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5 



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IV. Spacecraft Control 

GUIDANCE AND CONTROL DIVISION 



A. Advanced Horizon Scon Platform, H. H. Horiuchi 

1. Introduction 

The horizon scanner, mounted on the servo-controlled 
platform of a spacecraft, senses the local vertical of the 
target planet in view. The angle between the optical axis 
of the platform and the local veitical constitutes an error 
signal to rotate the platform toward the local vertical 

The scan platform consists of a planet sensor, actuator 
electronics, a gear train and the spacecraft dynamics 
(Fig. 1). The spacecraft dynamics includes the eflFect of 
platform supporting boom and the motor mounting struc- 
ture. The content of this study includes a survey of basic 
performance requirements, system analysis, computer 
simulation, breadboard design and testing. The advanced 
planet sensor is being developer!, and should be available 
in fiscal 1968. The planet sensor was simulated with ana- 
log circuits. 

2. Survey of Syttom Roquiromonts 

The basic system requirements have been summarized 
below from the past advanced studies, the technical re- 
ports, and the requirements for the current project. The 
following specifications have been assumed: 

(1) The minimum slewing rate of the platform is 
1 deg/sec. 



N67-i9i^5 



(2) The platform tracks the local vertical of the target 
planet with an accuracy of ±0.5 deg. 

(3) The platform follows the velocity input up to 0.2 
deg/sec. 

(4) The planet angular diameter varies from 4 to 60 deg. 



SCAN PLATFORM 
A'2 BOOM DYNAMICS 
GEAR TRAIN 



INPUT 




MOTOR 



MOTOR MOUNTING 
STRUCTURE 
DYNAMICS 



SPACECRAFT 



Fig. 1 . Scon ploff orm tysttm block di;igram 



JPL SPACE PJtOGilAMS SUMMARY 37-44, VOL. IV 



37 



INPUT 
O— - 



SENSOR 



/ 


/ 



i4r, 5 



A'2 r- 



5(5r^+i) 




SENSOR COMPENSATION AMPLIFIER 

DEAOBAND 



FEEDBACK 



MOTOR 



GEAR 
TRAIN 



BACKLASH 



Fig. 2. Platform functional block diagram 






82 ^9 

5^ + — 5+ — 

^2 ^Z 



BOOM DYNAMICS 
— ^- OUTPUT 



(5) The platform jitter \% below 500 ^rad in track modes 
of operation, 

3. System Description 

The system functional block diagram i% shown in Fig. 2. 
The planet sensor is shown as the summing point of the 
feedback system. The pulse-width mc 'ulated error signal 
from the sensor is integrated, amplified and drives the 
servo system. The boom structural dynamics is shown as a 
second-order resonance in the feedback loop. 

The sensor consists of two halves of semicircular scan- 
ning device, each of which has a view angle of 60 deg, and 
is made of discrete thermopile elements. The maximum 
sensor resolution then is determined by the angular sepa- 
ration between the two consecutive elements. As an elec- 
tronic sweep scans each element at 10 cps, the illuminated 
element (planet in view) produces a pulse. 

The first pulse produced in each sweep closes a sum- 
ming gate, which in turn produces a step output This 
step output remains constant until the end of each sweep. 

The step outputs from the two sensors are summed in 
the summing amplifier *o produce a pulse whose width is 
proportional to the diff'Tence between the two step out- 
puts. Hence the pulse width is proportional to the error 
angle between the platform and the planet local vertical. 

The sensor design approach with the discrete thermo- 
pile elements seems to provide several advantages over 
t!ie linear optical sensors. With this approach, the sensor 



gain may be kept constant over the wide range of inpul 
light intensity and also virtually eliminates the thermal 
drift from the sensor. 

4. System Analysis 

The Bode diagram of the system shows two struc- 
tural resonances (Fig. 3). The lower frequency resonance 
(15 rad/sec, damping factor of 0.07) represents the boom 
dynamics, and the higher frequency resonance (94 rad/sec, 
damping fa'^tor of 0.01) represents the motor base struc- 
tural dynamics. 

The preliminary system stability analysis was per- 
formed using a modified Bode diagram in which the 
normal loop gain was adjusted to accommodate the gain 



40 



20- 



a -20 

t 

2! -40 

I 

-60 

-80 
-KK) 



-FILTER BREAK 
I / FREQUENCY 




MOTOR BREAK 
FREQUENCY 

MOTOR 
MOUNT 
RESONANCE 



10'* I0-' 10® 

FREQUENCY, cps 

Fig. 3. Scon plotform iytt^m led« plot 



lO* 



38 



»l SMCE PffOGtAMS IMIAIAMCi 37-44, VOL IV 



variation as a function of signal level. The motor break 
frequency was approximately 4 cps, and the filter break 
frequency was set for 0.5 cps. 

a. Initial simulation. The control system shown in Fig. 2 
was simulated for dynamic response using the DSL-90 
program on the IBM 7094 digital computer. In this simula- 
tion, stiction was not included assuming that the motor 
delivers sufficient torque to the load. 

A permissible range of backlash was investigated to 
meet the noise requirenients. The eflFect of the backlash 
to the system noise is summarized in Table 1. The noise 
level is proportional to the amount of backlash introduced 
in the range of backlash considered, and the frequency of 
oscillation is the boom resonant frequency. 



Table 1 . Effect of backlash to scan system noise 



Backlosh. rad 


±0.0087 


±0.0017 


±0.00017 


+ 0.000017 


Noise, rod 


±0.02 


-+- 0.004 


±0.00035 


-+- 0.00004 


Noise frequency, 
rod/sec 


15 


15 


15 


}5 

1 



Base dynamics transfer function. 






and T is the applied torque. The values of the constants, 
i.e., the ^'s, ft\, ys and 8*s, are determined by the fs, Ks 
and B*s in the diagram. Because of a comparatively large 
assumed inertia of the spacecraft (170 slug-ft-), the effect 
of the spacecraft motion was neglected in the analysis. 

By using the RTL-90 program on the IBM 7094 digital 
computer, the system root-locus plots were obtained to 
study the transient and stability characteristics of the 
system. 

5. System Test 

a. Initial breadboard test. The primary purpose of this 
breadboard test was to determine the maximum jitter rate 
during the slew a!)d null mode when the control system 
is driven by a pulse-width modulated error signal. The test 
circuit is shown in Fig. 5. It was found that the tachometer 
feedback alone was not sufficient to smooth out the input 
pulses through the motor due to a relatively small motor 
time constant. 



I 



The tracking capability of the system was investigated 
by applying a ramp input of 0.0034 rad/sec. The system 
responds to the ramp input with the average error of 
0.0024 rad and with the maximum oscillating amplitude 
of ±:100 jLtrad. The frequency of oscillation is the boom 
resonant frequency. 

b. System analysis with structural dynamics. The struc- 
tural dynamics was included in the scan platform control 
system simulation. The modified section is shown in Fig. 4. 
The stiction, rolling friction, and backlash are also in- 
cluded in the diagram. The boom dynamics block is placed 
in series with the feedback loop, while the base dynamics 
block is placed in parallel with the loop. 

The structural dynamics was investigated by using a 
model which consists of three inertial wheels constrained 
by springs and dash pots (Fig. 1). The transfer runctions 
of the dynamics blocks were found as follows: 

Boom dyruimicM tramfer function^ 



T \S^ + PtS + pj S» 



A filter stage was provided for this purpose. The jitter 
amplitude referred to the platform was approximately 
±160 /Arad/sec at the motor threshold voltage. This indi- 
cated that it might be possible to achieve a low jitter 
operal-on using pulsewidth modulation. 

b. Transient test on the breadboard. The characteristics 
of the planet sensor were simulated using analog circuits 
as shown in Fig. 6. The planet sensor consists of a saw- 
tooth generator, two Schmitt triggers and a summing 
amplifier. 

The error signal is generated by manually rotating the 
potentiometer shaft for a desired error angle. The unbal- 
ance in the potentiometer position causes the Schmitt 
triggers to trigger at a different input voltage level, thus 
generating the error signal. The servo loop will move the 
potentiometer shaft back to its original position. The 
sensor characteristics as a function of potentiometer posi- 
tion are shown in Fig. 7. In this model, the sensor gain 
is approximately 10 V/deg within d:l deg range horn the 
null. The transient error response of the system was ob- 
served at test point 7T-1. The boom structural resonance 
was simulate! by a second-order resonant circuit. The 
general patterns of the transient response are shown in 
Fig. 8. 



jn sMCf nooMMs summahy 37-44, vol. fV 



39 



m. 



AMPLIFIER 



MOTOR 



r 










1 



FRtCTION 



BOOM 
DYNAMICS 



FEEDBACK 



<x 



BASE 
DYNAMICS 



DISTURBANCE 



.rr 



Fig. 4. Sean platform modd with dynamics 



W 



SCHMITT 
TRIGGERS 



ULTER DEAD9AN0 

AMPLIFIER 




Fig. 6. Scon platform system simulation 



UJ 

Q. 

-2 



/ 



A 



MOTOR 

SATURATION 
LEVEL. 10 V 



EXPANDED VIEW 
OF DEADBAND 



12 3 

ERROR ANGLE, deg 



PULSE 
GENERATOR 



400 cp« OSCILLATOR 

• f REFERENCE 



INTEGRATOR 



MODULATOR 



POWER 
AMPUFIER 



T\ 



W'NDING 

TACHOMETER 
OUTPUT 



or* 



MOTOR 

Fig. 5. Puisowidth moduiotion tost circuit 



40 








/- 


c "^ 






3U 
O 9A 






/ 










I 

rt lA 




A 


/lOTOR S 
LEVEL. 


ATURATI 


ON 






"2 lU 




/ 


tOv 














/ 

/ 




t 










-20 


/ 















-5 



5 K) 15 20 29 30 

ERROR ANGLE, deg 



Fig. 7. ChoroctorisNcs of simukitod oionot sonsor 







40 



in »ACB riOGIAAIS SUmUMMY 91-44, VOL IV 



- 



o 
q: 
tt: 

UJ 



.DEADBAND, 
iO 25 deq 




r 



WITHOUT BOOM 
RESONANCE 

AMPLIFIER 
SATURATION 
LEVEL, -10 V 



WITH SOOM 
RL'.ONANCE 



TIME 



SYSTEM TRANSIENT RESPONSE 



. ^ 



Z' -tif-ft: 




ii. ' 




> 



30 



Uj" 20 

Q 



10 




RESONANCE SIMULATOR 
CHARACTERISTICS 



Fig. 8. Platform breadboard test results 

The major souice of error (other than stiction) in point- 
ing accuracy comes Vom the planet sensor itself. The 
selected deadband o* ±0.25 deg corresponds to ±2.5 
volts at the ii^ut to the servo-amplifier. Therefore, all 
drift and offset voltages referred to the amplifier input 
(±8.5 mv maximum) becomes negligible. The circuit 
design ivas considerably simplified by using integratet^ 
circuit amplifiers. 



B. Calibration Procedure Us^d in the Attitude 
Control Nozzle Thrust Measuring Technique, 

J. 0. F^n^ra 

The intent of this report is to update a previous report 
entitletl "Attitude Control Thrust-Nozzle Measuring Tech- 
niques" by J. C. Randall (SPS 37-39, Vol. IV, pp. 4(M2). 
The cantilever beam and strain gage appkt)ach, as detailed 
in Randall's report, has been attempted. A flattened rod 
immersed in a pool of mercury has been added behind 
the valve nozzle combination in order that the amount 
of damping in the system can be easily varied (see Fig. 9). 
In using the strain gage approadi, considerable damping 
IS required to measure steady-state thrust levels; a mini- 
mum ount uf damping is required in determining the 
impulsi. bit as outlined in Randall's report. 



Fig. 9. Micrometer mounted in front of thrust nozzle 

The strain gage approach presented the following three 
problems: 

(1) The gages weve found to be very sensitive to 60- 
cycle noise and vibrations from the mechanical 
pump used in the vacuum station. 

(2) The gages were found to be sensitive to various 
temperature and pressure changes in and out of the 
vacuum jar which resulted in a zero shift, making 
any calibration procedure diiBcuIt. 

(3) The gages were found to be slightly sensitive to the 
flow of nitrogen gas in the cantilever beam feed 
tube. 

Because o' the problems encountered with the strain 
gages, a photoelectric pickotf was installed. A square- 
edged nlate was mounted to the thruster body such that 
motion of *he beam results in a varying amount of light 
passing thiMUgh a slot immediately above the photo- 
sensitive cell. This technique appears to work very well 
and is insensitive to the problems encountered with the 
strain gage. The photoelectric pickoff that is currently 
being used, however, is not shown in either Figs. 9 or 10. 

We are presently calibr^^ting the system using the fol- 
lowing method: By mounting a micrometer to the frame 
immediately in f ont of the nozzle cone (see Fig. ^J) and 
by advancing the micrometer in known increment's and 
recording the ^'oltage output of the photocell, a curve of 
thruster displaceirent versus output voltage can be gen- 
erated. A stainless stc^I ball is then suspended in front 
of the nozzle from a mici onieter attachment. This is shown 
in Figs. 10, 11. 1 r ^hxm th^ microui^ier is moved a distance 
X«, the beam, in equilibrium \\ith the applied r»rce, will 



JH SMCf rtOOMMS SUMMAHY 37-44, VOi. tV 



41 



I * 



^ \ 



■ ¥ 



^ % _ .^ 









rw 



r^\ 








Fig. 10. Sf«el ball suspended in front of thrust nozzft 

deflect an amount Xh\ Xj^ is determin(-d from the volt- 
age output versus thn^ster displacement curve described 
above. A study of Fig. 11 indicates that: 

f = (m)itane)=-^p^ (1) 

m ~ weight of ball, gm 

x~ \Xm "Xt|, in. 

I = length of wire plus one-half 
the diame* T of the ball, in. 

/ " applied force, gm. 

Since / is from 2 to 3 in. and x is on the order of 0.001 to 
0.003 in., / can be accurately determined by a simplifica- 
tion of Eq. (1): 



/- 



M(x) 
(0 



NEW POSITION 

ZFPO POSITION 




THRUSTER 



(2) 



Fig. 11. B«c<n diftploc«m«nt calibration proccduro 
for moasuring nozzle thrust 

Using Eq. (2) and the curve mentioned above, a curve 
can be plotted of the photocell voltage output versus the 
applied force (//, which is equivalent to the force pro- 
duced by a firing of the thruster. With this method, the 
thrust level can be very accurately determined. The im- 
pulse bit is determined a< outlined in Randall's report 
(SPS 37-39, Vol. IV, pp. 40-42). 

When this cahbration procedure is perfected, a test 
program will begin using Mariner type thrusters to verif> 
the results of a computer pro-am described in the SPS 
report, "Parametric Analysis for Steady-State Performance 
of AaituJ*? Control Thrusters," by J. D. Ferrer** and 
P. M. McKown (SPS 37-42, Vol IV, pp. 47^8). 



42 



in SMCE mOGRAMS SUMMARY 37*44, VOL. IV 






C. Generalized Cruise Gas Requirement Study, 

G. Paine 

1. Introduction 

The aim of this study is to provide an estimate of the 
cruise gas requirement for a general spacecraft as a func- 
tion of spacecraft weiglit. If a setter estimate is required, 
then the craft configuration must be specified since cruise 
gas consumption depends heavily on botli the spacecraft 
weight and configuration. This study can be used m an 
evaluation of the tradeoffs between various gas systems 
as well as of the tradeoffs between communication rates 
and powers and antenna pointing accuracy. 

The general spacecraft chosen has a cylindrical gf^om- 
etry (Fig. 12), a uniform mass density, and no solar panels 
The addition of solar panels can be allowed for easily; 
however, their addition at this stage complicates the 
design unnecessarily as their size depends on the power 
requirements and the mission trajectory. 

There are several types of gas attitude control cysi. ms 
that can be considered: cold gas, warmed ^as, and hot 
gas. The use of a dual thrust level system ran save con- 
siderable gas in a large spacecraft by providing a low Igp 
during cruise and a high I^^ during spacecraft maneuvers. 
The use of heaters in a cold gas system fulfills these 
requirements as the heaters may be turned on for space- 
craft maneuvers and left off during cruise. Such a system 
would be relatively simple and fail-safe. 

The attitude control system analyzed is a cold gas sys- 
tem in which the jets are sized for some angular accel- 
eration constant. The constant is determined by the 
maximum allowable initial acquisition time and the mid- 
course maneuver times. A dual level system was not ana- 
lyzed as they are not employed on current interplanetary 

0' WEIGHT DENSITY 



/? = 





spaciXTift. Likewise the minimum on-time for a jet was 
held at current levels, rather than being reduced to lower 
the cruise gas rt^quirements in a large spacecraft. 

The disturbance torques acting on the spacecraft have 
been lumped into two classes: In the first are those result- 
ing from valve leakage. The second includes all other 
sources, such as imbalanced solar pressure and gravity 
gradients. The second must be estimated for any particu- 
lar spacecraft configuration and mission. 

2. Conclusions 

The cRiise gas consumption for a 200-day mission is 
displayed in Fig. 13. These results include triple redun- 
dancy but no tankage factor. The parameters not appear- 
ing on the figure are from Mariner IV. The results can be 
scaled directly for missions lasting other than 200 days. 
The curve for zero disturbance torque limit cycie gas 
consumption can be scaled inversely to the limit cycle 
deadband. Less gas would be required with the addition 
of solar panels and with the attitude control jets located 
at their extremities since both the constant disturbance 
torque and the zero disturbance torque gas consumptions 
are inversely proportional to the jet lever arm. 

It is apparent that for the spacecraft geometry and 
other parameters assumed that cruise gas consumption 

)02 



10' 
^. 6 

I ' 

uj 

6 

4 

2 

I0-' 



























/ 


y 


























/ 


/ 


/ 


























/ 


/ 






CONSTANT DISTURBANCE 

SUM OF ALL THREE AXI 

1 ... 


TORQl 
ES-^ 


JE. 






/ 


/ 










^0 


dyn 


e- 


en 


1 


/f 






/ 


9 


/ 








^^ 


^0 






*-, 


^d 


/ / 




/ 




/ 


, 






- 








"S 


u 




/ 




/ w 


ORST-CASE 

VALVE 

LEAKAGE—! 


..^^ 


^ 


-^ 




.^ 


./ 


/ ^ 


/ 


^ 


/ 










^ 




S^ 


y^ 


/ 








^ 












A 


V 


r 


^ 






.. 6 cc/hr 

L ' ' 












/ 


y 












5 cc/hr J 












/ / 


\ 










^' 
















/ 


X 


"^ ZERO DISTURBANCE 






»/ 


»'4 


> 

mr 


/ 


/ 


A 


1 1 1 
*— TORQUE, *=l TO >t«4 











1 


/ 

























10* 2 4 6 10* 



4 6 10^ 



4 6 10* 



Fig* 12. Spoc^craft whost configuration 
\% o hollow cylindor 



SPACECRAFT WEIGHT, lb 

Fig. 13. Cruiso gos consumption during 
200-doy mitiion 



in SMCf UtOQUAm SUMHHAItY 37-44, VOL IV 



43 



depends more on disturbance torque level than it does 
on the zero disturbance torque situation for spacecraft 
weighing less than about 2000 lb. 

If the non-leakage disturbance torques are unspecified, 
it is reasonable in the light of the disturbance torques 
encountered on Manners II and IV to allow 2.0 lb of 
gas for cruise purposes for spacecraft weighing less than 
2000 lb. This is equivalent to about a 75 dyne-cm distur- 
bance torque on a 500-^b craft without solar panels or 
about 225 dyne-cm on a 500-lb cratt with them. On a 
200-day mission, an additional 0.75 lb of gas will cover 
the worst-case valve leakage. 

For missions where either the spacecraft is heavy or 
where the disturbance torques are expected to be small, 
the zero disturbance torque gas -^onsumption will be the 
dominating factor and the gas consumption will vary 
according to spacecraft weight raised to the 4/3 power. 

3. Ga$ Consumption 

As the spacecraft studied is not equipped with solar 
vanes which passively control the spacecraft, the gas con- 
sumption will depend directly on the disturbing torque. 
For a constant disturbing torque, Fig. 14 shows the rela- 
tionship between the magnitude of the torque and the 
weight of the gas consumed for a spacecraft which oscil- 
lates inside a symmetrical deadband. 

To the left of ro, the marginal torque in Fig. 14, the 
spacecraft oscillates and touches both sides of the dead- 




band. To the right, the spacecraft touches only one side 
of the deadband. 

If the maximum (worst-case) disturbance torque allowed 
is greater than ri, then the gas consumption is determined 
totally by the disturbance torque and the mission dura- 
tion. If the maximum disturbance torque is less than tu 
then the maximum gas consumption will occur with no 
disturbance torque present, and the gas consumption 
depends on a variety of vehicle parameters. 

4. Disturbance Torques 

A number of disturbances act on the spacecraft. They 
include externally caused torques such as those arising 
from solar pressure and gravity as well as internally gen- 
erated ones from valve and piping gas leakage, as well as 
other sources. 

Each disturbance torque can be related to a wide range 
of parameters. Here the parameters which depend on 
vehicle weight are called out exphcitly while the others 
are held constant. So, for instance, the solar torques will 
be taken as proportional to vehicle area, or vehicle weight 
raised to the 2/3 power, neglecting the many other factors 
influencing solar torques. Valve leakage, which can pro- 
duce a torque is taken as a worst-case constant, even 
though it varies from valve firing to valve firing. The 
same gas consumption will arise whether all twelve valves 
are leaking or whether only six are leaking and the other 
six must be fired to produce a restoring torque. 

The gas weights required to counterbalance each of 
these is shown in Fig. 15. The constant disturbing torque 



^^ 



X 
UJ 



5 




ZERO DISTURBANCE 
TORQUE LIMIT CYCLE 

DISTURBANCE TORQUE 
PROPORTIONAL TO AREA 



DISTURBANCE TORQUE 
PROPORTIONAL TO 
JET LEVER ARM 

CONSTANT DISTURBANCE 
TORQUE 



DISTURBANCE TORQUE AMPLITUDE 

Fig. 14. Mats of consumod got vorius 
disturbonco torquos 



SPACECRAFT WEIGHT 

Fig. 15. Cruiso gos consumpHon vortus spococraft 
wtiglit ond disturbonco torquo typ« 



Jn SMCf FROGMMS SUMMAItY 97^44, VOL IV 



r/ 



I" 



and the special case of zero disturbing torque gas con- 
sumptions are also included. In each case the counter- 
balancing torque is proportional to the lever arm, or 
spacecraft weight raised to the 1/3 power. Thus in the 
example of an area-dependrnt torque, the gas required 
increases only as the 1/3 power of the spacecraft weight 
although the disturbance increases by the 2/3 power. 

From Fig. 15 it can be seen that there are two distinct 
regions: one where the disturbance torques control the 
gas consumption, and another where the zero di^^turbance 
torque gas consumption dominates. The weight of the gas 
required depends in the former region on the predominant 
disturbance and will vary with spacecraft weight accord- 
ing to the nature of the largest disturbance. In the former 
region, the worst worst-case valve leakage gas consump- 
tion adds directly to the other gas weights. 

5. Mathematics of Gas Consumption 

As has been shown in many other places (Ee*^* 1 3 
and 4)^ the weight of the gas consumed depends on :». . e.al 
simple formulae,— making no allowances for either the 
triple gas redundancy or tankage weight factors. The 
formulae below give the gas consumption for each axis: 
The terminology is listed in Table 2. 

a. Zero disturbance torque gas consumption: 

I(aAt)H„, 



Table 2. Nomenclature 



Wo = 



4LI,,0oB 



b. Marginal disturbance torque gas consumption: Mar- 
ginal torque is the smallest torque for which the attitude 
control system touches only one side of the deadband. 



Wo = 



_ I(<xAt)H„ 



c. Torque balance gas consumption: 



Wo = 



u 



ep 



d* Largest disturbatuse torque wUh gas consumption 
equal to zero disturbance torque gas mnsumptiow 



4$, 



JDB 



'Also by H. K. Bouvier: A Field Guide to LimU Cycles, and 
Mariner TV Disturbance Torques, JPL-IOM 344*543. 



jn SMCI nOOHAm summary 37*44, VOL. IV 



Wo 


gas weight 




I 


moment of inertia of spacecraft 




L 


gas jet lever arm 




a 


acceleration constant 




M 


minimum on time 




tm 


mission time 




h. 


specific impulse 




OpB 


deadband 




To 


marginal torque 




Tl 


unbalance torque resulting in the 
sumption as no disturbing torque 


same gas con- 


P 


spacecraft weight density 




k 


spacecraft length to radius ratio 




g 


gravity 




We 


spacecraft weight 




R 


spacecraft radius 




I 


spacecraft length 




TD 


disturbance torque 





6. General Spacecraft Configuration 

The general spacecraft was chosen to be a hollow cylin- 
der (Fig. 12) with a uniform mass density. The gas jets 
are placed so that the roll jets have a lever arm (L) equal 
to the radius (R), and the pitch and yaw jets so that they 
have a lever arm equal to the greater of the radius (R) or 
half the vehicle length (/). In the following functions, k is 
the ratio of spacecraft length to radius, and / is the inertia. 

a. Pitch and yaw axis functior^. 

ForaUJfc 

3g U "^12/ 3g Vl6^12/ 



or 



, Wc(4WcY'/S fc'X 
^""4gV3W \4"^3; 



45 



Forfc^^2, L ^ i?, and 



I Spkir 

L" 16g 






For it > 2, L = //2, and 



L~ 8g Wfc-/ \4 3y 2% ^4"^ 3/ 



h. Roll axis function. 



/- 



2WrR' W,R' 5\V,R2 



3g 



24g 



8g 



For anv /c, L = R, and 



I _ IS^fcTT / 4W^Y '^ ^ 5W,R 
L ~ 32g VSiSfcTr/ "' 8g 



These factors, which enter directly into the zero dis- 
turbance torque gas consumption, are plotted in Fig. 16 
for some typical parameters: k = 1 and /S ~ 15 Ih/iV On 
Fig. 16, the actual (I/L) values are plotted for Mariner IV 
and for a proposed Venus- Mercury 1973 craft. The diflFer- 
ences are explained by the solar panels on these two craft. 
While the solar panels increase the inertia per unit space- 
craft weight significantly, they increase the jet moment 
arm even more rapidly, giving rise to a net reduction 
in i/L. 



7. Gas Weights and Results 

Even with the concepts stated earlier, more assump- 
tions are necessary before the gas requirements can be 
set down explicitly. Some of these concern the vehicle 
geometry, others concern the jet size and minimum 
on-time, while still others relate the disturbances and 
deadbands of the three axes. 

The gas weight shown is for three-axes stabilization 
and includes the triple redtmdancy factor but does not 
allow for tankage. The gas weights should be multiplied 
by 2.6 to obtain gas weight plus variable tankage weight. 



10* 

6 
4 



105 

s 

4 



10* 

6 

4 



10' 

6 

4 



lOO 

























, 


^ A 


r 


























/ 


/ 


























roll/^ 


"^ITCH, 






















A 


4- 


YAW 






















/ 


// 


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I02 2 4 6 10^ 2 4 6 10* 

SPACECRAFT WEIGHT, lb 



4 6 10' 



Fig. 16. Ratio of inertia to lever arm length 
versus spacecraft weight 

The gas weight factor to account for half-system perfor- 
mance is L5 rather than 3 in the case of valve leakage. 

In Fig. 13, gas consumption is shown as a function of 
spacecraft weight for A: = 1 to fc = 4, three levels of con- 
stant disturbance torque, two levels of gas leakage, and 
two deadband widths. 

In Fig. 13 the following additional constants, taken from 
Mariner IV (see Ref. 4), are assumed: 

Odb = same for all three axes 

a = 0.45 mr/sec* 
M = 0.02 sec 
hp " 60 sec 
1.1 = 200 days 

i8 = 15 lb/ft> 



46 



jn sPACff nooMm suiMAuir 97-44, vol iv 



References 



1. Project Report 4290, Barnes Engineering Co., 1966. 

2. Nicklas, J. C, and Vivian, H. C, Derived-Rate Increment Stabilization, TR 32-69, 
Jet Propulsion Laboratory, Pasadena, Calif., July 31, 1961. 

3. Turk, W., Ranger Block III Attitude Control System, TR 32-663, Jet Propulsion 
Laboratory, Pasadena, Calif., Nov. 15, 1964. 

4. Mariner-Mars 1964 Project Report: Mission and Spacecraft Development, Vol. I, 
TR 32-740, Jet Propulsion Laboratory, Pasadena, Calif., Mar. 1, 1965. 









m SMCC PROORAMS SUMMAItY 37-44, VOL IV 47