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1156 PROCEEDINGS OF THE IEEE. VOL. 71. NO. 10, OCTOBER 1983 Inertial Navigation Invited Paper Absrract-Inertial Navigation Systems have found universal application becomes more complex, involving coordinate systems, spheroids, bothmilitarilyand commercially. Theyare self-contained, nonradiating, ellipsoids, space, etc. Thesubset of navigation to bediscussed nonjammable, and sufficiently accurate to meet the requirements of users in a most satisfactory manner. An overview of inertial navigation is herein is that of inertial navigation. It is a class of implementa- provided, followed by several sections detailing aspecific, but different tionwhichaddressesNewton’slaws,theconstancy of momen- mechanization approach. A Ring Laser Gym (RLG) based navigation tum, of light, of stars, of gravity, etc. Thosephysicallaws of system design is reviewed with special emphasis directed at requirements nature dealing with acceleration ( F = mu), gravitational attrac- for navigation accuracy and alignment time. Along with discussions of the RLG unit, an introduction to a novel accelerometer approach, the Vibration tion ( F = G ( M , M , / R 2 ) ) , momentum ( H= k ) , velocity of light Beam Accelerometer (VBA), is provided. A gimballed, self-contained High ( c = fA), and the inertial sphererepresentedbythe stars are Accuracy Inertial Navigation System, denoted HAINS, represents one sensed and utilized to providethemeansbywhichthesimple approach toward achieving navigation capability of 0.2 nmi / h and an rms “where am I” question is answered. velocity of 1.5 ft / s per axis while retaining the form and fit and afforda- bility of standardinertialtactical flight navigators. The Stellar-Inertial There have been several excellent papers on the evolution of Navigation section illustrates the bounding of position and verticality errors navigation [l], [2]; hence this paper will not treat that aspect. This thus achieving exceptional accuracies. Two gyroscopic approaches, pre- paper will 1) presentacursoryexplanation of thetheory of sently in development arefinally discussed. The Fiber Optic Gyroscope inertial navigation; 2) illustrate implementationapproachesby (FOG) and Magnetic Resonance Gymscopes (MRG‘s) are of interest for several application examples; and 3) include some mathematical navigation because of their potential for low cost and excellent reliability. treatises and examples of inertial system technology so the reader can fully appreciate this very important aspect of navigation. The Authors paper will cover a) Introduction and Overview of Inertial Naviga- tion; b) StrapdownSystems; c) High-Accuracy Inertial Naviga- The authors are with the Kearfott Division, The Singer Com- tion; d) Stellar Inertial; and e) New Evolving Instrument Tech- pany, Wayne, NJ 07470. nologies. Apologies are offered to those agencies, companies, or Murray S . Goldstein other groups, or applications which are not sufficiently credited, Ivan A. Greenwood either due to limitation in space or byinadvertentomissions. Moms M. Kuritsky Also, we have attempted to write the group of papers in a way to Harold Lerman be meaningful to a broad spectrum of readers. James E. McCarthy 11. OVERVIEW Thomas Shanahan Marvin Silver One wayof providing an overview to self-contained inertial James H. Simpson navigation isto designsuch aninertial system and illustrate pitfalls, requirements, and some fundamental issues.We start with the basic requirement of navigation on our terrestrial body --Earth. We could have selected space or any other consistent referencesincethe fundamental issues are similar. Thegeneral problem which must be solved is that of three-dimensional navi- gation in an appropriate reference coordinate system. The refer- ence coordinate system selected for illustration is Earth’s North, West, Up triad. Self-contained inertial navigation starts withthedouble in- I. INTRODUCTION tegration of acceleration sensed in the Newtonian (Inertial) space frame. Several additional key physical laws must also be properly Navigation has many facets, many definitions, and many sub- utilized. Depending upon the mechanization approach, these laws sets. One begins by simply wanting to know “where am I?’ and involvetheconstancy of momentum or theconstancy of the then expands upon that simple statement with information de- speed of light, the existence of gravity, and the accurate measure- sired on how to get from “where I am” to “where I want to be.” ment of time or its equivalent. The question then arises “referenced to what?” and the problem The outputs of this system are to be a set of position coordi- nates for anytime ( t ) ; usuallyvelocity andattitudeare also provided. There are two coordinate systems which are most often Manuscript received April 20, 1983; revised May 1983. used. The first is -Earth-referenced as selected for illustration and M. M. Kuritskyand M. S. Goldstein are with theKearfottDivision,The Singer Company, Wayne, NJ 07470. providesposition in terms of latitude, longitude, and altitude. 0018-9219/83/1000-1156$01.00 01983 IEEE NAVIGATION KURITSKY AND GOLDSTEIN, EDS.: INERTIAL 1157 The second is Newtonian space or the stars to define an astro- Gyro (RLG)may not fit the classical gyroscopic definition, we nomical reference. When stellar observations are used, both frames continue to use that term. must be integrated consistently. Some instrument references pre- 5 ) Star sightings provide directional information. fer inertial space operation because of performance criteria inde- 6) Strapped down systems and gimballed systems operate simi- pendent of star utilization. larly with appropriate coordinate system definition, either elec- tromechanically or computationally developed. (Instrument error 111. SENSOR REFERENCE sources will propagate differently betweenthetwoapproaches and different error sources exist. These are covered in some detail The primarysensor in an inertial navigationsystemis the in subsequent sections.) accelerometer. This instrument producesprecise a output,in either analog or digital form, which is proportional to accelera- tion applied along the input axis of the sensor..Although we will V. MECHANIZATION limit our discussion to single-axis accelerometers, two-axis accel- Given that the initial position of a “plate” on the rotating erometers have been successfully built and used. If three single- axisaccelerometers are mounted so their input axesform an ,, Earth is latitude, Ao, and longitude, +, we can effectively stop orthogonal triad, any acceleration of this assembly will be resolved the rotation of that “plate” or measure and control its rate in to define an acceleration vector. It is necessary, however, that the with inertial space angle rate measuring devices called gyro- accelerometer assembly referenced be to a coordinate system scopes. A controlled or known horizontal rate of w,cos A and a which can be maintained or defined in a precise manner. It does vertical rate of we sin A is imposed upon the plate. This is accom- not matter how that accelerometer triad moves, as long as we plishedwiththreeaxes of gyroscopiccontrolviaa supporting keeptrack of its precise position, angularly and linearly. The gimbalarrangement or measured and computationallydefined. nature of this reference coordinate system depends on the nature Other rotations with respect to the reference frame due to motion of the vehicle and its mission. A manned aircraft or submarine or disturbances are either decoupled or measured and compensa- generallyuses an Earth referenced coordinate system.Aspace tionintroduced. Upon this same plate, accelerometers are vehicle is probably more concerned with a space-fixed or inertial mounted and the two horizontal sensing units are positioned to reference. Regardless of the coordinate system, it must be han- yield a null output, which effectively causes the gravity vector to dled consistently and information can be provided as needed. To be perpendicular to these horizontal accelerometer sensing axes. the accomplish orientation control of theaccelerometers, an The plate is now tangent to the Earth’s ellipsoid. (The undulative inertial sensor,thegyroscope,isused. The gyroscope has the geoiddue to gravityanomalies and somediscussion of exact required characteristic of being able to prescribe a reference in gravity representation is treated in the high-accuracy navigation inertial space. The three-axis reference may be obtained by the and stellar-inertial sections.) Now, if the plate moves linearly, use of either threesingle-degree-of-freedomgyros or twotwo- relative to space,theaccelerometerssensetheacceleration and degree-of-freedom gyros, or combinations thereof. Mounting the when properly the integrated, yieldsvelocity and distance accelerometers to the gyro reference package in turn provides a traversed. The Earth-induced acceleration components at the defined reference for the acceleration vector. point of interest must also be treated properly (the greater the The gyroscopes and accelerometers are generally mounted in a accuracy the greater the complexity in treating the actual geoid cluster arrangement which is then gimballed or strapped down to versus a simplified ellipsoid). measure vehicle motions about and along three orthogonal axes. Since we are dealingwithnavigation about a(moreorless) Gyros and gimbals are used in conjunction with electronics and spherical Earth, linear motion is very simply related to angular gimbaltorquers to createnull-seeking loops servo for the motion by means of the Earth’s radius, R , i.e., change in platform gimballed case. Any angular motion about the axes is sensed by attitude B = D / R where D is the linear distance from the depar- the corresponding gyro and via appropriate gimbal control main- ture point as derived from accelerometer information and mea- tains the cluster fixed within the reference frame. Output trans- along corresponding sured the great-circle course; similarly, ducers on the gimbals provide attitude output. Or, the mechanical D(rad/s) = Y(ft)/(s)/R(ft), where D = dO/dr. assembly of inertial sensors can be “strapped down” and compu- Thus the reference sensors are effectively precessed at a rate tationally a reference attitude matrix is determined,which is corresponding to the linear velocity of the aircraft. effectively the stabilized referencesystem.Thetwoapproaches Because the mechanization involves the double integration of produce similar results. acceleration to producedistance or angular traversal coupled with feedback from gravity to the accelerometer, a quasi-oscilla- tor is effectively produced. Since Earth’s radius is used, errors in IV. BASICTRUTHS theloop(accelerometer biases, gyro drifts, initial tilt, velocity Before embarking upon the mechanization of an inertial sys- temdesign,thereare several tenets whichshouldbeexamined errors, etc.) oscillate with the effective period T = 2 ~m 84.4 min. In fact, this is the famous Schuler period characteristi- or and remembered. These include the following: callyencounteredwhen an inertial systemnavigates relative to 1) Acceleration is an inertially derived vector. (Keep track of Earth. Aconsiderable amount of intuitionand mathematical the different rotational coordinate frames.) proof formedthe foundation of the above simple statement 2) The constancy of momentum or of the speed of light again before it was reduced to practice [3]. Refer to Table I for a brief is inertially derived. listing of error propagation characteristics. Note position error 3) The accelerometer, not the gyroscope, ties an object to increases with time, but many other errors are bounded in growth at E r h via gravitational mass attraction. and oscillate with the periodicity shown. 4) The gyroscope (no matter thetype)measuresangle (or Somereference has beenmade relative to errorpropagation angular rate) of the device upon which it is mounted relative to differencesbetweenagimballed and a strapdown system.Two inertial space. Applying a torque to the gyro causes a controlling examples of that difference are illustrated by the way the accel- rate reaction if the gyro uses a torquer. Although a Ring Laser erometer bias error and the East gyro bias drift error propagates. 1158 PROCEEDINGS OF THE IEEE VOL. 71, NO. 10, OCTOBER 1983 TABLE I PROPAGATION OF -ORs Position SourceError SourceError Accelerometer bias c a c ~a ( - cos w o t ) 1 instrument w0 sin w,t Initial velocity error c, e- " initial condition 00 Vertical gyro drift w, ( Rw, t - - - I instrument Initial vertical initial condition, - COS oat) RBo(1 alignment So calibration, instrument on initial azimuth Initial alignment q0 40VIct calibration, instrument Azimuth gyro drift wr [:' cos~0-1 1 u,VI, - + instrument where wo alternatelyused) Schulerfrequency (us. g gravity R Earth'sradius VIc inertial velocity, cross direction h latitude present t time RLG 1-AXIS 1.AXIS - \ RLGSENSOR BLOCK / - NAVIGATION DATA ATTITUDE Fig. 2. Typicalstrapdowninertialsystem the strapped down system, the errors decorrelate due to sensor motioncoincidentwiththevehicle,thusdecouplingthesensor reference and initial navigation coordinate reference. A typical gimballed system and a strapdown inertial system as well as their major component parts are shown in Figs. 1 and 2. Note that we have encountered two methods that Newtonian . Y . . I..... forces, or the mechanization as utilized, reduces or increases error \ ELECTRONICS 4 -GIMBAL PLATFORM growth: 1) the Schuler oscillation; and 2) the correlation/decor- relation of errors in a manner to self-cancel or to add dependent upon trajectory and mechanization. Another important error compensation effect is that caused by latitude-azimuthal cross coupling of errors. VI. SHIP'S INERTIAL SYSTEM The Ship's Inertial NavigationSystemoperatespurely iner- tially or in an aided mode, and is discussed in many papers [4]. It Fig. 1. Typicalaircraftgimballedinertialsystem long has a history of development and s i w c a n t successes. Although its implementation is accomplished in a muchex- On thegimballedsystem,both errors causecorrelated errors. pandedand neededcomplex manner, thebasic portion of its Accelerometer bias causes platform tilt with a corresponding null instrumentation includes the inertial reference cluster previously output and gyro bias drift causes an azimuthal error which then discussed. An understanding of the mentionedcross-coupling couples Earth rate into the system again to null the gyro bias. If error is required to understand the error propagation of such a the tilt and azimuthal error remain fixed in the sensor coordinate system or any inertial system over many hours. system, these particular errors do not propagate significantly. In The latitude, longitude, and azimuth errors are affected by the KURITSKY AND GOLDSTEIN. EDS.: INERTIAL NAVIGATION 1159 gyro drift in each channel by way of cross coupling. One of the INERTIALMEASUREMENTUNIT[MECHANICALSTABILIZATION) most significant cross-coupling effects propagates as follows: A North velocity error generates latitude error. As the latitude wRX error increases, the resolution of the inertial rate into the instru- VEHICLE ment reference coordinates becomes incorrect. The azimuth gyro ANGULAR RATE RELATIVE torquing error E ( W , )becomes TO X INERTIAL FRAME 4, COMMAND PRECESSION -i RATE OF (a, = Earth rate, V,, = Eastgroundvelocity, E = error, A = NAVIGATION GIMBAL LOOP BEHAVIOR FRAME P latitude) and it generatesazimuth error. The effect of azimuth RELATIVE TO error 6 on the North velocity error is the same as that of a , INERTIAL drift is - S , ( o , + 6) A. A FRAME hypotheticalEastgyrowhose cos North velocity error will thus be generated opposing the original TO PROCESSII4G velocity error. The long-term average azimuth error adjusts itself AX to drivethelong-termaverage North velocity error to zero. It VEHICLE follows that the latitude and azimuth errors remain bounded over P IS THE NAVIGATION (POSITIONIIJGI P FRAME REFERENCE the long term. Fig. 3. Use of gimballed gyros VII. INSTRUMENTS Acceleration causes mass unbalanced torque and gyro yields The final requirement in this simplified criterion for system on output as function of acceleration. design is the selection of inertial components and their instal- d) Proofmass supported electrostatically with appropriate lation. A wide array of components and installations is available readout capability. as illustrated in the following: A . GyroTypes C. Reference Platform Mechanization Approaches Single-degree-of-freedom,two-degree-of-freedom, free rotor, f Four gimbals utilized tosupport a cluster o gyrosand and solid-state including: accelerometers. I) Rotating Wheel: Three gimbals similarly used with attendant loss of gimbal a) Wheel within float in buoyant fluid freedom. Strapdown-the computer utilizes gyroscopic outputs to 1)jewelled bearing support computationally establish the reference desired coordinate 2) magnetically supported frame. 3) other exotic support. Hybrid strapdown-gimballed; a combination of gimballed b) Wheel supported by universal joint (hinge) electro-servo isolation and strapdown reference action. (One or two gimbals.) 1) torsional hinges Multifunction platform assembly-combination of inertial 2) flex hinges. and aided sensor operation (i.e., stellar-inertial). c) Wheel or rotor electrostatically supported. The actualdesign of the inertial systemobviouslydepends d) Momentum element support upon accuracy requirements, mission environments, usage time, reaction time, reliability, cost, and many other factors. Figs. 3 1)ball bearing and 4 describesome of thecoordinatesystemprocessing dif- 2) gas bearing ferences between strapdown and gimballed systems. 3) fluid bearing 4) electrostatic bearing. VIII. SELF-CONTAINED SYSTEM STELLAR-INERTIAL 2) Optical Gyro (Solid-state Gyro): a) Ring Laser Gyro (Passive and Active Resonator Cavity) Our overview continues self-contained into the realm of b) Fiber Optic Gyro (Phase or Frequency Detection). stellar-aided inertial navigation. Celestial bodies will remain one of the most accurate means of determining position. This fact is 3) Nuclear Magnetic Resonant Gyro. beingrecognized in thedevelopment of hybrid stellar-inertial 4) Multisensor- Combined Gyro and Accelerometer. navigation systems. In a manner completely analogous to a human navigator updating his position by taking star fixes, B. Accelerometer Types inertially determined position can be refined by auxiliary stellar information. Theseinclude single, dual axis, “freelysupported,” and the Aone-starfixpermitscorrection of headingmisalignment, following: whereas a two-star fix will permit determination of position. An I ) Proofmass Supported by Pendulum: excellent way of illustrating how a single star provides azimuthal a) Electromagnetic restoring loop (analog or digital). informationis to considerthefollowing: If a known vertical b) Proofmass supported by beam or string or tuning fork in exists at a reference point then a vertical plane determined by oscillator configuration: two lines: 1) that vertical and 2) the line-of-sight (LOS) to a star Oscillator system’s frequency changed by acceleration’s ef- is established. That plane’s intersect with Earth in a great circle fect on mass. (Further explanation of the vibrating beam determines azimuth. Two-star sightings yield positional informa- accelerometer is found in the strapdown section.) tion. The elevation angle to the first star is satisfied (assuming in c) Proofmass (effectively) supported by gyroscope: this case that a reference vertical exists) by any reference tracker 1160 PROCEEDINGS OF THE - 1 VOL. 71, NO. 10, OCTOBER 1983 GYROMEAWRESINERTIAL ANGULAR RATE *r COMPUTATION OF 0 MATRIX AN0 ACCELERATION (DATA STABILIZED] 41 (COMPUTED ANGULAR ~ RATE OFP TRIAOI ACCELEROMETER ATTITUDE VEHICLE VELOCITY ACCELERATION 1 A L B :x - TOTAL ACCELERATION INCLUDING - RATE OF R COORDINATE FRAME RELATIVE TO X INERTIAL COORDINATE FRAME, - EXPRESSED IN R COORDINATE TRANSFORMATIONOF R FRAME GRAVITY EXPRESSED I N P COORDINATES FRAME) TOP FRAME (LOCAL VERTICAL r (CPI - CROSS PRODUCT EXPRESSED AS A SKEW SYMMETRIC MATRIX w :x - RATEOFP (NAVIGATION FRAME1 RELATIVE TO X COORDINATEFRAME. EXPRESSED IN P COORDINATE Fig. 4. Gyroscope-strapdown application. on a circular locus on Earth whose tracker elevation angle is the same. A second star provides a second locus circle on Earth and thetwo intersect. The ambiguity of two intersections is easily handled. The reference point established by the vertical is at the location of one intersection. Fig. 17 (in Section IV)depicts the angles and directions dismsed. The stellar-inertial system is completely self-contained and is 5 REVOLUTIONS capable of high-accuracy long-range (longflight time) navigation. \ This system can also be ideal for tactical uses, when mobility is paramount and accurate initial alignment conditions cannot be established prior to flight. Using a stellar monitor, it is possible to determine azimuth (one-star fix) alignment by measuring the orientation of theplatformwithrespect to the starsin flight (assuming vertical and initial position are known). As indicated in the Introduction, thefollowingsections w li address in somewhat greater detail specific application of inertial navigation as well as the use of potentially more accurate p d / o r more reliable inertial sensors. W I O N PROFILE I1 5 MINUTEGROUNDALIGN (CLIME@ la) 3801 VELOCITY = 8u) F T M C RLG Strapdown System Navigation:A System DesignViewpoint Fig. 5 . Propagation of position error for given mission. Profile and budget of MARVIN SILVER Table 11. (a) Performance. (b) Individual error contributors. I. INTRODUCTION NAVIGATION ERROR 11. BASIC INERTIAL SYSTEM A RingLaser Gyro (RLG) based inertial navigationsystem ALLOCATION design is reviewed,continuingthetheme of this paper group. Special emphasis is directed at requirements for navigation accu- The quickest way for the system designer to gain a quantitative racy, alignment time, ambienttemperaturerange,shock,vibra- understanding of the RLG performance requirements is to review tion environments, and recalibration interval. These are quantita- the RLG error sources in relation to the total system error tively evaluated with respect to their system impact. budget. Table I1 presents a typical error budget for the 0.5-1.0- The ramifications of the listed basic system requirements on nmi/h class RLG navigation system. Fig. 5 details the contribu- RLG quality are shown bydetailing the components error budget tion of each error source to system navigation error for the given magnitude and sensitivity to each systemrequirement. FUG vehicle trajectory. While error propagation is greatly affected by lock-incircumvention, path length control, currentcontrolin trajectory, Fig. 5 still gives a quick view as to the relative impact addition to calibrationaccuracy and recalibration intervals are of various error sources.Reference[5]providestheinterested also treated. reader with a listing of many strapdown papers. KURITSKY AND GOLDSTEIN, EDS.: INERTIAL NAVIGATION 1161 TABLE I1 GYRO BEA FREQUENCY ICPS) ERROR TYPICAL BUDGETFOR K G AI Gyro Errors (Includes Electronics) Random walk 0.003 O /fi Bias stability error 0.004 O /h Bias thermal error over 0.004 O /h operating temperature range Short-term bias stability 0.003 /h Bias magnetic sensitivity 0.001 O / h / G Scale factor error 5 PPm Scale factor error thermal effect 5 PPm over operating range Scale factor asymmetry 1 PPm Scale factor nonlinearity 5 PPm over operating range Axis to axis orthogonality 3 arc-seconds Accelerometer Errors (Includes Electronics) ' LOCKINZONE IDEA0 BAND1 ' Bias stability error Fig. 6 . Gyro output versus input rate-illustration lock-in zone. Bias thermal error over operating range Short-term bias stability Scale factor error Scale factor thermal error 0.010 F Scale factor asymmetry - 0.008 Misalignment with respect gyro axis Magnetic field sensitivity F 5 Y - 0.006 a - E 0.004 111. RANDOM-WALK ERROR H 0.002 - 2 K = SCALE FACTOR, SEWCYCLE R L - LOCK IN RATE, D E G i H i RD = PEAK DITHER RATE, SEC/SEC The basic operation of the RLG is to excite and sustain two o v ' 4 0 1 0 0 m o 3 W u a 5 W 6 0 0 7 w oppositely directed traveling waves that can oscillate with differ- LOCK IN I'm1 lS2Ll ent magnitudes and frequencies. The frequencydifferencebe- Fig. 7. Random walk versuslock-in and ditheramplitude. tween the two traveling waves under ideal conditions is a direct function of the inertialrate perpendicular tothe plane of the traveling waves. The frequency difference, or beat frequency produces a rate about the gyro input axis that causes the gyro to expression is rapidly enter and leave the lock-in zone producing a random drift 4AOin error. Piezoelectric electric transducers provide the force to rotate Af= - the laser cavity block. LA The output of the fringe detection scheme over a time AT is where equal to a pulse of N pulses proportional to AB A L areaenclosedbythe laser cavity perimeter of cavity for the traveling wave AB = ftA'OiD dt + lr+*' random noise dt. A wavelength of emitted radiation Af beatfrequency (Hz) The random-noise term in the integral sum is the error in the Oin inertial rate perpendicular to cavity plane (input axis beat frequency caused by the dither rate going into and out of rate). lock-in with random phase. The integral of this random noise has the form of an angle error that increases as a function of \/6me This frequencydifference is measured optically viathetwo which is the classical random-walk function. The magnitude of lightwave interference patterns. As in any mechanical system that the random-walk error is related to the dither amplitude and the sustains two modes of oscillation, problems occur when the two lock-in rate per Fig. 7. Residual lock-in with randomized dither is frequencies approach each other. Energy is traded betweenthe a of at level 0.002O/h under and random-walk error of two modes and the frequencies tend to lock and become one and 0.00lo/6 to 0.003'/ 6 are being routinely achieved. A price (1) is violated at lock-in. This trading of energy or coupling is in has been paid, however,withrespect to additionalcomplexity by large part caused back-scattered from radiation imperfect and errors caused by the dither spring mechanism. These addi- mirrors. Loss producingmechanismswithinthecavitysuch as tional errors and their impact on system design will be addressed outgassing of epoxies also contribute to lock-in. Fig. 6 shows the in later paragraphs. relationship of gyro beat frequency versus input rotational rate, and shows the lock-in zone magnitude definition. IV. IMPACTOF RANDOM-WALK ERROR ON Current mirror technology produces lock-in magnitudes in the GYROCOMPASSALIGNMENT NAVIGATION AND vicinity of 100"/h, a far cry from the requirement of less than O.OlO/h for the 1-nmi/h class system. major The technique Random walk continues tobe, even at present achievable utilized, at this point in time to circumvent this lock-in has been levels of error, the major limitation in reducing the required time dubbed "the dither" technique. The technique consists of mecha- of groundgyrocompassingprior to system flight (unaidedsys- nically rocking the gyro through a stiff dither flexure suspension, tems). As previously described, the random-walk error produces which acts as a rotary spring, built into the gyro assembly, which anattitude error that buildsup as afunction of 6 . This 1162 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10,OCTOBER 1983 TABLE I11 GROUNDALIGNMENT HEADING ERRORS VERSUS RANDOM WALKAND ALIGNMENTTIME Time Random (Minutes)Alignment W a l k o / f2 i 1 9 3 8 4 75 6 10 0.001 2.5 0.89 1.3 1.8 0.95 1.4 1.0 1.1 0.84 0.79 Heading 0.003 7.5 5.3 4.3 3.7 3.4 3.1 2.8 2.7 2.5 2.4 Error 0.007 17.5 12.4 10.1 8.8 7.8 7.2 6.6 6.2 5.8 5.5 arc-min V. GYROBIASERROR Gyro bias error is defined as the difference between the true low-frequency gyro bias (period greater than mission time) and the calibrated gyro bias value loaded in the computer to com- ALL BODY AXES pensate for this error. As long as this term remains stable and the calibratedvalue is subtracted fromthegyro output, thecom- pensatedgyro output will indicatezero ratefor the zererate input condition.Oneyear stability of this error to better than ALL BODY AXES 0 . W 0 / h is achievable in production instruments to date. Instrument factors affecting gyro bias stability are as follows: 1) Stability of the mirror’s optical axis, andmirror surface erosion. 2) Outgassing of epoxy materials within the laser cavity. 3)Precise path lengthcontrol to correct for changes due to zoo0 4000 6wo 8000 lMXl0 TIME lmcondsl expansion, contraction, gyro and bending of the block Fig. 8. Navigationerrorversusrandom-walkmagnitude. (Zero vehicleveloc- material. Equation (1) shows the path length-beat frequency ity, alignment error due to randomwalk not included in navigation error.) relationship. 4) Control of the current required to sustainlasing of each beam with current differences to less than 50 nA. attitude error divided by the alignment time can be viewed as an 5) Sufficient control of the dither amplitude to maintain any erroneous drift rate causing a heading error equal to errors induced by dither tobe constant. The forces acting to = i heading error (radians) due to walk random gyrocompass during ground }0 (Rw) w,cos h (3) change dither amplitude are changes inthe piezoelectric element’s scale factor over temperature; dither change due to external vibration and dither cross coupling. where RW is the random-walk coefficient in ( ”/ 6) East along VI. IMPACT GYRO OF BIASERROR ON axis; T isalignmenttime(hours); and W,cos h thehorizontal GYROCOMPASS ALIGNMENT NAVIGATION AND components of Earth’s rate ( “/h). the Table I11 shows heading error as a function of the Heading error during ground gyrocompass alignment is given random-walk coefficient and alignment time at 45 O latitude. by Once in flight, the by heading error caused random walk during ground alignment produces a velocity error per (4) and (5). where ED,, is the East component of the total gyro bias error i North velocity error due to i vector during alignment (total drift vector formed by the vector EV;, = alignment ground by caused addition of the three gyro’s drift errors reflected in the East axis); walk random , D is the component of the total gyro bias error vector along the Up axis; and T is thealignmenttime.Thevelocity error is approximated by (7) EV, 5+ V E ~ ++ DU$W,cosA[ t 2 / 2 - w 2 ( 1 - C O S K ~ ) ] , i velocity East error due to random walk by i EV, = ground alignment error caused Equations (4) and (5) show the impact on velocity error due to = K*E+,, . (5) 1 + - DupVEt 2 heading error caused by random walk for short flight times. In addition, random-walk error causes a velocity error during flight where ED, is the error in North gyro bias and t is flight time: solely due to therandom drift integration intoattitudeerror t, = time after maneuver. during the flight. Fig. 8 shows the magnitude versus flight time As soon as the vehicle changes its orientation causing the gyro for this error as a function of various random-walk magnitudes axes to changefromtheorientation at groundalignment an (does not include velocityerror dueto alignment error). As additional velocity error in the form of R(l - cos Wsti)occurs. evidenced by Fig. 8 and (4) and (5), the random-walk error is a This is calledthe“decorrelation” effect of strapdown systems critical error source for asystem that requiresrapidground caused by the gyro drift vector changing with respect to the local alignment. level axes as discussed in the Overview (Section I>.Fig. 9 shows KURITSKY AND GOLDSTEIN. EDS.: INERTIAL NAVIGATION 1163 I 10.01" ALL BODY AXES IO.OM'/h) / ALL BODY AXES 2OW uxw) TIME lreoonds) Fig. 10. Impact of gyro randomness, first-order Markovian, T = 303 s (5-min lg , ai n 20-min nav, 180° turn-in azimuth, continue nav). 2000 uxw) TIME [SECONDS) drift Fig. 9. Gyro error navigationimpact. (Zero vehicle velocity; 5-min ground align, enter nav mode, turn 180 in azimuth after 20 min of nav.) O IX. SHORT-TERM BIASSTABILITY GYRO This error source is a catch-all that includes al effects of gyro l drift error that have a period substantially less than the mission the navigation error for various magnitudes of gyro drift error time. Jags and level shifts of various causes fall into this category. after a 5-min ground alignment, navigation for 20min at zero An approximation for the alignment error causesby this source is vehicle velocity, and then a 180" andturn, another 1 h of shown in (9). navigation. DEAVG + DCAVGTO VII. BIASTHERMAL GYRO ERROR = 2 RLG inertial navigation systems are usually designed without where D E A V G , DuAVG aretheaveragevalue of random drift heating or temperaturecontrol of thegyro and accelerometer. vector in the east and up directions over the alignment interval. The low thermal conductivity of the RLG and its sensitivity to Fig. 10 shows the navigation impact of short-term bias stability. thermal gradients makes heating of the inertial instruments coun- terproductive. A typical form for the thermal compensation model X. ERROR DRIFT GYRO MAGNETIC SENSITIVITY is shown in (8). The Earth's magnetic field isapproximately 1 G total, with Thermal Gyro Bias Compensation horizontal and vertical components latitude dependent. In addi- tion, magnetic fields can be created by other instrumentation in = B,e-"X + B , ( T , - T,) + B , ( T , - T , y (8) proximity to the system. One of the prime mechanisms for RLG magnetic sensitivity is the property of a magnetic field to change where the property of light that is not linearly polarized in such a way x constant self-heat time that achange in gyro drift occurs.Allelements of theoptical T gyro p-present, temperature; r-reference path are optimized to reduce this distortion of the desired linear B , , B , , B, calibrated thermal coefficients ( B , is a function of gyroshave polarized light. Unshielded can to errors up T , ; i-initial) 0.04"/h/G. Reduction of magnetic sensitivity of 60:l is achieva- t time from turn-on. ble with proper shielding, and values of O.OOl"/h/G sensitivity Factors affecting thermal drift and stability are similar to those are achievable. Error propagation is similar to thatdueto a listed in the gyro bias error paragraph, but now include thermal short-term drift change. are effects. Current systems achieving thermal modeling (dif- ference between true thermal drift and model compensated ther- XI. SCALE FACTOR GYRO ERROR mal drift) to under 0.004"/h over the thermal range of - 40 to Gyro scale factor is defined as the conversion of gyro output 190'F. pulse obtained by the fringe motion detector circuits into indi- cated angle. The basic gyro scale factor is dictated by the inertial VIII. IMPACT OF GYRO BIASTHERMAL ERROR ON of angle changeto produce one cycle motion of the beat frequency. From (l), Orad/cycle = L X / 4 A , where Orad is the inertial angle ALIGNMENT NAVIGATION AND change about the axis perpendicular to the cavity plane to For thermal drifterrors that are essentially constantduring produce one cycle of beatfrequency and onecycle of fringe alignment and navigation, error behavior can be approximated by motion. For a32-cm path equilateral triangle gyrowitha light lumping this error with the gyro bias error previously discussed. wavelength of 0.63 pm the nominal scale factor is 2.11 arc-sec- For environments that are harsher, the warmup error may follow onds/cycle. Utilization two of photodiodes spaced 1/4 of a an exponential.Even for this extreme case, thethermaltime fringe apart to detect the fringe motion allows a basic scale factor constant of the RLG is sufficiently large so as to allow approxi- resolution improvement of four times yielding an effective scale mation of the effect as a bias error for short flights. factor of 0.5275 arc-seconds/pulse for the example shown. 1164 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10, OCTOBER 1983 A number of factors effect the gyro scale factor magnitude and form of this error for a sinusoidal rate input is stability, however, the major influences are the frequency-depen- dent index of refraction of the gas, and the pulling of the cavity resonance toward the center of the atomic gain Both curve. effects are triggered by fluctuations in laser gain or oscillation where cDAsyM is the effective drift error caused by rectification frequency.Basicscale factor stability of 5 ppm one year after effect of gyro scale factor asymmetry, 8 is the peak magnitude of calibration and less than 15 ppm after five years are achievable sinusoidal rate input, and EKAWM is the scale factor asymmetry with current systems. error. Sensorblock rotation is causedbyvehicle linear vibration XII. GYROSCALE THERMAL FACTOR ERROR acting through unbalances in the isolation system. The linear-to- rotational transfer function is quadratic in nature with a resonant RLG technology has advanced to a point where scale factor peak. Sensorblockrocking amplitudes can be as high as 75 thermal stability isgoodenoughover a temperature range as arc-seconds under severe vibration environments (MIL-E-5400 large as -40 to l W ° F to makethe requirement for thermal vibration). For an isolator resonance at 80 H z , a peak rocking modelingunnecessary to meet thermal scale factor errors of 5 rate of 10°/s would be induced. Per (14), a 1-ppm asymmetry ppm. However, if thermal models are employed to shave off a few would produce a O.Olo/heffective drift. More realistic vibration parts per million of error or a desire to achieve 10 ppm over environments produce rocking at magnitudes of under l0/s, and several years they take the simplified form of asymmetry caused drift of O.OOlo/h acting as a gyro bias. Mea- sured scale factor asymmetry for production RLG systems are being held to well under 1 ppm. where So,, is the scale factor at reference temperature and linear thermal coefficient and q,I is the gyro temperature; present and reference. xv. ORTHOGONALITY/GYRO GYRO-TO-GYRO TO ORTHOGONALITY ACCELEROMETER XIII. IMPACT GYRO OF FACTOR SCALE ON The stability and knowledge of the gyro input axes relative to ALIGNMENT NAVIGATION AND each other and relative to theaccelerometeraxesis critical to strapdown navigation systems. This puts stringent requirements RLG scale factor error, because of its small values, leads to on: system design and system calibration; system mounting rigid- negligible error during ground alignment. Error occurs during ity; matched thermal coefficients of expansion between sensors vehicle attitude change. The mechanism for error can be quickly and sensorblock material; no heat application to avoid rapid approximated for simple maneuvers as thermal shock; isolators to avoid stresses due to body vibration (EK)(A$) (11) and shock; and instruments whose internal axes stability is consistent with the required overall stability. The present state of where cos, is the attitude error due to scale factor after vehicle the art for RLG navigation systems in this accuracy class is to maneuver and E Kis the scale factor error. have a combined calibration and stability error in the better than A 360° roll maneuver will produce an attitude error about the 3-arc-second class. roll axis of6.5 arc-seconds for a 5-ppm scale factor error. For a roll axis in the horizontal plane during the maneuver, the attitude error would be a tilt error with respect to local vertical (horizon- XVI. ALIGNMENT AND NAVIGATION DUE ERROR tal plane). Velocity error would propagate immediately after the ERROR TO SENSOR E S MISALIGNMENT AX maneuver in the form of The basic data flow for strapdown navigation is the transfer of Ev= sinw,t sensed body specific force acceleration measurements into a w, computational local level coordinate frame; compensate for local (terms are defined in Table I). gravity acceleration to obtain total acceleration; and integrate to The 6.5-arc-second attitude error of the example given would get velocity and position. Equation (15) shows this flow yield a peak velocity error of 0.81 ft/s. A second roll of 360 O in = (B)b(A)B +(g> (15) the same direction would double the velocity error. XIV. GYRO SCALE FACTORASYMMETRY total acceleration vector in local level coordinates; ERROR/VEHICLE VIBRATION direction cosine matrix that transforms body coordinate vectors to local-level coordinate vectors; This is another critical error source that is dramatically reduced specificforce as sensedby atriad of body mounted using RLG technology over mechanical gyros. The error isdefined accelerometers corrected for calibrated misalignment as the unknown difference between the scale factor for positive errors; rates and that for negative rates. local level gravity vector. The B matrix is initialized during the alignment phase (ground alignment for unaided systems). During this ground alignment, where K , is the scale factor error for positive rates and K - is the accelerometermisalignment errors contributeto errors in the scale factor error for negative rates. initialized ( B ) matrix. After initialization, the ( B ) matrix is A true input sinusoidal rate withzeromeanvalue will be computed by integrating the gyro outputs corrected for known transformed through this error with a nonzero mean or a net drift misalignment errors. Errors in gyro misalignments w cause a i l error (rectification effect) as long as the motion continues. The given vehicle rotation to be calculated by the system computer to KURITSKY AND GOLDSTEIN. EDS.: INERTIAL NAVIGATION 1165 0.01 ^^ 10.0r I 10 loo 1K 1 OK 10K 1K 10 100 FREQUENCY (Hz1 OF CONING MOTION FREQUENCY (Hz) OF SCULLING MOTION Fig. 11. Attitudematrixdriftsensitivitytoconingmotion. Fig. 12. Acceleration error sensitivity to scullingmotion. appearrotatedin spacebythenonorthogonality error of the sense a net bias acceleration. When the measurements of three gyro-to-gyro axes. For example, a Z-axis rate will cause an accelerometers are transformed, without distortion, into the refer- erroneous rate in the Y gyro of ($,,)bz where GrZ is the Y gyro enceframethenetbiasacceleration is canceled and, as with orthogonality error in the YZ plane. Thus the computer instead coning errors, do not accrue.However, if thesinusoidalaccel- of seeing the rate vector bzz sees the rotation vector bzZ erometer outputs are not transformed with fidelity into t e h + ( 8 , ) ( t ~ ~ ~ ) ~ a net error rotation of the rate vector of causing reference frame, the bias error is not canceled. This is the sculling $rz. An errorin the ( B ) matrix will result in attitudeerror error. Fig. 12 shows the relationship between sensor block rock- causing incorrect transformation of body accelerations and veloc- ing amplitude, linear acceleration, sculling error, and computer ity error. For the example given and for a 180' rotation about processing speed required in transforming accelerometer outputs the Z body axis will cause the gyro-to-gyro orthogonality error to through a sculling correction algorithm. This algorithm in con- propagate as an error velocity in the horizontal plane of junction with a coning correction algorithm, increases the effec- tive computation bandwidth of the computer to the processing speed of the algorithms. Vehicle linear vibrations,actingthroughunbalances in the A5-arc-secondnonorthogonality errorfor this casewould isolator structure, set up conditions for both coning and sculling cause a peak velocity error of 1.2 ft/s. Accelerometer misalign- motion. An important part of the design of the isolation system is ment errors in addition to causingalignment errors willcause to minimize vibration-induced rocking. improper transformation of sensedacceleration into the local Another cause of sensor block is self-induced rocking the level system. rocking caused by the dither mechanism. A good portion of this self-induced sensor block rockingdue to XVII. VEHICLE VIBRATION, CONING, AND dither may cause a coning of the block and a potential coning SCULLINGERRORS problem if the computation speed of the coning algorithm is not fast enough. Coning and sculling processing speeds of 2 to 6 kHz A sensor block motion or rocking that causes, with respect to will drop coning and sculling errors to less than O.OOlO/h and the reference frame, a sinusoidal Euler angle about one axis, and 5pg, respectively, for most flight vibration environments. a cosinusoidal Euler angle about an orthogonal axis will have the third axis transcribe a cone in space. A gyro along this third axis XVIII. ISOLATION CONSIDERATIONS SYSTEM will sense a net bias rate. Gyros along the other two wsense i l 0' sinusoidal rates 9 out of phase.Whenthegyro outputsare The isolation system for a dithered RLG navigation system is transformedfrom body space to thereferenceframewithout critical. The isolators must protect the instruments from shock by distortion, the two sinusoidal outputs are synthesized to buck out attenuating theshockspectrum to belowthecomponentmaxi- (cancel) the net bias rate and the system properly shows that the mum acceleration rating. High damping (low Q ) , and low reso- device is not spining in space. All is well. However, if the sin- have nant frequency minimum transmissibility to the higher usoidal gyro outputs are improperly transformed into the refer- frequencies of theusualshockspectrum.However,the isolator ence frame because the computer bandwidth (computation rate) systemmustbestiffenough to preventviolation of thesway is insufficient to reconstruct the sinewaves in the reference frame, space allocation; excessive attitude error with respect to vehicle thebiasconing rate will not be totally canceled out and the axes, and excessiverocking.Themaximumrockingis in most computer will erroneously compute a spinning or drift about the cases at the isolator systems rotational resonance. Care must be coning axis. This is coning error. Fig. 11 shows the relationship taken that this resonanceisnot too low so as to be in the between sensor block rocking amplitude, coning drift error, and autopilot's passband.Another critical consideration is to mini- computer processingspeed for theconingcorrection of gyro mizethe dither torquerequiredfromthe piezoelectric devices outputs. which have limited torque capability. Required torque to produce A sensor block rockingabout one axis with simultaneous linear a given dither amplitude and rate increases dramatically as the acceleration, in quadrature, about a second orthogonal axis will block natural frequency approaches the gyro's natural frequency. cause an accelerometer sensing along the third orthogonal axis to If the block frequency is made greater than the dither frequency 1166 PROCEEDINGSOF THE IEEE VOL. 71,NO.10, OCTOBER 1983 to obtainseparation, namely, a stiff mount, excitation of box resonance is a design danger plus the greater transmissibility of shock acceleration. If the block frequency is made lowerthan the dither frequency to achieve the separation, block rocking may be in the passband of the autopilot. The usual choice is to select the dither frequency for best lock-in circumvention, with the block frequency sigdicantly lower than the dither frequency to a point that balances the considerations of shock attenuation and maxi- mum torque efficiency to provide the required dither rate. Another dither-related problem is the fact that there are three dithering gyros on the same sensor block. A number of mechanisms have been developed to neutralize cross-talk problems, the simplest of which is to have the gyros dithering at different frequencies to Fig. 13. Vibrating beam accelerometer(VBA) schematic. avoid cross talk. Other techniques solve this problem without this required sexing of the gyros. In summary, RLG performance is well under control, and cost must be reduced to broaden the instruments acceptance. This will XIX. ALTERNATE ACCELEROMETER CONSIDERATION broaden the production base and further lower the cost. Recent trends in inertial navigation, especially in strapdown, have led to the need for new accelerometers that can operate in more demanding environments, yield improved performance and m reliability at a lower cost, and have a digital output. Quartz-crystal High-AccuracyInertial Navigation resonators have long demonstrated an inherent accuracy and stability in frequency control and time-keeping applications. Per- THOMAS SHANAHAN AND formance requirements are typically: range k 2 0 g ; bias stability JAMES E. MC CARTHY (one year one sigma) 20 pg; scale factor stability (one year, one sigma) 10 ppm; and, operating temperature range - 40 to + 70 O C. I. INTRODUCTION An approach inthe application of quartz crystals to accelerom- eters has been to use flexure-mode resonators in a beam config- The last two decades have seen the increased application, both urationto restrain the proofmass,hence a Vibrating Beam military and commercial, of inertial navigation equipment (com- Accelerometer (VBA). The behavior of a vibrating beam in monly referred to as an Inertial Navigation System or INS). The tension is somewhat like a string in tension whereby an increase marketplace is a crowded one characterized by acute competition in tension will causethe resonant frequency to increase.The in terms of product cost, performance, and reliability. beam,however, has thefollowing advantages overthe string: In characterizing the performance for aircraft applications, first, a beam requires no bias tension which is the major cause of I N S ’ S are generally sorted into two categories: bias instability in vibrating string instruments; second, a beam medium accuracy position error 1nmi/h (CEP) also responds tocompression. (Fig. 13 represents a VBA sche- velocity error 2.5 ft/s (rms) matic.) References [6] and [7] provide further details of the VBA. high accuracy position error 0.25 nmi/h (CEP) The described accelerometer approach is only one of several velocity error 1.5 ft/s (rms). that are usable in today’s strapdown systems. Others are listed in the Overview section. Significantly, more money has gone toward Gimballed platforms have been the norm for medium-accuracy the gyroscope over the years. However, the accelerometer is now applications for many years, although a growing trend for strap- demandingits day in court. Theaccuracy requirements enu- down is seen in this latter accuracy category. The higher accuracy merated are being met in the laboratory by the VBA and will bracket has until recently been a thin market populated by exotic soon undergo field testing. and relativelyexpensive equipment. One of themore accurate long-term inertial navigators in existence today is found in Fleet X X . THECHALLENGE Ballistics Missilesubmarines [4]. The instruments utilized are of a higher quality than those discussed herein, but they also contrib- The major system considerations of the RLG with respect to ute to a much larger, heavier, and more expensive system. The performance have been addressed; however, the most important gyroscopic instruments for this systemrange from the single- “error” facing RLG technology is cost. The cost is in error, it is degree-of-freedom floated units to electrostatic-supported gyro too high. The instrument is a natural for applications not only in rotor two-degree-of-freedom units. the high- and medium-accuracy class of navigation, but except Recent trends in the aircraft market, however, driven by mili- for cost, for attitudeand referencesystems ( A H R S ) , and for tary imperatives, disclose a significant growth in requirements for hybrid navigation systemsutilizing aids (2-5-nmi/h class). In- a high-accuracy aircraft compatible system. Concerns for precise dustry is actively working on ways and means of dropping this weapondelivery, reconnaisance, mapping, and survivability, to cost. Areas of significantcost reduction activity are in:auto- mention a few, have motivated a series of design initiatives aimed mated (robotic) manufacturing processes (RLG is ideally suited at upgrading the gimballed I N S ’ S to meet these requirements. The for this automation); mirror cost (dropping rapidly in cost from following discussion isa case study of the analysis and key design its present 25 percent of instrument cost); and more leverage in improvements employed to effect the transition from a Medium- reducing cost by reducing accuracy requirements. Accuracy INS to a High-Accuracy INS (HAINS). KURITSKY AND GOLDSTEIN. EDS.: INERTIAL NAVIGATION 1167 Fig. 14. SingerKearfott SKN-2400 inertial navigation system 11. PARAhETERS OF THE PROBLEM ity performance of the INS. Fortunately, the error models and error propagation for conventional gimballed navigators arequite In embarking on the HAINS design,which is essentially a wellunderstood and one can followatop-down analflcal ap- performance-improvement program, several of the following im- proachto assessingsystemperformance. Of interest hereare portant factors must be considered: 1) The Need for HAINS is Immediate. This eliminates consider- mission scenarios, which cases include where the HAINS is ation of esoteric design approaches. The design approaches must aligned on theground priorto flight as well as thosewhere be conservative with inherent low risk. alignment is performed in air using external references such as acceleration matching, velocity aiding, and position fixing. 2) The Environment is Highly Competitive. The objective is to The performancecapability of the Inertial Navigation Unit produce HAINS for a cost penalty of no more than 10 percent overaconventionalmedium-accuracy INS, andtofurther en- (INU) is evaluated utilizing an Error Analysis Simulator which hance reliability and maintainability features to maintain a low provides a statistical error analysis of aided (hybrid) and unaided life-cycle cost. INU performance. The simulator contains math models of the 3) A Broad Range of Applications Must be Met. Potential INU and sensor aid error sources. It also contains the dynamics aircraft requirements include both manned strategic and tactical required to simulate flight trajectories. Error sources considered missions, as well as cruise missiles. It is imperative that the include: HAINS supply the requisite accuracy without the volumetric and individual inertial instrument errors weight penalty of predecessor high-accuracy systems. system level sources With these general guidelines, a HAINS design approach was reference data errors (Doppler and position update) staked out which embraced the following features: environmental induced errors (warmup and gravity anomalies). Maintain the basic architecture and internal partitioningof the very mature and proven medium-accuracy INS (seeFig. 14). Mission scenarios, of course, vary with application, but several Incorporate essential performance improvements in the inertial benchmark trajectories have been established for evaluating per- sensor to ensure a produceable design to the more stringent formance for Tactical/Strike,Strategic/Standoff, and Penetra- requirements of HAINS. tion Missions. It is illuminating to look in detail at the analysis of Exploit advances that have materialized in airborne computer a typical strategic mission employing in-air alignment to identify technology to introduce improved software. the pressure points for HAINS performance. The aircraft trajec- Configure the HAINS along the lines of the USAF Standard tory for this case is shown in Fig. 15. Table IV summarizes the INS Specification (ref USAF Standard ENAC 77-1) to en- major error sources of the INS and the resultant position error sure a well-accepted form factor and interface capability. effects for this scenario and immediately points out the terms of concern for the desired HAINS performance. 111. MISSION/~RFORMANCE ANALYSIS After several iterations, includingconsideration of alternate mission scenarios, a revised error budget is established which can is to perform a series of The first step in this enterprise supply HAINS accuracy reasonable with margin. This error comprehensive mission simulations and to identify the individual budget is also shown in Table I V . performance parameters which drive the total position and veloc- In summary, the key design requirements for the HAINS focus 1168 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10,OCTOBER 1983 INITIAL LATITUDE * W T D +WW INITIAL LONGITUM -106 INITIAL HEADING t5Q FLIGHT OURATION Ohm GROUND SPEED YI) km ALTITUDE m . h a o ALIGNMENT: IN AIR ALIQNMENT DURATION: 11111 POSITION AIDING NO. OF FIXES 4 18 MIN SPACINQ ACCURACY ar,I t ICEPI X INOICATESPOStTlON FIX Fig. 15. Typical strategic mission trajectory. TABLE IV MAJOR ERROR SOURCES OF M S (6-h FLIGHT) Medium Accuracy High Accuracy ErrorValueCEPRate ErrorValue CEP Rate Error Source (1 sigma) (nmi/h) (1 sigma) (nmi/h) Gyroape Ru-Tc-RU Drift Vertical Gyro 0.010 /h 0.098 0.005O /h 0.081 Azimuth Gyro 0.015O /h 0.268 O.o06O/h 0.107 Random Drift Vertical Gyro 0.005O /h 0.257 0.00lO/h 0.051 Azimuth Gyro O.OlO/h 0.091 0.003O /h 0.027 Heading Dependent Drift Vertical Gyro 0.007O /h 0.360 0.00lO/h 0.051 Azimuth Gyro 0.015O /h 0.137 0.003O /h 0.027 Torquer Scale Factor 0.05% 0.127 0.015% 0.051 Accelerometer Scale Factor 0.05O /h 0.050 0.02% 0.021 Bias 50 P g 0.035 50 Pg 0.035 Asymmetry 150 PPM 0.042 NEG - Misalignments lmin 0.149 10 s 0.025 Other Errors* 0.234 - 0.078 Position Fix,Doppler, Gravity 0.110 - 0.110 Anomaly Errors* CEP Rate (Total) 0.65 - 0.21 *Inertial sensor nonlinear anisoelastic, mass unbalance effects. 0’ ; **Position fix, 6 0 Doppler, 0.1 percent; gravity anomaly, 25 p g . Geoid model/gravitational errors The HAINS software employs the WGS Geoid Model which represents the compo- nents of the gravity vector a a function of altitude and latitude.After implementation of s this model, the remaining gravitational influences on INS performance derive from anomalies and deflections of the vertical. C r e t o of these phenomena requires the use orcin of gravity survey data which must be stored in the INS nonvolatile memo?. Since the net effects of these phenomena if not compensated, arc normally less than 0.1 n m i f i , mapping or modeling of these terms is not presently incorporated for HAMS, but would obviously be needed when one approaches this latter requirement. in on the task of improving the inertial sensor in several signifi- ployed to realize the desired improvements andinclude a s u m m a r y C a n t areas: of test results. Gyroscope- restraint (long- term), random and heading sensitive drifts Iv. MAJOR PERFORMANCE IMPROVEMENTS - scale factor and mechanical stability In incorporating design improvements for gyro drift, which in Accelerometer- scale factor, asymmetry, .and mechanical sta- its various manifestations is the most important contributor to bility navigation system errors, several enhancements are required. A Component Alignments. key consideration in current low-cost tuned rotor gyroscopes is Thefollowing paragraphs elaborate on the approaches em- the support flexure or hingewhich connects thegyroscopicele- KURITSKY AND GOLDSTEIN, EDS.:INERTIAL NAVIGATION 1169 TABLE V GROUND ~UMhfARY TEST Test RPER (nmi/h) V,(KTS-ms) V,(KTS-ms) Six 10-h static navigationruns 0.113 (4-h avg) 0.166 (10-h avg) 0.140 (10-h avg) 0.126 (6-h avg) 0.140 (10-h avg) 0.27, Two 3-h Scorsby tests avg 0.048, avg 0.038, avg Four 84-min heading sens. tests 0.069, avg 0.147, avg 0.076, avg One 4.5-h simulated flight test 0.120 0.128 0.134 One 84-min sine vibrationat test 0.5 g 0.0% 0.085 0.105 ment to thegyro motor. Factors such as residual hinge spring Improved velocity error, however, is also a concern for HAINS rate, mechanical/kinematic phenomena which produce rectified applications. Of some importance then is the elimination of torques on thegyroscopicmass as a result of vibration, and asymmetry errors (Le., scale factors differing betweenpositive mechanical instabilities which result in mass shifts about the and negative acceleration inputs). Again, the availability of flight hinge a x i s are of paramount concern. Another significant contrib- the computers with additional throughput permiteffective utor to gyro drift is magnetic coupling between the gyro motor elimination of chis error. The computer 1/0 and software are and the gyroscopic mass. Thus gyro improvements for HAINS organized to process the positive and negative acceleration out- have concentrated on the following: puts independently. Strapdown navigation system applications are inherently more A reconfigured support flexure providing: dependent on accelerometer performance and pressure from this mechanical trimming to reduce anisoelasticity sector of the industry has resulted in development of instrument improved flexure axis orthogonalities enhancements which supply improved bias and scale factor sta- reduced sensitivity to acceleration bility. For these applications, Kearfott has developed improved improved flywheel to hinge stability. magnet materials and stiffer pendulum assemblies. The resulting An alternate motor control circuit and induction motor. accelerometersutilizingthese features havebeen incorporated A programmable speed controller. into HAINS. A. Heading Sensitive Drifts V. RESULTS This error manifestsitself as a change in gimbal drift as a function of platform case-to-cluster azimuth angle and is due to A number of prototype HAINS systems incorporating the magnetic, thermal, and vibrational nonsymmetries acting on the improvements noted have been assembled and subjected to ex- inertial components. In the HAINS platform, these effects have tensive laboratory and flight testing, including tests at govern- beenreduced by afactor of more than two fromstandard ment facilities. The evaluation consisted of a ground test phase medium accuracy navigators and have demonstrated high repeat- and a flight test phase. The ground test results are summarized in ability to permit further compensation by software. Table V. The HAINS was subjected to aprogram of twenty flights in C-130and C-141cargo aircraft. The flight trajectories B. ComponentMisalignment were especiallydesigned to give results which emphasize the errors inherent in the system. There were six E a s t p e s t trajecto- Misalignments between the sensitive axes of the inertial com- ries, six West/East, and six North/South. Two of each of the six ponents are a significant contributor to navigation error. In the flights were for 3 h and the others for 6 h. In addition, there were standard M U these misalignments are sufficiently small and two 10-h flights.Theaccuracywas determined by comparison controlled only by specified machining tolerances to keep naviga- with an accurate reference system, whose position and velocity tion error within allowable limits. These misalignments are stable, accuracieswere 12 ft (1 sigma) and 0 1 ft/s (1 sigma),respec- . however, and they can be accurately measured during calibration tively.The position error CEP for theensemble of flights was of the inertial sensors. The measured misalignments are stored in 0.18nmi/h (see Fig. 1 ) The rms velocity errors were 1.05 ft/s 6. the flight computer program. for V , and 1.25 ft/s for V,. C. Electronics Thermal Errors VI. SUMMARY Component tests demonstrate that the electronics modules which supply torquing pulses to thegyros exhibit scale factor A high-accuracy development program has been successfully temperature sensitivity. Theeffect (in the order of 1 to 20 0 completed which has resulted in introduction of improved ver- ppm/OC) is small for standard INU requirements, but becomes sions of the GYROFLEX" Gyro and Accelerometers. These significant for the HAINS. This has been overcome by mounting improved inertial instruments have been combined with several thesensitive pulse gyro torquing bridge circuits andcurrent higherlevel enhancements in platform assembly and shielding source within the temperature-controlled platform. and software-based compensations to produce an Inertial Navi- gator with significantly better error characteristics than predeces- D. Accelerometer Errors sor systems. In gimballed inertial navigators, the position error contribution from the accelerometers is generally of secondary importance. 'Registered in U.S. Patent Offifice. 1170 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10, OCTOBER 1983 ENSEMBLE POSITION ACCURACY 0 120 180 240 300 360 420 TIME IN NAV IMIN) Fig. 16. F l i g h t test. Iv STAR LINE OF Terrestrial Stellar-Inertial Navigation Systems HAROLD LERMAN, MEMBER, IEEE I. INTRODUCTION Inertial NavigationSystemsareself-contained, nonradiating, nonjammabledead-teckoningnavigationsystems.Theposition error propagation of an unaided inertial navigator unfortunately grows with time and distance. _... . Radio position aids (e.g., GPS, Loran, TACAN, etc.) can be EOUINOXI used to bound the position error of inertial navigators. However, Fig. 17. Observer position/star position relationship the integrated hybrid radio-inertial navigation system is no longer self-contained, nonradiating, nonjammable. For those situations, 11. CELESTIALNAVIGATION where the advantages of a self-contained highly accurate naviga- Celestial navigation is an ancient art. After the use of land- tion system are critical, a stellar-inertial navigation system satis- marks, it is one of the earliest means of position and heading fies the requirement. determination to update dead-reckoning navigation systems. Ter- The vertical error of an unaided inertial navigator is basically restrial positionisdefinedby latitude ( X ) and longitude (@). bounded. This was previously discussed in the Overview Section. Theseanglesdefinethedirections of the local vertical. Every In a stellar-inertial system, the position error growthis limited to position has a unique local vertical direction. In a similar manner, the bounded vertical error. (Position and/or velocity aids can be star positionis defiied bydeclination ( p ) and Sidereal Hour used to further reduce this error.) This bounding of error and additional (if any) improvements are maintained by the Angle (SHA). These angles define the direction of the star line of stellar-inertial system during self-containedoperation.The in- sight. tegration of the various sensors and the stellar observations are The relationshipbetweenthe rotating Earth primemeridan accomplished by an optimal filter operating within the airborne and the celestial space reference plane (vernal equinox) is called theGreenwich Hour Angle (GHA T ) and a isfunction of digital processor. Greenwich Mean Time. It is interesting to note that it was the This scope of this paper is limited to terrestrial stellar-inertial need for accurate celestial navigation which lead to the develop- navigationsystems for aircraft and cruise-vehicleapplications. ment of the chronometer. One second time error causes 1500-ft These systems are also used on various missiles [8] for position position error at the equator. fixes and/or heading fixes and for space applications like Space Fig. 17 showstherelationshipbetweenobserver’s terrestrial Shuttle. position and star position. Previously introduced terms: X , @, , p This section covers the following: SHA, and GHA T are shown. New terms: LHA, A , and E are brief description of celestial navigation basics; introduced. Local Hour Angle (LHA) is the hour angle difference stellar-inertial configuration, platforms, and system configura- the between observer position present and the star position. tions; Elevation ( E ) and azimuth ( A ) represent the star line of sight at performance, error discussions, and simulations. the observer’s present position. KURITSKY AND GOLDSTEIN, EDS.:INERTIAL NAVIGATION 1171 STAR SENSOR /' It is important to select two stars roughly 90 O apart in azimuth to obtain optimal position fix performance. The procedure for the celestial position fixis 1) star selection; 2) star pointing per (17)-(20); 3) star observation (measure residuals); and 4) position update per (22) and (23). In sophisticated stellar-inertial systems, both the star elevation and azimuth residuals are processed by a real-time Kalman filter RTIAL CLUSTER to both updateand calibrate the inertial system. The stellar-iner- tial system is configured to continuously observe multiple stars in sequence to obtain optimal performance. Fig. 18. Conventionalstellar-inertial measurement unit. 111. MECHANIZATIONS STELLAR-INERTIAL A star sensor isused to measurethe elevation and azimuth angles defined. The star sensormustbereferenced toa geo- A. Stellar-InertialPlatform graphic coordinate system at the observer's position. The equa- The star sensor of a stellar-inertial system must be mounted tions for the elevation and azimuth angles as a function of on a gimballed structure with known attitude (azimuth and observer position (A, (p) and star position ( p , SHA) and GHA T vertical). This permits aiming of the star sensor optical axis with are respect to N r h and the horizontal at the estimated star line of ot s i n A c o s E = -cospSin(LHA) (17) sight, without restricting vehicle operation. If this were nota constraint, the strapped-down mounting of a star sensor would COS A COS E sin COS X - COS psin XCOS (LHA) (18) bepermissible. Inertial systems,gimballed or strapped down, must know the attitude of their inertial sensors and thus serve as sin E = sinpsinx + cospcosXcos(LHA) (19) an ideal reference for celestial navigation star sensors. where State-of-the-art starsensors are generally solid state. They have LHA=SHA+GHAT-(p. (20) an m a y of pixelelements to sense and locate the star with respect to the telescope optical axis. Daylight sensing of stars is a Equations (17) to (20) can be used to directly solve for latitude requirement for aircraft stellar-inertial systems. To minimize the and longitude. However, latitude and longitude are known to the effect of sky background and maintain a reasonable number of accuracy of the dead-reckoning system; thus smallangle ap- pixel elements, a small star sensorfield of view is used. For proximations can be used resulting in the following: example, a 0.25O field ofview permits star acquisition without going through a search pattern for a maximum system error of AE - ( E X - 8,)cosA *7.5 mi. -(E(pcos A - e,) sin A (21) The telescope isof the folded orCassagrain design to achieve a reasonable form factor forgimbal mounting. Theoverall plat- where form assembly requires an optical window and gimbal freedom to permit a large sky field of view for adequate star selection/avail- E X , E+ error in latitudeand longitude of the dead-reckon- ability. A 95 O field of view is adequate and achievable. ing system; A gimballed inertial platform requires at least three axes (roll, e,, 8, vertical error about West andNorth axes of star- pitch, and heading) for operation utilizing isolation gyros. The sensor platform; inner cluster containing the inertial sensors (gyros and accelerom- AE difference between expected and measuredelevation eters) is normally maintained vertical and at a known angle with angle (residual). respect to North. An obvious approach is to mount a two-gimbal assembly containing a star sensor on the inertial cluster. Fig. 18 is Equation (21) shows not only the impact of position error on a mechanical schematic of such a device. It is sophisticated and the elevation residual, but also the impact of vertical error on the somewhat complex. The must fact that the star sensor have elevation residual. Vertical error directly limits the accuracy of optical access over a large field of view contributersto this the celestial position fix. This should come as no surprise, since a complexity. The inertial system operates as a conventional celestial position fix is accomplished by measuring the angular gimballed system neglectingstar sensor updates. direction of theobserver vertical. Notethat (21) isa line of Only two gimbals are required to position a star sensor to a positions in the vicinity of the observer, instead of the circle of commanded star line of sight. By mounting a strapdown inertial positions described in the Overview. Star azimuth is not usually system on a two-gimbal telescope cluster the azimuth and eleva- used to determine position. The star azimuth residual has a low tion of the telescope can be computed and controlled. A similar sensitivity to position error anda highsensitivityto attitude two-gimbal stellar-inertial system was successfully demonstrated error. A minimum of two star observations are required to on the T-16 missile used during test phases of the Assault Breaker determine a position fix. For zero vertical errorand two star missile program. This concept as well as the described conven- observations, (21) is used to solve for the position error of the tional approach can be considered for aircraft applications. The dead-reckoning system inertial system, Fig. 19, essentially behaves as a strapdown iner- EX - AElsinA2 - AEZsinA, sin (A2 - A,) tial system in computing position and velocity. Vehicle attitude is determined by combining data from the inertial system and the 1172 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10, OCTOBER 1983 STAR SENSOR AXIS / AXES ACCELEROMETER ATTITUDE GYROS OF: BEST ESTIMATES ACCELEROMETERS ATTITUDE VELOCITY UPDATES POSITION KALMAN FILTER GAINS KALMAN FILTER 19. stellar-inertial Fig. Two-gimbal unit. measurement OBSERVATION AVAILABILITY COVARIANCE (MODE CONTROL) MATRIX ACCELEROMETER 13) GAIN MATRIX GYRO STAR SENSOR I \ - - t + mechanization. Fig. 21. Stellar-inertial available until alignment is completed. The continuous availabil- ity of velocity data (even of low quality) is very beneficial to a stellar-inertial system. Velocity damping eliminates the Schuler oscillation and growth prevents of the vertical error due to gravity anomaly. The reduced vertical errors improves the posi- tion performance of the stellar-inertial system (refer to item 2, above). - \+-- a Relativelyfrequent star observations virtually eliminatethe effect of gyro drift bias and substantiallyreducethe effect of low-frequencyrandomgyro drift. Duringperiods of unaided Fig. 20. SIMU gimbal layout. inertial operation, the random gyro drift is a predominant error source. Gravity anomalies (affecting the vertical, refer to item 2, above) is another major error source. gimbaltransducers. Thestrapdown gyrospermitrapidreposi- To eliminate the effect of the larger random gyro drift of the tioning of the star sensor for multiple star operation. Fig. 20 is a strapdown in the two-gimbal stellar-inertial platform, a unique layout of this assembly. system approach has been configured. This approach is utilized The five-gimbal and two-gimbal stellar-inertial platforms are when redundant inertial systems are used. The integrated system functionally equivalent. The two-gimbal stellar-inertial platform uses both a two-gimbal stellar-inertial system and a high-accu- has theadvantage of simplicity, smallersize, and weight. The racy inertial navigation system, of the type previously described isolation gyros of the five-gimbal platform generally have better in Section 111. Both systems are modeled in a Kalman filter. This performance than thestrapdowngyros of thetwo-gimbal plat- configuration has the equivalent effect of physically mounting the form (given that the same kind of gyroscopic design is utilized). star sensor on the inertial cluster of thehigh-accuracy inertial navigationsystem. It isequivalent or better in function and B. Stellar-Inertial Navigation Systems performance to the five-gimbal stellar-inertial approach. It is important to point out that this approach does not depend Modem stellar-inertial navigationsystemshaveaconfigura- on any critical mechanical alignment between the two navigation tion similar to allhybrid-inertialnavigationsystems;they are systems. matching Velocity the systems the between two via integrated by a real-time Kalman filter. In fact, redundant posi- Kalman filter achieves the desired results. It is to be noted that tion and velocity observation can and should (when possible) be gravity is always an observable to both subsystems, permitting processed by thesameKalman filter. Fig. 21 is asimplified the all important vertical to be identical in the subsystems. mechanization block diagram of a generic stellar-inertial naviga- tion system. The inertial system continuously processes the iner- IV. PERFORMANCE tial sensor information. When stellar observations are present, the Kalman filter uses the data to update and calibrate the integrated The behavior of a stellar-inertial system can best be under- system. Based on a system error model using sensor a priori error stood by reviewing simulation results. A simple North-East flight statistics and real-time knowledge of vehicle motion, the K h a n wasselectedusinga0.2-mi/h (CEP) inertial navigator and a filter determines an optimal gain matrix. The productof the error stellar subsystem star sensor with 2-arc-second (1 sigma) observa- residuals (difference between integrated system outputs and ob- tion noise. servations) and theKalman fitler gain is used toupdateand S x runs are shown on Fig. 22 for different modes and condi- i calibrate the integrated system. The properties of an integrated tions. Runs identified with 1, 2, or 3 refer to the systems modes stellar-inertial systemare 1) effects of gyro drift bias are of 1) unaided inertial, 2) stellar-inertial, and 3 ) stellar-inertial eliminated; 2) the position error is bounded to the vertical error; withvelocitydamping. The velocity data errors are2-ft/s ( 1 and 3) initial conditions of velocity error and accelerometer bias sigma)bias and 0.5-ft/s (1 sigma)noiseaveragedover 100-s are not observable, and thus not correctable by a stellar-inertial period. Runs identified with the letter A or B refer to the system unless other aids are utilized. modeling of gravity anomaly errors. Runs 1.4, 2A, and 3 A In-air alignment of a stellar-inertial system is not possible (see contain a gravity anomaly random error source of 15pg (1 sigma) item 3, above) unless position and/or velocity observations are with an autocorrelation time constant of 120 s. Vehicle speed is KURITSKY AND GOLDSTEIN, EDS.: INERTIAL NAVIGATION 1173 I I 1A: PURE INERTIAL 18: PURE INERTIAL Z A : STELLAR4NERTIAL 5 ELECTRONICAST 26: STELLAR4NERTIAL 3A: STELLAR4NERTIAL VELOCITY RUNSA:GRAVITYANOMALYERROR RUNS 8: ZERO GRAVITY ANOMALY 2ww) t SPINNING MASS GYRO m c 0 l k ldoo zobo Fig. 23. High-performancegyroscopetechnologytrends, as forecast by Jeff Montgomery of Electronicast, RedwoodCity, CA. There is a widespread belief that FOG's will begin to impact I I I I I the RLG high-performance market late in the 1980's and take 0 1 m Jooo 4600 oo00 moo BOOQ lDwol2OOo TIME ISECONUSI over important segments of this market in the 1990's. Fig. 23 Fig. 22. Flight-modeerrorcomparison. shows the FOG forecast of J. Montgomery of Electronicast. This enthusiasm for the future of high-performance FOG'S has devel- oped since several research groups demonstrated FOG random- lo00 ft/s. Runs l B , 2 B , and 3B assumeperfectknowledge of walk coefficients of only a few millidegrees per root hour. Rate gravity. The stellar observation interval is 100 s alternating be- bias stability has been a major problem in FOG research. Early tween two stars. star First observation occurred 300 s after in 1983, inertial grade rate bias stability under laboratory condi- take-off. tions was demonstrated, reinforcing belief in the future of FOG's Refer to Fig. 22. Runs 1A and 1B serve as the reference for for high-performance applications. unaided inertial performance. The impact of the gravity anomaly The most popular FOG is a Sagnac interferometer. The basic error sourceis not significant for unaided inertial operation. idea is that light traveling in CW and CCW directions through a Runs 2 A and 2 B illustrate the caused improvement by fiber-optic coilemerges with a slighttimeandhence optical stellar-inertial operation. (100 ft is equivalent to 1 arc-second on phase difference between the two paths when the coil is rotating the Earth's surface.) The unaided inertial position error isre- about the axis of the coil. The Sagnac phase shift, At$, which is duced to the vertical error of the stellar-inertial system. There is the measure of rotation rate is s significant difference caused by the gravity anomaly error. A can be seen, the introduction of random gravity anomaly noise into At$ = 2nLDS1/Xoco (24) an inertial or stellar-inertial system causes the vertical errors to grow. Run 2 B shows the bounded position error (and, therefore, where L isthe fiber length, D isthecoil diameter, S1 isthe vertical error) for perfect modeling of gravity. The need for rotation rate, A. is the vacuum length of the light, and co is the gravity modeling to optimize stellar-inertial performance is obvi- velocity of light in vacuum. Ejekiel and Ardity [9] provide a ous. tutorial review to which the reader is referred for basic informa- Runs 3A and 3B show the advantage of velocity damping in a tion on FOG's. stellar-inertial system. Both the Schuler oscillation and the un- bounded vertical error growth are eliminated. Thesignificant 11. MAJOR PROBLEMS improvement is fromRun 2 A to 3A wheretheeffect of the gravityanomaly error isgreatlyreducedbyvelocity damping. Sagnac's interferometer experiment in 1913 produced a sensi- The velocity source can be Doppler radar, multimode radar, or tivity of 2 r/s. In 1976, Vaili and Shorthill [lo] demonstrated a true air speed (for low wind noise conditions). fiber-optic Sagnac interferometer. Between thatdateand the present, a number of major sources of both rate noise and bias v. SUMMARY instability wereidentified and ingenious approaches developed which have now brought laboratory FOG performance close to The stellar-inertial navigation system offers the user self-con- that predicted by theory. tained operation and excellent performance. Modular building block techniques exist for the simple installation and incorpora- A . Rayleigh Backscatter tion of stellar equipment with existing inertial navigation systems. In 1980, a major source of rate noise was identified by Cutler, Va Newton, and Shaw [ l l ] .This istheRayleigh backscatter noise Fiber-optic Gyroscopes from the fiber itself. Cures for this problemwere found to be optical source short correlation times (wide spectrum widths) and IVAN A. GREENWOOD, MEMBER, IEEE proper phase modulation within the fiber loop. I. INTRODUCTION B.Magnetic Effects Fiber-Optic Gyros (FOG's) havethe potential of being sim- An important contribution to bias drift,the effect of magnetic pler, more reliable, and lesscostly to build than Ring Laser fields acting on imperfect fibers, was described by Bohm, Peter- Gyros (RLG's) for many reasons, including: little or no lock-in mann, and Weidel [12]. With improved fiber design and simple phenomena; no plasma flow problems; no complicated base shielding, magnetic effects are now not a major problem. With block fabrication;and no critical mirror fabrication or aging perfect polarization-maintaining fiber, there wouldbe no mag- problems. netic sensitivity. 1174 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10, OCTOBER 1983 Dynamic C.Thermal Gradients 111. THEPRESENT STATE OF THE ART Another significant contribution to bias drift was identified in Several groups haddemonstrated FOG short-termrandom by 1980 Shupe 1131. This is the effect of changing thermal walk substantially as predicted by theory, about equal to RLG gradients along the length of the fiber coil. Once recognized, this performance, andadequateforstrapdown inertial navigation. has addressed symmetrical problem been by coil design and Inertial grade long-term rate bias stability had not been reported improved thermal gradient control. by these groups as of March, 1983. It was widely believed that polarization instability was the remaining major problem which D. Insertion Loss, Stability caused actual performance to differ from theoretical predictions over long periods of observation. Most early FOGS used discrete components. In 1981, Bergh, The results obtained by Burns et al. [25] confirmed this view, Lefevre, and Shaw[14]reportedthe first allsingle-mode fiber- and showed further that polarization-maintajning optical fiber is optic gyroscope. This wasmajor a advance in two respects. aviablesolution to theproblem at or veryclose to thelevel Insertion loss from source to detector was very small, leading to required for navigational quality strapdown inertial systems, at outstanding signal-to-noise ratio. Instability due to relative com- least under benign conditions. These observations close a period ponent motionwaseliminated. In addition, this approach may of intenseresearchproductivity the during which major and lead to lowerproduction costs, although this is not yetestab- mysterious besetting problems the original FOG experiments lished. All-fiber gyros have so far been reported only in connec- were identified and understood. tion with zero or small rate inputs. IV. THEFUTURE E. Rate Measurement The FOG is nowentering its development and engineering For strapdownapplications,very large rate dynamicranges, high phase. For applications not requiring performance, the scale good factor accuracies, and digital outputs usually are emphasis will be on finding adequate components of verylow required. These requirements ledto a search for means other than cost and assembly techniques which can be semi-automated. analog phase measurement for converting the Sagnac phase shift For navigational applications, much quality component devel- into a digital output representing inputrotation. Ezekiel and opment with reasonable cost objectives still lie ahead. The two Balsam0[15] in 1977describedaresonantringgyro in which most pressing needs are low insertion-loss frequency shifters for differential frequency represented input rate. Pool and Sellers [16] all-fiber gyros, and inertial sensor grade polarization-maintaining and Cahill and Udd [17] in 1978 filed patents on frequency shift optical fibers and couplers. For high-performance inertial sensors, phase-nulledSagnacgyros. In this approach, the output of the very the field will likely shift to the1.3-1.6-pm wavelength gyro becomes identical to a ring laser gyro of the same enclosed region,sincetheavailablelowerinsertionlosses are important area,independent of the number of turns on the fiber coil. hs and component research for t i wavelength region is progressing they many Although have problems, technical acoustooptical rapidly due to the demands of the telecommunications industries. frequency shifters are normally used in suchgyros.Fiber-optic frequency shifters for all-fiber gyros have not yet been reported. A frequency shifter described by Heismann and Ulrich [18] may be applicable to this pressing need. E Kerr Effect Early in 1982,Ezekiel,Davis, and Hellwarth [19] reported differential-intensity-induced nonreciprocal phase-shift errors in a narrow spectrum laser fiber-optic gyro. In only a few months, two solutions [lo], [21] to this problem were found. Square-wave modulation of the laser source in a Sagnac interferometer was Most fundamental particles, electrons, protons, neutrons, as shown to overcome the effect, and more generally, a broad well as nuclei, possess an intrinsic angular momentum. The use of spectrum source with appropriate statistics was shown to do as this angularmomentum for gyroscopicmeasurements has long well.Because of the Kerr effect, most fiber-optic Sagnac inter- been an intriguing idea [26]. Several approaches have been sug- ferometer gyros now use broad spectrum laser diode or super- gested for the design of instruments using this intrinsic angular luminescent diode light sources. momentum and performance approaching navigationrequire- mentshasrecentlybeenrealized in selected laboratory experi- ments. G. Polarization,PolarizationControl In all cases, the intrinsic angular momentum of the nucleus has Until recently, fiber-optic gyros a used combination of a been used in some form of Nuclear Magnetic Resonance (NMR). polarizer between the two couplers, and a polarization controller As in all NMR, anetnuclearmagnetization is established, in (or a depolarizer) in the coil loop path [22]-[24]. Most residual most cases by optical pumping, the exchange of angular momen- rate bias drifts were ascribed to lack of polarization control due tum between atoms in the vapor state, and circularly polarized to changes in fiber-optics properties over time and environmental resonant radiation [27]. conditions. In March 1983, Burns et al. [25] reported experiments The basic rotation information is obtained from observations (at zero input rate) with a gyro having a fiber-optic coil and its on the precessional motion of the net nuclear angular momentum associated coupler fabricated from polarization-maintaining fiber. inan appliedmagnetic field H,. The Larmorfrequency,the The opticalsourcewasasuperluminescentdiode.Over a 24-h frequency of precession about H,, is proportional to the magni- period they obtained a rate sigma of less than O.OlO/h with a tude of H,, oo = yH,, where the constant of proportionality y filter time constant of 125 s. the gyromagnetic ratio is a characteristic of the particular nuclide. KURITSKY AND GOLDSTEIN. EDS.: INERTIAL NAVIGATION 1175 PD - ANALYZER LINEAR BP - BREWSTER ANGLE POLARIZER Ha - DCMAGNETIC FIELD Hi - MERCURY ABSORPTION CELL L2 - READOUT LAMP L1 A14 - - PUMP LAMP QUARTER WAVE PLATE M - ALUMINUM MIRROR PD - PHOTODETECTOR FC - FILTER CELL A12 - HALF WAVE PLATE Fig. 24. Arrangement of opticalcomponents However, if the precession is observed from a coordinate frame rotatingat an angular rate or about thedirection of H,, the observed frequency will be shifted by a,, = yH, - or. For or o in the range required for a practical navigational gyro, measure- ment using this equation would impractically require precise knowledge of H,. This problem is overcome by using two differ- Fig. 25. Fused-silicamercury-vaporresonancecells. ent types of spin particles in the same magnetic field, resulting in twoobservables, L m o r frequencies, and two unknowns, the generated by a small coil near the resonant sample. The mercury magnetic field and the rate of rotation. is in the form of a low-pressure vapor, approximately 0.1 mtorr, - a = YIHO 1 or and is contained in a spherical cell about 1 cm in diameter that is and madefromhigh-purityfused silica, as shown in Fig. 25. The magnetic resonance is detected optically through the transverse rf a2 E - YZHO o r . Faraday effect. The readout beams for both cells are also derived One approach based on the implementation of these equations from a common source. employs noble gases as the resonant nuclei [28]. The net nuclear The ac magnetic fields for driving the magnetic resonances are magnetic moments are produced by spin exchange between opti- obtained byamplifymgthe outputs of thephotodetectors in a cally oriented rubidium atoms and the noble gases. The magnetic broad-band amplifier and feeding the output of the amplifier to resonances are also detected the through rubidium magnetic the drive coils to form a self-oscillating circuit. With a magnetic resonance. One way of looking at this phenomenon is to regard field of 1.3 G, the resonant frequencies of Hg’99 and Hg2” are theresonance in rubidium as a sensitive magnetometer that l o and 369 Hz, respectively. o0 detectsthenuclearmagnetizations.Anymethodbased on the The output of thegyro is obtained byformingthephase pair of equations above relies upon the constancy and knowledge O OH difference between the 1 O - zsignals and the phase difference of the ratio of gyromagnetic ratios y1/y2. between the 369-Hz signals from the two cells. An errorsignal for The requirement to know or control this ratio canbe eliminated control of one of the H, magnetic fields is formed by adding the byuse of two magnetic fields in opposite directions and a two phase differences. The gyro output is the difference of the resonance cell containing both nuclei in each magnetic field. This two phase differences and is the simply four times angle of second approach is shown schematically in Fig. 24. In this rotation. The gyro scale factor does not depend upon the geome- implementation, resonant the nuclei are the two odd stable try or size of the instrument. isotopes of mercury, Hg199 and Hg2”. The resonance radiation The performance of an MRG ischaracterizedbyanangle for optically pumping the samples has a wavelength of 253.7 nm. output noise, which depends on the signal-to-noise spectral den- Both pumping beams are derived from the same light source by sity ratio, an anglerandomwalk,whichdepends on theangle means of a polarizing beam splitter. The hyperfine structure of noise and the relaxation times of resonance, and a variety of bias the 253.7 nm line issuchthatthecomponentfromHg204is effects [29]. Recent laboratory tests havedemonstratedperfor- coincidentwith one of the components of each of the odd mance in the range of a few hundredths of a degree per hour. isotopes. 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