the vibration error of the fiber optic gyroscope rotation rate and methods of its suppression

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2013, Vol. 58, No. 8, pp. 840–846. © Pleiades Publishing, Inc., 2013.Original Russian Text © A.M. Kurbatov, R.A. Kurbatov, 2013, published in Radiotekhnika i Elektronika, 2013, Vol. 58, No. 8, pp. 842–849.

840

1 INTRODUCTION

A fiber�optic gyroscope (FOG) exhibits a numberof advantages over a mechanical gyroscope in theweight, size, cost, and so on. However, a FOG shouldbe designed with allowance for possible externaleffects, for example, produced by time�variable tem�perature fields and FOG support vibrations leading tothe rotation rate (RR) measurement error [1].

Below, an approximate analytical model of RRvibration errors is considered for open� and closed�loop FOGs. For both cases, a square�wave phase mod�ulation (PM) is used for light waves of a FOG ringinterferometer. In this case, the operation of a closed�loop FOG is explicitly described by an ordinary differ�ential equation (ODE) with coefficients varying intime with the vibration frequency. As a result, it isshown that, in this case, there exists an additional RRvibration error, which is not described in the literature.

1. THE VIBRATION ERROR IN AN OPEN�LOOP FIBER�OPTIC GYROSCOPE

In Fig. 1, the block diagram of an open�loop FOGis shown. It contains optical source 1, fiber coupler 2,integrated optic chip (IOC) 3, sensing coil 4, photode�tector (PD) 5, PD photo�current amplifier (PDA) 6,synchronous detector (SD) 7, and PM voltage gener�ator 8. The FOG sensitivity to small RRs is increasedwith the use of an additional PM. Consider the sim�plest square�wave PM with depth θ in the form θ(t) =±θ [2, 3], where the sign reverses each τ seconds (the

1 The article was translated by the authors.

time of light propagation over the coil). In this case,SD input signal has the following form:

(1)

Here, Q(t) = P(t)η(t)Z(t), P(t) is the light sourcepower, η(t) is the PD current sensitivity, Z(t) is PDAgain, and ΦS is the Sagnac phase difference. The firstterm on the right�hand side of (1) is referred to as aconstant component, and the second one (with the ±sign) is referred to as a rotation signal. Below, werestrict ourselves to the case ΦS � 1, because, for anopen�loop FOG with the square�wave modulation,the dynamic range extension is not topical, since thereis no way to stabilize the PM depth. Instead, we willconsider an open�loop FOG and demonstrate thenature of the RR vibration error, thus, making its fur�ther consideration for a closed�loop FOG more illus�trative.

The simplest demodulation of signal (1) is realizedthrough signal U(t) sampling on neighbor intervalswith length τ [2, 3]. Two signals of the form

(2a)

(2b)

can be constructed from these samples, where, opera�

tors acting on arbitrary function f(t) are determined

as = . In the case of slow varia�

( )[ ]{ }

( )[ ]S

S

( ) ( ) 1 cos cos

( )sin sin .

U t Q t t

Q t t

≈ + θ Φ

± θ Φ

( ) ( )

( ) ( ) ( ) ( )[ ]S1 cos sin ,

S t U t

Q t Q t t

−

− τ

− +

τ τ

= Δ

= Δ + θ + Δ Φ θ

( ) ( )

( ) ( ) ( ) ( )[ ]S1 cos sin ,

S t U t

Q t Q t t

+

+ τ

+ −

τ τ

= Δ

= Δ + θ + Δ Φ θ

±

τΔ

( )f tτ

±Δ ( )f t + τ ± ( )f t

The Vibration Error of the Fiber�Optic Gyroscope Rotation Rate and Methods of its Suppression1

A. M. Kurbatov and R. A. KurbatovThe Kuznetsov Research Institute of Applied Mechanics (a division of the Center for Ground�Based

Space Infrastructure Facilities Operation), ul. Aviamotornaya 55, Moscow, 111123 Russiae�mail: [email protected]

Received August 15, 2012

Abstract—The error of the fiber�optic gyroscope (FOG) rotation rate measurement is considered. This erroris induced by FOG vibrations (for open� and closed�loop FOGs). For a closed�loop FOG, a differentialequation describing the loop dynamics is derived. The coefficients of this equation contain terms varying intime with the vibration frequency. For the first time, it is shown that, in addition to the traditional rotationrate measurement error due to the superimposition of vibration�induced optical power oscillations and thephase difference in the FOG coil, there is one more error, which is due to vibration modulation of the loopbandwidth. Alternative methods of information processing are investigated, and, on the basis of them, a newcircuit is proposed for the suppression of vibration errors.

DOI: 10.1134/S1064226913070085

PHYSICAL PROCESSES IN ELECTRON DEVICES

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013

THE VIBRATION ERROR OF THE FIBER�OPTIC GYROSCOPE ROTATION RATE 841

tions of phase ΦS and circuit parameters, these signalscan be written in a simplified form:

(3a)

(3б)

Signal (3à) (the extracted rotation signal) containsinformation on ΦS, and signal (3b) (the extracted con�stant component) does not contain this information.The latter statement is valid if only ΦS � 1. Therefore,the simplest way of processing (i.e., of extracting infor�mation on ΦS) uses only signal (3a) (or more generally,signal (2a)). Let us call this method a conventionalprocessing technique. However, signal (3a) containstime variations of the Q(t) value that introduce the RRmeasurement error through the scale factor destabili�zation. For elimination of these variations, one canuse the signal [4]

(3c)

Below, it is shown that this signal, in addition to thescale factor stabilization, provides for the FOG vibra�tion sensitivity elimination. We call this processingmethod a dividing technique.

A. The Rotation Rate Vibration Error Source

Vibrations with frequency ω create, in addition tothe Sagnac phase, phase difference ΔΦcosωt (which,below, we call vibrational). For this reason, thereplacement

should be made in the rotation signal. The sources ofthe vibrational phase difference are as follows: (i) elas�tic waves in the coil [1, 5, 6] (through the photoelasticeffect changing the fiber refractive index at the vibra�tion frequency), (ii) periodic variations of the fiberlength [5], and (iii) the motion of fiber turns relative toeach other [5]. On the average, this phase difference iszero, and it is not treated as the RR error, because it

( ) ( )2 ( ) sin ,S t Q t t−

= Φ θS

( ) ( )2 ( ) 1 cos .S t Q t+

= + θ

( ) ( ) ( ) ( ) Stan 2 ( ).S t S t S t t− +

= = θ Φ

( ) ( ) ( ) ( )cos .t t t t tΦ →Φ = Φ + ΔΦ ωS S

does not lead to an RR long�term drift. Since thisphase difference varies in time with a rather high fre�quency, it should be treated as a noise component [1].However, it is known [1] that, along with the vibra�tional phase difference, the optical intensity modula�tion also exists:

Here, Р0 is the constant component of the opticalintensity in the absence of vibrations, Δр is the inten�sity oscillations at vibration frequency ω, and param�eter ε takes into account the fact that the vibrationalphase difference and intensity oscillations, generallyspeaking, are not in phase. This means that Q(t) =Q0 + ΔQcos(ωt + ε). These power variations are due tothe following time�periodic reasons: (i) time varia�tions of the mutual orientation of the anisotropy axesof the fiber and IOC polarizing waveguides [1] (thismechanism is considered as dominating at low vibra�tion frequencies [6]), (ii) losses at fiber componentmicrobends and at the points of junctions of fiberswith the optical source and the IOC [5], (iii) mechan�ical stresses in the IOC [5], and (iv) polarization cou�pling of modes leading to additional power branchingfrom the operating polarization mode of fiber compo�nents [5, 6].

B. The Vibration Error of the Rotation Rate in the Conventional Processing Circuit

Consider signal (2a). Its first term is a constantcomponent extracted at the demodulation frequency(1/2τ). For any function f(t) that slightly varies withintime intervals on order τ, the approximate relation�ships f(t + τ) – f(t) ≈ τf '(t) and f(t + τ) + f(t) ≈ 2f(t)+τf '(t) are valid, so that, for ΦS � 1, we have

(4)

( ) ( )0 cos .P t P p t= + Δ ω + ε

( ) ( )S S0 '2 ( ) sin

( ) sin cos .

S t Q t t

F t Q

−

⎡ ⎤= Φ + τΦ θ⎢ ⎥⎣ ⎦

+ + ΔΦΔ θ ε

1

2

3

4

5

6 7

8

Fig. 1. Open�loop FOG circuitcircuit: 1 is the light source, 2 is the fiber coupler, 3 is the integrated optic circuit, 4 is the sensingcoil, 5 is the photodetector, 6 is the preamplifier of the photodetector photo�current, 7 is the synchronous detector, 8 is the gen�erator of the phase�modulation voltage.

4

842

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013

A. M. KURBATOV, R. A. KURBATOV

Here, F(t) is a periodic function with the zero mean.In (4), of interest is the time�constant last term, yield�ing the RR error

(5)

where M = 4πRL/(λc) is the optical scale factor (R isthe coil radius, L is the coil fiber length, λ is the lightwavelength, and с is the light velocity in free space).Thus, in the open�loop FOG with the conventionalprocessing circuit, the RR vibration error is due to thesuperimposed intensity oscillations and vibrationalphase difference. Below, an analog of (5) will bederived for a closed�loop FOG.

C. Dividing Technique

In this technique, signal (3c) is used for obtaininginformation on the RR:

(6)

Expression (6) does not explicitly contain the lightintensity. Thus, it is possible to eliminate the influenceof its time variations, including its oscillations at thevibration frequency. As a result, only the time�periodicRR error with the zero mean is left.

At present, for an open�loop FOG, a sinusoidalPM is used [7]. In this case, for scale factor stabiliza�tion in the processing circuit, the FOG output firstharmonic amplitude is divided by the amplitude of thesecond harmonic and, for PM depth stabilization, thesecond harmonic amplitude is divided by the ampli�tude of fourth harmonic. As a result, with the scale fac�tor stabilization, a dividing techniquet eliminatingoptical intensity fluctuations and, in particular, theRR constant vibration error is obtained. Here, in con�trast to the open�loop FOG with the square�wavemodulation, it is reasonable to consider the problem ofmeasured RR dynamic range. This problem is success�fully solved with the help of additional measures [7].

( ) ( )02 cos ,M p PΔΩ = ΔΦ Δ ε

( ) ( ) ( ) ( ) ( )Stan tan2 2 cos .S t t t t≈ θ Φ + θ ΔΦ ω

1

2. A CLOSED�LOOP FOG CIRCUIT

In the case of a FOG with a closed feedback loop(FB), another way to extend the dynamic range bycompensating the Sagnac phase with the step sawtoothvoltage can be used. This voltage is applied to thephase modulator electrodes along with the PM voltage[2, 3]. Figure 2 shows the block diagram of such FOG,which, in addition to the block diagram from Fig. 1,contains a filter (integrator) and a step voltage genera�tor (SVG). In this situation, the PDA output voltage is

(7)

Here, Δϕ(t) = ΦS(t) + ΔΦ(t)cosωt – Φc(t), Φc(t) is thephase that compensates for the Sagnac phase and thevibrational phase difference and is introduced bythe step sawtooth voltage [2, 3]. Value Δϕ(t) is calledthe phase compensation error and the term with the± sign on the right�hand side of (7) is called the errorsignal (an analog of open�loop FOG rotation signal).

In the case of a closed�loop FOG, the time�constantvibration RR error is due to the superimposition ofvibrational intensity changes and the vibrational com�ponent of Δϕ(t) value. It is clear that, in the FB loopwith an infinite processing speed (bandwidth), weshould have Δϕ(t) = 0, so that there is no time�constantvibration RR error. In a real FB loop with a finite band�width, phase Φc(t) can be represented in the form [5]

where the first term compensates for the Sagnac phaseand the second one compensates for the vibrationalphase difference ΔФcosωt with the amplitude error(ΔΦ – ΔΦc) and phase delay δ, which are due to theFB loop finite speed. Hence, for the time�constantcomponent of the RR vibration error, we have an ana�log of expression (5):

(8)

For an infinite loop bandwidth, we have Φc → ΔΦand δ → 0 (the exact compensation for the vibrationalphase difference), so that ΔΩ → 0. However, as it will

( )( ) ( )(1 cos ) ( )sin .U t Q t Q t t≈ + θ ± θΔϕ

( ) ( ) ( )c c S c, cos ,t t tΦ = Φ + ΔΦ ω + δ

( ) ( ) ( )[ ]c01 2 cos cos .M p PΔΩ = Δ ΔΦ ε − ΔΦ ε − δ

1

2

3

5

6 7 8 9 10

4

Fig. 2. Closed�loop FOG circuitcircuit: 1 is the light source, 2 is the fiber coupler, 3 is the IOC, 4 is the sensing coil, 5 is the PD,6 is the PDA, 7 is the synchronous detector, 8 is the filter (integrator), 9 is the sawtooth step voltage generator, 10 is the generatorof the phase�modulation voltage.

4



be shown below, in a real FB loop with a substantiallyextended bandwidth, the value determined from (8),can be noticeably smaller than the value determinedfrom (5).

In the most general case, the fundamental limit ofthe FOG bandwidth is f0 = 1/(2τ). Thus, for coils withfiber lengths L ≤ 2000 m, we have f0 ≥ 50 kHz. This is alarge value, because the vibration frequency is withinthe range 0–2.5 kHz [5, 6, 8]. However, there are otherlimitations dictated by the characteristics of the FBloop and its components. Finally, the bandwidth isregulated by an integrator [6] and, in practice, it isnoticeably smaller than f0. For example, in [8], itis reported that the bandwidth can be extended to20 kHz as a result of special measures, which is one ofthe basic means for vibration RR error suppression.Let the value 20 kHz be the maximal FB loop band�width, which provides for the FB loop stability againstthe background of the values on the order f0 = 1/(2τ).

The qualitative description considered above leadsto (8) and does not explicitly take into account thedynamics of a closed FB loop but just uses the conceptof its bandwidth. In addition, (8) does not contain thedependence of the RR error on the vibration fre�quency and FB loop parameters. Below, the FB loopdynamics is explicitly described using the methoddeveloped in [3] for the simplest case of time�constantloop parameters.

As has been mentioned above, for FB loop closingin a FOG an integrator (accumulator) is applied. Itdrives the SVG, the step voltage is then applied to thephase modulator (below, referred to as modulator)electrodes. As a result, the compensating phase isdescribed by the equation

(9)

where h(t) is the integrator response and K is the coef�ficient of voltage conversion into the phase differenceon the modulator electrodes. For an ideal integrator,h(t) = const, and, by differentiating the (9) with allow�ance for (7), we can obtain a first�order ODE describ�ing the FOG FB loop dynamics:

(10)

where

(11)

The last term from (10) can be neglected due to itssmall value. We will also exclude from the consider�ation the noise component of power P(t).

Let us reveal the nature of parameter G0. Assumethat vibrations are absent and that the RR jumps at theinstant t = 0 from zero to a certain time�constantvalue. In this case, ODE (10) becomes the ODE for

( ) ( ) ( )c

0

,

t

t K duh t u S u−

Φ = −∫

( ) ( )[ ]

[ ]

c S

cot0 00 0

( ) ( ) cos

( ) ( )( ),

2 2 2

d G t t G t t tdt

P t tP tG G

P P

−−

ττ

⎡ ⎤+ Φ = Φ + ΔΦ ω⎢ ⎥⎣ ⎦

Δ ΔϕΔ θ+ +

0 02 sin ,G KhQ= θ [ ]0 0( ) ( ) .G t G P t P=

the FB loop derived in [3] for the simplest case oftime�constant loop parameters:

Hence, for the error of Sagnac phase ΦS compen�sation, we have the expression

Qualitatively, this solution fits that from [9] derivedfor a similar case by means of numerical solution(z�transformation). Thus, the exact phase ΦS com�pensation is possible only for times t 1/G0. Here, thetime value 1/G0 describes the FB loop processingspeed, and, therefore, G0 is the FB loop bandwidth(recall that G0 < 1/(2τ)). The latter, as it is seen from(11), depends on the light intensity. This, for example,means that, under vibrations, the bandwidth is modu�lated and value G0 transforms into G(t) (the secondexpression from (11)). The consequences of this factwill be considered below.

Consider again ODE (10). Under vibrations,according to (11), for the FB loop bandwidth, we haveG(t) = G0 + ΔGcos(ωt + ε) (where ΔG = G0Δp/P0). It isreasonable to split ODE (10) into three parts by virtueof its linearity:

(12)

(13)

(14)

The first one describes the Sagnac phase compen�sation, the second one describes the vibrational phasedifference compensation, and the third one describesthe compensation for vibrational variations of theintensity constant component. Consider these threeequations individually.

A. The Sagnac Phase Compensation

Using the solution to (12), we obtain the followingexpression for the Sagnac phase compensation error

It is seen that time�varying phase ΦS(t) can be com�pensated with an error that is zero only for G0 → ∞.However, for a bandwidththat is finite but wider thanmaximal RR frequency variations (~200 Hz [5]), thesum of quickly decreasing terms (i.e., compensationerror) is small.

B. Compensation for the Vibrational Phase Difference

Let us rewrite ODE (13) in the following form:

( ) c S0 0( ) .d dt G t G+ Φ = Φ

( ) ( ) ( )S c S 0exp .t t G tΔϕ = Φ − Φ = Φ −

�

[ ] ( )c,S S( ) ( ) ( ),d dt G t t G t t+ Φ = Φ

[ ] ( )c( ) ( ) cos ,d dt G t t G t t+ ΔΦ = ΔΦ ω

( )[ ] ( )

( ) ( )c

cot,

0.5 2 sin .Qd dt G t t

G tΔ+ Φ

≈ − τωΔ θ ω + ε

( ) ( )c,S c,S S S

1

( ) ( ) 1 ( ).( )

nn

n

dt t t tG t dt

∞

=

⎡ ⎤ΔΦ = Φ −Φ = − Φ⎢ ⎥⎣ ⎦

∑

2

( )[ ] ( )

( )c, 0 cos

0.5 cos 2 0.5 cos .

d dt G t t G t

G t GΔΦ+ Φ = ΔΦ ω

+ Δ ΔΦ ω + Δ ΔΦ ε

844



The first term on the right�hand side is due to thepresence of the vibrational phase difference and therest two terms are due to its superimposition on inten�sity oscillations, which lead to generation of the sec�ond harmonic at the vibration frequency (the secondterm on the right�hand side) and the time�constantterm (the third term). We will search for the time�con�stant component of Φc,ΔФ(t). For small intensity oscil�lations P(t), using the perturbation method, we canobtain an approximate analytical solution to ODE(13) by representing it in the following form:

where Φc,ΔФ(t) is independent of the light intensityoscillations. As a result, ODE (13) can be rewritten inthe form of two simpler ODEs:

Solving the first ODE and substituting the solutioninto the right�hand side of the second one, we obtainexpression (8) for the time�constant RR error whoseparameters, in this case, have the form

Thus, for G0 → ∞ we obtain ΔΦc → ΔΦ, δ → 0(i.e., we have exact compensation of the vibrationalphase difference). Figures 3a and 3b show frequencydependences of absolute values of the RR vibrationerror for ε = π/2 and ε = 0, respectively. Here, for cal�culations, the following parameters were used: Δp =

( )c c c, , ,0 0 , ,1( ) ( ) ( ),t t p P tΔΦ ΔΦ ΔΦΔΦ ≈ ΔΦ + Δ ΔΦ

( ) c0 , ,0 0( ) cos ,d dt G t G tΔΦ+ Φ ≈ ΔΦ ω

( ) ( )

( )c c0 , ,1 0 , ,0

0 0

( ) ( )cos

0.5 cos 2 0.5 cos .

d dt G t G t t

G t GΔΦ ΔΦ+ Φ ≈ − Φ ω + ε

+ ΔΦ ω + ΔΦ ε

c2 2

0 0 ,G GΔΦ = ΔΦ ω + tan 0 .Gδ = −ω

0.1P0, ΔΦ/M = 10 deg/h, and the bandwidth valuesG0 = 0.5, 2.0, 10.0, and 20.0 s–1. It is seen that broad�ening of the bandwidth of the FB loop up to the valuesseveral times larger than the maximum vibration fre�quency rather efficiently reduces the RR error for bothcases (i.e., for all remaining ε).

C. Vibration Error Due to Intensity Oscillations

Equation (14) describes the vibration RR error dueto intensity oscillations. It is independent of both thevibrational phase difference and ε and is not describedin the literature. Let us use again the perturbationmethod and rewrite ODE (14) in the form of two sim�pler ODEs:

Here, parameter ΔQ is the amplitude of oscillations offunction Q(t) at the vibration frequency (parameterQ(t) is determined in the course of explanation to for�mula (1)). Using these relationships, we obtain the fol�lowing expression for the time�constant RR error:

(15)

This expression is independent of coil fiber lengthL, because M ~ L and τ ~ L. In Fig. 4, frequencydependences of error (15) are presented in the range0–2000 s–1 for the bandwidth values G0 = 500, 2000,10000, and 20000 s–1. The remaining parameters are

( ) ( )

( ) ( )c

cot0 , ,0

0.5 2 sin ,Qd dt G t

G tΔ+ Φ

= − τωΔ θ ω + ε

( ) ( )c c0 , ,1 0 , ,0( ) ( )cos .Q Qd dt G t G t tΔ Δ+ Φ = − Φ ω + ε

( ) cot2 2

02 2

0 0

1 .2 2

QGp

M P GΔ

⎛ ⎞ τωΔ θΔΩ ω = ⎜ ⎟ω +⎝ ⎠

0.05

20000 1000

0.10

0.15

0.20

0.25

0.30

0.1

20000 1000

0.2

0.3

0.4

0.5

0.6

ΔΩ, deg/hΔΩ, deg/h

ω, s–1ω, s–1

1

2

34

5

1

2 3 4

(a) (b)

5

Fig. 3. Frequency dependences of the RR vibration error for (a) ε = π/2 and (b) ε = 0. Curves 1–4 correspond to G0: 500, 2000,10000, and 20000 s–1. Curve 5 is the error level in the open�loop FOG.



the following: Δp/P0 = 0.1, λ = 1.55 μm, L = 500 m,and diameter of the FOG coil is 100 mm. It is seenthat, here, broadening of the bandwidth of the FB loopexpanding has weaker effect on the value of the vibra�tion RR error than in the case of expression (8) and,for G0 = 2000 s–1, within the half of the considered fre�quency range, the RR error is larger than for G0 =500 s–1. However, for G0 = 10000 and 20000 s–1, theRR error is smaller than for narrower bands. We shouldnoted here that, under existing limitations on thecapabilities of broadening of the bandwidth of the FBloop (up to ~20000 s–1) this measure is scarcely critical,while it may support the reduction of this kind of the RRerror. In this case, according to (15), stabilization of con�stant component is a more efficient measure, because theRR error considered here is proportional to (Δp)2. Thisstabilization can be ensured either by means of compen�sation for intensity oscillations Δp with the help of pro�cessing electronics or by improving the FOG design(more rigid fiber splices to the source, IOC, etc.).

D. Dividing Circuit

Above, by the example of an open�loop FOG, thenature of the so�called dividing circuit was demonstrated.Here, the dividing circuit for a closed�loop FOG, whichalso uses signal (3c) is describedcircuit [4]. Carrying outthe same calculations as those used for derivation ofODE (10), we multiply signal (4c) by 2Q0sinθ in order toobtain correct dimension. Then, for a perfect integrator,we again obtain an ODE for the FB loop:

(16)

where G1 = G0tan(θ/2) is the new bandwidth of the FBloop. Here, first, there is no superimposition of inten�sity oscillations and the vibrational phase difference,and, second, there is no modulation of the loop band�width. A consequence of this feature is the absence ofterms with nonzero mean in parameter Φc value, i.e.,in fact, the absence of the RR vibration error. We againcan introduce three ODEs:

(17)

(18)

(19)

Obviously, ODEs (17) and (18) do not yield a time�constant vibration error. This is also true for Eq. (19),because it can be rewritten in the form

3

( ) ( )[ ]( )

[ ]

1 c

S

cot1

1

( )( ) 2

2 ( )

( ) cos ,

Q t Q td G t Gdt Q t

G t t

+ τ −+ Φ = θ

+ Φ + ΔΦ ω

( ) ( ) ( )c,S S1 1 ,d dt G t G t+ Φ = Φ

( ) c,1 1( ) cos ,d dt G t G tΔΦ+ Φ = ΔΦ ω

( ) ( ) ( )[ ]

( )cot1 1( ) .

2 2Q

Q t Q td G t Gdt Q t

Δ

+ τ − θ+ Φ =

( ) ( )

( )

cot1 10

00

( ) sin2 2

cos .

Q

nn

n

pd G t G tdt P

pt

P

Δ

∞

=

Δ θ+ Φ = τ ω + ε

⎛ ⎞Δ× − ω + ε⎜ ⎟

⎝ ⎠∑

Integrating the right�hand side over the vibrationperiod, we obtain zero. Thus, also do not obtain heretime�constant vibration RR errors, because the circuitis free of modulation of the bandwidth of the FB loopwith superimposition on the modulated constantcomponent (as it was above). Thus, the dividing circuitis an efficient mean for elimination of the time�con�stant vibration RR error, as it was for the open�loopFOG.

Note that, in (17)–(19), some small terms wereomitted, which may result in residual modulation ofthe bandwidth of the FB loop. However, the depth ofthe bandwidth modulation in ODE (16) is much lowerthan in (10), i.e., the time�constant vibration error isalso much lower than the same error in a conventionalprocessing circuit. For further suppression of this kindof the RR error, it is necessary to implement the abovestabilization of the constant component of the lightintensity.

3. DIVIDING CIRCUIT AND STABILIZATION OF THE CONSTANT

COMPONENT OF SIGNAL

Figure 5 shows one of possible variants of the elec�tronic circuitcircuit implementing the described set ofmeasures elimination of the RR vibration error bymeans of electronic data processing circuit (the divid�ing circuit and stabilization of the constant compo�nent). The circuit Circuitis designed on two boards 1and 2 containing analog and digital parts. Transitionsbetween them are realized by analog�to�digital anddigital�to�analog converters (ADCs and DACs). Thedigital part may be built on the basis of a microproces�sor or a field programmable gate array (FPGA). Notethat, here, SD 7 with two outputs is used. At the firstoutput, signal S–(t) is formed, and the second output

4

20000 1000

0.4

0.8

1.2

1.6

2.0

ω, s–1

ΔΩΔQ, deg/h

1

2

3

4

Fig. 4. Frequency dependences of the RR error caused byonly light intensity modulation (15). Curves 1–4 corre�spond to G0: 500, 2000, 10000, and 20000 s–1.

846



is used for generation of signal S+(t), which is neces�sary for the dividing circuit and is formed at point 9circuit.

The Circuitmain purpose of this circuit is full digitalreconstruction of the constant component at point 9,which is necessary for division operation. There is alsoanother important application: to abandon formationof large constant components of signal at DA 5(it replaces here the OA), in order to exclude distor�tions of the error signal caused by saturation (above, inthe description of the sources of the RR vibration, thisproblem was not considered).

For solution of these problems, it is proposed toconstruct an additional FB loop, which is used tobranch the reconstructed constant component frompoint 9 to both dividing unit 10 and unit 12 with a gainof 1/Z (in order to compensate for subsequent ampli�fication with coefficient Z in DA 5). This componentis then branched to DAC 13 and device 14 with thegain α < 1, and, afterwards, enters the second input ofDA 5. As a result, at the output of DA 5, we have theinitial constant component multiplied by (1 – α) 1(which prevents DA 5 from saturation), which is thenreconstructed to its initial value after device 6 with thegain 1/(1–α).

The signal from the DAC also enters circuitcircuit 15controlling the gain of PDA 4, compensating thevibrational modulation of the light intensity . As aresult, the circuitcircuit solves the problems of divisionand stabilization of the signal constant component,i.e. considerably suppresses the vibration error of RRmeasurement.

CONCLUSIONS

RR Vibration errors of the rotation rate in open�and closed�loop FOGs have been considered. For the

closed�loop FOG, an approximate analytical modelhas been developed. This model has revealed a newvibration error, which is associated only with lightpower oscillations and is independent of the vibra�tional phase difference. This error is caused by super�imposition of modulation of the bandwidth of the FBloop (due to intensity oscillations) on the demodu�lated constant component of signal (3à), which, in thepresence of vibrations, also contains a time�periodiccomponent. A circuitcircuit allowing considerablesuppression of the RR vibration error in the closed�loop FOG has been proposed.

REFERENCES

1. A. Ohno, S. Motohara, R. Usui, et al., Proc. SPIE1585, 82 (1991).

2. H. Lefevre, P. Martin, J. Morisse, et al., Proc. SPIE1367, 72 (1990).

3. G. Pavlath, Proc. SPIE 2837, 46 (1996).

4. T. C. Greening, US Pat, No. 2008/0079946 A1 (3 Apr.2008).

5. G. A. Sanders, R. C. Dankwort, A. W. Kaliszek, et al.,US Pat, No. 5923424 (13 Jul. 1999).

6. N. Song, C. Zhang, and X. Du, Proc. SPIE 4920, 115(2002).

7. K. Bohm, P. Marten, W. Weidel, and K. Petermann,Electron. Lett. 19, 997 (1983).

8. J. Honthaas, S. Ferrand, V. D. Pham, et al., in CD ROMProc. Symp. Gyro Technology. Inertial Components andIntegrated Systems, Karlsruhe, Sept. 16–17, 2008(Karlsruher Inst. fur Technologie, Inst. fur Theore�tische Elektrotechnik und Systemoptimierung,Karlsruhe, 2008); http://www.ite.kit.edu/ISS/2008/DGON_ITE_Gyro_Programme_2008.pdf.

9. M. Bielas, Proc. SPIE 2292, 240 (1994).

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4

4

4

1 3 2

45

15

14

13 12

6 78

9

10 11+

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Fig. 5. Circuit for division and stabilization of the signal constant componentcircuit: 1 and 2 are the boards with analog and digitalparts of the circuitcircuit, 3 is the PD, 4 is the PDA, 5 is the DA, 6 is the analog�to�digital converter (ADC), 7 is the two�outputSD, 8 is the device with the gain 1/(1 – α), 9 is the point of digital reconstruction of the constant component, 10 is the unit divid�ing one input signal by another, 11 is control device for the sawtooth step voltage generator, 12 and 14 are the devices with gains1/K and α, 13 is the digital�to�analog converter (DAC), 15 is the unit controlling the PDA gain.

4

SPELL: 1. techniquet, 2. bandwidththat, 3. describedcircuit, 4. circuitcircuit

the vibration error of the fiber optic gyroscope rotation rate and methods of its suppression

Engineering