Design criteria for a stable base gyrocompass : (unclassified title)

UlluLt\~~JtltU

at the

January) 1963 Mi~STER OF SCIENCE

QUIRElvIENTS FOR THE DEGHEE OF

CHUSETTS INSTITUTE OF TECI-INOLOGY

lVlassachus etts Institute of Technology ,-- /)

If

(Gl\"l'\ ...

i C)?,

It t-

t •. 'J ~ ~

(:. 960) CLASSIFIEDBV

_ltJ:p..1.C.b ..

C).~~

SUBJECT TO GENERAL DECLASSIFI~N---SCHEDULE OF EXECUTIiE ORDER 11652 AUTOMATICA~LY DOWNGRADED AT TWO YEAR

INTERVALS ON-OOlEMBER-91,o.a (I

":J

L TED IN P .i~~TIAL FULFILL?vIEI~T

dIF~fE

'

M

o

Z

b1)

-3

o

0:

0') Signature of f._uthor

Department of }~..eronautics and Astronautics) Januar,y 14) 1963

\ Certified by _ Thesis Supervisai' Accepted by Chairln~

DECLASSIFmll

UNCLASS!F1ED

DECLASSIFIED

ACI(NO\VLEDGEl\lIENT

This report was prepared under the auspices of DSR Proj ect 52 -156J sponsored by the Ballistic Missile Division of the Air Research and Devel-opment Command through US.AF contracts

AF 04(647)-303 and AF 04(694)-305.

This document contains inforlnation affecting the national defense of the United States within the meaning of the Espionage Laws J Title 18, U. S. C. J Sections 793 and 794J the transmis sion

or the revelation of which in any manner to an unauthorized person is prohibited by law.

The publication of this report does not constiJ~ute approval by the i\ir Force of the findings or -:he conclusions contained therein. It is published only for the exchange and stimulation of ideas.

DECLASSIl-1ED

-2-ACI(NO\VLEDGEMENT

DECLASSIFIED

The author is grateful for the association behveen the Department of Aeronautics and Astronautics and the Instrumentation Laboratory) which affords the opportunity to develop analytical concepts as well as an appreciation for an operating state-of-the-art system while performing , this investi~a tion.

The author wishes to acknowledge his appreciation to the members of the Skipper J. Group who developed the highly reliable system on which these data were recorded. In particul~r) the author wishes to thank the following persons.

Professor Vvin'ston R. Markey for his supervision) assistance) and criticisms of the initial draft while acting

,

as technical faculty advisor on this thesis.

Mr. Richard E. Marshall for suggesting the topic) guiding its direction) and markedly commenting on the initial draft.

Mr. Thomas E. Reed and Mr. William T. McDonald for their many stimulating discussions of the subject matter and the preliminary draft.

Mrs. Elaine Walsh for typing this paper.

~,'''-,,','-,,-'-, ",,'

~.

FORE\rVORD

DECLASSIFIED

This fore-v-rord describes the goals and results of the thesis and the general situation that surrounded this \,york. It is not considered a regular part of the thesis.

s

On October 1, 1962, the 2-16 system had been gyrocom-passing at the 1V1. 1.T. Skipper J Test Facility for 2000 hours. During t~1is tilne, stable base gyrocompassing data had been recorded on the system operating in a benign environment. E-1231 (Ref. 3) contains the pertinent results recorded on thegyrocompassing system hefore the fourth quarter of 1962. The results which are fundamental to and \vhich preceed the investigations performed in this thesis are as follows.

1. The 2 -16 gyrocompass demonstrated the existence of a memory-moqe gyrocompassing capability accurate to better than five seconds of arc rms error in azimuth for periods of hundreds of hours.

2. Incremental angles below the order of O. 01 seconds of arc about the horizontal axes can be consistently detected in the 2-16 gyrocompass operating in a benign environ-mente

3. Gimbal static and sliding friction nonlinearities did not appear to be a major problem in the oper~tion of a stable base gyrocompass.

D.urlng t e lrsth f.

DECLASSIPIED

quarrel" oI'-rTI"tf2prellmlnaryJ

..

expe

on a vrheel speed modulation technique of gyro error averaging demonstrated the feasibility

of

detecting very low level torque changes in a 2FBG-6F gyro with its input axis approximately east. _{Thes tests conducted on a single instrument} mounted on a test table are the subj ect of T-315 (Ref. 1). The results indicated the possibllity of utilizing wheel speed modulation to ach,ieve a fixed base azimuth indi.cating system for long opera-tional periods. First efforts on paper to devise an operational sample data procedure were carried on jointly by the author , and other members of the Skipper J Group (Ref. 2).

The program. for the 2 -16 system during the fourth quarter of 1962 had as a first goal the demonstration of system feasi-bility of the wheel speed modulation technique. The author, was in control of this effort as a research assistant.

The experin1.ental part of this thesis \vas to record data on system performance versus b<?-ndwidthto reveal characteristics of instrument uncertainties. These data forln one set of system inputs that limit frequency regions covered by practical solutions '(Chapter IV).

Equal importance vias attached to defining probable operating characteristics of such systems (Chapters II and III) and the development of adequate operational procedures (Chapter V).

Because the above aims were M¥=gW~m1?dent upon demonstrating feasibility of the vlhole technique, it became reasonable to report these results. Thus) a third goal of the thesis is to present organized theori~s of operation and resultsachie'ved for the wheel speed modulated gyrocompass.

s

.The value of having these related matters in a single docu-ment justifies the coverage which is often limited but manages at least one plausible solution in each area.

The -results of these investigations are as follovls. 1. The thesis analysis' of system performance versus

bandwidth reveals instrument uncertainties below the values previously anticipated. These data .extend the frequency regions of realistic system performance. (Chapter IV).

,2.• Wheel speed modulatio.n techniques aJ?e reported which appear to demonstrate the capability of detecting low level torque variations in the east gyro operating in a benign system environment (Chapter IV).

3.. rrhe thesis combines the data on the spectrum of system and instrument uncertainty with anticipated changes in system bandwidth and incremental angle measurements to define probable operating characteristics and gyro compensation procedures in a realistically perturbed environment (Chapter V).

4.

_{The thesis ,eports. stablDl:~SSImDmpaSsing}

charac~eristics and justifications for the simplifying assumptions \vhich describe this system with its unconventional closely coupled loops and superior instrument performance (Chapter II).

s .

5. The thesis' reports accuracy requirements on instru-ment aligninstru-ment and derives a possible alignment method (Chapter III) .

DECLASSIFIED

DESIGN CRITERIA FOR A STABLE BASE GYROCOMPASS

by

John M. Vergoz

Submitted to the Department of Aeronautics and Astronautics on January 14, 1963 in partial fulfillment for the degree of Master of Science.

ABSTRACT

The current status of self- azimuthing capabilities of inertial guidance systems in a fixed base missile is reported. Empiri-cal results and analyses are presented.

Data taken on an east gyre error averaging technique, wheel speed modulation, tentatively demonstrate the capability of detecting changes in the east gyro performance with sufficient accuracy to extend the operational period indefinitely.

The problems associated with absolute instrument alignment in a fixed base gyrocompass are discussed and one of the many possible solutions presented.

Data indicative of instrument uncertainty spectra in a benign environment are combined with estimates of the necessary changes in system bandwidth and recompensation techniques to operate the gyrocompassing system in an operational environment.

The data presented on gyro drift uncertainty spectra reveal that considerable latitude exists for optimizing system perfor-mance in the presence of horizontal acceleration inputs.

Thesis Supervisor: Title:

VVinston R. Markey Associate Professor

of Aeronautics and A 3tronautics

~i%~~;

_{~'j; ~,~4~4~'~~?1{~}

~j'~~~~~fi~~t~~~~,i;~~~tW~i*~~~~~t~~:

_{.~"~~~:~~<'H~P~s~~:~~~;~"~~~~~"~:::'\fZl~~}~~;t1l.L.-'~.~ir~} ,,'

-

...-~.,;.... - ..

--

--.-

--DECLASSIFIED

TABLE OF CONTENTS Page CI-IAPTER I CHi~.PTER II CIIAPTER III Intr,oduction . Gyrocompass Characteristics .

Laboratory Alignment and Calib-ratio 11 ••••••••••••••••••••••••••••

12

21

45 CI-IAPTER IV CI-Lt1.PTER V

System and Instrun1.ent Performance .. 70

Operational Syster.a. Evaluation .. 0 0 0 •• 105

.CI-IAPTER VI Conclusions

129

APPENDIX I

APPENDIX II

Non1.enclature .... 0 ••• 0 •••••• 0 •• 0 00 133

Spect:r'a of Gyrocompass Angles ... 139 '

DECLASSIFIED

LIST OF FIGURES

Figure 2 -1 Stable Base Gyrocompass Coordinates 23 Figure 2-2 Gyrocompass Functional Block Diagram 26

Figure 2-3 East AziInuth Gyrocolnpass 29

5

Figure 2 -4 East Azinluth Dynalnic Errors' 37

Figure 2 ~5 Azimuth Lateral Acceleration Errors 40

Figure 2-6 . Gyrocompass Step Response 43

Figure 3-1 Deviation of East Gyro IA From Geographic East .. 47

Figure 4-1 East Azimuth Gyrocompass 75

Figure 4-2 Uncertainty Frequency ' 77

Figure 4-3 2-16 Gyrocompass Short-Term Stability 83 Figur~ 4-4 2-16 Gyrocolnpass Effect or Varying Bandwidth .... 86 Figure 4-5 2-16 Gyrocolnpass Effect of Wheel Speed Modulation. 103

Figur..e 5 -1 North Gyroconlpass Loop 107

Figure 5-2 Qpen/Closed Loop - East i~.zimuth Gyrocompass 109

.,

Figure 5-3

Figure 5-4

Open Loop East Vertical Sensor Output for Incorrect

Azimuth Gyro COlnpensation 114

Open Loop East Vertical Sensor Output - East

Gyro Drift (one-half normal wheel speed) 122

Table 1-1 Table 2-1 Tab).e4-1a Table 4-1b Regions of Interest Error Coefficients 0 0 0 0 0 0 • 0 0 '. 0 • 0 0 0 •••••• '0

260 Hours - Low East/Azimuth. Coupling . 168 I-Iours - I-ligh East/Azimuth Coupling .

~

.: ...

,s

17

35

80

,DECLASSIFIED

I INTRODUCTION

The

effectiveness of an inertial guidance system is restricted by the accur,acy to :which 'it can be aligned in azimuth and

to

the,

5

vertical. Such guidance systems are presently aligned in azimuth with optical techI'!iques utilizing complex groundinstallatiqns and an accurately surveyed azimuth reference. The vertical is sensed and corrected with an accelerometer" pendulu~um, or other leveling' device operating with the system gyros.

Since inertial guidance of missiles became a reality, the possi-bilUy of alignment by gyrocompassing has been realized and, at tiriles',used. A deterrrent to gyrocompassing has been stringent alignment requirements using available instruments. Recent advflpces in the development of single-degree-of -freedom ,integraf-,ing gyros, coupled with system optimization techniques" have:'

developed the operational capability of self-aJigning missile guid-ance systems.

This paper discusses gyrocompassing theory and characteristics" as .well as laboratory alignment, calibration techniques" and opera-tional' concepts for a fixed base inertial guidance system. Datathat are'indicative of the uncertainty spectra of the gyrocompassing. sys-tern form the basis of the calibration and compensation techniques developed.

1. 1. 0 Gyrocompassing

_DECLASSIFIED

By definition) a gyrocompass includes any and all devic'es or systems that align the111Selves toa 'predetermined set"ofcoordinates) using the earthls angular velocity and direction of gravity as basic references. Instrun'lents which are sensitive to the earthls angular velocity) coupled \vith devices which determine the direction of gravity are) there-fore) the core or a gyrbcompassing system. The direction of gravity is .defined by connecting the vertical sensing devices (pendulums) electrolytic levels) accelerometers) etc.) to the 'stabilized member integrating drives. In the three-gyro system)

the gyro input axes are approximately vertical) north and east. Basically) the ideal east gyro torque sUl'nmation is only satisfied \vhen its input axis is parallel to geographic east. The azimuth' o,rientation of the gyrocolYlpassing systcr.o. defines the angular velocity applied to the east gyro input axis. By coupling rota tion8 about the east gyro input axis to the azimuth gyro torque generator via the east vertical sensor) a change in the angular velocity sum-mation of the east gyro 'will produce a new azimuth angle. This new azirnuth angle changes the horizontal con'lponent of earth rate applied to the east' gyro input axis.

Accuracy capabilities of the gyrocompassing system are primarily detern'lined by the stability of the east gyro. The closed loop gyrocompassing system cannot distinguish between

DECLASSIFIED

internal changes in the east gyro characteristics (uncertainty torques)) making the angular velocity summation no longer applicable) and a change in azimuth angle coupling a ne\v component of earth rate into the east gyro.

Gy:r6compassing techniques differ n1.ainly in the method of detern1.ining changes in performance of the east gyro with no external references. East-wrest averaging .. gyro wheel reversal) and wheel speed mo'dulation are three methods of east gyro evaluation which have been investigated in recent years. This paper limits attention to vrheel speed modulation in a system

environment because it has recently been us ed to determine low level torque variations on a single component basis.

1.2.0 Accuracy Requireluents

Accurate long-term gyrocompassing imposes severe demands on systelTI performance. The follovving discussion. is included to develo'p a feeling for the magnitude of the problem. Justification for these regions of interest in the gyrocompassing system are included in ,the body of this report.

As an example) assume that the absolute azimuth stability of the gyroconlpassing system shall be within

+

5 seconds of arc.' This defines the maximum allo\vable uncon1pensated east gyro drift uncertainty at medium latitudes to be 0.018 meru (0.00027 degrees per hour). Allowable north and azimuth gyro

uncertain-DECLASSIFIED

-

...• ..-:.... - ..

.

.

.-. '. '- '", ." ~

ties are a function of syst~.¥~~~hich is, a cOlnpromise between gyrocolnpassing uncertainties and lateral horizontal accelerations. In a benign environinent, north and azimuth gyro drift rates in the O. 1 to 0.5 meru region can be tolerated in. the_s closed loop gyrocompass. Changes in the perforlnance of the north and azimuth gyro can be con1.pensated for in the closed loop gyrocompass b~cause they produce vertical as well as azimuth errors.

To detect changes in the performance of the east gyro, the wheel speed Inodulation technique requires the n1.easurement of~small angles by the east vertical sensor. A highly stabiliza-tion servo gain and low values of gimba~ static friction will produce the r.nost desirable (i. e., most nearly linear) operation. In the system operating in a benign environment, stabilization servo gains of 50 ft.lbs/mr, coupled 'with gimbal static friction in the 1 in. oz. region, have combined to develop a system in which 0.01 seconds of arc can be detected and used to recompen-sate the east gyro. Since gyro drift produces angle variations in the gyrocolnpassing system, the time required to measure small angles defines the bandwidth of gyro drift detection.

Latitude relocations change the vertical and horizontal com-ponents of earth rate. If the east gyro input axis is not parallel to geographic east, latitude relocations will change the value of the earth rate applied to the east gyro. This change in earth rate

DECLASSIFIED

applied to the east gyro v/ould vary the ~zimuth angle of the gyrocompassing system. If the laboratory absolute azimuth reference and ope:ating area are separated by five latitude degrees" seven sec~nds of arc error in the laboratory know'-le?ge of:the east ~ro inpllt axis will produce one second of arc azimuth error at the nevI latitude location.

Thes e regions 6f interest are summarized in Table 1-1. The gyrocompassing system requirements are to maintain the angle stability over a maXllnUnl time interval vvhich, hopefully, can be extended to several years. The basis of thes e numbers is the subj ect of this report.

1.3. 0 Operational Integration

The unattended self-calibrating internal guidance system requiring no alignment or stability checks for several years after it leaves the laboratory is not a reality. How'.everJ

the guidance system under test has the ability to memorize a set of coordinates over several days vrith sufficient accuracy for all presently known requirenlents for missile and satellite launchings. If the gyros are periodically recompensated, the system can hold the reference for an extended period of time with no external-references. Integration of this system into a missile site containing optical references would immediately reduce the missile ts vulnerability to ground disturbances by

~:< Regions of Interest Angle Stability AZ,imuth Variation o 'North Variation East Varia tion

Uncompensated Gyro Drift North Aziriluth

<

+

5 seconds of

arc-<

+

1second of arc <

+

1 second of arc O. 1 to O. 5 meru 0.1 to 0.5 meru - East Component Alignment

<

0.018 meru Knowledge of laboratory-orientation of the east gyro input axis:

l\1iscellaneous

ST1.\.B Servo Gain Gimbal Friction Incremental P...ngles

better than 14 seconds of arc.

50 lb. ft/mr 1in. oz.

0.01 second of arc (benign environ-r,l1.ent).

0.1 second of arc (operational environment) .

Hepeatability oJ Changing

East Gyro Vvheel Speed: better than O. Olmeru

~~ 0

Table for t15 latitude.

.

DECtASS~D

nlemorizing the pr,oper g-Llidan'cesystem orientation, therefore, maintaining alignment after the optical reference is dislodged. The ,three main areas of interest for an operational system in the o~der of decreasing importance are (1) stability of the sys-tem ?-s an absolute reference, vlhich is primarily a function of gyro drift rates and the effectiveness of a recompensation pro-cedure; (2) sensitivity of the system to latitude relocations, vrhich depends on a knowledge of the absolute orientation of the

~ast gyro input axis; (3) response time of the system, which is dependent upon the 'nlaxin1.um allovvable system bandwidth.

_ The systeln, data, and concepts are applicable in any area \vhere accurate initial platform alignment by gyrocompassing techniques is required.

1.4. 0 Design Criteria For ...4,. Stable Base Gyrocompass

This report is intended to carry the reader from gyro ... compassing characteristics through operational concepts fora fixed base inertial guidance systen1.. A slight detour in Chapter IV presents pertinent data on an operating system. The following section sumlnarizes the organization of this report.

\•~

-Chapter II reduces the complexity of .the three-axis gyrocom-pass to a two-axis model suitable for analysis in ~he regions of interes't. This model is analyzed to determine the linear relations betvveen gyrocorn.pass gains, error coefficients response time,

and systen1 banchvidth.

Chapter III indi.cates the relation between knowledge of the east gyro input axis aligt1l11cnt and aziInuth accuracy at various latitudcB. A laboratory technique is developed to detern1.ine the orientation of the east gyro input axis. The wheel speed modulation concept of gyro error' averaging is analyzed and a laboratory optilnization technique presented. No change is anticipated in the labor'atory deterlnination of accelerolneter bias scale factor and input axis ,location.

Ch2pter IV detern1ines the spectrun1 of ,azilnuth and east gyro drift on the basis of several hundred hours of stable base gyrocompassing data. Two system bandwidths were used in the recording of these data s () that uncertainties in instrulnent performance can be properly identified. Hesults of vyheel speed modulation over two-hundred and sixty- eight hours of systen1 operation are also presented.

Chapter V develops a corl1pensation technique for a fixed base gyroco111passing internal guidance systeln. This tcch.nique is based on instrument perrornlance recorded in the previous chapter but anticipates the necessary changes in system band\vidth and increnlental angle measurements \vhen operating in a realistically perturbed enVirOn111ent.

Chapter VI sun1rl1.arizes the' difficult problen1s associated \~/ith

accurate stable base gyroconlpassing and the results achieved in this paper.

Appendix I contains a

Appendix II assumes that \vind gust spectrulu is the primary interferring function between the benign laboratory and the opera tional envirol1ru cnt. The' spcctrul1l. analysis analytically confirlns the orders of rnagnitude of the mean square azimuth as\vell as the vertical angles v/hich lTIUStbe detected in an operational gyro recompensation procedure.

DECLASSIFIED

II C"YTtOCOl\;JPASSING CHARACTERISTICS

Chapter II reduces the cOlllplexity of the three-axis gyro-c0111passing systeln to a nl0del suitable for analysis of systenl characteristics. All simplifying assulnptions are justified to produce a two-axis luodel which is used to develop the relation-ships between gyrocornpass gains, dynalnic and steady state errors, and response tinle of the system. In addition, the dcvelopluent considers the eHect of a vertical sensor with and without a first order lag in ihe region of interest. Although friction and angle transducer nonlinearities exist, the linear approach is justifia1?le becaus e it develops the necessary initial insight into stable bas e gyroconlpas sing characteristics. By referring to Appendix I for n0111enclature, Chapter II may be bypassed by the reader who is fan1.iliar v/ith gyrocon1.pass dynan1.ical relations with no loss in continuity, proceeding to "Laboratory Alignment and Calibration Techniques" of Chapter III.

2. 1. 0 Coordinates

Local geographic coordinates are designated Xct - north, b

yo.... east, and z - vertical. In this chapter, plat,fornl coordinates

b

g

are assumed to be perfectly aligned with instrument coordinates.

Sincc the inst':"L:ITlentcd coo~'clj:l~Jt(;'S and local geogr:':jphic cool~dir}atcs ar'c at all tiii.L~~3 nC::lrly coincident to a first approxin}~~tionJ ,_~rot.a_tiG:.! ~!bO~lt an. insLru111ented coordinat.e

is equivalent to c:' rotation Dbou La -geographic coordinate.

Since these rotations arc SEl8.llJ the angles developed by a

y'otai:ion about two coordinates add curnulatively to define a

nC'l.! oricntation of the third instrurn.ented coordinate. These approxilnations greatly's irnplify the development to follow and do not seriously affcc i.t.3 validity.

Figure 2 -1 i:~ a line SChC:::'Tlc~tic of ti.le gyroco111passing instrlunenJc ge()[~:l'aphic and .i.-;~s'l.rul~lcnted coordinates. The gyro orientat.ion is dcsi:;;:l~ltcd by th(~ ;''If;i)rOpriate location of its input axis (i, e., ec~sC&,':;TO iA is C:D.~=) ::). Tile vertical s

en-sorls sensitive 2xis is p~J..:.~2.11c:l to the icput axis of the gyro \vhich acceThs its output

0.

e.) east vertical SCllSO},IS sensitive

Using slnall angle approxirnations (cas f)

=

1 sin 8

=

0)J J~quation 2 -1 .indicates the total angular velocity input to each gyro.

w -x (J -y w ... z A_y + A_x U(')_{18 V} - A_Z W(l' e)h -A

+

A w,. ','

+

u,. ) z y tlC)l1 tIe v (2-1) ,<.

DECLASSIFIED

vV

(le)

h ::: \U

io

cas

L,L\T

r~J

OF<TH 'l (I 1\;;) V E f1 T I (' ,D•..L L.w '"r'l ..., -

-

\)

The steady state output of each vertical sensor is indicated in Equation 2 -1 . v\there Sand S are the sensitivities 01

xv yv

the north and east vertical sensors respectively nleasured in

iTO'.ts pe'Y' II 0" I

" ..t..) .1. b.~

North Vertical Sensor Output:

East Vertical Sensor Output:

r1 A S

b X xv

g

A S

y yv

(2-2)

The north gyro and vertical sensor are connected together in such a n1anner that they ideally constrain the input axis of the east gyro in the horizontal plane. The .east gyro torque sumu1.ation is basically only satisfied \vhen the input axis of the east gyro is east. If the east gyro input axis is not east, it will sense the horizontal component of earth rate, -A_Q w(; )h'

. '-' ...e where Az is the angle behveen geographic east an,d the east gyro input axis. This angular velocity will caus e the platform to tilt :-::.boutthe east axis, developing an angle, A . This

,

Y

tilt about the east axis will develop an east vertical sensor output equal to g Ay Syv' .which is coupled to the azilnuth gyro torque generatol" and commands an azin1.uth angular rate in such a direction as to reduce the angle between the east gyro input axis and the east geographic coordinate. For our ideal systen1} the east gyro torque sunlrnati6n (satisl'ied only when

l~itP

.. Itl' " .... ", ,': ...

, , .

-

.. " , ., ,'

..

the input axis of the eRf~~~~~.Qt), coupled with the torque Slllnrrlation in the azimuth gyro (satified only when the output of the east vertical sensor is zero) combine to produce a gyro-conlpassing systelu which uses the ,direction of gravity and earth rate as the only external reference for initiai platform alignnlent. A convergent solution is established by a coupling bchveen the east vertical sensor and the cast gyro.

L.,

2. 2. 0 Block Diagranl

Figure 2 -2 is a cOl"nplete functional block diagram of the gyrocon1.passing systen1. showing nlajar 111echanical, electrical, and earth rate couplings. In order to obtain a suitable v/orking model for stability analysis) determination of error coefficients, and response tilue, the follo'wing six assumptions., together v/ith their justification) are presented.

1. Typical stabilization servo tilne constants are O. 02 seconds. Gyro compass tilue constants are greater than one minute. The dynan1.ics of the stabilization se~vos may be neglected in the study of gyrocornpassing alignment dynarrlics and stability. (The platform is ahvays aligned with the input axis ,of its gyros.)

2. The gyro time constants of approximately 0.001 seconds do not affect the stability 01" the solution tin1.e of the

~~ ~~2:

<!:

nr""

\."....~ (!) \..-

<a:

r.:.::\ \'.,,,," ()

0

_~ ...J f(i

_...

..J

'~4:

_-.'"

e"_

0

•

...- _x

,

() ~

-

...

~~::

-

-

:::J

:J

LL /(f) Jf)

<J'

r'"

~_tq

::~~

C)

(...

)

0

f"'"p. ~

>-C!')

r;-'-'

15

I~

18

"'-'I,. Ct:

o

U) I~ I Z UJ (1) ....1

~

i"

L

If.:> +

f~

JO: x 0.. i Il:J> (f) t-> il ~ ~! ,....I !l J

(51

i. 12,...1_--"":'--.1I

r

.. if >, j ~) i I :;" ~ ; ~ ~--:-_¥ ! (\ I I :"'0 C' r.J I II a (!. -L!

3,' _{For this chapter}

DECLASSIFIED

_{static frictIon is assumed to be zero.} Static friction levels are approximately 1 in. oz. Typical stabiliz2..tion servo gains are 50 in. oz. per arc second. Therefore) 0.02 seconds of arc gyro signal generator error is sufficient to overCOlne static friction.

4. Alignment about north involves seeking the vertical onlYJ

and a sufficiently high gain loop can be closed so that north alignment errors have negligible effect on gyrocompassing

.'.

accuracy and solution Hnle. ',' The' upper boundary on the value of north gyrocompass loop gain is allowable system noise.

5 .. The horizontal component of earth rate coupled into the azilnuth gyro" due to a rotation about east .. is negligible compared to the azin'luth gyro angular rate comrnanded via the east vertical sensor and I(P,l..E for all practical values of I(li.E' (I{j\ T:"1

>

1 roeru per second of arc .. and

.L~.:....t:&

-gradient of earth rate at 45 degrees latitude equals 0.0034 roeru per second of arc. )

~:"

.A.t45 degrees latitude) steady state angle errors about the north gyro input axis .. il._x" produce an equal azirnuth angle, Az' of the stabilized nlernber. The steady state value of A.."can be easily kept belo.w one-half second of arc. The transient involved in north level-ing can be cOlnpleted faster than the gyrocolnpas sing transients.

Noise restrictions and vertical sensor dynaluics are the only restric-tions on the solution time of the north leveling loop.

G, Pel'fect co.r-t.h rate con1pensation is assulned, and gyro

...J .•

drift'" or instabilities in the gyro torquing circuits or' supply are included in total gyro drift rate, (u)w, , .

- , x,y,z

\tVith the previous assun1ptions J the diagran1 of Figure 2-2 reduces to that of Figure 2-3, It .is observed from Figure 2-3 that the azin'1uth gyro angular velocity summation is satisfied when (u)w_z

+

]\y K_AE equal zero. The east gyro angular velo-city sUlulnation is satisfied when --;;. w(")l

+-

(u)w -A I(~E

z Ie 1 y Y ~I

equal zero. Therefore, the steady state equilibrium azilnuth orientation of the gyrocon1passing system is

T(

1 LEE

-- [ (u)w

+ lr-

(u)W ] W(ie)h Y '---AE z

(2 - 3)

The solution to Equa t~on 2 - 3, in the abs ence of east gyro and azimuth gyro drift, is a gyrocompassing system 'with the instru-luented east axis parallel to geographic east. The open chain angular velocity sumlnation performed in the east gyro places a basic limitation on gyrocolupassing accuracy. Steady state east gyro drift only produces a change in the azimuth angle of this stabilized n'1'ember, Since the gyrocon'lpassing system

~::Gyr~ DrHt:- Throughout this paper the term gyro drift is used to designate total gyro drift (i. e., any internal change in the torque sumn1ing member and/ or torquing supply),

()

; 1...1) ~ :2-=

.

L0 J f _Cl) I l l

_I

I _J _~ i _<.( {

I

~ U _I \ -

,.

r-"", I _{I- Q..}

r~

~ 0.:: 1 w I ::-

_i

i

>

l- a v ~ J I- ii~

i

(f) 3 <t ~

i

~ t w

I

,

_i ~-' -.--.- n... i~ "Y' I' ~

j

<

.-.. ,:;} ... r0 I (\j I, I.LJ ~ ::) <D l1...

is the azinluth reference} in the clos ed loop system.

DECLASSIFIED

east gyro drift cannot be detected Azimuth _'-'Qyro drift} (U)LJ _z) Dro-_. duces an output from the east vertical sensor as \vell as an aZilTIuth angle error. The effect of aziInuth gyro drift on azirTIuth angle can be Ininimized by compensating the azimuth gyro and Ininimizing the output of the east vertical sensor.

2.3.0 Gains} Dynamic Errors} Response Time

Section 2:3. 0 \vill consider systen1 characteristics and ho\v these charac'teristics are affected by the use of a vert~cal sensor with,a time constant affecting the response thne of the gyrocolnpassing system. Stability considerations affecting gyrocolnpass gains \vill be considered before dynamic and steady state errors becaus e thes e errors dep'end on the gyrocompass gains. Response to lateral accelerations \vill be considered before transient respons.e because the transient response depends on the arnou~1t of filtering to be done.

Equal roots to the gyrocompass characteristic equation are assull1ed for two reasons. First} the case of equal roots is easier to l~andle analytically" yet provides the necessary insight into stability" dynamic errors,,' and response time of the second

,I.-as \vell ,I.-as the third order gyrocolnpass. ',' Second" in an

opera-,t....

"'SecondOrder Gyrocompass:- yertical sensor thne constant negligible cOlnpared to gyrocompass solution time.

Third Or'der Gyrocolnpass:- Vertical sensor thne constant greater than one-third the gyrocon1pass solution time.

DE

.'

--'-'--.

t" .

"Cr.AS~IJ."In'Dh

'

+ f "t""t"

'lng SYSteln, gains ViI11CU

s.~

L.e prO~1UCl. 0' senSl IVI les 01 electromechanical devices and aluplifications often acclunulate errors in excess of ten percent. Optinlun1 gain can only be deterrnined on the basis of system performance. The equal root solution gives the system designer an initial point fr01TI which gains can be optilnized.

Tll'e development to follow deterlnines system parameters for a critically damped system as a function of ~ysteln band-width. These parameters are next used to analytically deter-Inine closed loop system response to gyro drift and horizontal accelerations. Armed ,vith these analytical tools, the syste111' desibmer must next determine a realistic systen1 bandwidth based on design requirements, acceleration profile, gyro drift" and calibration techniques. Observing the operation of the critically. daJnped system, the designer adjusts bandwidth so the system will have steady state errors within the initial

design specifications. vVhen the maxilnuln allowable band\vidth for acceptable steady sta'te errors is determined" the designer can optll11ize further betw'een forced dynalnic errors ancllnini-.mUln response thne. T-l}.iswill" to a small degree" affect

steady state errors so that system bandwidth 111ayhave to be decreased to attain acceptable steady state perforn1ance again. The ,developn1ent that follows deterlnines gains" dynamic errors" and response ti:r11esof the gyrocompassing system \vith and

DECLASSIFIED

2.3.1 Gains

Figure 2-3 can be reduced to lVlatrix Equation 2-4.

I(EE _w(ie)h I.

(u)w

-(u)A K'_vE

A

I

y y .l!..:'., 1 +P(TP + f)

-p-

_y II -p- _{TTJ?'+ l)P} V

I

v

I

:: i I I I K_AE I (U)LJ _(u)A y I(AE I 1

A

! z - J?(T P +1) z I -p- _{-~p +}

_I1p

v

_I

v (2 -4)

The characteristic equation of Matrix 2-4 is

P2(TV P

+

1)

+

I(E" E P_{, ,} ;- 1<_AE w_(ie)h -_- 0

(2 -5)

1

.11'

T<

FI-~-'--- -. -~- ) a secondorder systemyery closely

approxi-v '\J ~AE LJ(ie)h

order system,

sensor to east gyro coupling fI<EE)) and the natural frequency is equal to the square root of the east vertical sensor to ,azilnuth gyro coupling times the horizontal component

0'£

earth r~tte)_,

::

(2 -6)

If the time constant of the vertical sensor can be chosen) RothTs stability criterion shov/s that Equation 2-5 is free from right ha)f-plane poles as long as I{EE

>

Ki\E w(ie)h TV. The choice of a T

v and I(AE' for this third order system depends on accuracy

requir.e-(

nlents) lateral acceleration pi-'ofile) and required response tilne of the systcln. If all roots of Equation 2-5 are equal (critical damping'), the following relations exist.

1 T3

=

1 3 T V :: ::: (2 -7)

. DECLASSlFIED

2. 3. 2 Dynan1ic Errors

DECLAS

The primary error sources in the operation of a stablc base gyrocompass arc gyro drift and uncertainties in vertical sensor operation. The vertical sensor uncertainties are usually a result of horizontal accelerations. lVIinimiza-tion of the 'effects of horizontal acceleration on stable base gyrocornpass performance is primarily a filter design problem. This section indicates the effect of gyro drift and uncertainties in vertical sensor operation on the closeclloop performance of the gyroco111passing system.

13)'

"taking the partial derivatives of Matrix 2 -4, with respect to gyro dri'ft and uncertainties ii1 vertical sensor operation, and

. ", . ~~

substituting the va~ue8 of T2 from Equation 2 -6, and T3 fron1

Equation 2 -7, Table 2-1 is obtained. The error coefficients in Taple 2-1 are expressed only as a function of the time constant of tl1e gyrocompassing system and the horizontal con1ponent of earth rate so that they are easily convertible to any convenient ,units of 111easure. This paper \vill use the Ineru, "which is equal

to .001 earth rate, as the angular velocity unit of 111casure, and the secondo£' arc as an angular measure. To becon1e familiar \vith the n1eru and to convert froln Table 2-1 into measurements other than reciprocal time, ~the follo\ving constants are noted.

,r_

",'

T2 is the time constant of the second order gyrocompass.

T3 is the time constant of the third order gyrocompass.

T.ABLE 2-1

ERROR COEFFICIENTS

Change in Azimuth Angle ~ A

_z

Change in East Angle D..A_y

Uncertainty in (u) Oncertainty in (u)

SECOND ORDER SYSTEM 1 East 2 'Gyro T 2 W(ie)h p 2 2 (u}w (P

+2:..)

_IP

++)

Y T 2 2 \ • ';. (P

+~)

Azimuth T 2 w(ie)h Gyro (P

+2:..)

2 (P

+2:..)

2 (u)W z T2 T2

THIRD ORDER SYSTEM

3 w(O )h (P +-) Ie T3 1 (P +-2-T 2 2 (P +2:..) T 2 2 T . 2

-35-P 2 T2 w(ie)h 2 (P

+2..)

T 2

East

Angle .... (u)A y

East

Gyro

(u)w

y

1

.n~r!T.Pt~~Tf'IED

()

_see

_{(1.5 x 10}-~~ · .₎

₌

_{arc second}meru

1 meru

=

1 meru

=

o.

015 degrees

hour

7.27 x 10-8 radians seconC1 (2-8)

,The error coefficients of Table 2-1 are plotted in Figure 2-4. Data recorded on system performance versus bandwidth (Chapter IV) indicates that gyro drift has a low frequency spectrum compared ,to the time constant of the gyrocompassing system. Therefore, (u)w

, . y

and (u)w primarily_.z affect the steady state performance of the gyro-compassing system. For both the second and third order systems ~ it is noted that ~zimuth gyro drift, (u),w

z' shows up as an .east verti-calsensoroutput as well as an azimuth angle. Figure 2-4 also indicates that the steady state output of the east ve'rtical sepsor is only a function of azimuth gyro drift. Recompensation of the azimuth gyro on the basis o! the output of the east vertical sensor isa prob-lem in compensat'ion techniques and will~ therefore, be coveredin

EAST

AZIMUTH

DYNAMIC

ERRORS

0.1 ~ Az

--

W(je)h

c\

(u)Wy I 0.1 I

Ib

1

Wo '1 100

1'--

0.11-I 0.1

i'Wo

I 100

'f

Wo. I 0.1

o

.112W(ie)h -I

ro

~Az ~(u)Wz I 0.1 .1Wo

'tWo

__ 2nd Order -37~ FIGU R E:.2 ~.4 , "" \. __ 3rd Order'

DECLASSIFIED

It is noted (Figure 2-4) that steady state east gyro drift,

(u)w , only produces azimuth angle errors.

Since the

elimi-y

nation of all external azimuth references .is the primary

objective in gyrocompassing,

this open-loop characteristic

of the east gyro places basic limitations on stable base

gyro-compassing.

With present state-of-the-art

calibration techniques,

the gyrocompass loops must be opened to detect east gyro drift

rates.

The detection and recompensation of the east gyro drift

will also be covered in Chapter V.

The uncertainties

in east vertical sensor operation which are

induced by horizontal accelerations

produce dynamic errors'.in

the closed-loop gyrocompass.

Each micro g of horizontal

acceleration which has a frequency below the breakpoint of the

vertical sensing device produces 0.206 seconds of arc error

in

the indicated vertical.

The coupling of these uncertainties

in

the east vertical sensor into the gyrocompassing system is a

function of the frequency of the disturbance,

the time constant of

the vertical sensing device, and the time constant of the

gyro-compassing system.

The effect of a vertical sensing device with

a time constant close to that of the gyrocompassing system is to

attenuate the higher frequencies of the horizontal accelerations.

This is evident in Figure 2-4 which indicates that uncertainties

in the east vertical sensor operation,

(u)A , are attenuated above

y

the break frequency of the gyrocompassing system by a slope of

DECLASSIFIED

minus two in the third order system and a slope of minus one

in the second order system.

In this section we are primarily

interested

in the effect of

horizontal accelerations

on the azimuth angle of the stable base

gyrocompassing system. 4* Figure

2

-5 indicates the magnitude

of dynamic azimuth errors

in the gyrocompassing system as a

function of the time constant of the gyrocompassing system and

frequency of horizontal accelerations.

This figure is obtained

8A

from

a

(u)~

of Figure 2-4 and the 0.206 seconds of arc east

y

vertical sensor uncertainty per micro g of horizontal

accelera-tion .

• *:.'<

The shortest allowable time constant

of the gyrocompassing

system can be determined from a knowledge of the predominant

frequencies

***

of lateral horizontal accelerationf? and accuracy

requirements

on the azimuth angle of the gyrocompassing system.

For example, assume the stable base on which the gyrocompass

is mounted is vibrating at ten micro g's with a frequency of O.1

radians per second.

The following azimuth errors

would be induced.

*

Superscript numerals denote references

in the Bibliography

*:.'c

Chapter V shows that the detection of gyro drift places a more

severe restriction

on the time constant of the east vertical

sen-sing device.

Consequently, the gyrocompass time constant is

defined for a critically

damped system .

.***

This analysis assumes horizontal acceleratio~

is sinusoidal.

A more sophisticated approach would involve a statistical

analysis

of acceleration

spectra.

Appendix II presents

pertinent results

of such an analysis.

AZIMUTH

LATERAL

ACCELERATION

ERRORS

4 ,/ / ,/ ,/ / / / / / / / / I 0.01 I 0.001 / / ,/ / / / ". / / / / / -3 "./ 4X I 0-2 4x 10 I 0.0001 4x 10 I

-5

.4 xlO

-z

o i=

«

-2

0:: 4x10-~ :I: I-::> -I :E 4 xl 0 N <t () 0::

«

lL. C'l

o

I

~ 0 Z 0::. o U ~ ~ en RADIANS

ACCELERATION FREQUENCY- SECONDS.

-40--- 2nd Order ---"3rdOrder FIGURE: 2-5

T

=

100

DECT)1~StFl'ED Azimuth Variation

2nd order system

40 seconds of arc

3rd order system

T

=

1000

2nd order system

3rd order system

2.3.3

Response Time

4 seconds of arc

Azimuth Variation

. 40 second of arc

. 004 second of arc

Although i~itial thermodynamic effects and final

non-linearities

in the small angle region make the

gyrocom-passing system response time deviate from that predicted by

linear theory,

certain insight into sequencing operations and

solution times can be extracted from the linear model.

Because

the initial output of the east vertical

sensor has a predominant

effect on the azimuth transient,

the most desirable response of

the gyrocompassing system consists of two phases:

(1)

pre-liminary leveling;

(2) gyrocompassing.

Preliminary

leveling

can be accomplished with a high loop gain, the'time constant of

which is only a function of the time constant of the vertical

sensing

device.

For the third order critically

damped system,

since

T

equals 3

T ,

one-for'th of the total alignment time will be us'ed for

v .

leveling.

If preliminary

leveling is not used, the east vertical

J

sensor output will often produce azimuth errors

in excess of the

initial azimuth- angle.

bECiAssfhi:Din

a system with an

east azimuth coupling of 100 meru per second of ,arc" an

initial east error of 10 seconds of arc in the wrong direction

,would produce over 300 seconds of arc azimuth error before

the output of the east vertical sensor changed sense.

Since

in'itial azimuth angles are seldom above 60 seconds of arc"

it is apparent that preliminary

leveling will" in most cas es"

reduce the transient response time.

The accuracy of pre~

liminary leveling is a function of lateral horizontal acceleration.

Since initial east error angles are around the 10 arc second

region, three time constants of the leveling loops should reduce

the output of the east vertical sensor to a small enough value

so that the gyrocompass loops may be clos ed.

By substituting A (0) for the uncertainties

_z

on the right hand

I

side of Matrix Equation

2

-4 and transforming

the results,

the

linear azimuth time response of the gyrocompassing: system is

obtained.

The response time for the second and third order

critically

damped systems with equal gyrocompass system time

constants are indicated in Figure 2-6.

It is noted that the

responses are 'quite similar and, also, that

if T

3 equals

three-quarters of

T

2

, ninety-five percent of the azimuth response

would be complete in both the second and third order systems in

an equal amount of time.

Referring to Figures 2-5 and 2-6, it

GYROCOMPASS

STEP

RESPONSE

1 10 I 9 I 8 I 7 I 6

AzU)

.

,_l.

( ) =

(.1

+

t )

e ~

Az

0 I 5 I 4 .I :3 I 2 Az (t) .t2 • _ t ! ~ Az(o)

:U+t+

213)e

1; ,.

0,

o

0.9- 0.8-

0.3-

0.7- 0.1- 0.4-

0.,6-

0.2-- 0 ';:;:; 0.5-<t

«

t

r

III -43-FIGURE: 2-6

DECLASSIFIED

third order system can have a wider bandwidth and will,

therefore,

have a much faster solution time than the

second order system.

DECLASSIFlEI>.

-44-'D~CT. ..1lc~r1-"T1m

III LABORATORY ALIGNMENT AND CALIBRATION

The knowledge required in an inertial guidance system to

permit accurate guidance is the position of the input axis of

the accelerometers

(i.

e.

I

the measurement

coordinatesL

relative to the local vertical and an azimuth reference in the

defined horizontal plane

(i.

e.

I

the geographic coordinates).

In sequence

l

one measures the position of the accelerometer

axes with respect to the vertical sensing device and azimuth

reference.

In the laboratory an azimuth optical reference line

can be surveyed in and used to erect and align the system.

A

mirror

attached to the stabilized member is designated as the

prime reference during the calibration sequence

l

and the

measu-)

rement coordinates are located in relation to the mirror.

If

absolute optical azimuth and vertical references

are available.

at the system's

operational sight

l

the measurement coordinates

are defined in terms of local geographic coordinates by slaving

the mirrored

surface to the optical reference.

Stable bas e gyrocompassing substitutes a gyro input axis

for the azimuth reference at the operational sight.

Assuming

that no absolute optical reference is available

l

the

predeter-mined laboratory orientation of the stabilized m.ember must

coincide with that orientation acquired by gyrocompassing at

DECLASSIFIED

I

laboratory optical reference,

one can write an equation

describing the angular velocity summation performed by the

east gyro of the gyrocompas sing system in terms of gyro

misalignments,

residual interferring

torques,

and

compen-sation.

(3-1)

Because of latitude coupling into Equation 3-1, the gyro

misalignment angles must be defined in the laboratory if the

laboratory and operational sights are at separated latitudes.

To determine the misalignment angle knowledge requirements

as a function of latitude, Equation 3-1 is differentiated,

holding

the residual term and compensation current constant.

*

A(h_y)

=

angle between horizontal plane and east gyro input

axis (Figur e 3-1) .

A(y _y) = angle between geographic east" and east gyro input

axisg(Figure 3-1)

DECLASSIFIED

DEVIATION OF EAST GYRO IA FROM GEOGRAPHIC

EAST

W{je}v

VERTICAL

NORTH INTO

PAGE

Y GYRO INPUT AXIS

A{h- y)

HORIZONTAL PLANE

-"----

---MIRROR YGYRO fA WEST-Yg

SOUTH - W(ie)h

__A(-Yg-m)-- _{GYROCOMPASS WE~T}

A(-Yg-y)

=

A(-Yg-m) ~A(m-y)

VERTICAL INTO PAGE

GYROCOMPASS EAST

Y GYRO fA EAST Yg

MIRROR

-47-FIGURE: 3-1

W(ie)[(A(h_y) cos Lat + A DECL~~mPd

Lat

(yg-y)

.

- cos Lat dA(y _y)] = 0

g

(3-2)

The angle between the horizontal plane and east gyro input

axis, A(h_y)' is only defined by the null* orientation of the

north gyrocompass loop and is independent of latitude.

The

change in azimuth angle of the gyrocompassing

system is

equal to the change in angle between geographic east and the

east gyro input axis.

Rearranging

Equation 3-2, the effects

of latitude change on the azimuth angle of the gyrocompassing

..

system are apparent.

dA

(y

g

-y)

=

d Lat

A(h_y) + A(y _y) tan Lat

_g

(3-3)

*

The north gyrocompass loop is always operated close to a

null.

When the system changes latitude,

the north gyro is

recompensated.

Section 2. 2.0 and Section 5. 1. 0 discuss

the north gyrocompass loop.

DECLASSIFIED

DECLASSIFIED'

Equation 3-3 indicates that the change in azimuth angle of

the system as a function of latitude only depends on the east

gyro's misalignment angles.

To develop a design

require-ment on the east gyro's misalignrequire-ment angles as a function of

latitude, assume that the calibration laboratory is located at

45 degrees latit~de.

For a

.:!:

5 degree latitude zone around

45 degrees, the rms a.zimuth error is to be less than two

seconds of arc.

Under these conditions, A(h_y) and A(y _y)

g

must be knowrito better than 14. 1 seconds of arc.

The remainder of this chapter presents a method which can

be used in the laboratory to determine the east gyro

misalign-ment angles.

Also presented is the initial laboratory optimization

of wheel speed modulation.

In order to determine these angles"

the laboratory should be equipped with a rate table and two auto-.

matic autocollimators referred

to geographic east and west.

In

addition, it is assumed that the system has one stationary mirror.

*

~(If the system permitted optical access to the east gyro case,

advantages could be gained by locating a mirror

:withrespect to

the gyro input axis on a single component basis and then observing

the position of the mirror when the system is gyrocompassing.

However, the development that follows assumes the more general

case of an arbitrary

azimuth mirror

on the stabilized member.

3.1.0 A(h_y)

_DECLASSIFIED

Several schemes have been proposed to reduce the

angle between the horizontal plane and the east gyro input axis.

One of these measures the azimuth orientation of the

gyrocom-passing system at several latitudes and determines A(h_y).

This method uses the effect to calibrate the cause but is

imprac-tical because of the large latitude separation of surveyed sights

required.

Also" this method assumes all other system

para-meters remain constant during the time required for latitude

change.

A second method rotates the stable member exactly

one-hundred and eighty degrees about a horizontal line" keeping

the east gyro input and output axis in a plane perpendicular

to

the line.

This method requires unrealistically

accurate

measure-ments of a large angle" and also assumes that changing the

orientation of the east gyro by one-hundred and eighty degrees

.in the gravity field will not affect its torque summation.

Method

number three artifically

applies a vertical rate to the stable

member of the gyrocompassing system.

When the east gyro is

insensitive to the component of vertical table rate along its input

axis" A(h_y) is equal to zero.

By rotating the stabilized member with a vertical table rate

before the gyrocompassing system is assembled" A(h_y) can be

measured to better than ten seconds of arc.

Once A(h_.y)is

DECLASSIFIED

-50-c

DEer.

It(!'C':I

known" it can be compen~~mD:>r

reduced,

depending on

the alignment flexibility

~'c

of the gyrocompass inptruments.

The deve~opmentthat follows assumes a method is available

for reducing A(h_y). This simplified the subs'equent

measure'-ments of A(y _y) and R" but does not reduce the validity of

g

the technique

if

A(h_y) can only be compensated for

(i.

e., not

reduced).

The vertical table rate is applied along an axis which is

nearly parallel with the east gyro output axis.

This table

rate couples unwanted torques to the gyro float through, other

axes.

The table angle changes the component of horizontal

earth rate applied to the east gyro input axis.

The accuracy

of measuring A(h_y) relies firmly on the table rate and angle.

The following detailed description of the calibration procedure

defines the accuracy of determining A(h_y) and the optimum

table rate and angle.

Mathematically" the moments about the output axis of a

single-degree-of-freedom

integrating gyro are as follows.

*

If the north vertical sensor null orientation with respect to the

stabilized member can be easily changed" A(h_y) can be reduced.

In practice,

machining tolerances will probably establish an.

A(h_y) of approximately 0.1 milliradians

and,

if

measured,

this

angle can be compensated for.

DECLASSIFIED

*

2

1;M(OA)

=

H wi-f(IA)

+

P Ig(OA) Ac ...f(OA)

(3-4)

Equation 3-4 assumes that the gyro angular momentum is

orthogonal to its input axis" the case and the float input axis

are perfectly aligned" and the spring force of the gyro float with

respect to its case about the output axis is equal to zero.

Solving

Equation 3-4 for the angle of the gyro float with respect to the

case about the output axis" Ac-f(OA)"

Ac-f(OA)

= 1 I P( g(OA)

Cd(OA)

STG

ic

+

---Cd(OA)

+

(u)M(OA) ] Cd(OA) (3 -5)

>.'c

Appendix I contains a definition of terms .

The, angular velocIty'Of

~fi~T~1J

float with respect to

inertial space about the gyro's input axis~ wi-f(IA)~ is

equal to the angular velocity of the gyro case with respect

to inertial space plus the angular velocity of the gyro float

with respect to the gyro case.

Wi-f(IA)

=

wi-c(IA)

+

wc-f(IA)

The angular velocity of the gyro case with respect to

(3-6)

inertial space~ wi-c(IA)~'is due to the angular velocities

external to the gyro case.

The angular velocity of the gyro

float with respect to the gyro case~ wc-f(IA)~ is produc~d by

inertial torques between the gyro float and case about the

gyro input axis.

With the gyro output axis nearly vertical~ a vertical table

rate,

wT~ is applied to the stable member.

If the gyro input

axis is not orthogonal to the vertical~ the table rate applies

WTA(h_y) ~o the gyro case. The problem is to minimize A(h_y)

by reducing the effect of the vertical

table rate on the gyro input

axis.

This table motion couples two unwa.nted chan~ing angUlar

velocities into the gyro input axis.

First,

the horizontal

com-ponent of earth rate depends on table angle.

Second, the table

DECLASSlflED

-53-_~.~;I .

DECI.ASSIPlED

rate being nearly coincident, with the gyro output axis, couples

a torque equal to

HW

_{T to the gyro float with respect to its case}

about the input axis.),'c The problem is to separate the angular

velocit:y applied to the gyro input axis due to A(h_y) from the

other effects and reduce A(h_Y).

To minimize the effects of varying earth rate applied to the

gyro with A(h_y) constant, the gyro input axis is aligned

approxi-1

11 1

h

),'c~'c

Th

'

..

1.

STG

mate y para

e to nort .

e gyro torqulng slgna ,

1 -H '

. c

is adjusted to cancel the northerly component of earth rate plus

any,residual bias in the gyro.

Under these conditions, when a

vertical table rate,

w_T'

is applied to the stabilized member,

the

varying angular velocities applied to the gyro case about its input

axis are

W

_i-c(IA)

=

(3-7)

),'cThehigh ratio of output axis damping to inertia in a '2FBG-6F gyro

forces the gyro float to very nearly follow the gyro case for output

axis rates below five hundred radians per second.

DECLASSIFIED

If the magnitude ofthe table angle" AT" is less than 0.03

radians and the table rate" w

T

" is greater than the vertical

component of earth rate" Equation

3-7

reduces to

(3 -8)

A sinusoidal table rate is chosen for- reasons which will

become clear as the alignment procedure proceeds.

A

=

B sin

W

T

T ' 0

w_{T -}-

B

w₀

cos

w₀

T

(3-9)

Substituting AT into Equation

3-8"

and remembering

that

sin2 x

=

1/2[ 1 - cos 2 x]" Equation

3-9

changes to

DECLASSIFIED

(3-10)

-55-DECLASSIFIEI>.

Equation 3-10 indicates that the frequency of the horizontal component of earth rate applied to the gyro is twice that of the .table rate. If the magnitude of these rates were equal, they

could be separated by observing the output of the gyro ducosyn with the table rate as a reference. If the magnitude of the change in horizontal earth rate applied to the gyro input axis is equal the vertical rate coupled into the gyro, the frequency and amplitude of the table rate are related in the following manner.

B=

W

4A 0

(h-y) w(ie)h (3-11)

Under the conditions of Equation 3-11, the effect of horizon-tal earth rate may be neglected because it can be separated out of the data .

. The torque of the gyro float with respect to its case about the input axis, H w_T' is produced by the table rate about an axis nearly coincident with the gyro output axis. Summing.

moments of the gyro float with respect to its case about the input axis, Equation 3-12 is obtained.

DECLASSIFIElJ

-56-~ M(IA)

=

DECLASSIFIED

2

wT H

+

P Ig(IA) Ac-f(IA)

+

P Cd(IA) Ac-f(IA)

+ Kc-f(IA) Ac-f(IA) = 0

(3 -12)

Equation 3-12 assumes that w_T is perpendicular to H, and the total table rate is transmitted from the gyro case to float

about the gyro's output axis.

*

Substituting wc-f(IA) for P Ac-f(IAL and rearranging, Equation 3-12 describes the angular velocity of the gyro float with respect to its case about the input axis.

wc-f(IA)

=

_P W _H__ T Ig(IA)

C

K

p2

+

P d(IA)

+

(IA) Ig(IA) Ig(IA) (3 -13)

The following numbers are applicable for a 2FBG-6F single-degree

*

_{Gyro float tracks the case for output axis rates below five-}

.

of-freedom

integrating

gyro.

H

=

2 x 106

DECLASSIFIEl>

2

gram em

second

dyne

em

radian _:.

/ second

dyne em

radian:,; / second

Ig(IA) Ig(OA)

=

2

x

103. .2

gram em

2'

gram em

dyne em

radian

Substituting numbers and combining Equations

3-5, 3-13,

and

3-6,

A

_e-f(OA)

=

1

[A

w P(10-3 P

+

1} (h-y) T 2 P w_{T 4 ~ 10} (u) M(OA)

---,,---

+.

]

PECLASSITI¥p

+

106p

+

0.2 x 106 2 x 106 (3-14)

-58-DECIJlSSIFIE~

Ac-f(OA) ~ W

T

-[A

P (h-y) P 2 x 10 -3

+

(u) M(OA)j 5 P

+

1 2 x 106 (3-15) W

Substituting AT for ':: from Equation 3-9"

A

cf(OA)

-(3-16)

Substituting the earth rate coupling constraint of Equation 3-11 . into Equation 3-16" . A cf(OA) -4 A(h ) W sin W T -y 0 0

[A

w(ie)h (h-y)

+

(u) M(OA)j 2 x 106 2

x

10-3 P 5P+1 (3 -17)

-59-DECLAsstFlED

The angle between the gyro float and case about its output axis" Ac-f(OA)" is sensed with the gyro ducosyn. In a benign laboratory" incremental angles of 0.1 seconds of arc (5 x 10-7 radians) can be measured with the gyro ducosyn. Therefore" from Equation 3-17" in order to record the component of vertical table rate coupled to the gyro input axis at 45 degrees latitude" 5 x 10-7 w(ie)h

>

4 w

=

o 6.6

x

10-12 w o (3-18)

But" from Equation 3-1 7" in order to separate the component of vertical table rate applied to the gyro case from the com-pc:>nentapplied to the gyro float about ~ts input axis"

A

.

(h-y)

>

2

x

10-3

_w

o

DECLASSrFlLtl

(3 -19)

-60-is

Substituting into Equation 3-18, _.

DECLASSlFIEIJ..

-6 1/3 2.

w

<

(1.7 x 10)

=

1.2 x 10- radIans

o second

(3-20)

Substituting into Equation 3-18, the minimum detectable A(h_y) .

A(h -y)minimum measurable

=

2.4 x 10-5 radians

=

4.9 arc seconds

(3-21)

Substituting into Equations 3- 9 and 3-11, the optimum table angle and angular rate are defined.

A_T

=

2. 3 x 10-2 sin 1. 2 x lOT-2

(3-22)

Since the maximum table angle achieved is 2.3 x 10-2 radians, the assumption of small table angles holds, and the expansion from Equation 3-7 to 3-8 is valid. The sinusoidal table rate allows the separation of the angular velocity components applied to the gyro