1959 IRE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY 47

Some Properties of the Gravitation Fiele and TheirPossible Application to Space Navigation*

J. C. CROWLEY,t S. S. KOLODKIN,t ANI) A. M. SCHN-EIDERt

Summary-This paper describes the principle of operation of a the gravitational field strength" is obtained. The namenew inertial instrument and the elements of a technique for making

g ( ,, .gmeasurements using this instrument. This device measures certain gravientometer (a condensation of gravitation gra-

spatial properties of the gravitational field, from which the direction dient meter") is proposed for the device which measuresof the vertical and altitude with respect to a nearby heavenly body this quantity.can be obtained. An elementary treatment of the mathematicsestablishes the theoretical foundations for the instrument and also From the vector gradient, information can be obtainedprovides an indication of the effects of various classes of disturbing from which to guide a craft. This information also includesinputs. Some of the problems associated with the reduction to range to the center of a heavenly body, altitude over thepractice of the basic transducer are considered, and gross feasibility body, and direction to the center, i.e., the direction of theis established.

geocentric vertical. It is also possible to separate andmeasure the local value of gravitation and the instaii-

INTRODUCTION taneous acceleration. The success of these methods doesFROM the principle of equivalence as stated in the not depend on maintaining a condition of preferred align-

general theory of relativity, it has been shown that ment established some time earlier, nor must the measure-an accelerometer, which is sensitive to both gravi- ments be made continuously. Therefore, a new geocentric

tation and acceleration, cannot be made to distinguish vertical can be established for each heavenly body en-between these two effects. This has important consequences countered in a trip from one body to another. Further-for inertial navigation, which relies on accelerometer more, a set of measurements at one instant of time providesmeasurements to define the motion of the craft.' These all the data needed to determine an instantaneous set ofconsequences include the inability to erect to the vertical values of the output quantities. The knowledge gained(i.e., to damp the Schuler oscillations) aboard a moving from the use of this fundamentally new inertial instrumentcraft without outside information, and the exponential working in concert with present inertial devices can resultdivergence of errors in the inertial measurement of altitude. in an entirely new approach to the guidance of orbitalIn space applications, accelerometers on orbital vehicles, and space vehicles.ballistic missiles, moon-probes, and other freely-fallingvehicles will read zero. Conventional inertial navigation THREEDIMENSIONAL PROPERTIES OF THE FIELI)systems used oni such vehicles do not obtain any new OF A MASS POINTnavigation information while the craft is in a free-fall To understand the principles of operation of the gravien-condition, hence inertial guidance effectively becomes dead tometer it is necessary to consider the properties of thereckoning after the termination of thrust and/or other gravitational field, in particular, those properties relatingnongravitational forces. to the space rate of change of the gravitational field

This paper shows that it is possible to distinguish gravi- strength, which can be called the "vector gradient of thetation effects from accelerations by purely inertial means. gravitational field strength," or simply, the "vectorThe basis of the method consists of measuring higher- gradient'.2order derivatives of the gravitational field. One way of Let us denote the gravitation potential as a function ofdoing this is to compare the readings of two conventional position by O(x, y, z), the gravitation field strength byaccelerometers separated by a known distance. The G(x, y, z), and the gravitationi gradient by the symbolacceleration of the support vehicle, being common to both do/dR (whose properties are yet to be defined). Weaccelerometers, affects each accelerometer by the same assume the existence of a right-handed rectangularamount, which is subtracted out in the comparison of the Cartesian reference frame xyz against which the field andreadings. Any difference in the readings must therefore its properties may be specified. The symbol R denotesbe due to the difference in the strength of the gravitational the position vector of a point in the field drawn from thefield between the locations occupied by the two accelerom- origin of coordinates. Let G and R represent the magflnitudeeters. From the difference in field strength at two points of G and RI, respectively.separated by a knownn amount, the "vector gradient of The potential f is a scalar function of position; at any

point in space, it is completely determined by a single*Manuscript received by the PGSET, November 14, 1958.t Radio Corp. of America, Burlington, Mass. 2 The vector gradient should not be confused with the gradient1 W. Wrigley, R. B. Woodbury, and J. Hovorka, "Inertial of the (scalar) field as conventionally, used. The term "gravitational

Guidance," Inst. of Aeronautical Sciences, Washington, D. C., field strength" or "gravitation" is used for the latter throughoutS.M.F. Fund Paper No. FF-16; January, 1957. this paper.

48 IRE TRANSACTIONS ON SPACE ELECTRONICS AiVD TELEMlIETRY March

number. The gravitational field strength G is a vector X kfunction. Three numbers, such as G., Gy, G,, specify itsvalue at any point in space. By extension, the space rate _ _ =of change of gravitation dGIdR is a tensor function of x dy &zposition, and nine numbers are required to specify this ox + fzfunction at a point.The nine numbers may be written in matrix form as -V) + i(¢. - c.) + k(&£- &V) 0. (7)

follows: Since each component of a zero vector must be zero, it is

7aGx aGx aGx evident thatax ay az frry = OYX

do aGy aGy aG| (1)X = 0,y (8)dR ax ay az 'pz = fr,

aGx aG, -a2G Hence, the matrix in (4) is symmetric.The previous relations between matrix elements are

Nine numbers are required because each component of general. Consider now a special case. Assume that thethe gravitation vector may undergo separate changes in field results from the presence of a single point of knownan incremental displacement, AR, depending on the direc- mass. It can be shown that only three of the nine elementstion of AR with respect to each of three coordinate axes. are independent, and measurement of an appropriate set

Since by definition of three completely determines the vector gradientfunction. Is there a one-to-one correspondence between

G- -Vt = - + + k+k ), (2) points in space and the vector gradient at those points?It can be shown that for the assumed conditions, the

the components of gravitation may be identified as vector gradient determines position relative to the pointGx = -ox I Gy -fOyl G, - _0Z. (3) mass within one of two symmetrically located points;

= ~, Q = ,-zG. = 4. (~) that is, range and bearing from the field point to the massThe G subscripts indicate the axis of a vector component; point are determined, except that the bearing has anthe 0 subscripts indicate the variable of partial differentia- ambiguity of 1800. To illustrate, let us calculate thetion. elements of the vector gradient for the case where theBy taking partial derivatives of (3), space derivatives of point of mass M is located at the origin, and the field

G can be rewritten as the second space derivatives of -0. point is located at P(x, y, z).Thus, (1) becomes The potential function5 is given by6

dG V -oxy -OX, yM _-'yM(9do fur -IY - Ivz (4) =-2 + = R

dl - -LI> where y is the constant of proportionality appearing inNewton's Law of Universal Gravitation. Taking partial

where the double subscripts indicate two partial differ- derivatives, it is found that the elements of (4) take theentiations. form:The nine elements of the matrix are not all independent. -R2 + 3X2 3xy 3x ]

Laplace's equation, valid for all points of the field not dG-3XyM 2occupied by mass,3 states that dGR -M 3xy -R + 3y 3yz (10)

V)-fXX + vYY + Ozz-O=(5) L 3xz 3yz -R2 + 3Zor in words, the sum of the elements of the principal Inspection of this result shows that three numbers (x, y, z)diagonal of the matrix equals zero. Also for any scalar completely determine the tensor. (Of course R =function4 of position 4, +x2 + y2 + z2, and yM is assumed known.) Conversely,

knowledge of three elements of the matrix, say the elementscurl (V4) = 0 (6)) of the last column, completely determine x, y, and z

or, expnding() into omponen form,except for the ambiguity previously discussed. That is,

3~~~~~~~~~~~~~~~~~~Ibid., p. 53. We use a different sign convention.0. D. Kellogg, "Foundations of Potential Theory," Dover 6 The same expression and the results deduced follow if the

Publications, Inc., New York, N. Y., p. 123; 1953. (Reissue of massive body has spatial extent, provided that its density varies1929 ed.) only as a function of distance from the center; the resulting equa-

IbiSd., p. 76. tions are valid outside the space occupied by the body.

1959 Crowley, et al.: Some Properties of the Gravitation Field 49

the triads (x,, y1, z1), (-x,, -y,, -z1) yield the same USE OF GRAVIENTOMETER TO MiEASURE THE VECTORvalue for each element of the matrix. GRADIENTTake the special case where the field point is on one of Having discussed the vector gradient as a property of

the coordinate axes, say the z axis. Then x = 0, y = 0, the field, the measurement of this quantity, or the elementsand z = R so that the matrix reduces to that characterize it, is now considered. Although it may

-1 0 ol be possible to build a single self-contained instrumentdo R0M that reads one element of the matrix directly, the measure-

dR? R3 O -1 o (I11) ment could be made using two conventional single-axis0 0 21 accelerometers arranged with their sensitive axes parallel

and their cases separated by a distance AR. AssumingMeasurement of the lower right element of this matrix, that the frame joining the two accelerometers is rigid,

given by any linear acceleration of the craft will influence the two

-yMlAl [2](12 accelerometers equally, and a subtraction of the readingzz =3 [2] (12) of one from the reading of the other will contribute zero

output. However, if the resulting difference is not zero,will suffice to determine R, assuming Mk is known. This the difference must be due to AG, the increment in thelast result is precisely what would be obtained in a single- gravitational field across the incremental distance AR.axis case, where scalar quantities instead of vectors can If the sensitive axes of the accelerometers are both parallelbe used. to the x axis, and if the direction of the relative displace-

It may be well to consider the single-axis case briefly ment Al is parallel to the y axis as shown in Fig. 1, thenbefore continuing with the three-dimensional situation. the measurement obtained isThe single-axis results will be useful in the error analyses AG, AGwhich follow. r=AX (18)

THE SINGLE-AXIS CASE

According to Newton's Law of Universal Gravitation,the gravitational field of a mass point is MSTABLE PLATFOR

G = R2 (13) SENSITIVE AXIS OFACCELEROMETER

where the minius sign indicates that the force is oppositethe displacement. By taking the derivative with respect Y

to R, and expressing results in terms of increments insteadof differentials, R

AG _2-yM3lAl? - ~~~~~~(14) .0,

which agrees with (12). Dividing (14) by (13) yields X ~G x

AG -2AR xa(-Ox) AGr reading of uniit 2-reading of unit 1

G R (1)~ay - yAFig. 1-Arrangement of two single-axis accelerometers as a gravien-

Substituting into (1 5) the expression for G given by (13) acceleration iS zero.and solving for R yields

R AG2MzR (16) Similarly, any of the elements of the matrix in (4) canA\G /be determined by orienting the sensitive axes of two

This equation allows us to determine a numerical value accelerometers parallel to the axis designated by the firstfor R if the increment in gravitation AG can be measured subscript, and separating them a distance AR in theover a known incremental distance AR, and -yM is known. direction designated by the second subscript. In view ofSimilarly, eliminating R from (13) and (15), and solving the symmetry of the matrix as expressed by (8), the direc-for G yields tions of the sensitive axes and the direction of the relative

-/ AG 2/3 displacement may be interchanged at will. The combina-G = "--VlM * (17) tion of two separated single-axis accelerometers together2AR ~~~~~~with a means for subtracting the output of one from the

This equation provides a solution for G if AR, AvG, and output of the other to obtain the output of the package'yM are known. constitutes a single-axis gravientometer capable of

50 IRE T'RANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY MIarch

measuring one element of the vector gradient matrix at a SYAXES OF THEMEASUREMENTpoinlt in space. LOCATION ARBITRARY INITIAL ORIENT-

MOVING CRAFTINSTRUMENTATION OF A

THREE-DIMENSIONAL SYSTEM P ( XX,yz)

The preceding formulation suggests how a set of threegravientometers may be used in a three-dimensionalsystem for the determination of range and bearing to anearby massive body. The following assumptions aremade: z

1) The gravitational field is due to a single spherically-symmetric massive body. AXES AT THE BODY

2) This body has been identified and its mass is known PARALLEL TO THEfrom outside information. AXES OF THE CRAFT

3) Enough outside information is available to resolvethe 1800 ambiguity. (It is possible to resolve this ,CENTER OFambiguity with the inertial system itself by acceler- - IVE;ating the vehicle for a short time.) ; //

4) Effects usually associated with the Special Theory \of Relativity are negligible.

Let the three gravientometers be mounted on a rigidrectangular reference system x', y', z', called the "measure-ment system," which is carried by the craft and stabilized Fig. 2-Reference axes at the craft and at the massive body.against rotation of the base in the manner of typicalstable platforms. Let the axes x, y, z at the heavenly z'body be parallel, respectively, to these axes (see Fig. 2).Then it makes no difference which set of axes are used asthe reference for the partial derivatives because -ox, =

-,Ox,, etc. 42As the three elements to be measured, let us select the

elements of the last column in the matrix of (4). A set ofgravientometers oriented as shown in Fig. 3 will accomplishthis.The system will be servoed to rotate the measurement a z'

reference frame x'y'z' about its origin (the xyz frame isrotated similarly about its own origin to maintain paral-lelism) until the xz and yz measurements are nulled, that is,until SAL

PLATFORM3-¢o,= 0, -4v, = 0. (19) IN VEHICLE '

Reference to (10) shows that when this condition is met,

x = 0, y = 0 (20) x

and hence the z and z' axes muist lie along the line joining 1, 2 measure-4zz 3, 4 measure- yz 5, 6 measure- x,the craft and the massive body, as in Fig. 4. Before the Fig. 3-Arrangement of gravientometers in a measurement system.z' axis has been made to point directly to the body, theoutputs of the xz and yz gravientometers provide signalsapproximately proportional to the two components of described previously, and (13) through (17) are valid.angular deviation. These signals can be used as the servo Thus, both range and direction to the center of thecommands to rotate the measurement frame about the heavenly body as wvell as the local value of G can be ob-.x' and y' axes. When the gravientometer signals are tamned.nulled, z' points to the massive body and the reading Essentially the same nulling scheme can be instrumentedsbz of the z'z' gravientometer can be used in a computer with four instead of six accelerometers using the methodwhich determines range by solving (12). By this means shown in Fig. 5. In this method the same three matrixthe problem has been reduced to the single-axis case elements are measured.

19.59 Crowley, et al.: Some Properties of the Gravitation Field 51

z SEPARATING GRAVITATION ANDACCELERATION BY INERTIAL MEANS

In general, the use of gravientometers permits an inde-X ;y. pendent inertial determination of gravitation and accelera-

p Ption. Conventional accelerometers measure the specificforce f which in turn is related to gravitation G and craftinertial acceleration a by the relation

xi zf=G-a. (21)

The vector G is readily found from the gravientometermeasurements. The vector d is then the only unknownin the equation and can be determined.The instrumentation scheme of Fig. 3 has the advantage

that the three components of f are obtained directly,being measured by the set of accelerometers clustered at

I0r - - - the origin.0A ,* | EXTERNAL DISTURBANCES AS SOURCES OF ERROR

The gravientometer has been conceived as operating,' K / -in the presence of a single spherically-symmetric body;

no disturbances or instrument errors have thus far beenx

considered. In this section, the error in vector gradient isFig. 4-Conditions when measuremenit system has been ciulled, derived in the normalized form (E)AG/AG, where AG is

the true value of the increment in gravitation in theZ' direction of the massive body, and (E)A G is the error in

4 its measurement.The measured value of AG is used to calculate range R

and gravitation G according to (16) and (17), respectively.Standard error analysis techniques when applied to theseequations indicate that an error (e)/A G in the measurementof AG causes a corresponding error (E)R in calculated range

STABLE and an error (E)G in calculated gravitation given byPLATFORMIN VEHICLE__ (E-)1R (E)AG (,E)G 2 (-),AGVAZ R 3AG' G 3 AG (22)

Errors in y, M, and AR are neglected.l l These external disturbances logically divide themselvesh1 3 into two categories: effects due to motion and effects due,Ir ,| y' to mass distributions.

Motion Disturbances

12 /A Y "Motion" disturbances can be shown to result fromresidual rotation of the sensor device, but not from

Ai ,translation. If we assume that Newton's Law applies (norelativity effects), then translation including all its deriva-tives will have the same effect on both accelerometers ofthe single-axis-gravientometer since they are connectedby a rigid rod; thus translation ideally does not produce

1, 2 measure-0xa 1, 3 measure-k,y 1, 4 measure-oz2 any (erroneous) reading in the output. On the other hand,Fig. 5-Alternative arrangement of instruments requiringff only rotation will produce a difference reading between the

four accelerometers,two accelerometers. If it iS assumed that both accelerom-eters are rotating at a rate co about an axis perpendicular

It is not essential to rotate the measurement axes; a to their input axes then the maximum difference accelera-scheme which determines range and bearing using in- tion Aa sensed isertially nonrotating axes has been evolved but is not A\a = w2AR. (23)described in this paper.

52 IRE T'RANSA(CTIONS ON SPACE ELECTRONICS AND TELEMETRY March

If this difference acceleration is regarded as a measurement 0o01error (E)AG, then replacing Aa by (E)AG in (23) andddividing by (14) yields

(E)AG co>2R3 co2Ro(R (24) 3 45)1AG 2-yM 2Go IRo

where Ro and Go are the values of R and G, respectively,at the surface of the massive body. This function is plotted z

in Fig. 6 for several values of angular rotation rate. //(Wherever the plotted curves and examples require the :ouse of numerical values, values appropriate to the earth W /are adopted.) .ooooil

i 2 4 10 20 40 10OMass Disturbances NORMALIZED RANGE R

03 R"Mass" disturbing effects are in part concerned with

the distortion of the radial field of a mass point resulting Fig. 6-Effect of residual rotation.from the distributed masses in the vehicle which carriesthe instrumentation. It can be shown that everywhereinside a homogeneous spherical lamina the gravitation dfield strength G resulting from that lamina is zero; ther--- rsame applies for the vector gradient.7 The vector gradientresulting from certain other symmetrical mass distribu- A R MASStions is also zero. In a practical case, it is likely that a large GRAVIENTOMETER POINTportion of the mass of the vehicle would fall into one of

Fig. 7-Pictorial representation of a localized asymmetric massthese categories and would not affect the measurement. point and the gravientometer.A rough approximation of the effect of the remainingmass can be derived by considering a mass point m locateda distance d from the center of the gravientometer along othe accelerometer input axes, as shown in Fig. 7. < R -lThe difference acceleration measured by the two acceler- - CI 0R

ometers due to the presence of the mass is '0-

0R000Aa

d _ym yn(

2dARym (25) E 2ymN 0f<a^ = dR3 (25) 001lAi? M 2

__22

tob eemnd . Thi fucto is plote fo eea

Asfsuming Aa(e )a AG, and normalizin,g with respect to tg0 b ,a:AG by dividing by (14) yields er

dR3~~~~~~~~~-.00001L-(e)AG=cn dR (26raI 0 s00 w000AG Al1 -_R9) DISTANCE (METERS)d

is~~~~~~~<orceo oaie rn \ D R

where M is the mass of the heavenly body whose range is Fig. S-Disturbance effect of nearby point mass source.to be determined. This function is plotted for severalvalues of m in Fig. 8. The curves are calculated for acraft located at the earth's surface; i.e., RnR, assuming that the given body, the disturbing body, and the instru-AR=u 2 meters. ment all lie on the same straight line. Since the distances

Since the disturbance due to the mass of the vehicle to the bodies are much greater than AR, the normalizeditself is fixed relative to the measuring irstruments, its error iseffects cani be calibrated the same way a magnetic compass (-) AG _MD /R ~3

1959 Crowley, et al.: Some Properties of the Gravitation Field 53

radius, is shown plotted in Fig. 9 for both solar and lunareffects. Note that lunar effects are much greater than solar o|effects within the moon's orbit and both are small at the <earth's surface. .01Up to this point, it has been assumed that the body

whose field is to be measured is spherically symmetric. NIn the case of the earth, the shape more closely approxi- a/mates an ellipsoid of revolution. It is interesting to examine 2 001

0the error resulting from the assumption that the earth is z

a homogeneous sphere. The error in vector gradient c lmeasure may be derived from the potential function of an °/ /ellipsoid and that of the equivalent sphere.8 It can be 00oo/shown that maximum error occurs along the polar axis;if given numerically I/

2~~~~~( pvG =0.0079( R) . (28) a:2 2 4 10 20 40AG (28 .001I

NORMALIZED RANGE R

The equatorial error is numerically one half the polarerror. Of course, the effects of ellipticity can be compen- Fig. 9-Maximum error resulting from neglecting the moon orsated in much the same way that one corrects a conven- the sun.

tional inertial navigation system for these effects.

INSTRUMENT ERRORS

The major source of instrument error is in the accelerom-eters themselves. If it is assumed that an absolute un- /certainty, U, exists in measurement of difference . lacceleration, the normalized error in the AG measurement °assuming U _ (E),AG is N l

4

(e)AG = UR (R- (29) a: .00lAG 2GoAR Ro/ 0z

where AG is given by (14). This function is shown plotted | /0in Fig. 10 for various values of U/G0AR. CC.o000_ / 0

INSTRUMENT REQUIREMENTS, DESIGN, AND FEASIBILITY fV-z

The absolute requirements of the instrument in terms aof a specific application are now examined. An application e oooo lis the determination of the altitude of a vehicle which is

C 1 2 4 10 20 40 100in orbit around the earth. From (16) it can be shown that NORMALIZED RANGE R

(E)R =6A_G)R() AG. (30) Fig. 10-Effect of instrument error on the gradient measurement.

It is now stipulated that the instrument must measurealtitude to within 0.35 per cent when R is 9000 kilometers. The total value of AG may be found from (14):This represents approximately 0.1 per cent error in the AG = 1.1 X 10-6Go (32)measurement of R. It is assumed that errors producedfrom sources other than accelerometer error may be A comparison of these figures with those obtainableeliminated through calibration. Further, it is assumed from accelerometers currently available shows that whilethat AR = 10 meters and may be easily maintained at the percentage accuracy, approximately 0.3 per cent,this value to within any required accuracy. Insertion of may be easily realized, the absolute level of the quantitiesthese value.s into (30) yields involved is considerably below that which can be measured

e(AG) = 3.5 X 10-9Go. (31) by current instruments. However, several factors must beconsidered to extrapolate present performance to meet therequirements established for this instrument. These are:

8 E. J. Routh, "Dynamics of a System of Rigid Bodies," Dover 1) Present instruments are almost always designed toPublications, Inc., New York, N. Y., pt. 2, p. 340; 1955. (Reissueof 6th ed. pulblished by The M\acmillanl Co., London, Eng.; 1905.) measure a full scale reading of several "gees." A typical

54 IRE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMIETRY March

example is one designed to measure from 10-4G, to 1OG0. 2) Performance of present instruments is always meas-These instruments may be resealed to shift the maximum ured in at least a one-gee field. In the absence of this onevalue while maintaining the ratio of maximum input to gee, many of the sources of error, such as mass shifts dueminimum detectable signal. For example, if the proof to thermal gradients, convection currenits and shift of themass of a conventional electromagnetically-restrained sensitive axis due to base mounting shifts, would disappear.pendulous accelerometer were to be increased by some From these considerations, it is feasible to predict afactor, n, the acceleration input which produces the error design based upon present practice which will meet thelimit of the restraint will be reduced by the same factor. requirements of this instrument.The effect on the absolute value of the null uncertainty is Another approach to the design of a gravientometer ismuch more difficult to predict. However, the sources of to consider radical departures from the present inistru-this null uncertainty can be divided into two categories mentation. For example if a mass were to be placed atand examined separately to determine their effects. The some distance corresponding to aR from the vehicle,first source is that portion which will be uneffected in observation of the relative motion between the two wouldabsolute magnitude of equivalent torque by this change yield information on the magnitude of the vector gradienlt.in proof mass. These sources include electrical effects, A second example would be to consider the use of thelead-in wires, and stray magnetic fields. With the increase radiant pressure of light beams as a suspension system.in proof mass, these torques will now produce an uncer- Some preliminary calculations indicate such a suspensiontainty in acceleration that is reduced by the same factor to be feasible.that increases the proof mass. The second source must nowinclude all other effects. Because these torques are sensitive ACKNOWLEDGMENTto the magnitude of the proof mass, it can be deduced The authors wish to thank their many coworkers atthat they will be considerably reduced when the instru- RCA who contributed to the thoughts and presentationment is in a free-fall condition. of this report.

The "Proceedings of the 1958 National Symposium on Telemetering" canbe obtained from the IRE, 1 East 79th Street, New York 21, N. Y., at $5.00per copy.

Reprints of individual papers presented at the Symposium are also avail-able at $0.25 per copy.