PH I LI PSji

IJ~1

TECHNICAL REVIEW

VOLUME,29, 1968, No. 3/4

The RAMP inertial navigation systemF. Hector'

Ever since travel began man has been faced with the problem of finding his location. Simplealignment with respect to known landmarks such as those of a coastline cannot be used indarkness andfog, and is useless for crossing the oceans. Many attempts have been made tofind a navigation system which is reliable under all circumstances. At the beginning of thiscentury navigation at sea was based upon sextant, magnetic compass and chronometer.In the first decades of this century the gyro-compass and radio-navigation systems wereintroduced. Still more recent are sophisticated navigation systems based on the applicationof inertial forces and making use of gyros and accelerometers. In this paper the RAlvlPsystem, a new inertial navigation system, is described. It uses a pendulum which indicates thechange in the direction of the vertical in a moving vehicle.

Introduetion

The problem of navigation is as old as man. In thebeginning landmarks were used both for land and seanavigation, a method which was hampered by visibilitylimitations in darkness and fog. Later, navigation wascarried out by observation of the sun, the moon andthe stars. Although this method enabled mariners tomake fairly accurate measurements of latitude, theascertainment of longitude remained extremely difficultfor a long time. In 1637 the Government of the UnitedNetherlands awarded a prize to Galileo Galilei for hisproposal of a navigation system. This was based on ob-servations of the eclipses of the satellites of Jupiter, andthe times of these eclipses for the meridian of Venicewere given in tables. The system was ingenious, but com-plicated. The situation became easier when ChristiaanHuygens and others invented the chronometer. Fromthe 13th century onwards the magnetic compass wasused for determining direction. Better results still wereobtained with the gyro-compass which first came intouse at the beginning ofthe 20th century.

Radio navigation systems, introduced at the endof the 'twenties, solved many of the problems of vis-ibility and long-range navigation. Although their con-tribution to the reliability and safety of shippingand air traffic is unquestionable, some of the draw-bac~s of these systems are nowadays becoming more

F. Hector, M. A. Physics, is with the Swedish Philips Co., Ltd,Stockholm.

and more perceptible. Radio navigation is possible onlyin regions where accurate and reliable measurements ofdirections to radio transmitters are feasible or wherelandmarks can be observed by radar. Again, this is notthe case for the greater part of the distance ships andaircraft have to cover when crossing the oceans, andthere can also be difficulties in war-time when trans-mitters can be jammed. Higher speeds of travel havealso resulted in a heavy demand for navigation tech-niques which operate automatically. For this purposemore sophisticated systems have been developed, suchas Doppler radar and automatic star trackers, but eventhese have been found inadequate for the aircraftwhich are now being planned for travel at supersonicspeeds. .'During the last two decades navigation methods

have been evolved whichmake use ofinertial forces andfor this reason have been termed "inertial navigation".These systems have the remarkable feature of being in-dependent, of any external source of reference.(except forgravity) and are therefore independent of the vagariesof the weather and of enemy interference in war time.This paper presents a new inertial navigation system,

developed by the Swedish Philips Co., Ltd, calledthe RAMP system (RAMP standing for Rate andAcceleration Measuring Pendulum). It was developedprimarily as a navigator for airborne vehicles. The gen-eral. specification was written for speedsup to 1000

70 PHILlPS TECHNICAL REVIEW VOLUME 29

nautical miles [1] per hour, corresponding to supersonicconditions (Mach 2). Another requirement was that itcould also be used' during long-duration flights ofabout 9 or 10 hours with an accuracy corresponding toarrival within the terminal area, i.e. within 15 or 20nautical miles, after a transoceanic flight.

Before describing RAMP we shall deal with thefundamental principles of inertial navigation andconsider the most important characteristics of pen-dulums and gyroscopes.

The principle of inertial navigation

Most inertial navigation systems are based on meas-urement and double integration of the linear acceler-ation of the vehicle. In order to be able to determinethe displacement in a horizontal plane, it is necessaryto measure the ·acceleration in two perpendicular hori-zontal directions. For this purpose two accelerome-ters are mounted in a frame, generally called a platform,which is kept in a horizontal position. This means thatfor a vehicle moving on or parallel to the Earth's sur-face, the platform has to be kept perpendicular to thelocal vertical (it should therefore follow the curvatureof the Earth), or a correction should be made for thedeviation.The accelerometers have to meet very severe require-

ments for their horizontal alignment. If they are tiltedwith respect to the horizontal plane, they record an"acceleration" due to gravity which, if no correctionis made, is added to the measured horizontal accelera-tion. As the effect of gravity is continuous and gives amuch greater acceleration than the motion of an air-craft (except for take-off and very sharp turns), thismisalignment may cause considerable errors. If, forinstance, an accelerometer senses 0.1% of the gravityfield, double integration of the corresponding accele-ration results in a position error which, after one minute,will amount to 18rn, and after ten minutes to 1800m.For longer time intervals the errors would be cata-strophic.The local direction ofthe vertical can be indicated by

means of a plumb-bob. This, however, is only possibleunder the condition of no motion. Linear accelerationand circular motions of the pivot will influence the di-rection ofthe plumb-bob. In 1923M. SchuIer [2] provedtheoretically that at the surface of the Earth a pendu-lum device with an oscillation time of 84.4 min willmaintain a vertical indication independent of motion.As this is the principle on which the RAMP systemis based, we shall return to it later.

Measurement ofthe change ofthe vertical

If the direction of the local vertical is ascertained bymeans of a "Schuler tuned" pendulum and the initial

direction of the vertical (at the starting point of thevehicle) can be stored, the accelerometers mentionedabove are no longer necessary. It is evident that thechange of position can be found from the change of thedirection of the vertical (fig. 1) [3]. Since a pendulumsuspended from an axial pivot has only one degree offreedom, full position inforrriation should be obtainedby measuring the change of the vertical in two orthogo-nal planes. Corrections for the rotation of the Earthhave to be applied, and also the appropriate trans-formation from great circles to latitude and longitude.

R

bd----~~ ,

<,-,

"'\ '\,\\\\\\\

~

Fig. I. The change of position of a vehicle moving from a to bcan be found from the change ra of direction of the local vertical.R is the radius of the Earth. The RAMP system is based onthis principle.

The vertical deviates from the radial direction because thegravity field is caused not only by the attraction force, but alsoby the centrifugal force of the rotating Earth. The angle betweenthe vertical and the equatorial plane is known as the astronomicallatitude (jig.2). For position indication, however, the geographicallatitude is required; this is the angle between the equatorial planeand thenormal to the reference ellipsoid ofthe Earth. Thedifferencebetween the two latitudes is called the (local) deflection of thevertical. This deflection varies, but its magnitude seldom exceeds10 seconds of are. In this presentation it will be neglected .:

Pendulums

Aphysical (or compound) pendulum is a rigid bodywith an axial pivot above its centre of gravity. Its oscil-lation time for small deviations is:

T= 2nV Imgh '

(I)

where m is the mass ofthe body, fits moment ofinertia

1968, No. 3/4 lNERTIAL NAVIGATION SYSTEM 71

/////

normal to referenceei/ipsoid

local vertical-/////I/II

centrifugal acceleration vector

surface of the Earthattractionvector reference ellipsoid

gravity vector

/

I/astronomlcal latitude

geographical latitude equatorial plane

Fig. 2. Relationship between the gravity vector and its corn-ponents, and geographical and astronomical latitudes.

about the axis of suspension, h is the distance betweenthe pivot and the centre of gravity and g is the accelera-tion due to gravity.

A special case is the mathematical (or simple) pendu-lum. This consists of a particle of mass m, suspendedfrom a fixed point by a weightless inextensible string.In this case the distance h between the pivot and thecentre of gravity is equal to the length of the string.Therefore I = mhè, and the oscillation time is:

VhT= 2n .

g

A pendulum which is not in motion with respect tothe Earth's surface shows the direction of the localvertical. It can, in fact, be considered as a plumb-bob.When the pivot is part of a vehicle performing anaccelerated horizontal motion the direction will de-viate from the vertical. However, as we have alreadynoted, a "Schuler pendulum", with an oscillation timeof 84.4 minutes, maintains a vertical indication, inde-pendent of the motion of the point of suspension. Onthe other hand, as SchuIer himself emphasized, a purelymechanical pendulum fulfilling this requirement can-not be made. A mathematical pendulum of the oscilla-tion time quoted would have a length h equal to theEarth's radius, whereas for a physical pendulum with afeasible mass, the distance between pivot and centre ofgravity would be a few nanometers, which is of coursequite unrealistic.

This can be shown by considering a pendulum moving in ameridian plane over the surface of the Earth (fig. 3). (The Earthis taken to be a perfect sphere.) This pendulum will continue tocoincide with a vertical line if the angular acceleration of the pen-dulum equals the angular acceleration 6i ofthe radialline. Underthis condition the linear acceleration of the pendulum is Rë andtherefore the torque acting on the pendulum body, due to thisacceleration, is mhRë. To obtain an angular acceleration ë thistorque should be equal to lë. The moment of inertia shouldtherefore be:

1= mhR,

and according to (I) the oscillation time is:

A comparison with equation (2) shows that a rnathernatical pen-dulum with a length equal to the radius of the Earth would meetthe requirement. Putting R = 6.37x 106 m and g = 9.81 m/s",we get T"", 84.4 min. Taking some realistic values for the physi-cal pendulum, m = I kg and [= 10-2 kg m", we obtain h "'"1.6 X 10-9 m.

Fig. 3. fndication of the local vertical with a physical pendulum.R Earth's radius; g latitude; J pivot; 2 centre of gravity; hdistance between 1 and 2.

(2) Although a purely mechanical realization of aSchuIer pendulum is impossible, solutions using serv0-techniques can meet the requirement of a long oscilla-tion time. One of these solutions will now be described.

Electrical feedback

In fig. 4 a pendulum IS shown which is equippedwith an electrical feedback system. Attached to theshaft PAthere is a generator SGp, delivering a signalwhich is fed after amplification to an electrodynamicmagnet system TGp, called a torque generator (or tor-quer). This exerts on the shaft a signal which dependson the applied signal eT.

[1] As usual in navigation the distances in this article are given innautical miles. I nautical mile = 1851.9 m.

[2] M. Schuier, Die Störung von Pendel- und Kreiselapparatendurch die Beschleunigung des Fahrzeuges, Phys. Z. 24,344-350, 1923.

[3] This method is fundamenrally the same as navigation bycelestial observations.

72 PHILIPS TECHNICAL REVIEW VOLUME 29

A pendulum with an electrical feedbacksystem as shown is the principle underly-ing a type of accelerometer called the re-balanced pendulous accelerometer. The gen-erator SGp in this case supplies a signalproportional to the angular deviation ofthe pendulum from the vertical. If the ampli-fication is sufficiently large, any deviationfrom this position is then reduced nearlyto zero by the torque generator: the feed-back system acts as an "artificial spring".The period of oscillation in this case is verysmall. When the pendulum is acceleratedin a horizontal plane, in a direction per-pendicular to the shaft PA, the torque-generator voltage eT required to maintainthe vertical position ("rebalancing") is adirect indication of the acceleration a.Neglecting transient phenomena, a fol-lows from the equation:

giving:eTST = mha,

STa=-eT

mh '

where ST is the sensitivity of the torq uegenerator (newton-metres per volt).

Instead of a generator that delivers a sig-nal proportional to the angular deviation,a type may be used that supplies a signalproportional to the angular acceleration.In this case the feedback brings about anapparent increase ofthe moment of inertiaof the pendulum, which can be shown inthe following way.

The equation ofmotion of the pendulumIS now:

.. ..Ie + mhgi) + SGAsTe = 0,

I

TRAITE~·DE LA LVMIERE.

Ou font expliquées

Lt.! CtZufts dtee !JuiJuyarriveDans la REFLEXION, & dans Ia

REFRACTION.Et particu/ierement

Dans retrang~ REFRACTION

DV CRISTAL D'ISLAN DE.Par C. H. D. Z.

Avec un Vifèo1lrs de J4 CaufoDE LA PESANTEVJl.

~ LEI D E~Chez PIE R RE v AND E 1. A A , Marchand Libraire.

:MDCXC.

where e is the deviation from the vertical, SG is thesensitivity of the signal generator (volts per unit ofangular acceleration), A the amplification of the am-plifier and ST is again the sensitivity of the torquegenerator. From this equation we get the oscillationtime:

1Ir + B

T= 2n /--.mgh

--------------------------------Fig. 4. Schematic diagram of a pendulum with electrical feedback.PA pendulum shaft. m pendulum mass. h length of the pendulum.SGp signal generator. A amplifier. TGp torque generator. er signalfed to TGp. If SGp delivers a signal which is proportional tothe angle of rotation, the device will operate as a rebalancedpendulous accelerometer, with "zero" deflection (and a veryhigh natural frequency). However, if SGp delivers a signalwhich is proportional to the angular acceleration, the oscilla-tion time compared to that of the free pendulum will be in-creased, giving in one special case the "Schuier period".

A

(3)

INERTIAL NAVIGATION SYSTEM 731968, No. 3/4

. DEL APE SAN T EU R. I4-Jplomb pe(~ autant yers Ie bas, qu'elle peferoit vers te ha. ut, fi »demeurant a la mefme diftance du centre de la Terre, elle tour-noit auteur avec autant de viteffe que fait la matiere flui-de. Mais je trouvé par ma Theorie du mouvement Circulaires·qui s'accorde parfairement avec l'experience , qu'un corps·roumant en cercle , Ii on veur que fon effort à s'éloigner ducentre, égale juftement l'effort de fa fimple pefanteur, il fautqu'il faffe chaque tour en autant de temps) qu'un Pendule, dela longueur do demi diametre.de ce cercle, en emploie à faire·deux allées, 11 fàuc done voir en combien de tem ps un pen-è!u1e, de la longueur du demidiamerre de la Terre, feroit ces·deux allées. Ce qui eft aifé par la proprieté connue des pendu-.les , &: par la longueur de celuy qui bat les Secondes) qui eftde 3 pieds Si lignes, mefure de Paris. Et je trouve qu'il fau-droit pöur ces deux vibrations 1heure l+~minutes; en fup-

o pofant ,fuivant l'exaêe dimënûonde Mr. Picard , le demidis-metre de la Terre de 1961~800 pieds de lamefme mefure, Laviteffe done de la matiere fluide , à I'endroit de la furface de laTerre. doit eftre égale à celle d'un corps qui feroitle tour de laTerre dans ce temps de I heure , 14-~minutes. Laquelle vite£-fe eft, à fort peu pres, 17 fois plus grande que ceUed'un pointfous l'Equateur; qui fait Ie rnefme tour, à l'égard des Etoilesfixes, comme on doit le prendre icy, en 1J heures , ~6 ininu-·res, ce qui paroir par la proportion entte ce temps &celuy d'une heure 1+; minutes, qui eft tres pres comme de17 à I.Jefçayqttecette rapidité femblera étrange à quila voudra

... comparer avec les mouvemens qui Ie veient icy parmy nous.·Mais cela ne doit point faire de difficulté j &mefme , par ra-port Ma fphere, ou à la grandeur de la Terre ;eUe.ne 'paroitra'point extraordinaire. Car ft , par exemple , en rega.rdant $'Glebe Terretre ,dcceuxqu'on fait potll:l'ufagede laGe~ra-.; _ . V l- phie ,

The "Schuier period" of 84.4 minutes, whichplays such a prominent part in F. Hector's ar-ticle in this issue, is there shown to be equal to theoscillation time of a simple pendulum whoselength is equal to the radius of the Earth. Thisoscillation time had already [been calculated byChristiaan Huygens in the seventeenth century- witness the page reproduced here from histreatise "Discours de la Cause de la Pesanteur"published at Leyden in 1690.

Huygens was interested in the value of thisoscillation time because it is equal - as he hadshown - to the orbital period of an objectmoving near the surface of the Earth at a speedsuch that the centrifugal force balances thegravitational force. This is the well-known con-dition which holds for Sputnik and the innumer-able subsequent satellites that now circle ourplanet and which, when orbiting at a height of nomore than about 200 km (just outside the regionof atmospheric friction), have in fact an orbitalperiod of about I± hours. It may come as asurprise to many who have followed these satel-lite developments that this basic orbital period foran object circling the Earth had already beencalculated correctly and quite accurately in theseventeenth century.

Huygens can hardly have dreamed of artificialsatellites, nor was he concerned with the naviga-tional significanee of the Schuier period. Hemade these particular calculations in an effortto support a curious theory of gravity. This theory(the "vortex theory") was devised by Descartesin an attempt to do away with the assurnptionof an ac/ion at a distance, on which Newton'stheory of gravity was based and which manyscientists of those times found difficult to accept.

where

is the apparent increase of the moment of inertia. Bymaking B sufficiently large, a pendulum can be madewith an oscillation time equal to the "Schuler period"of 84.4 min (about 5000 s). If the period of the physicalpendulum proper has a reasonable value, e.g. 0.5 s, thiscan be done by an apparent increase of the moment ofinertia by a factor of 108.

The signal generator SGp, which is required to deliver a signalproportional to the angular acceleration, will be discussed later.It may be noted, however, that a generator could be used whichdelivers a signal proportional to the angular velocity (angularrate) or to the angular displacement (as noted above). In thesecases, in order to obtain the required artificial increase of themoment of inertia, the signal should undergo a single or doubledifferentiation.

(4) Gyroscopes

The main part of a gyroscope is a wheel spinning at avery high angular velocity w. The angular velocityabout axes perpendicular to the "spin axis" must bevery low compared to w. The wheel is mounted on ashaft which is supported by a set of gimbals. By apply-ing Newton's laws of motion the fundamental charac-teristic of the gyro can be explained, i.e. its reactionagainst any change of direction of its spin axis. There-fore, if a spinning gyro-wheel is supported by a set of"frictionless" gimbals (fig. 5), with axes going throughthe centre of the wheel, it will maintain its direction inspace, irrespective of movements of the supportingframe.It also follows from Newton's laws that a torque

about one of the gimbal axes results in a rotation about

74 PHILIPS TECHNICAL REVIEW VOLUME29

Fig. 5. Schematic diagram of a free gyroscope. SA spin axis ofthe wheel W, which has an angular velocity os. I and IJ are thegimbal axes.

the other axis. This phenomenon is èalled precession.The directions of rotation and torque are such that thespin momentum of the gyrowheel attempts to alignitself with the torque. The basic equation underlyingthe precession is:

To = He.Here H is the spin momentum of the wheel, that is, itsmoment of inertia fw multiplied by its angular veloci-ty w:

H= [ww.

Furthermore e is the angular displacement about oneof the gimbal axes (e.g. the axis I; è is thus the angu-lar velocity about this axis), and To is the torque aboutthe other axis (IJ).

Gyro with spring restraint

An angular velocity of the gyro about one of thegimbal axes can bemeasured by determining the torqueabout the other axis. This can be done by applying alinear elastic restraint, e.g. a set of springs (fig. 6).The angular displacement cp about the latter axis isnow proportional to the torque To:

To = Ksp,

where K is the spring constant. The gyro now has onlyone degree of freedom; the corresponding gimbal axisis generally called the input axis (IA), the other axis,where the torque is measured, being the output axis(OA). From (5) and (7) it follows that:

H.cp =K e, (8)

and the angular displacement about the output axis isthus proportional to the angular velocity (rate) aboutthe input axis. This rate therefore can be read from acalibrated scale about the output axis. A gyro equippedwith a spring restraint about one of its axes is thereforecalled a rate gyro.

Fig. 6. Schematic diagram of a gyro with a spring restraint aboutone of the axes. The deviation cp about this axis (the outputaxis OA), which can be read from the scale, is now proportionalto the angular velocity é of the gyro-wheel about the otheraxis (the input axis IA). This device is known as a rate gyro.

(5)

In some types of gyro a restraint of another kind is used. Thegyro-gimbal is then cylindrical in shape and immersed in a liquid.The torque given by this viscous restraint is proportional to theangular velocity about the appropriate axis. If now the outputaxis has a viscous restraint, the torque which accompanies anangular velocity about the input axis must be caused by an angu-lar velocity about the output axis. It follows from (5) that in thiscase these velocities are proportional, so that the total displace-ment angle about the output axis is proportional to the integralof the angular velocity (rate) about the input axis. For this reasona gyro equipped with a viscous restraint is called a rate-integratinggyro, or simply an integrating gyro.(6)

Gyro with electrical feedback

Instead of a spring restraint a gyro can be equippedwith an electrical feedback system, consisting of asignal generator, an amplifier and a torque generator.Fig. 7 shows a gyro which has been provided withsuch a feedback system about the output axis. The gen-erator SGo is required to deliver a signal proportionalto the angular displacement cp about this axis.Therefore

(7)!SA

Fig. 7. Schematic diagram of a rate gyro with an electrical feed-back system. SGo generator, delivering a signal proportional to therotation angle cp about the output axis OA. A amplifier. TGotorque generator. In this case also cp is proportional to the angu-lar velocity e about the input axis IA.

1968, No. 3/4~ V. PHILtP~' r.WfllAM

INERTIAL NAVIGATION SYSTEIvfl PENFABRIEKEIt 75

the torque due to the torque generator TGo is alsoproportional to this angle and the feedback system canbe considered as an "artificial spring", the "spring con-stant" being:

where SGO is the sensitivity of the signal generator (involts per radian), A is the amplification of the amplifier,and STO the sensitivity of the torque generator. Thegyro in fig. 7 will have the same characteristics as theone shown in fig. 6: the rotation angle cp about theoutput axis is proportional to the angular velocity ëabout the input axis. In fig. 7 a scale about the outputaxis is not needed; the reading can be obtained bymeasuring the signal fed to the torque generator. If thetorque exerted is proportional to this signal, the readingon the signal meter is proportional to the angular velo-city about the input axis.If the electrical circuit is modified a signal can be

obtained which is proportional to the angular acceler-ation about the input axis. For this purpose an inte-grator is inserted in the feedback loop (fig. 8). In thiscase the torque exerted by the torque generator is:

To = SGoA ~ STol cp dt,

where 0 is the time constant of the integrator. From(5) and (10) it follows that:

Hocp= ----ë.

SGo A STO

Thus the angular displacement about the output axisis proportional to the angular acceleration about theinput axis. The signal from the generator can be used asan indication; after amplification this can be read fromthe meter shown in the figure.

!SA

Fig. 8. By inserting an integrator lilt in the feedback IOQptheangle of rotation rp about the output axis OA, and thus also thesignal read on the meter, becomes proportional to the angularacceleration ë about the input axis1A.

(9)

The RAMP, a combination of a pendulum and a gyroSince a gyro with an integrating feedback system for

one of its axes delivers a signal proportional to the angu-lar acceleration about the other axis, it can be used asa signal generator for artificially increasing the momentof inertia of a pendulum, in accordance with the prin-ciple illustrated in fig. 4. A diagram of this device isshown infig. 9. The outer gimbal ofthe gyro forms thependulum and is therefore loaded with a mass m. Theouter gimbal axis (input axis) serves as pendulum axis,PA. The outer gimbal is equipped with an integrating

(10)

(11)

Fig. 9. Schematic diagram of a Rate and Acceleration MeasuringPendulum (RAMP). The pendulum is formed by the outer gimbalof the gyro, loaded with the mass m: its angle of rotation is e.The output axis OA ofthe gyro has an integrating feedback loopas in fig. 8. The signals D, Eand Fare proportional to ë, e ande respectively (indicated by the bracketed terms), i.e. to theacceleration, the velocity and the displacement of the vehicle.SGp signal generator for the servomotor of the slaved platform.

feedback loop, as in fig. 8. Apart from being used forthe feedback of the gyro, the amplified signal D fromthe generator SGo, which is proportional to the angularacceleration ë about the pendulum axis, is nowalsoused for controlling the torque generator TGp for thisaxis. From (4) and (11) itfollows that the artificial in-, crease of the moment of inertia of the pendulum in thiscase is:

(12)

where STP and STO arethe sensitivities of thetorque gen-

76 PHILlPS TECHNICAL REVIEW VOLUME 29

erators TGp and TGo respectively. By making B suffi-ciently large, the pendulum can be "Schuier tuned":it indicates the vertical, independently of movements ofthe pivots. When mounted in a vehicle moving overthe Earth's surface in a direction perpendicular to thependulum axisPA, the pendulum changes its direction inspace, because of the curvature of the Earth's surface.The angular velocity é is proportional to the horizon-tal velocity of the vehicle (v = RÉJ) and, as indicated in(5), this velocity is also proportional to the torque ex-erted by the torque generator TGo. The signal Eapplied to the torque generator therefore indicates thevehicle's horizontal velocity in the direction perpendicu-lar to the pendulum axis.

As the signals E and D, delivered by the device areproportional to the velocity (rate) and the acceleration,this combination of a pendulum and a gyro is calledRAMP (Rate and Acceleration Measuring Pendulum).By the addition of another integrator, Int', a signal Fis obtained which is proportional to e, and hence tothe displacement of the vehicle in the direction men-tioned. We have thus determined the displacement fromthe change of direction of the vertical without storing

B

Fig. JO. Layout ofa RAMP with slaved platform. The pendulumaxis (input axis of the gyro) is perpendicular to the plane of thefigure. B base. The pendulum is carried on a frame Q on a plat-form P, which follows the pendulum with the aid of the signalgenerator (pick-off) PO, the amplifier Af and the servomotor M.

Fig. 11. Cut-away picture of a floating gyro with a single degree of freedom.

the direction of the vertical at the starting point of thevehicle. We measure directly the change of direction.

In an equipment the RAMP is mounted on a "plat-form", which is not rigidly fixed to the vehicle, but

follows the movements of the pendulum (it is "slaved"to the pendulum). The pendulum thus only has tomake very small movements with respect to this plat-form, which simplifies the construction. The plat-

1968, No. 3/4 INERTIAL NAVIGATION SYSTEM 77

form is therefore mounted in gimbals and equippedwith a servomotor. This motor is controlled by theamplified signal from a generator SGp, shown on theinput axis in fig. 9, which delivers a signal proportionalto the deviation between the directions of the pendulumand the platform.

An approach to a realistic arrangement is shown infig. 10. The signal generator SGp for controlling theplatform has been designed as a differential transformercalled a "pick-off" (PO). An important characteristicof this arrangement is that the closed feedback loopscan be achieved without external connections via slip-rings. By the use of micro-electronics the correspondingelectronic equipment can be built into the RAMP unititself.

The RAMP pendulum bearing is a "compensatedspring" bearing and is therefore "stictionfree ", thusimproving the gyro environment in comparison with aconventional platform. In the present design the gyrois a miniature floating type [41. The wheel and the spin-axis assembly are enclosed in a sealed cylindrical can,

I

which is immersed in a liquid whose density is equal

Fig. 12. Complete pendulum for the RAM P navigation system.

to the effective density ofthe can. The angular momen-tum of the wheel is 105 gcrn-s :". A photograph ofsuch a floating gyro is shown infig. If, andfig. 12 showsa complete RAMP.

As the pendulum and gyro form one unit, assemblyand replacement in the platform is very simple [51. Thealignment of the RAMP is further simplified by thefact that the dimensions are so chosen that clean-roomconditions are not necessary.

The dynamics and the stability of the RAMP withits slaved platform can be studied in a signal-flowdiagram. The transfer functions are then given by theirLaplace transforms. Fig. 13 (on page 78) shows such adiagram, but we shall not go into further detail here.

The generallayout

As navigation req uires the determination of displace-ment in two perpendicular directions, two RAMPs,the X-RAMP and the Y-RAMP, are mounted on aplatform (an "inertial platform") which is slaved toboth of them. As a reference for the heading of theaircraft, a directional gyro, the Z-gyro, is also mountedon the platform. (The input axis of this gyro pointsvertically downwards.) A diagram of the arrangementis shown in fig. 14. In practical use the x-direction,which corresponds to the direction of the pendulum

x SA

y

GA

z

Fig. 14. Arrangement oftwo RA MP units and a gyro as mountedon the platform in a RA MP navigation system. SA spin axes.IA input axes. OA output axes. The input axes correspond to theco-ordinate system x, y, z (cf. fig. 26).

[4) Introduced by C. S. Draper at the Massachusetts InstituteofTechnology.

(5) In a conventional inertial navigation system both the accel-erometers and the gyros have to be carefully aligned.

78 PHILIPS TECHNICAL REVIEW VOLUME 29

Fig. 13. Signal flow diagram in a RAMP with a slaved platform. The transfer functions areindicated by their Laplace transforms. s Laplace operator. mh mass unbalance of the pendu-lum. g gravity acceleration. R radius of the Earth. IR moment of inertia of the RA MP pendu-lum. H gyro angular momentum. Ig moment of inertia of gyro gim bal. Dg gyro viscous damp-ing constant. lp moment of inertia of slaved platform. K; ... K4 constants. F; ... F4 Laplacetransforms of transfer functions of correction networks. D, E, F output signals (cf. fig. 9).

Fig. IS. The complete inertial platform, with two RA MP units and a directional gyro.

1968, No. 3/4 INERTIAL NAVLGATI.ON SYSTEM 79

axis of the X-RAMP, points north and the y-direction(pendulum axis of the Y-RAMP) points east (cf. fig. 26).

Complete freedom of the platform with respect tothe vehicle can be obtained by a three-gimbal system.However, if the attitude of the vehicle with respect tothe platform is such that one gimbal axis coincides withone of the other axes, one degree of angular freedom islost. This is known as "gimbal-lock". I t can be avoidedby introducing a fourth gimbal, which is equipped witha torque generator, controlled by a signal from a gener-ator on the second gimbal. This is a known techniqueand will not be explained here.

A complete RAMP platform is shown in fig. IS. Ablock diagram of the complete navigation equipmentis given in fig. 16. The output signals from the RAMPplatform are proportional to the acceleration and thevelocity in two perpendicular directions (D and E in

computerinertial and other display

platform electronic unitfunctions

I tI

mode control powerunit supply

Fig. 16. Block diagram ofthe RAMP navigation equipment.

figs. 9, 10 and 13); signals giving information about theattitude of the aircraft (heading, roll and pitch) are alsoobtained. These signals are fed to the computer and elec-tronie unit. This unit comprises an analogue computerwhich covers a number of functions: co-ordinate trans-formation (great-circle angular velocities to latitudeand longitude velocities), integration of velocity toposition (in the integrator Int', see figs. 9 and 10), cor-rection-term computations (cf. Appendix) and compu-tations of the velocity vector from its two components.This unit also comprises signal convelters and cir-cuits for bias-setting for the Earth's rotation and corn-pensation of the gyro drift.

The computer and electronic unit is connected to thedisplay unit, which presents latitude, longitude, heading,roll and pitch. A photograph of this unit is shown infig. 17. It is equipped with two controls for settinginitial latitude and longitude before the flight.

The mode control unit serves for the switching ofseveral functions. It contains switches for "Stand-by",

"Navigation" and "Off", and controls to compensatefor the Earth's rotation and gyro drift.

Preparation of the equipment for use starts with switching onthe heating of the gyros to bring the fluid in which the floatinggimbals are immersed to the right density for operation; theheating time is about 20 or 30 minutes. In the meantime theRA M Ps are roughly aligned to enable the platform to find thelocal vertical direction. For this purpose the electronic tuning isswitched to a pendulum period of 3 minutes and afterwards to aperiod of 8.4 minutes. Through heavy damping which is switchedon during this period the platform now finds the vertical. Whenthe amplitude of the oscillations is sufficiently small, the Schuierperiod of 84.4 minutes is switched on.

By rotating the platform around its vertical axis from therough initial alignment (where the y-axis is pointing east) to theopposite direction (y-axis pointing west), the gyro drift of theY-RAMP and true East are established. The gyro drift of theX-RAMP is measured by rotating the platform through 90°.The Z-gyro drift is measured when the platform is accuratelyaligned.

The power supply converts the aircraft mains voltage,115 V, 400 Hz, three-phase, to stabilized d.c. voltagesand accurate a.c. voltages. The gyro power is suppliedby a 26 V, 400 Hz, three-phase generator with a Ire-

Fig. 17. The display unit, which indicates latitude, longitude,heading, roll and pitch of the aircraft.

-------------------------------------~~---~-~--~------------

80 PHILlPS TECHNICAL REVIEW VOLUME29

quency precision of 10-5 and a total distortion and am- 6',----,---,;__-,--,.----,---,--,----.---,

plitude error of about 10-3.

Analogue computer investigations

A mathematical analysis of the complete RAMPsystem is given in the Appendix. One of the principalresults of this analysis is that the introduetion ofcorrection terms effects a coupling between the X-RAMP and the Y-RAMP. This coupling inducestwo kinds of oscillations of the system; one with aperiod equal to the Earth's rotation time and one withthe Schuler period (84.4 min). The latter oscillationsare modulated; the modulation frequency correspondsto the speed of rotation of a Foucault pendulum at thecorresponding latitude.

In order to check these characteristics of the systema model of the inforrnation flow, which is shown infig. 27 in the Appendix, was made in an analogue com-puter. In this case the main object was to deduce therelation between the gyro drift and the errors in positionand ·velocity. This was done by applying a step func-tionforthe gyro drift ofO.01 o/hto each gyro individual-ly. The corresponding position and attitude deviationsare shown infig. 18. As will be seen from these curves,the dominant errors have a 24-hour period while theerrors having the Schuler period can hardly be seen.To give some idea of the corresponding position errorswe note that 1minute of arc longitude at a latitude of60° corresponds to 0.5 nautical mile.

Laboratory tests

In order to investigate the behaviour of the inertialsystem over longer periods of operation it was run forperiods of 12 and 24 hours in the laboratory. Duringthese times normal drift tests of gyros and alignmentwere made as in real flight tests. The absence of shocksand accelerations did of course result in smoothercurves and also in considerably smaller total errors.Fig. 19 shows the positional errors, latitude and Ion-

6nout.

5

D 4

Î ~-1-2-3-4

D

j-00 2 4 6 e 10 12j 16 ."ttl I 11'1D, ::lJ

I I16 18h

Fig. 18. Recordings of analogue computer investigations on theinertial navigation system. The recordings show the longitudeerror Dl, the latitude error DL, and the direction error Dz, allresulting from the application of a step-function gyro drift of0.01 °th successively to the X-gyro, the Y-gyro and theZ-gyro, asa function of the time t (in hours). The roll and pitch errors werevery small.

Fig. 19. Errors D in longitude (a) and latitude (b) during a l2-hour laboratory test. Thecurves show a half period of a 24-hour oscillation and Schuier oscillations (period 84.4 min).These oscillations are superimposed on a linear gyro drift, which amounts to about 0.5 nauti-cal miles per hour in longitude and 0.3 nautical miles per hour in latitude. .

1968, No. 3/4 INERTIAL NAVIGATION SYSTEM 81

I '

~.

I a I, II II II . ,

gitude, obtained in a typical 12-hour test. The curvesshow Schuler oscillations and a 24-hour oscillation,superimposed on a linear gyro drift of about 0.5 naut.m.p.h. for the longitude and of about 0.3 naut.m.p.h.for the latitude.

Fig. 20 shows a recording ofthe velocity signals fromthe two RAMPs when an oscillation of about I' ampli-

check points at a low altitude (50 to 100metres),thus ensuring an accurate position indication. Whenhe gave the signal, the position - latitude andlongitude - indicated on the display was photo-graphed or recorded manually together with the time.Altogether 36 flights were made during this period.

In the beginning the results varied due to teething

Fig. 20. Part of a recording of the velocity signals from the X-RAMP (a) and from the Y-RAMP (b) at an initiated oscillation of about I' amplitude during a 12-hour laboratory test.

tude was initiated. It was found from the recordingthat very smooth Schuler oscillations remain, with aharmonic distortion as low as about 5%, which corre-sponds to second-order disturbances smaller than0.01 a/hour. Furthermore the Foucault effect can berecognized as a modulation of the SchuIer oscillations.

Flight testsTwo flight test programmes were carried out. The

first one was made in one of the twin-engine aircraftowned by Philips, a_De HaviIland Dove. The second

troubles in the electronic and mechanical equipment.At the end of the flight test period, however, the re-sults became fairly reproducible. The results of theflights number 32, 33 and 35 are shown infig. 22. Theaverage uncertainty amounted to 2-3 nautical milesper hour. This corresponds to the estimated accuracy,based on measured gyro drift.

Flight test in the De8

The same equipment that was tested in the De Havil-land Dove was mounted in a special rack, adapted to

DJ]DJ]

Fig. 21. InstaIIation of the test equipment in the De HaviIIand Dove. a platform. b electronicunits. c power converters. d mode control units. e display unit. f auxiliary equipment. g record-er. h oscilloscope.

programme was carried out in a DC8 aircraft, belongingto Scandinavian Airline Systems, during flights overthe European continent and over the North Atlantic.

Testflights with the De Havilland Dove

The installation ofthe equipment is shown infig. 21.Three different tracks were used: two East -West tracksand one North-South track. The pilot flew over the

one of the seats in the lounge of the DC8. Beforethe flights (normal passenger flights) the equipmentwas warmed up and aligned in a service bus onthe ground, after which, operating on its own battery,'it was installed in the aircraft and connected to itsmains supply. During the first flights, which were.carried out over the European continent (Copenhagen-Teheran, 3000nautical miles, flight time about 9 hours

82 PHILlPS TECHNICAL REVIEW VOLUME29

with stops), the position was checked with the ordinarynavigation system on board. Accurate check pointswere reported, particularly when passing over the radiobeacons. Over the North Atlantic (Copenhagen-NewYork, 3000 nautical miles, flight time about 8.5 hours)the Loran C network was used which, at the beginningand at the end ofthe flights, offered a fairly good check.The radio beacons along the Scandinavian and Ameri-can Atlantic coast were of course also used.The results of these long-flight tests showed that the

recordings from the flights over the Continent weresmoother and more accurate. A typical recording isshown infig. 23, where the maximum error correspondsto 3.5 naut.m.p.h. and the average to 2.2 naut.m.p.h.The recordings of the transatlantic flights are more

lûnaui.m:

D 8

t 64

30 2 ~-D 00

~t

J Fig.22

20

10

3

ments for inertial navigation equipment. Thus, thegyros were fire control components of high accuracy,improved so as to come close to inertial navigation ac-curacy. Although gyros were specified with a drift ofless than 0.01-0.02 "[u, gyro drifts of several tenths of adegree per hour were often measured. For practicalreasons an analogue computer constructed from elec-tro-mechanical components and electronic amplifiersand networks was chosen. Important sources of in-accuracy were the lack of linearity and the uncontrolleddrift of the integrators, both of which amounted toabout 0.1%. Even more serious errors come from theco-ordinate transforming process. Here it was ratherdifficult to maintain an accuracy of 0.1%. Finally themechanical unit of the pendulum system in the RAMP

4 5 6_t

7 9h

Fig.23

Fig. 22. Results obtained during three of the later test flights with the De HavillandDove (flight numbers 32, 33 and 35). The solid lines show the position error D, in nauticalmiles, as a function of time. The dotted lines a and b correspond to errors of 2 and4 naut.m.p.h.

Fig.23. Recording of the longitude error (a) and the latitude error (b) in nautical miles as afunction of time during a flight over the European continent.

disturbed, due probably to the checking method. Thedistribution of the longitude and latitude errors aftersix hours' flight time with the DC8 is shown infig. 24.The equipment is shown fitted into the DC8 in fig. 25.

Comments on the test results

When discussing the test results it should be pointedout that in order to lower the cost some componentswere used which did not fulfil the advanced require-

had some not unexpected teething troubles like ther-mal drift. The total error was expected to be of theorder of a few nautical miles per hour and it was en-couraging to. note that the flight test results were veryclose to this expected accuracy. Introduetion of bettergyros, a digital computer and an improved mechanicaldesign of the pendulum system would very probablylead to an accuracy of the order of 1 nautical mile perhour.

Appendix: Analysis of the complete RAMP navigation system

For an analysis of the complete navigation system we mustknow I) the location on the Earth's surface of the vehicle inwhich the system is installed; 2) the forces, motions and theirmathematical relations. Referring to fig. 26 we establish a three-axis co-ordinate system, which is geographically orientated: thex-axis points north, the y-axis east and the z-axis downwards

1968, No. 3/4

20

N

I

a

iNERTIAL NAVIGATiON SYSTEM

/ 20 I/ //

/ // N I/ I I/ I/ .1/ I

·1 I·1 I·j /

I. 10 /"../ .//

/. Y/ /l I. ./ ./.l / b

I. / ./ .1

/ I./ ./ 00

/

10 20 JO 40 noul.m. 10 20 JO 40 nout.m.-Dm -Dm

10

Fig. 24. Distribution of the errors in longitude (a) and in latitude (b) indicated by the RAM Pequipment during six-hour flights over the European continent and over the Atlantic Ocean.The dots show the number of flights N with a maximum error less than Dm (in na ut. miles).Errors larger than about 30 naut. miles occurred only in the beginning.

along the vertical. The co-ordinate system thus rotates with theEarth. The angular velocities imparted to the platform by thisrotation and by the movement of the vehicle follow the equations:

WIz (WIE + i) COS L, (13)

Wry -L, (14)

WIz -(WIE + i) sin L, (15)

(hence Wiz -Wiz tan L), (16)

r,Fig. 25. Installation of the test equipment in the De8.

83

84 PHILlPS TECHNICAL REVIEW VOLUME29

Fig. 26. Geographically orientated co-ordinate system x, y, z,pointing north, east and downwards. I longitude. L latitude. WIErotation speed of the Earth.

where WIE is the frequency of the Earth's rotation, I is the longi--tude and L the latitude. '

The mathematical derivation of the accelerations acting on theplatform along the three axes gives an expression whose princi-pal terms are:

ap", = -R(l - 2e cosê L)wry - 2RwlY -

-Rwr",wlz - eR sin 2L + ,apv R(orz + 2Rwr", - Rw2r", - RWlywlz + ,apt -R' + R(W2rx + (J)2IY)+ ... ,

where R is the Earth's radius and e is the eccentricity of the refer-ence ellipsoid of the Earth. These accelerations will generatetorques about the pendulum axes. The torque equations for twosingle-axis pendulums each contain a dominant term:

Ms", -mhRwI"',Msy = -mhRwlY (I - 2e cos- L).

They also contain terms which are small compared with the mainterms and therefore may be treated as correction terms; these are:

!~siti~n-'-'-'-'-'-'-'-'-'-'!. IL ~~.~ ..

L._. ._._._. ._._. ._.

Fig. 27:"Signal flow diagram ofthe complete RAMP navigation system. The bold lines correspond to mechanical connections, thethin lines to electrical connections.The hatched blocks represent the X-RAMP and the Y-RAMP. In the block marked "position" the output signals of the RAMPs

undergo the mathematical operations represented by the equations (I3) and (I4). After correction for the Earth's rotation, integrationand introduetion of the geographicallatitude and longitude at the starting point of the flight, Lo and 10, the latitude and longitude ofthe actuallocation of the airplane are found.

In the block "-tanL" the computation of equation (16) is carried out. This yields a signal proportional to the z-component ofthe Earth's angular velocity, which is used for correcting the Z-gyro for Earth's rotation.The blocks to the right represent the servo system for keeping the platform in a horizontal position. The servomotors are con-

trolled by the amplified error signals ex, Ey and ez of the RAMPs and the Z-gyro. The signalof the Z-gyro, which has to controlonly one servomotor, is fed directly to this motor. The error signals of the RAMPs however have to be resolved into two components,which must be fed to the other two motors in a certain ratio (depending on the position of the gimbals). This is done in the blocksmarked "resolver".

1968, No. 3/4 INERTIAL NAVIGATION SYSTEM 85

Mkz = -mlz(2Rwlz - RWlyWlz),

Mky = -mlz(Rwlzwlz + Rw2IE sin L cos L ++ 2RWlY + eR sin 2L).

A block diagram of the signal flow in a complete RAMP sys-tem is shown in fig. 27. The correction torques are calculated in acomputer and added to the total torques. As the equations indi-cate, there is a coupling between the two RAMPs, which willlead to induced oscillations.

A three-axis system analysis (which will not be presented here)leads to the following characteristic equation:

(P2 + W2IE){p4 + 2(g/R + 2W21Esin2L)p2 + {g/R)2} = 0,

where p is the linear time derivative operator d/dt. We see thatthis equation is of the 6th order. The solution consists of termscontaining the following factors:

sin WIEt,

cos WIEt,

sin (ws +WIEsinL)t,cos (ws + WIE sin L)t,sin (ws - WIE sin L)t,cos (ws -.WIE sin L)/,

where Wscorresponds to the frequency of a Schuier pendulum.As the last four terms reveal, these "Schuier oscillations" aremodulated; the modulation frequency, WIEsin L, corresponds toFoucault oscillations. Consequently the RAMP system will besubject to oscillations at the "Earth's frequency" (period 24hours) and to modulated Schuier oscillations (period 84.4 min).

Further literature:

C. S. Draper, W. Wrigley and L. R. Grohe, The floating integrat-ing gyro and its application to geometrical stabilization problemson moving bases, Institute of Aeronautical Sciences, S. M. F.Fund Paper No. FF-13, New York, 1955.W. Wrigley, F. E. Houston and H. R. Whitman, Indication ofthevertical from moving bases, Institute of Navigation Paper, Massa-chusetts Institute of Technology, Cambridge, Mass., 1956.F. Hector and K. .J. Äström, .Sy,;_<;.dishPat. 189 939, applicationmade 24 September 1959, application granted 16th January 1964.K. J. Àström and F. Hector, Vertical indication with a physicalpendulum based on eleëtromechanical synthesis of a highmoment of inertia, Report 590802 of-the TTN-Group, August1959, reprinted in the Journalof the Swedish Association ofElectrical Engineers, Eleknik 8, 1965, No. 4.W. R. Markey and J. Hovorka, fhè mechanics of inertial posi-tion and heading indication, Methuen, London 1961.

Summary, Description of an inertial navigation system, devel-oped by the Swedish Philips Co., Ltd, primarily for aircraftnavigation. The system is based on indication of the change of thevertical with a Schuier pendulum, whose oscillation time is84.4 min. This device, which contains a gyroscope with integratingfeedback, has been named "RAMP" (Rate and AccelerationMeasuring Pendulum). The complete system comprises twoRAMP units, mounted on an inertial platform, which is "slaved"to the pendulums. For indication of heading, a directional gyrois mounted on the same platform. The equipment was testedboth in the laboratory and during a number of test flights, in-cluding flights over the European continent and transoceanicflights. The average uncertainty varied between 2 and 5 nauticalmiles per hour.