This document discusses vibration errors in fiber optic gyroscopes (FOGs) that occur due to external vibrations. It analyzes vibration errors in both open-loop and closed-loop FOGs. For closed-loop FOGs, it derives a differential equation to describe the loop dynamics, showing that vibration causes time-varying coefficients in this equation. It identifies an additional vibration-induced rotation rate error for closed-loop FOGs beyond those traditionally described. It also analyzes alternative information processing methods that can help suppress vibration errors, such as dividing signals to eliminate intensity fluctuations.

1. 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 number
of advantages over a mechanical gyroscope in the
weight, size, cost, and so on. However, a FOG should
be designed with allowance for possible external
effects, for example, produced by time variable tem
perature fields and FOG support vibrations leading to
the rotation rate (RR) measurement error [1].
Below, an approximate analytical model of RR
vibration 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 ring
interferometer. In this case, the operation of a closed
loop FOG is explicitly described by an ordinary differ
ential equation (ODE) with coefficients varying in
time with the vibration frequency. As a result, it is
shown that, in this case, there exists an additional RR
vibration 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 FOG
is 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 increased
with 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 source
power, η(t) is the PD current sensitivity, Z(t) is PDA
gain, and ΦS is the Sagnac phase difference. The first
term on the right hand side of (1) is referred to as a
constant component, and the second one (with the ±
sign) is referred to as a rotation signal. Below, we
restrict ourselves to the case ΦS Ӷ 1, because, for an
open loop FOG with the square wave modulation,
the dynamic range extension is not topical, since there
is no way to stabilize the PM depth. Instead, we will
consider an open loop FOG and demonstrate the
nature 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 realized
through signal U(t) sampling on neighbor intervals
with 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. Kurbatov
The Kuznetsov Research Institute of Applied Mechanics (a division of the Center for Ground Based
Space Infrastructure Facilities Operation), ul. Aviamotornaya 55, Moscow, 111123 Russia
e mail: akurbatov54@mail.ru
Received August 15, 2012
Abstract—The error of the fiber optic gyroscope (FOG) rotation rate measurement is considered. This error
is induced by FOG vibrations (for open and closed loop FOGs). For a closed loop FOG, a differential
equation describing the loop dynamics is derived. The coefficients of this equation contain terms varying in
time with the vibration frequency. For the first time, it is shown that, in addition to the traditional rotation
rate measurement error due to the superimposition of vibration induced optical power oscillations and the
phase difference in the FOG coil, there is one more error, which is due to vibration modulation of the loop
bandwidth. Alternative methods of information processing are investigated, and, on the basis of them, a new
circuit is proposed for the suppression of vibration errors.
DOI: 10.1134/S1064226913070085
PHYSICAL PROCESSES
IN ELECTRON DEVICES

2. 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 signals
can be written in a simplified form:
(3a)
(3б)
Signal (3à) (the extracted rotation signal) contains
information on ΦS, and signal (3b) (the extracted con
stant component) does not contain this information.
The latter statement is valid if only ΦS Ӷ 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 conventional
processing technique. However, signal (3a) contains
time variations of the Q(t) value that introduce the RR
measurement error through the scale factor destabili
zation. For elimination of these variations, one can
use the signal [4]
(3c)
Below, it is shown that this signal, in addition to the
scale factor stabilization, provides for the FOG vibra
tion sensitivity elimination. We call this processing
method a dividing technique.
A. The Rotation Rate Vibration Error Source
Vibrations with frequency ω create, in addition to
the Sagnac phase, phase difference ΔΦcosωt (which,
below, we call vibrational). For this reason, the
replacement
should be made in the rotation signal. The sources of
the vibrational phase difference are as follows: (i) elas
tic waves in the coil [1, 5, 6] (through the photoelastic
effect changing the fiber refractive index at the vibra
tion frequency), (ii) periodic variations of the fiber
length [5], and (iii) the motion of fiber turns relative to
each other [5]. On the average, this phase difference is
zero, 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 this
phase 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 optical
intensity 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 vibrational
phase difference and intensity oscillations, generally
speaking, are not in phase. This means that Q(t) =
Q0 + ΔQcos(ωt + ε). These power variations are due to
the following time periodic reasons: (i) time varia
tions of the mutual orientation of the anisotropy axes
of the fiber and IOC polarizing waveguides [1] (this
mechanism is considered as dominating at low vibra
tion frequencies [6]), (ii) losses at fiber component
microbends and at the points of junctions of fibers
with 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 branching
from 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 constant
component extracted at the demodulation frequency
(1/2τ). For any function f(t) that slightly varies within
time 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 Ӷ 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 sensing
coil, 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

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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 is
the coil radius, L is the coil fiber length, λ is the light
wavelength, and с is the light velocity in free space).
Thus, in the open loop FOG with the conventional
processing circuit, the RR vibration error is due to the
superimposed intensity oscillations and vibrational
phase difference. Below, an analog of (5) will be
derived for a closed loop FOG.
C. Dividing Technique
In this technique, signal (3c) is used for obtaining
information on the RR:
(6)
Expression (6) does not explicitly contain the light
intensity. Thus, it is possible to eliminate the influence
of its time variations, including its oscillations at the
vibration frequency. As a result, only the time periodic
RR error with the zero mean is left.
At present, for an open loop FOG, a sinusoidal
PM is used [7]. In this case, for scale factor stabiliza
tion in the processing circuit, the FOG output first
harmonic amplitude is divided by the amplitude of the
second harmonic and, for PM depth stabilization, the
second harmonic amplitude is divided by the ampli
tude of fourth harmonic. As a result, with the scale fac
tor stabilization, a dividing techniquet eliminating
optical intensity fluctuations and, in particular, the
RR constant vibration error is obtained. Here, in con
trast to the open loop FOG with the square wave
modulation, it is reasonable to consider the problem of
measured 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 by
compensating the Sagnac phase with the step sawtooth
voltage can be used. This voltage is applied to the
phase 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 the
phase that compensates for the Sagnac phase and the
vibrational phase difference and is introduced by
the step sawtooth voltage [2, 3]. Value Δϕ(t) is called
the phase compensation error and the term with the
± sign on the right hand side of (7) is called the error
signal (an analog of open loop FOG rotation signal).
In the case of a closed loop FOG, the time constant
vibration RR error is due to the superimposition of
vibrational intensity changes and the vibrational com
ponent of Δϕ(t) value. It is clear that, in the FB loop
with an infinite processing speed (bandwidth), we
should have Δϕ(t) = 0, so that there is no time constant
vibration 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 phase
and the second one compensates for the vibrational
phase difference ΔФcosωt with the amplitude error
(ΔΦ – ΔΦc) and phase delay δ, which are due to the
FB loop finite speed. Hence, for the time constant
component of the RR vibration error, we have an ana
log of expression (5):
(8)
For an infinite loop bandwidth, we have Φc → ΔΦ
and δ → 0 (the exact compensation for the vibrational
phase 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 generator
of the phase modulation voltage.
4

4. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013
THE VIBRATION ERROR OF THE FIBER OPTIC GYROSCOPE ROTATION RATE 843
be shown below, in a real FB loop with a substantially
extended bandwidth, the value determined from (8),
can be noticeably smaller than the value determined
from (5).
In the most general case, the fundamental limit of
the FOG bandwidth is f0 = 1/(2τ). Thus, for coils with
fiber lengths L ≤ 2000 m, we have f0 ≥ 50 kHz. This is a
large value, because the vibration frequency is within
the range 0–2.5 kHz [5, 6, 8]. However, there are other
limitations dictated by the characteristics of the FB
loop and its components. Finally, the bandwidth is
regulated by an integrator [6] and, in practice, it is
noticeably smaller than f0. For example, in [8], it
is reported that the bandwidth can be extended to
20 kHz as a result of special measures, which is one of
the 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 against
the background of the values on the order f0 = 1/(2τ).
The qualitative description considered above leads
to (8) and does not explicitly take into account the
dynamics of a closed FB loop but just uses the concept
of its bandwidth. In addition, (8) does not contain the
dependence of the RR error on the vibration fre
quency and FB loop parameters. Below, the FB loop
dynamics is explicitly described using the method
developed in [3] for the simplest case of time constant
loop parameters.
As has been mentioned above, for FB loop closing
in a FOG an integrator (accumulator) is applied. It
drives the SVG, the step voltage is then applied to the
phase modulator (below, referred to as modulator)
electrodes. As a result, the compensating phase is
described by the equation
(9)
where h(t) is the integrator response and K is the coef
ficient of voltage conversion into the phase difference
on 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 its
small value. We will also exclude from the consider
ation the noise component of power P(t).
Let us reveal the nature of parameter G0. Assume
that vibrations are absent and that the RR jumps at the
instant t = 0 from zero to a certain time constant
value. In this case, ODE (10) becomes the ODE for
( ) ( ) ( )c
0
,
t
t K duh t u S u−Φ = −∫
( ) ( )[ ]
[ ]
c S
cot0 0
0 0
( ) ( ) cos
( ) ( )( )
,
2 2 2
d G t t G t t t
dt
P t tP t
G 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 of
time constant loop parameters:
Hence, for the error of Sagnac phase ΦS compen
sation, we have the expression
Qualitatively, this solution fits that from [9] derived
for 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, the
time value 1/G0 describes the FB loop processing
speed, 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 second
expression from (11)). The consequences of this fact
will be considered below.
Consider again ODE (10). Under vibrations,
according to (11), for the FB loop bandwidth, we have
G(t) = G0 + ΔGcos(ωt + ε) (where ΔG = G0Δp/P0). It is
reasonable to split ODE (10) into three parts by virtue
of its linearity:
(12)
(13)
(14)
The first one describes the Sagnac phase compen
sation, the second one describes the vibrational phase
difference compensation, and the third one describes
the compensation for vibrational variations of the
intensity constant component. Consider these three
equations individually.
A. The Sagnac Phase Compensation
Using the solution to (12), we obtain the following
expression 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 than
maximal RR frequency variations (~200 Hz [5]), the
sum of quickly decreasing terms (i.e., compensation
error) 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 −
ӷ
[ ] ( )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 t
Δ+ Φ
≈ − τωΔ θ ω + ε
( ) ( )c,S c,S S S
1
( ) ( ) 1 ( ).
( )
n
n
n
dt t t t
G 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 G
ΔΦ+ Φ = ΔΦ ω
+ Δ ΔΦ ω + Δ ΔΦ ε

5. 844
JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013
A. M. KURBATOV, R. A. KURBATOV
The first term on the right hand side is due to the
presence of the vibrational phase difference and the
rest 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 second
term on the right hand side) and the time constant
term (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 can
obtain an approximate analytical solution to ODE
(13) by representing it in the following form:
where Φc,ΔФ(t) is independent of the light intensity
oscillations. As a result, ODE (13) can be rewritten in
the form of two simpler ODEs:
Solving the first ODE and substituting the solution
into the right hand side of the second one, we obtain
expression (8) for the time constant RR error whose
parameters, in this case, have the form
Thus, for G0 → ∞ we obtain ΔΦc → ΔΦ, δ → 0
(i.e., we have exact compensation of the vibrational
phase difference). Figures 3a and 3b show frequency
dependences of absolute values of the RR vibration
error 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 G
ΔΦ ΔΦ+ Φ ≈ − Φ ω + ε
+ ΔΦ ω + ΔΦ ε
c
2 2
0 0 ,G GΔΦ = ΔΦ ω + tan 0 .Gδ = −ω
0.1P0, ΔΦ/M = 10 deg/h, and the bandwidth values
G0 = 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 values
several times larger than the maximum vibration fre
quency rather efficiently reduces the RR error for both
cases (i.e., for all remaining ε).
C. Vibration Error Due to Intensity Oscillations
Equation (14) describes the vibration RR error due
to intensity oscillations. It is independent of both the
vibrational phase difference and ε and is not described
in the literature. Let us use again the perturbation
method and rewrite ODE (14) in the form of two sim
pler ODEs:
Here, parameter ΔQ is the amplitude of oscillations of
function Q(t) at the vibration frequency (parameter
Q(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 length
L, because M ~ L and τ ~ L. In Fig. 4, frequency
dependences of error (15) are presented in the range
0–2000 s–1
for the bandwidth values G0 = 500, 2000,
10000, and 20000 s–1
. The remaining parameters are
( ) ( )
( ) ( )
c
cot
0 , ,0
0.5 2 sin ,
Qd dt G t
G t
Δ+ Φ
= − τωΔ θ ω + ε
( ) ( )c c0 , ,1 0 , ,0( ) ( )cos .Q Qd dt G t G t tΔ Δ+ Φ = − Φ ω + ε
( ) cot
2 2
0
2 2
0 0
1 .
2 2
Q
Gp
M P G
Δ
⎛ ⎞ 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–1
ω, s–1
1
2
3
4
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.

6. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013
THE VIBRATION ERROR OF THE FIBER OPTIC GYROSCOPE ROTATION RATE 845
the following: Δp/P0 = 0.1, λ = 1.55 μm, L = 500 m,
and diameter of the FOG coil is 100 mm. It is seen
that, here, broadening of the bandwidth of the FB loop
expanding 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
, the
RR error is smaller than for narrower bands. We should
noted here that, under existing limitations on the
capabilities of broadening of the bandwidth of the FB
loop (up to ~20000 s–1) this measure is scarcely critical,
while it may support the reduction of this kind of the RR
error. In this case, according to (15), stabilization of con
stant component is a more efficient measure, because the
RR error considered here is proportional to (Δp)2. This
stabilization 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, the
natureoftheso calleddividingcircuitwasdemonstrated.
Here, the dividing circuit for a closed loop FOG, which
also uses signal (3c) is describedcircuit [4]. Carrying out
the same calculations as those used for derivation of
ODE (10), we multiply signal (4c) by 2Q0sinθ in order to
obtain 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 FB
loop. 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 of
terms with nonzero mean in parameter Φc value, i.e.,
in fact, the absence of the RR vibration error. We again
can 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 G
dt 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 2
Q
Q t Q td G t G
dt Q t
Δ
+ τ − θ+ Φ =
( ) ( )
( )
cot1 1
0
00
( ) sin
2 2
cos .
Q
n
n
n
pd G t G t
dt P
p
t
P
Δ
∞
=
Δ θ+ Φ = τ ω + ε
⎛ ⎞Δ
× − ω + ε⎜ ⎟
⎝ ⎠
∑
Integrating the right hand side over the vibration
period, we obtain zero. Thus, also do not obtain here
time constant vibration RR errors, because the circuit
is free of modulation of the bandwidth of the FB loop
with superimposition on the modulated constant
component (as it was above). Thus, the dividing circuit
is an efficient mean for elimination of the time con
stant vibration RR error, as it was for the open loop
FOG.
Note that, in (17)–(19), some small terms were
omitted, which may result in residual modulation of
the bandwidth of the FB loop. However, the depth of
the bandwidth modulation in ODE (16) is much lower
than in (10), i.e., the time constant vibration error is
also much lower than the same error in a conventional
processing circuit. For further suppression of this kind
of the RR error, it is necessary to implement the above
stabilization of the constant component of the light
intensity.
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 of
measures elimination of the RR vibration error by
means of electronic data processing circuit (the divid
ing circuit and stabilization of the constant compo
nent). The circuit Circuitis designed on two boards 1
and 2 containing analog and digital parts. Transitions
between them are realized by analog to digital and
digital to analog converters (ADCs and DACs). The
digital part may be built on the basis of a microproces
sor or a field programmable gate array (FPGA). Note
that, here, SD 7 with two outputs is used. At the first
output, 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 by
only light intensity modulation (15). Curves 1–4 corre
spond to G0: 500, 2000, 10000, and 20000 s–1
.

7. 846
JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 8 2013
A. M. KURBATOV, R. A. KURBATOV
is used for generation of signal S+(t), which is neces
sary for the dividing circuit and is formed at point 9
circuit.
The Circuitmain purpose of this circuit is full digital
reconstruction of the constant component at point 9,
which is necessary for division operation. There is also
another important application: to abandon formation
of 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, in
the description of the sources of the RR vibration, this
problem was not considered).
For solution of these problems, it is proposed to
construct an additional FB loop, which is used to
branch the reconstructed constant component from
point 9 to both dividing unit 10 and unit 12 with a gain
of 1/Z (in order to compensate for subsequent ampli
fication with coefficient Z in DA 5). This component
is then branched to DAC 13 and device 14 with the
gain α < 1, and, afterwards, enters the second input of
DA 5. As a result, at the output of DA 5, we have the
initial constant component multiplied by (1 – α) 1
(which prevents DA 5 from saturation), which is then
reconstructed to its initial value after device 6 with the
gain 1/(1–α).
The signal from the DAC also enters circuitcircuit 15
controlling the gain of PDA 4, compensating the
vibrational modulation of the light intensity . As a
result, the circuitcircuit solves the problems of division
and stabilization of the signal constant component,
i.e. considerably suppresses the vibration error of RR
measurement.
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 model
has been developed. This model has revealed a new
vibration error, which is associated only with light
power oscillations and is independent of the vibra
tional phase difference. This error is caused by super
imposition of modulation of the bandwidth of the FB
loop (due to intensity oscillations) on the demodu
lated constant component of signal (3à), which, in the
presence of vibrations, also contains a time periodic
component. A circuitcircuit allowing considerable
suppression of the RR vibration error in the closed
loop FOG has been proposed.
REFERENCES
1. A. Ohno, S. Motohara, R. Usui, et al., Proc. SPIE
1585, 82 (1991).
2. H. Lefevre, P. Martin, J. Morisse, et al., Proc. SPIE
1367, 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 ROM
Proc. Symp. Gyro Technology. Inertial Components and
Integrated 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).
Ӷ
4
4
4
1 3 2
4
5
15
14
13 12
6 7
8
9
10 11+
–
Fig. 5. Circuit for division and stabilization of the signal constant componentcircuit: 1 and 2 are the boards with analog and digital
parts 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 output
SD, 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 gains
1/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