Abstract

RECENT GRAVITY GRADIOMETER DEVELOMPENTS Ernest H. Metzger

~ ~ s o s p a c s u m m l U~v~s~on of Textron Inc

Buffalo. New York

Three different instrurnentat~on approaches measuring gravity gradients aboard a moving vehicle are in active develop- ment. The interest in and need for moving base gravity gradient measurements are briefly reviewed. The instrumentation pro- blems associated with real time gravity gradient measurements in a dynamic environment are outlined. A discmion of the develop- ment status of the rotating accelerometer gravity gradiometer covers the nlajor portion of the paper, Included are a description of the approach, a summary of the development problems en- countered, and solutions implemented. A presentation of the latest test data concludes the paper.

1. Introduction

Over the last few years the interest in moving base gravity gradient measurements aboard vehicles such as aircraft, submarines, ships, e tc., has increased significantly. The latest inertial instru- ments, gyros and accelerometers, have attained a performance level where the gravity anomalies are the largest remaining error mechanism in high performance inertial navigation systems. The horizontal gravity anomalies, sometimes also referred to as the deflection of the vertical, impact on inertial navigation systems very much like accelerometer errors. As the vehicle traverses through the anomalous gravity field, the Schuler loop is excited and velocity and position errors are generated which increase with time. One possible solution t o this problem is real time measure- ments of the elements of the gravity gradient tensor, and the derivation of the gravity anomalies from this data for correction of the inertial navigation system.

Another potential application of gravity gradiometers is real time mapping of the components of the gravity anomalres aboard moving vehicles. Present day gravity surveys are conducted with gravimeters which are sensitive to vertical accelerations and altitude and measure only the total gravity vector. The horizontal gravity anomalies have to be extrdcted by post mission analysis using the Mening Veinesz Technique. The addition of gravity gndiometers to existing gravity survey systems will permit high speed, real time surveys of the horizontal and vertical components of gravity aboard the aircraft. A large amount of data about the structure of the earth crust and'topography is of vital interest to exploration organizations and various agencies of DoD.

It is not the intent of this paper to discuss the nature of the gravity anomalies and how the horizontal gravity anomalies impact inertial navigation systems. Nor are integration methods of the gravity gradiometer with other systems part of this paper. €hese subjects have been studied by a number of organizations including Bell, TASC, Draper Labs, and TRW, etc., and the find- ings discussed in the literature or in technical papers presented at previous technical meetings. Some of these reports are included in the list of references.

The findings of these analyses agree that gravity gradio- meters of sufficient accuracy would solve these problems well and effectively.

The standard unit of gravity gradient is called the Eotvos Uoit in honor of Baron Roland von Eotvos who conducted gravity

gradient measurements in the early 1900's with the torsion bar. The Eotvos Unit corresponds to an acceleration difference of 1U9 cmIrsec' per cm of linear distance or approximately IU1 'g at two points I0 cm apart. The required gravity gradient measurement accuracy depends on the specific mission characteristics, vehicle velocity and altitude being important factors. There is general agreement that the gravity gradiometer noise power spectral density should be in the order of 10 EU2 per rad/sec for the high-speed aircraft and thai it can be as high as 4000 EU2 per radlsec for low-speed ship. This corresponds to a randomness of about 1 EU and 20 EU, respectively, as measured through a double-section, t en-second filter. These noise figures include con- tributions from all error mechanisms, instrument self generated as well as noise induced by acceleration, magnetic and tempera- ture sensitivites in conjunctiori with environmental disturbances. The bandwidth over which gravity gradients have to be measured depends on the applications. For correction of inertial navigation systems, the narrow bandwidth about the Schuler frequency of 0.00 1 2 rad/sec is important. For gravity mapping purposes, gravity gradient measurements as high as 1 rad/sec may be required.

Three different gravity gradiometer concepts are presently in active development in the U.S. A tloated sphere is being devel- oped at the C.S. Draper Laboratories by Mr. Tregasser. This ap- proach atttempts to splve the problem by attaining high degree of mass balances, temperature control and material stabilities. The development of Dr. Forward's elastically coupled rotating dumbbells is being pursued at Hughes Research. The succecs of this approach hinges on noise free bearings, a highly isoelastic structure and excellent mass balances and material stabilities. The discussion of these concepts is left to the two respective developers.

The Bell rotating acceleronleter gravity gradionieter dif- fers from the other two approaches in the following aspects:

One line, automatic correction techniques of error mechanisms are implemented rather than resorting to electromagnetic, electromechanical and electronic design improvements beyond the present state of the art.

Only proven instrument concepts are used

Specific force rather than torque measurements are made.

11. Rotating Accelerometer Gravity Gradiometer Concept

Detection of acceleration differences of 1U"g at two points 10 cm apart in the presence of gravity and vehicle accelera- tion at first galance appears t o pose an unsurmountable instru- mentation problem. A typical null output of a high-performancr force rebalance accelerometer such as the Bell Model VII as il- lustrated on Figure l , exhibits a bias in the order of 20 micro g's. a low frequency meandering of a few tenths of a micro g and typically a bias trend in the order of a micro g per day.

De:cction of 1U" g in this mountain of noise millions of times larger does not appear encouraging.

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ACCELEROMETER BIAS - MICRO "g"

1 .O 2.0 3.0 4.0 TlME - HOUR

Figure 1. Typical Accelerometer Null Bias

A power spectral density plot of a long null bias run (Figure 2) reveals that the power spectral density is high at very low frequencies and at higher frequencies of a few cycles per minute the null bias noise approximates the thermal noise level. If the gravity gradient o r higher input information to be measured could be shifted t o a frequency of cycle per minute, then the detection of very low level acceleration inputs becomes a possibil- ity. In fact, at very low frequencies, inertial instruments are thermometers and indicators of aging charaiteristics of materids and hence the high power spectral densities.

0.01 0.1 1 .o 10.0 100.0 CPH

Figure 2. Typical Accelerometer Null Bias Power Spectrum

Mounting four accelerometers to a rotating fixture with their input axis aligned tangentially as shown on Figure 3 ac- complishes this. Each accelerometer exhibits its null bias charac- teristic in the low frequency domain. The linear acclcrations per- pendicular to the rotation axis are modulated at the rotation fre- quency and the gravity gradients are modulated a t twice spin fre- quency becarrse both input axis and the distance from the center of rotation are modulated at the rotation frequency. The desired gravity gradient information has therefore been separated from input acceleratinns and null bias by the spin and twice spin frequency. The outputs of the diametrically opposed accelero- meter (a, ,and a, and (a, and a, ) are summed to reject linear accelerations perpendicular to the rotation axis within the scale factor balance of the two instrumerrts in each pair. Excellent scale factor balance between pairs of accelerometers in each pair is critical to minimize gravity gradiome ter sensitivity t o linear

t4R /WXy) COS 2at

Figure 3. Schematic Illustration of Rotating Fixture in Shake Structure

input acceleration at or near the rotation frequency, since these are modulated by rotation to a twice spin speed signal. The summed output of a second pair of acceleron~etcrs (a, and a4 ) mounted in space quadrature is subtracted from the sum of ( a , and d l , to reject angular acceleration about the spin axis within the scale f x t o r balance of the sumnied pairs. Angular per- t u r t u n ~ ~ * s about the spin axis a! twict' spin spccd are gencrated by the best bearing and drive mechanism to the extent that these are not cancelled by scale factor balance of summed pairs of accelerometers, gravity gradient errors develop.

As a side benefit, four accelerometers multiply the gravity signal measured by one instrument by a factor of four. The difference of the inline gravity gradients are measured at the sin 252 t phase and the cross gradients at the cos 212 t phase as in- dicated by the equations on Figure 3. ?'he orientation of the spin axis of the gravity gradiometer with respect t o thc reference coordinate axes determines which elements of the gravity gradient tensor are measured. Three gravity gradionzeters and the application of the LaPlace relationship are required to measure the six elements of the gravity gradient tensor for derivation of the three orthogonal components of tlie gravity anomalies. -nit. spln axes o f the gravity gradiomcters do not necessarily have to bc aligned with the principal navigation coordinate axes, nor is it essential that the spin axes are mutually perpendicular even through this is a preferred condition for optimum performance. As illustrated on the block diagram, Figure 4, the summed accelero- meter output signals are amplified and demodulated a t sin 2St t and cos 252 t t o yield the inline and cross gravity gradient informa- tion. Square wave demodulators are used a t present. A Data General Eclipse computer has been purchased and the programming carried out to convert to digital sin and cos demodulation to ex- tract the gravity gradient signals. Thi3 will prevent signals at odd harmonics of the gravity gradient frequency from corrupting tlie data.

111. Scale Factor Balance Loops

Online automatic: loops have been imp1ementt.d wi th the rotating acclerometer gravity gradiolnetcr which maintain excellent scale factor balances of acclerometers. They are typical examples of the system techniqi~es applied t o enable gravity gradient measurements with existing i~s t rurnent con- cepts. Three such loops are in operation.

A. Scale Factor Balance Loops for Accelerometers in Each Pair -

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COS at REF

REF SIN COS 2 a t 2Qt

AMPLIFIER

L I

I2 + - DEMOD

-

t SIN WEF

'4 SF ADJ COIL.

G2's' - DEMOD

wxx - w yy 2

GRADIOMETER OUTPUT SIGNALS

Figure 4. GGI Signal Process Block Diagram Including Scale Factor Balance L o o p

The rejection of vehicle axelera tion pc.riwndicular tu the spin axis is proportional to the dcgrec of scdle factor balance between acceleromt*tcrs In euch pair 3s exprcsscd by equation 111-1.

A1 7 K I . AS4 K are scale Factor unbalanucs between accclerorneters denoted by sub- scripts

is base aicelcration perpendicular to spin axis

is spin frequency

is the sunirned output produced by scale factor unbalances and acceler- ations perpendicular to spin akis.

Base input accelerations perpendicular t o the spin axis at or near the rotation frequency will generate acceleration outputs at the gradiorneter frequency and hence an out put error. ExceHent scale factor balances in the order of 1 0-a g/g are desirable to tolerate a reasonable level ol' random base input accclerations. Expcnence with inertial grade accelerometers has shown that accelerometer scale factor adjustments t o much better than lo-" g/g are difficult to maintain over prolonged periods of time because of aging characteristics and temperature coefficients of materials. Fortunately, the gravity gradiometer output infonns us continuously as to the state of scale factor unbalances by signals at the spin frequency (rather than the gradient signal at twice spin frequency) proportional to the component of gravity perpendicular to the spin axis. The signals at the spin frequency are effectively used to maintain the scalc factors balanced to a high degree of accuracy, online and automatically, as illustrated by block diagram Figure 4. The gravity gradiorneter signal is demodulated at the sin and cos phase of the spin frequency, ~2 the resultant signals integrated and a current injected into coils wound about the permanent magnets of the Model VII acceleron~eters. These currents strengthen or weaken the accelerometer magnetic fields of one accelerometer in each pair until scale factor balances are obtained. The scale factor balance loops are operating satisfactorily and have substantially contributed t o the development progress. The reaction time of these loops is a fraction of an hour from a cold s tar t and they track

scale factor changes with temperature. The gravity gradiometel- is not temperature controlled beyond the + 2 O F environment provided by the laboratory. Test data indicates that scale factor balances in the ordel of' 10-a g/g are being maintained.

B. Scale Factor Balance Loops Betwe~n Summed Pairs of Accelerometers

I

Angular accclcration about the spin axis o r near twice rotation frequency result in an output signill at the gravity gradicnt frequency ( 2 S 2 ) proportional tu the difference of the summed scale factors of pairs of' accelerometers as expressed by equation 111-2,

Where (L:1 , -2 ;34)Ki istl~edifl'crenceofthesummedscale factors of summed pairs of accelero- meter proofmasses t o spin axis

is the radial distance of accelerometer proofmasses to spin axis

is the angular acceleratio~i about the spin axis

is the sunin~ed ou tpu t proportional to angular accelerations a b ~ u t spin axis and differences in scale factors of sunmed accelerometers.

Angular accclerations at twice spin speed, measured by a pair of accelerometers as lugli as lo-' g, have been observed. A scale factor balance L.' summed pairs of accelerometers in the order of t 0-6 g/g is therefore required. Scale f'actor balance of scale factors of surn~ned pairs of acceleronietcrs is achieved by an additional control loop. An angular oscillation at a frequency non-harmonically rehted to the spin speed is imposed about the s ; ~ n axis through the torque motor. Tile summed accelero- meter signal is deniodulated a t that frequency, subsequently ~ntcgratcd and a currcnt injected into the scale factor adjust- ment of a third accelerometer imtil the desired balance is achieved, as illustrated on block diagram Figure 3. In effect, the scale factors of three accelerometers are continuously ad- justed t o that of the fourth. The time constant of the third scale factor balance loop is longer than the other two t o mini- mize interaction. This scale factor balance loop is operating t o better than g/g and imperfection in the bearing drive system achievable with state-of-the-art components are of no concern.

IV. Mudcl VII Accelerometer

Four Bell Model VI1 accelerometers are the basic sensing m i t s for the rotating accelerometer gravity gradiorneter, and a short description o f the Model VI1 operating features and charac- teristics are in order for a cfiswssion of pertinent error mechanism tor this application. Figure 5 sliows a cross-sectional view of the standard Model VII. The aluminum spool proof mass is s~~ppo r t ed by a flexure suspe~ision system perpendicular to the input (or sensitive) axis. The input axis is parallel to the longitudinal axis of the conical Alnico magnet in the dkection of maximum com- pliance of the suspension system. The position of the proof mass is detected by a ciifferential capacitive bridge formed by capacitive pickoff rings on either side of the proof mass spool ends. When the accelxotneter is accelerated along the input axis, the proof mass has a tendency t o deflect about the output axis (in line with Newton's law of motion) and unbalance the capacitive bridge. The proof mass pickoff signal is amplified, demodulated and used

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MAGNET MAGNET HOUSING

PROOF MASS

SPRING CLAMPS

- PEN DU LO US ' d' AXIS

Figure 5. Model VII Accelcrorneter

MODEL VII G PROOF MASS PICK OFF

EXCITATION ClRCUlTS

Figure 6. Model V11 C Constrainment Loop

A. Thermal Brownian Noist.

Table 1. Comparjson of Standard Model VI I and Gravity Gradiorneter Model V I I Parameters

----- - A - -- Gradlometer

Stantldi ct Vers~on Pdrameter Un 17s

-- -- - - - Model V I I Model VI I G - -- -- - - --

Torque r n a l q 0 23 30 Conc;tant

Pendulos~ty tlyn cmig 1 50 1200

Spr 1r1 y qirad 3 2 0 4 Cons tan t

VISCOUS dy n cmiratllsec 3 50 .,'a 0 1 Damp~ng

Moment of grm cm2 0 14 1 1 Inertia

1 orque Co~l ohms 300 1 G Res~stance

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B. Effect ot h e n Order h o r Coe t t~c~en t s 01 Model V l l Un Gravity Gradiometer Performance

With an off vertical horizontal rotation axis orientation, Model VII a , ~ *i(rometers input and output axis are tumbled at sin S2t or cos it ,%-ith respect t o gravity. This generates output errors at twice spin frequency through the even order error ro- efficients of the instruments. These even order error coefficients have been improved over the years to the level of a few micro gjg2 where they are of little concern for most applications. For the gravity gradiometer application, even order error coefficients of this magnitude will produce a bias error of many thousands of EU, and an undesirable high sensitivity to base input accelerations. The mechanisms contributing t o the even order error coefficicnts are well understood and attempting t o improve them by two to three orders of magnitude is deemed mrealistic and unlikely to be successful. Tecliniclues t o detect the even order coefficients on line and methods t o introduce signals to the acceleronietcrs con- tinuously t o reduce or null these coefficients are being developed and indications are that these will be successful.

The performance of a force rebalance accelerometer can be expressed by its model equation, an abbreviated version of which is given by equation IV-2. This model equation is useful in dis- cussing this problem and also describing the selected approach to reduce the effect of the even order coefficicnts for the gravity gradiometer application.

where

aind

'out

ai

a P

K1

KO

indicated acceleraticm

accelerometer o ~ ~ t p u t (current 1

applied acceleration parallel t o the input axis

applied acceleration parallel t o the output axis

applied acceleration parallel t o the pendulous axis

scale factor (output units per g )

Bias (g)

Kz . K4, KS , K6, K7 , K B = second order error coefficients (gfg2 )

Substituting the appropriate input and output accelerations for each of the four accelerometers with the horizontal rotation axis orientation results in gravity gradiometer errors as given by equations IV-3 and IV4

Error introduced by even order error coefficients K, , Ks and K6 are:

E E U in-line channel =

E E U in-line channel and EEU cross channel are the gradiome ter output errors in response to rven order error coeffi- cients K 2 , Ks and K, in EU's

ZTK2 , ZTKs and ZTK6 are the algebraic sum of the even order coefficients of the four accelerometers in g/g2

g is gravity

q v and all+ arc tltc low frequency base accelerations per- pendicular t o the rotation axis along the vertical and hori- zon tal, respectively

GW is the conversion factor from g to EU (4 x IU" gJEU)

To reduce the even order error coefficients, it is necessary to measure them and then to have sonie means to reduce or null theni. The detcction of the appropriate sunimation of cven order error uoel't'icients is zlchicvcd by shilking the gravity gradiometor along the vertical a t a discretc f rcq~~cncy I~arrnonicaily urirelalccl t o rotation I'requc.ncy. 'flils yrclds tlic c v e ~ orc1t.r crror coet'ficicnts clctcction signals as given by equations IV-5 and V1-6.

Correction signals for rven order error coefficients K , , K g and K6 are:

a, I S thc amplituctc of' the shake parallel t o gravity ( in g )

All other terms have heen defi~wd previously.

By dernodulation, the summed acceleronleter output signals at sin a r t followed by demodulation at sin 2 a t , cos 2 a t , the de- sired sunimation of even order coefficients art. det ic ted. I t is m e n tial that the shaker mecltanism introduce "purely" linear motion wi t l~out generating angular rates; (rate}2 inputs about the gravity gradionleter input axis produce gradiometer output errors.

The even order error coefficient K, , solnetimes called second order linearity, is equivalent to a scalc factor change pro- portional to the input acceleration. The scale factor adjustment coils wound about the Alnico V lnagnets fill a second purpose. A current proportional to the output of accelerometer (a l ) is in- jected into its scale factor aciji~stment coil until the detection signal dven by equation IV-5 is nulled. The cven order error coefficient K, is equivalent to a scale factor cllangc proportional to the ac- celeration along the output axis of an accelerometer. This informa- tion is available from an orthogonal accelerometer mounted on the gravity gradiometer. A current proportionat t o the output of ac- celerometer (a,) or (a , ) is injected into the scale factor adjustment coil o f accelerom~ter (a , ) until the even order error coefficient signal expressed by equation IV-6 is nulled. These even order error coefficients correction loops are operating very well s t present with manual gain adjustment. No difficulty has been experienced in

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im-rnnentxl ;i;l!::<tments oi t O - ' O g l ?? . i i o w e ~ er- ;IU t 0111;111~ on - line ajustments are required fo r long-term stability of t h e adJust- ments. The automatic adjustment loops are. ~IIustrated by bloc-h diagram Figure 6 and will be set up In the near future.

A total o f nine ( 9 ) on-line and a u t o l i ~ a t ~ c ~ ' O I I ~ ~ ~ I I ~ ~ ~ I O I I

loops are being planned for scale factor a d j u s t m c ~ ~ t t ' ~ t ' ~ ~ or1lc.r error coefficient c o ~ l i ~ ~ r n s a t ~ o n and s e n s ~ t i v ~ axls dllgtlnlen ts. 1 !!c four loops not discussed il: t h ~ s paper arc. slrnilar to thc tlvC clt8-

scribed here but control c o n l l x ~ i a t ~ o n o1 the evc.~i order crws- co:lpIing coefficrents I(, and K, and ~ n p u t axi5 al~gnmrtnts.

V. Development S t a t ~ ~ s of Rotating Accelerometer Gravity Gracltomctcr

A feasibility muclcl of' tltc rotat~lig ;~ccclcronlctcr g r ~ k l t ) grad,orneter has been under rjevelopmtb~~t for SAMSO. USA1 s ~ n c c Aprd o f 1974. The primary ot;ject~vc of the p r o g r m is t o d t m o n - strate that moving base gravity gradient mc.as~lrc.nicnts can be made in the laboratory t o an accuracy ot 1 1:LI In a ten-wconrl measurement period. Only proven ins t rument~ t ion concepts and n o ma term1 stability requircnleli t s and tolcranc~c\ hcyond t l ~ e prcQ- sent state of the art hiivr b w n 114clil. I t i \ d f'irm C ~ I I ~ I I ~ I ~ I I ~ L ~ I ~ ~ to adhere tu this guiclel~nr In the t i ~ t u r ~ I. lc.u~h~llt y 111 ~ r l c o r p o r ~ ~ t ~ t ~ g tnocl~fications a\ ~ n c l l c ~ t ~ i l by ~~lvt. \ t~g,itr\ t+ t t b 4 f \ c t ~ l ~ l ,111dy\i\ I \

stressed in the t l c s ~ p at t h ~ \ t1111c r-atlicr 111,111 ~ 1 . 1 1 1 \ l / c * . ~ r l ( l M C ' I ~ I I I

Excellent progress lids Iwcn ~ n ~ t l t - towarcl r e ~ c h ~ r i p thc goal of acquiring a pract~cal gravity gradlometer suitahlc lor opt-rational appl~cat ions . The c o n f i d e n ~ e level that t h ~ ' ~ I ~ J C C ~ I V C will he achieved has increased with t h v conscciit~vt~ rnllcstoncs mt-t.

A cutaway d~agrdm ot the rotdt~ng L~~cc l t ' r c )~ncb tc r grab~ty gradiometer is illustrated o n F~gi i re 7 !-our Moclel VII accclcr- omrters are mounted t o a square instrument block and ~ n d ~ v ~ d u a l l y shielded magneticdlly. The instrument block I, s ~ ~ p p o r t e c l on either side by hydros ta t~c gas branng\ The r t l td t~ng ~.It.ctrunrcs. consisting of the cons t ra~nment loop\ tor the four a c ~ ~ ~ l c m l n e t c r \ , the surnmlng amp~~;' i .*r tor the four acc.eleronlett.r currt'nts dncl compensat~on loop t.le,-tron~cs, are packdgtnd on vector bodrds w ~ t h every component acces ~ b l e tor rnon~tonng signal5 as rrqutrcd by investigat~ve tests.

The instrument block 1s clnven by an eclcly current mot or, and the rotation speed controlled by the s ~ ~ m n e c l uirtput ot LI

pair of the accelerometcrs and a phase lock loop using 2' lnarkers as seen o n the right-hand side of the cutaway diagrani. Commercial slip ring assemblies o n elther of these bring out the signals for gravity gradlent measurenlrnts and connect powtlr for the ~ n s t r u - ment block dssembly and command signals for the correction loops. Derivat~on of the correction loop c o n m a n d signals and the extractions ot' the gravity gradlent signals is carried ou t by ,malog demodulation with electronics housed 1rl standard electronics racks The cylindrical gravrty gradiomet t r assembly is suspencied In an 18 inch c t d x by a shake mechanism using linear magntltic actua- tors and linear hydrostatic gas bearings. The shake frequency can be varied between 1.0 and 15 H L with the nominal value ct'3.C H7

a t a n amplitude of 0: 20 g.

n* I . IIU~UL~U.... -I--+.;* .-

Rotating Shaft Encoder Electronics 7

- Rotating / Mounting Block

Eddy Curren Motor

Rotar Air \,.---//- Bearings (21 L Mnrlel V I I Accelerometer (41

Figure 8. Rotating Accelerometer Gravity Grabiornctcr

- I Iw gravity gradient inducer is locatcd 111 tlie hachgroimcl

of Figure 8. It cons~sts o f a 100 kgrn luad hall w h ~ c h can be tra- versed on arl overhead trolley with an e lezt r~c motor . The post- tion r radout along the track can be read out t o JII x c i ~ r ~ ~ c y o f 0 1 mni fhroi~gtl a microscope scanning a prec~sion steel ri lkr. The gravity gradient inducer ha\ b w n very useful ln est,ihlisli~ng that

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The rotat~rag ,~ccc.lerorrit.tc.r grclt I t y gr.~cI~onlc.tcr t lc\cl~)lj- rnc.111 p r o y r m liar hr.er~ organ1,'cd 10 tackle t h ~ c r l t ~ ~ ~ i l I I ~ I I C \ ~ ~ I I C ~

Figure 9. Linear Motion Inducer

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I . Method of Dsts Recordine arid I h t n Rcctuctinq. Mt1c.11 of the subsequent discussion pertains to performance lev&<t t air1 t ci so far. The standard defii~ition of bias, scale factor o r environ- mental sensitivities apply. However. random noise depends the method of data acquisition and data processing, and an cxplana- tion is required.

All mcasure~nents are made through a c lo~~lde see ti011

ten-second time constant filter. A data point from eac.11 ol' the two gravity gradiorneter channels is recorded every ten sccol~ds. After relording 128 suck data points (approxiniately 7 0 minutes) a bias value. a linear change of bias with tinic ant1 the one sigma randomness about this best straight line is computed with an HP330A desk top computer. This process is continued adto- matically for the length of the run, ~~sua l l y overnight and somc- times for entire weekends. From this data. PSD's are c o n ~ ~ u t ~ d over various record lengths. A considerably morc versitile latn recording and processing system is being set u p at prcscnt ilsing a Data General Eclipse Computer.

2. Non-Rotating, S ta th ia ry Model VII-G Accelcron~cter Noise. The non-rotating stationary noise of four modif~etl Model - VII-G accelerometers is the factor lin1iti:lg pc.rf'ormancc Icwi of the gravity gradiometer, as all other noise so~lrces are acldltlvc. At a demodulation frecluency of 1 i 2 fit (~yi~ ivc l lcn~ to 1 '4 spill q ~ . ~ * t l )

the mea i r ed noise is in the order of 0.8 1. I]. It is a con5i+tall1 measur~*rnc~;t from 20-minute to 20-mlni~tc pvsiod arid is wll~tc 111

value. Thc mechanisms con tributi~ig to this ~ioisc- Icc cl arc Ilst t ~ l i ) i t

Table 2 and thermal iloise is only a snlall contributor to thc total.

Table 2. Model VII G Accelerometer Noise Sources --

Mechanism EU -1 sigma E U ~ per/rad/sec

Theoretical Thermal 0.20 0.41 Constrainmerlt Lcrop Electronics 0.4 1 1.68 Suspension Spring Damping 0.55 3.03 Detectior. Electronics 0.33 1 .09

Total - 0.79 6.21

3 . Rotating Gravity Gradlometer Pertorrnancc - Vertical Spin Axis - Static Base. The vertical spin axis orientation tests conducted with a slight off vertical tilt of' about 20 arc ~nln~ltc*\ to provide a sufficient component cf gravity coupling into the input axes of the accelerometers for operation of the scale factor balance ioops. The spin frequency is 1/4 Hz. The twenty-m~nute itncor- rected averages of a typical day's run for the ~nline and cross gradient o i~ tput channels are illustrated on F I ~ L I S ~ 1 1 . The per- formance is summarized on Table 3. The i~ncorrrctetl bias of under 50 EU includes the gradients induced by equipment in thc labora- tory. The inability to detect a bias tempcraturc cvefficient with a 4 O F laboratory temperature change is indicative of thc sclunt1nes.i of the approacLi. The short term randomness over twenty-minutcl

periods is consistantly under 2 EU for both channels Tlic long period randomness is illustrated by the power spectral dcnsrty plot on Figure 12. The power spectral densrty rises f'rorn 30 t.U2 pCr rad/sec at 0.1 rad/sec to about 180 EU2 rad;'stsc a t the SL. I I~I IC~ frequencies. A statistical earth gravity gradient moclcl a\ cb\- perienced by a ship traveling at I 0 knots and ,.m aircral't t l q ~ n ~ ~t 350 knots at an altitude are also shown for comparison. Even at the present development stage, this is quite adequate for the low- speed sea level case. It must be emphasized that this is only an interim report and that achieving a performance level 01' 10 EU2 per radlsec over the entire frequency regions of interest remains the goal. A number of modifications are being incorporated in the feasibility model at this time and should result in substantial im- provemen ts.

IN-LINE VSA NO CORRECTIONS

CROSS VSA -10

25.28 HOURS

Figure 1 1. Performance Runs - VSA lnline arid Cross Gradient Channels 20 Minute Means

Table 3. Vertical Rotation Axis Performance

Bias (both channels) Under 50 E U € 3 : ~ Temp Coefficient ( l t2 '~ . Temp) Not Detectable Bias Trend Under 0.1 EU/hr Short-term Randomness - 20 Minutes

( 19 sec average) 2 E U Power Spectral Density under 40 €u2/rad/sec

Reaction Time from Cold Start Less Than % hour

EU*/RAD/SEC EU

100,000

10,000

1,000

100

0 0 0 10-4 1 0 ' ~ 0.1

FREQUENCY IN RADISEC

Figure 1'2. Power Spectral Densities - lnline and Cross Gradient in VSA and HSA Orientation 25-Hour Runs

I'ht- grad~c~nts ~nrluced ancl mcas~~rcd t'rom a ma\s t lyby of' thC 100 kg111 Icad ball at a track d~\tance ot 38 cm 16 rccorclecl on I-1gi1rc 13. I t shows thtl cIiar3ctcr1stic patterri 0 1 fllc gr;f~lient+ grtneratccl by a p3int mass travermg along a track.

4. Gravity Gradiometer Performance - Horimntnl Rota- tion Axis - Static Base. At this t i~ne. the even order error co-

p-

efficient compensation loops for the coefficients K , . Ks and K, are operating, but with a manual gain adjust. The gain for these loops is set at the beginning of a run. Thc spin speed is 1 i4 H/ and the automatic scale factor loops are operatins.

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pAPCR!2PEs? = zh*WIMiM

SCALE FACTOR = 7.0 EUIMM 10 SEC TC OUTPUT F l LTER ROTATION AXIS VERTICAL

5. Preliminary .- Acceleration - - - Sensitivities. T o obtain a pre- Liminary assessment, the rotatlng accelerometer grdvi~y ~I~UIUIUL;~ ,

was exposed t o accelerations at the critical frequencies, namely the spin frequency and its harmonics. The results are listed on Table 5 asd are most encouraging. The sensitivity to all harmonics except the first, is under 500 EU/g with no measurable response above the 9th. The sensitivity to the acceleration at all frequencies will be further explored with the correction loops in operation.

Table 5. Sensitivity to Accelerations at Critical Frequencies

Sensitivity t o After Accelerations Adjustment

at EU/g

Figure 13. Mass Fly-By (100 kg -38 cm Track Distance)

Table 4 summarizes the performance with a horizontal spin axis orientation. No meaningful bias data is available because the manual even order error coefficient adjustments were made to null the cross gradient and inline gradient signals at the heginning of a run. The automatic compensation loops which are being imple- mented a t this time will null the even order error coefficients and the gradiometer output will then read bias as well as the earth gradients.

The 450 EU/"C cross gradient temperature coefficient could be associated with the temperature coefficient ol' K, and K , . A substantial in~provement is expected with incorporation 01' these even order coefficient loops. The low temperature coefficient of K6 is surprising.

Tsble 4. Performance - Horizontal Spin Axis Orientation

Bias: lnline Gradient Channel No Meaningful Data Cross Gradientchannel No Meaningful Data Bias Temperature Coefficient: I n Line Gradient Channel Not Detectable Cross Gradient Channel 450 EU/CO Short Term Randomness (20 Minute) Both Channels 1 Sigma Under 2 EU

Power Spectral Density Under 40 ~ ~ * / r a d / s e c

The short-term randomness is as good if not somewhat better than in the vertical spin axis orientation. The twenty-minute means for a five-hour run are illustrated on Figure 14 and the cor- responding PSD plot also shown on Figure 12 . The power spectral density a t the Schuler frequency is in the order of 300 EU2 per radlsec. Much improvement in this data is expected when the auto- matic order error coefficient compensation is implemented.

Figure 14. Performance Run MI", - Inline and Cross Cradien t Channel 1 6 -Minute Aveiages

after Adjustment far Moving Base Operations

la 5,000 E U 2n < so0 3a < 500 4S2 € 500 5!2 < 500 6a 600 752 < 500 852 < 500 9a and above n o measurable

sensitivity

VI. Sununary and Conclusions

The results achieved so far and 1 1 1 ~ progrcss triadc in the developtnent of the rotating accelerometer gravity gradionieter show that the approach has an excellent chance to solve an im- portant and difficult instrunie~itation problem. It p r~mise s to lead t o a very useable device with short reaction time from a :old start, low temperature and acceleration sensitivities and will ac- complish this with instrumentation concepts well wit hiti the pre- sents state of tl,: wt .

Table 6 sumniarizes the accomplishments so far. Table 7 lists the tasks to be accomplishsd in thc future. It is not pre- dictable with certainty when these niilestones will be achieved. Experience has shown that months of investigative testing has been required at times to identify an error mechanism. As in the past, the policy of clear identification of an error mechanism and a positive modification t o reduce or eliminate it will be adhered to.

Table 6. Summary of Accomplishments

Table 7. Tasks to be Accomplished

Short Term Randomness (20 Minutes) Bo th VSA and HSA Low Absolute Bias - VSA Low Bias Temperature Coefficient - VSA

Close Automatic Even Order Compensation Loops Reduce PSD at Schuler Frequency Reduce Power Under PSD Over Frequency Range o f Interest HSA Implement Axral Shake Close Additional Automatic Eden Order and Alignment Loops Verify MovingBase Capability on Linear Motion Inducer Verify Ternperatwe and Magnetic Serlsitivities VSA and HSA

Under 2 E U Under 5 0 EU Under 1 EUI'C

Reasonably Low Power Under PSD Over Frequency of lntecest - VSA Preliminary Moving Base Tests Encouraging HSA 10.019 t o 15 Hz) Preliminary dc Acceleration Tests Encouraging Shake Mechanism with Low Jitter Functioning Jitter Compensation Feasibility Demonstrated Using Gyro Three Automatic Scale Factor Balance Loops Operating Short Reacting Time VSA and HSA Less than 112 Hour

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References

1. Jordan S.K., "Moving-Base Gravity Gradiorneter Surveys and hterpretation", 46th Annual International Meeting Socie ty 5. of Exploration Geophysicists, 27 Oct. 1976.

2. Heller, W.G., Jordan S.K., "Error Analysis of Two New 6. Grediometer-Aided Inertial Navigation Systems", AIAA J . Spacecraft - Vol. 13 No. 6, June 1 976.

- I .

3. Metzger, E.H., Jiricitano, A., "Inertial Navigation Perfor- mance Improvement Using Gravity Gradient Mat.ching Techniques", AIAA Journal of Spacecraft Val. 13 No. 6, June 1976.

Vehicles", International Symposium on Applications of Mrrrinr r . v t j r l r c \ i , E.l!td-lle i n c t l t ~ n t r I 1 1 y Y ! U7Lt

Moritz, H., "Kinematical Geodesy 11", Report No. 165 Dept. of Geodetic Science, Ohio State University, 197 1

Moritz, H., "Lecture Notes on Kinematical Geodesy ", Bolletino Di Geodesia E. Science Affini No. 2, t 975.

Sjogrcn. W . L . , Anderson, J.D., Philhps. R.J., and Trask, D.W., "Gravity Fields", IEEE Transactions on Geoscience Eleo- tronics, Vol. GE-14 No. 3 July 1976.

4. Metzger, E.H., Jiricitano, A., "Analysis of Real Time Mapping of Horizontal Gravity Anomalies Aboard Moving

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