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Hemispherical Resonator Gyro Technology. Problems and Possible Ways of their Solutions.

(May 1999)

E.A. Izmailov, M.M. Kolesnik, A.M. Osipov, A.V. Akimov. The Moscow Institute of Electromechanics and Automatics

5 Aviatzionny per. Moscow, 1253 19 Russia

Phone: 155-06-47. Fax: 152-26-3 1

Abstract Key words: hemispherical resonator gyro,

fundamental wave, quadrature wave.

Hemispherical Resonator Gyro (HRG) technology is now considered as one of the promising gyroscope technologies. However, to implement HRG potential capabilities one should solve a number of problems connected both with manufacturing an isotropic resonator having a high mechanical Q-factor and with conserving its parameters within the gyro.

The article. basing on the experience of developing HRGs of two standard sizes, presents some of the mentioned problems and the results of mock-up tests.

Introduction

All known gyro types are based on using either the law of conservation of momentum or moment of momentum of solid bodies, material points or fundamental particles or on using main statements of the theory of relativity. In any case, gyro quality is determined by the conservatism level of its proper sensing element and a pickup system. The more energy dissipates within the sensing element and the pickup system under other similar conditions, the lower the gyro accuracy and reliability are. From this point of view the HRG has good potential characteristics. Using inertial properties of a standing wave in a high Q-factor resonator, which is an axis-symmetrical shell, in combination with an electrostatic system for excitation. support and control of vibrations on a resonance frequency, and with a similar pick off system, the HRG is characterised by extremely low specific energy losses.

The above mentioned HRG feature is also a basis of its unique property - to preserve information on angular movement of an object during momentary interruptions in power if the HRG is used as an integrating gyro (IG). Depending upon tolerable error, this interruption can be of several dozens of seconds.

The same sensing element, depending upon switching external electronic circuits, can be used both as an IG and an angular rate sensor (ARS) [ 11.

As the majority of gyros, existing and being developed, the HRG is an integral system consisting of a sensing element and functional electronics which not only generates an output information signal but also provides conditions for sensing element operation. Depending upon the conditions of HRG application, this electronics can be of various complexity ( also from the point of view of implemented functions), however, there are common problems and without their solving there is no sense to speak about the complexity of electronics types.

Simple design of the HRG sensing element consisting of 2-3 rigidly connected parts, and potentially low cost are also attractive. The current HRG development experience allows to predict its cost which is at least five times less than that of a laser gyro provided they have similar accuracy parameters.

Practical works on HRG creation proved the validity of main statements of fundamental investigations conducted in this field [2], [3], [4], [5], and revealed some application matters presented in this article.

Sensing Element

The HRG’s sensing element is a hemispherical resonator in which a standing wave is excited by electrostatic forces. As the HRG is based on inertial properties of the standing wave, it is clear, that minimisation of energetic connection of this wave with structural elements moving in the inertial space is the main way to increase HRG accuracy. The value of this connection is subdivided into two components: dissipation of the resonator’s vibratory energy and position force action on the vibrating shell.

In case of an ideally made resonator, the first component includes: energy losses in the resonator and energy losses in the electrostatic pick off and wave control systems. This means, that the resonator material and the whole process of resonator manufacture must finally provide a vibratory loop with minimum possible damping. As the resonator is a spatial shell in which the presence of “precedence” directions is prohibitive, only isotropic material can be used for its manufacturing. In addition, this material must feature minimum possible losses for internal friction. Fused quartz most fully

Paper presented at the RTO SC1 International Conference on “Integrated Navigation Systems”, held at “Elektropribor”, St. Petersburg, Russia, 24-26 May 1999, and published in RTO MP-43.

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meets these requirements. However to retain potentially high characteristics of the material in a manufactured resonator one must solve such technological problems as full removal of internal stress and a defective layer. These defects appear in the process of mechanical treatment, mainly in the process of shaping. It means that in these processes the forces appearing in the treatment zone shall be tangentially directed to the surface being treated. However, in this case internal stress and a defective layer also appear. Therefore, chemical and thermal treatment is the necessary condition to obtain a high-quality resonator.

Creation of the technological process for manufacturing a resonator on the basis of the above requirements allowed to produce resonators with the mechanical Q- factor of (12-13~10~ for a 60-mm diameter and (6- 9)x lo6 for a 20-mm diameter (Figure 1).

As for electrostatic systems, there are problems connected with pick off channel optimisation and damping minimisation due to currents induced in the resonator metal coating. The latter problem is not in conflict with the requirement to minimum vibration damping in the resonator because it complies with the requirement to minimum thickness of the film provided it retains its homogeneity. But this implies additional requirements to the quality of treatment and preparation of the resonator surface to applying the metal film as well as to the process of film applying. In particular. surface quality shall comply with the optical class of cleanness for which chemical and mechanical treatment is used at the final stage.

Fused quartz features the presence of nonsaturated radicals on the surface which causes absorption of different substances. including water. from the atmosphere by the surface layer. This phenomenon not only reduces the Q of the resonator but also negatively influences the quality of the applied metal film which finally deteriorates Q as well. Therefore, not only surface layer passivation is necessary but observance of technical hygiene as well.

In case of electrovacuum application of the metal film, the most promising method is the one which potentially provides maximum possible plasma homogeneity provided it has enough energetic parameters. In this case, deterioration of the resonator reference Q after metal film application does not exceed (5lo)%.

It is apparent, that the ideal resonator can not be made. As it was mentioned in well-known publications [6], this leads to resonance frequency difference and vibrations of resonator center of mass. The methodology for eliminating difference in resonance frequencies is fully elaborated at present. It includes both adjustment operations at the phase of sensing element

manufacturing and structural compensation methods implemented in functional electronics. However, it should be taken into consideration that the emergence of the compensation loop of the resonance frequency residual difference requires solving the task of phase error minimisation. Elimination of the resonator mass center vibrations is a more complicated task. Solving this task is necessary because when the resonator mass center is vibrating the vibratory energy is substantially dissipated at the resonator fastening points and in body members including metallic ones. As far as this process is principally instable, the standing wave energetic connection with structural elements moving in the inertial space also becomes instable. The experience of development and investigation of 60-mm diameter and 20-mm diameter HRG mock-ups shows that with the reduction of the resonator diameter the influence degree of the resonator mass center vibrations becomes prevailing.

As it is well-known, the process of resonator symmetrization consists in mass elimination along the resonator rim perimeter. Availability or nonavailability of teeth on the resonator rim depends upon its manufacturing cost and mass elimination procedure as well, which may be much more expensive than the resonator manufacturing. If the resonator does not have teeth, mass elimination shall not cause change in material structure (at the place of elimination) and emergence of internal stress. For example, in the course of comparative experiment minor effect of the COZ laser radiation in the balancing zone of the resonator without teeth caused anisotropy of its mechanical Q-factor of SQ - 40% under simultaneous decrease of maximum value of - 20%. Similar effect on the resonator with teeth practically did not influence its reference Q. Thus, if toothless resonator is used, mass must be eliminated only by chemical or ion-chemical processes. The resonator with teeth can be effectively symmetrized using COZ laser radiation.

The second component in the mentioned case emerges due to: force electrode system asymmeq, phase errors of control signal generation, and induced “wandering” potentials. Symmetrization of the force electrode system is implemented both at the design stage and by selecting appropriate procedures for sensing element assembly and adjustment. As the residual asymmetry (if the above procedures are rationally selected) is a small and stable magnitude, it can be eliminated algorithmically. “Wandering” potentials can be eliminated both by implementing equipotentiality for the resonator metal film and stability of permanent potentials on all electrodes. The most complicated problem is minimisation of phase errors of control signal generation. As it is connected not only with the electronic circuits errors and their instability but with noise in the data channel as well, the best decision can

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be made only when control action generation algorithms, functional electronics building structure and component capabilities are considered as a whole.

Pick off Electrode System

The pick off electrode system generates both the necessary control actions on the resonator and the HRG output signal as well. It contains 8 electrodes symmetrically located relative to the resonator; the electrodes together with the resonator surface form the same number of capacitors (Figure 2). Of course, in reality neither absolute symmetry of electrodes nor their absolute identity are feasible. In addition, a real gap between electrodes and the resonator surface is not similar either. Therefore, each capacitor will have its own C, magnitude and its deviation from an ideal angular position vi, where i = 1,2,..., 8. Let US write Ci in the form of an equivalent flat capacitor:

(1)

where: d - dielectric penetrability; S, - an equivalent area; 6i - an equivalent gap.

Here and further the second resonator vibration mode is considered. Assuming that resonator vibration amplitude is A<<Fi, the change of the capacitor magnitude can be presented in the form of:

Ai=Kix Al, (2)

where: K, = -5; 4

A, - gap change under resonator vibrations.

--.,_-. - - - - Fundamental Wave

..--... Quadrature Wave

Figure 2. The pick off electrode system.

Let us assume that there is only the fundamental wave in the resonator and that the pick off signal is generated due to constant voltage U applied to the resonator surface. Assuming that all capacitors are connected to similar active resistors and there is no relation between them, output signals over sin and cos channels can be written with an accuracy of up to the scale factor:

u,=UAr(EIcos28+E2sin28)cosot; (3) ~=UAAE&QB-E4cos2t3)cosot,

where: E1=K1~os~1+K3~os~3+K~cos~5+K~cos~~; E2=KI sincp, -K3sincp3-K5sincp5+K7sincp7; E3=K2~os~2+&~os~4+K~cos~6+K~cos~~; E4=K2sin~-K&in~4-K&in(P6fK-8Sin(Ps; o - angular frequency of resonator vibrations; t - time.

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Signals (3) are used to control resonator vibration amplitude and to determine angular position of the wave. An expression for estimation of the amplitude square will be obtained in the form of

A; = O,ZSU’A~[B, + B2(40 - c$)] (4)

where: B, =Ef +Ei +E,’ +Ef;

B, =&* +E;)* +(E; +E$ +2(&!$ -E,E,)* ;

6, = urctg 2(W, - EdA)

E;+E,2--;-Et’

Thus, in case of pick off channel non-identity the amplitude stabilisation loop will produce a false signal with a period of 48.

It is not dilficult to show that an evaluation error of wave angular position is determined by the relation:

8’ = 4 5arctg 4 + B4 sin@ + 6-I) B, +B,cos(48+&)’

(5)

where: B3=E2+E4;

(E, - E,)2 + (Ed - E2)2 ;

B5=E, +E3;

5, = arctg E4 -E2 . E, -E,

Expression (5) describes the HRG scale factor error which is also a periodic function of 48. In addition, there is a certain constant shift defined by a relative angular error of electrode location.

Now let us consider the display of the same asymmetry under parallel existence of fundamental and quadrature waves having different angular frequencies of or and oq vibrations. It is evident that such a mode can exist only in case of orientation of the mentioned waves over orthogonal axes of the resonator frequency anisotropy. Subject to this factor, expression (3) can be rewritten in the following form:

~=U[A,AE1cos28+E2sin28)coswfi-A@, sin28- E+os2O)sino,t]; (6) &=U[At(E&20- E4cos29)cosoft+A,(E3cos2B+E4sin29)sino,t].

Let us introduce an instant angular frequency o defined by the relation:

WflOIW,, then: of = o + wl and q,=o - 02. (7)

Subject to (7) equation (6) is transformed into:

b=U(z,cosot-z2sinot); (8) K=U(z+osot-z4sinot),

where: zl =(A& cosol t- A,E2sino2t)cos28+(A&2cos~,t+A~E,sinW)sin28; z2=(A& sinw I t- A,E2cosW2t)cos28+(Af2sinol t+A?\,E, coso2t)sin28; z3=(Arl$coso1 t-A, E&ino2t)sin28- (A&cos0,t+A~E~sin0~t)cos2% z4=(AfE3sino1 t-d, E4cosw2t)sin28- (A&&ino, t+AqE3cosa2t)cos20;

As for the real resonator m <Cl, z, are slowly w

changed periodic time functions.

Phase shift between u, and uC signals is obtained from (8) as an angle between corresponding vectors in coswt, sinwt planes (Figure 3).

Y =urctg z 2A,A$3, *a, - $ >t

(A, +A;)& +[(A; -A;‘,,& -2B&JsW+ -@1sfi4@

-[(A; -A;)B, +2B8A,A4sin(w, -wJ]cas4B’ (9)

where: B6=E1E3+E2E4; B7=E2E3-E,Ed; B8=E, E3-E2E4; Bs=E, E4+E2Es;

It follows from (9) that \v is a nonshifted estimation of A4 for the considered asymmetry of pick off electrodes and therefore is suitable to be used in an appropriate control loop. Using v as a control signal of the electronic balancing loop (quadrature wave suppression), equation (9) can be simplified subject to (o&J+O:

2A,A,B, y = arctg (A> + Ai)B, + (A> - At)B,, sin(48 - 4)

> (10)

where: B,, = (E; + E,~)(E: + E,f) ;

4 & = arctg-- 4

It should be mentioned, that in case of pick off electrode system symmetry i.e. E2=E4=0 and E,=E3, the \v function described by the relation:

T = arctg 2A,A,B6

(A’f -A’,) sin48

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Substituting corresponding symbols, we obtain:

Figure 3.

has the first order break v =5/2 either in case of ATA, or under A#A, in case of 8=(n-l)n/4. In real conditions full symmetry of pick off electrodes is slightly probable, but it is necessary to take into consideration the mentioned feature of the ,l~ function when the control loop is being formed.

Let us consider free vibrations of the resonator described by equations (8). It is not difficult to show that in the us, uc plane (Figure 4) they describe ellipse [7]:

YT(uC) ,

Figure 4.

b2{ (A:+As2)B, ,+[(A: -Aq2)B,2+4ArA,B,5 sin(c+ o,)t]cos48-2[(A:-A,z)B,5-AfAqB,2sin(o,- o,)t]sin48}+g2{(A:+Aq2)B,3+[(A?-Aq2)B,.I- 4AfAqB,6sin(o,-o,)t]cos48+2[(Af2- A,2)B,6+AfAqB,4sin(o~o,)t]sin48)- 2u&{ (At+A,2)B,7+[(A: -A,*)Bp-2ArA,B9sin(wt- o,)t]sin48-[(Afz-Aq2)B9+2ArAqBgsin(wf o,)t]cos48}=2U2Af2A,2B,j2cos2(o,-Qt. (11)

where: B, ,=E32+E42; B,2=E42-E32; B, 3=E, 2+E22;

2 2 B,4=E, -E2 ; BI~=E& B,ts=Ed%;

It follows from (11) that coefficients which define ellipse parameters are periodic time functions. Hence, we can watch some ellipse evolution’s in the us, uc plane (for example, the screen of a double-beam oscillograph). For the purpose of simplification, the further consideration will be provided for symmetric electrode system. Taking into account. that fundamental axes of the ellipsoid are orthogonal we can determine them by evaluating the expression for one of them. In this case the desired relation is as follows:

4 = Okrctg (A~-A~)sin48-2A,A,sin(wf-w,)tcos4e,(12)

(A;-A;)cos4f?

Thus, fundamental axes of the ellipse will make angular movements. In this case, at the moments of t=(n- l)d(ofw,J cp will exactly correspond to the angular position of the fundamental wave ( in the us, tic plane (p=2e).

It follows from (11) that at the moments of t=(2n-l)rr/2, the ellipse degenerates into direct lines:

AfA‘7 sin(48 - nrctg ~

A;-A;’ ; (13) 24, = A’r -A2 Us

1 + sin(48 + arcfg +) / 4

sin( 48 + arctg ~ A; -A;)

u, = A; -A; Us

1 + sin(48 - arctg r) / 4

Thus, ellipse evolution consists in its angular vibrations, accompanied by its ellipticity change and degeneration at the boundaries into direct lines (13). Ellipse vibration period precisely equals CO,-W,, and an angular amplitude of its vibrations is determined by the amplitude

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relationship Aif and A4. If AFA, at the moments of t=(n- l)~/(o~oJ the ellipse is changed into a circumference. In this case direct lines (13) exactly coincide with angular position of fundamental and quadrature waves,

It follows from the above, that availability of two different resonance frequencies under free vibrations of the resonator do not cause wave drift. This fully agrees with conclusions of the publication [2]. In addition, the obtained result is an information basis when the resonator is balanced for elimination of the 4-th harmonic of the resonator elasticity and mass inhomogeneity. We should mention here that it is not difficult to provide practically ideal symmetry of the pick off electrode system in the production equipment.

In case of generating an pick off signal by applying alternating voltage of angular frequency R, to the sin channel electrodes and frequency 0, to the cos channel electrodes [S], the signals taken from the hemisphere to the purely resistive load for each frequency can be represented in the form of:

u,=UIC-21 [E~+A~E,cos2Cl+E2sin28)cosoft-A,(E2sin28- El cos28)sino,t]sinR, t; u,=U2R2[Es+Ar(E3sm28- E~cos20)coso~+A,(E~cos28+E4sin2B)sino,t]sin~~~. (14)

where: E5=K,+K5-Kj-K7; Eg= K2+K6-K4-Kg.

Technically It is not difficult to provide U,!&=U&& condition. Having provided an ideal synchronous detection of signals (14) in accordance with corresponding carrier frequencies we will obtain signals different from (6) by availability of permanent members Es and E6 which express modulation deviation from 100%. Upon liquidation of these members, we obtain the above considered expression (6). Thus. ignoring nonideality of the above mentioned operations we can make a conclusion that errors which occur from asymmetry of the pick off electrode system do not depend upon pick off signal generation procedure.

Force Electrode System

The force electrode system includes a ring electrode and 16 discrete electrodes. The ring electrode is used for parameter stabilisation of the resonator vibration amplitude. The discrete electrodes opposite the ring one are combined into pairs and provide resonator excitation on its resonance frequency. electronic balancing of the resonator, and wave position control in case of using the HRG as an ARS. As the force is proportional to the square of potential difference between the resonator surface and an electrode, control with the help of alternating potentials will generate a certain permanent force.

Practically, the force electrode system features asymmetry similar to that of the pick off electrode system. An exception is the ring electrode the asymmetry of which is generated by gap irregularity. In the result, permanent force acting on the resonator shell from the side of vibration amplitude preserve loop will be the function of 0. As this force changes shell elasticity, the resonator elasticity-mass parameters will also become a certain function of 8. which can be expanded into a Fourier series according to 8. It can be easily shown that the most characteristic gap irregularities of eccentricity and ovality type generate just the first four expansion members or first four inhomogeneity harmonics. Thus. even in case of absolutely symmetrical resonator the functioning of the vibration amplitude preservation loop under asymmetric gap will cause two resonance frequencies and occurrence of resonator mass center vibrations, which is more unpleasant [6]. The latter will inevitably cause losses of the resonator vibrating energy in the HRG structural elements and consequently deterioration of the resonator Q and its asymmetry. It is well known [2] that the resonator SQ is the reason of HRG drift. Thus, ring electrode asymmetry is the source of J3RG additional drift. If to take into account the error of generating information signal of the vibration amplitude preservation loop (4), the drift caused by this loop will be the function of 40.

An additional drift is proportional to the square of potential difference (necessruy to preserve vibrations). vibration amplitude, relative asymmetry of the gap and in inverse proportion to Q. Hence, Q increase and vibration amplitude decrease will cause decrease of the additional drift. Decrease of relative asymmetry of the gap under other similar conditions is possible only by increasing its nominal value which will require increase of potential difference. In addition, gap increase and vibration amplitude decrease requires increase of hemisphere potential to keep the required signal-to- noise ratio in the pick off channel. Thus. there is a task of rational selection of the said parameters.

Figure 5 and Figure 6 present drift velocity dependencies of one of the HRG mock-ups with a 20- mm diameter resonator. These dependencies were obtained in different conditions: the results shown on Figure 6 were obtained for a reduced vibration amplitude and an increased potential on the hemisphere. As vividly seen from the comparison of the presented curves, they comply with the said above.

Electronic balancing is made by providing appropriate constant potential difference between the hemisphere and two pairs of orthogonal discrete electrodes. In case

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Figure 5. Case-Oriented Drift as a Function of Pattern Angle. I - vertical component of Earth rotation.

Jw I- I ! I I/ I ------__ _--_ _________ ___ ____________ i I I 1 r :

d I I

I I i 1 / / I I ! I

I I I I I I I I I

-I I I I I 1 I 1

c 2% ml

-20 0 20 40 60 80 100 120 140 160 180 b

Figure 6. Case-Oriented Drifr as a Function of Pattern Angle. I - vertical component of Earth rotation.

of full symmetry, these potentials create the 4-th harmonic of resonator elasticity-mass parameter inhomogeneity which compensates for its initial inhomogeneity which becomes apparent in availability of two resonance frequencies. Subject to discrete electrode system asymmetry the mentioned potentials will also create lower harmonics of inhomogeneity, i.e. we shall obtain the result described above.

Equation (9) obtained before, which is an information signal of electronic balancing loop (quadrature wave suppression) does not take into account phase errors of electronic circuits which generate w assessment. Availability of these errors will cause the shift of the assessment. In the result, in the resonator there will be kept two resonance frequencies, the difference of which is determined by the value and the sign of the \v

assessment shift. Availability of vibration amplitude stabilisation loop under these conditions will lead to the drift independent of 8 [2]. By this fact we can explain availability of constant shift of the curves on Figure 5 and 6 relative to the Earth angular velocity. As phase errors of the electronic circuits can not be considered as stable unlike electrode system asymmetry, full or partial calibration of these circuits is expected to become apparent in the drift characteristic. Unfortunately, in the course of experiments it was possible to calibrate only electronic circuits of input amplifiers of pick off signals u, and LL. Figure 7 shows drift plots of the said above mock-up. The plots are drawn by wave return into the initial point by the mock-up body turn. No calibrations were made. Figure 8 shows similar plots, but each time calibration of input amplifiers in initial state was made. The results obtained are not in conflict with the said above.

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I 1 7 13 1925313743495561 6773 79859197103 109115121127133139145151157 163

Figure 7. Three 0.5h dr&?i lines. The calibration was not pe$orrned.

Trend%,[dgr/h]: I-10.13; II-10.09; III-IO.04

Figure 8. Two 0.5h drift lines. Two calibrations was performed.

Trend 28, [dgr/h] : I - 10.13; II - IO.11

Summary system asymmetry. Experimental data comply with the statements given above.

Based on the development experience of HRG mock- ups of two standard sizes, some essential problems of HRG technology are raised and methods of their solving are proposed. Errors in pick off signal generation and assessments of vibration parameters used in control loops have been analysed. It is shown that free vibrations of the resonator having two resonance frequencies do not cause drift. A model of the drift additional components is given which is defined both by pick off electrode system symmetry and force electrode

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References

l.Lynch D.D., Matthews A., Dual-Mode Hemispherical Resonator Gyro Operating Characteristics, 3-rd Saint Petersburg International Conference on Integrated Navigation Systems, 1996, part 1, pp. 37-44. 2. XypameB B.D., KJIHMOB A.M., BOJIHOBO~~

Tsepnorexbmrii mpoc~ocon, M., Hayrq 1985, c. 123. 3. Zhuravlev V.F., Theoretical Foundation of Hemispherical Resonator Gyro (HRG), Izv. RAN, Mekhanika Tverdogo Tela, Allerton Press Inc., Vol. 28, N3, 1993, pp. 3-15. 4. Lynch D.D., Vibratory Gyro Analysis by the Method of Averaging, 2nd Saint Petersburg International Conference on Gyroscopic Technology and Navigation, 1995, part II, pp. 26-34. 5. Zhuravlev V.F., Lynch D.D., Electric Model of a Hemispherical Resonator Gyro, Izv. RAN, Mechanika Tverdogo Tela, Allerton Press Inc., Vol. 30, N5, pp. 12- 24. 6. Zhbanov Yu.K., Theoretical Aspects of Balancing the Hemispherical Resonator Gyro, 2nd Saint Petersburg International Conference on Gyroscopic Technology and Navigation, 1995, part II, p. 88. 7. Shatalov M.Y., Elements on General Control of Vibratory Gyroscopes, 5-th Saint Petersburg International Conference on Integrated Navigation Systems, 1998, pp. 204-213. 8. Lynch D.D., Matthews A., Varty G.T., Transfer of Sensor Technology from Oil-Drilling to Space Application, 5-th Saint Petersburg International Conference on Integrated Navigation Systems, 1998, pp. 27-36.