,

LIMlT MAGNITUDE CX 1 COMMANDED MIDDU

IMBAl ANGU TO 700

COMMANDE D G I MBAL ANGUS

I I I

Rm��: l RATES

PERM !TIED

lAG ANGLE I

REFERENCE I G�:

L�L I

INCREMENTS ! I I I

REF E RE NCE G I MBAL ANGLES I ! I� � 19 � 15 � � � �� � �� § I: � � � l g � � � � � �

MEASURED G I MBAL

ANGLES

Figure 1 4 LM Powered - flight Attitude C ontrol

DAP

COMMANDS

SPACECRAFT ATTITUDE

inte rface. Two c omputational c oordinate frame s are introduced. F rom guidance

and n avigation inputs, ATT compute s a c ommanded-body frame (tag C B) to represent

the commanded attitude inherent in the input vectors . F rom ATT inputs, the DAP

com putes refe rence gimbal angles to compare with measured gimbal angle s for

c omputing the attitude errors . The reference gimbal angles define a reference- body

fram e (tag R B). ATT computes that the attitude errors are zero when these two

c omputational coordinate fram e s coincide. Of course, there m ay be DAP control

errors undetected by ATT, but any thrust pointing error is detected in the ste ady

state by a thrust- direction filte r, and corrected.

The guidance and navigation inputs to ATT, shown in F igure 1 4, consist of a

unit thrust comm and, a unit window command, and a thrust- ac cele r ation measurement.

,\TT proce sses the thrust- ac celeration measurement in a thrust- direction filter to

dete rmine an e stimated unit th rust vector with respect to the reference- body fr ame.

C orrecting for the offset of th e e stimated unit thrust vector with respect to the

X R B- axi s, ATT uses the unit thrust command and the unit window comm and to erect

the commanded-body frame. F rom the commanded-body frame m atrix, ATT extracts

commanded gimbal angle s which it compares with the reference gimbal angles to

generate inputs to the D A P . Ten times per second, the D1\P update s the refe rence

attitude and gene rate s the corre sponding control command s . The dyn amic respon s e

i s sufficiently fast and tight that the refe rence attitude is a good me asure of

instantaneous spacecraft attitude.

A feature of thi s configuration is that, although ATT runs at a s ample r ate of

2 seconds, clo s e to the fuel- slosh resonant frequency at certain points in the mis s ian,

it avoids exciting fuel slosh by avoiding all coupling with the actual spacecraft attitude

except th rough the slow thrust- direction filter.

F igure 1 5 details the Powered-flight Attitude - m aneuve r routine. The thrust­

di rection filter compute s the thrust - ac celeration measurement in reference-body

coordinates by constructing the required transformation from the reference gimbal

angles ( Eq . ( 1 5 . 1 ) ) . The change in thrust direction is limited on each cycle to 7 - mr

(Eqs . ( 1 5 . 3 ) and ( 1 5 . 4 ) ) , the m aximum travel of the trim gimbal in 2 seconds. The

total excursion of the estimated unit thrust vector is limited to 1 29 - mr (Eqs. ( 1 5 . 5 )

and ( 1 5 . 6 )) , the mechanical excursion limit o f the trim gimbal 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 provides a unit w indow command too c lo sely alined with

the unit thrust command to ade quately dete rmine the attitude o rientation about the

39

L

I

I 'Jf't· l '. : 1 "' 11 TH�t ·)J CtlM��"-jO U�CP ! 'NIT W INlHl'll I (l�Y.ANO UM'fCP TIIRI • 'll�A.U l l t R II, TiON W A <., U Rf W NT MP !?Hi R! "''rl C IMfi:A.l .l.N�;[ f ') RGAx .

�GA y, 1\�'A I I Afl l ! A.l'\ \ t f ll ii, I ION,Ri r i � ! �'i

AllflY 1 nnR�l t NAH '. _g_ R8

BEG IN

1 THRliS 140 I R�Cl iON f 1. Tf R

�fR8 · C I R t; A ); Rf.A.y R G A f l�fP �·�Af R � · I I� I I I �fR8 I

t. l l 'f R By · l l � t l [ D l l lJ�f RBy u,.,-RB y I 0 001] 6 U"' R s7 • I I Y. l l [ 0 � ! liNAf RB l Uti-"RBz I 0 001] 1 1"11- RBy · 1 1'-IIT [1 Ur,ifRBy • t..U�RBy I. 0 12'] t1NF RB1 · : I M I T [t L'Nfli'Sz � AU�R:8z I , 0 12'9]

�-------- --- - · - •

llGA z

' I I 111 m· 141

II' 161

l 1R; r�-Ac-f t lr'ol t t W'f t r-.oow (OMMANo tr wiTHIN ts0 or uNn THRUST coMMAtfD o\N[) � : \ I f VV I�Oll'W H A,f;

! � � ,!!,t ' • 1 If I :J\'Wt r - UNH P f. <OS<'

1 ,0 OR If P�NWC P • £8Pz FlAG o(l Ill l l f � �"-Wi'f'

_·-�NI :· p �

_:_ .. �05 ,' I "I c.l • .!l"MC P • - 'BPx

- -- --

---f Rf r 1 ! ����MA.�Of 0 \JBD;( · \.!_M I- r

·-- -

- ---- - --

-- -�l•OY I RAMI

�CRP,/ llr-.11 l �f\IW( p x !.;.CBPXI

�,- R P ,l I ':::<'RPX )( ��l B P� L........_ __________

L --·

I

191

1101

I l l \

I Ill

�-���;ti;-��MM-ANOI [) . J. - -----------

BODY FRA.W: f RfCTION FOR THRUST OffSfT

• UN II I Ct:RP� UNfRB y C.CBPYI U>HB z I,C BPi I I �C BPX 1 13 1

I �� B P 'I" · �i'!H') X �\BPX: 11•1

I

l i,._C R P/ · �rRP)( )( �f.BPy I I II

� t W�,,( T Cll�MAND£D G I MBAL "NGLES

1 (.!> ( · ARCH� I(, I C C B Pz y CCBPn I flO)

J (,tl \ · ,!I,P[TP )f, l Cf BP)( l CC BPXX I U T I I I j i ,A , A�\TR IC ! C C BP )(Y y I r CBP)(� ) UBI

I

--l i MIT MIDDLE COMMA NOLO G IMBAL AM;lE SAVING ORIGINAl VM U[

FOR POSSIBL� AlARM Q • CGAz ()91

CGA1 • l iMIT I CGA z ro• 1 1101

� I SSut G I M BAL-l OC K ALARM COOl

J I t I COMPUTE Ulll IMITEO �EFE�EM:E G I MBAL ANGlE CHANGES US I NG

MODULAR SUBT�ACTION

L6URGA • CGA M RGA 1211

r---- ·- l RES£1 W I NDOW HAG If COMMAND lNG [)(CESS lYE Y OR Z G I MBAL ANGLE CHANCES If I �URGAy I >41� FLAG . 0 1221

1 f I AURGA.z l >4�� FLAG . 0 i2}t

+ 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

l'lRGA z • l iMIT (ll,URGA z 10° 1 1241

l'lRGA y · L I MIT 1/j.URGAY COS RGAZ 20°

1 I COS RCA l ll�\

l'lATT X · L IMO [ll.uRGAX t.. URGA y S I N RGAz I, 'lff J ;.; ' �'j If HAG • 0 . 6 ATT X • 0 Q1\ fj,RGAX • l'l ATTX · l'lRGA y S I N �GAl i281

-+ II SUE OAP COMMANDS

6 ftGA • l:.ftGA I 10 1291

. [ � S I N RGAz l i N ;GAj " R B COS RGA z COS RGA X 6B_GA I 2 SlC llOI

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

I 4>RB X · L IMIT ( o.� R B X J �o� R B X I I 2 a R B X 10° 1 l $RB Y · l iMIT I �J RB y i OI R B y Jl � o R By 10° I 01 1

�•ez · L I M I T I • RBz I • R Bz ! 1 1 o RB1 . 10° 1

OUTPUTS : REFfRENC[ G IMBAL ANGI( INfRI MFNTS � REn RfNCE All ITUOf RATI S PfRM InfO I AG ANGI • :, �

END

Figure 1 5 Pow e red - flight Attitude - maneuve r Routine

40

RGA iie RB

XCB -axis, or 2 ) the guidance program i s P66 (which provides no unit window com ­

mand ), then ATT p rovides a unit window command suitable for e rection of the

c ommanded - body frame and resets a flag to indicate that no attitude rotation i s

allowed about the XC B -axis . A T T first p rovid e s the current Z B -axis ( Eq. ( 1 5 . 8 ) ) .

But this choice may also b e nearly collinear w ith the unit thrust command, s o a

second poss ibility, the current negative X B- axis , is also offe red (Eq. ( 1 5 . 9 ) ) .

Because the Z B - and XB -axes cannot both parallel the unit thru 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

vecto r s exp r e s s ed in platform coordinate s, is computed to s atisfy the unit thrust

command, the unit window comm and, and the thrust offset (th e angular displacement

between the estim ated unit thrust vector and the X R B - axis ) . C C B P is computed in

two steps as illustrated in F igure 1 6 . The first step ( E q s . ( 1 5 . 1 0 ) - ( 1 5. 1 2 ) ) uses

the unit thrust com m and and the unit window command but fails to account for thrust

offset. The second step ( E q s . ( 1 5 . 1 3 )-( 1 5 . 1 5 ) ) corrects for thrust-offset components

U N F R By and UNF R B Z . Since the s e corrections are s m all, no unit need be taken

in J::q. ( 1 5 . 1 4 ) . A small window pointing e r ror, shown in F igur e 1 6, is introduced

by the thrust- offset correction. Define d as 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 - axis. Although the

trim gimbal h as a m aximum displacement of 6°, the m aximum thrust offset during

descent is 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 edge of the LM window.

Becau s e the m atrix e e B P is the transformation from platform to commanded­

body coordinates, it c an be expres sed in terms of the I M U gimbal angles which

would place the body axe s in the commanded direction s . Therefore, commanded

gimbal angle s c an be extracted from the commanded-body m atrix. E xpre s s ing e e BP

as the p roduct of the three m atrices that correspond to rotations about the three

gimbal axes yields

[+C Z

eeBP = -ex +SX

ey sz e y sz e y

: +sz I

I

+ sx SY : +ex I

+ ex SY : - sx I

: -e z SY I I e z : +e x sz SY + sx C Yl I

c z : - sx sz SY + e x ey I

( 1 3 )

where S and e indi c ate s ine and cosine, and X , Y , and Z ind i c ate the commanded X ,

Y , and Z gimb al angl e s . F rom Eq. ( 1 3 ), i t i s apparent that the commanded gimb al

angle s are extracted from the elements of eeBP by E q s . ( 1 5 . 1 6 ) - ( 1 5 , 1 8 ), with

AR e TR IG defined as follows .

41

- �N F R B y

YC B �C B Py

- U N FR B y ---

c c s P 1 = 1 - X U N F C P J ( U N I T T H R UST

COMMAND) - U N FR B z

UNWC P ( U N I T W I NDOW

COMMAND )

- U N F R Bz

Z C B fC B Pz

Figure 1 6 Geometry of E re ction of C ommand e d - body Frame Viewed on a LM - centered Unit Sphe re

4 2

The AR C T H IG function of two arguments yields the angle whose tangent is

the ratio of the first and second argument s . ARC TR IG extracts the angle anywhere

in the ci rcle by u s ing the ratio of the s m alle r - m agnitude argument to the larger ­

magnitude argument a s the tangent of the angle o r its complement, and b y u sing the

s igns of the arguments to dete rmine the quadr ant of the angle. Equations ( 1 5 , 1 6 )

and ( 1 5 . 1 7) yield the outer and inner commanded gimbal angle s anywhere in the

c i rcle. B ecau s e the second argument i s always positive, implying a pos itive c o sine,

l·: q . 0 5. 1 8 ) yields the m i ddle commanded gimbal angle between ±9 0° .

To p reclude commanding gimbal lock, Eq. ( 1 5 , 2 0 ) limits the m iddle com m anded

gimbal angle to 70 ° m agnitude. Becau s e the unlim ite d v alue lay between ±9 0° and

the oute r and inner commanded gimbal angles were computed consi stent w ith the

mi ddle com m anded gimbal angle range, no quadrant switching of the outer o r inner

comm ands is r equired by gimbal- lock limiting. If limiting changes the middle

command, the guidance is commanding gimbal lock, and the gimbal- lock alarm code

is i s s ued.

U nlimited reference gimbal angle c h anges are the changes which would be

req11ired to bring the D ,; p ' s reference gimbal angle s into coincidence w ith the

commanded gimbal angles . The se are computed by subtr acting, modularly, the

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

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

If a Y o r Z gimbal angle change greater than 4 5° i s requi red, t h e flag i s

r e s et indicating n o attitude rotation is allowed about the XC B - axis . T h i s i s nece s s ary

to p r event false starts about the XC B - axi s as derived in the appendix of refe rence

1 2 .

Equations ( 1 5. 2 4 ) - ( 1 5. 28 ) yield the reference gimbal angle changes oy limiting

th e magnitude of the attitude changes to 20° in 2 seconds (1 0° / s e c ) about e ach of

th ree o rthogonal axe s; one axis i s coinc ident with the XC B- axi s and the other axes

l ie in the YC B , ZC B plane. This permits an angular - r ate vector of length 1 0 {3 deg ; sec . N ote that if the flag is reset, the attitude rotation about the XC B - axis is

made zero, resulting in an outer gimbal angle change to offset the inner gimbal

angle change (Eq. ( 1 5 . 2 8 ) ) .

The DAP comm ands cons ist of the reference gimbal angle increments to be

appl ied by the D .\P each 1 I 10 second, the reference attitude r ates, and the p e rmitted

43

l ag angle s . The refe rence g i m b al angle increments are the referen c e gimbal angle

c h ange s m ultiplied by the r atio of the I L\ P and ATT s ample interval s ( Eq . ( 1 5 . 2 9 ) ) .

T h e r·efe rence attitude r ates a r e computed b y the nonorthogonal transfo rmation o f

th e r · cfe rence gim b al angle changes shown in E q . ( 1 5 . 3 0 ) , The p ermitted lag angles,

w h i c h account for the angle s b y which the attitude will lag behind a ramp angular

c om m an d due to the finite ac c ele r ation s available, are computed using the available

a c c e le r· ati on gR B, and then individually magnitude limited (Eq. ( 1 5 , 3 1 ) ) . The DAP

avoici s attitude - r ate ove r shoot by permitting lagging attitude errors equal to the

p e r m itted lag angle s .

4 4

T H HO T T LE HOUTIN E

The throttle routine connects the currently oper ating guidance algorithm to

the DPS, as illustrated in F igure 1 7 . For ease of understanding, all thrust levels

are represented as per centages of the DPS rated thrust of 4 6 706 newton s . T HROT

generates th rust inc rement commands to drive the input thrust-acceler ation

m e asurement into c oincidence with the input thrust - acceleration comm and wheneve r

the resulting thrust would lie within the illustrated pe rmitted-th rust region. When

the re sulting thrust would lie below or above the p ermitted-thrust region, THROT

c auses m inimum o r m aximum thrust. The hysteresis-like region from 5 7 to 6 5 o/o

th rust a\'oids frequent alternation between the m aximum- thrust point and the

permitted-thrust region when the thrust command dwells at the boundary between

the permitted-thrust and forbidden-thrust regions.

:\ digital- to- analog interface between the LGC and the DPS is p rovided by the

descent engine control assembly (DEC A). E ach guidance cycle (once per two seconds,

except once per second for P 6 6 ) T HRO T generates the thrust increment comm and

�FC % which is conve rted to a pulse train and is sued to the DEC A. E ach pulse

causes about 1 2 . 5 newtons thrust change, and the pulse rate i s 3 200 I second. F ollowing

i s suance of a thrust inc rement command, the thrust therefore changes at the r ate

of 4 0 , 0 0 0 newtons I s econd ( 8 5% of rated thrust per second ) until the thrust inc rement

is achieved. \Vith a guidance cycle as short as one second and an engine re sponse

time which m ay be a substantial fraction of one second, it is nec e s s ary for P66 and

T H ROT to account for this transport delay.

c\s illustrated in the rightmost box of F igure 1 7, in the region from 1 1 to 9 3 %

the DPS th rust i s a nearly linear function of the pulse count ac cumulated by the

DEC :\. :\i ot shown in Figure 1 7 is the m anual throttle command, which is summed

w ith the DEC A output command by the DPS and p rovides the minimum 1 1 o/o thrust

when the D I,:C A com m and is zero. The DPS contains a mechanic al stop at typic ally

9 3 o/o rated th rust. This thrust level minimizes p ropellant consumption on a nominal

descent, considering the los s of specific impulse at higher thrust. To ensure that

the D P S is driven to the mechanical stop, the DEC A satur ates at a substantially

higher thrust level (about 9 9 o/o ) and the throttle routine drive s the DEC A into

saturation whenever maximum thrust is required.

� onlinearities in response and unce rtainties in DEC A and DPS scale factor s

a r e ove rcome b y the thrust inc rement command concept. N ominally, THROT p rovide s

dead-beat response to step inputs , but with downstream nonlinearities and s c ale-factor

er rors THROT drives the thrust- acceleration error to zero in the ste ady state .

45

*"" Ol

Af(P

THRUST ACCEURATIQN

C<Y>I""''«l

MA S ) f S TIMATI_

Af P THRIJSHCCUERATI0N

MEASUREMENT

I � I ;:: I � � � I § Z I O< ; I @ � � 3 1 � � I

COMPUH THJlU�T COMMA'NO

COMf'UH lHRUST ,'v\£ASURfW"NT

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

O�HtALL SYSTE\1 RlSPO�;Sf

� 9),...,.... __ _

� � 0

THRUST COMMAt«l

Figure 1 7

MC• �ccu .. u c � • c , '" " ' , "

C�.��:-A T I V l THRUST CCY>IMANO 1� T H RU S T I FCC\ fLC'J �6fC'I ll<J � fCC� � qq,. INCRfMENT

C.-AND '---------_l I ' I I I 1 ,. I � � � � I g :S

!Oi l � � ;:: I � � � I � a.: � I � � � I � � I o o

DPS Thrust Control

DECA AND D PS RfSPONSl

:;; 'l'l '\ -/, -, � OPS \'-' _,/ 1

Mt:_(HANJC M 0 "' > STOP

C U M U l AT IVE THRUST COMMA"..O

m

OEI IVE RI D

THRUST

Figure 1 8 illustrates the Throttle Routine computations . The thrust comman d

and thrust measurement are computed using the input m as s e stimate (Eqs .

( 1 8 . 1 )- ( 1 8 . 2 ) ). The input thrust- accele ration measurement is the aver age over the

p receding sample interval, du ring which a thrust inc rement com m and was is sued

producing an instantaneous th rust profile as illustrated in F igure 1 9 . Therefore,

to obtain the current sample- instant thrust, Eq. ( 1 8 . 3 ) corrects the thrust

m easurement by adding the th rust c orrection inc rement computed the previous cycle.

The th rust- control logic for p roviding the r equired overall system respons e

illustrated in Figure 1 7 is t o pick one o f four pos s ible thrusting policies acc ording

to the regions of the p receding and present thrust commands ( FCOLD% and FC %).

and to reset the th rust command if nec e s s ary to satisfy DPS constraints. Equation

( 1 8 . 4 ) or ( 1 8 . 8 ) re sets the thrust command to the thrust actually anticip ated . A

th rust c ommand augment ( F C AUG%) i s computed that e ither drives the DEC A into

s aturation if the policy is to initiate or retain maximum thrust ( Eq. ( 1 8 . 5 ) or ( 1 8 . 9 )) ,

o r corre cts for the region between the DPS mechanical stop and the DEC A s aturation

value if the policy is to initi ate thrusting within the permitted-thrust region ( Eq.

( 1 8 . 7 )) . No thrust command augment is required when the policy is to continue

th rusting within the perm itted-thrust region. No equivalent thrust- control logic is

needed at the minimum-thrust point because minimum thrust would occur only if

the com m ander could is sue five or more downward ROD commands within a s ingle

P 6 6 guidance s ample interval, p ractically impossible.

The th rust increment command ( Eq. ( 1 8 . 1 2 )) is composed of the actual thrust

increment �F Ao/o, plus the thrust command augment FC AUG% to drive the DEC A in

or out of s aturation, when required.

P reparatory to computing the thrust correction increment for the succeeding

pas s , Eq. ( 1 8 . 1 3 ) computes the total effective transport lag. The terms in the effective

transport lag are 1 ) the c omputation duration t - tSI, 2 ) the e stimated DPS time

c onstant of 0 . 0 8 s econd, and 3) the effective DEC A delay equal to hal1 the time

requi red to output the thrust increment command pulse train at 8 5% thrust change

per second. As long as the actual thrust inc rement AF A ( Figure 1 9 ) is c ontaine d

enti rely within the sample interval At, i t is clear that. a s LAG approaches zero,

the thrust measurement ( obtained by differencing accelerometer r e adings at the

s ample instants ) app roaches the sample- instant thrust F . Similarly, as L AG

approaches the s ample interval At, the thrust m e asurement must be augmented by

an amount approaching the actual thrust increment AF A to obtain the s ample- in stant

th rust F . F rom this heuristic argument, i t i s apparent that the thrust correction

increment which must be added to the thrust measurement to yield the s ample- instant

4 7

I N I T I AH

MAX I MU M

THRUST

I NPUTS

THRUST ACCH I RATiflN COM\IAND Af C P

THRlJST ACI H I RAT ION Mf A SU RI '.II NT Af P C l OC K - T I MI

SAMPll I NSTANT Cl CURRENT

GU ID ANCf C YCl E l S I

C U R RI N! MASS f S T I MAIT M

BI G I N

COMPUIT TH RU S T COMMAND A N D THRUST MlASUREMENT

IOO Af C P M

FC'I. 46-i�-Niwror.iS 100 Af P M

�� 4blliil-NfWTONS

I] I

COMPUH SAMPlf I NSTMIT THR \ I S T BY ADD INC TH> ACTUAl THRUST THF PREC f D INC CYCI I

{ R! SH THRUST COMMAND & } S £ 1 TH RU S T COMMAND AUGMENT

COMPUTE ACTUAl THRUST I �C RIMFNT AND SAVl

THRUST COMMAND FOR S U C CH D lNG PASS

I'. I A'' I C'' F�

r ca o ' r c,, I 101 1 1 1 1

THRUST CORRfCTION I N C R!Mf �T f OR P 66 VfRT I C A! �OD}G U I DANCl ALGORI THM

b f A b t A • 46 706 NI WTONS I IOO

Figure 1 8

OUTPUTS,

I N il

THRUST I NC RFMENT COMMAND M C '\

THRUST CORRfC T ION I NC R f MoNT b f A

Throttle Routine

4 8

l ] l l

1 141

..,. co

�--- Guidance Sampl e Interval �t

Computation Del ay Effective .

P l us D P S Time r-; DECA Del ay 1 nstantaneous Thrust P rof i l e Constant • .1�J

_ Sampl e - 1 nstant

i Actual Thrust Increment 6FA 14---- LA£ ------.1

P revious Sample Instan t

Th rust Correction Increment o F A = NA LAt 6t

C ur rent Sample Instan t

Figure 1 9 Thrust Dynamic s Within a Single Guidance Sample Interval

Thrust F

thrust is proportional to L AG as c omputed by Eq. ( 1 8 . 1 4 ). A r igorous derivation of

this re sult is presented in Appendix A of reference 1 1 . The sole purpo se of E q.

( 1 8 . 1 5 ) is to inte rface the P 6 6 Vertic al ( ROD) Guidance Algorithm.

\V ith the thrust command FC % either within the permitted- thrust region o r

reset to the value which will actually be achieved, A F Ao/o is an accurate p rediction

of the actual thrust increment, and 8F Ao/o or c5F A is an accurate thrust correction

inc rement . aFAo/o or aFA is slightly in e rror w hen initiating thrusting w ithin the

permitted -thrust region. The slight e rror is due to ne glecting FCAUG% in the

computation of LAG.

50

BR AK IN G - P H ASE AN D A P P RO AC H - P H ASE TAR G E TING P ROG R AM

The targeting program gene rates mis sion- dependent data for the P 6 3 Ignition

Algorithm and for the P 63, P 64 Guidan c e Algorithm. All d ata are exp r e s s e d in guidance coordinate s . The ignition algorithm requires nominal initial altitude, r ange,

and speed data that determine ignition time and indirectly determine the throttle­

c ontrol duration. The guidance algorithm requires tar gets for P 6 3 that p rovide an

efficient transfer and targets for P 6 4 that provide a traj ectory m eeting s everal

c onstraints on geometry, visibility, and thrust. Des c r ibed in detail in referenc e 5,

the P 6 4 constraints provide a fast shallow app roach phase more akin to an airplane

approach than a helicopter app roach, although the termin al- descent phase i s

e s s entially vertical i n helicopter fashion. T h e landing site m u s t b e appr o ached

along a nearly straight- line path depre s se d typic ally 1 6° from hori zontal, terminating

typic ally at 30 m altitude 1 1 m ground r ange. The l anding site must be visible

c ontinuously until a few seconds before approach- ph ase terminus, and the DPS thrust

must begin at around 5 7% and must lie c ontinuously in the 1 1 to 65% region.

Geometry, visibility, and thrust during approac h c annot be specified explicitly.

Visibility depends upon the pos ition and attitude profile s , and the s e profiles (with

the th rust and mass p rofiles ) are constrained to s atisfy the law s of phy s i c s . The

guidance algorithm will p rovide the t r an sfer from any arbitrary initial state (within

bound s ) without regard to any visibility or thrust constraint s . The task of the tar geting

p r ogram is to set up the P 6 4 initial state and guidance targets such that suitable

vis ibility and thrust profiles are r e alized implicitly.

During the final portion of P 6 3 , and throughout P 64, the guidance algorithm

will gene rate a traj e ctory whose pos 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

polynomials and 2) extracting the guidance targets as the position vector and its

derivative s at a target point, lying on the polynomial, substantially beyond phase

terminu s .

T h e P 6 4 targeting concept i s to con st ruct the approach-ph ase quartic b y

imposing nec e s s ary and sufficient constraints . With quartic degree, five independent

c on straints m ay be imposed in each of three axes . The nomin al t r aj ectory i s

arbitrarily m ade planar, requiring the Y - components i n guidance coordinates o f

position and all its derivative s t o be zero and leaving two axes t o specify . Bec ause

the initial state c an be controlled by the preceding b r aking- phase guidance, all five

constraints in each 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 onstraint s e t - completely determine the

5 1

(J1 N

IGNI T I O N NON OUAR l iC

Figure 2 0

T o - 1 8 0 SEC T H ROTTLE

CONTROL

RECOVERY

T o -60SEC T o - 1 56 S E C BRAKING- APPROACH-

PHASE PHASE TERMINUS INCEPTION

'\.. /

. -· -

T o 0 SEC

BRAKING­

P H A S E TARGET POINT

Composition of the Lunar-de scent T rajectory

P 64 traj ectory, the geometry and vis ibility p rofiles c an be determined in closed

form, and the th rust profile c an be determined from a prior knowledge of m as s .

Thus P 6 4 targeting consists of generating closed-form solutions for a number of

P 6 4 constraint sets and picking one which provide s adequate visibility and thrust.

Specification of P 6 4 con straint sets i s reduced to a two- dimen s ion al s e arch, as

w ill be desc ribed in the following section .

The P63 targeting proce s s is not so clean . Because the engine must be run

at fixed maximum th rust for most of the phase, the guidance commands are not

s atisfied, and the refore, as shown in F igure 20, the m aximum- th rust portion of

P 6 3 i s not quartic . W hen throttle control i s recovered, generation of a quartic is

begun. But the throttle recovery point is not clo s e to any target point. Therefore

the state vector at this point c annot be controlled, and we have no closed-form

solution for it . Since the initial position and velocity on the braking-phase quartic

must be free, there remain only three constraints, in each of two axes, which c an

be imposed arbitrarily. The guidance algorithm permits a fourth constraint i n one

axis by solving for the current target- referenced time such as to s atisfy a con st r aint

on the ZG - component of j e rk . Thus a P 6 3 c on straint set composed of seven constraints

is spec ified arbitrarily, and the rem aining three conditions required to define the

b r aking-phase quartic are determined iteratively by simulation. Three or four

ite rations are generally required bec ause there i s bilateral inter action b etween the

targets and the simulation.

C onstraints

The P64 constraint s et i s constructed as follows :

1 . Fou r constraints at a specified target- refe renced terminal time T APF : Two

te r m in al vertical constraints, specified by the m i s s ion comm ander, are the termin al

altitude ( R APFGX = 30-m typic ally ) and altitude r ate ( V AP FGX = - 1 - m / sec

typically) . Two termin al horizontal constraints , imposed by the choice d effective

P 6 6 horizontal time constant T, are that the terminal position, velocity, and

ac celeration shall be related by R AP FG z = AAPFGz r 2, V APFG z = - AAPFG z r.

These P 6 6 compatibility constraints c ause the pitch comman d s at P 6 4 terminus

and P 6 6 inception to be identical ( avoiding a p itch transient at the phasic interface )

and c ause the P 6 6 algorithm t o null the horizontal position error a s well a s the

hori zontal velocity error, without position feedback. Bec ause the P 6 6 hori zontal

algorithm feeds back the prior acceleration command, and because of the tran sport

delay, an effective r of 8 seconds h as been found s atisfactory r ather than the 5

seconds used by the P 6 6 algorithm.

5 3

2 . Four c on straints at an unspecified target - referenced m idpoint time T AP M :

T h e midpoint constraints are specified by the commander according t o h i s sense of

s afety and compatability with a pos sible m anual transition to P 6 6 . Typic ally, he

m ay specify - 5- m / sec altitude rate at 1 50 - m altitude, and a 1 6° s lope, completely

determ ining the m idpoint state R AP MG ,..Y_APMG given R AP MGy = V APMGy = 0.

3 . Two con straints at an unspecified target- referenced initial time T API: The

initial position is arbitrarily spec ified to lie on the 1 6° p ath and to provide an approach

phase of typic ally 7 . 5 -km length, dete rmining the initial position vector R AP IG given

R AP IGy = 0 .

This completes the P 6 4 constraint set except for specifying the time s T AP M

and T API at which the midpoint and initial constraints apply. These times are

determined by running the Approach -phase Targeting Routine over the two­

dimensional sweep of value s of T APM and T API. F rom the c as e s run, one is p icked

that exh ibits suitable attitude and thrust behavior ( b ased on an a-priori P 64 initi al

m as s e stimate ). If subs equent simulation p rove s the mass estimate exc e s s ively in

e r ror, the initial thrust will be uns atisfactory, and an altern ate c ase must be picked.

The seven P 6 3 constraints are specified as follow s : Four constraints are

specified by c omp atibility of the terminal state on the braking-phase quartic with

the initial state on the approach- phase quartic . Two constraints are imposed on

terminal acceleration by requiring the termin al thrust to be 57% and by specifying

the term inal pitch angle . The final constraint is imposed on the hori zontal component

of termin al j erk by requiring zero rate of change of thrust at terminus, The term in al

pitch angle, typic ally around 60°, is chosen by trial and error to m inimize propellant

c onsumption.

App roach- phase T argeting

F igure 2 1 illustrate s the Approach- phase T argeting Routine. Normally, this

routine is first run separately in search of targets for the approach phase, and

then run jointly with the Braking-phase T argeting Routine ( F igure 2 2 ) to determine

targets for the entire lunar descent.

In the XG - axis ( altitude), the terminal acceleration, j erk, and snap are computed

by Eq. ( 2 1 . 2 ), which is obtained immediately from

54

RAPMGX 1 TMF TMF2/2 TMF3/ 6 TMF4/24 RAPFGX

VAPMGX 0 1 TMF TMF2/2 TMF :3/ 6 VAPFGX

RAPIGX 1 TI F TIF2 /2 TIF3/ 6 TIF4/24 AAPFGX ( 14)

JAPFGX

SAPFGX

where Tl\TF and TIF are the m idpo int and initial te rm inus - referenced time s computed

by Eqs . (2 1 . 1 ) .

I n the YG - axi s, pos ition and all its derivative s are zero t o produ c e a planar

t i· aj ectory.

In the ZG - axis , Eq. ( 2 1 . 4 ) is obtained by substituting the P66 comp atibility

constraints H AP FG Z = A. APFG z r 2, V APFG Z = - AAPFG z r into th e ZG-axi s ve r s ion

of Eq. ( 1 4 ) and inverting. Equations ( 2 1 . 5 ) and ( 2 1 . 6 ) complete the definition of the

app r oach-phase quartic . It remains to compute the approach-phase t argets as the

position vector and its der ivatives at the target point on the quartic .

F o r a quartic polynomial, a 5 x 5 state t r an s ition m atrix ID ( T 1 , T 0 ) c an b e

defined by

R1 1 (T 1 -T 0 ) 2 (T 1 -T 0 ) / 2 3 (T 1 -T 0 ) / 6 4 (T 1 -T0) / 2 4 Ro Y.1 0 1 (T 1 -T 0) 2 (T 1 -To > / 2 3 (T 1 -T 0) / 6 .Yo

2 = m(T l ,TO)XO, X = A l 0 0 (T 1 -T 0) (T 1 -T0) / 2 Ao 1 I.1 0 0 0 1 (T 1 -T 0 ) I.o �1 0 0 0 0 1 �

whe re H . to S . are row vecto r s . _m (T 1 , T 0 ) can b e derived using linear system-s -1 -1

theory,

R r� 0 1 0 0 0 R

v 0 0 1 0 0 y . X A li 0 0 0 1 0 A 0: X

j 0 0 0 0 1 J � 0 0 0 0 0 .s._

55

w ith the solution

where I is the 5 x 5 identity m atrix. The exponential series is zero after the fifth

term because o i = 0 for i > 5 . All the p roperties of state tran s ition m atrices c an

be applied t o s c alar and vector polynomial s .

Equations ( 2 1 . 7 ) yie ld the comp lete target and initial states by u s ing state

tran sition m atrices and the definition of target- referenced target time as zero.

B r aking- phase Targeting

To target P 6 3 , we must completely determine the b r aking-phase quartic shown

in F igure 2 0 . Seven of the ten nece s s ary c onditions are determined in closed 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 remaining three c onditions nec e s s ary to define the quartic are

determined iteratively by simulation. The terminal pitch angle IJ P BR F is a fixed

input to the B r aking- phase T argeting R outine.

F i gure 22 illustrates the routine. Four conditions are specified by s etting *

the P 6 3 termin al position and velocity equal to the P 6 3 initial state (Eqs. ( 2 2 . 1 ) ) .

A unit vector in the term inal- th rust direction is computed from the terminal pitch

angle OPBRF ( Eq. ( 2 2 . 2 ) ), and the terminal ac celeration i s c alculated by Eq. ( 2 2 . 3 )

using the terminal thrust F BR F , the P 6 3 term inal m as s e stim ate MBRF, and allowing

fo r lunar gravity GM. The XG - component of terminal j erk must be determin e d by

s imulation and is therefore set to zero for the first iter ation (Eq. ( 2 2 . 5 ) ) . The

ZG - component of termin al j erk is computed by Eq. ( 2 2 . 5 ) to p roduce zero r ate of

change of thrust at terminus , accounting for the e stimated terminal mass flow r ate

c omputed by Eq. ( 2 2 . 4 ) ; the j erk coeffic ient KJ, typically 1 . 2 , accounts for the

XG - component of thrust. The te rmin al snap must be determined by simulation and

is the refore set to zero (Eq. ( 2 2 . 6 ) ) for the first iteration. This completes the

fi rst- ite ration definition of the braking- phase quartic.

* N ot shown i s the c apability of the targeting program to set the P 6 3 terminal state

to a backwards extrapolation of the P 6 4 initial state to allow for a short transition du ring which the acceleration is as sumed to change line arly with time. Thi s c ap ability is not alw ay s used, and to show it would unn e c e s sarily complicate the p resentation of F igure 2 2 .

5 6

B r aking-phase targets are c omputed by Eq. ( 2 2 . 7 ), using the state transition

m atrix and the definition of target - referenced target time as zero. U sing the

c om puted targets, a s imulation is run to produce corrected d ata.

The nominal initial r ange used by the ignition algorithm is corrected by Eq.

( 2 2 . 8) to correct the error in the target- referenced time of throttle control recovery.

The simulation produces a braking-phase quartic s atisfying the target values

of pos ition, velocity, acceleration, and Z G - component of j erk. The remaining

conditions nece s s ary to define the quartic c an be obtained from the cur rent state

on the last pass of the b r aking-phase simulation. The equation for the current

state,

1 [ :: l [: T

T

RBRTG

Y:BRTG

ABRTG

.J.BRTG

§BRTG

is re adily solved to yield the achieved target j e rk and snap according to Eq. ( 2 2 . 9 ).

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

computation of T by the guidance algorithm; agreement between achieved and input

values is typic ally to s even plac e s .

I n preparation for correcting e stimates at the terminus, the complete state

at terminus is computed by Eq. ( 2 2 . 1 0 ). E quation ( 2 2 . 1 0 ) yields a term in al state at

the specified terminal time TBRF p recisely, whe reas the state RG, Y:G applies at

the time T which m ay differ from T BR F by up to the 2 - second granularity.

E quation ( 2 2 . 1 1 ) corrects the P 6 3 terminal mas s estimate using the rocket

equation. Equation s ( 2 2 . 1 2 ) - ( 2 2 . 1 4 ) correct the termin al acceleration, j erk, and

snap using the corrected P 6 3 termin al mass e stimate, the achieved XG - component

of terminal j erk, and the achieved XG - and ZG- components of terminal snap .

F inally, state convergence test quantities are computed b y E q s . ( 2 2 . 1 5)

( 2 2 . 1 7 ) . Since only three conditions (JBRFGA:x_ , SBRFG AX ' and SBRFG Az ) defining

the b r aking-ph ase quartic are sought ite r atively, only three convergence c riter i a

5 7

I NPUTS: TERMINAL 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 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

COMPUTE TERM I NUS-REFERENCED T I ME'> f-----

BEG I N

"IMF • TAPM - TAPF, T I F · TAP I - TAPF

TAPF RAPFGx VAPFG X T TAP M RAPMG, \IAPMG TAP I BAPIG

(\)

TERM I NAL X COMPONENTS

TMF 3 1 6

TMF 21 2

TIF 3 1 6

- TMF - 1

-TIF

O] '� o ;:::�: 1 I I RAPMG IZI

X VAPMGX RAPIG X

TERMINAL Y COMPON NTS

RAPFG y • 0, VAPFGy • 0, AAPFGy • 0, JAPFGy • 0, SAPFGy • 0

.-----TERM I f.IAI 7 �" '"""c"'

JAPFGz � - T + TMF

[ "'"1 [ 'L, '" ' "" l/ 2

SAPFG z l t 2- t TIF + T I F 21 2

R A PFG l • AAPFGz t 2

VAPFG z •-.\APFGz T

TMF 3 1 6 TMF 2 1 2

T I F 3 1 6

• w '' T [ ""ffiz l TMF 31 6 VAPMGz

TIF 4 1 24 R A P I Gz

COMPUTE COMPLHE TARGET AND I N I T IAL STATES [ '""] r '"'" VAPTG VAPFG �APTG ·¢ 10, TAPFI l �APFG JA PTG JAPFG SAPTG SAPFG - - . l'""l r· l VAPIG VAPTG �APIG · ¢> 1TA P I , O I fAPTG

JAP IG JAPTG �APIG -APTG

OUTPUT S :

141

151 161

171

COMPLHE TARGET AND I N IT IAL STATES

f N O

['""] [ '" "j VAPTG VAPIG AAPTG AA P I G JAPTG JAP IG SAPTG SAPIG - , -

Figure 2 1 Approach - phase T argeting Routine

58

c.n CD

I NPUTS Pb4 IN TERM I TfRMI TERM I P63 TE DPS EX J£RK Cl MJMINA

FO COEFFI REOU I I CONVl' CONVl'

l�l Sl�TE ft�PIG. ��PIG l T�RGET-REHREN:ED T IME TBRf I PilCH A�GlE CotiSTRAINT 6 PBRF

.L lHRUS.l C ONSTR� I NT FBRF I IN�l MASS ESTIMATE MBRF �UST Vl'LOCITY ESTIMATE fOR P6l TERMINUS VEXBRF JF ICIENT TO �CCOUNT fOR THE Vl'RTIC�l COMPONENT Of THRUST KJ . INITI�l AniTUDE, RANGE, ANO SPEED THE IGNITION ALGOR ITHM

RBRIGX , RBRIGz

ENT fOR CORRECTING MJMINAl iNITIAl RANGE RBRIGz KlG D TARGET REHRFN:ED T IME Of THROTILE RECOVl'RY ITHROT :ENGE CRITERION fOR TAR<.ET ·REFERENCED T IM£ OF lHROTILE RECOVERY f TIHROT :EN:E C R I TERION FOR ITRMINAl STAT£ ERRORS f

SEG IN

COMPUTE FIRST ITERATION Of P6l COMPLETE TERMINAl STATE

RBRFG · RAP I G. YBRFG • YAPIG Ill YNFBRFG • I COS 8PBRF , O, - S I N B PBRF I 111 ABRFG · I FBRFI MBRF I UM'BRFG · I GM. O. 0 I Ill �BRf • - FBRFI VEXBRF I( I lBRFG • I 0. 0, - KJ �BRFGz MBRFI MBRF I 151 �BRFG · I D,O.O I 161

t COMPUTE COMPl£1£ TARGET STATE [�BRTG [ RBRFG]

VBRTG VBRFG ABRTG · 4> 1 0. TBRf I ABRFG Ill JIRTG JBRFG SIRTG LimG

+ S IMULATE DESCENT USING TARGETS Of CURRENT ITERATION

S I MULATION OUTPUT DATA USED FOR CORRECTING IGNITION ALGORITHM PARAMEHRS! ACTUAl TARGET ·REFEREM:ED liME Of THROTILE CONTROL RECOVl'RY ITHROTA ACTUAL STATE ON FIRST GUIDAM:E PASS BBRIGA. YBRIGA

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

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

• RESET NOMINAl INITIAL RANGE TO CORRECT THE Tfft!OTILE CONTROL DURATION

""" ''z " """ '"z KlG TIHROTA · TIHROT I

t COMPUTE JERK AND SNAP AT TARGET POINT ACHIEVl'D 8Y S llllJLATED TRAJECTORY [!�"W'' " ' ' " w •' •l�l }.IRTGA nt T 4 411 Tl 111 r ·nt T4 141 TJ �IRTG

�BRTG !!.G yc

I

Ill

191

VBRIG

---

CDMPUH COMPtm SI Alf Al llRMINUS ACHIEVED BY 5 1 MlllATED TRAJECTORY

r'·l ['""'' J VBRFGA VBRlG �BRfGA ·+ (18RF,OJ XBRTG

JBRIGA IBRT<.Ax JBRTGAy JBRTGz BRfC.A �BRTC.A

t CORRICT Al l ESTIMATES Al TfRMINUS

MBRF • M rxPI t lyBRFG I 1 �G p/VfxBRF l �BRI G · l fBRf l MBRf I �M' BRfG f GM, Q. O l lBRfG · I JBRFGA x , 0, KJ ABRFGz MBRFI MBRF I 2BRFr; - j SBRfGA X . 0. SBRFGAz l

t COMPUTE I lATE CONVERC.fNCE TEST QUANT IT IES

Ul " I ' AOKIGA z . A B R f G 7 I I I �BRfG\1 02 · I I ABRfGA x OJ • )1 VBRfGA X

YES

ABRfC. X I I I ABRfG I I VBRIG X I I IYBRf G I I

YES 01 · E

NO

YES 01 , r

NO

YES OJ · f

NO

i nHROTA TIHROT I , E TIHROl

TNo

i !lOI

1 1 1 1 1111 <Ill 114)

bl 1161

!171

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

RBRIGX · RBRIGA X . R8R1G z · RBRIGA Z , VBRIG · I YBRIGAI 1181

OUTPUTS : COMPl£TE TARGn STATE.

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

MJMINAl INITIAL SPUD END

Figure 2 2 Braking - phase T argeting Routine

RBRTG VBRlG �8RTG JBRTG �8RTG

RBRIG X . R B R l GZ

VBRIG

are needed. The three criteria chosen are important for guidance performance

and are related nons ingularly to three conditions sought. If any one of the state

c onvergence tests fails, or if the throttle control recovery time convergence test

fails, the braking- phase targets are corrected and another simulation is run; otherwise

the targeting is concluded by correcting the ignition algorithm inputs per Eq. ( 2 2 . 1 8 ).

6 0

R E F E R E N C E S

1 . K riegsman, B. A. , " R adar- Updated Inertial N avigation of a C ontinuous ly­

Powered Space Vehicle' ' , IEEE Aerospace Systems C onference, S e attle,

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

2 . Widn all, W . S . , " Lunar Module Digital Autopilot", Journ al o f Spacec r aft an d

Rocket s , Vol. 8 , N o . 1 , J anuary 1 9 7 1 .

3 . C herry, G . W . , 1 1 E Guidance - A General Expli c it, Optimizing Guidan c e Law

for Rocket- P ropelled Spac e c r aft" , M I T Instrumentation Laboratory R eport

R - 456, August 1 9 6 4 .

4 . Klumpp, A. R . , 1 1 A M anually Retargeted Automatic Des c ent and Landing System

for LE M1 1 , MIT Instrumentation Laboratory Report R - 5 3 9 , M arch 1 9 6 6 ,

5 . Klumpp, A. R . , ' ' A M anu ally R etargeted Automatic L anding System for the Lunar

Module (LM) 11 , Journal of Spacec r aft and R ockets, F ebruary 1 9 68 ,

6 . ?vloore, T. E . , G . G . M cSwain, and J . D . Montgomery, ' 'Guidance Laws for

C ontrolling Off- Nom inal LM Powered Des cent Traj ectorie s B ack to the

I\omin al1 1 , Inte rnal Note MSC - E G - 69 - 9 , P roj ect Apollo, N AS A, M anned

Sp ace c raft C enter, Houston, Texas, F ebruary 2 8 , 1 9 6 9 ,

7 . l\lcSwain, G . G. and T . E . Moore, 1 1 F alse High G ate Targeting for L M Powe red

Des cent" , Internal Note MSC - 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 . , 1 1 A T argeting Scheme for Fuel Optimal Rocket T r ajectories -

W ith Applic ations to the LM De scent B r aking Phase" , Technic al Memo r an dum

T M - 7 1 - 20 1 4- 1 , Bellcomm January 22, 1 9 7 1 .

9 . MIT Charle s Stark Draper Laboratory R eport R - 5 6 7, "Guidance System

Operations P l an for M anned LM E arth Orbital and Lunar M i s sions U s ing

P rogram Luminary 1 D' . . Sect ion 5, Guidance Equation s , R ev. 9 , December

1 9 70.

6 1

1 0 . Eyles, D. E . , 1 1 Apollo LM Guidance and P ilot- Assistan c e During the F inal Stage

of Lunar De scent - Software C onsiderations " , Fourth IF AC Symposium on

Automatic C ontrol in Space, Dubrovnik, Yugoslavia, September 6- 1 0, 1 9 7 1 .

1 1 . Klumpp, A. R. and G . R . K alan, " Elimin ation of N oise and E nhancement of

Stability and Dynamic R esponse of the Apollo LM R ate-of- Des c ent P rogram",

MIT C h arles Stark Dr aper Laboratory R eport E - 2 543, October 1 9 7 0 .

1 2 . Klumpp, A. R . , " F IN DC DU W - Guidance Autopilot Interface Routine", MIT

Instrumentation Laboratory LUMIN ARY Memo No. 2 7, Rev. 1 , September 2 6 ,

1 9 6 8 .

1 3 . Bennett, F . V . , " Lunar De scent and Ascent T r aj ectories " , AIAA Eighth

Aerospace Sciences Meeting, 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 Descent and Ascent",

to be published as a N AS A Techni c al Note.

6 2

Internal:

R - 6 9 5

DIST RIBUTION LIST

P. Adler

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R . Battin

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P. Mimno

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J . Scanlon

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P. Wei s sman

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