t
9
SURVEYOR TERMINAL R . K . Cheng2
GUIDANCE^
, -
ABSTRACT
Under the auspices of the J e t Propulsion Laboratory and the National Aeronautics and Space Administration, Hughes A i r c r a f t Company
e
I
i s performing the design and development of the unmanned, lunar softlanding spacecraft system for the Surveyor project. This paper d e s c r i b e s
the basic concept and implementation of the t e r m i n a l guidance s y s t e m for the landing phase. The t e r m i n a l descent profile consists of a solid rocket deboost phase with earth-based commands for thrust orientation as well as p a r t i a l control of ignition. This is followed by a closed-loop v e r n i e r guidance phase with The automatic v e r n i e r guidance concept consists
h
throttleable liquid engines.
D
ICODEI
(NASA C R
O R T M X OR AD NUMBER)
CATPQORI)
J
D
1 T h i s work was performed in pursuance -of Contract 950056 with the Jet
Propulsion Laboratory, California Institute of Technology, under Cont
No. NAS 7 - 100 sponsored by the National Aeroqautics and Space
Administration. 'Associate Manager , Systems Analysis Laboratory, Surveyor P r o g r a m ,
Hughes A i r c r a f t Company, El Segundo, Califorma, U. S. A.
of (1) velocity magnitude control i n accordance with a velocity v e r s u s slant
range descent law and (2) gravity t u r n steering. The differential equations
of motion are solved and implementation requirements obtained for anticipated variations i n t r a j e c t o r y and site slope conditions. The on-line Earth-
based guidance p r o g r a m for solid rocket t h r u s t attitude and ignition control
is a l s o discussed.
- 3 Surveyor Terminal Guidance -__ 1
R. K. Cheng
I.
2
INTRODUCTION
The Surveyor is the f i r s t U.S. project t o softland a n uninanncd vchiclc and i t s payload on the moon with the primary purpose of discovering those surface and environmental c h a r a c t e r i s t i c s important to future manned lunar exploration as well as the understanding of the universe. The spacecraft, a
model of which is shown in F i g u r e 1, w i l l be launched into a translunar t r a j e c t o r y with a planned midcourse correction to nullify the effects of injection e r r o r s such that the approach to the moon is a direct impact a t a p r e -selected landing site. The terminal slowdown begins a t approximately
60 m i l e s above the lunar surface when a pulsed r a d a r generates a marking
signal at a p r e s e t range, An earth-commanded time delay ensues and is
followed b y the ignition of a high thrust, solid propellant engine called the
*
'This work w a s performed in pursuance of Contract 950056 with the J e t Propulsion Laboratory, California Institute of Technoloty, under Contract
No. N A S 7 - 100 sponsored by the National Aeronautics and Space
Admini s t ration. 2Associate Manager, Systems Analysis Laboratory. Surveyor P r o g r a m . Hughes Aircraft Company, El Segundo, California, U.S . A .
-Irmain retro. During the main r e t r o phase, constant attitude is maintained by
a set of t h r e e differentially throttling liquid v e r n i e r engines located symmetrically about the roll o r t h r u s t axis. After burnout and staging of the main
r e t r o , these same verniersare then used for removal of the remainder of the approach velocity i n a closed-loop guidance mechanization involving a t h r e e beam doppler r a d a r for the measurement of vehicle velocity and a fourth beam for measuring slant range to the lunar surface. - A t a r a t h e r low altitude and a correspondingly low velocity, the v e r n i e r engines a r e shut off and the vehicle f r e e falls t o the surface. The landing is made with a nominally
v e r t i c a l attitude r e g a r d l e s s of the slope condition. The design of the spacecraft emphasizes simplicity and reliability. In
the t e r m i n a l descent and guidance system f o r example, the on-board r a d a r s a r e body-fixed, the main r e t r o engine employs no thrust termination device, and the delicate task of soft-landing is to be achieved without the use of a digital computer.
A p r i o r paper, Reference 1, discussed some of the p e r t i +-i~:v
e
nent design considerations especially concerning approach t r a j e c t o r y and main r e t r o burnout constraints a r i s i n g due to propulsion and sensing limi-
tations , and their implications with r e g a r d t o fuel and guidance requirements.
t
In t h i s paper, the guidance mechanization, both in t e r m s of the on-board as
well as earth-based portions, is discussed and c e r t a i n characterietics of interes a r e obtained through the solution of felatively c l h p l e though idealized equations of motion.
uiibrakec! impact velocity
in the flight control p r o g r a m
other guidance
A
thrusting attitude, is accomplished by szquentially torquins a set of strapped-
..
celestial s e n s o r s .
Since thc
L,,
-..-:i.>-;
<.- , - ~ c t i a : 11-i ,
inertial coordinates is
c r a f t rotation angles is a strai,.
will be sent t o and executed by
_ c
c3,;--pctation. :;--,-C
s
The commands
:.--L
sA;,accL -
~ , xt~ : i yminutes before the
of the s u r L c c ?being approached xx2-y oe t r a n s m l t t e d back to e a r t h for
future analysis. Main Reti-0 C'liase The principal features
1.
1.-
:e
1 ; -
~
;;G
---#-:--5
.,base are:
Constant attitude
3.
Burning to fuel deple:lon
-
:'i-.x.s:
termination
T h e only on-board guidance i ~ . ~ : r u i ~ e n t a , l o - ,s l d e f r o m the strappedc down g y r o s needed f o r sensing ~ ~ z t i r u d e> ~ ; ; ~ and causing the v e r n i e r c es engines to throttle differential::;
:<, coui"c:-~c~ such
changes, i s a crude
longitudinal acceleration switch w 3 c h seilses the decay of t h r u s t and initiates a fixed-duration separ&,tionsec;;z:>ce.
The conditions at the em5 ; the sc;>zr&tionperiod, a l s o called the i
bul-nout cor?ditions, determine
t
1 0
a large 2Czree the satisfactory operation
0-
t h e subsequent v e r n i e r ciescent j>'nasc, dssuming that no ixalfur,ctior,
occurs.
Tlic chief constraints o;^i the zltic;lLe a s well a s tkc niag:>,tude
and direction of the velocity vect.2: a r i s z due to p ~ o p u l s i o a~l l t e r m i r d i i s e n s o r limitations as d i s c u s s e c i;i 3 e f e r e m e 1. The altitude constraint i s
iunc;io::
of velocity, the l o w e r l i m i t
being further dzte rniined by tlie c z s i n n i m t h r u s t /weight capability and tne
.-
I
upper limit Sy the amount of p r o ; ; ~ . ' . ~ aT t
i s limited approximately to the ir,t;:-va!
:~:ni;zg.
. .
The velocity magnitude The flight
ol - 3 0 t o 650 f t / s e c .
z,r.dlc pzth is r e s t r i c t e d to a cone of 4 5 O . L . ~ - c ~ , ~ : c
__
t b o u t the local vertical
in o r d e r t o e n s u r e proper incidcc<-ci :
ti;c
r a d a r beams on the
lunar surface once the gravity turr. steering begins.
The actual burnout
t velocity for a given flight even w i ~ kh e d f e c t s of the variability i n the
translunar trajectory and midcour;e corrccrion and dispersions i n the m a i n - r e t r o phase taken into accoixr, still has a v e r y high probability of beisg within the indicated allowatlc Yange. T h e actual burnout altitude i s determined by the earth-commanded setting of the ignition delay and r e t r o phase dispersions.
I f there were
110 iiia1-y-
.
c::spci-sions,
the burnout altitai.:
would be a t a value specified by a :x:ciio;>,
cali~ci tiic 'Noininal Burnout
Locus", of the magnitude of the prellctcd .iominal burnout velocity.
The
locus i s designed a s a compromise bctweec 5x4 consumption and clearance above a minimum altitude which cay: b e handlc72 by the limited acceleration capability of the v e r n i e r engines, allowieg any given operating point on the locus.
: J : -
< h e random dispersions about
A graph o i the burnout locus,
d i s p e r s i o n ellipses and constraint contours wlllch illustrate the design
p,
problem i s shown i n Figure 3 .
Verr-ier P t a s e
The h e a r t of the vernier g4Lc..-A-Lcz -:-ecF.r,a-',zai.ion is the RADVS (RLdar -4;r'Lmeter Doppler Velocity S e n s o - ) s - y s : ~ . - ~ - -:-z .
*:. s5own in Figure 4 T h e r e a r e two z;-,:i: ;>z: .
GCL.X
-
geometry of which i s
t ~ . ~ z : ; = ~placed n e a r the bottom as
of the vehicle basic framework.
Antenna i supplies one of the t h r e e doppler
beams as well as the altitude be=
rhe thrust direction).
<:-ctre a??roprlazely, the slant range along
The
.
Antenna 2 s ~ r p p l i e s the other two doppler beams.
C ::: OZ LS
do?pler b e a m s a r e directed a t thre;:
of a square and all a r e located at
The shift in frequency of the r e -
an eqiual asgie 9 away f r o m the t h r c c t axis.
t u r n irom that of the transmitted sigzal is ? r o p r t , i o n a l to the spacecraft veiocity multiplied by the cosine of the angle betwee3 the velocity vector and the b e a m direction. Referring t o F i g u r e 4.
= chfl = f
O
-
V., s i n
i .
-.'
: I -A
I _
V.- sin -- -_->' Jsz-
9
+ vz cos
8
From Equztions ( 1 ), ( 2 ) and (3),
%e
velocixy connponents along the vehicle
fixed coordinate s y s t e m m a y b e salved
- 9 -
I t may b e seen that this p a r t i c L z r ha::--
gec:xetry r e s u l t s i n the algebraic
summing o r differencing of only two quantities for each of the t h r e e cornponents of velocity along x, y, z .
Thus, it p e r m i t s d i r e c t frequency mix-
ing of-the r e t u r n signals i n p a i r s , a distinct simplification i n mechanization : than a the c a s e where the bezms are symmeLrically located (at 120" a z i -
muth separation) around the t h r u s t a s l s . T h e altimeter t r a n s m i t s a C W s l p a l modulated i n frequency by a triangular wave. equal to at=2R
C
The r e t u r n wLve s:-lz-,e i s shifted i n time by an amount
(7)
If the two frequencies a r e difzerericed, the resuit is a wave represented by
the dotted function in Figure 5. ?he portion above the reference a b s c i s s a
axis h a s a height, i n the case of zero relative velocity between the r a d a r
and the target, proportional to the range along the beam. In the actual
situation, the doppler shift adis an additioriai frequency shift proportional to
Vz.
This, however, is easlly removed ir, the data processing circuitry
z
sirice V , i s itself obtainable by mixing two of the three doppler beam
f requeccies.
The outputs of the RAZIVS a r e Vx, V
Y'
V
z
and 3, i:; the form of ana-
log voltages.
The f i r s t two of k c s e a r e used to generate pitch arid yaw
s t e e r i n g commands and the Last x m fo: thrust z.cce:eration coxtrol.
The flight control s y s t e m :>:rick c..~,;--:xi showing all of the important elements active during the v e r c i c r C c s c e z t -,base i s shown in F i g u r e 6 . The
V a d V signals a r e used t o tor:*Le the stra??ec-down yaw and pitch ratex Y integrating gyros. The slant range 5 i g n d R is fed into a function generator
which generates a corresponding required velocity V twee:i V and V, is defined a s the veiocity e r r o r . Et L
R '
The difference be-
It is amplified and limited The output of a
b e f o r e becorning the thrust accolel-ation c c ~ . ~ . m x - . signal. d
lor,gitudinal z c c e l e r o m e t e r i s comsz,red w5t5 this signal and the difference is ever,:ually arr,ount.
used to raise o r lower L:-.e -*rx- ' u - L -
-
r
-L , .- t h r e e engines by the same s
Returning to the pitch and y a w channels, the r a t e e r r o r s a r e inte-
grated by the gyros, then amplified 2 n d fed i n t o a mixing network which a l s o
accepts the t h r u s t acceleration e r r o r signal.
The t h r e e iaputL: ?-J;?~~%'$I~.I.!+?&C
determine the t h r e e outputs which a r e the thrust commands to the engines. Finally, the vehicle dynamical geometrical relazions complete the feedbad<
;
t o the RADVS and the accelerometer.
e
111.
V E R M E R G U I 3A X Z E
C Z\ 3A C ‘EXIS TI CS i ‘
Basic Guidance Concept T h e b a s i c concept useC iz :;?e v e r r . i z r 2;cidance s y s t e m involves;
(1) gravity t u r n steering throughor;: the dczccnt immediately following RADVS
I C
acquisition of the velocity vector and (2) a minimum acceleration phase followed by guiding along a nominal Czscent contour o V v s R for the generation f of velocity e r r o r which in t u r n is used to control the t h r u s t acceleration.
The u s e o gravity t u r n steering has s e v e r a l outstanding advantages. f
First, because of the fact that the t h r c s t a x i s is required t o be colinear with the velocity vector which, as was shown e a r l i e r , is easily obtained in body coordinates, t h e r e is no requirement f o r the knowledge of another direction. In a n y other steering scheme, the knowledge of the direction of local v e r t i c a l
is generally n e c e s s a r y , either explicitly cr unexplicitly, r e s d t i n g in more
complex instrumentation.
SeconZly, the Zravity t u r n descent h a s a very
d e s i r a b l e property that,as the L-cLocity a p r o a c h e s zero, the flight path (thus a l s o t h r u s t direction) tends towards the vertical. maneuver i s needed p r i o r t o touchdown.
No violent reorientation
Thirdly, a gravity t u r n with suit-
able t h r u s t acceleration control i s a n eiiicient t r a j e c t o r y fuel wise for a wide
:’
range of initial flight path angles r z c g i a g from n e a r horizontal to vertical. In the t h r u s t acceleratio.-L c z z t r c l e k a z n e l , the use of a nominal V v s Fz descent t r a j e c t o r y perm-its near maximurriu:ilization capability, of the v e r n i e r thrust
The transition from Surnou: t o t;iis m o d e of l e s c e n t is via a min-
imum acceleration descent r a t h e r t’mn a complete shutoff of the v e r n i e r engines because of the r,ecessity si mzintaizing attitude stabilization for proper sensor operation.
l
Doppler and range a c q u l - -.c.,L , :.-e ZAS_3VSs anticipated during the .. , . i minimum acceleration phase. represented a s a point in t h e
is^>
>:ig
.
L)
;he x x a s u r e d state of the vehicle,
VC-GC-,:~-T~TL-,C
-
?:“,ase plane, is well above t h e
a i a level determined by the min-
descent contour, the acceleratiox i s cc,:stant
imum t h r u s t capability of the engines a s well a s the anticipated weight of the spac ec r a f t . The m a j o r portion of the de.-cer,t carLtouris a n approximation of the
c
idealized parabolic relation:
which, in the c a s e of v e r t i c a l desee;::, equal t o ng.
r e q - < i r e s a constant t h r u s t acceleration
5
Later it will be show:: .*hat fo;.
f a i r l y wide range of flight path
as well as average surface slope condition, the acceleration requirement does not differ appreciably f r o m ng.
This i s conslstent with the general d e s i r e t o
find a simple guidance law which c o d d coae with a wide range of anticipated conditions while, at the s a m e time, yield m a r l y maximum performance within the available thrust con s t r a i n t
.
The contour n e a r the origin ~f the R-V $ z x e deviates f r o m that given by Equation 8,
A constant low veloci:y d e s c ~ r . : :sha se i s incorporated which
s e r v e s t o a b s o r b altitude errors w5en the s2zzecraft velocity r e a c h e s this Trzlue. Finally, engize shutoff i s x:ade
w k e r , the r s z g e beam indicates a specific vL,l-ie
(different f r o m z e r o ) , in o r d e r t o avoid possible detrimental effects resulting f r o m the impingement of rocket e x l z z s t or, The surface.
Diifereztial Equations Referring to F i g u r e 7, t k e k;szc:
.
c.i;:srential
-.*e
equations f o r a gravity
turn descent with an a r b i t r a r y t b z u s t acceleration (not n e c e s s a r i l y constant) are
-+'= u c
c?v
v =-
-
a f g cos
I#
(9)
These equations may or may not y i LId c ; a ~ z i !form solutions depending on the nature of the accelerations o r indirectly, the p a r t i c u l a r longitudinal guidance law chosen. Xinimum Acceleration P h a s e During the minimum acceleraticn p h a s e which links the operation along the descent contour with m a i n - r e t r o burnoat, the behavior is v e r y simply solved by noting that in Equation (91
Thus, by dividing ( 9 ) by (10) w e o b ~ a l r ,
(12)
In the i n t e r e s t of conserving f u e l , n a i n is chosen t o be somewhat l e s s than
unity. for F o r @ lying somewhere S e t w e e n z e r o and 90" i t is therefor2 possible
o r ne:-;zL:ive.
dV - t o b e e i t h e r ?ositive
If initizily cos $is l e s s than n
min This t r e n d i n most c a s e s however is
V tends to d e c r e a s e with t i m e a t f i r s t .
dv
-
- 1
L-;
;C will
soon cause dV t o change sign. dt
r e v e r s e d because the decreasir,g value of
-
Integration of Equation ( 1 2) r e s u l t s in
-1
1
0
C \? rxin
i l
s e c 2& 2 sec
tan
2
2 Jro
2
where the subscript "0" denotes ini:iai v a l m s .
The factor
which may be called the "velocity integra?" i s ?lotted in F i g u r e 8 f o r v a r i o u s values of n mi n. o c c u r s at gmin zero. The Surveyor design presently h a s n = 0.9. As a r e s u l t min Jr = cos -1 nmin = 2 5 . 8 " , the ?aim a t which 3 passes'through dt The phase plane behavior JS illustrated i n F i g u r e 9 f o r values of $ on
0
either side of cos-' n min. Intersection with Descent Conto-zr
.
The minimum a c c e l e r a t i m phase i s terminated when the m e a s u r e d slant The altitude ha at this point i s a
r a n g e r e a c h e s a value given b y E5;iztion ( S ) .
" 7
function not only of velocity but a l s o of f l i g h t path angle
Va.
I
2
E , =
a
2(n
-
1)g
cos $ a
-
15
-
The exact point of intersection i s s a i v a k k ky the iterative f o r n u l a
1 2(1-n
min
)
111111
A
I
which m a y be used with (13) and (15;for t h e complete determination of the intersection point. In general, $ a
i s always less than
Jr 0 due
t o the effect of
0'
the gravity t u r n steering while V x a y b e either higher o r lower than V
b
In
the c a s e of Surveyor, t h e r e is a high probability that V
a
will not differ f r o m
SO
V o by m o r e than 50 feet p e r second or so, due t o the fact that nmin is
close t o being unity. Idealized Behavior Along Descent Contocr -
The idealized behavior along the Descerit Contour may be found by the following procedure. that
P
Noting f i r s t f r o x :he geometry shown in F i g u r e 7
R = k? s e c I $
we differentiate Equatior, (17; with respect t o time,
b *
0
R = h sec $ t h
) I
s e c $ tan
I$
1.-
.
But,
Also,
I .
h =
v
2(n- l)g
cos
Jr
T htl s
which, in view of Equation ( l o ) ,
comes
Now, if Equation (8) is differentlztcc, w:::? Fespect to time
7 -
Equations (22) and (23) a r e combined t o e ; i r A h a w R
0
V =
-
L
-
i 1) g + T s i r ,
L
)I t a n (I
-
1
If Equation (24) is divided by E q ~ a ; i m
* *
n-ie c a m m y variable time is
.
eliminated and we have the k e y t i f i e r e n t i d equation relazing velocity and flight path angle
i
which m a y be integrated to yield :>e general solution
- 17 a result differing f r o m the constant-aceeleratior, gravity-turn solution only i n the exponent of the secant tern?. The numerator of the right-hand side of Equation (26) is plotted in F i g u r e 10 for s e v e r a l representative values of n including the case of n = 2, v e r y nearly the Surveyor characteristic. here.
KO startling behavior is exhibited
It is seen that velocity d e c r e a s e s monotonically as
Jr
becomes smaller.
Of g r e a t e r interest is the t h r u s t acceleration required t o follow the
Descent Contour.
F r o m Equation ( 9 )
a = gcos $
-V
Q
*
But V is given by Equation (24), thus
a = ng
+ (COS
t
(I +-sin
2
$ tan
-
1) g = ng
+
(1 c o s JI)' 2 c o s Jr
-
Equation ( 2 8 ) shows that the requiyecl thrxsr zcceleration depends on Q only. The second term in the r i g h t - h a d side, pio:ti:d in F i g u r e 11, is the e x c e s s a c c e l e r a t i o n r e q u i r e d above the nominal v a l w of n. The interesting and
i m p o r t a n t property of this t e r m is its negligibly small magnitude compared t o g, for
I '
IJ ranging f r o m z e r o up t o even a s high a s 4 5 " . Tkcls, f o r all practical
purposes, the thrust acceleration required m a y be regarded as a constant. In the design of the Surveyor, the Descent Contour i s therefore made t o c o r r e s p o n d to a value of t h r u s t acceleration just slightly l e s s than the
maximum capability.
In the c a s e where a s u r f a c e s l o p
e x i s t s in the direction of travel,
the acceleration required m a y b e shown to 'be
a = ng
+
i
cos \ir
s T7 i n
1
1
tan (il;
- rJ.) ) is 1
g
(29)
.
where a positive
indicates a n =?hill approach.
Thus, the acceleration
negative.
required tends t o be greater f o r the dciwnhill c a s e where
It
is not difficult t o see f r o m physical interpretation that this m u s t be true. As the gravity t u r n bends the traSectory towards the vertical, the slant
range tends t o d e c r e a s e m o r e ra3idly in the downhill than i n the uphill sit-aation. Correspondingly in t h e I'ormer c a s e a higher t n r u s t acceleration
i s demanded of the system.
The choice of n, the nominal descent contour a c c e l e r a t i o n level, given that t h e r e is a maximum capability "slope-flight path" capability equation
T ,
max'
is determined by a
2
1) (!+2n
rxa x
COS
i- cos 2 (I)
- 4 (nmax 2
4 n rnax
- 1rr?aX
2 In
-
1) (2 nmax + c o s $)
F i g u r e 1 2 is a plot of G Lersiis $ f o r n
L -
= 2.35 and various values 'of.n.
It
n, the max acceleration m a r g i n between tk,c maximum capability and the norcinal descent a c c e l e r a t i o n (in the vertical c a s e ) . In the actual design, aliowance m u s t a l s o be made f o r s e n s o r and electronic e r r o r s which m a y be lumped together
a s a n equivalent d i s p e r s i o n i n n.
t u r n s out that the capability curve is a s t r o n g function of n
-
A margin of 0. 25 to 0. 3 0 lunar g is found t o be
r e q u i r e d f o r the Surveyor system.
ing r a t e
rg
From Equations (10) and (261,
Another interesting property of t h e t r a j e c t o r y is the manner the turn-
4 v a r i e s as V approaches
zcTo.
since V and
Jr
approaches z e r o together,
v+ 0
lim
: . 9
-
$! $jO
lim
‘
where c is a constant.
Thus, t h e r e iire three p o s s i b i l i t i e s :
1. 2.
If n < 2, $ a p p r o a c h e s zero
If n = 2, $ approaches a non-zero constant
If n > 2,
3.
f
diverges
The last possibility p r e s e x t s a problem a t l e a s t in the idealized s e n s e
if we a t t e m p t t o follow the gravity turn t o z e r o velocity.
In practice, how-
e v e r , two things can be done t o aLeviate the situation (a) use of a lower a c c e l e r a t i o n level near V = 0 o r , { S ) incorporate i n e r t i a l hold at some finite
l velocity and accept a s m a l l r e ~ l d u a horizor,ta? velocity component.
Both a r e
implemented in the Surveyor de ,Izn altr*ough not p r i m a r i l y f o r the above reasor. Straight-Line Approximation of De scent Contour Equation (8) is not exactly implemented i n the spacecraft. Rather, a
4-segment straight line approximation of this equation, shown ir, Figure 1 3 and actually furnished by a diode function generator, is used instead.
Because of the finite number of segmerLts,t'r..ere a r e some deviations i n behavior f r o m those characteri z i r L gthe ic!ez:j
zdd c a s e , the chief difference
being the presence of t h r u s t accsLs.-ation s;tcration periods when the system attempts t o follow the Descent C o r z o u r .
":-As 'sehavior, discussed i n
Reference 1, has the beneficial effect of shielding the longitudinal guidance channel f r o m r a d a r noise at l e a s t during most of the acceleration- saturated portions. However, towards the bottom end of each segment, close tracking
should and does occur. Taken a s a whole, the straight-A;np mechanization yields practically the s a m e r e s u l t s with r e g a r d t o f u e l cons-x-nption and flight path angle reduction as the idealized case. The actual t h r u s t acceleration command i s derived as shown in F i g u r e 14, with the resultant chayacteristic of Figure 15.
-.
(A
l i n e a r region of operation
3 .ax ,
V
The width.of the
I (
i s of the o r d e r of a few feet per second.
F i n a l Segment and Touchdown The operation along the f i n a l segxent to actual touchdown consists of s e v e r a l important f e a t u r e s w:~:chdese:ve some discussion.
: The f i n a l segment desigr, h a s i passing through the origin i n the
R
-
V plane a s shown in Figure 15.
However, the portion near the origin
is not used f o r guidance, a s it is ea5y to ;how that following a s t r a i s h t
line t o the origin i s not p!iysicaXy ~ c ~ ~ lZrom the ~ ; ~ ~ i z ~ fuel requirement
1 .
standpoint.
Instead, a constant-velocity (of nominally V
C
= 5 f:/sec)
This also means
descent subphase i s implemented prior t o engine shutoff.
that altitude d i s p e r s i o n s tzhich exist a t tke b e g i m i n g of t h i s sub?hase w i l l
produce a
*
21
A;--c,:?.~rciiscrete change i n
negligible effect on the landing veioc ixy.
implementation f r o m the p u ~ (,Y ..'.' :;--:zrA-':-.-.ade is the u s e of inertial attitude hold at a slightly higher velccity
1; 7
U
af say 10 f t l s e c , thus
effectively removing the r a d a r inpzts to the l a t e r a l channel a t these v e r y low velocities where r a d a r noise might produce jerky attitude changes. Simultaneously the velocity command is switched t o the constant value of
V
C'
When the vehicle state r e a c h e s P _ the t o p : of the final segment,
u
the flight path angle in general has b e e n reclzced t o a small enough value
(< 2 0 " )
SO
that s m a l l angle approximations x a y be used t o solve for the
T h e thrust acceleration is saturated a t
behavior f r o m this point onward.
f i r s t with a value equal to nmax g and the velocity-flight path relationship
is approximately
(33)
Also,
. (34)
P
At the point where the t h r i ; a t acce:$rctioc c x x e s out of saturation, the t r a j e c t o r y "acquires" the straight 1;ze z:-.-! b e g i n s t o t r a c k it. If b is defined
a s the slope of the segacnt, o r
b =- R V
(35)
satisfy the equation
22
-
i t m a y be noted f r o m Equation (34) that t h e acqcisition velocity V a m u s t
v 02 - va 2
Solving for Va
=2g(n
XXaX
-1) (bV
0
- bVa)
Va = 2bg
I
(2.rLiax1 )
-vo
(37)
V
a
m u s t i n practice be a value somewhat higher than Vb' the point at which
T h i s implies that b m u s t be
the commanded velocity is switched t o V C' l a r g e r than s o m e positive constar,t
On the other hand, it m a y be shown that the fuel requirement f r o m Va t o
Vb i s AV =
va - v L + b g i n -
V
a
%
(39)
T h e gravity l o s s term i s thus directly proportional to b , which m u s t not there-
-_
L*
f o r e b e excessively l a r g e . The value of b a l s o influences the spacecraft flight path angle a t V 5 ' Since ideally the attitude coincides w i t h t h e f:iz'nt path, this angle m u s t be kept small s o that the subsequent t h r u s t velocity Increment does not give rise t o a n e x c e s s i v e horlzontal velocity cs,x?oner,t.
- 23
by the following differential equatior,
-
During the "trackhg:"phase f r o m V,. t o Vb, the behavior is described c
the solution of which is
v
cos
Jr=
cos $ v c o s $a 1 tan 3, a 1 + In 1 tan 9 , %
I
v
F o r small angles, Equation (41) ,say be shown to 3e equivalent t o
/v-
- v',
bg
Thus, the value of b needs t o be la; evaluated at V
- B ir,
c,.-c!e: that the above ratio,
?CJSSIL?.
b
be as small as practica1;y
.- -
The above considerationslcz2 to a ckoice of b which is a compromise between fuel c o s t on the one hand and fixzl f l i g h t path off vertical.on the other. When the state r e a c h e s B with a ve:ac:ty
r *
of V b' the switching of the
commanded velocity t o a conskint v-.lue cf V C momentarily c a u s e s saturation of t h r u s t a c c e l e r a t i o n of the system, k a t w i t L , i a v e r y short altitude decrement the velocity of V c is essentially a:t,;ned.
T h e subsequent descent t o point C
r e m o v e s any initial altitude dispe~5sion except f o r that component which i s common t o the engine cutoff mechanization.
r-
r
'
I
Engine cutoff i s commanded w h z a :he Rk!.I)VS indicates a p r e s e t range
of Rc.
The choice of Rc (equal t o : 3 f e e t iz -;'ne actual mechanization) i s a
compromise between rocket exhzust i n t e r a c t i o n and touchdown velocity cons ide r a ti on s. The vehicle lands with a vertical velocity determined p r i m a r i l y by the cutoff altitude and secondarily by the actual velocity during the constant= velocity descent. Both these pazameters, aside from t h e i r obvious dependence
on the nominal settings, a r e cl=ieiiy determined by the RADVS measurement. Likewise, i n the l a t e r a l channel the horizontal velocity component is p r i m a r i l y caused by t h r e e sources:
1.
Residual attitude, differing f r o m v e r t i c a l due t o r a d a r noise effect, at V b'
2 Residual attitude due t o terminating gravity t u r n steering .
at finite velocity.
3.
Thrust t o RADVS misalignment.
The noise effect t u r n s out t o be the most im7ortant as shown by analyses.
- 25 IV. COXCLUDING REiMARKS
The principal c h a r a c t e r i s t i c s of the Surveyor Terminal Guidance System, discussed in the preceding sections, have shown that a simple concept with a resulting simple mechanization can be used to solve what might appear a t f i r s t to be a cornplzx problem. The fact that a number
of the characteristics of interes: can b e I s l r ; c &
in siinple, analytical: forms
is of enormous help in actually s2ecifyi:lg z u n e r i c a l performance
requirements f o r the subsystems.
It is a firA= belief of the author,
biased it may be, that a concept r e s d t i n g in mathematically simple behavior h a s a definite advantage over those which can only be analyzed numerically on a computer.
It i s fortunate that such a concept is
available f o r the design of the Surveyor.
S om en cl a tu r e
v l , v2‘ v3
Af,,
C
velocity along dopTfer beams 1, 2 and 3 ;e s;2ectively .
,“i-zcA;enc.; shifts corresponding to above
AfZ, A f 3
velocity oi light transmitted frequency
body-iixel Cartesian coordinates
:
-
f
0
x, Y, z,
vx’
vy’ v z
velocity aIa:ig the X, Y, and Z axes
bczm to t h x i a ; zxis angle
a
R
V
g
magnitude of velocity
l u n a r surfacz 51-zvitational attraction
n
nor*inal t h r u s t to lunar weight r a t i o
r h r u s t a c c e I e Tat i on
~
fligkt path angle, with r e s p e c t to v e r t i c a l
n
c-
min
t o lunar weight ratio during minimum ti: 111s t pha s
L-.Y;;s~
u
n max
C C
surface slope r?lasin=urr, t h r u s t to lunar weight r a t i o . nominal c a t oif velocity
n o x i n a l ciiz 03 aititude
I .
I
V
R
I
-
b
s:o?e of fixal segment
I
,
-
References :
1.
27
-
R. K. Cheng and D. - , C o n r z d : D?:aign Considerations f o r the I.
Surveyor Terminal S z s c e z t S y s t t m .
AIAA P a p e r No. 64-644,
Guidance and
presented at the .PL~.. -,‘lGK -4sr::orlynamics
Control Conference, L o s Angclos, California, August, 1964.
2.
R. K. Cheng and I. P i e f f e r : T e r m i n a l Guidance System f o r
Soft Lunar Laridii-,;,
Gzic
-
. -3. -.cc 2x2 Control, R. E. Roberson .-
and J . S. F a r r i o r , e d i t o r s , i c a d e m i c Press, New York, 1962. 3. B.A. Kriegsman z;nd M. XI. X e i s s : T e r m i n a l Guidance and Control Techniques f o r Soit L;;r,ar Landing. March 1962. 4.
W. G. Green: Logarithr-ic Savigation f o r P r e c i s e Guidance
of Space Vehicles.
ARS Journal,
I X Z T r a n s a c t i o z s on Aerospace and
Navigational Electronics, 2 i x e 1961.
5.
, .. S. J. Citron, S. E. aLinin, and E. F. :;!issing2:’:
.‘ m . - ~ A A i i n a l
., -*--.-’
Guidance Technique far S u a r Landing. M a r c h 1964.
AIAA Journal,
6.
I
.:
Z
n rn
n
0 13 ;
Z
0
9
-
n
Z
0
f
60,000
50,009
I
I
40,000
99?h D!S?ERSION ELLIPSE FGR
MAXIMUM MIDCOURSE
I
I
/--\
J
30 000
20 000
f
10,000
,
-
-
I
!
j
I
I
I
i
i
0
FIGURE 3. VERNIER DESCENT PHASE
ANTENKC, 2
FIGURE 4. RASVS XA?4 G2Ohr,fTRY
.-
.
1
.
FR EQUEN CY
f = RECElVED FREQU2NC
r
0-
-J
I I
1 :
I .
L J
I
-.
-
..
b
THRUST ACCELERATION
a
a
c.9.
OF VES!CLE
FIGURE 7 . G?AVITY
GEOMETRY
1.90
I
I
i
L
i
1.80
i
7 Y
,
i
i
1.70
1.60
=4a
N
u
Q,
v)
c I
1.50
c
0
Y c
1.40
II
v
s
0-
\
1.30
1.20
1.10
1 .oo
i
2
60
0
10
20
30
40
50
FLIGHT PATH OFF VERTICAL @,DEGREES FIGURE 8. VELOCITY INTEGRAL F R MINIMUM ACCELERATION PiiASE O
nmin = 0.9
FIGURE 9 TWO POSSIBLE ZEKAVIORS 3i-2 . ING MINIMUM ACCZLERAT!O,\: WASE
1.0
08 .
06 .
,
0.4
1
-
02 .
" 9 v
A
0.1
P
0.08
0 -06
i
i
10
;t I
0.04
i
0.02
0
20
c
-
\;J
15
50
6' i
FLIGHT ? A I S OFF VERT!CAL
+, DEG2IFE
FIGURE 10. VELOCITY INTEGRAL F R CONSTANT O
- V2/k
GRAVITY TUZN
l
e
v ,
w
n
I
-
I
t 4
e
w
v)
X
.
0
c
E 7
Z
0
v
I
03 0
9
cv
0
0
0
20°
1OD
. .
0"
-10"
-20"
-30"
-40"
-SO0
-60"
-70'
-80"
-90"
0"
10 "
20°
30"
4° 0
5 0 '
60"
70°
8° 0
9c
FL!GHT PATH O F VERTICAL,$ F FIGURE 12. SLOPE-FL:Gi-rT PATH CAPAE!LIN VERSUS NOMINAL CONTOUR ACCELERATION LEVEL F R 2.35 g ACCELERATiON L1,MIT O
("ma =
2.35)
R
RANGE
n
ACTUAL DESCENT CONTOG?
IDEALIZED DESCENT CONTO U R
I
I
I
V
VELOCITY
FIGURE 13. 4-SEGMENT APPXOXIMATlON OF PARABOLIC DESCENT CONTOUR
.
a D
0
n
70
C
5
m
L
P
z0
-4
r rn
G
n
>
Z
3 3 9 Z
0
8
3 m n
I
Z N
>
Z
z' 0
9
M
b
.
THRUST ACCELERATION COMMAND
- A
A
A
VELOCITY ERROR, qZ-V
FIGURE 15. THRUST ACCEtERAT!ON COMMAND CHARACTER iST I C
l
L
RO
i
,
i
s Z
< a i
pc
I I I I
1I
4
5
v)
1
I ! I I
!
I
1
I
I
I
Rb RC
0
VELOCITY V FIGURE 16. TRAJECTORY ALONG FINAL SEGMENT
I
I
vO