Proceedings of the 1999 IEEE Canadian Conference on Electrical and Computer Engineering. Shaw Conference Center, Edmonton, Alberta, Canada May 9-12 ...
Proceedings of the 1999 IEEE Canadian Conference on Electrical and Computer Engineering Shaw Conference Center, Edmonton, Alberta, Canada May 9-12 1999
Computer Modelling Of Microelectronic Closed Loop Fiber Optic Gyroscope A. Noureldm', M. Mintched, D. Irvine-Halliday', H. Table? Department of Electrical and Computer Engineering, University of Calgary', and International Downhole Equipment, La2, Edmonton, Alberta, Canada Abstract both counter-propagating beams [2]. Thus, the output signal from the photo-detector becomes an odd sinusoidal function of the Sagnac phase shift from which the direction of rotation can be detected. For low values of the Sagnac phase shift, this relationship can be considered linear [3]. In order to improve the dynamic range at high input rotation rates, a closed loop configuration was proposed using a serrodyne modulator which generates a saw-tooth signal to null out the Sagnac phase shift resulting from the input rotation rate [3]. The frequency of the saw-tooth signal is directly proportional to the rotation rate. Previous models studied the FOG system without any emphasis on the different parameters affecting its performance and with no analysis of the effect of the external environmental factors [4], [ 5 ] . The aim of this study was to suggest new avenues for appropriate FOG testing based on computer modelling.
Computer simulation of the Fiber Optic Gyroscope (FOG) has not been well elucidated in existing studies. In this paper, we present a comprehensive method for FOG modelling. Different parameters affecting the performance of the FOG and its stability and sensitivity are discussed and the performance of the FOG with respect to vibration and shock is analyzed. The simulation results show that: (1) both the sensitivity and the stability of the FOG are improved with increasing both the length of the fiber coil and the gain of the electronics signal processing system, (2) the modeled FOG can have immunity to vibration with resistance against shock up to 9% N.
Keywords : Fiber optic gyroscope, Modelling, Inertial Navigation System.
2. System Analysis
Acknowledgement: This study was supported in part
The closed loop FOG, as shown on Figure 1, consists of the following parts:
by International Downhole Equipment LTD (Edmonton) and Micronet Centers of Excellence (Toronto).
1. Introduction The Fiber Optic Gyroscope (FOG) is a single axis rotation sensor which is currently employed in inertial navigation systems. The operation of the FOG is based on the Sagnac effect stating that two counter-propagating coherent light waves exhibit a relative phase difference on a complete trip around a rotating closed path [l]. This phase difference is proportional to the rotation rate. A directional coupler splits the input optical beam produced by a superluminescent diode light source. into two counter-propagating beams which travel around a fiber coil. Upon rotation a phase difference is exhibited between the two counter-propagating beams. This phase difference (Sagnac phase shift) is proportional to the system platform rotation rate with the proportionality constant dependent on the dimensions of the fiber coil and the wavelength of the optical beam. The receiver averages the instantaneous intensity obtained by the photo-detector and produces an output in the form of an even cosinusoidal voltage, which is a function of the input rotation rate. Thus, the direction of rotation is not detectable. In order to determine the direction of rotation, an AC-bias modulator is used to introduce phase modulation at one of the coil leads for
0-7803-5579-2/99/$10.00 01999 IEEE
Source
I
Directional Coupler
I
Optical
1
Shifter
T
AC- Bias Modulator
Serrodyne Modulator
-b N
V b Figure 1. Closed loop Fiber Optic Gyroscope A single-mode polarization-maintainingfiber optic coil of length L and diameter D is used. The length L of the coil affects the accuracy desired [6]. A superluminescent diode with its narrower spectral width and high emitting
633
power is usually recommended for FOG operation [ 6 ] .An optical directional coupler is used to divide and carry the optical source beam to the opposite ends of the fiber coil. The coupler adds the two returning counter-propagating beams together. A photodetector analyzes the resulting interference beam and provides a voltage signal that is proportional to the intensity of this beam. An AC-bias modulator activates the phase shifter to phase modulate the counter-propagating beams. This phase modulation improves the sensitivity of the FOG and produces direction-sensitive output [2]. A serrodyne phase modulator is used to filter the Sagnac signal and to drive a voltage control phase shifter which cancels out the Sagnac phase shift. The rotation rate can be extracted from the input to the voltage controlled optical shifter [2]. The FOG w i l l be modelled and analyzed as a closed loop system with a rotation rate input and an output signal depending on this rotation rate. Unlike other FOG systems described before [4],[5], this system depends on all FOG parameters. In addition, all factors affecting the overall performance can be studied. When rotating, the FOG sensing coil is an accelerating frame of reference [3]. Due to Sagnac effect, a phase difference between the two counter-propagating beams is produced. This phase difference is known as Sagnac plase shift and related to the rotation rate as follows [3]:
where (ps is the Sagnac phase shift, .
n is
N
Y
? = qs7 where
on Figure 1 to be
8
where; is the power of the main optical beam, A is the amplitude of the sinusoidal signal produced by the A G bias modulator activating the phase shifter and J , (A) is the 1" order ofthe Bessel function of 1'' kind [3].
Unfortunately, this simple open loop structure fails, if the produced Sagnac phase shift becomes large with high input rotation rates. In addition, the open loop system suffers from the non-linearity of the sinusoidal function at high rotation rates. The serrodyne modulator was proposed as a solution to this problem [7]. The serrodyne modulator generates a saw tooth signal which has a maximum value of Z . The frequency and the slope of the saw tooth signal are controlled by the output w signal The serrodyne modulator is introduced to
v.
cancel out the Sagnac phase shift (psproduced by the rotation rate
a. The output signal from the open loop w
v
configuration is fed back to the serrodyne modulator (see Figure 1). This signal is integrated and fed to the voltage control oscillator (VCO). The signal used to drive the VCO is found to be directly proportional to the input rotation rate [7]. Thus, the FOG output signal is taken from the input to the VCO and can be written as [7]:
(4)
As the serrodyne modulator forces the phase shift between the counter propagating beam to be of a very low value, the AGbias phase shifter can be modeled as a gain block in the forward path of the system. The serrodyne modulator is modeled as an integrator block, which integrates the output of the AGbias phase shifter delivering the output of the FOG system. This output is fed back through a gain block to deliver an estimate of the
-
Sagnac phase shift (p3,which is used to cancel the actual Sagnac phase shift produced by the input rotation rate so that the loop is kept locked. The overall system block diagram describing the FOG as a closed loop system is shown on Figure 2. The input/output transfer function described in S-domain is given as
KG H (s) =
N
The output signal is maximized by using A=1.8 which gives J , (1.8) = 0.5815 [3]. Thus, the sinusoidal AC bias modulator produces a direction-
0-7803-SS79-2/99/$10.000 1999 IEEE
(3)
K is a constant equal to poJ,( A ) .
the refractive
index of the fiber coil, 1 is the peak wave length of the optical beam, and c i s the speed of light in free space. Considering the rotation rate R as an input to the FOG system, a gain block can describe the relation between it and the output Sagnac phase shift. This gain block depends on the different optical parameters of the FOG. The phase shift produced by the AC-bias modulator causes the output signal given as [3]:
sensitive output. For low values of the rotation rate and respectively low values of the Sagnac phase shift, . The output voltage in this sin (ps can be expressed as case is directly proportional to the Sagnac phase shift and consequently to the rotation rate.
S +
634
[ E] (%) 27r
2nKG
(5)
I--I I Coil I
I-----I AC-Bias
----
1
Modulator
'!
I L
--VCO
I
1.2 1
1
i
1
b
a¶ U)
I
I
qs
0.8
I
p.
System , I Electronic L,,,,, I
v)
2 0.6 Q
aa
Z 0.4
v
0
Figure 2. The FOG as a closed loop
1 0
A unit step input will be considered for the input rotation rate and the step response will be analyzed. In our model we assumed that the FOG had the following configuration: L =loOOm, D = 0.1 m, A = 0.85 pm, n = 1.48, K4.5815 and G (electronic gain) = 1000. The step response of this FOG model is shown on Figure 3. For the above configuration, the system rise time was found to be approximately 50 seconds. The system is stable since it has only one pole located on the negative part of the real axis of the S-plane close to the origin.
3. Performance Characteristics The performance of the FOG is determined by three major variables:( 1) the bias drift, (2) the angle random walk and (3) the scale factor.
3.1. The Bias Drift Sometimes the bias drift is also called bias instability [8]. It is defined as the deviation in the measured rotation rate at constant temperature. The bias drift is highly affected by the electronic gain G and the length of the fiber coil L. Increasing either L or G reduces the bias drift, increases the degree of stability of the system, reduces the rise time and increases the sensitivity. The step response of the FOG system for different values of L and G is shown on Figures 4 and 5 respectively.
0-7803-5579-2/99/$10.0001999 IEEE
0.5
1
t (hours) Figure 3. The FOG step response. Li1000m; D=O.lm; G=lOOO;A=0.85pm; n=1.48; K=0.5815; In general, the gain of the electronic system is usually kept large enough to ensure small rise time and a tight loop closure [7]. However, the length of the fiber coil can not be increased over 1000 m since the attenuation and dispersion problems related to the optical beam become evident [3]. Typical fiber,coil lengths in practical FOG applications range from 160 m to 1000m.
3.2. The angle random walk The output of the FOG contains a broad band random noise component (white noise) which results either from the shot noise or the thermal noise in the photodetector. This noise is known as the angle random walk [3]. This effect is modeled as a white noise input to a separate system. The transfer function of this system has the same denominator as the original system [I 11. However, the numerator of this transfer function is a constant gain equal to the multiplication of the two gain blocks K and G in the forward path of the original system, since the output of the detector is affected by the AC-bias demodulation and the electronic gain before delivering th'e output of the FOG (see Figure 2). Considering the same FOG configuration used on Figure 3, the effect of the angle random walk is depicted on Figure 6 .
635
1
i
Om2
0
1
o !
Om2 O !
0
d
0.2
0.4
0.8
0.6
1
0.2
0.4
0.6
0.8
t (hours)
1
t (hours)
Figure 6. The FOG step response with the effect of the angle random walk at G = 1000
Figure 4. The FOG step response for different values of L. G = 500. The rise time is 100,200,400 seconds for L =1000, 500,250 meters respectively.
1
1
1J
@ -
@
2
I.* 1
U)
0.8
0.8
0
n
U)
G=1000
0.6
E'n 0.6
CI
0.2
0.4
G=500
0.4
G=250
V
0.2
0 4 0
1
0.1
0.2
0.3
0
0.4
0.2
t (hours)
0-7803-5579-2/99/$10.000 1999 IEEE
0.6
0.8
t (hours)
Figure 5. The FOG step response for different values of G. L = 1000 m. The rise time is 50, 100, 200 seconds for G = 1000, 500,250 respectively. The standard deviation from the steady state value at the output of the FOG (Figure 6) was equal to 0.145 deg.kr. Reducing the electronic gain G reduced the effect of the angle random walk on the FOG output. However, this affected the system sensitivity and increased the bias drift. The FOG output using G = 500 is shown on Figure 7. The standard deviation from the steady state value at the output of the FOG (Figure 7) was equal to 0.112 deghr.
0.4
Figure 7. The FOG step response with the effect of the angle random walk at G = 500
3.3. Scale Factor: The scale factor is the constant of proportionality between the actual rotation rate and the FOG output signal. This relation is given as
ci=*[$]
636
D
The instability of the scale factor (-)
;Lz
1
1.2
is due to the
temperature variation which changes the value of the peak wavelength In general, most of the commercially available FOG products can survive up to temperatures of 80°C [lo].
U)
S
A.
0.8 ul
f!
Q
0.6
0)
t;; 0.4
4. The effect of vibration and shock
Y
*r
The effect of vibration and shock on the operation of the FOG depends on the environment and the application in which the FOG is used. For a particular application in well-drilling for the oil industry, the major source of vibration is the mud flow. Mud circulation is used to carry the measurements from downhole to the surface [ll]. A sinusoidal wave with a frequency of 1.7 Hz was previously used to simulate this vibration effect [l 11. In our FOG model, this vibration impact was added to the white noise and processed the same way. The effect of vibration is shown on Figures 8.a and 8.b for G = 500 and G = 1000 respectively. It is clear that high electronic gain G increases the FOG sensitivity to vibration. In both curves, it can be noted that the FOG signal containing the bias drift and the angle random walk manifests itself as an envelope to the vibration signal. The standard deviations from the steady state value at the output of the FOG for the data displayed on Figure 8.a and 8.b were 0.141 deg./hr and 0.178 deg./hr respectively.
{
0.2
I
0
0
0.2
0.4
0.6
t (hours) Figure 8.a. The FOG step response with the effect of vibration at G = 500
I
1
ho.21
The shock was simulated as a pulse of a very short duration. This pulse was applied to our model at arbitrary time and the effect was analyzed. In general, we considered that the shock effect was acceptable if it caused a steady state error of less than 10 % [12],[13]. Different pulse amplitudes simulating different shock forces were used. It was found that the FOG in our configuration could perform properly up to a shock of 9Og [Newton]; where g is the gravitational acceleration (9.8 m/sec2). This value is similar to those found in the FOG standards of different producers [12], [13]. Figure 9 depicts the shock effect together with the effect of vibration. The configuration used at his simulation is exactly similar to that of Figure 3 with G =500. The time to recover the steady-state after shock was found to be similar to the rise time.
,
,
I
/ 0
t (hours)
Figure 8.b. The FOG step response with the effect of vibration at G = 1000
1.2
1
----, I
I
I
1
c ' U)
0.8 U)
0.6
P
-2 '
0.4
0 0 I
0.2
0.4
0.6
t (hours)
I
Figure 9. The FOG step response with the effect of shock of 90g N. 0-7803-5579-2/99/$10.0001999 IEEE
637
5. Conclusion 4. The system analysis showed that FOGS of particular configurations and dimensions can be modelled as a first order closed loop system. The simulation helps to gain insight into the concepts and the principle of operation of the FOG with a closed loop configuration. Moreover, the effect of all FOG parameters on the overall performance can be easily analyzed. Our modelling indicated that the diameter and the refractive index of the fiber coil together with the wavelength of the optical beam affected only the scale factor. Any nonlinearly in this scale factor can be attributed to the temperature variations. The increase in the length of the fiber cable andlor the gain from the FOG electronic system improved the overall sensitivity and reduced the rise time and consequently the bias drift. However, the increase of the electronic gain increased the angle random walk and made the FOG more sensitive to vibration effects. In addition, the simulation suggested that the FOG could perform adequately (reading errors less than 10%) when submitted to shock forces up to 9Og
5.
6.
7. 8. 9.
10.
N. 6. References Bennett S.M., Emge S. and Dyott R.B.; “Fiber optic gyroscopes for vehicular use”; IEEE Conference on Intelligent Transportation Systems, Boston, Nov. 9-12,1997. 2. Kim B.Y.; “Optical fiber rotation sensing”; Academic press, Inc, N.Y. (USA) 1994. 3. J. M. Aein; “An optical signal processing model for the Interferometric Fiber Optic Gyroscope”; RAND, Santa Monica, C.A. (USA), 1995.
11.
1.
0-7803-5579-2/99/$10.000 1999 IEEE
12.
13.
638
Lin J., Kuo K. and Yeh C.; “Phase domain model of a second order fiber optic gyroscope dynamic system”; IEEE Instrumentation and Measurement Technology Conference; Taiwan, May 10-12,1994. Tanaka R., Kurokawa A., Sat0 Y., Magome T., Hayakawa Y.., Nakatani I., and Kawaguchi J.; “Signal processing for F O G ; Journal of the SPIE on Fiber optic and laser sensors, vol. 2292: 192-202; 1994. Perlmutter M.S.; “A tactical fiber optic gyro with all digital signal processing”; IEEE position location and navigation symposium, Apr. 11-15, 1994. Merhav S.; “Aerospace sensor systems and applications”;Springer,N.Y. (USA), 1993. H. Lefevre; ‘‘ The Fiber optic gyroscope”; ArtechHouse, (USA), 1993. G.F. Franklin, J. D. Powell and M. L. Workman; “Digital control of dynamic systems”; AddisonWesley, Menlo Park, C.A. (USA), 1990. Perlmutter M.S., Reynolds C.1 and Yahalom R.; “Initial production results of a new family of fiber optic gyroscope”; Gyro Technology Symposium; Stuttgart, Germany, Sept. 16-17; 1997. Girard A. and Mintchev M.P.; “Testing non-linear adaptive compensators in non-ideal noise environments”; dh IEEE International Workshop on Intelligent Signal Processing and Communication Melbourne, Australia, Systems (ISPACS’98), NOV4-6, 1998. KVH Industries Inc; “KVH Ccore 2000 FOG’; KVH, Rhode Island, N.Y. (USA) 1998. Fibersense Technology corporation; “FOG 200/ 45 family”; Fibersense, Canton, M.A. (USA), 1998.
