IInertial Navigation by ashi7790

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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
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.
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-

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
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
                SourceError SourceError
                                         Accelerometer bias c a          c
                                                                         ~a ( - cos w o t )
                                                                                1                        instrument
                                                                                    sin w,t
                                         Initial velocity error
                                                              c,               e-
                                                                                "                        initial

                                         Vertical gyro drift w,                (
                                                                        Rw, t - -
                                                                                              -   I
                                         Initial vertical initial condition, - COS oat)
                                           alignment So                                                     calibration, instrument
on   initial        azimuth              Initial
                                           alignment q0                      40VIct                         calibration,
                                         Azimuth gyro drift wr            [:' cos~0-1 1
                                                                   u,VI, - +                             instrument

                                          wo                        alternatelyused)
                                                Schulerfrequency (us.
                                          g     gravity
                                          R   Earth'sradius
                                          VIc inertial velocity, cross direction
                                          h        latitude
                                          t     time

                                                                                    RLG                     1-AXIS                               1.AXIS

                                                                                                                     \     RLGSENSOR BLOCK   /


                                                                                                               Fig. 2. Typicalstrapdowninertialsystem

                                                                                              the strapped down system, the errors decorrelate due to sensor
                                                                                              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   .   .
                                                                                              forces, or the mechanization as utilized, reduces or increases error
        \              ELECTRONICS                     4 -GIMBAL
                                                                                              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

                                                                           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
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
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
  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
  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


               (COMPUTED ANGULAR                    ~

               RATE OFP TRIAOI

                         ACCELEROMETER                       ATTITUDE

            VEHICLE                                             VELOCITY




                TRANSFORMATIONOF R
                                         GRAVITY EXPRESSED I N P COORDINATES

                                            TOP FRAME (LOCAL VERTICAL

                      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
                   TYPICAL    BUDGETFOR K G
                   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
        Scale factor asymmetry                                                        -
        Misalignment with respect
          gyro axis
        Magnetic field sensitivity

                                                                              a       -
                                                                              E    0.004

                  111.   RANDOM-WALK        ERROR
                                                                              H 0.002 -
                                                                                                                           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.
                                                                              The output of the fringe detection scheme over a time AT is
where                                                                       equal to a pulse of N pulses proportional to AB
      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
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
   Current mirror technology produces lock-in magnitudes in the
                                                                                     GYROCOMPASSALIGNMENT NAVIGATION
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
                                   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
                                                          1.1                                              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
                                                                                             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

                 heading error (radians) due to
                                during ground                  }0      (Rw)
                                                                       w,cos h
                                                                                                 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
axis; T isalignmenttime(hours);     and W,cos h thehorizontal                                      GYROCOMPASS
                                                                                                             ALIGNMENT NAVIGATION
components of Earth’s rate ( “/h).
  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
                                                                                          where ED,, is the East component of the total gyro bias error

                  North velocity error due to

                                                                                          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 ~ ) ]

              East           error due to

        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

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

                                                            ALL BODY AXES

                                                        /   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)

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
                                                                                      This error source is a catch-all that includes al effects of gyro
                                                                                   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).
                                                                                                                 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
                                                                                               GYRO       MAGNETIC
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
            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.
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
                                                                                   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
                                                                                   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
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
                     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
                     OF         FACTOR
                            SCALE    ON                                    The stability and knowledge of the gyro input axes relative to
                        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
tal plane). Velocity error would propagate immediately after the                                             ERROR
                                                                                    TO SENSOR E S MISALIGNMENT
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;
                          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
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

               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
 + ( 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
will sense a net bias     rate. Gyros along the other two wsense
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
                                           Source                   (1 sigma)  (nmi/h)     (1 sigma)    (nmi/h)
                                   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
                                   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
                                 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
                                 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
                                                                                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                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)
                          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

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
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
           HAROLD LERMAN, MEMBER,                    IEEE

                       I. INTRODUCTION
   Inertial NavigationSystemsareself-contained,         nonradiating,
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
                                                                              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.
                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
   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
             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


                                                                                                    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,
                 \+-- 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-
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

                  RUNS 8: ZERO GRAVITY ANOMALY

       2ww)   t                                                                                      SPINNING MASS GYRO
                                                                                         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$, which is
duced to the vertical error of the stellar-inertial system. There is         the measure of rotation rate is
significant difference caused by the gravity anomaly error. A can
be seen, the introduction of random gravity anomaly noise into                                          At$    = 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

Gradients                                                                          111. THEPRESENT
                                                                                           STATE                  OF THE   ART
  Another significant contribution to bias drift was identified in        Several groups haddemonstrated FOG short-termrandom
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
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-
G. Polarization,PolarizationControl
                                                                          In all cases, the intrinsic angular momentum of the nucleus has
   Until recently, fiber-optic gyros a
                                        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


                                       -        ANALYZER
                                 BP    -   BREWSTER ANGLE POLARIZER
                                 Ha    -   DCMAGNETIC FIELD
                                 Hi    -   MERCURY ABSORPTION CELL
                                 L2    -   READOUT LAMP
                                           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
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.
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.                                                              Tests on experimental models and developments          of theory and
   The magneticresonance is drivenby anac magnetic field               new approaches continue.
1176                                                                                     PROCEEDINGS OF THE IEEE, VOL. 71, NO. 10, OCTOBER 1983

                                                                                       May 1, 1977.
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       H. Marc, “Navigation, the government and industry: An ancient                   490, Nov. 10, 1981.
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