lacoste and romberg stabilized platform shipboard gravity meter

GEOPHYSICS, VOL. XXII, IUO. 1 (FEBRUARY, 1967), PP. 09-104, 7 1*1(X, 2 TABLESLACOSTE AND ROMBERG STABILIZED PLATFORMSHIPBOARD GRAVITY METERtLUCIEN LACOSTE*, NEAT, CLARKSON*, GEORGE HAMILTON*The LaCoste and Romberg gravity meter designed for operation in gimbals was redesigned Ior satisfactorystabilized platform operation and a suitable stabilized platform was made. A major modification of the gravity meterwas redesign of the suspension to more nearly restrict motion to a single degree of freedom. Errors due to crosscoupling between horizontal and vertical accelerations and an unsuspected type of cross coupling clue to gravitymeter imperfections are important but correctable.The stabilized platform gravity meter has been tested at sea. Compared to the gimbal type of gravity meter, it issimpler to operate; it will operate in rough weather; and it is considerably more accurate.INTRODUCTIONLaCoste and Romberg has recently completedand tested a stabilized platform shipboard (orairplane) gravity meter. Earlier L & R shipboardgravity meters were hung from gimbals and correctionswere made ior the swinging of the gimbals.There were several reasons for changing to astabilized platform. An important reason was theavailability of inertial quality gyros with estimatedlives of l+ years. This is a much longer lifethan the 1000-hour or 7-week lives given in specificationspreviously known to L & R.A second reason for changing to a stabilizedplatform was that experience showed that it wasfrequently desirable to operate gravity meters ataccelerations much greater than the *SO galaccelerations L & R gimbal gravity meters weredesigned for. At greater accelerations, the horizontalacceleration correction (or Browne correction)required with gimbal gravity meters becomesso large that a digital computer would berequired for adequate accuracy. This would sogreatly increase the cost and complexity of thegravity meter that it appeared to be necessary touse a stabilized platform.A third reason is that inertial quality gyroshave more than adequate accuracy for shipboardgravity meter operation while the overall performanceof the long period pendulums used withL & R gimbal supported gravity meters was marginal.The long period pendulums also requiredappreciable field testing and adjusting by operators.Before describing the details of the new modelgravity meter, the requirements for stabilizedplatform operation will be discussed a comparisonbetween these requirements antI those for agimbal gravity meter will be made; and themethod of satisfying the requirements will begiven.THE STABILIZED PLATFORMIn order to determine what accuracy is requiredin the stabilized platform, it is necessary to findout how the gravity meter reading is affected byerrors in verticality (LaCoste, lY5Ya1. This can bedone by a consideration of Figure 1, which showsa gravity meter on a stabilized 1)latform. Thesensitive axis of the gravity meter is shown to beoff vertical by an angle e. The error in measuringgravity is thene, = g cos e - ah sin e ~- g. (I)when g= gravity and ah= the horizontal accelerationin the direction of the error angle e. Equation(1) can be approximated bye, G - ahe - ge”/‘2. (2)The last term in (2) is the static error thatexists even when there is no motion. For a onemgal accuracy, the second term requires a verticalaccuracy of 5 minutes, which is not lrard to attain.f Portions of this paper were presented at the 1965 Regional SEG meeting in Bakersfield, Calif., and at the 1966Regional SEG meeting in Long Beach, Calif. Manuscript received by the editor 22 July 1966.* LaCoste & Romberg, Inc., Austin, Texas99

Stabilized Platform Shipboard Gravimeter-Design 101IIGIMBALJOINT0 =-ah/g- GRAVITY METER.g =ga cose=g, -&3IACCELERATIONg’=g,-j--(Q - e)2Error=g’-g=g0e-ge2 .r=-ahe-g$FIG. 2. Gravity meter suspended by gimbals.Thereforeg = g, cos 8 h g, - g02/2. (3)The last term in (3) is known as the Browne correctionor horizontal acceleration correction.If the periods of the horizontal accelerations arenot long compared to the natural period of thegimbal suspension, then (3) does not in generalhold. However it can be shown to hold for allperiods if the gravity meter is placed at the correctdistance from the gimbal joint and if averagevalues are used (LaCoste, 1959b). This correctdistance is the length of a simple pentlulum whichhas the same period as the gimbal suspendedgravity meter. In order to show that equation (3)holds in this case, the gimbal suspended gravitymeter is treated as an undamped pendulum subjectedto a sinusoidal horizontal accclcration or toa Fourier series representing an arbitrary horizontalacceleration. The approximations are alsomade that the average values of sin? 8 and cos2 0are f and the average values of sin B, cos 13 cos 0,

102 LaCoste, Clarkson, and Hamiltonand cos 0 sin 0 are 0. These are the average valuesior an integral number of cycles and thereforethey are very close approximations for any timeinterval of many cycles duration.In order to compute the Hrowne correctiongiven in (3) it is necessary to measure the value of0, and this measurement rcquircs a stabilizedreference If there is an error e in the stabilizedreference, then the measured value of 0 will beO-e and the measured value of gravity will beg’ = g, - g (0 - e)“/2. (4)The error in g can then be obtained from (3) and(4); it ise, = g’ - g = g& - ge”/2. (5’1Equation (5) can be simplified by making use ofthe differential equation of the gimbal suspendedgravity meter. It iswhere I= the length of a simple pendulum havingthe same period as the gimbal suspended instrument.Equation (6) can be written as(6)-go = &, + Iti. (7)The right side of (7) however is the horizontalacceleration a’h at the gravity meter which is adistance 1 below the gimbal joint. Therefore (5)becomesey = - aA’e - (@)/2. (8)If equations (8) and (2) are compared and it isnoted that the acceleration at the gravity meter in(2) is denoted by ah and in (8) it is denoted bya’h, it can be seen that the two equations are identical.The gimbal supported gravity meter istherefore just as sensitive to errors in its stabilizedreference as the stabilized platform mountedgravity meter is. There is no preference in thisrespect. Also it is obvious that gyros and accelerometerscould be substituted for the long periodpendulums in the gimbal system, which wouldresult in an improvement in performance.In order to attain the vertical accuracy requiredby equation (2), a satisfactory gyro erection systemhad to be designed for the stabilized platform.A diagram of the erection system is included inthe block diagram of Figure 3. There are twogyros, only one of which is shown. Each gyrocontrols its torque motor to make the stabilizedplatform follow the gyro. This however, is notsufficient to insure verticality of a reference lineon the platform because: (1) the reference linemight not be vertical to begin with; (2) gyros havesome drift; and (3j the earth rotates and gyrostend to remain fixed in space. In order to attainverticality, accelerometers or levels are mountedon the platform as shown. Ii the accelerometers donot indicate level, they put out. error signals. Eacherror signal plus a constant times its integral is fedto the corresponding gyro to gradually precess itto bring the reference line on the platiorm tovertical.One of the main reasons ior using an integralterm in the feedback from the accelerometers isthat it eliminates an error that woul~l otherwise bepresent because of the rotation of the earth. Theearth’s rotation requires an equal precession rateof the gyro in order to keep the platform vertical.This precession rate would require a constant errorsignal from the acceleromctcr if there were nointegral term in the feedback.The use of both error and integral of error in thefeedback gives a second order differential equation,and therefore the stabilized platform behavesexactly like a long period damped pendulum.The amount of integral feedback determinesthe natural period of oscillation of the stabilizedplatform and the amount of ordinary feedbackdetermines its damping.The L & R gimbal supported shipboard gravitymeters use long-period pendulums as verticalreferences; so again it is found that there is mathematicalequivalence between the old gimbal andnew stabilized platform meters. In the old modelssatisfactory results were obtained with pendulumsof about two minute periods. If the periods are tooshort, errors are introduced because of fishtailingof the ship and because of long period wave motions.If the periods are too long, the long-periodpendulums tend to drift, and transients introducedafter a ship turn became objectionable.Because of the greater stability of gyros ascompared to the long-period pendulums, it ispossible to use periods much longer than the previouslyused 2-minute periods. It is also possible toreduce the time of objectionable transients after aship turn by providing slewing switches to manuallybring the platform to approximate verticalafter the turn. Such slewing switches are providedin the new model and periods of either 4 or 6 min-

Stabilized Platform Shipboard Gravimeter-Design1031. GRAVITY METER UNIT2. STABLE TABLE3. GYRO4. HORIZONTAL ACCELEROMETERAUTOMATIC READERGRAVITY C~OMPUTERrSf? TENSIONCONTROLr-z2 CROSS: COUPLINGCOMPUTER5aiti$s ;-01r % 00 a00 w0aRDER1tI -SHORT PERIODFIG. 3. Block diagram of stabilized platform gravity meter.utes are made available merely by turning aselector switch.Since Z-minute periods were satisfactory in thepast, it was thought that 4-and 6.minute periodswould certainly be adequate if the ship’s trackwas reasonably straight. Furthermore, the availabilityof both 4- and 6-minute periods made possiblea test of the adequacy of the periods. If thesame values of gravity are obtained with bothperiods, then either period is adequate. Actual seatests showed the periods were adequate.Another requirement of the stabilized platformis that it be shock mounted in order to filter outship vibrations that might cause resonances in thegravity meter unit itself. Shockmounting a gimbalsupported gravity meter is simple because the

104 LaCoste, Clarkson, and Hamiltongimbal suspension itself filters out horizontalvibrations and there is no problem in adding aspring or shockcords to filter out vertical vibrations.However shockmounting a stabilized platformcauses some difficulties because the servomotors controlling the platform need to bemounted on a firm base in order to act quickly. Ashockmounted base makes them act sluggishlyand causes them to hunt.In order to overcome this problem, parallellinkages were installed between the shockmountedbase and the supporting frame fixed to the deck ofthe ship. These linkages are shown in the photographof the stabilized platform gravity meter,Figure 4. The linkages allow translation of theshockmounted base in any direction but preventrotation of it relative to the ship. The base thereforeprovides a good support for the servo motors.FIG. 4. Stabilized platform gravity meter.

106 LaCoste, Clarkson, and HamiltonFIG. 5. Diagram showinginherent typeof cross coupling.withstanding rougher handling without gettingout of adjustment.A third improvement in the new gravity meterwas to increase its damping to such an extent thatit can withstand vertical accelerations of f 0.5g ata period of 7 set without having interference betweenfixed and movable parts of the ‘gravitymeter. At a period of 3.5 set, the gravity meterwill withstand vertical accelerations of t- lg. Thedamping was increased by a factor of 2 or 3 overthat of the earlier model.The increased damping also decreased theordinary cross coupling effect by the same factorbecause it reduced the motion of the beam by thatfactor.A fourth improvement was design and adjustmentchanges to reduce certain types of crosscoupling error that have not been considered todate. In order to understand what was done, adiscussion of the cross coupling problem will begiven.One type of cross coupling has been known(LaCoste and Harrison, 1961) for several years.Although it has been described in the literature itis probably worthwhile to give an example of howit can cause an error in a gravity meter. Figure 5 isa diagram of a gravity meter being moved in acircular path in a vertical plane. If the gravitymeter is highly damped, its beam deflection willbe proportional to the negative of the verticalvelocity to which the gravity meter is subjected.The positions of the gravity meter beam willtherefore be as shown in the figure. The beam isassumed to be horizontal at the top and bottomwhere the vertical velocity is zero.The effect of the horizontal accelerations willnow be considered. At the right there is a centrifugalforce to the right, and since the gravity meterweight is above center, this force will produce aclockwise torque. At the left the centrifugal forceis to the left; but the weight is below center, andtherefore the torque is again clockwise. Consequentlythe torque does not average out butmakes the beam move in a clockwise direction. Ifthe direction of the circular motion is reversed,the error will also be reversed.This cross coupling error is present even if thegravity meter is not overdamped, but in this casethe error occurs with a ramp motion rather thanwith a circular motion. The error can be eliminatedby keeping the beam accurately nulled orby using a gravity meter that has symmetryabout a vertical axis; otherwise the error is inherentin a stabilized platform gravity meter. Across coupling computer was therefore providedwith the new model gravity meter; it is shown inthe block diagram of Figure 3. It receives inputsof beam position and horizontal acceleration alongthe beam and it computes their product, which isa measure of the required inherent cross couplingcorrection. The output appears on the recorder sothat it can be corrected for. The correction canalso be made automatically to the gravity readingin the automatic reader if desired.It is interesting to note that a gimbal supportedgravity meter is not subject to cross couplingerrors if the gravity meter is placed at the correctdistance below the gimbal joint. This “correctdistance” is the same distance that makes equation(3) correct for all periods. This adjustmentavoids cross coupling errors by eliminating allforces on the gravity meter except a force along itssensitive axis (LaCoste, 195Yb). The absence ofcross coupling effects in gimbal supported gravitymeters is a real advantage over stabilized platformgravity meters, but the cross coupling effectis usually less than 20 mg and therefore a simplecross coupling computer can make adequatecorrections.The previously described cross coupling effectwill be called the inherent type, because it is inherentin certain types of gravity meters. Thetypes of cross coupling about to be described aredue to imperfections in the gravity meter and

Stabilized Platform Shipboard Gravimeter-Design 107Ayj-\_ 3-at(LAverage Measured Gravity= (l/z)(L-Ai)(g-a)+(L+AL)(g+a1= Lg+ALaFIG. 6. Diagram showing an imperfection type of cross coupling.therefore will be called imperfection types. Theyoften have even greater magnitudes than theinherent type. On the other hand it is possible tomake the imperfection type negligible by carefuldesign and adjustment of the gravity meter.There are many ways in which imperfections cancause cross coupling errors, and it is often difficultto determine which is the major cause of error. Inorder to show how they can cause errors, twoexamples will be given.Figure 6 shows a gravity meter with a hingedbeam being moved up and down a r-amp. Therewill always be some elasticity in the Ilearn and inthe wires providing a hinge for it. This elasticitywill allow the center of gravity of t hc beam tomove horizontally in response to horizontal accelerations,thereby changing the moment arm ofthe beam. The moment arm on the right of thefigure can be denoted by L+AL and on the left itcan be denoted by L-AL. Also because of verticalacceleration the vertical force on the I)eam at theright will be g+a, where a is the vertical accelera-

Stabilized Platform Shipboard Gravimeter-Design 109Table 2. Main features of new L & Rshipboard gravity meter unit1. Reduced yielding to lateral accelerations by a factorof 40 compared to earlier model.2. Linearity improved to reduce vertical accelerationerrors td less-than one mgal at vertical accelerationsof *0.1&?.3. Damping increased to permit operation at +O..5g ata 7 set period without interference between fixed andmovable parts.4. Cross coupling computer for correcting inherent typeof cross coupling.5. Imperfection type of cross coupling made negligibleby careful design and adjustment.6. Simpler to operate and maintain than earlier modelNote: The accuracy at sea is discussed in the followingarticle by Nettleton and Lal’ehr.parts is as follows. A servo in the automaticreadercontrols the spring tension S to approximatelynull the beam position B of the gravitymeter. Theservo is made slow so that it will not followwave accelerations very much. Gravity is com-puted in the automatic reader from the springtension and the derivativetheof the beam positionafter some filtering. Gravity is then recorded on astrip chart recorder or tape recorder or both. Formonitoring, strip chart records arc made of thehorizontal accelerations and the Ilearn positionboth with and withoutfiltering.Tests of the new L & R stabilized platformgravity meter have been made b!. GeophysicalAssociates International, by the LT. S. SavalOceanographic Office and by Dr. Roland vonHuene of the Naval Ordnance l‘c,st Station atChina Lake, California. The results of the testsmade by Geophysical Associates internationalappear in the article following this article. Itappears that the new model give-. considerablybetter accuracy than the old model in addition tobeing able to operate in much rouglrcsr weather. Itis also very much simpler to opera t (a and to keepin adjustment.REFERENCESLaCoste, L. J. B., 1959a, Surface ship gravity measurementson the Texas A and M college ship, the“Hidalgo”: Geophysics, v. 24, p. 309-322.---, 1959b, U. S. Patent 2,899,826.LaCoste, L. J. B., 1961, U. S. Patent 2,9i7, 799.LaCoste, L. J. B., and Harrison, J. (‘., 1961, Sometheoretical considerations in the measurement ofgravity at sea: Geophys. Jour. Royal AstronomicalSociety, v. 5, p. 89-103.