UlluLt\~~JtltU lVlassachus etts Institute It,--t- /) If (Gl\"l'\'J ...~i of Technology C) t •. ?, ~ _ltJ:p..1.C.b .. CLASSIFIED BV C).~~ SUBJECT TO GENERAL DECLASSIFI~N----- (:. 960) SCHEDULE OF EXECUTIiE ORDER 11652 AUTOMATICA~LY DOWNGRADED AT TWO YEAR INTERVALS dIF~fE TED IN P .i~~TIAL FULFILL?vIEI~T ON -OOlEMBER-91,o.a QUIRElvIENTS FOR THE DEGHEE OF Mi~STER OF SCIENCE at the CHUSETTS M o INSTITUTE January) Z OF TECI-INOLOGY 1963 b1) -3 o 0: 0') Signature of f._uthor Department of }~ ..eronautics and Astronautics) Januar,y 14) 1963 \ Certified by _ Thesis Accepted Supervisai' by Chairln~ DECLASSIFmll UNCLASS!F1ED (I ' ":J L DECLASSIFIED ACI(NO\VLEDGEl\lIENT This report was prepared DSR Proj ect 52 -156 Missile J under the auspices sponsored of by the Ballistic 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 the national inforlnation affecting defense of the United States within the meaning of the Espionage Laws U. S. C. J J Sections or the revelation unauthorized the transmis is prohibited of this report by the i\ir Force conclusions Title 18, of which in any manner person The publication approval 793 and 794 J to an by law. does not constiJ~ute of the findings or -:he contained therein. It is published only for the exchange and stimulation DECLASSIl-1ED -2- sion of ideas. ACI(NO\VLEDGEMENT DECLASSIFIED The author is grateful Department for the association of Aeronautics Instrumentation and Astronautics Laboratory) to develop analytical for an operating which affords concepts behveen the and the the opportunity as well as an appreciation state-of-the-art system while performing , this investi~a tion. The author wishes to acknowledge his appreciation to of the Skipper J. Group who developed the the members highly reliable system In particul~r) on which these data were recorded. the author wishes to thank the following persons. Professor assistance) Vvin'ston R. Markey for his supervision) and criticisms of the initial draft while acting , as technical faculty advisor Mr. Richard its direction) E. Marshall and markedly on this thesis. for suggesting commenting on the initial Mr. Thomas E. Reed and Mr. William their many stimulating the preliminary Mrs. discussions the topic) guiding draft. T. McDonald for of the subject matter draft. Elaine Walsh for typing this paper. DECLASSIFIED and ~,'''-,,','-,,-'-, ",,' ~. FORE\rVORD DECLASSIFIED This fore-v-rord describes thesis and the general It is not considered the goals and results situation a regular that surrounded of the this \,york. part of the thesis. s On October passing 1, 1962, the 2-16 system at the 1V1. 1. T. Skipper J Test Facility During t~1is tilne, recorded E-1231 stable base gyrocompassing on the system operating (Ref. 3) contains thegyrocompassing The results performed to better recorded the the existence capability of accurate of arc rms error in azimuth of hours. angles below the order in the 2-16 gyrocompass of 1962. are as follows. gyrocompassing arc about the horizontal on to and \vhich preceed demonstrated of hundreds Incremental results in this thesis than five seconds for periods data had been hefore the fourth quarter The 2 -16 gyrocompass a memory-moqe 2. the pertinent system for 2000 hours. in a benign environment. which are fundamental investigations 1. had been gyrocom- of O. 01 seconds axes can be consistently operating of detected in a benign environ- mente 3. Gimbal static and sliding friction appear to be a major problem nonlinearities in the oper~tion base gyrocompass. DECLASSIFIED did not of a stable . h f. Durlng t e lrst DECLASSIPIED .. quarrel" oI'-rTI"tf2prellmlnary expe J on a vrheel speed modulation technique of gyro error demonstrated of the feasibility averaging detecting very low level torque changes in a 2FBG-6F gyro with its input axis approximately east. Thes tests conducted on a single instrument a test table are the subj ect of T-315 (Ref. 1). indicated the possibllity mounted on The results of utilizing wheel speed modulation to ach,ieve a fixed base azimuth indi.cating system for long operational periods. sample First efforts data procedure , and other members on paper to devise an operational were carried on jointly by the author of the Skipper J Group (Ref. 2). The program. for the 2 -16 system of 1962 had as a first during the fourth quarter goal the demonstration bility of the wheel speed modulation of system technique. in control of this effort as a research The author, was assistant. The experin1.ental part of this thesis \vas to record system performance of instrument versus uncertainties. inputs that limit frequency feasi- b<?-ndwidthto reveal data on characteristics These data forln one set of system regions covered by practical solutions '(Chapter IV). Equal importance characteristics development vias attached to defining probable of such systems (Chapters of adequate operational operating II and III) and the procedures (Chapter V). Because the above aims were M¥=gW~m1?dent demonstrating reasonable the thesis feasibility to report these results. is to present resultsachie'ved s of the vlhole technique, organized theori~s of operation for the wheel speed modulated matters and gyrocompass. in a single docu- the coverage which is often limited but manages at least one plausible solution in each area. The -results of these investigations 1. it became Thus) a third goal of .The value of having these related ment justifies upon are as follovls. The thesis analysis' of system performance versus bandwidth reveals instrument below the values previously anticipated. frequency uncertainties regions of realistic These data .extend the system performance. (Chapter IV). ,2.• Wheel speed modulatio.n techniques appear to demonstrate the capability level torque variations in the east gyro operating benign system 3.. aJ?e reported rrhe thesis environment of detecting low uncertainty with anticipated system bandwidth and incremental to define probable operating compensation procedures environment (Chapter V). in a (Chapter IV). combines the data on the spectrum and instrument which of system changes in angle measurements characteristics in a realistically and gyro perturbed 4. The thesis ,eports. stablDl:~SSImDmpaSsing charac~eristics assumptions and justifications \vhich describe unconventional instrument s 5. closely performance . The thesis' reports ment alignment (Chapter III) . for the simplifying this system with its coupled loops and superior (Chapter II). accuracy and derives requirements a possible on instru- alignment method 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. Empirical 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 performance in the presence of horizontal acceleration inputs. Thesis Supervisor: Title: VVinston R. Markey Associate Professor of Aeronautics and A 3tronautics DECLlissrflED ~i%~~; ~j'~~~~~fi~~t~~~~,i;~~~tW~i*~~~~~t~~: ~'j; ~,~4~4~'~~?1{~ .~"~~~:~~<'H~P~s~~:~~~;~"~~~~~"~:::'\fZl~~}~~;t1l.L.-'~.~ir~ ...... ... - .. -- - ,,' ~.,;. -- --.- DECLASSIFIED TABLE OF CONTENTS Page CI-IAPTER I Intr,oduction CHi~.PTER II Gyrocompass CIIAPTER Laboratory III ratio Characteristics Alignment . 21 45 11 •••••••••••••••••••••••••••• System and Instrun1.ent Performance CI-Lt1.PTER V Operational .CI-IAPTER VI Conclusions APPENDIX I Non1.enclature .... APPENDIX II Spect:r'a of Gyrocompass ~ 12 and Calib- CI-IAPTER IV BIBLIOGRP.PHY . Syster.a. Evaluation .. .. 0 0 0 •• 70 105 129 0 ••••••••• 0 ••• 0 0 •••••• 0 •• 0 00 Angles ..... ••• 0 • 0 • 0 •• 0 133 139 ' •• 145 DECLASSIFIED LIST OF FIGURES Figure 2 -1 Stable Base Gyrocompass Figure 2 -2 Gyrocompass Figure 2-3 Functional Coordinates 23 Block Diagram 26 East AziInuth Gyrocolnpass 29 37 5 Figure 2 -4 East Azinluth Dynalnic Errors' Figure 2 ~5 Azimuth Lateral Figure 2-6 Figure 3-1 Deviation of East Gyro IA From Geographic Figure 4-1 East Azimuth Gyrocompass Figure 4-2 Uncertainty Figure 4-3 2-16 Gyrocompass Short-Term Figur~ 4-4 2-16 Gyrocolnpass Effect or Varying Bandwidth Figure 2-16 Gyrocolnpass Effect of Wheel Speed Modulation. 4-5 . Gyrocompass Acceleration Frequency North Gyroconlpass Figure Qpen/Closed Figure 5-3 ., 40 Step Response Figur..e 5-1 5-2 Errors Loop 43 East .. 75 ' 77 Stability 83 5-4 - East i~.zimuth Gyrocompass Open Loop East Vertical Gyro Drift (one-half 86 103 107 109 Sensor Output for Incorrect Azimuth Gyro COlnpensation Figure .... Loop Open Loop East Vertical 47 Sensor Output normal wheel speed) DECLASSIFIED 114 - East 122 Table 1-1 Regions of Interest Table 2-1 Error Tab).e4-1a 260 Hours - Low East/Azimuth. Coupling . 80 Table 4-1b 168 I-Iours - I-ligh East/Azimuth Coupling . 81 Coefficients 17 0 0 0 0 0 .: ... ,s ~ 0 • 0 0 '. 0 • 0 0 0 •••••• 35 '0 ,DECLASSIFIED I The effectiveness INTRODUCTION 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 with optical techI'!iques utilizing are presently complex groundinstallatiqns an accurately surveyed azimuth reference. and corrected with an accelerometer" device operating with the system Since inertial aligned in azimuth The vertical pendulu~um, and is sensed or other leveling' gyros. guidance of missiles became a reality, the possi- bilUy of alignment by gyrocompassing has been realized and, at tiriles',used. alignment A deterrrent requirements to gyrocompassing using available advflpces in the development ,ing gyros, instruments. of single-degree-of coupled with system developed the operational has been stringent optimization capability Recent -freedom techniques" of self-aJigning ,integrafhave:' missile guid- ance systems. This paper discusses as .well as laboratory gyrocompassing alignment, calibration tional' concepts for a fixed base inertial are'indicative of the uncertainty theory and characteristics" spectra tern form the basis of the calibration techniques" and opera- guidance system. Datathat of the gyrocompassing. and compensation developed. DECLASSIF!EO sys- techniques 1. 1. 0 Gyrocompassing By definition) DECLASSIFIED a gyrocompass includes any and all devic'es or systems that align the111S elves toa 'predetermined set"ofcoordinates) using the earthls direction angular velocity and of gravity as basic references. are sensitive to the earthls devices which determine Instrun'lents angular velocity) the direction fore) the core or a gyrbcompassing coupled \vith of gravity are) system. 'stabilized electrolytic member levels) integrating devices etc.) to the In the three-gyro the gyro input axes are approximately Basically) sensing accelerometers) drives. there- The direction of gravity is .defined by connecting the vertical (pendulums) which vertical) system) north and east. 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. new azirnuth angle changes the horizontal This con'lponent of earth rate applied to the east' gyro input axis. Accuracy primarily capabilities of the gyrocompassing detern'lined by the stability closed loop gyrocompassing system system of the east gyro. cannot distinguish DECLASSIFIED are The between DECLASSIFIED internal changes in the east gyro characteristics torques)) making the angular applicable) (uncertainty velocity summation no longer 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 external references. East-wrest of the east gyro with no averaging .. gyro wheel reversal) and wheel speed mo'dulation are three methods evaluation which have been investigated paper limits attention environment it has recently level torque variations 1.2.0 Accuracy in recent to vrheel speed modulation because in a system been us ed to determine long-term gyrocompassing for these regions imposes low severe The follovving discussion. included to develo'p a feeling for the magnitude of interest is of the problem. in the gyrocompassing are included in ,the body of this report. As an example) the gyroconlpassing assume that the absolute drift uncertainty allo\vable uncon1pensated at medium latitudes per hour). azimuth stability shall be within + 5 seconds system This defines the maximum degrees This on a single component basis. demands on systelTI performance. system years. Requireluents Accurate Justification of east gyro of of arc.' east gyro to be 0.018 meru (0.00027 Allowable north and azimuth gyro uncertain- DECLASSIFIED . . ........• . - . ~- '. '", '- , ..-:.... ." t .. ~ ties are a function of syst~.¥~~~hich between gyrocolnpassing is, a cOlnpromise uncertainties and lateral horizontal accelerations. In a benign environinent, gyro drift rates in the O. 1 to 0.5 meru region can be tolerated in. thes closed loop gyrocompass. north and azimuth 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 wheel speed Inodulation technique of~small angles by the east vertical of the east gyro, requires the n1.easurement sensor. A highly stabiliza- tion servo gain and low values of gimba~ static friction produce the r.nost desirable In the system operating (i. e., in the 1 in. oz. region, most nearly linear) in a benign environment, servo gains of 50 ft.lbs/mr, the will operation. stabilization coupled 'with gimbal static friction have combined to develop a system in which 0.01 seconds of arc can be detected and used to recompensate the east gyro. Since gyro drift produces in the gyrocolnpassing system, angle variations the time required to measure small angles defines the bandwidth of gyro drift detection. Latitude relocations ponents of earth rate. change the vertical and horizontal com- If the east gyro input axis is not parallel to geographic east, latitude relocations the earth rate applied to the east gyro. DECLASSIFIED will change the value of This change in earth rate DECLASSIFIED applied to the east gyro v/ould vary the ~zimuth angle of the gyrocompassing reference degrees" system. and ope:ating If the laboratory absolute azimuth area are separated by five latitude seven sec~nds of arc error le?ge of:the east ~ro arc azimuth error know'- inpllt axis will produce one second of at the nevI latitude location. Thes e regions 6f interest The gyrocompassing angle stability in the laboratory are summarized system requirements are to maintain the over a maXllnUnl time interval can be extended to several years. in Table 1-1. vvhich, hopefully, The basis of thes e numbers is the subj ect of this report. 1.3. 0 Operational Integration The unattended self-calibrating system requiring no alignment or stability years after it leaves the laboratory internal guidance checks for several is not a reality. How'.ever J the guidance system under test has the ability to memorize set of coordinates for all presently launchings. system over several for missile If the gyros are periodically with no external-references. missile days vrith sufficient accuracy known requirenlents can hold the reference t and satellite recompensated, the for an extended period of time Integration site containing optical references reduce the missile a s vulnerability of this system into a would immediately to ground disturbances DECLASSIFIED by ~:< Regions of Interest Angle Stability AZ,imuth Variation < + 5 seconds of arc- 'North Variation < + 1 second of arc East Varia tion < + 1 second of arc o Uncompensated Gyro Drift North O. 1 to O.5 meru Aziriluth 0.1 to 0.5 meru - East < 0.018 meru better than 14 seconds Component Alignment Knowledge of laboratoryorientation of the east gyro input axis: of arc. l\1iscellaneous 50 lb. ft/mr ST1.\.B Servo Gain 1 in. oz. Gimbal Friction Incremental 0.01 second of arc (benign environr,l1.ent). P...ngles 0.1 Hepeatability oJ Changing East Gyro Vvheel Speed: ~~ Table for t15 0 better second of arc (operational environment) . than O.Olmeru latitude. DECLASSIFIED . DECtASS~D nlemorizing the pr,oper g-Llidan'cesystem maintaining alignment after the optical reference The ,three main areas of interest the o~der of decreasing cedure; (2) sensitivity vlhich is primarily of the system (3) response in of the sys- a function of of a recompensation pro- to latitude relocations, vrhich depends on a knowledge of the absolute ~ast gyro input axis; system are (1) stability and the effectiveness therefore, is dislodged. for an operational importance tem ?-s an absolute reference, gyro drift rates orientation, orientation time of the system, of the which is dependent upon the 'nlaxin1.um allovvable system bandwidth. _ The systeln, \vhere accurate techniques data, and concepts are applicable initial platform This report section ~\ 4,. Stable Base Gyrocompass For ... is intended to carry characteristics fixed base inertial IV presents by gyrocompassing is required. 1.4. 0 Design Criteria compassing alignment in any area through operational guidance systen1.. pertinent sumlnarizes the reader concepts fora A slight detour in Chapter data on an operating the organization from gyro ... system. The following of this report. • Chapter II reduces pass to a two-axis interes't. the complexity model suitable of .the three-axis for analysis This model is analyzed to determine betvveen gyrocorn.pass gains, error coefficients DECL-liSSIFlED gyrocom- in ~he regions of the linear response relations time, and systen1 banchvidth. Chapter III indi.cates the relation east gyro input axis aligt1l11cnt A laboratory latitudcB. the orientation between and aziInuth technique knowledge accuracy of the at various is developed to detern1.ine The wheel speed of the east gyro input axis. modulation concept of gyro error' averaging laboratory optilnization anticipated in the labor'atory is analyzed technique presented. deterlnination and a No change is of accelerolneter bias scale factor and input axis ,location. Ch2pter IV detern1ines the spectrun1 gyro drift on the basis of several gyrocompassing the recording operation Two system bandwidths were used in can be properly over two-hundred identified. Hesults in instrulnent of vyheel speed and sixty- eight hours of systen1 are also presented. Chapter V develops a corl1pensation technique for a fixed base gyroco111passing internal based on instrument but anticipates increnlental perturbed guidance systeln. perrornlance the necessary recorded This tcch.nique is in the previous changes in system angle measurements \vhen operating chapter band\vidth and in a realistically enVirOn111ent. Chapter accurate hundred hours of stable base of these data s () that uncertainties performance modulation data. of ,azilnuth and east VI sun1rl1.arizes the' difficult problen1s stable base gyroconlpassing this paper. associated and the results \~/ith achieved in Appendix I contains a Appendix II assumes interferring that \vind gust spectrulu function between the benign laboratory opera tional envirol1ru cnt. confirlns as\vell the orders as the vertical operational is the primary The' spcctrul1l. analysis and the analytically of rnagnitude of the mean square azimuth angles v/hich lTIUStbe detected in an gyro recompensation procedure. DECLASSIFIED DECLASSIFIED II Chapter C"YTtOCOl\;JPASSING CHARACTERISTICS II reduces c0111passing systeln characteristics. produce the cOlllplexity of the three-axis to a nl0del suitable All simplifying a two-axis and response dcvelopluent friction are justified to dynalnic and steady state In addition, the eHect of a vertical sensor order lag in ihe region of interest. and angle transducer approach gains, tinle of the system. considers without a first assulnptions of systenl luodel which is used to develop the relation- ships between gyrocornpass errors, for analysis gyro- nonlinearities exist, the with and Although the linear is justifia1?le becaus e it develops the necessary initial insight into stable bas e gyroconlpas sing characteristics. referring to Appendix I for n0111enclature, bypassed by the reader dynan1.ical relations "Laboratory Chapter II may be who is fan1.iliar v/ith gyrocon1.pass with no loss in continuity, Alignment By and Calibration proceeding Techniques" to of Chapter III. 2. 1. 0 Coordinates Local geographic coordinates are designated Xct - north, b yo.... east, b and z g - vertical. are assumed to be perfectly In this chapter, plat,fornl coordinates aligned with instrument DECLASSIFIED coordinates. Sincc the inst':"L:ITlentcd cool~dir}atcs ar'c at all tiii.L~~3 nC::lrly approxin}~~tionJ ,_~rot.a_tiG:.! is equivalent Since to these y'otai:ion coo~'clj:l~Jt(;'S and local rotation c:' rotations about ~!bO~lt arc of the third approxilnations greatly's instrurn.ented by a developed add curnulatively irnplify affcc coordinate. to define coordinate. the development a These to follow i.t.3 validity. ge()[~:l'aphic and .i.-;~s'l.rul~lcnted coordinates. axis ec~sC &,':;TO iA is (i, e., sorls sensitive \vhich acceThs by th(~ ;''If;i)rOpriate is dcsi:;;:l~ltcd orientat.ion its input the angles coordinat.e 2 -1 i:~ a line SChC:::'Tlc~tic of ti.le gyroco111passing instrlunenJc gyro to a first an. insLru111ented SEl8.llJ nC'l.! oricntation Figure coincident Dbou L a -geographic two coordinates and do not seriously geogr:':jphic 2xis is p~J..:.~2.11c:l to the 0. its output east e.) location Tile vertical C:D.~=) ::). icput axis of the vertical The SCllSO},IS of s en- gyro sensitive Using slnall angle approxirnations (cas the total angular f) = velocity 1 sin 8 input = 0) J J~quation 2 -1 .indicates to each gyro. wx - (J w y z - A y + A x U(')18 ... A z V (2-1) - A Z W(l' e)h - + A y w,.tlC)l1 ',' + u,. ) tIe v ,<. "'A glossary 01' symbols ~~L.AS.srlfi~;endix 1. DECLASSIFIED r~JOF<TH 'l (I 1\;;) vV(le) h ::: \U io cas V E f1 T I ('..., ,D•..L - L.- \) w '" r'l L,L\T The steady state output of each vertical in Equation 2 -1 . v\there Sand the north and east vertical iTO'.ts " ..t..) p e'Y' .1. S xv yv sensors sensor is indicated are the sensitivities respectively 01 nleasured in II b.~ 0" I North Vertical East Vertical Sensor Output: r1 b A X S xv gA y S yv Sensor Output: (2-2) The north gyro and vertical sensor are connected together in such a n1anner that they ideally constrain the east gyro in the horizontal sumu1.ation is basically east gyro is east. plane. The .east gyro torque only satisfied \vhen the input axis of the If the east gyro input axis is not east, will sense the horizontal component of earth rate, where Az is the angle behveen geographic gyro input axis. -A w(; )h' . '-' ...e Q east an,d the east developing an angle, A . Y tilt about the east axis will develop an east vertical This sensor output equal to g Ay Syv' .which is coupled to the azilnuth torque generatol" and commands such a direction it This angular velocity will caus e the platform to tilt :-::.boutthe east axis, , the input axis of an azin1.uth angular gyro rate in as to reduce the angle between the east gyro input axis and the east geographic coordinate. systen1} the east gyro torque sunlrnati6n - .. l~itP .. Itl' , r~ , .. .,..1 " . .... :#__ ", (satisl'ied ,': , " . t~ ., For our ideal .. ....... ,' ._,,' ," only when the input axis of the eRf~~~~~.Qt), coupled Slllnrrlation in the azimuth gyro (satified of the east vertical conlpassing sensor only when the output is zero) combine to produce a gyro- systelu which uses the ,direction earth rate as the only external alignnlent. A convergent bchveen the east vertical with the torque reference of gravity and for initiai platform solution is established sensor by a coupling 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. model for stability and response tilue, their justification) 1. In order to obtain a suitable analysis) determination the follo'wing six assumptions., coefficients, together v/ith are presented. Typical stabilization servo tilne constants Gyro compass tilue constants are greater The dynan1.ics of the stabilization in the study of gyrocornpassing stability. of error v/orking (The platform are O. 02 seconds. than one minute. se~vos may be neglected alignment dynarrlics and is ahvays aligned with the input axis ,of its gyros.) 2. The gyro time constants do not affect the stability con1.passing system. of approximately 01" 0.001 seconds the solution tin1.e of the gyro- ~~ ~~2: >, <!: nr""~ \." j ~) iI :;" ~ ;~--:-_¥~ .... (!) <a: ! \ ..- r.:.::\ \'.,,,," () 0~ ......J f(i ...... (\ I ..J :"'0 r.J '~4: -.'" I (!. - L! -~~:: () x ~ ... :::J r;-'-' :J LL /(f) Jf) <J' r'" ~_tq II a e"_ 0 •,...- I C' 15 I~ 18 ::~~ C) (...) Ct: o 0 f"'"p. Z >- UJ (1) U) ~ "'-'I,. I~ I C!') ~ L ....1 i" If.:> JO: l:J> I x (f) + f~ 0.. i t> li ! ~! ~,....I J l (51 12 ,.......... 1_- -"":'--.1I i. r .. if 3,' For this chapter Static friction DECLASSIFIED static frictIon is assumed levels are approximately to be zero. 1 in. oz. Typical stabiliz2..tion servo gains are 50 in. oz. per arc second. Therefore) error 4. 0.02 seconds of arc gyro signal generator is sufficient to overCOlne static friction. Alignment about north involves seeking the vertical onlYJ and a sufficiently high gain loop can be closed so that north alignment have negligible accuracy errors effect on gyrocompassing .'. 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 compared to the azin'luth gyro angular via the east vertical values of I(li.E' gradient about east .. is negligible sensor (I{j\ T:"1 rate comrnanded and I(P,l..E for all practical > 1 roeru per second of arc .. and .L~.:....t:& - 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 leveling can be cOlnpleted faster than the gyrocolnpas sing transients. Noise restrictions and vertical sensor dynaluics are the only restrictions 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 supply are included - , in total gyro drift rate, \tVith the previous reduces assun1ptions to that of Figure 2-3, that the azin'1uth gyro angular equal zero. orientation is satisfied Therefore, It .is observed velocity the steady azimuth state of the gyrocon1passing The solution T( LEE + lr'---AE velo- +- (u)wy -A Y I(~E ~I equilibrium system 2-3 azilnuth is (2 - 3) (u)W ] z to Equa t~on 2 - 3, in the abs ence of east gyro and gyro drift, is a gyrocompassing luented east axis parallel angular velocity to geographic sumlnation a basic limitation performed on gyrocolupassing east gyro drift only produces this stabilized Ie 1 2-2 is satisfied The east gyro angular z -[ (u)w W(ie)h Y (u)w, , . x,y,z from Figure summation when --;;. w(")l 1 or' the diagran1 of Figure J when (u)wz + ]\y KAE equal zero. city sUlulnation circuits n'1'ember, system east. 'with the instru- The open chain in the east gyro places accuracy. Steady state a change in the azimuth Since the gyrocon'lpassing angle of 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), () r0 I (\j I, I.LJ ~ ::) <D l1... ; .~ f l I i { 1...1) :2-= L0 Cl) J l I ~ <.( \ - ,. ~ 0.:: Q.. ::- ~ I 1 I Ir-"", _J U r~ I I- w i l- i > a ~ii ~ 3 ~~ v J I- I i <t i w (f) , t ~-' -.--.i~ "Y' ~ < .-.. ,:;} ...... I' j n....... Ii is the azinluth DECLASSIFIED reference} east gyro drift cannot be detected in the clos ed loop system. Azimuth '-' Qyro drift} (U)LJ z ) Dro. duces an output from the east vertical aZilTIuth angle error. sensor as \vell as an 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 2.3.0 Gains} Dynamic Errors} Response Section 2:3. 0 \vill consider and ho\v these charac'teristics vert~cal sensor affecting gyrocolnpass response are affected by the use of a gains. system. Stability becaus e thes e errors Response before transient to lateral assull1ed for two reasons. First} to l~andle analytically" dynamic errors,,' before accelerations depends on the arnou~1t of filtering into stability" considerations dynamic dep'end on the respons.e because Equal roots to the gyrocompass easier systen1 characteristics gains \vill be considered and steady state errors considered Time with,a time constant affecting the response thne of the gyrocolnpassing gyrocompass sensor. \vill be the transient to be done. characteristic equation are the case of equal roots is yet provides the necessary and response insight time of the second ,I.- as \vell 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. DECLASSIFIED, DE .' " . "Cr.AS~IJ."In'Dh t'lng SYSteln, gains ViI11CU s.~ electromechanical errors L.e on the basis root solution --'-'--. f senSl "t""t" IVI les + 0' devices and aluplifications in excess of ten percent. deterrnined ' prO~1UCl. often acclunulate Optinlun1 gain can only be of system performance. gives the system 01 designer The equal an initial point fr01TI which gains can be optilnized. Tll'e development for a critically width. to follow deterlnines damped system These parameters Inine closed loop system accelerations. daJnped system, a realistic acceleration Observing the designer for acceptable can optll11ize further .mUln response thne. steady state errors decreased adjusts so that system gyro drift" of the bandwidth so within the initial allowable band\vidth the designer dynalnic errors T-l}.iswill" to a small degree" to attain acceptable ancllniniaffect bandwidth 111ayhave to be steady state perforn1ance The ,developn1ent that follows deterlnines and response profile, is determined" betw'een forced the syste111' the operation vVhen the maxilnuln steady sta'te errors tools, systen1 bandwidth the system will have steady state errors design specifications. deter- to gyro drift and horizontal Armed ,vith these analytical based on design requirements, critically. as a function of ~ysteln band- response techniques. parameters are next used to analytically desibmer must next determine and calibration system ti:r11esof the gyrocompassing gains" again. dynamic errors" system \vith and DECLASSIFIED 2.3.1 Gains Figure 2-3 can be reduced to lVlatrix Equation 2-4. I. I(EE 1 +P(TP + f) w(ie)h -p- V A y (u)wy II -p- I -(u)A y K'.l!..:'., vE TTJ?'+ l)P v I Ii :: I I I II KAE 1 - J?(T P +1) A z !I (U)LJ z -p- I v (u)Ay I(AE + I1p -~p v (2 -4) The characteristic equation of Matrix 2-4 is + 1) + I(E", E, P ;- 1<AE w (ie)h P2(TV P -- 0 (2 -5) 1 .11' T< v FI-~-'--- -. -~~AE LJ(ie)h '\J lnates the system ) a secondorder characteristics. systemyery For a critically closely approxi- damped second order system, sensor to east gyro coupling fI<EE)) and the natural is equal to the square root of the east vertical gyro coupling times the horizontal sensor frequency to ,azilnuth component 0'£ earth r~tte) , :: (2 -6) If the time constant stability criterion of the vertical sensor shov/s that Equation system RothTs 2-5 is free from right ha)f- plane poles as long as I{EE > Ki\E w(ie)h and I(AE' for this third order can be chosen) The choice of a TV. depends on accuracy T v requir.e- ( nlents) lateral the systcln. acceleration pi-'ofile) and required If all roots of Equation the following relations 1 T3 = 1 3 response 2-5 are equal (critical tilne of damping'), exist. :: ::: T V (2 -7) . DECLASSlFIED DECLAS 2. 3. 2 Dynan1ic Errors The primary error stablc base gyrocompass vertical sensor The vertical of horizontal tion of the 'effects of horizontal gyrocornpass performance This section indicates in vertical in the operation of a arc gyro drift and uncertainties operation. are usually a result sources sensor uncertainties accelerations. acceleration is primarily in lVIinimiza- on stable base a filter design problem. the effect of gyro drift and uncertainties sensor operation on the closeclloop performance of the gyroco111passing system. 13)' "taking the partial derivatives ii1 vertical to gyro dri'ft and uncertainties . of Matrix 2 -4, with respect the va~ue8 of T2 from Equation Equation 2 -7, Table 2 -1 is obtained. Taple 2-1 are expressed of tl1e gyrocompassing 2 -6, and The error of 111easure. to . 001 earth rate, system con1ponent of to any convenient as the angular velocity unit of 111casure, and measure. To becon1e familiar froln Table 2-1 into measurements time, ~the follo\ving constants are noted. ,r_ T 2 is the time constant T3 in This paper \vill use the Ineru, "which is equal \vith the n1eru and to convert ",' fron1 coefficients and the horizontal the secondo£' arc as an angular other than reciprocal ~~ T3 and only as a function of the time constant earth rate so that they are easily convertible ,units operation, . ", substituting sensor is the time constant of the second order gyrocompass. of the third order gyrocompass. D~CLASSIFIED T.ABLE 2-1 ERROR COEFFICIENTS Change in Azimuth Uncertainty Angle ~ Az Change in (u) in East Angle Oncertainty D.. A y in (u) SECOND ORDER SYSTEM 1 East T 'Gyro (u}w Y \ • 2 ';. Azimuth (P 2 +2:..) IP ++) T 2 2 (P +~) T w(ie)h 2 2 2 Gyro (u)W z p W(ie)h 2 2 (P +2:..) (P T +2:..) T 2 2 P 1 (P +-2T 2 2 T2 (P w(ie)h 2 2 T . 2 +2..) T 2 (P +2:..) T 2 2 THIRD ORDER SYSTEM East Gyro (u)wy w(O East Angle .... (u)A y -35- 3 Ie )h (P +-) T3 1 () see (1.5 1 meru = 1 meru = x 10 .n~r!T.Pt~~Tf'IED meru -~~ · . ) = o. 015 7.27 arc second degrees hour x 10-8 radians seconC1 (2-8) ,The error coefficients Data recorded indicates on system of Table 2-1 are plotted in Figure performance versus that gyro drift has a low frequency bandwidth spectrum 2-4. (Chapter IV) compared ,to the time constant of the gyrocompassing system. Therefore, (u)w , . y and (u)w.z primarily affect the steady state performance of the gyrocompassing system. For both the second and third order it is noted that ~zimuth gyro drift, calsensoroutput indicates angle. Figure 2-4 also that the steady state output of the east ve'rtical gyro on the basis V. gyro drift. Recompensation o! the output of the east vertical lem in compensat'ion Chapter (u),w ' shows up as an .east vertiz as well as an azimuth only a function of azimuth techniques systems ~ and will~ therefore, sepsor is of the azimuth sensor isa prob- be coveredin EAST DYNAMIC AZIMUTH ERRORS 1'-- -0.1 ~ Az W(je)h c\ (u) Wy I 0.11- I 0.1 Ib '1 I 100 0.1 i'Wo 1Wo 2 o .11 ~Az W(i e) h - ~(u)Wz I I ro I 0.1 100 'f Wo. I 0.1 'tWo .1Wo __ -37~ , __ FIGU R E:.2 ~.4 2nd Order "" 3rd Order' \. DECLASSIFIED It is noted (Figure 2-4) that steady state east gyro drift, (u)w , only produces azimuth angle errors. y Since the elimi- 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 gyrocompassing. 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 the closed-loop gyrocompass. produce dynamic errors'.in 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 gyrocompassing 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 D£ClJiSSIP1ED 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 acceleration . • The shortest allowable time constant *:.'< of the gyrocompassing system can be determined from a knowledge of the predominant frequencies *** requirements of lateral horizontal accelerationf? and accuracy 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 sensing 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. DECLASSII1ED AZIMUTH .4 xlO LATERAL ACCELERATION ERRORS -5/ () / 0:: ,/ « lL. C'l o I ~ Z / ,/ / / / / 0 en / ,/ ". / / 0::. o U ~ ~ ,/ / / / / / / / / -3 "./ 4 X I 0- / z o i= « 0:: -2 4x10- ~ :I: I- ::> :E -I 4 xl 0 N <t 4 4x 10 I 2 4x 10 I 0.0001 I I 0.001 0.01 ACCELERATION FREQUENCY- RADIANS SECONDS. ---- -40FIGURE: 2-5 2nd Order ---"3rdOrder = T 100 DECT)1~StFl'ED Azimuth Variation 2nd order system 40 seconds of arc 3rd order system 4 seconds of arc = T 1000 Azimuth Variation 2nd order system . 40 second of arc 3rd order system . 004 second of arc 2.3.3 Response Time 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: liminary leveling; (2) gyrocompassing. Preliminary (1) preleveling 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, equals 3 T leveling. v , since T one-for'th of the total alignment time will be us'ed for . If preliminary leveling is not used, the east vertical J sensor output will often produce azimuth errors DECLASSlf1ED 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 Az(0) for the uncertainties 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 T2, 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 is noted that for equal lateral horizontal acceleration errors the GYROCOMPASS RESPONSE STEP 0.90.8- . t2 Az (t) Az(o) ~ •_ t ! 1; ,. :U+t+213)e 0.70.,6- 0 <t « ';:;:; 0.50.4- AzU) Az (0) 0.3- = (.1 + t . ,_l. )e ~ 0.20.1- 0, o I 2 .I :3 I I I I I I 1 4 5 6 7 8 9 10 t r -43FIGURE: 2-6 III 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- III 'D~CT. ..1l c~r1-"T1m ALIGNMENT AND CALIBRATION LABORATORY 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. In sequence l I the geographic coordinates). 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. that no absolute optical reference is available l Assuming the predeter- mined laboratory orientation of the stabilized m.ember must coincide with that orientation acquired by gyrocompassing at DECLASSIFIED 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 -46:: DEVIATION OF EAST GYRO IA FROM GEOGRAPHIC EAST W{je}v VERTICAL Y GYRO INPUT A{h- -"---- y) HORIZONTAL NORTH INTO PAGE AXIS PLANE -----------------MIRROR YGYRO fA WEST-Yg __A(-Yg-m)-- GYROCOMPASS A(-Yg-y) VERTICAL WE~T = A(-Yg-m) INTO ~A(m-y) PAGE SOUTH - W(ie)h GYROCOMPASS Y GYRO fA EAST Yg MIRROR -47FIGURE: 3-1 EAST W(ie)[(A(h_y) cos Lat + A DECL~~mPd (yg-y) Lat . - 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 -48- . 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 misalignment 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 misalignment 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. DECLASSlTIED' 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 gyrocompassing system at several latitudes and determines A(h_y). This method uses the effect to calibrate the cause but is impractical 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. DECLASSIFIED -50- Once A(h_.y)is DEer. It (!'C':I known" it can be compen~~mD:>r c 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. DECLASSIF7RIl . * 1; M(OA) = DECLASSIFIED H wi-f(IA) + P 2 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)" 1 Ac-f(OA) = I P( g(OA) Cd(OA) STG ic + --- Cd(OA) >.'c + (u)M(OA) ] Cd(OA) Appendix I contains a definition of terms . .P~CLliSSlE1ED (3 -5) 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)= (3-6) wi-c(IA) + wc-f(IA) The angular velocity of the gyro case with respect to 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 HWT 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),'c~'c Th ' .. 1. STG mate 1y para 11e1 to nort h . . e gyro torqulng slgna , 1c -H ' 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, wT' 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. ),'c*The horizontal gradient of earth rate around north is a minimum. DECLASSIFIED If the magnitude ofthe table angle" AT" is less than 0.03 radians and the table rate" w " is greater than the vertical T 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. AT = w T -- B' sin W 0 T B w 0 cos w 0 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 -55- (3-10) 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 could be separated of these rates were equal, by observing they 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. W B= 4A (3-11) 0 (h-y) w(ie)h Under the conditions of Equation 3-11, the effect of horizontal earth rate may be neglected because it can be separated out of the data . . The torque of the gyro float with respect the input axis, axis nearly moments axis, to its case about H wT' is produced by the table rate about an coincident with the gyro output axis. of the gyro float with respect Equation 3-12 is obtained. DECLASSIFIElJ -56- Summing. to its case about the input DECLASSIFIED ~ M(IA) = wT H 2 + 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 wT is perpendicular the total table rate is transmitted about the gyro's output axis. and rearranging, Substituting = wc-f(IA) for P Ac-f(IAL the angular velocity _H __ W T Ig(IA) p2 + P The following numbers of to its case about the input axis. _P * Gyro from the gyro case to float Equation 3-12 describes the gyro float with respect wc-f(IA) * to H, and C (3 -13) K d(IA) + (IA) Ig(IA) Ig(IA) are applicable for a 2FBG-6F single-degree . float tracks the case for output axis rates below fivehundred radians per second as previously noted. DECLASSIFIEl> of-freedom integrating gyro. gram em second = 2 x 106 H 2 dyne em radian _:. / second dyne em radian:,;/ second .2 gram em Ig(IA) Ig(OA) = 3 2 x 10 . gram em 2' dyne em radian Substituting numbers and combining Equations 3-5, 3-13, and 3-6, A e-f(OA) = 1 P(10-3 P + [A 1} (h-y) w T 2 P wT 4 ~ 10 --------,,-------PECLASSITI¥p + 106p + 0.2 x 106 +. (u) M(OA) 2 x 106 ] (3-14) -58- DECIJlSSIFIE~ W Ac-f(OA) P 2 x 10 -3 T -[A P (h-y) ~ + (u) M(OA)j 2 x 106 5P + 1 (3-15) W Substituting A AT for ':: from Equation 3-9" c-f(OA) - (3-16) Substituting into Equation A c-f(OA) - the earth rate coupling constraint of Equation 3-11 . 3-16" . 4 A(h -y ) W 0 sin W 0 w(ie)h + T [A (h-y) (u) M(OA)j 2 x 10-3 P 5P+1 (3 -17) 2 x 106 -59- DECLAsstFlED The angle between the gyro float and case about its output axis" Ac-f(OA)" is sensed with the gyro ducosyn. laboratory" radians) incremental can be measured vertical of arc (5 x 10-7 angles of 0.1 seconds with the gyro ducosyn. from Equation 3-17" in order to record In a benign Therefore" the component of 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 of vertical to separate the component 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 -60- (3 -19) Substituting into Equation 3-18, _. DECLASSlFIEIJ.. w o < (1.7 -6 1/3 x 10) = 1.2 x 10- 2. radIans second (3-20) Substituting into Equation 3-18, the minimum detectable A(h_y) . is 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 -2 sin 1. 2 x lOT (3-22) Since the maximum table angle achieved is 2.3 x 10-2 radians, the assumption of small table angles holds, Equation 3-7 to 3-8 is valid. separation and the expansion from The sinusoidal of the angular velocity components -61- table rate allows the applied to the gyro DECLASSIFIED input axis. The fact that the change in horizontal coupling is the second harmonic gyro coupling is in quadrature amplitude earth rate of table rate and the internal with table rate allows the of these effects to be made equal to the component of table rate proportional to A(h _y)' and yet they can be separated. The equations indicate that A(h_y) can be reduced to less than five seconds of arc by this method. minimized, the north vertical sensor null, or if this is not possible, vertical sensor 3-1 for a known latitude 3.2.0 A( yg-y should be brought to a the angle output of the north can be recorded. be used to recompensate When A(h _y) is This is A(h_y) and should the east gyro on the basis of Equation relocation. ) The angle between geographic input axis can be determined with the gyrocompassing can east-west region. by east-west system. average with accuracies east and west, averaging system to determine gyros in the low arc second system should be able to yielding the same coefficients the east seeking gyro as a single component. that follows applied east-west techniques Single state-of-the-art A complete gyrocompassing gyrocompass east and the east ,gyro averaging A(y _y) and R. ,g, . The development techniques Numerical " DECLASSIFIED on to an entire calculations are DECLASSIFIED not presented because the process depends mainly on the sta.bility of the east seeldng gyro during the averaging val. Mathematically, A(y _y) is determined inter- as follows. g Since A(h _y) is small, nearly in the horizontal mation performed system forces the input axis of the east gyro is plane. The angular velocity sum- in the east gyro of the gyrocompassing its input axis to deviate from geographic by A(y _y), due to residualllR" torques. east ~athematically, g the east gyro's angular velocity summation = - A(y _y) W(ie)h + ~ g By observing approximately east, is operating (3-23) 0 the absolute orientation when the gyrocompass is of a mirror surface with the east gyro input axis and then obs erving the same mirror when the gyro input axis is approximately west, surface A(y _y) is g .determined. With the system on the rate table" to reach a steady state orientation. tion 3-23" the east equilibrium system the gyrocompass From Figure orientation is allowed 3-1 and Equa- of the gyrocompassing is . - w(ie)h A(m_y) - w(ie)h A(y~~) DECLASSJ.'J: J..t&D R + H = 0 (3-24) The system is next. chan~~~ffi~r~Dgyrocompass mode to position mode of operation and rotated approximately hundred and eighty degrees. The system is then stabilized, KAE reversed, to the gyrocompass and returned mode. one- From ) Figure 3-1 and Equation equilibrium 3-23, it is seen that the east gyro equation when gyrocompassing west is (3-25) Since A(m_y). is a constant, added to yield (y -m) 2 Substituting A(m _y). east, +A g ~ into Equation In order ] (-y g -m) 3-24 or Equation (3-26) 3-25 yields to align the east gyro input axis with ge~graphic a compensation generator 3-24 and 3-25 may be ~. W(ie)h[A " Equations current is applied to the east gyro torque to reduce A(y _y) to zero . . g ~ECLASSIFIED -64- DECLASSIFIED = R (3-27) -H With the co:r:npensationcurrent calculated by Equation 3-27 applied to the y gyro torque generator" the east-west averaging technique of Section 3. 2. 0 may be repeated to be sure that " A(y g_y) is withln the design requirements. '3.3.0 Laboratory Optimization of Wheel Speed Modulation In an o~erational system" wheel speed modulation is I used with sample data techniques to determine low level torque changes in the east gyro and to recompensate the gyro. This technique depends on a low value of torque uncertainty being introduced when the east gyro wheel speed is cut in half. The wheel speed modulation technique depends on an initiallaboratory calibration primarily optimizing gyro wheel power at both wheel speeds. This section discusses the initial laboratory optimization procedure. Section 4. 3. 0 presents data recorded on a gyrocompassing system using wheel speed modulation. Section 5. 2. 0 indicates a possible operational procedure which makes use of wheel speed modulation.: If the steady state gyrocompass east gyro wheel speed is . DECLASSIFIEn -65- DEe cut in half, . Equation ~~Imf:&ges to Equation 3-28. (3-28) W m is the drift rate introduced in the east gyro due to the gyro wheel speed being cut in half. Am w(ie)h is the horizon- tal component of earth rate necessary and i c have not changed, autocollimator orientation = W m the angle A for the gyrocompas optimized zero. • Since R m (3-29) m can be measured which s ens es the variation will be almost m w(ie)h in azimuth mirror equalized at both speeds East gyro wheel conditions if the gyrocompass with the s at both gyro wheel speeds. Gyro wheel power can be properly W W it is seen that Am In the laboratory, to cancel orientation so are does not vary when the east gyro whe~l speed is cut in half. To see how wheel speed modulation can be used to determine DECLASSIFIED -66- Equation 3-30 describes the closed loop angular velocity summation performed by the east gyro in the gyrocompass mode at normal wheel speeds. (3-30) If R changes to R + AR, the following equation describes the closed-loop gyrocompass performance. S R + 6. R' +. H lC TG fI -.A (yg-y) w (ie)h = 0 (3-31) The 6. R only produces an azimuth error in the gyrocompass system and, therefore, cannot be detected in a closed-loop basis. If the gyrocompass loops are opened and east gyro wheel speed cut in half, Equation 3-32 describes the angular velocity summation performed in the east gyro. DECLASSIFIED -67- DBCLASSIPIED + ~R) . Z{R H ... 2. i STG C ~ - A( n Yg-Y ) w(' Ie )h ... W my = 0 (3-32) If the azimuth gyro is properly constant. Combining Equations compensated, 3-30, that the open-loop east gyro drift, w By recompensating = Yg -y ) is 3-31, and 3-32 shows ' my at half wheel speed is equal to the change in the east gyro residual wmy A( term AR II (3-33) the east gyro on the basis of open-loop .. STG t . STG + A R' th drl.ft rate an d ch anglng lC lr 0 lC lr -rr-' e varIa. t'Ion in the east gyro residual zero, A(y _y) returns to g and Equation 3-30 again applies to the closed-loop east gyro. performance. term is compensated, It is observed be detected and compensated 3. 4. 0 Accelerometer that changes in ic and STG ca~ also for in this manner. Location The location of accelerometer input axis, scale !actor, and bias determination can all be evaluated in the laboratory conventional methods. With the autocollimator DECLASSIFIED -68- by defining the azimuth ~ECLASSIFIED mirror orientation .. a four-point will determine calibration in the laboratory the beta matrix which is stored computer for future reference. DtC1.ASSIl:TED -69- in the guidance DECLASSIFIED IV SYSTEM AND INSTRUMENT Representative PERFORMANCE data on the stable base performance of a gyrocompassing system c,onsisting of three 2FBG-6F singledegree-of-freedom gyros and one Kearfott two-axis pendulum is analyzed in Chapter IV. These, data. are presented to permit an evaluation of gyrocompassing un certainties as a: function of system bandwidth and to determine the effectiveness of the' wheel speed modulation technique. Once gyrocompassing uncertainties as a function of system bandwidth are defined in a benign environ( ment, an operatidnal gyro drift detection and recompensation I procedure can be derived for a more realistic environment. These data also permit an evaluation of the individual instruments in a system environment. Two samples of data will be analyzed, (1) four-hundred and , twenty-eight hours of steady state stable base data at two system bandwidths; (2) two-hundred and sixt:y-eight hours of steady state data utilizing wheel speed modulation to update the east gyro compensation. 4. 0.1 Environment Before proceeding to the analytical background and .t)ECLASSIFIEI>. -70-' DECLA analysis of data" the environment" which was chosen in order to determine component capabilities" must be defined. For the recording of these data" the system was operating on a seventeen-ton concrete slab experiencing lateral accelerations in the micro g region. * The system is designed to provide the proper environment for the operation of the 2FBG-6F gyros and Kearfott pendulums. uncertainties The region of interest is torque of O. 0012 dyne-centimeters" of 0.01 meru" O. 00015 degrees to angular velocity uncertainties per hour. which correspond In this area" gyro performance is a function of the electronics used to power and control the instrument as well as the accelerations and temperature present. Over-all system accuracy is a measure of the performance of the instruments the controlled system environment. in System data such as that of a stable base gyrocompass can often be separated to give an indication of the individual instrument's performance. These data" in turn, can be correlated with instrument history" age, and variations in environment to provide the basis for changes in instrument and'system design. The data that follow were recorded on the 2-16 gyrocompass where instrument temperature degree Fahrenheit. is controlled to better than O. 04 The east gyro is operating with its input * Measurements with wide bandwidth accelerometers defined the accelerations r en DECLASSIFIED -71- have DECLASSIFIED axis approximately five meru. vertical east and a total compensation The fact that the east gyro's makes this instrument's sitive to gravity torques. of the three-gyro performance output ,axis is nearly torque summation less sen- It is a necessity gyrocompass requirements of less than in the operation that the east gyro, whose are two orders of mcagnitude better than the north or azimuth gyro, be positioned in the optimum orientation system. to act as the basic reference in the gyrocompassing The a'zimuth gyro is compensated which is approximately acceleration seven-hundred is acting perpendicular for vertical meru. earth;rate One gravity to the azimuth gyro's spin and output axes" which are horizontal. The north gyro's uncertainties they do not seriously system. are not analyzed because affect the accuracy of the gyrocompassing * 4. 1. 0 Bandwidth" Instruments, and Environment The design of an effective gyrocompass a'compromise or optimization uncertainties and the acceleration The- interpret~tion and estimation ~c . between the system's of instrument environment errors of specific spectral this statement. D:ECLASSIPlED instrument during operation. on an uncertainty characteristics l Section 5. 1. 0 supports is largely basis provide one I>ECLASSIFIED set of bounds defining the lower limit on system bandwidth. "Acceleration enviroment in a fixed base produces uncertain- .. ln angu Iar measurement t les ~c in system operation which define the upper frequency on system bandwidth. estimate of lateral The data presented acceleration ** •• and lnduces dynamlc errors inputs J limits together with an are used to -identify J an area of general design interest. I In a system operating . sample data techniques periodically J and an integrator recompensate recompensation process over a prolonged period of time, the gyros. procedure *** The frequency depends on the accuracy the bandwidth qf the gyrocompassing the spectrum instrument of instrument uncertainties J on system performance in this chapter. uncertainties. to of the of the system and J The spectrum of and the effect of these uncertainties as a function of bandwidth J Chapter V derives are defined a gyro recompensation cedure which could be used in a more realistic update the compensation are required to gyros with similar environment spectral proto charac- teristics. ,,'< AppendixII defines the spectrum of angular measurement uncertainties as a function of lateral accelerations. *~'< Chapter V indicates that uncertainties in angular measurement. which limit the bandwidth of gyro drift evaluation place more severe restrictions on system bandwidth than dynamic angle uncertainties. The uncertainties in angle measurement define the time constant of the vertical sensing device (i. e. the critically damped gyrocompass time constant) .. J J *"c ..... The integrator's low sensitivEY does not affect the stability the gyrocompassing system. ~Cl.~~~TFTR " "l" of 4. 1. 1 Instrument Uncertainties The effect of instrument azimuth angles can be determined uncertainties on east and from Figure 4-1. steady state azimuth angular velocity summation wh~n (u)wz + KAE Ay = = compassing O. Therefore" system z = is satisfied only when -A z w(. )h + (u)w le y the equilibrium angles of the gyro- are A A The 0., The steady state east gyro's angular velocity is satisfied , - KEE Ay * y = 1 (u)wz KAE (4-1) KEE --[ (u)w + -K (u)w] w(ie)h y AE z Equation 4-1 indicates (4-2) that the output of the east vertical \ sensor is proportional to azimuth gyro drift. in the output of the east vertical frequency sensor range of the gyrocompassing Therefore" changes which occur withi.n the system are proportional ~!< · Chapter II: contains the justification of Figure 4-1. In addition" Chapter II gives a detailed development of the dynamic as well as the steady state changes in east and azimuth angles due to instrument uncertainties. (Equation 4-2 is a repeat of Equation 2-3. ) DECLASSIFIED -74- r--------P I >-------l I ~t 0:: 0 ~ ~ -I -I 0 ~t 0 0 I- :::> <{ i ~I~ - o lO.ciE . ~ I a 0:: 0 0 ~ I I I I I IL ______ _10.. N -3'; :c .- :J ~ N « ::s 00. , -Ia. L€..3 + ..J 0:: W l.1.. W -I 0... I- o lO..ciE I I I I llJ w 1 J 0:: 0:: ~I'- .J:: Ci) ~ ~ <{ ~ J <{ <t W. a:: 0 C/) Z W C/) ...J <1: U i= 0:: W > IC/) <{ W W - <{ .. -75- ::s W W I q- DECLASSIFIED to uncertai~ties in azimu~h gyro performance. As previously formance noted, uncertainties in the east gyro per- only produce azimuth angle errors compassing system. in the gyro- By combining Equations 4-1 and 4-2, 'it is noted that steady state east gyro performance extracted from an appropriate summation can be of azimuth and east angle variations. (4-3) 4.'1.2. Spectrum of Instrument Uncertainty Section 4. 1. 1 has only discussed angular velocity summations east gyros. instrument in the azimuth and The words "steady state" in the analysis uncertainties to uncertainties frequency performed the steady state in the gyrocompassing system refer which occur at a frequency below the break of the system. For data analysis, to divide the total spectrum ties into three regions Region 1: of of possible it is instructive instrument uncertain- (Figure 4-2). Instrument Uncertainty When the frequency is above the natural frequency Frequency> of instrument of the gyrocompassing DECLASSIFIEP W o uncertainty system, S I Q) ~.ro ro~ CI.l H CI.l ~ ~2 ~ c.> tf.lor-! I-+J (]) S ~ ~"O :;:jro Q) -+J ~-+J ~Q) ~ ~CI.l ro H .~"O Q) 0.. ro CI.l• :>0.. +>(])(]) o H ro roor-! ~o ro c.> +>:;:j tf.l Q) ~c.> CI.l 0 S::~~Q) (]) H or-!+> ::s or-! ~ r-i+>o~ c.>b.O ro ro s:: Q) S::CH :;:jtf.ls:: H Q) 0 "0 OJ ro CH . ::s ~-+J. c.> Q) (]) ro S Z 0 - (.!) w 0: >- (]) !:l Hor-! () Z W ~& ::> . Q) b.O W s:: ro H Ct: (\J ~ c.> >- I- Z « a.. ~ CI.l "0 !:lroor-! ~ ~Q) H+>. 0 -+J 3 C\J CJ LL 0>:0 or-! !:l"O en en Z 0 (.!) w 0: s:: (]) & (]) H tH S::s ~ a:: or-! "'0 W (]) ~ (.) ro ro Q) roQ)r-f Q)b.Ob.O or-! ctls:: s:: 1jro or-! ~ ...c roc.>+> tc.>::S Q)or-! S c.> Sor-! ~ tiS N ::1 s:: ctl +' ~"'O ~"'O s:: rQ) ro E~+> ::s :;:j tf.l H"O ctl t;oQ) s:: H s:: H 0.. or-! z - 0 (!) W 0: 0 -r-! b.O Q) H ~. c.> s:: (]) ::s eJi Q) H CH ~ 0 ~ l1.. 0 0 0 >- ..J HH~ +>ctl+> 0 .~ CI.l CI.l s:: oS Q) or-! r-fc.>Q) ro~+> :;:j ro ctl "'OSH or-! ctl :>Ho.. or-! 0 (j) "'OtHro s:: H HQ)O o..+> U Z 0 S ::s c.> UJ i 0 H::s ~ -r-! +>~b.O UJ ctl Q) .5c.>H -77- ~ 0 lJJ a ~ :::> 0:::: l1.. IJ.. w 3 C\J I ~ W ::::> • S H"OQ)O +>O+'H .5roH~g0.. co 0 ::s Q)o~ ~ UJ Q)Q)...c ro ctl r-!or-! or-! +> +' (]) ~~ s::+> s::UJs::+> -r-! ~ or-! CI.l -r-! s:: ctl r-i ro UJ ~ ~ or-! +'r-f H"O H ro H Q) 0.-1 0 b.O Q) Q) c.>+, c.>+'HS:: s:: ctl s:: s:: Hor-! :;:j H :;:j <U CI.l ro Q) UJ +>ctl +>roQ)ctl s:: Q) r-i 0.. s::o.. Q) Q) OJ b.O Q) s:: S ~ :;:jc.>tiSO S (.!) ~ <t C\J W 0 0: CD "'0 Q) j'() 0 >- ~ ~ u 0: +> r-fci ::s 0 c.>or-! +> (]) !:l ~CH bJJ Q)0r-! H s"O • Z :::> 0 0 0 0 a:.: <!:> DECLASSIFIED the effects will be attenuated These uncertainties in the closed-loop do not seriously gyrocompass. affect the performance of the system as long as the amplitude of the variations duced are acceptable. pro- (The only exception to this is the east gyro drift rate which is detected in an open -loop basis in a short time interval. Region 2: ) Inst~ument Uncertainty When the frequency is near the breakpoint difficult to instrument" determin~ region" the system if any noted disturbance If the variations it is is due to a single of these. In this to uncertain- the uncertainties The total effect of all uncertainties above the breakpoint by observing repeatable in this data over which is sampled at a frequency of the gyrocompassing between instrument compass performance report. ) uncertainty system" responding of system performance" system time constants dynamic relations o and environments. region can be determined several w in east and azimuth angles are within the design requirements are acceptable. or a combination is simultaneously ties in all instruments of instrument of the gyrocompassing environment" ~ Frequency system.- performance are covered in Section 2.3.2 (The and gyroof this DECLASSIFIED Region 3: Instrument Uncertainty Frequency In the range o'f frequencies point of the gyrocompassing system, is obtained to all disturbances. uncertainties errors o below the break- a steady state solution In this region, instrument produce steady state east and azimuth angle as indicated in Equations analysis < w 4-1, 4-2, and 4-3. of data in this frequency the individual instrument's quency of instrument of The highest fre- observable one-half the sample frequency. to d-c values of instrument range allows separation performance. uncertainty The is equal to This region extends down uncertainties. 4.2. 0 Steady State Data Table 4-1 summarizes parameters for the recording the important of two bandwidths state data which will be analyzed. of arc. coupling, of steady The first two-hundred and sixty hours of data were recorded sensor-azimuth-gyro system with an east-vertical- KAE' of six meru per second This produced an over-damped, 0.0021 radian per ,~ second system, 8 A z 8 (u)w z = highly sensitive KEE K AE to azimuth gyro drift. "." The 1 w(ie)h' the higher the ratio of KEE to KAE' the more sensitive the azimuth angle of the gyrocompassing system is to azimuth gyro drift. DECLASSIFIED DECLASSIFIED TABLE 4-1a 260 HOURS - LOW EAST/AZIMUTH COUPLING 6 meru per second of arc. 2 meru per second of arc. Roots W b.A (u)w b.A ~ z = y K AE z z KEE = = o = 2.1 x 10 1 -W-(i-e)-h ~ = -3 radians per second 100 seconds of arc per meru 286 seconds of arc per meru - 0.165 seconds of arc per meru DEctASSIFIE~ -80- DECLASSIFIED TABLE 168 HOURS 4-1b - HIGH EAST/AZIMUTH COUPLING 60 meru per second of arc. 2 meru per second of arc. Roots PI" P2 = 1.5 x 10 = 6.7 x 10 w o. AA ~ z z AA z ~ y -3 [-1 + radians 1 - 2.1 x 10 ':'1 per second 1.0 seconds of arc per meru = = -2 = 286 seconds of arc per meru = - 0.0165 seconds of arc per meru -81- ] - 'DECL.ASSlflEL second one-hundred recorded and sixty-eight with the east vertical hours of data were sensor azimuth gyro coupling of sixty meru per second of arc. lighter damped system with an order of magnitude in sensitivity state data will be analyzed in two sections. one typical two-hour period of data at both system bandwidths will be studied to determine of all uncertainties two-hundred increased and sixty hours of narrower effect Second" the bandwidth data will and sixty-eight hours of bandwidth data. Short-Term Figure Stability 4-3 is a typical two-hour azimuth orientation of the stabilized of the east vertical sensor. bandwidth. Short-term peak amplitude 3 member of the and the output are a function of system oscillations damped system of Figure recording As one would expect" the short- term azimuth and east variations of 2 x 10- the short-term on system performance. be compared with the one-hundred 4.2.1 reduction to azimuth gyro drift. The steady First" This produced a 4-3a. are apparent These oscill~tions in the lighter have a of five seconds of arc and occur at a frequency radians per .second" slightly below the breakpoint of the gyrocompassing system. Due to the large number of dynamic solutions taking place simultaneously DECLASSIFIEU -82- in the frequency ~• ~ r-f 0.-1 ~ .;,..) tI) s~ (l) .;,..) I ~ 0 .t:: tI) tI) t/l (f) ~ ::> .0 rt> I o 0r;:- (\J u:: :x: ~ ro ~ S O () 0 ~ ~ Cj (0 ..... I N M I ~ ~ -,... Ii. DECLASSIFIED range, it is difficult to attribute particular Figure instrument. these oscillations However, 4-3 (unless uncertainties to any from Table 4-1 an.d cancel during this period), by observing the total angle change over the two-hour an upper limit can be placed on instrument interval uncertainties. Figure 4-3a (u)wz < 0.3 meru (u)wy < 0.01 meru (u)wz < 0.03 meru (u)w < 0.01 meru Figure 4-3b y ,. The high value of azimuth gyro uncertainty, 4-3a), is due to the uncertainty (u)w (Figure z in east angular measurement. 0.3 meru corresponds to an output of 0.005 second of arc from the east vertical sensor in the wide bandwidth system with an east azimuth coupling of sixty meru per second of arc. ' There is nothing unique about Figure hou~s of stable base gyrocompassing Figure 4-3 have been recorded. we are primarily interested 4-3. data similar stability DECLASSIF1BJ1 .. of to that of The two conclusions in from Figure east and azimuth gyros short-term Thousands 4-3 are, which (1) the for both system bandwidths are- wItliTri""'ihe initia 1-1; design criteria of Table (2) the higher bandwidth system has larger short- term dynamic uncertainties. 4.2.2 Long-Term Stability Data similar averaged to that indicated over a forty-two minute time interval hundred and twenty-eight azimuth orientation hours. .The first two-hundred sensor system and the are plotted in Figure 4-4 . under The second one-hundred hours were recorded , ha? a marked values of the and sixty hours were recorded the conditions of Table 4-1a. Table 4-1b. 4-3 was for four- These average of the gyrocompassing output of the east vertical sixty-eight in Figure It is noted (Figure under the conditions and of 4-4) that system bandwidth effect on the long-term stability of the gyro- compass. The first two-hundred strong correlation sensor and sixty hours of data indicate a between the output of the east vertical and azimuth angle of the gyrocompassing period of these disturbances is approximately with an east vertical amplitude sensor mately 0.02 second of arc. system. 10.6 hours, variation With an east vertical gyrq coupling of six meru per second of arc, The of appr~xisensor aZUnuth 'an azimuth gyro drift of 0..12 meru is large enough to produce the variations DECLASSIFlEI} DECLASSIFIED M U) ...... .. Cl ct U) ,., U. ii: 0 ~>- a:: c: 9 0 ::I: m ::l !a "': .. (ll 0 ::I: . II: 3 ::l :< tf.) ~ .t:: ~ .ro~ . s: roQ (lj ..0 b.O Q .~ >, ~ (lj > ~ 0 t: ~t) ii: Cl 0 Q) c:: l; ~ ~ Q) ::l tf.) :'.! N UJ (lj < ~ S0 u 0 ~ >, M U) ...... 0 < c:: U) 'P 2 0 a:: ::l IC ::I: 0 tJ CO ...... I C\] U) N {= C\I ~ I ~ . .~b.O ~ D CLASSIFIED o 8o~S~8 d :10 SONO:>3S NOIJ.'1U!'1J\ HlnvtlZ'l :;)H'I d ~ d O. d ~ SONO:>3 J1!) d I CO CO I indicated in the first ~C~A1d 4-4. Some of the probable sixty hours of Figure causes of, an 0.12 meru azimuth gyro shift are indicated as follows: Torquing supply variation: Torque generator 0.010/0 sensitivity.change: ' 0.020/0' Output axis torque variation: 0.015 dyne-em Since the period of these variations any known disturbance, do not correlate they are assumed change in the torque summation with to be caused by a inside the azimuth gyro. The azimuth gyro output and spin axis are both in the horizontal plane. The torque summation is sensitive to mass the spin axis due to shifts in the spin axis bearing shifts along and/ or the ball retainer. When these data were recorded, five-thousand operating hours. the gyro wheel had over Equation 4-4 indicates the magnitude of the 257 gram gyro wheel shift along the spin referenc'e axis, which would result . t' in a 0.015 dyne-centimeters . (5) torque varIa Ion. d = 1.5 x 10-2 (2. 57)(9.80) x 104 = 5.9 'DECLASSI~D -87- x 10 -8 centimeters angstroms (4-4) DECLASSIFIED These periodic shifts which are apparent system bandwidth are quite small. here as a point of interest at the lower They are reported to show how powerful a tool the variation in system bandwidth is for determining performance of a gyrocompassing in a benign environment. the system and its instruments Whether or not the variations in azimuth angle noted at the reduced system bandwidth will .. affect the system r s operation depends on the restrictions system bandwidth as a result realistic environment. of lateral in a In any ev~nt, since thes e shifts show up as an output of the east vertical angle; accelerations on sensor as well as azimuth their effect on the azimuth orientation of the gyrocom- passing system can be reduced by sampling the output of the east vertical sensor pe:-iodically and updating the compensation to the azimuth gyro. It is noted (Figure 4-4) that the peak variations angle over the first two-hundred + 11 seconds of arc. and sixty hours are less than The total azimuth gyro drift recorded during this period was less than 0.25 meru. determine It is difficult to an east .gyro drift during this time because large variations of the in azimuth angle. The second one-hundred Figure in azimuth 4-4 were recorded and sixty-eight hours of data in under the conditions It is obvious that increasing of Table 4-1b. the coupling between the east DECLASSIFIEtl -88- DECLASSIFIED . vertical sensor and azimuth gyro reduced the amplitude of the periodic variations previously noted. This is the expected result if. these variations are a result of uncertainties in the performance of the azimuth gyro. The higher KAE, the less tilt about east is necessary to compensate for a change in the torque summation inside the azimuth gyro. For equal east vertical sensor to east gyro gains, the lower the tilt about east, the smaller the change inazim~th angle. The azimuth variations were smaller in amplitude for the wider bandwidth system, but an essentially constant azimuth angle variation of two and one-half seconds of arc per day is evident. This could not be' a result of the ~zimuth gyro. because a total azimuth gyro drift of 1.8 meru ~s required at this system bandwidth to produce eighteen seconds o~arc azimuth error. With 8:neast azimuth coupling of sixty meru per second of arc, a 1. 8 meru azimuth gyro drift would result in an 0.030 second of arc output of the east vertical sensor. The d-c value of the output of the east vertical sensor did not vary by more than 0.005 second of arc, while the azimuth angle changed by eighteen seconds of arc. Since the output of the east vertical sensor remained essentially constant while the azimuth angle had the constant variation noted, it is reasonable to assume that an east gyro DECLASSIFIED !)EC ~ _.~ - ''- -,.-.-_.>4._ ...... """"._,., ...'" A-'''- --.--.,-.-- drift produced the azrmutt-\~SJn~riati~n ~,~.~'".".'<_~~-'~ - ~ .•. during the second ,Ie one-hundred and sixty-eight hours. 'I From Table. 4-1" it is noted that an east gyro drift of less than 0.01 meru per day is large enough to produce this constant azimuth angle change. The puzzling point from a comparison samples of data in Figure the east gyro drift, of the two 4-4 is that there is no reason why or the effects of this drift" should be a function of system bandwidth. A computer analysis two-hundred of the data indicated that the first and sixty hours of data had a constant variation of 0.4 seconds of arc per day. In addition, a constant vari- ation of O.0028 seconds of arc per day was noted in the output of the east vertical sensor. The sense of the constant change ~ in the output of the east vertical sensor was in such a direc- . tion that it canceled part of the constant azimuth variation above O.4 seconds of arc per day. Using the azimuth east angle coupling of Table 4-1a and adding these effects" it is observed that O.68 seconds of arc per day azimuth variation been realized remained if the output of the east vertical constant .. This was in disagreement would have sensor with the 2. 5 seconds of arc per day data of the second one-hundred sixty-eight hours and corresponds to a possible had and east gyro ,Ie 'I Equations 4-1 and 4-2 ma~.~~_atically make this same statement. constant variation of O.0024 meru per day. In spite of I this apparent and frequency discrepancy, of instrument from the data of Figure point, conclusions uncertainties 4-4. spectrum From an instrument view- bandwidth system. The low to be, east gyro drift < O.01 meru of what appears per day indicates that many hours can elapse between sub- sequent recoinpensations of this gyro. Wheel Speed Modulation Wheel speed modulation over a two-hundred ,passing system. alternately hour interval The east gyro 'compensation easily be programed Section 4. 3. 1 contains method may proceed The procedure of S"ections 4.3.1 the analytical us ed to background and 4.3.2. of how was used in the benign laboratory detect changes in the torque who is primarily which could to the east gyro and data recorded is the subject wheel speed modulation was updated following a procedure into a computer., update the compensation to detect of the east gyro in the gyrocom- by two engineers during this period was used periodically and sixty-eight changes in the performance reader can be obtained it is noted that the azimuth gyro should be recompen- sated hourly in the narrower 4.3.0 on the amplitude summation interested in the east gyro. The in data and not background di~~ctly, to Section 4.3.2 DECLASSIFIED to with no loss in DECLASSIFIED the results obtained via wheel speed modulation This test appears to indicate that wheel speed modu- lation can detect changes in the performance gyro in the gyrocompassing system. was conducted in an unrealistically the transfer of this system condition is a major step. background for the transfer 4.3.1 technique. of the east However, the test benign environment and and technique to an .operational The preliminary analytical is covered in Chapter V. Wheel Speed Modulation Theory and Test Before the test was started" east gyro compensation was adjusted so that the azimuth angle of the gyrocompassing system did not vary when the east gyro wheel speed was cut in half. >:~ Under these condit.ions" Equation 4-5 describes angular velocity summation the of the east gyro. Full East Gyro Wheel Speed (4-5a) ~:~ Section 3.3.0 discusses the initial laboratory optimization of wheel speed modulation which this condition establishes. , . DLCLASSIFlEIJ nECT.ll . East Gyro Half Wheel "Speea.SSIFIED 2 R - A zo w(ie)h 0 + 2' + --nIC S TG --n- +w m = 0 (4-5b) Gyro wheel power was initially speeds so that nearly W m ~. equalized the modulation at both wheel induced drift rate~ is zero. In the closed loop gyrocompass~ residual term produces a change in the east gyro's a change in azimuth angle. (4-6) The following procedure sixty-eight was used over two-hundred hour$ to determine if wheel speed modulation could detect changes in the east gyro residual term in the benign environment. During the test~ an automatic collimator the azimuth orientation member; monitored however, this independent not as a source recompensate gy.zECLASSlFIE -93- auto- of the stabilized data source was used as a check on the test, th~ east and of information to Step #1 Two times per day the output of the east vertical was forced to zero by recompensating sensor the azimuth gyro. Step #2 After the azimuth gyro was recompensated, transferred from the gyrocompass mode of. operation. wheel speed, the system was mode to the gyro stabilized* With the east gyro operating at normal the loops were left open for T 1 seconds. end of T 1 seconds the output of the east vertical l At the sensor is equal to T1 2 1. 5 x 10- -+ i ~ (-A~l w(ie)h + ~1 S TG)dt sw H + ic S~G 2 T1 T1 (1.5 x 10)-2 5S o Wz w(ie)h dt 2 0 (4-7) ,I, "." Gyro stabilized. There is no connection between the vertical sensors and the gyro torque generators in the gyro stabilized mode of operation, but all gyro drift and compensation.'remain as inputs to the gyros. ** w(ie)h = 2~6 me~: per second Meru x 1. 5 x 10 of arc at 45 degrees latitude. = seconds of arc per second of time. DECLASSIFIED" The first integral is a result east gyro at time zero. of angular velocity in the The second double integral is derived from the open -loop azimuth gyro drift coupling a horizontal component of earth rate into the east gyro. is, principally, the uncertainty east gyro torque generator loop gyrocompass. Prior Eq:uation 4-6 described in the signal applied to the, between the closed and opento opening the gyrocompass the angular velocity summation formed in the east gyro, where A R o isw zl = A Zo loops, per- + AA , and Rl Z = + AR. Substituting Equation 4-6 into Equation 4-7, T 1. 5 x 1 0 -25"~ 1 sw STG dt ,------H o (4-8) For a constant azimuth gyro drift rate during T 1 and a constant change in the east gyro due to opening the gyrocom- DECLASSIFIE14 pass loops" (4-9) Equation 4-9 describes the output of the east vertical sensor after T 1 seconds . At normal wheel speed" this angle is the result of an angular velocity component switched in when the sys- tem is transferred from a closed-loop to an open-loop system and an angular velocity component porportional to the azimuth gyro drift. Step #3 After T 1 seconds were used to evaluate the open-loop output of the east vertical sensor at normal wheel speed" a ~'c tight leveling loop was closed between the east vertical and the east gyro. The gyro wheel speed was cut in half and T 2 seconds were required rium condition. reference until the east gyro reached sensor an equilib- Since the azimuth gyro has been the azimuth during Steps #2 and #3" a change in azimuth angle has coupled a change in the horizontal earth rate applied to the ~~ Leveling Loop: east gyro torque generator signal derived from the east vertical sensor. Azimuth gyro open-loop. P.a:. \"L.n~SIFIED. east gyro. At the end of T 2" the total change in azimuth DECLASSIFIED angle is Az(T2> = 2 1. 5 x 10- T1+T2 S W (4-10) z dt o Evaluation of Equation 4-10 for constant azimuth gyro drift is (4-11) Step #4 At time T2" the east leveling loop is opened. east gyro operating describes at half wheel speed, the output of the east vertical With the Equation 4-12 sensor. DECLASSIFIEl1 (4-12) DECLASSIFIED Now A J z3 is equal to the azimuth angle at the beginning of the wheel speed modulation test plus any angle accumu- lated since the test started. (4-13) Also" if any change in the east gyro's occurred since Equatlons - (A z ) w o term 4-5b and 4-6 applied" 2 R + :(ie)h residual -<r""'T 0 n S + 2 1" + w TG c .H = m 0 (4-14) Remembering .6.. R" Equations H that A zl = A Zo + 6.A , and that ~A z z we.1e)h = 4-11 a~d 4-14 combine to T S + 2 i sw TG)dt H - (1. 5 2 x 10- ) 2 . 3 SS 2 Wz w(ie)h dt T1+~2 (4-15) DECLASSIFIED Equation 4-15, and making (T3-(T 1 + T 2» equal Integrating I to T 1) (i.e~ I times to evaluate east angle at both wheel speeds ar e equal)" -2 -2 1.5 x 10 [(-1.5 x 10 w(ie)h Wz (T1 + T2)T1 (4-16) A (T3) is the output of the east vertical E sensor when the east gyro is operating at half wheel speed and the leveling loop is open for ~3 seconds. AE(T 1) is the output of the east vertical sensor when the east gyro is operating loop is open for T 1 seconds. at full wheel speed and the leveling Adding -2 AE(T 1) to AE(T 3), Equation 4-1 7 is obtained. + AR T H 1 + DECLASSIFIEQ -99- 2 -2 ' T1, 1.5 x 10 Wz W(ie)h -2-)] (4-17) Rearranging Equation 4-17, DECLASSIFIED + 1.5x10 -2 .6 R (4-18) trT1 Since we are trying to detect a change in the gyro residual t~rm" ~ R" it is seen from Equation 4-18 that by adding two times the angle measured measured at the full wheel speed to the angle at half east gyro wheel speed" the switching rent has canceled. Also" the coupling of horizontal cur- earth rate into the east gyro" due to azimuth gyro drift" is partially canceled. The sum of the angles indicated in Equation 4-18 , propor .t'lona It 0---:fr' .6.R' IS Step #5 On the basis of the angles measured in Equation 4-18, the east gyro cOlnpensation was changed by ~ R . speed changed to normal This technique and the gyrocompass cannot distinguish east gyro residual term, , . loops closed. between changes in the .6 R, and the earth rate coupled into the east gyro due to the azimuth gyro drift. 4-18" the mInImum East gyro wheel d.PtCLA~S111En etec1aoTh1r"~S f From Equation . a unctlon 0 f th e open loop azimuth gyro drift.J;attJ . DE CLASS w .can be calculated . lr1 . Z .6.R Hw z = (4-19) Typical values of T l' are three minutes T 2' and T3 in a benign environment each .. Substituting x 10-2 1.42 The possibility of utilizing into Equation 4-19, this method in an environment other than the benign laboratory depends up.on'the open-loop azimuth gyro drift as ,well as uncertainty ments. 4.3.2 (4-20) Thes e are the subj ects of Chapter in angular measureV. Wheel Speed Modulation Data Section 4.3.1 developed the wheel speed modula- tion technique used for the recording the wheel sp'eed modulation test, of these data. During the system was operating under the wide bandwidth conditions indicated in ,Table 4-1b. DECLASSIFIED and the parameters . DECLASSInED' Figure . gyrocompassing system speed modulation test. recompensation modulation of the during the eleven day wheel of 2: had a six seconds of arc due to uncertainties. However, east gyro recompensation the wheel speed procedure the steady state azimuth angle variation did remove previously noted. 4-5b is a plot of the total east gyro compensation change applied as a result test. ' It is noted that the system peak azimuth variation Figure < 4-5a ~s a grapn of the azimuth variation of the whe~l speed modulation It is noted that this compensation meru per day, which is similar performance previously system data of Figure to the steady state east gyro predicted 4-4. has a slope of 0.007 from, the wide bandwidth These data are redrawn Figure 4-5c as a source of data comparison angle variation of the gyrocompassing in of the azimuth system with and with- out wheel speed modulation. It is noted'that wheel speed modulation removed the steady state compon'ent of azimuth angle variation. the' recompensation variations technique introduced more short-term in azimuth angle. This was the first wheel speed modulation on a system basis, test att,empted and it is believed that future tests reduce the azimuth variations. a vertical However, These future tests will utilize sensor w'ith a longer time constant, DECLASSIFIED -102- can reducing 'the c 0 tn L.f") I ~ .. Z 0 ~ <! ....J ::> I I I I ~ Q I 0 w W 0- en -' w DECLASSIFIED .. L:J W ~ a:: ::>- ::> ::> C!) e.!) (!) I LL 1J... I z L1.. 0 t- .. <1 I )( I I ..J I I I I 0 LtJ fZ I v ::> 0 0 ...,.. I 0 LrJ 2 lLJ 0- 0 OO t- 1 « I IJJ ::> w I c :z: 3: '0 2 1 IU W ::> 0 I ..J I W :c ...J ..J 3 0.. <! W a. 3: t1.. en ....<! z en <t 0.. -2 0 I Z W 0 f<t a.. f- c:: u- Z :?: 0 0 ~ :c u >- l- 0 ~ >- (!) <:{ .... -v I I I J: I f::> I :?E I N W <t I <t N I I en f- -It)~ I ~ e.!) ::> 0.:: I <t a:: ~ 0 > Cl w 0 en <t I >- -(D~ :J: z (f) en I w z 2 0 -,..... I en u.. I I 0- W ::> I w w ;';;;'.0) I 0 l1J 0 .-- en I ~. e.!) W I 0 0 ::> ...J 0 I CJ) en IJ_ ..-« - W fl. ':;> ;.;> I .2 IJJ :c z c.!) Cl Lu I r ..J ..J 0 , ~ -rt> -C\J I CD -I I I C\J. I I 0 C\J .+ JHtI 0 C\J I .::!O SONO::>3S ,..... ,..... 0 0 0 I 0 + n~3tAJ DECLASSIFIED . 0 C\J + 0 f 0 (\J I ::>H'iI :10 SONO::>35 -0 DECLASSIFIE uncertainty . in angular measurPment. computer will replace the engineers test, with a resultant reduction In addition, used for this initial in many uncertainties. The initial wheel speed modulation test appears demonstrate variations the capability a to of detecting low level torque in the east gyro of a gyrocompassing operating in a benign laboratory. D!:CLASSIFlEIl, -104- system DECLASSIFIED V OPERATIONAL SYSTEM EVALUATION In order to use the gyrocompassing refer'ence" gyro drift must be detected and compensated an operational system. Chapter IV has presented indicate that the short-term stability does not produce appreciable "c errors for in data which of state-of-the-art gyros in the gyrocompassing - system." Therefore" reference over a tim'e interval the system can be used as an absolute 01 a few days with no gyro recom- However" since the proposed system Is operational pensation. requirements long-term system as a long-term could be extended over a period of several gyro drift must be compensated years J for in the gyrocom- passing system. Compensation for gyro drift is primarily the gyro drift by incrementa~ measurements the gyrocompassing vertical sensing devices such as pendulums and electrolytic The sensitivity zontal accelerations * of sI?-all angles. In system operating in a benign environment" are capable of incremental of arc. a problem of detecting angle measurements of these vertical will increase levels below 0.01 seconds sensing devices to hori- the noise in the gyrocompassing If the system bandwidth is such that the steady state azimuth gyro drift - azimuth angle coupling is less.than one-hundred seconds of arc per meru. DECLASSIFIEP -105- system when operating increase in a perturbed 'environment. the size of an angle necessary lent signal to noise ratio. A vertical This will to produce an equivasensing device which has a first order lag will be used to filter out the higher frequencies of lateral acceleration. If the time constant of the vertical sensing device is one-third that of the gyrocompassingsystem" a third order critically damped system closely approximates system characteristics. "e:: Typical gyrocompass "I the time constants between 100 and 500 seconds will be investigated.,*:.:e:: Chapter V discusses the parameters development of a recompensation ment. The chapter provides to be considered procedure in a realistic insight i:nto the problem" f~rm solution is not obtained for two reasons" the low frequency spectrum able; perturbed a of the gyrocompassing environment the firmest I data on is not availavailable; s performance in the solution to this problem. North Gyro Compensation Figure 5-1 ;'c provides system analyzer environ- b':lt a closed (1) sufficient accelerations (2) the system is the best environment opt~ization 5.1. of horizontal in the *** is a simplified ' drawing of the north loop Chapter II develops this theory. ~(~( Vertical sensors with time constants seconds are commercially available. *~* · ... Throughout . between O.1 and 500 this entire paper we have assumed that the north .gyro is used prima:r:-ily for leveling purposes and" therefore" is of negligible importance in the ,analysis of the gyrocompassing properties of the system. ----, 1----- I 12 I I 10 Ii= 1« I I~ e l-l lroo <! I > > (.!) 11-0:: I CJ) W I I ::I: I I10:: I I a. - .r:: en:: D- o o -' (f) (J) 0 I~j~/U~ 'J Z I' 1- --~ -1- __ I' 1 3: -- _ ---'--'j ::J 1 ~ ~ z z o () o ... W 0::. ~ a::: >(!) ::> C> -----~ :c t- ___ o:: o :z 0:: o CJ) z w en -l « (..), + 'I- a.. 0:: , w > > '\:- :I: r- !r: o 2 a. c -107- .__ I -_-1 ~geLASSIFIEJ) of the stable base gyrocompassing that the vertical system. The figure assumes component of earth rate coupled into the north gyro due to rotations about east is negligible. Since rotations about east will always be less than ten seconds of arc" KNN can . , be adjusted so that the 0.03 meru of earth rate coupled into the north gyro at medium latitudes will not produce appreciable rotation about the north gyro input axis" A . . Stability requirements on the north loop indicate that KNN must be less than one-quarter half plane. x T for the poles to be in the left v North gyro recompensation change in the output of the north vertical signal to zero py recompensating instruments and instrumentation involves detecting a sensor and forcing this the north gyro. techniques" kept below one-half second of arc. With available * Ax can At middle latitudes" about the north gyro input axis" Ax , produce a nearly angle change 6f the stabilized easily be member. rotations equal azimuth . Since A one-half second of arc" the azimuth variations x can be kept below which are a result of north gyro drift are negligible. 5. 2. 0 East Azimuth Loop Figure 5-2 is a simplified diagram of the east azimuth ~~ A sample data technique and a lo\v sensitivity integrator could be used. The low sensitivity of the integrator does not affect the stability of the system. N <t ::l ~ 0) I r (\.---.. r---I . Iz I}- 1 r----- I I I I I IS N :~=1 I~ <! A L _ I I I ___ J \... >. (!) en0 1 1(- W I 1(1) I 1<1: IW I I I I L IL- ~ ,...., 0 I- lLJ 1(1) U) J N - I I I I~O:: I (I) I Ij:: 1<1: I::Jo 1- > I I (f)W I 10 N I 1«> I- 0: I 'M r-i cd -10.. -0 a"""'" I Q) 'M I ----+ JI s:: +II -M r-l ro-o-> o..H 0..0:> I ___ J s:: rnb.O 0) C 'M I 0 "0 M 0) 0.. 0 r-! X L!':l Q) :> ......... Q) tv) "0 -or-! + Ul 0 Q)"O \I ~c C'M b.Q 'M Q) :J - a rn .-I 0..00 0 ror-la r-l ' ro ro H C 0) 0 ~ l"\J 0.. ~Q) ,N 'M I- 00.. ~o 0 -0->,.83 C Q) 3- 0) Ul 3 o 1 I I I I N 0) .!:: '.-4 S 0) l"\J - o H 0 :>'a b.Oo ~ H Ul:>' rob.O S0S N-o I 3......... ::l • 0"00 H Ul o cd ~o.. .!::b.O ~ o..::l ;j 0 H . M 3 0) o a H 0 :>,H b.O:>' .-l t':l C Q) b.O 8S ~o - - --1 I 1 Iz I I I~ I:J Imo H 0) Q) 00 '~ Ul ~ ro 0"0.. I 12 I :r: --.., S cd . s:: -0-> 'M ,ror-i r.l1f:r1 : Q) ~<e: rn~ -2- II II p:; N b.OH 'M 0 ..'!. 3 ~ 1 r~ ___ II C\J N :>. >:>, :>:>, N I LO .. W 0:: ::::> <!> l1.. 00 00 ro 0.. So () o (f) 'H :>, en (t a. o o 0 >- (- (!) (f)'I q ~ ...... ::> u "'"' ........ c::::. Z g: <! w .-l X L1.) 00 ~ C o <.> M o o U 0:: lLJ o > I- U) « "01 w ~01 ,.- N <i o Ien o ~ s:: ~ro 0 -1 <t w l"\J I ~ 0 --1 ~ £:) II CJ o.. .- 0 -109- II II II II l- t- t- t- loops of the gyrocompassing system. Gyro drift can be detected by observing the output of the east vertical switches 8 1 and 82 open. The accuracy ation depends upon getting a clearly presence of acceleration technique permitting error A m , with of the gyro drift evalu- . defined. angle indication in the induced angle errors. the system indication is obtained, sensor, The sample drift while a relatively clear depends directly upon the other drifts and effects occurring during the sample period. Chapter IV indicate that the frequency pared to what will turn out to bea The data of of gyro drift is low com- sample period. Both east and azimuth gyro drift are evaluated by observing the output of the east vertical 'open. sensor, The angular velocity summation Am' with switches Sl and 82 in the east gyro is always . ;' satisfied by inputs other than those derived from the east vertical sensor; therefore, of the east vertical ever, 8 1 when the switches are open, the ideal output sensor is only due to azimuth gyro drift .. How- when the east gyro wheel speed is cut in half and the switches and 82 are open, the output of the east vertical proportional to east as well as azimuth gyro drift. . sensor is The optimiza- I . tion of gyro drift compensation involves choosing a system time constant and sampling interval so that in spite of limitations .angular measurements the uncertainty in gyro compensation minimum effect on the azimuth angle of the gyrocompassing in has a system. By writing a general vertical . sensor (Figure 5-2) I be developed. I A m I expression for the output of the east after switches 81 and 82 are opened the relative importance For this analysis" of the parameters it is assumed will that the gyro- compassing, system has reached the steady state when the switches are opened. In addition it is assumed that gyro drift l ...~ does not change while the switches are open. 'I When switches S1 and S2 are opened" the east and azimuth gyro torque generators are shorted out so. a step change in angular velocity equal to V is applied to the azimuth gyro and -V z Equation 5-2 describes y to the east gyro. the output qf the east vertical sensor after switches Sl and 82 are opened and the signals applied to the east V and azimuth gyro torque generators respectively are - {, and V z +P' A (P) = m a. (P) 1 (TVP + 1) + P2(T v -2 V 2 2 V w( . )h y + (1. 5 x 10 -) Z 1e P + 1) p3(T P + 1) 1. 5 x 10 v (5 -2) ~:c Chapter IV indicates that the highest frequency of important gyro drift is in azimuth which has a period of approximately ten hours. Since the loops will be open for less than one-half hour this is a D~C' good assumption. J tJtsSIPl£~ Am{P) = ai{P) -r:::--TP~ T T \IV I- -2 2 + (1.5 'x 10 ) 3 w(')h le J.) T (u)w [-2---Z P (T P v + 1) (5 -3) Transforming Equation 5-3 to the time domain and expressing time constant in terms of the gyrocompass 2 +..!.. 9 time constant" T = 3 the TV" - 3t (I - e T)} (5 -4) Substituting K{t)T for t" Equation 5-5 is obtained. A m = (t) K(t) _ 1) + K(t) [K(t) + .!.(e-3 3 6 Contribution 2 _ K(t) + ~ (1 _ e -3 K(t»] -927 from V y (east gyro torque genera- gyro torque generator) tor) when 8 is opened. 2 is opened. Contribution from V z (azimuth when 8 1 (5 -5) Equation 5':'5 indicates east vertical sensor torque generator generator that the signal appearing is composed of the component at the east gyro. at t = 0 and a component at the azimuth gyro torque at t = O. Assuming the average . plotted in Figure at the output of the a.(t) 1 = OJ Equation 5 -5 is 5- 3. Three points should be noted from Equation 5-5 and Figure (1) after the gyrocompass cal sensor is ideally proportional (2) the amplitude square ordinate loops are opened J the output of the east verti- to uncompensated of the output of the east vertical of the time 'constant of the gyrocompassing which varies gyro drift J narrower linearly azimuth gyro drift; sensor system (for a given uncompensated bandwidth system produces alarger paCLASSIFIEQ 5-3: varies as the for the time azimuth output from (- --' 0 <! cr: z >- ~0:: (.!) (.!) W z C/) :r: Z ::> 0 0.. ~ f- q ::> Cf) 0 cr: z. W CL ", " "- - <! to I- <t , LO '\. en " '\. 0::: -I <t zoO:: ~e.:: w (f)>- Z w lJ.. (.!) (!) IJJ ~ 0... .07: 0 0 0 <.) --' Z I- o ~~ 0:: '\. r<> '\. '\. '\. . '\. " w 0:: 0 I- 0::« '\. N """- C1 WI- 0 l.L. "- '\. , '\. to l,Q LO LO JQ x 10 x LO !O nJaW 'JJ\/ r<> 0 0 .<1> I LO C/) w -lI- , ::> =>w CJ) 0 ~ '\. '\. ~ « () "0 c L' I- <l C/) 1<:( Z uJ 0- :E C/) w :.?E W N W .- N ~ W (.!) '\. >--' (!) <! () :c > ::> 0::: ""- , 0::: ::J I- lL. 0 => 0 2 00 0 Z () W 0 (f) rD.:: 0:: 0 lO spuo'JaS -114- z,LzM ( n.) w'V ",\ 0 -. e.:: => (!) lL. the east vertical relative sensor in an equal time' interval}; DECLASSIFIEIl shape of the contribution to the output of the east vertical (3) the of the east anaazimuth sensor time constant of the gyrocompassing is independent system loops of the (for reasons which will become clear later" ~ = 3.3 will be chosen to ~valuate the T output of the east vertical 5.2.1 sensor). Azimuth Gyro Drift Compensation From Section 5. 2. 0" it is noted that after the gyrocom- pass loops are opened" the output of the east vertical proportional Figure to the uncompensated sensor is azimuth gyro drift" (u)w. From z 5-3" the output of the east vertical sensor at t= 3. 3 T is equal to A m = (u) W z T 2 (3. 5 x 10- 6 + 7. a x 10-6) (5 -6) The uncertainty in azimuth gyro drift detection is found from Equation 5-6 to be A (u)Wz (measurement , uncertainty) m ( .. -2- mlnrmum T x detectable) (5 -7) DBCLASSIFIE~ = For...!.T uncertainty 3.3, Equation 5-7 defines the relation between the in azimuth gyro drift in meru" minimum angle in seconds of arc, As a realistic is assumed. and the time constant of the gyrocompass. example, a vertical For a critically gyrocompassing detectable sensor with a 110 second lag damped system, this requires system time constant of 330 seconds. the value into Equation 5-7 and'remembering Substituting that the gyrocompass loops are open for 3. 3...!."the following relations T Time to evaluate azimuth gyro drift: East Vertical Sensor Output Azimuth Gyro Drift Change in Azimuth Angle Azimuth Gyro Drift With the system operating are apparent. 16.6 minutes 1 second of arc meru 15 seconds of arc meru in a benign environment" 0.01 seconds of arc is easily detected at the output of the east vertical defines the vertical ferred sensorrs to a perturbed by lateral accelerations capability. environment" a sensor. This When the system is trans- it is assumed that noise introduced will change this angle to close to O. 1 seconds ,~ of arc, averaged over the total time the gyrocompass loops were opened. >:c Appendix II presents the results of a spectrum analysis of gyrocompass error angles' which indicate that detecting O. 1 seconds of arc output from tl;1evertical sensor is in agreement with less than a five second of arc mean square azimuth angle error. DECLASSIFIED "'l" This would produce an uncertainty -gyro compensation the gyrocompass of O.1- meru in the azimuth and a 1. 5' seconds of arc azimuth error while loops were opened. Sample data techniques and a low gain integrator to bypass KAE and recompens,ate its open loop drift rate. can be used the azimuth gyro on the basis, of There are many variations on the method used to detect drift in the azimuth gyro and update its compensation. The three most important a recompensation procedure are simplicity racy" and speed of convergence. extremely interesting this paper. things to consider of implementation" Although these subjects and tempting" This chapter when determining illustrates accu- are they are beyond the scope of a method which could be used to detect gyro drift and leaves the resultant recompensation of the gyros as a subj ect for future discussions. B~fore we proceed to the detection of east gyro drift" the ef~ect of O.1 meru azimuth gyro uncertainty in the closed loop, gyrocompass on steady state azimuth angle must be considered. K EE between azimuth gyro drift and azimuth angle is -K AE From Figure 5-2" this is equal to 4.5 x_ 10-2 time constant of 330 seconds" T (u)w.z O.1 meru uncompensated The coupling (u)w z. w(ie)h For a system azimuth gyro drift would produce a steady state 1. 5 seconds of arc azimuth angle error. ~CLAs.SIFrED DECLASSIFIED 5. 2~,2 East Gyro Drift C~mpensation In the steady state closed loop gyrocompass" angular velocity summation satisfied. performed by the east gyro is always East gyro drift is counteracted azimuth orientation of the stabilized the via a change in the member coupling a new hori- zontal component of earth rate to the east gyro input axis. speed modulation reduces the sensitivity new component of earth rate. Wheel of the east gyro to this If the gyrocompass loops are opened and the east gyro wheel speed cut in half" the east gyro drift rate is equal to the change in earth rate applied to the east gyro input axis. Under the aforementioned of the east vertical of the east gyro. sensor conditio,ns" observing yields data on the change in performance U~dating the east gyrd compensation of these data and closing the gyrocompass stabilized member Sections 3.3.0" optimization to its original 4.3.1" loops should return the have developed a laboratory of the wheel speed modulation method which was used to determine These sections on the basis azimuth orientation. and 4.3.2 speed n1.odulation on a system the output technique the effectiveness operating as well as a of wheel in a benign environment. show that when the azimuth angle does not change and the east gyro wheel speed is cut in half" the east gyro drift rate is equal to the change in the residual {u)WE(half wheel speed) term inside the east gyro. = D~CLAsSIF1EI>. {5-8} DECLASSIFIED In this section we shall see how well'.6. R can be determined H in an environment where 0.1 seconds of arc is the minimum detectable The following sequence illustrates angle. the proce- dure for evaluating the east gyro drift rate by opening switches Sl and S2 (Figure 5-2) and cutting the east gyro wheel speed in half. Step #1 Disconnect the east vertical (open switch Sl ~ Figure sensor azimuth gyro coupling 5-2). Step 112 Increase ,the east vertical sensor east gyro coupling" KEE" * and reduce the east gyro wheel speed to half its normal value. (Less than three minutes are necessary for this transient to settle. ) Step #3 Open the east vertical and record sensor east gyro coupling" S2 (Figure the output of the east vertical mum three minutes' required sensor. During the maxi- to reduce the east gyro wheel speed" the azimuth gyro has been the azimuth reference. -gyro has changed the horizontal the east gyro. 5 -2)" Drift of the azimuth component of earth rate applied to The loops will be opened for over fifteen minutes :>:C Reducing the east gyro wheel speed couples an inertia reaction torque to the gyro output axis. To minimize the effects of this transient" the east vertical sensor east gyro coupling is increased .l:1lI~!ASSIFl.ED m!~~ . during which time the azimut~wgfh¥:lfduces an output of the east vertical sensor as indicated in Section 5.2.1. the gyrocompass loops are opened for over fifteen minutes the three minutes necessary be neglected, Under the aforementioned considerably conditions, Equation 5-9 describes sensor and the east gyro is operating at ~alf wheel speed. (P)= m ~ (T a.(P) ~ + 1) v and intro- error. the output of the east vertical A time, to change east gyro wheel speed can simplifying the calculations ducing less than a ten percent Because after 81 and 82 are opened -2 2 + (1.5 x 10 ) w(ie)h (u)wz (5 -9) This equation is exactly the same as Equation 5-3 which was used to evaluate the azimuth gyro drift except for a term porportional the east gyro drift, (u)wy. ,The term proportional here because the east gyro is operating speed. We want to determine . to to (u)Wyis included at half its normal wheel (u)w from the output of the east verti- y 1)!;CLASSIF1Bl) ca1 sensor described DECLASSIFIED by Equation 5-9. Assuming that the aver,age interferring equal to zero, acceleration the part of Equation 5-9 proportional gyro drift has been previously plotted in Figure term' of Equation 5- 9 \vhich is proportional reduced wheel speed is plotted in Figure It should be noted (Figures of the gyrocompassing term is to the azimuth 5-3. The last to east gyro drift at the . 5-4. 5-3 and 5-4) that the time constant system is the only system parameter used to evaluate the east and azimuth gyro open loop drifts. In order to show how these figures of Section 5. 2. 1 is continued. In this example, was detected and recompensated gyrocompass should be used, azimu,th gyro drift for by opening the 330 second loops and observing the. output of the east vertical It was noted that if O. 1 seconds of arc is the minimum angle, there would be an uncertainty vertical sensor detectable the azimuth gyro drift of O. 1 seconds of arc output of the east in 16.5 minutes azimuth gyro is immediately sity of recompensating sensor. of less than + 0.1 meru of 'azimuth gyro compensation .. Under these conditions, would produce a maximum the example of time. If the recompensation followed by a test to determine the east gyro, it is assumed .that of the the neces- azimuth gyro oJ, drift does not vary appreciably during this time intervaL .....Therefore, ~:c This is a good assumption.because the highest period of measurable azimuth gyro drift was over ten hours (Chapter IV). -121- -r-- I::J D- I:) 0 0:: 0 en -w l.!J Q.. (1) ....J -LO I.u Z W liJ ..!- (j) -<.0 0 "'0 ;: c: ....J -1 <t <! ~ 0 -~ f- a cc. 2 , lJ.J -IN ...- -f'i') > ,r(f) !- L.. <r ~ D_ o 0 <..9 ...... 1- w Cl -C\J cc 0 >...J ~ W CL 0 0 (/) <:! w 0 C\JI '0 C\J I 10 >< J.{) C\J n..lC)W 'JJ'tj 10 spuo:>C)S -.122- LO .. w Q) a:: (f) => -Il- ~ , ~ (/) (!) lJ.. 0.1 seconds of arc is trie maXlmum output of the east vertical sensor due to azimuth gyro drift durJ!~a~sfm~nterval required to evaluate the east gyro. From Figure 5-4" it is noted that the output of the east vertical sensor produced by the east gyro drift during 3. 3.!. is equal to T Am (east gyro drift) ~ y 15 seconds of arc meru = (5 -8) During this test the total output of the east vertical to be produc'ed by the east gyro drift. seconds of arc angle measurment sensor" the rms uncertainty assuming sensor is assumed is an uncertainty and an uncertainty of arc output from the east vertical Therefore" There sensor of O. 1 of O.1 seconds due to azimuth gyro drift. in the output of the east vertical it is all a result of the east gyro drift" is equal to Am (uncertainty when evaluating the east gyro drift) O. 14 seconds of arc (5 -9) Combining Equations gyro drift measurement 5-8 and 5-9'~~cLRSsiFIttiy in the east is obtained . . (u)wy (uncertainty 0.01 meru = in measurement) (5 -1 0) .At 45 degrees error latitude, this corresponds to a steady state azimuth of 2.9 seconds of arc. Step #4 After the gyrocompass time, loops have been opened for 16. 5 minutes the east gyro is compensated the east vertical pensation sensor is indicated on the basis (Equation 5-8) . of of the angle output of (The uncertainty in this com- by Equation 5-10.) Step #5 A tight east leveling loop is closed while the east gyro wheel speed is returned to its normal value. Step 116 After waiting approximately settle, the gyrocompa$s for all transients to loops are closed. During this entire procedure reference. four minutes The compensation the azimuth procedure gyro has been the azimuth relies firmly on an open loop DECLASSIFIED azimuth gyro drift rate of less than O.1 'meru. number Using this it is noted that a total azimuth angle of less than 2.2 J seconds of arc developed during the 23.5 minutes the gyro- compass loops were opened for east gyro recompensation. 5. 3. 0 Compensation Sequence Before a compensation data must b~ available and the accuracy gyro has, on the spectrum of the recompensation tive data on spectrum been presented sequence can be developed, of instrument process. Representa- of azimuth and east gyro uncertainties in Chapter IV. essentially, a maximum steady state uncertainty Since 0.01 meru corresponds of arc azimuth error at the medium latitudes, accuracy system. uncertainty of O.2 meru periodic ten hours. For the 330 second system, uncertainty corresponds to three But, the east gyro recompensation The accuracy because the east gyro could effect on the The azimuth gyro has an with a period of approximately 0.2 meru azimuth gyro seconds of arc azimuth error. procedure requires an azimuth of less than O. 1 meru. of the recompensation process is difficult to define of the large number of variabJffi involved. recompensation of to three seconds every twelve hours with very little of the gyrocompassing gyro uncertainty has These data indicate that the east 0.01 meru per day. be recompensated uncertainty accuracies From the expected of Section 5 ~2.1 and Section 5. 2. 2, the rms azimutn. error due to east and azimuth gyro uncertainty is t>ECLASSlFIED K (u)A z ;; (u)w (__EE K.AE z) 2 (u)w +( W (ie)h y) 2 W (ie)h ;; 3. 2 seconds of arc (5-11) Gyro drift is the maj or source base gyrocompassing . system. of azimuth it is representative with azimuth gyro uncertainties ties of O.01 mer-u. assumptions of the expected rms azimuth of the performance which would result of 0.1 meru and east gyro uncertain- If properly should keep the uncertainty in the stable Although many simplifying have been employed in the derivation uncertainty" error executed" the compensation procedure in gyro drift below the aforementioned value over the maximum period of time. It is important system that it may be transferred operating with sufficient from the gyrocompass accuracy to the gyro stabilized the maximum time interval. mode of operation It was noted in the previous that large azimuth angle errors did not result Therefore" gyro recompensation over sections when the gyrocompass loops were opened and the output of the east vertical to detect changes in the performance to have the sensor monitored of the azimuth and east gyros. can be a continual process because " it does not affect the operational readine,ss of the system. Armed with the data of Chapter :t\Egh~~~~pensation procedures previously sequence is suggested. developed" the following compensation Each time the gyrocompass opened" the sample data system, averages vertical sensor the output of the east and" at the end of the indicated applies a compensation time interval" to the azimuth or east gyro. data technique applies azimuth gyro compensation steps" loops are east gyro compensation in O.1 meru in 0.0066 meru steps. 'PECLASSJJ;'.LBD The sample " "'l.1t~S~. COMPENS ... J\ TI~~QUENCE Test Number Time 1 00: 00: 00 Event Open east-azimuth evaluate (u)w . gyrocompass loops z 00:16:30 Record A m and compensate azimuth gyro (0. 1 seconds of arc = O. 1 meru). Close gyrocompass loops. 2 to 11 01:00:00 Reneat of Test #1 at one hour intervals. 02:00:00 If the azimuth gyro drift in Test #11 is J. < 0.1 meru go on to Test #12. l If the azim1;1thgyro drift is > O. 1 meru l repeat Test #11 until azimuth gyro drift is < O. 1 meru. 12 11:'00:00 'Open east azimuth loop. Tighten east vertical sensor east gyro loop. Reduce east gyro wheel speed to one-half normal value. 11:03:00 Open east vertical 11:19:30 Record A m and compensate = secQnds of arc 13 sensor east gyro loop. east gyro (0.1 0.0066 meru). 11:20:00 Close tight 'east vertical sensor east gyro loop. Change east gyro wheel speed to normal. 11:24:00 Close normal 12:00:00 Repeat Test #1 gyrocompass 1 etc. . DE CLASSlTIEDf -128- loops £Ct.AS~ VI CON~USIONS One year ago a preliminary analysis of the arduous problems involved in stable base gyrocompassing indicated that present could not perform uncertainty instruments, the job. systems, First, of all instruments with a three-gyro system. effects of short-term operating in the system instrument uncertainties instruments changes in instrument attempting high accuracy The second deterrent angles which must be consistently compassing system to recompensate siderable seconds of arc. ..I.. ...... was the minute problem Shori.'-term: all time up to approximately the gyrocompassing system. the gyro- long-term references, nonlinearities.l angular measurements The third formidable the to detected,in the primary and angle transducer doubt existed regarding angles it is to steady state stable base gyrocompassing Because of friction to re'duce the But, the narrower incremental gyros. in the is to change the the more susceptible performance. environment and gyrocompassing the bandwidth of the system. bandwidth of the system, ~~ dynamic errors The only method available coupling between the various by decreasing and instrumentation, the combined short-term must be small enough to 'produce acceptable gyrocompassing system con- below O. 1 was recompensation five time constants of of the east gyro. Regardless of the gyR~~~~Wtnethod I employ'ed (east-west averaging,J gyro wheel reversal, speed m9dulation,J etc.), dyne-centimeters,7 of the east gyro. 0.01 meru,J approximately is 'of paramount Because importance tests performed' on a single component,J this paper was interested environment. be consistently nonlinearities in the repeat- technique in a system Although gyros are temperature \vheel hysteresis 0.0015 in the evaluation of recent preceeding ability of the wheel speed modulation wheel sensitive and gyro exist,J the gyro wheel speed must changed from 12,7000 to 6, 000 rpm,7 introducing less than 0.0015 dyne-centimeters of uncertainty in the torque sum+nation. Laboratory testing of a prototype gyrocompassingsystem , produced data on, system performance which removes problems. many of the previous These tests produced system parameters,J test program,J system ficiallimits and systems. seemingly evaluations on the performance of state-of-the-art Many of the answers of prototype Four conclusions uncertainties,7 A flexible of proposed can remove the previously to future system obtained only from a combined analytical analysis insurmountable system performance. coupled with realistic characteristics,J in a benign environment data on instrument and over-all has . assumed initial arti- instruments designs can be design and laboratory models .. can be derived from the experimental data and ;, analytical 1.' concepts pres ented in this paper. f)!:CLASSIFIED Superio'r gyro performance gyrocompass (Chapter IV) in the 2-16 is the primary gyrocompassing is currently for the improved performance single-degree-of-freedom superior to any previous power supplies, Two reasons (1) the 2FBG-6F gyro design is gyro design in its class; the proper temperature circuitry (2) con- to derive from its instruments. to be ap. east gyro torque uncertainty in the 2-16 gyrocompass was a ramp with a slope of less than 0.0013 dyne-centimeters gyro torque are, and supporting the optimun:- ,performance What appeared realizable. integrating the 2-16 system provides trol, reason that stable base per day. The azimuth u,ncertainty was approximately a pe:riod of ten hours and an amplitude period~c with of O.026 dyne- centimeters. 2. Two'-hundr,edand sixty-eight hours of data indicate wheel speed modulation to be a usable technique in the 2-16 gyrocompass. This initial test on the feasiblity modulation (Chapter IV) produced tests encouraging results; future over a longer time duration employing refined procedures are necessary before any firm committments the operational 3. of wheel speed The transfe~ring capability can be made on of the technique. of the stable base gyrocompassing DEClJ!.SSlF1E:o system , .DECLASSIF1ED from the benign environment to a more realistic rounding is a major change. analysis environment. Absolute instrument accomplished that the system has capability to permit its being adapted to a more realistic 4. However ~ a preliminary (Chapter V) indicates considerable sur- alignment and calibration with available instrumentation so that reasonable latitude relocations absolute accuracy of the gyrocompassing Data recorded techniques do not affect the sys,tem. in the benign environment that the gyrocompassing'system can be indicate is capable of being adapted to an op.erational environment with acceptable degradation in performance. tested in the laboratory extending the. accuracy The preliminary concepts also indicate the possibility of the self-azimuthing system over an indefinite period of tim,e. system is actually tested in a realistic is dangerous to extrapolate of guidance Until the environment~ it these data~ yet~ unrealistic to ignore the data and possibilites it implies. DECLASSIFIEQ' NOMENCLATURE x g, 0 '.0 •• y goo: z () 0 0 0 0 0 0 0 • 0 0 0 • 0 0 0 0 • 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 • 0 0 0 0 0 • 0 0 0 • Geographic Geographic Direction North East of Gravity b ~x . 0 0 0 0 0 (u}Wx . W .. y 0 (u)W y W z o. 0 0 0 x 0 Ay . A z . (u)A • 0 0 • 0 • 0 ••• 0 0 0 0 •••• 0 0 • 0 0 0 • 0 0 0 • 0 0 0 •• 0 0 • 0 0 . 0 0 0 •• •• 0 0 • Angular 0 • • 0 '0 0 • 0 • 0 0 0 •••••• 0 0 0 0 0 0 0 0 • 0 0 0 0 •• 0 0 y • 0000000.00.00 0 0 0 (u) Woox A 0 0 0 •• 0 0 o. 0 0 0 • 0 0 0 0 0 0 0 • 0 • 0 •••••• 0 • 0 • 0 0 00 0 0 0 •• 0 0 • 0 0 0 0 • 0 0 • 0 0 •• 0 0 0 0 0 0 .0 0 0 • 0 0 into North Gyro North Gyro Drift . Angular • •••• 0 0 0 0 Velocity Velocity into East Gyro East Gyro Drift Angular Velocity Azimuth ' Gyro Drift Rotation About x Gyro Input Axis Rotation About y Gyro Input Axis Rotation About z Gyro Input Axis Uncertainty -133- into Azimuth in East Angle Gyro DECLASSIFIED NOMENCLATURE W(ie)h Horizontal W(ie)v ',' Vertical g Magnitude of Gravity. S xv ' K EE '!(AE . '" , T Component of Earth Rate Sensitivity of North Vertical volls/g Sensitivity of East Vertical volis/g , S yv ••••••••••••••••••••• v' , Component of Earth Rate Sensor Sensor East Vertical Sensor - East Gyro Coupling" meru/arc second East Vertical Sensor - Azimuth Gyro Coupling" meru/arc second Verticai Sensor Time Constant T2 Characteristic Gyrocompass Time of Second Order T3 Characteristic Gyrocompass Time of Third Order A(h _y) Angle Between the Horizontal East Gyro Input Axis bE CLAS SIFIErJ. 'J> Plane and APPENDIX'I DECLASSIFIED NOMENCLATURE A' A (Y g,-Y) Angle Between the Horizontal East Gyro Input Axis (-Y Angle Between Geographic East Gyro Input Axis g -y) East Gyro Torquing Command STG Torque Generator Sensitivity meruf ampere' '.' A(y -m) g (Compensation) Full Speed Angular Momenta of Gyro Wheel , R and West and ic' H Plane ' All non -acceleration in the East Gyro .; Dependent Torques Angle Between Geographic Mirror Normal East and A (-y -m) g Angle Between Geographic Mirror Normal West and A(~_y) Angle Between Mirror Gyro Input Axis (u)A(h~y) Uncertainty in Measurement of A(h_y) (u)R Uncertainty in Measurement of R DECLASSIFIED ~ Normal and East ' DECLASSIFIED NOMENCLATURE (u)A(y _y) ~ : Uncertainty in Measurement g ~R Change in R .6. ic Change in ic .6. STG ~.. ~ W m W my Change in Torque Modulation Generator Induced Drift Rate W(ie) cos Lat Horizontal.Component 'W(ie) sin Lat Vertical Component . Input Axis Angular Float With Respect Wi -f (IA ) Output Axis Inertia Resp~ct to Cas e C d(OA) Sensitivity East Gyro Drift Rate of Normal Speed. ••••••••••••••••••• Ac-f(OA) of A(y _y) , ' of Earth of Earth Wheel Rate Rate Velocity of Gyro to Inertial Space of Gyro' Float With Output Axis Angle of Gyro Float With Respect to Case . Output Axis Damping of Gyro Float With Respect to Case APPENDIX' I NOl\tIENCLATURE (u)M(OA) ~ Wi-c(IA) 0 W c -f{IA) W T ... AT W • o 13 0 0 • 0 0 0 0 0 •• . 0 0 0 0 •• 0 0 •••• ••• 0 ••• • 0 0 0 •• 0 0 0 • 0 • 0 0 0 • 0 0 0 •• 0 • 0 0 0 0 0 0 • • 0 0 0 ••• 0 • 0 •• 0 ••••• 0 0 • 0 ••• 0 0 0 • 0 0 • 0 0 0 Output Axis Torque Input Axis Angular Case With Respect •• 0 • 0 •• 0 0 0 0 0 0 •• Table Rate 0 0 •••• 0 Velocity of Gyro to Inertial Space Input Axis Angular Velocity of Gyro Float With Respect to Cas e • Table Angle o. 0 Uncertainty • •• Sinusoidal Frequency of Table Angle Amplitude of Table Angle Input Axis Total Moments With Respect to Case Ig(IA) .. 0 '00 0 0 0 0 • 0 • 0 • 0 0 Input _t\xis Inertia Respect to Case •• i Ac-f(IA) . 0 • 0 •• 0 • '0 0 0 •• 0 • , C d(IA) ... 0 ••• 0 Kc -f(IA) .... 0 , ••••• ••••• ' 0 0 of Gyro Float With Input Axis Angle of Gyro Float With Respect to Case Input Axis Damping of Gyro Float With Respect to Case •••• 0 of Gyro Float •• 0 Input Axis Spring Constant 'Nith Respect to Case of Gyro Float APPENDIX I ft DBCtASSl'F1.Bw: NOMENCLATURE Azo · Ro . · Initial Azimuth Angle of Gyrocompassing System .. Initial East Gyro Residual D. A Change in Azimuth Angle of Gyrocompassing System z Am Ou tpu t .. vz y a.(P) ........•.......... of Eas t Vertical Sens or Z Gyro Angular Velocity Command Proportional to Uncompensated Azimuth Gyro Drift v 1 Term ) Y Gyro Angular Velocity Com+nand Praportional to Uncompensated Azimuth Gyro Drift Interferring Acceleration PECLASSlfIELl APPENDIX'II DECLASSIFIED SPECTRA OF GYROCOMPASS When the s~able base gyrocompass benign laboratory to an operational angle uncertainties arise. is transferred environment" produced primarily In the course of this paper) the rel~tive pass error orders error of this analysis indicate that in this paper are realizable. of stable base acceleration frequencies orders frequencies .. Reference (6) provides = a source which over all of experimental data of wind velocity which may be written spectra for our case. 'Y (32 S2 7T( _S2 was assumed system) of uniform unit transmissibility on the power density spectra of acceleration of gyrocompass spectra to be impress,ed on the stable base gyrocompassing is mounted in a structure are poorly in driving functions and l?ase ,In order to derive the relative n(s) spectra stable base gyro- angles) wind power density acceleration in terms accelerations of magnitude of mean square value gyrocom- defined because of the vvide variations dynamics. of wind gust acceleration The results angles assumed The spectra from the questions by lateral were assumed to be acting upon the two-axis compass of Figure 5-2. ANGLES + 'Y (II -1) 2) \bLCLASSIE1Etl ~~;!t~~.. ..fkIY m:~ :.. " ,',.'l..;-:'. (~ 2 ~ ' . is th~ amplitude the average of the \vind autocor~WisSFpJEDonl frequency square (Figure values 5 -2) and Parceval's of the following output of east vertical angle. compass From error is of wind occurrence. This function was used in conjunction function and'Y sensor this analysis with the syst~m Theorem gyrocompass I (2) A y I to derive error east anglel the mean square transfer the mean angle (1) A (3) A z I value of these m I azimuth gyro- angles were found to be T KAE w(ie)h K EE v TV (II -2) 'Y - KEE + 'Y 1 (II -3) DECLASSIFIELl DE~SStFIED EE 2 13 [ .....2 'v [ [1:. + [ I{AE w(ie)h TV [ KEE (II -4) These equations all have the same denominator ,feedback in the gyrocompassing system. function due to The mean square value of azimuth erro~ angle, A.AA (0)., is only proportional of the' east azimuth cou~ling squared, value of the east error One is proportional squared, ang:e, Ayy~OL to east vertical (KEE)2 ,and second term can be neglected is composed of two terms. sensor "east gyro coupling for most ratios. of K AE s~nsor, functioI?- of the first sensor plus a feedback via the east- gyro loop. negligible to the coupling earth rate squared. output of the east vertical produced by the transfer portional The mean square the other is proportional via the azimuth gyro and horizontal The mean square (KAE)~ to K O. 4 radians second < 'V . EE AMM(O), The term proloop is and has been disregarded. For typical wind gusts, This order lag vertical to a feedback via the azimuth gyrocompass in magnitude to the value < 3 radians I DECLASSIFIEP second' is Therefore, .L1. so, submittillit..J-,ke critically DECLASSJJ:'1L1J. damped third order, gyrocompass relations and assuming that the system is operating from Equation 2-7 at a medium latitude, the equations reduce to 1'0 {32 1 3 (-) 3 'Y T - - (II -6) (II -7) Equations 11-5, 11-6, and 11-7 express the gyrocompass error the third order critically that the assumption the mean square value of angles as a function of the time constant of damped gyrocompass. It should be note~ that 'Y » I(EE has reduced Equation 11-4 to Equation II -7 by neglecting the feedback to the east vertical via the east loop and is, therefore" pass loops are opened in Figure applicable 5-2. -1 _I- -.-For 100< T < 500, K EE < < 1 s econ d. sensor when the gyrocom- Equations II-5 and 11-7 are 3 for any third order critically square value of thes'e error mE€JIiASSl~y be used damped gyrocompass. The mean angles can be evaluated from a knowledge of the amplitude, of the wind autocorrelation function 3 ? i'" 'and average frequency of wind occurrance {32 know 2" or 'Y and they can only be determined 3 experimentally. 3 However 3 for the system time constants 100 seconds < T < 500 seconds. nitude ot thes e error T 3 2" x = orders T :: 10-2 f3 2 < 'Y in this paper 3 of mag- angles turn out to be 100 1 x 10-6 {3 3" discussed The relative We do not 'Y. 3 < 2 1 -3 x 5 "2 x 'Y 500 10-8 J3 2 'Y < The import,ant point to be noted from this analysis is that the mean square value of azimuth angle and the output of the east DECLASSIFIED' , ":'143- , 'f"'I:£D iEC'L~~Slr ve'rtical sensor appear to be consistent'v/ith throughout this paper. those ~alues Ayy(O) is so small compared angles that it is belo\v our region of interest. this paper a design criteria five seconds of arc. assuming cal sensor to the other In the course has been rms azimuth. variations A compensation procedure that vie could detect an average of 0.1 seconds of arc. J. assumed of of was developed by output of the east verti- Since A.M1v.J:(O) is between one and two orders of magnitude below A. that reasonable relative AA (OL this analysis values have been assumed DECLASSIFIED -144- indicates for these angles. UNCLASSIFiED BIBLIOGRAPfIY DBCL.1{SSIPIEJ) 1. Williamson" R. P." Technique" achusetts (Unclassified Institute oratory" p.~ Gyrocon1.pass Error TitleL Report T-315" of Technology" Cambridge" Averaging (lVIass- Instrumentation Massachusetts" June" Lab- 1~62). CONFIDENTIAL 2. Reed" T. E." '~Theel Speed Modulation" TitleL (lV£assachusetts Institute mentation.Laboratory" August, 3. 1962). Cambridge, Vergoz, OnA Self-Azimuth October Cannon, R. H., J. lVI:., Some Test Results (Massachusetts nology, Instrumentation 4. Laboratory, 1962). 5 .. Schesser, R. J., Alignment bridge, 6. Inc., (Journal T-149" Instrumentation August, Feedback of the Aero- 1961). l'vIicrostability Report Newton, G. C., Gould, L. A., Design Of Linear Mass- Guidance Systems November" Smith" A. J., lVlassachusetts, of Tech- Cambridge, Th:?ry, New York, of Technology" Institute of Inertial scope Spin Axis Bearings, Institute (Unclassified SECRET By Gyrocompassing--Linear 'space Sciences, Instru- Massachusetts, ICBM Guidance System, Report E-1231, achusetts, of Technology" CONFIDENTIAL l\tIarshall,' R. E., Title), (Unclassified of Gyro- (Massachusetts Laboratory, Cam- ,1957). Kaiser, J. F., Controls,(John Analytical Wiley & Sons, 1957). ntCLASSIFIED UNCLASSIFiED