BALLOON-BORNE THREE-METER TELESCOPE FOR FAR-INFRARED 

AND SUBMILLIMETER ASTRONOMY 

' The Smithsonian Astrophysical Observatory 

is a member of the 

L Harvard-Smithsonian Center for Astrophysics 

Grant NAGW - 50 9 

Semiannual Status Report No. 6 

For the period 1 March 1986 through 31 August 1986 

Principal Investigator 

Dr. Giovanni G. Eazio 

December 1986 

IELZSCOPE (hRS A-CE-1733 FOR 45) FAR-INFhAEEG L ALLCCN-EGG8 APL E SUEMILLIMETER 3-YETBR N87- 14220 

AEXEONORY 

Ciservatcrp) 53 F CSCL 03A G3/89 43612 

Semiaricual Status Fegcrt, 1 Bar, 

- 31 Aug. 1966 (Snithscnian Astrophysical Ui1clas 

Prepared for 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

Smithsonian Institution 

Astrophysical Observatory 

Cambridge, Massachusetts 02138 

I 

The NASA Technical Officer for this grant is Dr. Nancy Boggess, Code EZ-7, 

National Aeronautics and Space Administration, Washington, D.C. 20546 
BALLOON-BORNE THREE-METER TELESCOPE FOR FAR-INFRARED 

AND SUBMILLIMETER ASTRONOMY 

Grant NAGW -50 9 

+/- 

Semiannual Status Report No. 6 

For the period 1 March 1986 through 31 August 1986 

Principal Investigator 

Dr. Giovanni G. Fazio 

December 1986 

Prepared for 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

Smithsonian Institution 

Astrophysical Observatory 

Cambridge, Massachusetts 02138 

The Smithsonian Astrophysical Observatory 

is a member of the 

Harvard-Smithsonian Center for Astrophysics 

The NASA Technical Officer for this grant is Dr. Nancy Boggess, Code EZ-7, 

National Aeronautics and Space Administration, Washington, D.C. 20546 
CONTENTS 

Page 

SECTION 1.0 INTRODUCTION .................... 1 

2.0 BALL-BEARING GIMBAL DESIGN AND ANALYSIS ...... 2 

2.1 Candidate Design Evaluation ........... 2 

2.2 Ball-Bearing Gimbal Mechanical Design ...... 3 

2.3 Ball-Bearing Gimbal Servo Concept ........ 5 

2.3.1.2 Selected Components ............... a 

2.3.1.3 Full System Model ................ 12 

2.3.2 Azimuth Servo (Coarse Loop) ........... 20 

2.3.3 Summary ..................... 21 

3.0 SYSTEM ERROR ANALYSIS ............... 23 

3.1 Configurations .................. 23 

3.1.1 Co-Aligned Star Tracker ............. 23 

3.1.2 Beamsplitter Feed - Focal Plane Star Tracker ... 24 

3.1.3 Instrument Star Tracker ............. 25 

3.2 Error Sources .................. 25 

3.2.1 FixedErrors ................... 25 

3.2.2 Operational Errors ................ 26 

3.3 Determination Of The Relationship Between Secondary 

Misplacement And Image Motion .......... 29 

4.0 RESEARCH PLANNED FOR NEXT REPORTING PERIOD ..... 31 

2.3.1 Inertial Mode (Fine Position Servo) ....... 6 

2.3.1.1 Description of Fine Position Servo System .... 6 

2.3.1.4 System Response ................. 17 

APPENDIX A Pointing Control System Servomechanism Analysis 

F I GURE S 

4 

2-2 Nonlinear System Block Diagram ........... 13 

2-3 Fine-Pointing Control System Response to a Sinusoidal 

Command Signal .................. 18 

2 -4 19 

FIGURE 2-1 Gimbal Assembly .................. 

Telescope Point Response to a SxIO-’ Nms Torque Pulse 

2 -5 Block Diagram of Azimuth and Cross-Elevation Servos 22 

i 
CONTENTS (Cont . ) 

TABLES 

Page 

TABLE 2-1 

3-1 

Torque Sensor Specifications ............ 11 

System Error Performance for Three Star Tracker 

Configurations .................. 20 

ii 
Page 1 

1.0 INTRODUCTION 

This is the sixth Semiannual Report submitted under Grant NAGW-509 for 

the development of a Balloon-Borne Three-Meter Telescope for Far-Infrared 

and Submillimeter Astronomy. It covers the period 1 March 1986 through 

31 August 1986. 

The Three-Meter Balloon Borne Telescope is a joint program of the 

Smithsonian Astrophysical Observatory (SAO), the .University of Arizona and 

the University of Chicago. 

Effort during this reporting period focused on studying and revising 

the gimbal design to eliminate the alignment and limited rotation problems 

inherent in the flex-pivot design. A new design using ball-bearings to 

replace the flex-pivots has been defined and its performance analyzed. An 

error analysis for the entire gondola pointing system was also prepared. 

Arizona continued its mirror development and test program using mirror test 

blanks from Dornier and other sources under separate funding and will 

report on its effort independently. SAO, Arizona, and Chicago held 

telephone conferences during this period to coordinate activity and discuss 

technical issues. 
Page 2 

2.0 BALL-BEARING GIMBAL DESIGN AND ANALYSIS 

2.1 Candidate Design Evaluation 

Two types of gimbal designs have been evaluated. The first is the 

baseline flex-pivot design proposed last year in the Preliminary Design 

Report (PDR)*; the second is the active gimbal concept proposed in the 

last Semiannual Report (#5, April 1986) which uses ball-bearings to support 

the telescope. 

The advantage of the flex-plvot designs were detailed in the PDR. 

Further examination of the concept revealed several desJgn weaknesses all 

of which relate to the limited rotation range of the flex-pivot. A long 

period of undisrupted tracking requires a large stiff flex-pivot which is 

difficult and expensive to fabrlcate. The flex-pivot design also requires 

a secondary actuation system to unwind the flex-pivots accumulated rotation 

due to tracking and to move the telescope through large angles when 

acquiring a new source. Recent experience on an SA0 balloon borne 

spectrometer also revealed that a flex-pivot based design would require 

precise alignment across the telescope frame at a level difficult to 

maintain in the field. 

The ball-bearing design provides unlimited angulation without the need 

for a secondary drive system. The graininess and slip-stick which they 

exhibit in slow-speed servos is compensated for by using a feed-forward 

path in the torque motor servo loop driven by bearing torque measurements. 

Balloon-Borne Three-Meter Telescope for Far Infrared 8nd Submillimeter Astronomy Preliminary 

Design Report, Semiannual Report #4. Grant NAGW2-509. October 1985. 
Page 3 

The torque sensor, which has an appropriate torsional spring rate, permits 

telescope rotation while the ball-bearings are locked by graininess or 

slipstick friction. While this design does have certain inherent 

non-linearities which the baseline design avoided, the feed forward servo 

incorporated in the design can compensate for them more than well enough to 

achieve the desired pointing accuracy and stability. 

2.2 Ball-Bearing Gimbal Mechanical Design 

The mechanical design is shown in Figure 2-1. 

A backplate acts as a pedestal on which the servo components are mounted. 

The torque sensor is rigidly attached to the backplate on one end and is 

machined on the other to mount a matched pair of angular contact bearings. 

These bearings provide the required rotation via a spindle which has the 

servo motor stator and position resolver armature mounted to it. It also 

provides the bolt pattern for attachment to either the telescope or X-elev 

frame. The backplate also has provisions for mounting the servo motor 

armature and position resolver stator. 

The torque sensor is a commercial device which has the appropriate 

structural load capacity for all normal operational situations. During 

severe load applications (parachute deployment and crash landing for 

example) the torque sensor will deflect until the,diameters of the 

servomotor mount and bearing spindle come into contact. This contact will 

then carry all such severe loads with a factor of safety 23 insuring that 

rupture and/or separation between the telescope and gondola cannot occur. 

This design allows complete assembly and functional checkout of the 

gimbal to occur at the sub-assembly level. Two completed gimbals are then 
Page 4 

Figure 2-1. Gimbal Assembly 
Page 5 

installed between the Telescope X-elevation frame and gondola X-elevation 

frame (one per side) to form the telescope elevation mount. This 

installation is easily accomplished by centering the telescope in the frame 

and attaching the gimbal assemblies to each by using the appropriate 

hardware. All hardware required is accessible from the outside to 

facilitate this attachment. The same gimbal design is also used in the 

cross-elevation servo. 

2.3 Ball-Bearing Gimbal Servo Concept 

The Servo System operates in two modes: 

1) Magnetometer Mode - Celestial position acquisition controlled by 

the course servo loops, and 

2) Inertial Mode - Inertial tracking via the fine position servos. 

During initial source acquisition (Magnetometer Mode) the coarse loops 

in both axes are active. During inertial tracking the azimuth coarse loop 

acts only to dump excess momentum accumulated in the reaction wheels and to 

keep the telescope centered within the gimbal travel. 
Page 6 

2.3.1 Inertial Mode (Fine Position Servo) - 

2.3.1.1 Description of Fine Position Servo System - 

During inertial tracking the control loop consists of four elements*: 

The prime attitude control loop, 

The gyro update arm, 

The momentum management loop, and 

The active bearing assembly. 

The fine control loops in both elevation and cross-elevation take into 

account telescope inertia, reaction wheel behavior, gyroscope behavior and 

designed-in control laws. A simple proportional plus derivative controller 

was selected for the inertial loop. Its projected bandwidth is 2 Hz with a 

.7 damping ratio and it will be implemented in software. Integral feedback 

is not required as steady state errors are implicitly corrected within the 

momentum management loop. Space Telescope wheels were selected for the 

reaction wheels. Gyroscopes are used as the primary inertial reference and 

are drift corrected by a coaligned startracker. These positional updates 

take place every 1 to 5 seconds, too slowly to use this information 

directly within the fine control loops, however, more than fast enough for 

updating the gyros. 

The reaction wheel momentum is transferred from the telescope 

structure to the gondola by the gimbals' brushless torque motor. The 

See also Section 2.4 of the PDR 
Page 7 

reaction wheel velocity is monitored and is used to generate a proportional 

torque from the torque motor in the same direction. This causes the 

telescope to angulate. The change in position is noted by the prime 

inertial loop which, while compensating for this motion, causes the 

reaction wheel to slow down. The operation of the momentum management 

system is mathematically similar to integral feedback and therefore acts to 

correct steady state errors. 

Finally, the active bearing assembly operates as a torque-free gimbal 

for the telescope. The gimbal consists of a ball-bearing, torque sensor, 

and a brushless torque motor (the same one used in the momentum management 

system). As the telescope rotates, the bearings generate friction torque, 

which is measured and compensating torques fed into the gimbal by the 

torque motor. This is not a closed loop system and the torque motor 

reaction time is important. By keeping this time as short as possible the 

effects of bearing "graininess" or slip-stick behavior are minimized and 

the chances of an interaction between the telescope attitude control loop 

and the active bearing are avoided. In order to successfully cancel the 

torque generated by the bearing the torque motor constants must be well 

understood. The motor will be calibrated at all orientations and 

conditions of expected operation. If necessary, a look-up table of torque 

constants will be used in demanding torque from the motor. 

Each gimbal assembly cancels the torque created by its' bearing. This 

prevents the bearing torque from distorting the mirror cell. Which gimbal 

assembly to select for momentum management is still an open question. Both 

are equally capable of dealing with the momentum transfer, but whether to 

split the load equally between the gimbals assemblies or give the duty to 

one entirely has not yet been determined. 
Page 8 

The cross elevation fine loop is virtually identical to the elevation 

fine loop, however its range of travel and interaction with the gondola and 

coarse loop are different. 

motion. The gimbal is kept centered within this range by occasional 

activation of the azimuth coarse loop. The cross-elevation gimbal, as it 

moves, changes orientation with respect to the gond,ola, since it is the 

inner gimbal. Therefore, when cross elevation momentum is dumped into the 

gondola it will excite a complex gondola motion which is not necessarily 

around an axis parallel to the cross elevation gimbal. Passive methods for 

damping this gondola momentum will be required. 

The cross elevation axis has less than +4' of 

2.3.1.2 Selected Components - 

(a) Gimbal Motor 

The d-c gimbal motor is a high-performance design from Sperry's 

Aerospace and Marine Group for use on the Space Telescope and is designated 

as their model 2940616-2. The motor's key features which recommend it 

strongly in this application are: 

1) It is a brushless motor. 

2) It has been specifically designed to have an extremely low ripple 

torque. Sperry specifies its ripple as 0.5% maximum, but states 

that this typically tests out to less than 0.25%. This is a 

percentage of the applied torque, which in our case is expected to 

be very low (no more than the break-free torque of the gimbal 

bearing, - .l NM) . 

3) Cogging torque is also designed to be very low. In this model, it 
Page 9 

is specified as 0.4 oz-in. maximum (0.8 oz-in, peak-to-peak) . The 

spatial frequency of the cogging torque is 48 cycles per rotor 

revolution (equal to the number of stator teeth). 

4) The rotor-to-stator air gap is large (.120 in.) and the torque 

characteristics of the motor are relatively unaffected by 

de-centering of the rotor by as much as .030 in. This allows for 

sag in the torque sensors which support the full weight of the 

telescope in elevation, and eases the alignment requirements on 

the gimbal assemblies. 

Note that in the final application, it may be necessary to accurately 

"map" the fine-scale torque variations as a function of rotation angle for 

each individual motor. This information will be used to correct the 

torque-cancelling current fed forward across the gimbal, increasing the 

precision of the cancellation, and therefore reducing the demands placed on 

the outer servo position loop. 

(b) Reaction Wheels 

Space telescope reaction wheels or their equivalent have been selected 

based on the points below: 

1) Their 0.8 Nm torque capacity meets our requirements. 

2) They have been thoroughly characterized during the space telescope 

bui 1 dup . 

3) They have been selected for a number of up coming missions (AXAF 

and GRO, among others) thus making them readily available. 
Page 10 

(c) Torque Sensor 

The torque sensor selected is one of a class produced by Brewer 

Engineering, a subsidiary of Teledyne. The sensor element consists of four 

rectangular cross section posts machined in a cylindrical piece of metal. 

The cross sectional area and cylinder diameter is varied to achieved 

different torque sensitivities and load carrying capacities. Though Brewer 

Engineering provides these elements fully packaged, we would only use the 

bare element to reduce the gimbal's size. 

elements is given in Table 2-1 taken from a Brewer data sheet. We would 

use an A-OS type device. 

The specification of the package 

(d) Position Encoder (Resolver) 

The telescope's position with respect to the gondola must be known 

with sufficient accuracy to compensate for gimbal motor torque anomalies. 

Also, the elevation servo loop requires this information when operating in 

Magnetometer Mode in order to generate an appropriate error signal. Both 

requirements call for a readout device with absolute position accuracy of 

zt5 arcminutes. 

Many of the same considerations involved in the choice of the gimbal 

torque motor enter here also. The sensor must not introduce any 

significant disturbing torque across the gimbal, it should mount on a 

hollow shaft (to allow cable pass-through) and its performance must be 

relatively insensitive to decentering. 

One device which meets all these requirements is a brushless resolver. 

In particular Sperry has supplied on special order position resolvers with 

extremely wide radial air gaps, accommodating radial translation up to 
Page 11 

Type A Torque Sensors 

Capacities 

10,20,50,100,200,500.1,000,2,000 

5,000. 12,000 and 30.000 in-lbs 

(larger capacities available) 

Performance 

Calibration accuracy. ............. .0.25% R.T.. CW or CCW 

Nonlinearity 

Output at rated torque (R.T.). ...... .1.5 mv/V 

10-50 in-lbs ................... .O.lO% R.T. 

all others ..................... .0.05% R.T. 

Repeatability .................... .0.05% R.T. 

Hysteresis 

10-50 in-bs ................... .0.15% R.T. 

all others ................. ;. .. .0.05% R.T. 

Temperature 

Temperature range, safe. ........... -50 to 140' F 

Temp. effect on rated torque. ...... .0.005% Reading per . F 

compensated. .. .O to 140" F 

Temp. effect on zero balance ...... .0.0025% R.T. per . F 

Electrical 

Excitation. recommended ......... .12 V, AC or DC 

maximum.. ......... : . .25V, AC or DC 

Zero balance @ 72*3'F .......... .2.5% R. T. 

Terminal resistance, input ......... .350 f 1.5 ohms 

output ........ .350 f 3 ohms 

Termination h63102A- 18-8s-A105 

connector with mating 

15-ft. cable assembly 

Insulation resistance 

' bridge to ground .............. .2,OOO megohms . shield to ground. .............. .l,OOO megohms 

Overload Ratings 

Safe ............................ 120% 

Electrical failure ................. .300% 

options: 

Consult factory for minor dimensional 

mdificxtion and available WM x" IlSOiS 

Dynzmic Characteristics 

Type A 

Nom The tabulated data pertains to the torque pickup only and does not include the effects of cou 

plings or other rotating masses that would be pan of a dynamic system. 

Table 2-1. Torque Sensor Specifications 
Page 12 

f.200 inch, and having an accuracy of fl minute of arc. A modification of 

one of their standard space-qualified designs has been tentatively adopted 

by us for the present application, and is shown in the gimbal mechanical 

assembly (Figure 2-1). 

2.3.1.3 Full System Model - 

A nonlinear system block diagram is shown in Figure 2-2. This is 

similar to Figure 2.4-3 on page 53 of the PDR. The notable differences are 

the inclusion of the ball-bearing in the flex-pivot/ball-bearing block, the 

ball-bearing deflection, its accumulation, and the explicit representation 

of the gimbal motor's time constant. The nature of the ball-bearing 

deflection, which acts to reduce the quantity and thus the torque across 

the ball-bearing and sensor is not clearly definable. The deflection must 

. be assumed to be nonlinear and therefore beyond the capabilities of our 

present computer model. Fortunately, however, linear modelling does 

provide useful results. 

(a) The Linear Model Configuration 

Telescope angulation for most fine-pointing operations is absorbed by 

the torque sensor. The torque sensor acts as a flexure since for small 

angles the ball-bearing will be locked by friction. When the stored torque 

in the flexure reaches the bearing release friction the bearing will 

rotate. While there are many scenarios for the behavior of the bearing at 

the time of release, the anticipated worst case is a snap that will 

recenter the torque sensor (i.e., to zero torque). This can be seen by the 

following scenario: If the bearing were to relieve slowly the torque 

sensor would follow the relaxation and demand less torque from the gimbal 
Page 13 

I eRw 

&- 

I t+ I 

I-_ .I TELESCOPE: e, 

I \ -w I 

I I I I 1-4-a U 

l 

a i 

Figure 2-2. Nonlinear System Block Diagram 
Page 

motor. Ideally this would be transparent to the telescope as the gimbal 

motor would always track the bearing torque. However, if the bearing 

snapped back to the torque sensor’s center point, the torque sensor and 

14 

motor would not have time to follow and the net torque on the telescope and 

gondola would be that applied by the motor. The motor torque would decay 

to zero as the sensor reading and motor driver electronics overcome the 

motor time constant. Under these conditions, the telescope would be 

subjected to a torque pulse approximately .2-.5 Nm in height and about 10 

ms (four times the motor time constant) in duration. 

A linear model can be constructed from these.assumptions. It yields a 

model of the pointing system from which worst case performance can be 

judged. In this model the telescope is suspended on a torque sensor, has 

two controlled torque sources, the reaction wheel and the gimbal, and is 

subjected to occasional noise pulses of SxlO-’ Nms. The noise pulses model 

the bearing snap. The results of this analysis are given below (Section 

2.3.1.4). This linear analysis obviously does not predict limit cycle or 

other characteristic nonlinear behavior. 

(b) Nonlinear Model 

In a nonlinear model the torque sensor would be compared to the 

break-free frictional torque of the bearing. When these torques were equal 

the nonlinear representation of the bearing motion, would be triggered. The 

boxed-in node in the block diagram representation of the system depicted in 

Figure 2-2 is where this non-linear behaviour is modeled. The torque felt 

by the telescope would be given by the time-dependent equation: 
Page 15 

for as long as the ball-bearing deflected. The exact nature of the model 

nonlinearity (as opposed to the true system nonlinearity) would be twofold: 

The comparator operation between the sensor torque and the 

break-free frictional torque when it results in the switch (i.e., 

the bearing breaks away and its symmetric operation to freeze the 

bearing motion) , and 

The nature of the function A Bgi (t). 

However, the rest of the model and the equations are the same as the 

linear representation. This nonlinearity only presents itself when the 

telescope has moved sufficiently to cause the flexure or sensor to develop 

a torque equal to that of the bearing break-free torque. The required 

angle for break-free torque is about lo and depends on the sensor spring 

constant and the bearing properties. 

The telescope's prime operation is tracking stars, which rotate at a 

maximum rate of 15O/hr. In operation, the bearing nonlinearity will be 

excited on the order of once every four minutes, which is infrequent in 

light of the 2 Hz control bandwidth. 

The linear model does not show any limit cycle behavior which might 

result from the bearing's stiction, however, we can determine if a limit 

cycle may result by looking at the system response to the bearing noise 

pulse. If the telescope motion is large it may excite a further bearing 

release (not shown by the linear model) and thus enter a limit cycle; If 

it is small no limit cycling occurs. The linear simulation indicates that 

in fact the telescope motion is small enough to avoid exciting limit cycle 

behavior. 
Page 16 

(c) Development of System Equations 

For the development of the linear equations please refer to Appendix A 

in which Appendix B of the PDR is reproduced. 

see that in the new system model the figure of the torque summing junction 

is still valid and the same as that shown in Figure 2-2 in the dashed 

Referring to page A-14 we 

enclosure. In fact equations (1)+(2) of the appendix are still valid, as 

far as they go. The model for the proposed design only changes from the 

baseline in its control law for the gimbal motor. Thus equation (5) on 

page A-16 of the appendix will become: 

where: TFP = the torque-sensor torque 

(note a typo lists 8, as &w in the original text.) 

By adding equation (1) here to equation (1) of the appendix (page 

A-14) the result is removal of the dependency on KFK and thus cancellation 

of the effect of the flex-pivot, or spring constant of the torque sensor. 

Therefore, if we follow this through to page A-17, equations 8-13 

remain valid if we set Kpp to zero. 

The ball-bearing shows up as an impulse under TN, noise torque, to 

which we must determine the system response. 
Page 17 

2.3.1.4 System Response - 

System behavior was simulated using the computer program "TF" written 

at Stanford University. This program provided a Bode plot (see Figure 2-3) 

and an impulse response (Figure 2-4) for the modelled system. The impulse 

assumed for the sake of the analyses was SxlO-) Nms as mentioned above, 

applied at T=O, it resulted in a maximum predicted excursion of the 

telescope from its commanded position of .025 arcsec. At such a small 

angle the system behavior begins to be quantized by the digital-to-analog 

converters and cannot be considered strictly linear, so the response shown 

in Figure 2-4 may not be a realistic prediction of system response. 

However, we do see that the "snapping" of the ball-bearing has little 

effect on the telescope pointing. 

The rest of the model behavior is nearly identical to the system 

described in the PDR. The new gimbal model predicts that the telescope is 

completely isolated from the gondola. This is different from the earlier 

system model in which the telescope was explicitly coupled to the gondola 

behavior by the flex-pivot. In practice, the gondola will excite some 

telescope motion through the nonlinearities in the gimbal. 
Page 18 

Magnitude 

of 

Response 

10 

1 

.1 

8 telescope 

Bcommand .. 

.Ol 

.OOl 

. 000 1 

1 10 100 

Frequency Rad/sec 

1000 

Figure 2-3. Fine-Pointing Control System Response to a Sinusoidal Command 

Signal 
Page 19 

.025 

.020 ;, 

.015 

.010 

.005 

0 

-. 005 

-. 011) 

Figure 2-4. Telescope Point Response to a 5x10-' Nms Torque Pulse 
Page 20 

2.3.2 Azimuth Servo (Coarse Loop) - 

The Azimuth Servo is the outermost control loop. It is a &oarse 

pointing loop, which contains the cross-elevation, or fine loop within it. 

The functions of this outer loop are threefold: 

1) In Magnetometer Mode, the Azimuth Servo is used for the initial 

acquisition of a targeted celestial object, and points the gondola 

to within f30 arcminutes of the commanded location. This is 

close enough to allow recognition of the star field in the aspect 

TV camera. Azimuth reference in this mode is the earth's magnetic 

field, as sensed by a Schonstedt flux-gate magnetometer. The 

inner gimbal is driven to its center position and locked while the 

target is being acquired. 

In Inertial Tracking Mode, the error input which drives the 

Azimuth Servo is switched from the magnetometer to the inner 

gimbal position sensor and the inner gimbal is unlocked, allowing 

the cross-elevation fine servo to become active. The Azimuth 

Servo then maintains coarse orientation of the gondola so that the 

target object remains well within the limited range of the 

cross-elevation gimbal. Our present intention is to include a 

threshold, or dead-banding, feature so that the outer loop becomes 

active only when the inner gimbal moves away from its center 

position more than some minimum angle. This will reduce the 

amount of activity in the Azimuth servo, and result in fewer 

disturbance torques for the pointing control to cope with. 

3) The coarse Azimuth servo also serves as a means of dissipating 
Page 21 

momentum built up in the cross-elevation reaction wheel. When 

this momentum is dumped (by torquing the cross-elevation gimbal 

motor) a component of it, proporfional to the cosine of the 

elevation angle, is coupled into the gondola azimuth axis. The 

azimuth reaction wheel responds by taking on a velocity and this 

energy is removed by the azimuth loop twisting against the 

suspension lines. 

A full block diagram of the combined Cross-elevation and Azimuth 

servos, showing their areas of interaction is shown in Figure 2-5. 

A more complete explanation of the overall system concept, and 

descriptions of the subsystem elements, is contained in the PDR, Section 

2.4.4, "Pointing System Control Laws and Predicted Performance" . 

2.3.3 Summary - 

The ball-bearing gimbal design presented in this report will satisfy 

the pointing requirements of the telescope without the initial alignment, 

secondary drive system, and limited angulation inherent in the earlier 

design. 
Page 22 

- REACTION WHEEL OR., 

WHEEL e I wREACTION 

KIP MOTOR JRr,C . 

r, s+l - 4 

81 TELESCOPE 

SENSOR * e - I 7- 

81 

J, $2 

+ 

(1 ' KFP - 

I SENSOR I 

I. 8PlOl. 

COMPENSATION -e 

MOTOR e - - J, s2 

a SUSPENSION 

SENSOR 

@ea I loo" 

Figure 2-5. Block Diagram of Azimuth and Cross-Elevation Servos 
Page 23 

3.0 SYSTEM ERROR ANALYSTS 

An error analysis has been performed for three different star tracker 

configurations: 

1) Co-aligned but outside the focal plane of the telescope; 

2) Beamsplitter feed at the focal plane; 

3) Direct IR focal plane measurement by a high resolution instrument. 

These results are based on error magnitudes that have been taken from 

the PDR or if unavailable these, assigned on the basis of best engineering 

judgment. The values of 1.0, 0.61 and 0.30 arcsecs RMS were calculated for 

image position precisions for the three star tracker configurations 

respectively. 

3.1 Configurations 

3.1.1 Co-Aligned Star Tracker - 

The co-aligned star tracker is the present baseline design as 

described in the PDR. 
Page 24 

3.1.2 Beamsplitter Feed - Focal Plane Star Tracker - 

The beamsplitter feed star tracker consists of a star tracker like the 

one proposed for the baseline, viewing a target or guide star image folded 

from the main telescope image. This arrangement has several advantages 

over the baseline: 

1. The star tracker can compensate for secondary motions which would 

otherwise result in image motion at the focal plane; 

2. The structural separation between the tracker and the focal plane 

is smaller and thus the tracker is less likely to move relative to 

the focal plane; 

3. The star tracker can compensate for image displacement resulting 

from relative motion between the primary mirror and its housing. 

The disadvantages of this arrangement are: 

1. The mirrors must be of optical quality in the visible wavelengths 

to yield an image usable to the star tracker; 

2. There is a beamsplitter between the secondary and the infrared 

experiment which results in light loss to both the star tracker 

and instrument. 

3. The beamsplitter star tracker leg of the star tracker optical path 

is still separate from what is being stabilized (i.e. the focal 

plane image), thus motions in that leg or in the focal plane would 

result in an image error. 
Page 25 

3.1.3 Instrument Star Tracker - 

The third arrangement examined was one using a high resolution camera 

in the focal plane to yield a tracking signal. This arrangement has all 

the advantages of the previous one with none of its listed disadvantages. 

Furthermore the system does not have to be boresighted as the star tracker 

and the telescope line-of-sight (LOS) are identical. 

3.2 Error Sources ’ 

The possible error sources are split into two categories: Fixed and 

Operational. The fixed errors are those which either cannot be removed 

during boresighting or do not vary with time (e.g. those resulting from 

imprecise knowledge). The operational errors are those which occur during 

flight. 

3.2.1 Fixed Errors - 

The fixed errors are the base irreducible system errors . In the two 

boresighted arrangements these are the boresighting accuracies, in the 

other, the only fixed error is measurement error in determining the center 

of the focal plane. 

The error sources inherent in the boresighted designs are: 

1. Control system pointing performance 

2. Star Camera boresight uncertainty 

3. Focal Plane CCD boresight uncertainty 
Page 26 

4. Unremovable secondary decenter precision, and 

5. Focal plane boresight uncertainty 

The control system and star camera values were taken from the PDR, the 

others are best estimates. The focal plane boresight accuracy was assumed 

to be equal to that of the star camera. The unremovable decenter tolerance 

requires that the secondary mirror be placeable to f.0008 cm, which can be 

considered a specification for that system. The focal plane boresight 

uncertainty requires that the focal plane geometry be known to f.002 cm; 

neither of these requirements are difficult to achieve. 

3.2.2 Operational Errors - 

The operational errors are: 

1. Gravitationally induced truss deflection, 

2. Star tracker misalignment, 

3. Star tracker centroid accuracy, 

4. Control system noise, 

5. Truss thermal deflection, 

6. Secondary decenter, 

7. Secondary misalignment, and 

8. Mirrorfiousing relative motion. 

These all effect the coaligned star tracker arrangement; 2, 3, and 4 

effect the beamsplitter system and only 3 and 4 effect the instrument star 

tracker. 
Page 27 

The gravitationally induced truss deflection results from the fact 

that the viewing and the boresighting elevation angles are different and 

the gravity gradient across the truss therefore changes at the sidereal 

rate. The star tracker misalignment is taken from the PDR for the 

co-aligned arrangement and assigned for the beamsplitter case. It should 

be noted that a co-alignment of .6 arcseconds between the star tracker and 

the telescope is achievable, but is considered a difficult specification. 

Star tracker centroid and control system errors come directly from the PDR. 

The thermal deflection of the truss, the secondary misalignment and 

misplacement, and the mirrorfiousing relative motions each cause focal 

plane image motion through two coupled mechanisms. 1) secondary decenter, 

and 2) secondary misalignment. The mirrorfiousing motion does not cause 

true secondary motion but it can be treated as if it does. 

All calculations assume that the system is boresighted prior to each 

observation. The contribution of image blur to the image position error 

was reviewed, found to be minor and thus not included. The two errors 

resulting from each source are first added together algebraically before 

root-square-summing these values with errors resulting from other 

uncorrelated sources. The precisions assigned to the secondary mechanisms 

and the primary mirror to housing motion are estimates. We conclude that 

it will not be possible to achieve one arcsecond total pointing error 

without the use of a focal plane guider. 

The various R.S.S. of the errors are presented in the lower part of 

Table 2-2 along with the total fixed and operational error. 
Page 20 

4 

. b 

0 

0 

b 

0 

. e 

+)cyea . . 0 

e. NelHd 4 . 0 0 0 0 

G * 

. 

* 

rl 

00 

el 0 

rl He4 

0 

Q, 

3 

4 

a 

c 

0 

0 

w 

e 

u 

aJ 

c 

a 

rl 

pc 

rl 

a 

0 

0 

crc 

aJ 

C 

m 

r( 

a 

4 

w 

W 

0 

rl 

a 

W 

4 

E 

& 

aJ 

.- 

a 

'CI 

c 

0 

rl 

a 

4J 

8 

- 

0 

aJ 

cn 

Table 3-1. System Error Performance for Three Star Tracker Configurations 
Page 29 

3.3 Determination Of The Relationship Between Secondary Misplacement And 

Image Motion 

The alignment tolerance shown in Table 2.2-2 of the PDR were used as 

the basis for establishing the relationship between secondary decenter and 

misalignment, and image motion. The values in Table 2.2-2 of the PDR are 

given as allowable errors in secondary placement for a 1 arcsec apparent 

image displacement in the field. As such these values are not, strictly 

speaking, ratios between secondary motion and apparent ,image angular 

displacement. However, for this error analysis they have been used that 

way. In defense of this approach all the image motions resulting from 

secondary displacement are near and less than 1 arcsec thus treating the 

relationship as linear is not going to be far wrong. 

An example of how these calculations are done, in the case of the 

effect of mirrorfiousing relative motion on image stability, is given 

below. 

First the primary mirror relative motion is translated into apparent 

secondary decenter and misalignment. 

Secondary Decenter: 

where: is the apparent displacement of the 

secondary - (cm) 

#M is the actual mirror motion -(radian) 

CT is the interoptic spacing -(cm) 
Page 30 

Secondary Misalignment: 

From Table 2.2-2 the secondary decenter tolerance for a 1 arcsec image 

stability is -.0022 cm while the misalignment tolerance is .06647 arcmin 

for the same image stability. The resulting image motion from a given #M 

is then found as follows: 

Secondary Decenter:' 

-&pa rent FED = .0022 

or from eg. (1) 

-4M eT FED = - 

.0022 

(3) 

(4) 

where: FED is the apparent image motion or field effect due 

to displacement. 

Secondary Misalignment: 

,, 

where: (57 0 60) is to convert from Radians to arcmins. 

The total image motion is given by: 

and 

a~~T = -4MeT + 4M(57b60) 

0.0022 0.06647 
Page 31 

4.0 RESEARCH PLANNED FOR NEXT REPORTING PERIOD 

Engineering analysis and design under this grant is complete. Effort 

during the next reporting period will follow research presently underway in 

producing lightweight mirrors and we will continue our discussions with 

foreign groups or possible collaborators on this project. 
APPENDIX A 

Pointing Control System Servomechanism Analysis 
Page A-1 

Glossary of Symbols 

0 

C 

0 

m 

m 

E 

JW 

T 

TM 

m 

ext 

J 

T 

k 

TM 

JH 

H 

0 

H iAl 

S 

W 

K' 

AK 

6 

T 

T1 

T2 

Angular command (desired pointing) of mirror assembly with respect to 

inertial space 

Angular response (actual pointing) of mirror assembly with respect to 

inertial space 

Angular pointing error of mirror assembly 

Moment of inertia of momentum wheel 

Torque developed by momentum wheel torque motor 

Moment of inertia of mirror assembly 

External torque applied to mirror assembly 

Spring rate of flex-pivot suspension of mirror assembly !torque units 

per radian] 

Torque developed by ancillary torque motor 

Angular position of Horseshoe gimbal with respect to local vertical 

Moment of inertia of Horseshoe gimbal 

Frequency of angular vibration of Horseshoe gimbal [rad/sec] 

Laplace operator 

Angular velocity of momentum wheel with respect to inertial space 

Ancillary torque motor conversion gain [torque units per rad/sec] 

Amplifier-Torque motor constant [torque units per radian] 

Lead time constant of leadllag network 

Lag time constant of lead/lag network 

'applied Total systemic torque applied to the pendulous Horseshoe gimbal (reaction 

of ancillary torque motor and flex-pivot) 

T Torque Noise (bearing torque noise, motor cogging, etc.) V 
Page A-2 

.- 

€/ 

VERTICAL 

6 wkk s +WH 

-1 

M T 

7 FLEX PIVOT 

TORQUE 

TIS+ 1 1 'm 

- T2S+1 T Jm S2 w 

* A- K 'm 

'm 

. 

I 

T I 

'W 

(Bm+ iw) 3 ew 

- 

1 

TACHOMETER MEASURE OF 

MOMENTUM WHEEL SPEED 
Page A-3 

System Dynamics Equation is: 

-[+-A. TIS+l c] 

T2S+1 t JwS 

.- 

Om 

d 

OH 

Determinant is: 

f 

+ Jmk + JHk A(S) = itmJHS 5 + (JmJHuH 

TIS+l 

+- T2S+1 AKT [(S2+u;) tHS + 7) +k{ 

which can be recast in form of: 

Stable! 
Page A-4 

. 

Full System Response to Pointinq Command 

TIS+l 

T2S+1 AKT ( 1+- :is)% + 'ext 

-- TIS+l K' 

T2S+1 AKT 5 'c W 

- 

-k 

-___ Final value of mirror pointing 1s: 
Page ~-5 

0 

m1s.s. 

J 

1 K'J 

- 

K'J 

H~~T 

'2 

w HT 

withzerofinal error even with torsional 

flex-pivots, external torque, and disturbing 

horseshoe-motion coupling. 
Page A-6 

FREQUENCY DOMAIN ERROR ANALYSIS 

_. 

1 

JH 

s +WH 

22 

m 

e - 

kr I . 
, 

-. 

-_ 

Page A-3 

Mirror spectral sensitivity to Horseshoe Gimbal motion is given by: 

where 

K4 = T2JW 

K3 = Jw 

AK, K ' 

K1 = (2 k + - 

Jm 

Note: *Zero at origin reduces steady-state sensitivity to zero. 

.Sensitivity at "horseshoe" gimbal frequency is given by 

above equation 
Page A-8 

Frc 

0 

\\ 

@J\ \\ 

+\\ \ 

\\ \ \T 
Page A-9 

Because "horseshoe" gimbal characteristic frequency, w is very 

low, the low frequency asymptote of spectral sensitivity 

H' 

expression may be used: 

0 k Jw w 

2% 

A K, K' OH 

, rad/sec where w = du; f - k 

JH 

k = flex-pivot constant, [torque units per radian] 

Jw = momentum wheel moment of inertia 

2 

[torque units per 

radian/sec 1 

AK, = amplifier torque motor constant [torque units per radian] 

K' = ancillary torque motor conversion gain [torque units 

per rad/sec.] 

JH = horseshoe moment of inertia [torque units per rad/sec 2 1 
System Spectral Sensitivity to Torque Noise 

2 2k 

0 

-(SI = 

S[T2S+11[S +W H +- J~~~H 

m 

T V K6S 6 +K S 5 +K4S 4 +K S 3 +K S 2 +K S+KO 

5 3 2 1 

where: 

K6=JJT 

Kg - J~JH 

mH2 

- 

.K4 = [(JmJHWH+Jmk+JHk)T2+AK 2 J T ] TH1 

K3 = J J w 2 +Jmk+JHk+AK mHH i 

2 2 K'JH + u:JHj 

K1 T 

(W 

HH 

J +k)+AK -rH1Jw w T - 

- 

KtJH 2 Kg = AKTuH 

Jw 
Page A-11 

SPECTRAL TORQUE-NOISE SENSITIVITY 

wkJH + E 7 

I I 

\ 

\ 

DENOM 

\ 

\ 

\ 
Page A-12 

The - low frequency asymptote may be approximated by: 

0 

2% 

'I 

V 

(low freq.) 

w 

V 

where wV is the circular frequency of the torque-noise in rad/sec. 
Page A-13 

Generic Control 

E 

l 

-. 

The controlled system can be written as: 

where 

If we 

.. 'c KFC xT = -K /J X + Tc/JT + - 

JT 

FP T T 

X is the angle being controlled T 

is a spring constant KFP 

J is an inqrtia 

T is the control torque 

X is the command input 

select a simple proportional plus rate controller we have: 

T 

C 

C 

T = -K (i T + X,) 

C gT 

where: T is the zero location of the controller 

K is the control gain 

g 

Combining (2) and (1) and rewriting we get: 

x KFP =- KT (KFP + K ) 

JT JT JT C 

XT + iT + 

The generic equation for a 2nd order system like this is: 

.. 9 ,-J 

unL xc 

x + 25Wn ;: + WL x = n 

where: 5 is the damping ratio 

is the natural frequancy n 

If we select Wn, 5 we can match coefficients between (4) & (3): 

2 K +Kf 

P=u 

n JT 0 

KT 

0 

In this way we solve for K and T and get our control. 

g 

(2) 

(3) 
Page A-14 

________) TGm 

-. 

Telescope 

The individual loops are then combined into a system of equations and solved. 

Some adjustments are made for the ignored coupling. 

Elevation Axis Equation. 

TF P 

I 

The equation of motion of the telescope is: 

JTBT e K (0 -0 + TGm + T FP g T m + Tn 

where: J is the telescope inertia T 

T 0 is the telescope angular position 

KFP is the flex-pivot sprint constant 

0 is the gondola's angular position 

g 

is the gimbal motor torque TGm 

T is the reaction wheel motor 

T is noise torque 

m 

n 

The equation of motion for the gondola is 

where : w is the compound pendulum natural frequency. P 
Page A-15 

, 

Elevation Axis Equation (continued): 

Equation of motion of the Reaction Wheel is: 

Jrw Grn = -T 

m 

The control law states: 

& 

If we put these equations into state space form as the following: 

x2 = GT 

x3 = 

x4 = og 

where: 0 is used to define a state existing in the control law or: 1 

(3) 

T =K 0 

m 11 (7) 
Page A-16 

Elevation Axis Equations: 

Eqs. (11, (21, (3) & (4) with eqs. (6) can be put into 6 first order differential 

equations: 

x* = - -K X +K X +T '1 + .Fp '3 2 5 1 6 n JT 

x3 = x4 

'6 

- T2 x2 - x6 + x61 i6 - T 1 [-x1 

2 

Where: X is the commanded signal for the telescope. These equations 6 

can be placed in matrix form as 

where: [A] is the matrix of coefficients of the x 's 

i 
Page A-17 

In this form the system was run through a program called 

Function) written at Stanford University to generate system responses. 

"TF" (Transfer