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https://ntrs.nasa.gov/search.jsp?R=19760010102 2020-05-09T10:56:47+00:00Z

FINAL TECHNICAL REPORTO Tp

f PRECISION ATTITUDE DETERMINATION SYSTEM

13900-6016-RU-00

1 JULY 1974 1

PADS SYSTEM DESIGN AND ANALYSIS

(SINGLE-AXIS GIMBAL STAR TRACKER)

(%TASA-CF-144 7 26) P ECISION ATTITUD'S N76-17190PBT7PM-rri TI0N SYSTFM (PAOS) SfSTFM DESIGN'AND ANALYSIS: ")1VGLF-AXIS GIMDA.L STA?TFACKLP Final Technical Report ( T ?td SystQms Unclas

Group) 91 p tiC 1; 5.00 CSCL 14 B G3/19 14790

`., FEB 1976a

' REC^tV^^ ^^Contract No. NAS5-21111

^'^-? NASA STI FAG1-° INPUT 5W

Prepared for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONGODDARD SPACE FLIGHT CENTER

Greenbelt, Maryland 20771 ' A,i

r

TRWSYSTEMS GROUP

ONE SPACE PARK + REDONDO BEACH, CALIFORNIA 90278

r

FINAL TECHNICAL REPORT

PRECISION AT'T'ITUDE DETERMINATION SYSTEM

13900-6016-RU-00 1 JULY 1974

PADS SYSTEM DESIGN AND ANALYSIS

( SINGLE-AXIS GIMBAL STAR TRACKER)

a! t

_11 1 Contract No. NAS5-21111

Prepared for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONGODDARD SPACE FLIGHT CENTER

Greenbelt, Maryland 20771

TRWSYSTEMS GROUP

ONE SPACE PARK • REDONDO BEACH, CALIFORNIA 90278i

I,

'Se};.: . k M ark _ .... _.. .... ... .-...: ...... .. ..

rx -

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1

y

TABLE OF CONTENTS

Page

1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1-1

2.0 SYSTEM DESIGN. . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1 System Configuration/Definition . . . . . . . . . . . . . 2-12.2 Design Evaluation . . . . . . . . . . . . . . . . . . 2-3

3.0 SYSTEM ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1 Star Availability Analysis . . . . . . . . . . . . . . . . 3-13.2 Error Analysis. . . . . . . . . . . . 3-1

3.2.1 Star Tracker Errors. . . . . . . . . . . . . . 3-63.2.2 Gyro Errors . . . . . . . . . . . . . . . . . . . . 3-6

3.3 Covariance Analysis . . . . . . . . . . . . . . . . . 3-133.3.1 Equation Development . . . . . . . . . 3-133.3.2 Performance Evaluation . . . . . . . . . . . 3-23

4.0 REFERENCES . . . . . . . . . . . . . . . . . . . . . . 4-1

Appendix A - Computer Program for Star Availability . . . . . . . . A-1w

A.1 Program LOOK1.

.

. . . . . . A-2

A.2 Yale Catalog of Bright Starsto 4th Magnitude . . . . . . A-6A.3 Ordered Working Star Catalog. . . . . . . . . . . . . A-17

Appendix B - Computer Program for Single-Gimbal PADS CovarianceAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.1 Program PADSCO. . . . . . . . . . . . . . . . . B-2B.2 Summary of Test Cases . . . . . . . . . . . . . . . B-8

r

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F^

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3

1.0 INTRODUCTION

TRW has conducted design and development of Precision Attitude Deter-

mination System (PADS) technology for NASA Goddard Space Flight Center

(GSFC) under contract NAS5-21111. This report documents the feasibility

evaluation -of an evolutionary development for use of a single-axis gimbal

star tracker from prior two-axis gimbal star tracker based system applica-

tions. Detailed evaluation of the star tracker gimbal encoder is also

addressed.

The report provides a brief system description, including the aspects

of tracker evolution and encoder evaluation. System analysis includes eval-

uation of star availability and mounting constraints for the geosynchronous

orbit application, and a covariance simulation analysis to evaluate perfor-

mance potential. Star availability and covariance analysis digital computer

programs are included as Appendices.

,,

tg"

t

1-1

ka.

2.0 SYSTEM DESIGN

The Precision Attitude Determination System (PADS) employs star

trackers and gyros to determine inertial-referenced attitude to arc-second

level accuracy. PADS system design employing a two-axis gimbal star tracker,

has been previously documented [1]. Key system design differences in em-

ploying a single-gimbal star tracker involve both hardware (e.g., tracker

and associated electronics) and software. These differences and potential

system design and performance tradeoffs are discussed in this section.

2.1 astem Configuration/Definition

The PADS star tracker, electronics, gyros, and computer/software shown

schematically in Figure 2-1, are functionally and operationally similar to

previous design [l]. The single-gimballed Star Tracker Assembly (STA) is

used to track stars and provide a measure of the star line-of-sight (LOS)

angle. The Sensor Electronics Assembly (SEA) provides the gimbal servo/

drive electronics and gimbal readout processing. The Gyro Reference Assem-

bly (GRA) incorporates a configuration of strapdown pulse rebalanced rate

integrating gyros plus electronics. A Reference Block Assembly (RBA) pro-

vides a stable thermo-mechanical interface with she spacecraft to which the

sensors are mounted and aligned relative to each other. The Digital Pro-

cessor Assembly (DPA) comprises the computer - I/O to implement system soft-

ware.

The gyros provide high-bandwidth data which is compensated to account

for systematic gyro errors and used to derive attitude. Attitude and gyro

drift (bias) corrections are periodically estimated via star tracker mea-

surements processed in a Kalman filter.

The gimballed tracker is operated to intercept defined stars of oppor-

tunity. The gimbal freedom is such as to provide a single axis of mechani-

cal scan. The tracker is mounted in the spacecraft such that the FOV moves

in a plane generally defined as normal to the orbit plane, with gimbal free-

dom chosen to assure adequate star sightings. Based upon a series of such

F^ 2-1

STAR

TRACKERASSEMBLY

GYROREFERENCE

ASSEMBLY

REFERENCEBLOCK

ASSEMBLY

R

f

TRACKER

PROCESSING/MEASUREMENT

ERROR

STARTRACKER

CONTROL

CATALOGPROCESSING/STAR SELECT

DIGITAL PROCESSOR ASSEMBLY

GYRO ATTITUDEPROCESSING ALGORITHM

KALMAN

FILTER

Figure 2-1. PADS Functional Block Diagram

-2

2-3

i

dam,_

;Al

sightings, the attitude estimate will converge to an accuracy consistent

with system alignment and bias uncertainties; typically, steady-state is

reached within 10 sightings or less. During this period, the system will

reach thermal equilibrium. In-flight calibration may be utilized to deter-

mine compensation for systematic errors. Observables include gyro input

axis misalignments, scale factor uncertainties, and tracker misalignments.,

The configurated PADS system physical characteristics are summarized

in Table 2-1. The single-axis gimballed tracker utilizes the identical Star

Sensor Unit (SSU) and one-axis of the gimbal suspension/drive assembly of

the two-axis tracker design [1]. Likewise, the Sensor Electronics Assembly

is of identical design, although it again provides drive and readout elec-

tronics for only a single channel. Revised size, weight, and power charac-

teristics have been derived accordingly.

2.2 Design'Evaluation

Prior PADS star tracker development., although oriented toward a two-

axis gimbal design, was limited in hardware development to a single-axis

engineering model gimbal. This single-axis design is assumed used directly

for the current PADS application. During the PADS star tracker development

and test [2], however, problems with the encoding electronics were noted but

not fully defined. In the following discussion, these problems are identi-

fied and design modifications recommended.

In testing the PADS system, certain anomalous behavior was noted in the

breadboard Inductosyn angle encoding. Investigation was confined to the

single speed channel, which was modified to read out all 14 available bits.

There were three basic forms of anomaly as shown by the error curve in

Figure 2-2.

(1) A wide-range cyclic error which appeared to be in excess of 20

peak-to-peak over the available 90 0 of gimbal travel.

1

..,rte i.

2 _

Table 2-1. Summary of PADS Physical Characteristicsk.

Wt (lb) Power (w.) Envelope in.) r

Star Tracker Assembly (STA)* 26 7 9.6 x 20.0 x

Sensor Electronics Assembly (SEA) 4 8 6 x 8 x 4

Gyro Reference Assembly (GRA) (1) 9 10 6 x 6 x 5

Reference Block Assembly (RBA) 10 -- --

Digital Processor Assembly (DPA.) 17 27 11.5 x 10.5 x 6.5a

Integration Hardware 4 -- --

N

Total System 70 lb 52 w 2111 26" x 20"(3)

* Single-axis gimbal design(1) Triad, Unheated(2) Typical a

(3) Approximate total envelope occupied

r

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i

!c.40i

i

^i

-0.60

f

-i.24n

rFigure 2-2. Single-S peed Error Data

ERROR VERSUS ANGLE-1-20

0.0 ;0 .0 40.0 50.0 60.0 10 .0 60.0 X0.0

iU TL^ f CE )

d ^'.

(2) A fine-scale cyclic spatial modulation of the error function,

with amplitude approximately 0.1° and periodicity about 1.4°.

(3) Noise in the output data of about 4 bits peak-to-peak.

The investigation of these problems was initiated, with emphasis on

diagnosis. The system was set up, including the SEA breadboard, the gimbal,

and the test set. An optical cube mounted on the Star Tracker case permit-

ted the use of a T3 autocollimator. A rotary table enabled rotation of the

gimbal base with a resolution in the region of a fraction of a minute, theR

gimbal being manually counter-rotated to the Autocollimator null for each

table setting. Note that because the original null is relatively arbitrary,

accuracy is here to be understood in the sense of linearity. Absolute ac-

curacy (bias) depends on the initial Inductosyn mechanical alignment in ro-

tation, and on total system stability thereafter. As a first step, the test

findings were essentially verified in all three respects: a gross non-

linearity, fine-scale linear ity r i pple, and excessive output data jitterY^ Y pP p J

of about 16 counts.

The spatial ripple error was suspected to be due to stray coupling from

the 256-speed channel because of its periodicity. The amplitude-to-phase

breadboard was a poor layout, with a considerable amount of physical inter-

leaving of the single-speed and multi-speed elements. To check this, all

the integrated circuits of the multi-speed channel were removed, although

the preamps were left operating. The ripple error was reduced from about

0.1° peak-to-peak to a value indiscernable_under existing conditions.

b

The wide-range error was tracked to a misw_ired capacitor in the

amplitude-to-phase converter, Correction of this yielded a linearity error

of approximately 0.5 degree peak-to-peak (Figure 2-3). This is similar to

the parrand data for the single-speed Inductosyn, although the shape of the

developed error curve cannot be matched with that for the Inductosyn because

of unavailability of Farrand`s definitions of electrical zero, positive

angular rotation, and positive error.

n

2-6

I

l0 X 10 TO "2 INCH 46 1323^^^ 7 K 10 INCHES ..or .. u f .. .

KEtI ►►EL O ESSER CO.

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In an attempt to characterize the noise behavior, random data samplesb

at a given gimbal angle were recorded and plotted. A typical histogram of

100 samples is shown in Figure 2-4. The horizontal quantization is one

least-significant count, 2 -14 of an electrical revolution. In the 256-speed

channel, a quanta would be 0.309 arc-seconds.

The analog processing chain, consisting of the Preamplifiers, the

Buffers, and the Amplitude-to-Phase Converters appears to be generating AM

---°'

noise. This chain has a relatively high gain and uses uA715's, which are

not notable as low-noise devices. This AM noise in the sinusoidal phase

signals would be mapped into phase noise by the Zero-Crossing Detectors. A

1

scope examination of these signals did Iddeed show some 200 to 300 ns of

phase jitter of the logic transitions of the lag signal with respect to the

lead.

To check this, a passive RC phase-shifting network was constructed and

a

excited directly with the excitation voltage, so as to give two sinusoidal

outputs, one slightly lagging the other. These were fed directly to the

Zero-Crossing Detectors, bypassing the analog front-end elements. The phases

were now extremely stable, but the encoded output data was still quite noisy.

To obtain a noise comp"arison, the output data was D-A converted and the

result displayed on a scope and photographed with a slow sweep. Figure 2-5

illustrates the resultant noise indication. Each dot represents a successive

data sample. Each row of dots is associated with a distinct data quantiza-

tion la,vel of one least significant (2 -14) unit.

The data for the "noiseless" phase-shifter appears to have a width of

16 quanta and a relatively sharp-peaked distribution. That for the normal

system has width about 20 quanta wide and a somewhat broader distribution

peak.

2-8

a rrti^ ,

j

1

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ttS

,

X

XX

X

X XX XX X

< X X

X X XX X XX X X

1X1 X X X X

X X X X X X

X X X X X X X X X

X X X X X X X X XX X X X X X X XX XX X X X X X X X X X X

. — LX X X X X X X X X X X X Xr-iX X X X X X X X X X X X X X X

Quanta ---=

i^

Passive Network("Noiseless" Phase-Shifter)

p

Ncrmal System

Figure 2-5. Encoding Data Noise

Vertical: 20 MV/CM

""^' r:n ,^ fad+

Horizontal: 0. 2 Sec/CM

2-10

rr {

w ^3

The conclusion must be that there is a second noise source associated

with the encoding process itself. That was tracked to the VCO within the

Excitation Section. In order to obtain 214

resolution with a moderately

r

high (ti10 kHz) excitation frequency and a manageable clock frequency (2.8

mHz), phase determination is done over 15 excitation cycles. The excitation

is generated as 15/2048 of the clock frequency. This has the effect of lin-

d

ing up the clock pulses in successive excitation cycles, as illustrated in

Figure 2-6. Each excitation cycle contains 273 1/15 clock pulses, leading

i

to the 1/15 precession shown.

A phaselock loop is used to obtain the integer 15 portion of theas

excitation-to-clock frequency relationship. Considerable short-term phase

jitter was observed in the VCO output, The phenomenon of output data noise

due to this can be related as follows. Consider the vertical lines in Fig-

ure 2-6 to represent a time gate which admits clock pulses to a counter.

As shown, the gate opens and closes at precisely the same phase each excita-

tion cycle. A count of one or zero might be registered depending on the

arbitrary choice of mean phase.

Widening the gate with the same mean phase would represent the result

of Inductosyn rotation. A random phase jitter of both gate edges, with re-

spect to one another from cycle to cycle would represent the result of the

phase noise previously described. There would be a corresponding variation

in the count from one block of 15 cycles to another.

Now assume that the gate width remains precisely constant, as would be

the case for the passive network, but that its mean position jittered ran-

domly from cycle-to-cycle. It is likely that at times it could sweep along

at just the proper rate to gather up 15 clock pulses. At cther times, it

might intersect none. At still other times it may sweep up some inter-

mediate number.

c

Clock Pulse2.8 mHz

Successive I

Cycles

355p..,

I

= r

I6 Z

f

7 Z '

8f Z

9 Z I

I

10 ---j-

^I

12 t I I13 Z

^

fI _.

115 Z

V_.Figure 2-6. Clock Phases in Successive E. ati on, Cycles

r

2-12

A

Hence, with a completely "noiseless" analog processing front end, but

a phase-jittering excitation, noise is still possible. This phenomenon is

a hazard peculiar only to multi-cycle encoding, since a single-cycle encoder

would not use a phaselock loop.

Basically, the PADS encoding technique appears to have been rather suc-

cessful. During design, it was necessary to make certain practical compro-

mises in the interests of working out and demonstrating the basic concepts.

Primary stress was placed on accuracy. This is still not assessed because

of our present inability to subtract the Inductosyn errors. But even if

one assigns the whole ±0.2 degree linearity error of Figure 2-3 to the

electronics alone, this is equivalent to ±3 arc-seconds at the multi-speed

level. This could probably be improved to the ±1 arc-second vicinity.

Little attention was paid to circuit shielding in PADS, partly because

of its expense, but mostly because of the desirability of exposed circuitry

in a developmental effort. It seems that a packaging style using a number

of enclosed shielding compartments will be necessary for the analog front-

end elern:-^nts, at least for the multi-speed channels.

Noise performance tended to be ignored in the PADS design. The choice

of the PA715 for the front end elements was dictated by a desired for a wide

bandwidth. Gain precision, and hence large excess gain at 10 kHz, is essen-

tial to accuracy. This device is not characterized for noise. Newer ampli-

fiers are available with reasonable bandwidths and noise characterizations.

Although they do not have the amount of open-loop 10 kHz gain, it is possi-

ble that a more judicious gain distribution can be used.

The requirement for a very quite phaselock loop was unsuspected, partly

because of the subtlety of the noise phenomenon previously described. Future

design will probably require a crystal VCO.

The present noise performance of the PADS gimbal readout appears to be

of the order of ±8 to 10 quanta, corresponding to ±2.5 to 3 arc-seconds,

multi-speed. Assuming that this is composed of approximately equal parts of

2-13

G

,r

analog front-end noise and VCO noise, that the front-end noise could be re-

duced by 5 using quieter amplifiers, and that the VCO noise could be elimi-

nated by using a crystal VCO, it is likely that the noise performance could

be reduced to the order of ±1.4 quanta, a fraction of an arc-second. Note,

however, that this estimate is made without consideration for those sorts

of non-linear sampling effects which may occur when noise and quantization

approach each other.

Evolution of the PADS software for the single-gimbal tracker system

is limited to those elements which relate to models, measurement, and con-

trol of the star tracker. All other software remains unchanged. Thus the

only elements effected are the catalog processing (star sort, selection),

computation of the measurement residual, and derivation of the measurement

matrix. The catalog processing reflects logic to acquire and track stars

of opportunity (rather than at periodic intervals), and the measurement re-

lated algorithms previously derived are, in this case, constrained by the

`` assumption of zero outer gimbal angle.

v l

2-14

R

i^

r

3.0 SYSTEM ANALYSIS

3.1 Star Availability Analysis

Assessment was made of the tradeoffs in geometry and mounting constraints

for use of the existing single-gimbal tracker in geosynchronous orbit appli-

cation. The key tracker constraints that must be satisfied include:

• Star magnitudes brighter than m y +3.5

o Tracker gimbal freedom < ±60 degrees

• Sun avoidance angle of 45 degrees

From a spacecraft systems application standpoint, certain other constraints

and/or design guidelines were established, viz.

• Only one single-gimbal tracker need be active

• Minimize FOV geometry interface requirements

• Configuration well suited to operation during all-times of year

A star availability digital computer program was developed which orders

stars (by orbit angle) in sequence of observation as constrained by the

gimbal FOV, The orbit angle between successive star sightings becomes a

measure of star availability, The digital program (Appendix A) includes

specification of orbit parameters, tracker performance and mounting para-to

meters, and input from a separate master star catalog file. Table 3-1 sum- Yti

marizes the results of a series of computer runs.

From the tabulated results, it appears that the best mounting geometry

is for the nominal boresight (zero-gimbal angle) to be directed at one of

the two poles. This minimizes the effects of sun interference (greatest for

boresight N in summer, for boresight S in winter). Some flexibility in star

mounting and selection appears achievable. Although choice of ±45° gimbal

3-1

Table 3-1, Single-Axis Gimbal Tracker Star Availability

(Geosynchronous Orbit)

Lir^

SoresightMounting

StarMagnitude(minimum)

GimbalFreedom(deg)

Time ofYear

EvaluatedResults/Comments

Zenith 3.5 Significant sun interference problem, particularly near equinoxes

45 6N 3.5 30"'

Spring/Summer

Spring - 69 stars, gap of 40 0 (about :gun)Summer - 53 stars,, gap of 119° (about sun)

45°N 3.5 45 Spring/Sumer

.Spring - 86 stars, gap of 400Summer - 67 stars, gap of 106°

H 3,5 45 Spring/Sumner

Spring - 58 stars, maximum gap of 270Sumner - 48 stars, maximum gap of 27°

N 3.5 60 Summer 74 stars, few scattered gaps of 12- 150

PI 4.0 45 Sumer 78 stars, scattered gaps of 10-12'

45^-5 3.5 30 Minter 80 stars, maximum gap of 750 (about sun), other scattered gaps of 10-22`

i 450S 3.5 45 I Winter 196 stars, maximum gap of 71° (about sun), other scattered gaps of 10-22"

S 3.5 30 All 40 stars, scattered gaps of 10- 230

S 3-.5 45 Spring 104 stars, scattered gaps of 10-17°

a

i

"'k A freedom appears reasonable, opening the freedom to ±60° improves performance

potential whereas reduction to ±30 0 , while potentially degrading performance

somewhat, minimizes spacecraft; FOV constraints. It was also noted that se-

lection of slightly dimmer stairs (e.g., m y = +4.0) did, in fact, somewhat

increase performance potential.

Figure 3-1 is a histogram which summarizes tradeoffs for cases of bore

sight South. A baseline system configuration is selected with the boresight

South and gimbal freedom of ±45 degrees. In geosynchronous orbit, 15 degrees

of orbit motion corresponds to an hour. Thus, the worst case time between

stars for the given orbit will be somewhat greater than one hour.

3.2 Error Analysis

PADS using a single-axis gimbal tracker will have performance character-

istics which differ somewhat from a system usin g a two-axis gimbal tracker.

This comes about primarily from the imposed limitation to utilize stars of

opportunity rather than being able to seek stars at periodic (arbitrary)

intervals. As a result, performance becomes more dependent upon gyro be-

havior, since updates will not be available as often (as noted through the

star availability analysis of the preceding section). TRW, in recognizing

this, has done considerable work in the modelling and analysis of gyro ref-

erence systems 133, results of which may be applied directly. This section

summarizes the system and component errors - the system error budget is

outlined in Table 3•-2. The star tracker errors and gyro data for drift and

bias uncertainty are based upon data and models developed in the sections

which follow. Gyro alignment and scale factor errors assume the full un-

certainty is realized with a uniform rate of change over 20 minutes. The

average update period of 20 minutes implicit in the results for gyro error

is consistent with the star availability in the preceding section. The RSS

of these error sources betters the goal of 0.001 degree (= 3.6 arc-second),

la, per axis.

I ^ t

}y

5

ZENITH

r

` ZERO BORESIGHT ANGLE OF ZERO BORESIGHTFRO?,1 ZENITH

.., . Sl.'.!ATH OFvBSERVED STARS \

GIMBAL FREEDOM = ±7

ORBIT DIRECTION OFNORMAL ORBITAL MOTION -

I LOCALVERTICAL

SUN!iMARY OF REPRESENTATIVE CASES.' GEOSYNCHRONOUS ORBIT, ZEROBORESIGHT SOUTH (ALONG ORBIT NORMAL) a

g0 GIMBAL FREEDOM, 45 DEGacw

Q 60

z . GIMBAL FREEDOM,.30 DEG0 40

w w ^co 20

0 1 t

0 5 10 15 20 25

ORBIT ANGLE SET%'IEEN STARS (DEGREES) I

*DATA AVAILABLE FOR MANY CASES. INDIVIDUAL STARS AND PERIOD a

BETWEEN SIGHTING IDENTI; t'--­D.

Figure 3--1

3-4

SUBTOTAL, RSS

COMPUTATION

2. 54 sec

1.0 se

TOTAL, RSS 3.29 sec

(1) Based upon uniform change over 20 minutes

ERROR SOURCE/BUDGET

STAR TRACKER

BIAS 0.44 e (1 Q)

RANDOM 1.31 sec (1 Q)

NOISE 1.2 sec

w GYRO REFERENCEu,

RANDOM DRIFT

BIAS UNCERTAINTY

ALIGNMENT STABILITY ,

SCALE FACTOR STABILITY,

PULSE QUANTIZATION ,

CONTRIBUTION (1 a, per axis)

0.44 se

1.31 sec

1.2 sec

SUBTOTAL, RSS

1.83 sec

< 2 sec over 20 Minutes 2 sec

0. 001°/Hr

1.2 "sec

10 sec

0.45 sec (1)

100 ppin 0.9 sec (1)

0.01 se

•- 3.2.1 Star Tracker Errors

The star tracker errors have been derived from analysis and test data

associated with the single-gimbal engineering model tracker [ 'Vl. The

tracker and electronics errors are summarized in Table 3-3.

3.2.2 Gyro Errors

The development of gyro errors is based upon a gyro model having the

characteristics as shown in Figure 3-2. The input axis rate, w, is gyro-

scopically converted to a torque acting upon the viscously damped float.

Angular motion of the latter about the gyro output axis is converted by the

signal generator to an electrical output whose average value is essentially

linear with respect to float motion. This output might simply be a contin-

uous analog signal; or else a train of frequency modulated pulses, with the

pulse rate dependent upon the float motion. The gyro electrical output is

fed back to a magnetic torquer in order to rebalance the float. As indi-

cated in Figure 3-2, appropriate selection of the gyro parameters usually

results in either a rate gyro or a rate-integrating gyro. The former are

generally characterized by fairly high bandwidths (e.g., 5 Hz or so)

whereas the latter employ low bandwidths (e.g., 0.004 Hz) in order to effect

their integration process. Clearly, in order to obtain angular outputs with

a rate gyro, an additional integration must be performed. With frequency

modulated pulse signal generators, this integration is easily accomplished

for PASS by use of a counter whose contents are read and reset to zero each

integration period, e.g., 200 ms. Thus the gyro output used for PADS is a

measure of the incremental change in angle measured about the gyro input

axis over the (short-term) integration interval.

Three basic uncorrelated gyro error sources have been postulated.

Electronic noise can exist at the signal generator input (or equivalently,

its output). Additionally, the various gyro mechanical imperfections give

rise to a torque noise, nv/H, acting directly upon the gyro float. Both

these noise sources will be assumed zero mean and white. The fact that some

self-correlation does exist in these noise sources is of little consequence

r^

3-6

r

.14 arc-sec

,46 arc-sec

.2 arc-sec

.5 arc-sec

.5 arc-sec

.2 arc-sec

.2 arc-sec

.7 arc-sec

.5 arc-sec

.13 arc-sec

.4 arc-sec

,27 arc-sec

.27 arc-sec

.13 arc-sec

.13 arc-sec

0.3 arc-sec

k

Table 3-3. Single-Gimbal Star Tracker and Electronics Error Characteristics

SENSOR

Electronic (Bias)

Thermal-Mesh (Random) (30°C Range)

Noise Equivalent Angle

GIMBAL AND ENCODER

Gimbal Alignment (Rando,^^)

Runout

Perpendicularity

Gimbal Thermal-Mech (Random)

Stress Relaxation

Thermal

Inductosyn (Random)

e

256 e

READOUT ELECTRONICS

Bias

Zero Crossing Offset

Phase Stability

Random

256 e Excitation Harmonics

512 e

Cross Coupling

Gain Unbalance

A-e Converter

Noise-Quantization

3-7

l

. `

^^^

= `°

`'

m ^^ n_

^ .^ b

`p

^ nV

^ .- ` FLOAT SIGNAL

DYNAMICS GENERATOR/` | | ^ l /

' , |. ' U S^ |

n -- —^~} -- ^^ 8

, ^

` |

^

. rPULSE COUNTER USED IN

DEFINITIONS TORQUE RATE GYRO APPLICATION^^ REBALANCE

'' ^H = GYRO ROTOR MOMENTUM

C. = GYRO DAMPING CONSTANT RATE GYRO: K. = H, T << l.' o ^^

'^ g

'

u, K = SIGNAL GENERATOR GAIN

/

no RATE INTEGRATING GYRO: K = C /^ ^ >> IK. = TORQUE REBALANCE GAIN

`d' '

g

m = GYRO INPUT RATE

b = FIXED GYRO DRIFT RATE

' p = GYRO OUTPUT

8 = INTEGRAL OF GYRO OUTPUT FOR RATE GYROS

z = C /KK- o 8/t -t \' ^='g d'^-.^ ^ ^ I 2 / " «

2

n_ =

WHITE SIGNAL GENERATOR NOISE ' E[O /t\] = 0 E[n /t \n /t \] = c i` ' ^` l / 2/

nyl

= WHITE FLOAT TORQUE NOISF/^ ^^^ ^ ^ f` — '8

\ ^'=^

"

! n = WHITE TORQUE DERIVATIVE NOI3F/H "J "V '

Figure 3-2. Block Diagram of Gyro Model and Definition of Parameters

`` ^_

;I

as their bandwidths normally extend far beyond that of the gyro loop itself.

Finally, there exists a gyro drift rate bias, by far the most significant

error source unless appropriately compensated. While it is tempting to

suppose that this drift rate holds constant at some value, b, it must be

recognized that this bias may, itself, vary over the long time intervals

normally encountered in attitude determination applications due to thermal

and other effects.. The model treats the drift rate as a random walk pro-

cess, biased to the value b at some initial time, and wandering thereafter

as a result of the zero mean white noise influence, n u . The fact that thin

drift rate model implies an unbounded gyro output variance with time is of

little concern here, as In u l is numerically so small that for the typical

time intervals indicative of attitude determination applications, a real-

istic drift behavior is indicated. Experimental justification of this en-

tire gyro model is demonstrated at a later point.

The characteristics of the integral of rate gy,-o output is summarized

in Table 3-4. Noting the definitions introduced in Figure 3-2, Table 3-4

shows how the integrated rate output, e, depends upon the various error

sources previously discussed. Transfer functions are first; presented re-

lating these outputs to the error sources; and then the output standard

deviation of each error is shown. Letting "E" denote expectation and g(t),

the pertinent transfer function impulse response, the standard. deviations

were computed by noting that

t t ,

6 e 2 (t) = E lei = EJ

n i (T) g (t-T)dT ni( X)g (t-^)da

0

t t tf

g (t-T) f E(n i (T)n i (X)J g (tr-;)dX dT = 6i 2 f g2(t-T)d`T0 0 L 0

i

Table 3-4. Characteristics of the Integral of Rate Gyro Outputs

w

0

OUTPUT TRANSFER FUNCTION OUTPUT STANDARD DEVIATION AT TIME t

GYRO ERROR SOURCE To LIMIT AST ^ <,NOISE SOURCE T > 0 g

g (IDEAL MATE GYRO)

WHITE FLOAT TORQUEs CT

la l^rz^^p /a

_^1-e'tht^l!~a'e/^s^ c^ ^Nti /2& l t t

NOISE/H

DRIFT RATE DERIVATIVE 1 1T9J

/Z[3iTs!' _ /l t! 2 a t3 /2

FLOAT TORQUE $NOISE/ H

[^8 s+11i r

c_2

-^/Tt _ 2c/^t ^tt 2 U

' 1

WHITE SIGNAL T8+1){T -21-e 2t/=91

0

GENERATOR NOISE Be

UNCERTAINTY, Ilt THE 1DRIFT RATE BIAS AT

s(z S

S+15 FTg(1-e_t/=g) a''b #1

!

ab

t-0

, . _, tab t® the standard deviation of the gyro bias uncertainty at t'0

3-11

s

Note that only the signal generator noise in the rate gyro integral applica-

tion results in a stationary process. This is because the noise is high

pass filtered by the gyro loop dynamics, and thus the gyro output has zero

noise power density at d.c.

Assuming the rate gyro time constant to be small but f%nite, the sepa-

rate effects of Table 3-4 can be root-sum-squared to yeild the dependence

of the output standard deviation, a,, to all the error sources:

2 1

6e + a t += 2 ab t +2 2 1 a 2 3 2-^=- v ^' u

t

It is seen that 68 2 contains terms dependent upon t 0 , t 1 , t2 , and t3. It

will be later observed that a minimum limit in ab , dependent upon av and au,

exists if the drift rate bias is to be calibrated using the Kalman filter

estimation.

Figure 3-3 illustrates the results of data taken on two gas bearing

gyros typically used in attitude determination applications - the Nortronics

K7G and the Honeywell GG334. Fixing a specific sampling (or integration)

time, t, the standard deviation a,, of a large number of samples (with the

drift rate bias removed) was experimentally established. This was done for

integration times ranging from 0.1 second to 106 seconds. It can be seen

that Figure 3-3 supports the model which was derived. For 0.1 < t < 10 sec-

onds, the data indicates no strong time dependency and thus can be charac-

terized as signal generator nose. For 10 < t < 2000 seconds, the slope of

the data on this log-log plot is about 1/2, indicative of the noise source

nv . As the sampling interval increases, the slope grows more steep, until

finally it reaches 3/2, matching that of the drift rate variation noise

source, n u . The analytical model derived for the data of Figure 3-3 is ex-

pected to improve with use of current gyros. For example, based upon lim-

ited data, av = 0.015 appears a good choice for the Bendix .64-PM-RIG gyros.

A GG334Q K7 GQ ANALYTICAL MODEL

0-y = 0.05;C k = 1.6x10-5 iJ

a o

SLOPE0

o

SLOPE _ ,^ o

,a

i1

10-1 i 0 10^ 10^ 103 104. 10' 106 107

SAt,1PLING TIME (SECONDS)

i

3

Figure 3-3. Test Data and Modelled Gyro Characteristics

sad4

3-12

Y

I

I

a

Performance tradeoff curves were derived which relate the modelled

gyro performance to the (attitude and gyro bias) estimation process. Error

is normalized to (attitude update) sensor random noise. Figure 3-4 is a

series of curves for estimated bias including gyro performance representative

of the state of the art. Typically, 0.1 < S.v < 1.0 and 0.05 < Su < 0.5 for

1 sec < an < 5 sec and update period of 10-20 minutes. Estimates of gyro

bias are on the order of 0.002 degree/hour - limited by the (ideal) case of

a perfect attitude sensor:

1

2Qu 2 T2

ab u rQv + 12 1

3.3 Covariance Analysis

A PADS covariance analysis digital computer program was developed to

evaluate PADS performance using the single-gimbal star tracker. The covar-

iance analysis establishes a lower bound on performance under the essential

assumption that the statistical models employed in the Kalman filter accu-

rately represent the real world.

3.3.1 Equation Development

The equations are developed to establish the covariance associated with

estimation of attitude and gyro bias. Thus the state vector, x, comprises

three attitude variables (e.g., e i ) and three gyro biases. The state error

covariance, E[xx T 1, is propagated according to the Kalman filter equations:

S j+1 Oj+1

P J+1 + aj+1

-1Kj+1 - Sj+1 Mj+1 IMj+1 Sj+1

MT +1 + Rj+1

Pj+1 = [I - Kj+1 Mj+1 I Sj+1

3-13

.01 .02 .04 .06 .08 .10 .20 .40 .60 .80 1.4 2.0 4.0 6.0 8.0 10.0

Q t2/2S = v

un

Figure 3-4. Performance Estimate Tradeoff Curves

/

° Cr t3/2

du ui

n ^i

du = 0.05,.-K = 10—

au = 0.5, K=

f

LL

"v%TQ = Ks n

w

Cra

v

N

ON

OO

00

Pj = E[xxT ] at time tj ,just after an update

Sj E[xxTa at time tj just before an update

It is necessary to derive the state transition matrix, the state noise

covariance matrix, Q, the measurement matrix, M, and the measurement noises4

covariance matrix, R, based upon the system model.r

The spacecraft is assumed to be three-axis stabilized to an earth

pointing, local vertical reference -Frame rotating at orbit rate, w o . The

attitude of the spacecraft [b- frame] relative to the reference [r°-frame] is

._ given by:

j xb 1 03 -0 xrI

yb = -0 3 1 a1 Yr

fr ! Z 02 01

1zr

PADS is assumed Configured with gyro measured rates (attitude) in three or-

thogonal axes (b-frame]. The single-gimbal star tracker is mounted to pro-

vide mechanical scan in a plane normal to the orbit plane (shown previously

in Figure 3-1), with tracker axes_ [ j -frame] related to the body axes [b-

frame] as

xj 1 0 0 xb

yj - 0 0 r1 yb

zj 0 1 0zb

l 1h,

1

where

x^ = axis of gimbal rotational freedom

yj = nominal sensor boresight at zero gimbal rotation

z^ = form right-handed orthogonal frame with x j , y3

From Section 3.2.2, each gyro is assumed to satisfy a mathematical

model given by

Wm = w^ + w' + v^ i = 1,2,3

where v^ is a white noise component; wl is the random walk gyro drift rate;

and

W = e l - w o 63

W = 8 2 - wo

w303 + wo el

Therefore:

wm = e l - wo e3 + w l + vl

wm 62 - wo + w2 + v2

For estimates:

1 wm = al- 11063 +w

2wm = e 2 - wo

wm = 0 3 + wo 0 1 + 113

.' 3-16

__ f4

t • .^

A

^j^

ell

i'.

Setting 60 i = 0 - O i and 6W^ = w^ - w^

66 1 = W0 60 3 - 6w 1 - v 1

6® 2 = -6w2 - v'^

60 3 = -`oo 60 1 - 6w3 - v2

Also, as w is a random walk

w' = u' , w^ = 0

and therefore:

6w^ wi - w^ - u^

where u i is random white noise.

Since pitch and roll/yaw decouple kinematically, separate solutions can

be obtained for pitch and roll/yaw. For pitch

662 1 -Tj +1 662- tj+1 -v2-(T-s)u

6w2 01 6w2J + u2 ds

+1 j tj

For roil/yaw:

60 1 0 -1 tuo 0 6e -vl

6W10 0 0 0 6W1u^- +

60 3-coo 0 0 -1 663-v3

6w30 0 0 0 6w3u3

t

6w i (t) = 6w'(t^) +

t i u ds; i = 1,3; t t < tj+1

_:l

-

K ./'

68 = -«u^ - d ^S + m, 68 - V

~''" 3

! ` t( ~^

.^` i^

^ ^3 f 3 3

66w 68 - 8w - V ds - V3 u l'

/^ u ^^ t^^ ' J

for which the hODOg8n8nUS solution is developed a3:^-.`

'(t- ' o^^ m, ~ z^. ` ^Jfl' u '+l '+l v ^fl

/t \ O l U) ,_ ^ ^^fI° =- ^

/t' -sin m_ T. 0 cos m~ T

' `^` 'J+l' u J+1 v '+l^ .

` Uxu"/t )U ^ U O U1fI

^^^ ` ^^^ °^

The (full) state trOASit10D U0t p iK is sumDOrized:. i

"^-

'

|

'

/0 `t )

- /t \^^

/- T^t'fl^i\

1 j 0nw~(t )J

^` C -^

jf| O O 3 O

' O l O O 0 0

^ O O l -z.Jfl

O OW =

' ~ O O 0 l D O

'/

S O O O C -T.J+l

KO^'

U O O 0 I 8_^

U^y ^, where: C = cos w^ ^^+l,^. - .

S = S1O w,- T^f1~

3-l8

,``

Al

1.

The state noise covariance matrix for roll/yaw is developed from the

particular solutions:

^. cosw0s scosw0 s -sinwo s -ssinwo s -v

t 0 1 0 0 ul q2x(tj+1 ) B(t) J sinw s ssinw t cosw s ssinw s -v3 ds = 33 q r

K 0 0 0 14

where B(t) is the matrix for the homo1geneous so ution,

After very considerable algebraic manipulation,

E [q'J E [q3] Qu- T 3 - 2- T^ (1-cos w0 T) Q^ TW 0

E Lg2J_ E ^g41 = Cyu T

E [qlq2J

=E[g2g1]=

E [g 3g 4] =1-cos w T

E Cg4g3]= u 0 T2

JJJ

2L

wo

E =[q l q 4] 1rE Cg4g1J=

E rg 2 g 3] =L J [q3

lau(sin wo TE

g2J=

w0 .\T -

w0 -1 \

E Lg1g3J = E [g 3 g 11 = E L12g4J = E Lg4g2J = 0

Thus, the full state noise covariance matrix, Q, has the form:

ate--*

6u sinwoT2 1-coswoT 2

u -/

0 0 0 6u ---^-- T

w0 \ w0 / w0

6uT

2

l

4 3 2T 2 -cosw T 0 Qu S 1 nW0T

2u

1-Costa T^ +6T 620

20 " w T w0 L` 0 V u^ 2 0

w

w0 0

2

2.1-coswOT 2 2 0 0 ^u T- sinwOT

6 2 — T ^uT W0 wu 0w0

0 0 QVT 0 0

Q

0 0 0Q2T 0

WNO

0

0

0

Q2sines T 11

0 u T- 0 0 0 6u[3 T 3 - wT

1- COSw 0T 1

J FQvT

w0 w0 0

2

I-

COSwOT2

Q u —r, — - T

0

I

i

f 1

6

{

i<

-^a

r

R

{

<I. To establish the measurement equation, results previously developed

for the two-gimbal tracker are used directly with the constraint that the

inner gimbal is "locked". Recalling the measurement equation as:

d YI -u1 0 - 1-u 1a'1M - n1

Yu3 + _ u2 0 a M - n

y° 1-u 1-u2 2

i 1 1

where

y iis the sine of the inner gimbal

yois the sine of the outer gimbal

a1Mga2Mare the star sensor outputs

} n1,n2 is the sensor noise

and s''

x

u ius yj i = 1,2,3

zj

where us is the star unit vector defined in the reference axes as:

r,

u s = a 1 x + a2 Yr + a 3 zr

The key constraint is applied by assuring that a 1 is first order, a 3 < 0

and z a i l = 1

3-21

P"

Therefore:

u1 = a l + a2 e 3 - a3 e2 (1st order)

lu2 = -a l e 2 + a2 e l - a3

xu3 = -al

e3 + a2 + a 3 el

Substitution for u in the prior equation yields:

..-j 3 p'

YI 0 a3 -a2 e l-a1 -a2M + n2

Y = = e2 +

Yoa3 0 0 e3 a2 + a3 a1M - a3 nl

0 a3 -a2 e l-a1 -a2M

y = e2 +^

a30 0 e3 a2 + aM

3 a1

and

0 a3 -a2 n2

by = y-Y = de +

a 30 0 -a3 nl

Therefore, the measurement matrix, M, is given as

.: 3-22

°i,i

0 0 1-a22 0 -a20

Mj+1

- 1-a2`0 0 0 0 0

The measurement noise covariance matrix, R, is given as:

1 0_ 2

Rj+1 2 an0 a3 j+1

3..3.2 Performance Evaluation

The covariance analysis digital computer program was used to evaluate

performance potential and tradeoffs. Those elements evaluated for tradeoffs

include star sensor noise and gyro random drift characteristics. The com-

puter program provides a time history of the error covariance. Summary data,

used as the performance measure, includes the average estimation error covar-

iance and final value of gyro bias estimation error covariance.

The program was run with several sets of initial conditions to evalu-

ate both the convergence properties and steady-state performance. Conver-

gence to steady-state accuracies from initial attitude uncertainty of

0.1 degree (1a) and gyro bias uncertainty of 0.1 degree/hour (la) was

achieved within four (4) star sightings. Cases for evaluation of steady-

state performance had initial conditions of 10 arc-seconds (1a) in attitude

uncertainty and 0.05 degree/hour (1a) in gyro bias uncertainty. Small ini-

tial transients were noted in the attitude covariance.

Table 3-5 summarizes the observed covariance analysis results as sum-

marized by the computer program performance measures. The gyro parameters

are consistent with a range of values based upon observed test data (see

Figure 3-3). The star tracker errors are consistent with those developed

in the error analysis of Table 3-3. Evaluation of performance goal of 0.001

degree (= 3.6 arc-seconds) is achievable in the high altitude (geosynchronous)

y3-23

, r

wN

2

2

2

3

3

3

3

3

Attitude Estimatea e ( c)l

4.6/6.1/6.2

3.2/4.2/4.1

2.6/3.1/3.1

5.0/6.4/6.5

3.5/4.4/4-.3

3.0/3.5/3.4

3.1/3.8/3.7

2.5/2.7/2.7

Trucker NoiseGyro Random Drift

( es c) (sec/sec) (sec/sect)6n^v 1/2 6u 1/2

(rad/sec) (rad/sec)

.115 3.33x10-5

.05 3.33x10-5

.05 1.6x10-5

.115 3.33x10-5

.05 3.2x10-5

.05 1.6x10-5

.015 3.2x10-5

.015 1.6x10-5

Gyro Bias EstimateQb (deg/hr)

1

.0024/.0026/.0025

.002/.0022/.0021

.0011/.00121/.0012

.0024/.0026/.0025

.0016/.002/.0018

.001/.0012/.0011

.0014/.0018/.0016

.0009/.001/.0009

R;

i

±e,

Table 3-5. Table of Performance Results (1)

( ' ) All cases - geosynchronous orbit, real stars brighter than m y = +3.5,

star tracker boresight South, gimbal freedom ±45 degrees

application using the current PADS single-gimbal star tracker and available

state-of-the art strapdown gyros. Certain general conclusions may also be

developed, namely:

• Small star tracker noise variations are observed to have little

effect on overall performance

• Gyro drift characteristics have the largest effect on performance

s Changes in the gyro drift term, a v , have a moderate effect on

attitude and bias estimation

• Changes in gyro drift term, a u , have (almost) a direct effect on

the bias estimation

A typical time history of the attitude estimation performance is shown

in Figure 3-5. The time-history plotted is the square-root of the trace of

the attitude error covariance matrix. Selected case history output data is

included in Appendix B.

3-25

• d.

.' Imi

0 1 2 3 4 5 v 7 8 9 10 11 12TIME (HOURS)

k

GE OSYNCHRONOUS ORBITTR/`+CEEB BORESIGHT SOUTH,GI^P,IBAL FREEDOM {45 DEG,REAL STARS mv> 3.5STAR TRACKER NOISE = 2 SECGYRO NOISE: ca = 0.05,

1.6 x 10-5

r

i

w t°

r^rn (D

W

c0cc

ul

z

0 16

Q U^L'J

12w^t--LIJ

° 8ul X°a

ua

e 4

0

y

t

L

y'^^ . s r

(r-4,E

4.0 REFERENCES

(1) 13900-6014-RU-00, PADS System Design and Analysis (Two-AxisGimbal Star Tracker), 1 July 1973.

(2) 13900-6015-RU-00, PADS Star Tracker Test, 1 July 1973.

(3) Farrenkopf, R. L., Generalized Results and Other Considera-tions in Attitude Determination Involving Gyros,TRW IOC 74.7530.3-11, 21 January 1974.

APPENDIX A

COMPUTER PROGRAM FOR STAR AVAILABILITY

A=1

r

A.1 Program LOOK1

Program LOOK1 is a FORTRAN program to determine and order stars ob-a

served by a star sensor mounted on an earth orbiting spacecraft. The

program works from stars in a general catalog (input file TAPE1) which in-

cludes RA, DEC, magnitude, and catalog number. Each star is tested in

1 turn for the position in orbit that observation may be made including tests

for sun interference, earth interference, and constraints of orbit, time of

year, and sensor/spacecraft geometry. If observation is determined, the

star is ordered in a working catalog on the basis of orbit angle from as-

cending node at the position of observation. The star LOS is defined by

orbit angle and angle measured from zenith at the indicated orbit angle.

The input file used for current investigations is given in A.2 and the

output file of ordered stars used for covariance analysis given in A.3.

r;

A- 2

77

00310 C TEST ON STAR MAGNTTUIE'00320 IF(MV. GT. MVLTM)GO TO 500U330 C FORM STAR DIRECTION COSINES00340 RA=DEGRAD*RA00350 DEC=DF(s' RAO*DFC00360 _____ . A; =CO S DEC)*COS(RA)00370 A2=rOSf DEC) WSTN(RA)00380 A3=SIRIDEC)

I,00390 -C----, -UNTEST-5 -.lNT-EPfEF-EkLr-,E-00400 SI=GOS(S)00410 S2=STN(S)*COS(.408)B042n 332:-7 SlIf IS) *

'

_S IbU A_4=j00430 DOTSUN=Ai*Si+A2*S2+A3*,-z 3400440 lF(DOTSUN.GTsCOS(StJNLTM))GO TO 500100450 NN!.3

00460 FLAG=0.80470 C DETERMINE REF PLANF-ORBIT PLANE INTERSFOTTON00480 --

iG0490--

92=-SIN(TNCL)*COS(OMEG)00500 83=COS(INrL)

illilaft ..-PROGRAM LOOKI(IMPUT ,OUTPUT, T.APFI=INPVT-,-T-AP-E?,T-AP.E3Doila C TAPE3 CONTAINS MASTER STAP CATALOG00120 DIMENSION NNS(400)gNUS(400),GAM(400)gALPH(403),H-i-32 REAL. TNCL,.MV2'00140 REWIND 3:00150

DATA OMEG,INCLvS/0-,0.v0./__DATA MVl-IMs-S(l-NLl-M:3-B--T

00170 NAMFLIST/DATA/OMEG,INCL,S,MVLIM,SUNL-IM,9ETA,GAMLIM00180 READ(IIDATA)

---r)l-aPLA Y -*PIKE,13 .9-1 N(^Lia.5-=^ I N

00195 DISPLAY *MVLTMvSUNL T-Mv EFTA, GAMLTM= *vMVLIM,SlJNLIM,BFT4,GAMLIM0 702 '0 0 DFGRAD=3.i415927/180.

-G-PAa—.00220 SUMLIM=DEGRAD*SUNLTM60230 GAMLTM=DEGRAD*GAMLIM00240

-D LQ-(IMEG= EGRA,04- -MU-..-V3250 TNCL=DEGRAD*TNCL100260 S=DFGRAD*SiGn27 -0_-_- ACCEPT,00289 ^ L=100290 w no 500 T=I,NS

-A-C '3) SL v 30 Ml!.-C -EPT C _N' -NF -iPA,DE('

t

n

00520 ,rV2=A3*B'-Ai*R300530 V3=Al*B2-A2*8100,5A0 --- -- W1-R2*V3-133-*V2.^00550 W2=@3*V1-91*V3;00560 W3=B1{V2-32{V1^00570 ._- _-___ 4(=50RT ,(Wl 'WI+W.2-*W2±N3^`W3)__.—

_-00580 W1=W1/W00590 W2=W'?/W00600 W3=W3/W

160610 c COMPUTE ANGLE FROM ZENITH FOR STAF LOS00620 GAMMA=ACOS(Wi*AI+W2;A2+i43*A3)0063.0 ._ _ _-TF4 (Ai*$1±A2*RZ±A_3*B3).(,-T.C?_.,)GO_TO__100 e00640 GO TO 200 f00 65 0 100 GAMMA=-GAMMA

_20000670

-(GMMA--GAMMA*BETA) .,CT GAMLTM- ) ;0 . _ TOGAM (L) =A

00680 C COMPUTE OR9TT ANGLE FOR REF PLANE INTERSECT sj00690 ANI=COS (O'(E(-,)00700 AN2=SIN(OMFG)00710 A SALpHAO=B3+(-AN2*Wi+ANI*W2)+(81*AN2-B2*AN1)*W,3 t00 .720_ - _._._ CALPHAO=AN1{W1+AN2{N2'00730 AlPHAO=ATAN2(SALPHAC,CALPHAO)00740 ALPH(L)=RADDEG*ALPHAO,00750 IF(ALPH(L).LT.O.)ALPH(L)=ALPH(L)+360.00760 C SORT ON BASIS OF 099TT ANGLE00770 250 NUS(L)=L00780 ITFNP2=NUS (1);00790 60 300 J=1,L.00800 M=NUS(J)00 Q __ _- TF_QL_PH(1,)_.LT. A_L HTl ) !: __T0 4U.n_--- -00820 ITEMP2=NUS (J+t)00830 300 CONTINUE

00 8 5 0 00 450 K= J, Lj0860 ITEMPI=ZTEMP2

10, 870 _-___---00880 NUS (K +i) = ITF'MPi00890 450 CONTINUE0 0 900 0__Cr..___.._COMpL-ETE SORT

—_------ -- --0905 C TEST FOR ALPHA +1800910 IF(FLAG.GT.1.)GO TO 480

3

_.-___^—_-----_--

00930 IF (,a (180.—GAM (L)—BETA) .CT.GAMLIM RADPFG) G rl TO 48000940 L=L+1

.^ 100960 GAM (L) =180. —GAM (L-1)

i ;00970 ALPH(L)=ALPH(L-1)+180.:fl1^.0_.__ ___ .IF.s1^L.PH.LL)^_G.S^3:5.17_._L _ALPt-}(L)=AL_PH(L)-3^^___.-_____- ___. _^— :_ ^•

00990- GO TO 250

01000 480 CONTINUE

01020 500 CONTINUE

01030 C END OF SORT TTERATION LOOP

01050 L=L-1

01055 DISPLAY (2) L

j !01070 J=NUS(I);01080 DISPLAY (2) I,NNS(J),ALPH(J),GAM(J)

01100 FNDcn

f

a

I

j

4 f

-. ' + ^_ _^_",_ ^^i III YI II x^ob^slaLYdNi °a"' A

P ! `a

,

A.2 Yale Catalog of Bright Stars to 4th Magnitude

Data includes (from L to R):

i • Working catalog order (ordered by magnitude)

• Yal Catl og Number

• Right Ascension

• Declination

• Magnitude (S-20)r

A-6

r

F1 2491 100.91355 -16.67130 --1.46002 4210 160.93650 -59.51870 -1.00003 -2325 .^_ __^95+ 491Z_._.-5: ,-66 ra30- -.5754

^ Y4 7001 278.94600 38.74430 .04005 1713 78.22530 -8.23967 .0487

i 6 47.2_ -2.4: 000 925997 5267 210.35700 -60.20230 .34408 1708 76.;54330 45.96600 .6530

_ _10 5056 200.84850 -10.97430 .79i9

1 11 5340 213.52800 19.35900 .7217t._iz _ _Z^i64_._ _^9Z^..312 f.0_ .._. 32..9.3170 .7525

13 5459 219.32250 -60.69170 .835614 4853 19t.43600 -59.51300 .9698

r 15 Z5 937 -- .- . .._.^ . R . 77 6 n it X71 216 3982 151.63950 12.13t00 1.242017 4730 186.17700 -62.91300 1.2480

-343.344 51._ _ --2 1g ag 800 1.249019,D 2518 104.32395 -28.92130 1.254020" 2061 88.33125 7.39433 1.3320

_ 2 7 4 ? 4 _ __ n .Il 5 8..0.4 . -. _ 45,_L4.73..0 _._-1,349022 6527 262.82400 -37.07730 1.350023 1790 80.82720 6.32167 1.3820

__ 2Sz_.- 140 --8-3. 6-0-3-0. ._-182.22.53_ i. 4a0025 1457 68.4918g 16.43200 1.500026 1948 84.76245 -1.9F700 1.5050

28 3207 122.12130 -47.24800 1.525029 8429 331.52250 -47.13100 1.58903 n 2 89-0 _ _._..113 t-1-0715-- i . 611031 5191 20E.54850 49.48670 1.638032 6134 246.83250 -26.35970 1.6380

-_13^,._2la 6.5 5 - tom°• 57500 1.701034 7790 305.73750 -56.84100 107080,

! 35 2294 95.30100 -17.93300 1.7100!_ 36 4gU _19,,3jj4Q0 56..13700 1.7390

37 2990 115.80930 28.10170 1.730038 6879 275.47800 -34.40030 1.8090

._39-.4131_ _....1$6:1$150__ -E2. 95300 1.832040 2004 86.53785 -9.67800 1.842041 7121 283.2900 0 -26.33970 1.8780

f_,f

9

42 15 1.65666 28.89630 1.913043 2421 98.93760 16.42830 1.930044 2088 89.215005 44.94430 1.931045 3165 1.955046 1852 82.56930 -.32267 1.966047 3485 i3D.9440lk '-54.59200 1.991045 5958 239.52000 26,01300 2.000049 2891 113.10735 31.95700 2.021050 5132 204.43200 -53.29100 2.042051 5469

-219.91800 -47.25270

--2.0540

936 -4-6. 4. 9191) 40.81970 2.087053 5440 218.33550 -42.00270 2.128054 4819 189.91059 -48.77970 2.13905, 5 ' -6,5, 8--o 2- 6- 5 . 0- 3 3-5 0 -9 -.3- 0 16 3 0 2.152056 1017 90.475611 49.73100 2.192057 5953 239.58000 -22.52930 2.202053 5793 233.31150 26.83009 2.209059 6556 263.34000 12.c;8930 2.225060 6553

--176.83350263.7J800

- -- --42.97730 2.2280

61 404 ' 14.75930 2.2?9062& 4763 187.32009 -56.92409 2.244063 - 37.34 - 140.2-6.56 0 ,54.86930- 2.268064 5231 208.35300 -47.1330O 2.282065 5054 200.64000 55.10900 2.290066 5460 219.32250 -60.69170 2.317067 2827 110.68515 -29.23200 2.326068 4295 164.94900 56.56470 2.339069 2693 106.75425 -26.34330 2.3550

F---70 264 13.66217 60.53530 2.404071 5571 224.07609 -42.99730 2.412072 3699 139.04685 -59.12500 2.412073 4301 165.40950 61.93130 2.43?074 8781 345.76800 15.01870 2.437075 -4554-.--- 1 53,88700 2,440076 6508 262.11300 -37.27170 2.454077 4662 183.51600 -17.34630 2.4820

LI-8--ti,931 33.44185 --.5. 93933 2.487079 4798 188.78850 -68.94630 2.488080 6378 257.10900 -15.67700 2.4990

-9.?5867 2.4920f 82 1956 84.60510 -34.10030 2.5010

83 424 31.96755 89.08570 2.5080

1 _tf v

__._B^_.^.X07.__,_125^_4:^4gQ_..__-.5:2,^9^32._ __.2,^1^__^. ---_-•. ---- -------_.__._-___ _---- ----_ .__---- ___.___.-____ ______85 4199 160.43400 -64.20770 2.514086 5776 233.22150 -41.05330 2.524081 4F 21_._._1$1:?.0 Q_,----,5D.52970 2.5340

F84 2095 89.34975 37.20000_ ___

2.5450.- ___^.-__.__.__ .._-•- --------._.._._--

89 5984 240.86400 -19.70930 2.545090 4b5fz_.^3$ 33 6-00.__..- 5$5-15 3 00 2.5500 ---- ----_.._-^-.-91 21 1.83834 511.96300 2.561092 6217 .251.26500 -68.97100 2.564093- 39-. 87051 330 2.572094 6175 248.82000 -10.49870 2.580095 8162 319.44300 62.44170 2.6260

T96.._^a_7__^_15§.:2700_ - ._20.,_02000 2.6350_-97 337 16.95735 35.43530 2.6450

98 3748 141.47790 -8.51933 2.6470 99_ 211031_.312110 __ --- 23.29130 2.6550

100 5944 239.19750 -26.62030 2.6700'101 188 10.46942 -18.17030 2.6740 3iA O? 4357 16 8103, 719 4 __. 28 5 .11250 - 2 9.917 7 0 2.6800 - --• --- ,,-. _______.^ ^.. _. _ _. - ---_---:. _- _ _._._ _ ._ . -------__---- _

10 4'0 5288 211.17000 -36.21330 2.61340105- _ 1220- 58.89165 39.90370. 2.6930106 8636 340.15950 -47.05900 2.7190

-

107 5563 222..79650 74.29i70 2.7200__1Q9_. 2845_^ 111.32700 ______ 3.35133 2.7330

109 7796 305.25000 40.14230 2.7410110 6510 262.30500 -49.85500 2.74201.11 4915 193.60950 38.49800 2.7610

1.12 603 30.44955 42.16900 2.7510_-

113 1165 56.36505 24.00900 2.764028 ..19220 - ._- 29._6357 . 0 2.7770

115 11165 82.80795 -17.85600 2.7970_

116 2282 94.74915 -30.04970 2.7980_117___,5028__ 1.9°_.6.68Q0 ._ _ . 77.3:6..535.3 -- ^ 2.8010118 403 20.89740 €0.05770 2.8070-ii9 3534 136.68690 -43.29730 2.8i90

_x,20.___.4844__.. 191.04900 -67.92_970 __-2.8300121 1122 55.12425 47.67570 2.8380122 7528 ?95.97700 45.04930 2.8490

^.123 y . 1,910 - _ 83.9.0385 21.12730;124 5576 224.23500 -41.96400 2.8500

'^ - ° 125 6705 268.9530'0 51.48900 2.8610

i

Yq'

i

-12-6-,,. 5 3 327.96900 -37.52530..._- _2•.8710__-_._127 5671 228.93450 -6.9.55870 2.8900128 5531 222.25059 -15.90330 2.8950129 8 238

'130 -

168 32.9.64608 76.345302.

.8980`--131 6247 252.39150 -37.99330 2.9060

._132 4757 - 187. 02300 -.16. 3 297 0_ __.?...-9080133 1E66 76.54455 -5.12867 2.9270134 6241 251. 99100 -34.23770 2.9350135 5695 229.78350 -40.52530 2.9520136 2653 105.40275 -23.78230 2.9550137 3445 129.87645 -46.53100 2.9570

_138__ -. 12- 03,_ 57. 49g 3r0 _ ._3t. 7 813 -0 _- 2.9570-139 1641 76.03275 41.18800 2.9720140 7235 285.96300 1381570 2.9900141 ._5:505 . 220.87500__ __._27^•?2500..___ 2.9,93,0. `_142 6084 244.7835o -25.50400 - - - _3.0260

_._..- --- - ----__ _-

143 897 44.24085 -40.43600 3.0270144 - 99 6.15150. -. 42.48700 3.0360145 y 8322 326.28900 -15.28630 3.0460146 F!, 6453 259. 97853 -24.96600 3,0460 ^Y14704825_ 1 , 8.9._98550 _._..-1 . 26 3 00 -3-_0490_.__.__-143 2451 99.18105 -43.15500 3.0520149150_._

639683.08..

257„17500325.. 62750

65.756309.72467

3.05203.0530 `.

i51 6165-_. _

248.44050__. .-28.14870 3.0610 CZ

152 4467 173.55300 -62.82970 3.0670153_ - • 7039 _ .280._88550___ -27.03400 3.0?3fl154 794-9 31-1-1.20950 33.84200 3.0870'155 5735 230.24550 71.95230 3.1000-155- - 5897._._ .23 $.. t -'A'U9 -_ - -63.32570 ._ 3. 1 TO157 591 29.,42385 -61.73100 3.1250158 622 31.17755 34.82470 3.1270

!160 1788 80.69295 -2.41733 3.1400i161 5948 239.46900 -38.30370 3.14201._._1.62,_....5190 206.8650a -41.51330 _.._3.1420________ -_____163 5708 230.08950 -44..57530 3.1500164 5235 208.26600 18.57000 3.1530165_ - ..- 14 5 5 6 8.3' 5,85 -_55.11800 _3.1 53{1_-_....__._- -- ^....;166 1702 77.85090 -16.23970 3.1520!167 8775 345.53400 27.89630 3.1680

t

-

^^ r

169 6462 260.03400 -56.34930 3.1780170 1879 83.31585 9.x'1067 3.1920

121. 52475- _-Z4 21137 Q172 7178 284.41650 32.63800 3.1970173 5435 217.67850 38.46400 3.2110

.--------175 7264 286.93659 -21.07330 3.219n176 6410 25e.40800 24.873D0 3.25!00

_iZ7..--_-365_g _.137.51775: 178 4037 153.23550 -69.86330 3.22501 179 5193 206.89200 -42.31330 3.2360_i_8^ __ _ _3117__ _ . _ ii 8^..4:Z910 __ _-.^2_^:.8.9 2.7 Q-- -- ^.2_S Qom_--- -_ _ _ ____ -----._- _. _ _ __ ____ --- --- _-_--_-•-- ----_..- --- __ . _ _ . ,.^---181 4786 188.14350 -23.21300 3.2600182 98 5.99126 -77.44270 3.2750,

,_.1.8.3..- _..22.9... 59.5 9.4.0 _ _--i_ 3710, 3.2 8- --- --- - ----------- _ _ --- -- -- _ - - - ... _-___ -_ - -184 8634 339.939D0 10.65770 3.2330185 4216 161.32650 -49.25200 3.2830

X 186 1573 .---- -73.696,5._..___ 33.09900 3.2910_----____.__._.--.187n 4359 i68.1i550 15.62030 3,3000188 6897 276.11550 -45.98370 3.30208^^ 550_ . _ 220..87500 __---27.22500 3.3030199 5854 235.55150 6.52433 3.3080

3569j 191 134.22195 48.36930 3.3120X142___6252.. 252.50850.-37-,96000 3.3140

193 E212 25D.00050, 31.66230 3.3150194 2773 108.98715 -37.03770 3.316095_ 5354 . ___-214.30650 -45.90800 3.3300

--- - -196 7236 286.11150 -4.93433 3.3330197 8709 343.21200 -15.99300 3.3400

_ 195 rat 32_._.._245,._^^5^_-51..3-793, 3.3460, -- -- -- -- -- -199 6056 243.13800 -3,59833 3.3490200 7525 296.15850 10.53170 3.3510Zn . -6.85-9. _. 74.,3:0550__. _- a :3370 .-3.3600

- - --- _-- --

202 6148 247.18800 21.55700 3.3850'203 153 8.75851 53.71300 3.3880

1-U-4- _4560,-.__ .18. 57.22 030 3.39 00205 4140 157.70250 -£1.50770 3.3930205 5463 219.93750 -64.81930 3.3940

3890,.. _145. ,_56245 -64.90- 8 00 -3 .3990-^238 3447 129.83130 -52.79770 3.4000209 6615

i

266.30100 -40.i0530 3.4070

^ -'<x _.. e...:..a..w . ....__ - •. :;s. y.St... _,.;.,;- ... _ .......^...,x...n.._...,:-. ._ .. --. :.::. _... _:. ,,,. .^_-:_.^ s^'.-° _""'_ ^ -_'wwiMBUUI..'sa'.rti"s'_- ^wddYM'l'

J

211 1829 81.69140 -20.77830 3.4160212 7106 282.20400 33.32700 3.4200

213-.--674

- -33.82185 -51.67530

-3.421-fl ----

2 14 134-7 - -- --' -- - -----64.15335 33.88500 3.4210215 660 .3 265.44759 4.58367 3.4270

Ljj5 _.-1:.5,05 8.81 05. _. ..-_43,7_8,23 0 --!* X 00 - .. _. __----- -- -- _ _ --- _-- - _ -217 3940 148.91775 -54.41900 3.4340218 915 45.58380 53.36970 3.43902 1 9 4932- _ t95.12150 11.14800 3.4550220 2550 ig1.96220 -61.89930 3.4590221 3975 151.36950 16.93100 3.4590

L22Z---I-Z!5 --7 8 -3ATZ 5- --=6.e84QQ 3.4610223 2286 95.22735 22.53370 3.4680224 51V 203.24109 -,42433 3.4680225 8232 . 3 2 2 _. 444 5 0_- 1 -- 5. 73 0 6 7 3.4710

1225 581 2 234.14400 -29.67000 3.4730227 6913 27E.46800 -25.45030 34790228 _ 1931 8.4. . 26115 6 17 0 0- 3.4800229 -4 d3 3' 153.76200 43.08E70 3.4310230 r3468 130.55445 -33.07530 3-4820

-?-3 t ^ 4 -r2 1 -- - - -2 51jL^l T5 0 3,491Q232 5881 236.96100 -3.31467 3.4970233 6500 262.90650 -60.65500 3.5030

72.- 350 85 5..54333 3.5030235 8502 334.04250 -60.42000 3.50190236 1567 73.11915 2.39333 3.1510023 7____ 838 __ 41.99580 27.12500 3.5230

;238 8762 345.08850 42.13530 3.5240239 3573 145.98075 23.92530 3,5280

40__11 7a -56, 78595 .k,910-23-d - -UQ -.1. 5 3!iO--241 8650 340.35150 30.04100 3.5430242 8414 331.00950 -.48100 3.5550

- . - 804 4-D - 38-k3 5 -_ - 3,-03t,57- 3,5530- ' 244 8675 341.62500 -51.49800 3.5600'245 6743 270.99603 -50.10000 3.5640246 3775 142.64445 51.83630 3.5690247 1142 55*71240 24.00900 3.5720248 1543 72.00060 6.89333 3.5720249 5?48 209.04900 -41.93570 3.5740

! 25G 2553 102.27225 -50.57700 3.1378051 1412 66.68070 15.79300 3.5320

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1_244.__ 5511 _- 2.21:•13300 21,0-4167 3.7400295 7417 292.33650 27.89300 3.7440296 3663 137.62395 -62.17500 3.7500297 3547,_ _ 133.40,085 _..__._6.080.33 3.750.0298 2484 100.84755 12.93400 3.7520299 6832 273.83250 -36.77800 3.7530

_.3.0 0__. 3803_ 142.54635 _ -96.88600 3.7530..301 4520 17E.99950 -66.52970 3.7650302 2040 87.44130 -35.77800 3.7670303 3705 139.74540 34.541711 3.7750304 8518 334.97400 -1.55333 3.7770305 7950 311.46000 -9.62467 3.7800

_306 8585 -__337._47150 _ ._50.10770_-- 3.7800_-_v307 6561 263.91300 -15.37730 3.7810

308 4232 1.61.98500 -16.00770 3.78203 ^2.__.._4^3 8__ -1,.?+ 575 Q _ _ _ -52-,17-9 T A _.-3.785QL ^. _ _. - __ --- --- ------1310 1149 55.94835 24.25900 3.7860 a5311 6629 266.54850 2.72800 3.7910X312 5471 219.96300 -_- -37.6. 527.0 - 3 802 0

313 x6779 271.55250 28.7E100 3.8090314 -6418 258.46650 36.83970 3.8140

:-_315 _365-__18,14970- _- _2._^45t^33__. ,. x.17.0_ --•-- - - _____-_ __ _316 6771 271.43550 9.561.00 3.8190

1317 4898 193.15050 -55.99630 3.8220=318 _ . 2777 -._10 9. 52205 ___-- 22.114570-- 3x8240

--1 -- ----319 779 39. 43455 . 18b 00 3.13240320 7337 290.04750 -44.52900 3.824032 1 1.654 76.0968^0 -_ _.-2.2.-.41200- 3:.8300i 322 8115 317.87250 30.04170 3.8300

'323 5477 219.88050 13.86400 3.8300324 _- 338 -16. 7388 0. -55.43130_ 3. 8330---- -325 545 2.7.91305 19.11900 3.8380326 5849 235.33050 26.40770 3.8400327 6 149 . 247.3.005. 0 2.- 005703 .940 0328

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;329 4679 184.14750 -63.81300 3.8420j 33Q _ 7348_ 290. 38500 _ _ _401.67900 3.8420

331 6299 254.02500 9.43433 3.8470332 544 27.78630 29.41900 3.13490333 56 25 22E. , 63650 _,45.15300 3. , 520334

_-5892 237.28050 4.58533 3.8550

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1335 1520 70.9500.0 -3.31233 7.8560

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i 346 1131 55.54650 32.17570 3.8800!347 3690 139.17990 36.95830 3.8800

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352 322 16.14120 -46.89800 3.9000353 3487 131.21655 -45.92530 3.9000

j 354--46.8 9 184.. 5 4 0 5.13__3550 3129 119.31465 -49.14270 3.9020356- 5287 211.11150 -26.51900 3.9020

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_3-6.0_ _--6324_ 254.7495Q _ _3q-;-.96779 _ _3..910-4361 4802 138.96100 -48.34630 3.9100362 7352 307.89750 11.18670 3.9110

--153 _. - 219 11.76075_ _ ..57. 53530-- 30- -3. 91---364 6623 26E.28300 27.75030 3.9140365 5744 231.04350 59.08570 3.9180

i. 3-66___-U5Z. 144. 83415 _ . _ 1 0_e 05300 . _ 3.9 190367 7590 299.16450 -72.99070 3.9190368 2216 93.20385 22.51130 3.91903.59:_ _125-1 _ 60._3355___371 8597 338.40150 -.29233 3.9240

;371 7420 92.21050 51.65970 3.9260i 7 _ 3.2-3 12_6.86.i1fi -._ - 50.8300. 0_ . ._ 3.01430373 1336 63.49515 -62.55170 3.9460374 1949 84.76245 -1.96700 3.9520

375-__4A?.,5.__. -. 89_.98550. __ .- .-1.2630. 0 .__3.9530376 3757 142.21320 63.21400 3.9540

Y 377 8278 324.55050 -16.81970 3.9560

'r

a.

- -^. '..

i

of

1378 1756 79.50060 -13.21730 3.9600379 8028 313.97850 41.03630 3.9600380 1922 83.33055 -62.50600 3.9680381

-- 580 -1 5

-- - -1--- -1 - -6-0.0,33152.252307

- ---- -- --- - 3.9709 --- - - - ---- -- - - -

i382 666ii- 2 .28600 -9-77767 3.97003.97001 383 2657 105.55350 - -15.58230 3.97-10,384 7234 286.20300 -27.71770 3.9780

385 6095 245.10750 7.q.22930 3.9810386 8465 332.41809 158.03000 3.9840387 4386 169.84500 C-.22033 3.9$360:3 88 269 13.71575 38.31300 3.9880

i389 1273 61.54905 47.62600 3.9890390-_67 ?6 93333 3,9900

391 921 45.75045 38.70300 3.9950392 6927 275.68350 72.71630 3.9950393 B5._76250 -22.4f130 3.9960394 2085 88.71330 -14.17230 3.91360

:t

I I 395 5367 214.62000 -.37.74130 3.9980396 5987 241.08750 -36.70930 4.0000

rn

F- -- - --- - -- , , -, - --

L, .11.

_

i

A

A-17

A.3 Ordered Working Star Catalog

Shown is the ordered working catalog derived from Program LOOK1 for

single-gimbal tracker, boresight South, ±45° FOV gimbal, my = +3.5. Data

output includes:

• Local catalog order (ordered by orbit angle measured from

ascending node)

• Working catalog number (see A.2)

• Orbit angle of observation

• Angle from zenith

Note: Output is provided on file named TAPE2

r f{

k

E

104 _1 87 1.647005318 1.294703E+02 l

2 90 3.336005318 1.21437E+02^^`^ -tis (mss^J

3 182 5.99126 7.74427E+014 17 6.177005318 1.17087E+025 39 6.181505318 1.17087E+026 62.. - 7.320005318 1.2307.6E±027 79 8.788505318 1.110537E++028 54 9.910505318 1.312213E+029 120 1.104900532E+Q1 i.1207 03F+02

{10 14 1.143600532E+01 1.20487E+02':11 6 2.410905E+01 5.742E+01

12 5_0._ --2.443200.532E±e_1 `.. 1.25709E+0213 64 2.835300532E+01 1.3287E+C214 157 2. 1342385E+01 6.1731F+91 r` r15 7 3.035700532E+0 1 1.i97977E+0216 3.382285E+01 5.16753E+0117 195 3.430650532E+Qi 1.34092E+-0218 13 3.932250532E+01 1 .i93Q83E+02i9v 66 3.932250532E+01 1.193083E+0220 51 3.991800532E+01 1.327473E+0221_ ....206 -__ - 3.9937 .50532E±O_i . l._1518Q 7E+02

i22 127 4..993450532E+Qi 1.114413E+0223 156 5.803800532E+01 1.166743E+0224.___..155 ..__:6.831.585E±Di-5.5118E+6125 92 7.126500532E+01 1.11029E+0226 231 8.061750532E+01 1.245007E+0227 169 8.063400532E+01 1.236507E+0228 110 8.230500532E+Oi 1.30145E+02

!29 3 9.579915E+01 5.26663E+013Q__ ._ .x.8. 8 __ ^fi1155063- 2E-t0 i .. _ _ 1.30163E+0231 220 1.019622E+02 6.18993E+0132 180 1.189791E+02 5.28927E+01

^33 .- _ 28 .. __-1.. 2 2.123.-3 +D2 .._ 4.7248E+0134 84 1.254549E+02 5. 93923E+01.35 34 1.257375053E+02 i..23159E+021 36 208 1.298313E+D2 5.27977E+01 -37 137 1.2987645E+02 4.6531E+Oi38 47 1.30944E+02 5.4592E+0139 177 1.3751775E+02 5.8825E+01

140 33 1.3820655E+02 6.9575E+0141 72 1.3904685E+02 5.91.25E+Oi

r,

is

^^ y

3

I"^c

♦ _tie-,:

j

43 207 1.4656245E+02 6.4908E+0144 217 1.4891775E+02 5.4419E+01

1 46 178 1.532355E+02 6.98633E+01i47 205 1. 1577025E+02 • E*15077E+01_4A_ 1 nr,

49 85 1.60434E+02 6.42077E+0150 2 1.699365E+02 15.95187E+01

-_ 1-85 I, -6_i32 -65Et0 Z_ k,4Z5_2E+0 iX 522 152 1.73553E+02 6.28297E+91

53 87 1.316470053E+02 5.05297E+D11,8333-60053E+02 - -5-a72-63-f-+-U-

55 182 1.8599126E+02 1.025573E+0256 17 1.861770053E+0? 6.2913E+91

--5-7- - 3-9- I-A61815Q53.E+B-2 6-4L2

tw

?58 62 1.873 200053E+02 5.6924E+01

X59 79 1.887885053F-+02 6.89463E+01_6D__-_5.-4..-_ 1.. 89ql.O 5D53E+,Q Z -49-8779TE. ,±0 i61 T' 120 1.910 ,490053F+02 6.79297F+0162 1- 14 1.914360053E+02 5.9513E+01

fn , 2.,841092-5EtU2

64 50 2.044320053E+02 5.3291F+01

165 64 2.083530053E+02 4.713E+gl-65- 157. - 2,01-42345EtU - 1,p,lAME±-QZ-

67 7 2.103570053E+02 6.02023E+0i68 213 2.1382185E+02 1.283247E+02

3Ft^0 2 4.5908E+0170 13 2.193225053E+02 6.06917E+01

171 66 2.193225053E+02 6*06917F+01

73 206 2.199375053E+02 6.48193F+01

74 127 2.289345053E+02 6.85597F+01-Z5--156-

76 165 2.4831585E+02 1.24882E+02

i

77 92 2.512650053E+82 6.8971E+01Lzi- 2 s50 j51.75053F+02 5.54993E+01

79 169 2.606340053E+02 5.63493E+01

80 lio 2.623050053E+02 4.9855F+01Al- -3-- ___2._7579915E+92 1.273337E+02

; 82 188 2.761155053E+92 4.59837E+01

1 83 220 2.819622E+02 1.181007E+02

p

C

F

84- . _1 0 .. _2- .98.9741E+-02 1,_271D73E+02 _...85 28 3.021213E+02 1.32752E+0286 84 3,054549E+02 1.206077E+02-37 - --_34--.- 3.0. 573-75.0,53E+0? .___56841E+O1_.--_88 20f3 3.09 83313E+02 1.272023E+02__.--

.89 137 3.0987645E+02 1.33469E+02_90 _._47_..._ 3..10944E+02_ 1.25408E +02

91 177 3.1751775E+02 1.21175F+0292 33 3.1820655E+02 1.10425E+029 3 7 2 3.19D4685E+02 1.2 0875E+02

X94 03 3.2112656E+02 1.251307E+C2X 95 207 3.2656245E+02 1.15092E+02

96 - ,_2173.- 2891775E+02 _ 1 .25581E+02 __-97 29 3.315225053E+02 4.7131E+0198 178 3.332355E+02 i.iD1367E+02

r 9-._^2Q5_- _3T._ -377.025E±02_1.1849.23 _2

10D 105 3.40i595053c+02 4.7059E+01101 85 3.40434E+02 1.157923E+02

`102 ^ 2 __. 3.4D 9365_E.±O.2 -- 1.204913E±0 2-- --------__ -- ---.-

103 185 3.413265E+02 1.30748F+02104 a 152 3.53553E+02 1.171703E+02

i

i

^k

a

i

i.

i

c.

fI

;' l

APPENDIX B

COMPUTER PROGRAM FOR SINGLE-GIMBAL

PADS COVARIANCE ANALYSIS

t

f ^^

B-1

T

y

B.1 Program PADSCO

A FORTRAN digital computer program, Program PADSCO, was developed

to implement the covariance analysis algorithms derived in Section 3.3.

The program operated using star data from file TAPE2 and input data de-

fining orbit and filter characteristics. A source program listing is

provided, followed by a summary of output data in B.2.w

C 4

l'

f4

i

100100 PROGRAM PADECO(INPUT,OUTP(IT,TAPf1,TADE2,TAPE5=INPUT)00105 C TAPE2 CONTAINS STAR. DATA00110 DIMENSION P(6,51,S(6,6f,SSt3,Z)00120 DIMENSION ALP4(400),GAM(400),N(400),NNS(400),A(3)99121 COMMON WO00125 REWIND 2

i 00130 NAMELIST/PARAM/Ti,T2,T3,W,SN,SV9SU,H,TF,TG00140 DATA Ti,T2,T3/10.,10.,10./00150 DATA W/.05/00160 DATA SN,SU,SU/3.,.115,.333E-4/00170 DATA 'H,TF,TG/19300.,86400.,216P0./00180 READ(5,PARAN)00190 WRITE(i,2) Ti, t2,T 3,W,SN,SV,SU,H {<t00200 PRINT 2,T1,T2, T3,W,SN,SV,SU,H t,^00210 2 F0RNAT(X,llHT1,T2,T3 = ,3(F6.1,2X)/X,4HW = pE9.2/XpllHSN?SV T SU = ,00220 13(E9.2,2X)/X,4HH = ,F7.1/)00230 WR IT= (1,3)00240 3 FCRMAT 14X,1H T ,7X,,2HS1,6X,2HS2,6X,2HS3,6X,2HPS,6X,2HRS, r-00250 16X,2HT1,6X,2HT296X92HT3,6X97—HPT,6X92HRT/)400260 , ACCEPT (2)NSTR`00270 `' ACCEPT (2) (N(I),NNS(I),ALPH(I),GAM(I),I=i,NSTR)00280 0EG9AD=3,1415927/180.06290 IR=O00300 C INITIALIZE TIME

100310 T=T9=0.00320 C INITIALIZE PERFORMANCE MEASURE00330 00 ti I=i,30034. 0 11 SS(1,1)=SS(I,2)=0.00350 C INITIALIZE ERROR COVARIANCE MATRIX00360 DO 10 I=1,600370 DO 10 J=196

10 P (I,J) =0,.

P(1,1)=Ti**2P(3,3)=T?**2Nc5,5^=isTt^P (2,2)=P(4,4)=P (6,6)=W**2WO=SORT(1.875E5/3./(3440.+H)**3)

20 RT=SQRT(Tim+2+T2**2+T3**2)RS=SQRT(Si**2+S2;42+S3**2)PS=SQRT (Sl**2+S?**2)PT=SQRT(Tlf}2+T2**2)PRINT 8,T, RS, RT

a=

003800039000400004100042000430004400045000460UU4tU00472

Lkl... _._ ... ^... .._^.... _i_ . .^..A_...^..e.... __.u.._._...a._.3.._.•4..

^^ a

i

I

r ^^

00474 8 FORMAT(X,F7.1,2(?X,F6.1))00480 WRITE(194) T,S1,S29S3,PS,RS,Tt,T2,T39PT,RT00490 4 FORM=AT(X9F7.1,10(2X,F6.1))00495 C COMPUTE TIME TO NEXT UPDATE,DT00500 IR=IR+i00502 IF(IR.GT.t) GO TO 2100504 OT=ALPH(1)*0EGRAO/WO00506 GO TO 2400510 21 IF(IR.Lr.NSTP) GO TC 2200520 IR=i00530 DT=(360.-ALFH(NSTR)+ALPH(t))*DEGRAO/W000540 GO TO 2400550 22 DT=(ALPH(IR)-ALPH(IR-1))*DEGRAD/WO00555 C COMPUTE STAR COORDINATES00560 24 A(1)=0.00570 A(2)=SIN(GAM(IR))00580 A (3)=-COS (GAM (IR)00590 C TEST/COMPUTE CUM PERFORMANCE MEASURE00600 IF(T.LT.TG) GO TO 3100610 W 00 40 I=1,300620 J=2*T_-100630 K=J+t00640 A=P(K,K)00650 a=SV**2-2.*P(J,K)00660 C=P(J,J)00670 SS(2,2)=SS(I,2)+A*DT**3/3.+B*QT**2/2.+C *0T00680 RR=(4.*A*C-E**?)/8./A**1.500690 CC=-9/4./A*SORT(C)-RR*ALOG(ASS(2.*SORT(A*(.)+R))00700 X=A*DT-`"2+8*DT+C00710 R1=(2.*A*DT+g) /4. /A'*SQRT(X) +RR*ALOG(A`9S (2.*SQRT (A*X)80720 1+2.*A*DT+8)100730 SS fi l l) =SS(I,1)+Rt+CC00740 40 CONTINUE00750 T9=T9+DT00760 C INCREMENT TIME TO NEXT STAR -00770 30 T=T+OT00780 C TEST FOR END OF RUN,TF00790 IF (T. GT.T F ) GO TO i00 --00800 CALL FILTER(Q,S,DT,SN,SV,SU,A)00810 Si=SQRT(S(i,l))1)0820 S 2 = S 0 R T ( S ( 3 , 3 ) )D0830 S3=SQRT(S(5,5))

j

A ;`4

r

l (''

00840 Ti=SQRT (P(i,i) )00850 T2=SQRT (P (3,3) )00860 T3=SQRT(P(5,5))00870 GO TO 2000880 C COMPUTE FOR PER P. MEASURE;MEAN,SIGM00890 100 CONTINUE89900 DO 50 I=1,3009i0

SS(I,1) =SS (T,'t)/T900920 .SS (I,2) =SQRT (SS (I, 2) /T9—SS (I,1) **2)00930

50 CONTINUE00940 PRINT 500950 5 FORMAT (5X,4HME_AN,4Y,5HSIGM;Q/)00960 00 60 I=i, 3 Fa ,00970 60 P RINT 6,SS(I,i)9SS(I,2)00980 C ESTIMATED GYRO R-LASES00990 PPJ=SQRT(P(2,?)) *,'^01000 PP2-SQRT (P (4, 4))01010 PP3=SQRT(P(6,F,))01020 PRINT 7,PPi,PP2,PP301030 '1'0 7 FORMAT(X,i4HGYR0 BIASES _ ,3(EiG.3,2X)) C01040 L" 6 FORMAT (2PX, 17 5.2) )01050 END

SUBROUTINEE FILTER(P,S,OT,SN,SV,SU,Y)DIMENSION P(6,6),S(E,6),Q(6,6),R(2,2),RM(2,61,PHI(6,6),A(6,61,iH(6,2),C'(2,2),D(2,2),G(6,2).H(6,6),Y(31

GCMMON WOZERO MATRIX ELEMENTSDO 10 J=1,6RM(I,J)=RM(2,J)=0.DO 10 I=i,6

0 Q(I,J)=PHI(I,J)=0.MEASUREMENT NOISE COVARIANCE ?MATRIX

R(2,?)=(SN*Y(3))**?STATE TRANSITTON MATRIX,PHIDO 20 I=1,F

0 PHI(I,I)=1.SB=SIPS! ;v` T)

01060 C01070 C01.080 C01.0900110 001110011ii01120 C01130OSi4001150011600117 6-C,011800119 00120001210 C012200123001240

^rr ^'

V-

012411 C 8 = C_o S ( iii nw*_D.l_).___._.--- --_ -01242 PHIL1111=pHI(515)=C801243 PHT(,5)=S801244 PHI (5, 1) =-SB01245 PHI(1,2)=PHI(3,4)=PHI(5,6)=-nT01250 C STATE NOISE COVARIANCE MATRIX,QOiz60 T4=4. *DT**7/3_.-2_.*OT* (I,-C8) /WD*.V 201261 Ot3,3f=DT*SV**?+SU**2*DT**3/3.01262 Q(1,1)=0(5,5)=DT*SV**2+T4*SU**2D1263 0(1,Z)=Q (2,_11=Q(5,E)=Ot6,5)=( (1.-C8)/^IO^`*_?-OT**2)*SU**201261, Q(1,6)=0(6,1)=0(2,5)=0(5,2)=SU**2*(OT-S8/W0)%WO01265 O(2,2)=Q(4,4)=Q(6,E)=DT*SU**201266 Q('„4)=Q(4,3)=-(SU*CT)**2/2.01280 C MEASUREMENT MATPTX,PM01290 RM(t,3)=R.M(2,1)=Y(3)01300 RM (1, 5)=-Y (2'01310 C PROPAGATE ERROR COVARIANCE,S=PHI*P*PHIT+Q -01320 DO 30 I=1,601330 00 30 J=1,E01340 A (I , J) = 0 .

7)

01350 DO 30 K=1,601360 30 A(I,J)=A(I,J)f-P(I,K) *PHI (J,K) - _ _-01370 DO 40 I=1,6 -

-

01380 00 40 J=i,E01390 S( Ili )=Q(I,J)01400 DO 40 K=1,6

- -_

01410 49 S(I,J)=S(I,J)+PHI(I,K)*A(K,J) F^01420 C COMPUTE OPTIMAL GAIN,G01430 DO 50 1=1,6(11440 00 50 J=1,201450 8(I,J)=0.01460 DO 50 K =1, 601470 50 B(I,J)=B(I,J)+S(I,K)*PM(J,K)01480 DO 60 I=1,26490 00 60 J=1,201500 C(I,J)=R(I,J)01510 DO 60 K=1,601520 60 C ( I, J) =C ( I, J) +RM4 I, K) *8 (K,J) --^---01530 DEL=C(1,1)*C(2,2)-C(1,?)*C(2,1)01540 D(1i1)=C(2,2)/DEL01550 D(i,2)=-G(1,2) /DEL___-_--01560 0(2,1)=-C(2,1)/DEL

^y

_t

M_

tw

[011570 0(2,2)=C(1,1)/DFLOi580 DO 70 I=1,601590 00 70 J=1,201500 G (I,J)=0.01610 90 70 K=1,201620 70 G(I,J)=G(I,J)+B(I,K)*D(K,J)01630 G UPDATE" ERROR COVAPIANCE,P01640 DO J31 I=1,6 -01650 00 81 J=1,601650 81 H(T,.))=-G(I,i)*PM(i,J)-G( T,2)*RM(2,J )01670 80 H(I,I^=H(I,I)+1.91680 00 90 I=1,601690 00 90 J=1,601700 A(I,J)=0.01710 DO 90 K=1, 601720 90 A(I,J)=A(I,J)+H(I.K)*S(K,J)Oi730 C ASSURE POS. DEF.01740 00 100 I=1,F01750 DO i00 J=1,E91760 100 P(I,Ji=IA(I,J)+A(J,1))*.501.770 RETURN01780 END

ti +` {4t

B.2 Summary of Test Cases

Star Data Used From Appendix A

Geosynchronous Orbit

Tracker - Nominal Boresight S, Gimbal Freedom ±450

my Brighter than +3.5, Real Stars

Initial Ai+. Initial Gyro Bias Sensor Noise Gyro NoiseCovariance Covariance SN SvT1/T2/T3 w(arc-sec) (deg/hr) (arc-sec)

10/10/10 .05 3 .115(2 sec in5 minutes)

Gyro NoiseSU

3.33x10-5(.002 deg/hr/hr)

10/10/10 .05 3 .05 3.2x10 5

.05 3 .05 1.6x10-5

.05 3 .015 3.2x10-5

.05 3 .015 1.6x10-5

360/360/360 .05 3 .05 1.6x10-5

10/10/10 0.1 3 .05 1.6c10-5

360/360/360 0,1 3 .05 1.6x10-5

B-8

T Si S2 S3 PS RS

0.8 0.0 0.0 0.0 0.0 0.0394.3 22.2 22.2 22.2 31.4 38.5798.6 11.0 26.7 33.1 28.9 43.9

1434.3 8.9 22.1 14.7 23.8 27.91478.8 3.1 22.9 11.9 23.1 26.01479.8 2.2 2.0 1.5 2.9 3.3

1.2 302 3.41.5 4.2 4.47.8 7.7 10.93.6 5.3 6.55.6 7.9 9.6

15.2 9.2 17.86.6 2.9 7.3697 3.4 7.5

1752.4 3.1 3.2210 4. 0 3.8 3.82372.6 3.5 5.02645.1 3.3 31.122737,B 2.6 3.45771.7 11.9 12.1584 9. 0 W 3. 1 9.66787.7 10 4.9 lo. i7044.1 3.3 10.1

^7267.4 -3.0 3.45096.9 4.5 4.28213.0 209 4.5

2.9 4.5 5.3 2.2 2.2

4.6 5.4 7.0 2.3 3.8

3.7 6.1 7.1 2.3 2.1

2.6 413 5.0

11.4 17.0 20.48.1 10.1 12.97.5 11.2 13.57.7 10.6 13.22.5 445 5014.6 6..2 7.7211 5.3 5.7

24.2 3.12. 0 2. 12.9 9.42.2 7.72.6 9.52.2 2.72.1 1.62.5 4.22.1 2.7

9413.8 5.4 5.9 5.1 8.0 9.5 2.6 1.59413.8 2.6 1.5 5.1 3.0 6.0 2.0 1.19556.3 2.5 1.8 5.5 3.1 6.3 1.9 1.69561.0 199 1.6 203 2.5 3.4 1.6 1.3

11714.9 7.5 7.4 7.4 10.5 12.9 2.8 7.413894.3 8.2 .13.0 7.5 15.4 17:1 2.8 3.816354.8 9.0 9.5 13.7 13.1 18.9 2.8 8.317060.8 4.7 10.1 4.0 11.1 11.8 2.5 5.31929908 8.3 10.E 8.5 13.4 15.9 2.8 8.019303.7 2.8 8.0 3.+6 815 902 2.1 2.5

i 19703.8 3.3 3.6 2.7 4.9 5.6 2.2 3.52293 4 * 3 1006 11.6 10.2 15.7 18.7 2.9 7.123010.0 3.1 7.3 7.9 7.9 11.2 2.2 4.924409.7 5.9 8.1 6.9 10.0 12.2 2.7 7.428483.5 13.1 1t.7 15.3 22.0 26.8 2.9 8,.82923508 4.9 10.1 17.3 11.3 20.6 2.5 1.530033.8 4.7 4.1 8.3 603 10.4 2.5 2.3

{

T1 T2 T3 = :10.0 10.0 10.0N 5.90E-02SN,SV,SU = 3.00E+00 1.15E-01 3.33E-05H = 19300.0

Ti

T2 T3

10.0 10.0 10.0

3.0 13.8 17.5

2.9

9.0 4.911.41.41.42.82.52.12.21.37.95.37. 11.62.21.63.55.15.12.31.31.67.41.92.83.6

2.8 2200

2.1 2.0

1.7 1.7

PT RT

14.1 17.314.1 22. 59.5 .10.7

22.2 24.92.9 3.32.5 2083.1 4.14.5 5.13.1 3.8348 4.4 r-2.9 3.29.8 12.68.0 9.5908 12.1

-3.5 3.92.6 3.44.9 5.13.4 3.73.0 6.02.3 5.62.5 3.42.1 2.47.9 BOB4.7 3.88.8 9.05.9 6.58.5 9.2

i. 3

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