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Physics Procedia 33 (2012) 1817 – 1824

2012 International Conference on Medical Physics and Biomedical Engineering

Application of the Scaled Unscented Kalman Filter with Augmented State in the Fiber Optic Gyroscope Zhongxiao Ji1,2, Caiwen Ma1 1

Xi'an Institute of Optics and Precision Mechanics of the Chinese Academy of Sciences, Xi'an, Shaanxi, China, 710119 2 Graduate University of the Chinese Academy of Sciences, Beijing, China, 10039 E-mail: [email protected], E-mail: [email protected]

Abstract The interferometric fiber optic gyroscope (IFOG) is a nonlinear response system. The noise, which results from the optic and electrical parts of the gyro for the environmental disturbance, includes both nonlinear noise and additive noise, and degrades the bias stability of the gyro. In this paper, the nonlinear state equation about the closed-loop IFOG is deduced from analyzing the sources, characteristic and effect of noise, and then the scaled unscented Kalman filter (SUKF) with augmented state is presented to reduce the noise. The experiment showed that: by adjusting SUKF parameters, it evidently decreases the standard deviation and Allan variance of the bias stability. Therefore this algorithm achieves better performance of filtering and improves effectively the bias stability of the gyro.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Open access under CC BY-NC-ND license. Key words: fiber optic gyroscope, nonlinear noise, Kalman filter , bias stability

Allan variance

1. Introduction The interferometric fiber optic gyroscope (IFOG) is a kind of the new type all-solid-state inertial instrument, which is made up of the superluminescent diode (SLD), fiber coupler, Y wave-guide, fiber coil, the photo detector and the circuit parts [1]. When IFOG is affected by the environmental change such as temperature, electromagnetic interference, vibration, thermal stress and circuit noise, there is the non-reciprocal random phase shift induced from the fiber coil. Due to IFOG is a nonlinear response system, the response to this random phase shift is also nonlinear and regarded as the nonlinear noise. The noise is characteristic of white noise and flicker noise (so called 1/fr noise) [2]. Flicker noise is a nonstationary random process and is mainly reason to degrade the bias stability of the gyro. The scaled unscented Kalman filter (SUKF) is a nonlinear filter, which improves the performance of unscented Kalman filter (UKF). This improvement overcomes dimensional scaling effects by calculating

1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. Open access under CC BY-NC-ND license. doi:10.1016/j.phpro.2012.05.290

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Zhongxiao Ji and Caiwen Ma / Physics Procedia 33 (2012) 1817 – 1824

the transformation of a scaled set of sigma points, at the same time the filtering accuracy can be maintained [3]. Because of nonlinear noise and additive noise in the gyro, the conventional SUKF is unsuitable for the noise of IFOG. So in this study the state vector of the standard SUKF is augmented so as to be represented of the noise. Finally the experiment showed that SUKF with augmented state can enhance effectively the bias stability of the gyro. 2. Nonlinear Response of IFOG IFOG is a kind of the inertial instrument based on Sagnac effect. When the gyro rotates, IFOG produces interference in the fiber coil, and then induces the Sagnac phase shift which is proportional to the rotation rate of the gyro [1].The response to the rotation rate of the gyro is: P P0 [1 cos( R )] (1) R Where P0 is the light source power; is the Sagnac phase shift. Equation (1) shows that the response R is a raised cosine function, as shown in Fig.1. So IFOG is a nonlinear response system.

Figure 1.

The nonlinear response of IFOG

At the zero point of the response, its slope is zero approximately. So when the gyro rotates slowly, the response is insensitive to the gyro rotation. To get high sensitivity and judge the rotational direction, IFOG employs the phase biasing modulation so that IFOG operates at the point with a nonzero response slope. A stable phase bias is applied in the gyro’s signal [1]. Then the response to the rotation rate of the gyro is: P P0 [1 cos( R b )] R 2 =P0 [1 sin( R )] Where b is the phase bias and is generally equal to /2, the response to the gyro becomes a sine function with a stable zero point, as shown in Fig.2.

Figure 2.

The effect of stable bias vs. unstable bias

While the Sagnac phase shift changes in a small range, the response to the rotation rate is approximately linearity. 3. Nonlinear Noise in IFOG

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The Y wave-guide in the IFOG is a linear phase modulating integration optics based on a lithium niobate (LiNbO3) substrate, which is sensitive to the voltage signal. The phase bias modulating voltage signal, which drives the Y wave-guide, becomes unstable for white noise and flicker noise in the gyro’s circuit, which causes the unstable phase biasing modulation [4], as shown in Fig.2. The consequence yields random phase bias drift, which mainly results from the random flicker noise of the circuit. In addition to the transient non-reciprocities there are two additional sources of true non-reciprocity: the magneto optic Faraday effect and Kerr effect. Faraday effect is induced with polarization maintaining fiber. Due to the random rotation of the birefringence axes of practical polarization maintaining fibers, which yields a state of polarization with a small random ellipticity along the fiber coil [5]. Faraday effect induces the random phase error which is a flicker noise. The second non-reciprocal effect is Kerr effect. An imbalance in the power levels of the counter-propagating light waves can produce a small phase difference for nonlinearity of the refractive index induced by the high power density in the fiber. Therefore slow variations in the splitting ratio in the fiber coil may translate directly into bias drift [1]. The temperature and the stress, which act on the fiber coil of the gyro, cause the refractive index changing alone the fiber. The effect yields the random birefringence in the fiber, which results in the difference between the time that two light waves spending in the fiber. The effect produces the random phase bias error [6]. At the same time, the influence of vibration and the D/A in the circuit may be equivalent to the random phase shift which is characteristic of flicker noise. Consequently the nonlinear response of the gyro is as follows: P K[1 cos( R )] W (3) R b is the total random phase shift which is the response to the disturbance in the gyro, K is the Where circuit gain and the light power detected in the photo detector, w is white noise of the circuit. 4. Nonlinear State Equation of IFOG The closed-loop IFOG employs the square wave biasing modulation: the fixed phase bias is applied periodically in the positive and negative half-periods of the square wave. The gyro’s response in the positive half-periods is: P PP K[1 cos( R )] (4) R b P W and the response in the negative half-periods is: PN

R

K[1 cos(

R

b

N

)]

N W

(5)

The difference between the responses in the positive and negative half-periods is regarded as the gyro’s output signal: (6) P 2K sin( R R a ) cos( s) w Where a

(

s

(

w

N N N w

P

)/2

P

)/2

P w

Due to using the polarization maintaining fiber in the fiber coil and SLD, the random phase shift is reduced in a certain extent, and its change between the positive and negative half-periods is very small. So (6) may be simplified by (7). P ( R ) 2K sin( R (7) a) w Therefore the discrete time nonlinear state equation of the closed-loop IFOG may be represented by (8):

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yk

Where xk

R

k , vk

w

k ,

k

a

xk xk 1 wk 2K sin( xk ) vk k

(8)

k ,

wk is the process noise, E ( wk ) 0 , Pw cov( wk ) , vk is the observation noise, E (vk ) 0 , Pv cov(vk ) , is the random noise, E ( k ) 0 , P cov( k ) , k and that vk , wk and k are irrelevant to each other.

5. SUKF with Augmented State In (8) the system state equation is linearity and the measurement equation is nonlinearity. Furthermore the measurement equation includes not only nonlinear noise term but also additive noise term. So the standard SUKF is unfit for (8), and the random state vector needs to be augmented as follows: xa

T

x , w , v ,

(9)

The augmented state dimension is: N nx nw nv n (10) Similarly, the augmented state covariance matrix is built from the covariance matrices of x, w, v and : Px 0 0 0 0 Pw 0 0 (11) Pa 0 0 Pv 0 0 0 0 P As to the discrete nonlinear state equation of the closed-loop IFOG, the step of SUKF with augmented state is as follows [7-9]: 1) Initialization xˆ0 E[ x0 ] Px0

E[( x0

xˆ0 )( x0

xˆ0a

E xa

E xˆ0

xˆ0 )T ] 0 0 0

T

Px P0a

E

x0a

xˆ0a

x0a

a T 0

xˆ

0 0 0

0 Pw 0

0 0 Pv

0

0

0 0 0 P

2) Computing Sigma points a 0

xˆ0a

i

a i ,k 1

xˆka 1

N

a i ,k 1

xˆka 1

N

Pka 1 Pka 1

0

i 1,......, N i

i i

N 1,......, 2 N

(12)

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a i,k 1

[ 2

x i ,k 1

N

L

w i,k 1

v i,k 1

i,k 1

W0

m

W0

c

Wi

m

N N Wi

c

2

1 1 2 N

i

0

i

0

(13)

i 1,......, 2 N

]T

N

Where (0 1) and L( L 0) are tuning parameters of the scaling, is a non-negative weighting parameter introduced to affect the weighting of the zeroth sigma point and can be used to incorporate knowledge of the higher order moments of the distribution. For a Gaussian prior the optimal choice is 2 . Wi m and Wi c is the weights [9]. 3) Time update Transform the sigma points through the state update function. x x w i =0,......,2N (14) i ,k / k 1 i , k -1 k -1 Calculate the apriori state estimate and apriori covariance: 2N

xˆk

Wi m

x i,k k 1

(15)

i 0 2N

Pxk

c

Wi

x i,k / k 1

x i,k / k 1

xˆk

T

xˆk

(16)

i 0

4) Measurement update Transform the sigma points through the measurement update function: Yi , k / k 1 2K sin ix, k / k 1

v k 1

k 1

i =0,......,2N (17)

and the mean and covariance of the measurement vector is calculated: 2N

yˆ k

Wi m Yi .k / k

(18)

1

i 0 2N

Pyk

Wi

c

Yi , k / k

1

yˆ k

Yi , k / k

yˆ k

1

T

(19)

i 0

The cross covariance is calculated according to 2N

Pxk yk

Wi

c

x i,k / k 1

xˆk

Yi , k / k

1

yˆ k

T

(20)

i 0

The Kalman gain is given by, Kk

Pxk yk Pyk

1

and the SUKF estimate and its covariance are computed from the standard Kalman update equations, xˆk xˆk K k yk yˆ k Pxk

Pxk

K k Pyk K kT

(21)

(22)

6. Experiment

In the experiment, the static data of a digital closed-loop IFOG were filtered by SUKF with augmented state. And then the standard deviation and Allan variance was applied to examine the effect of the filter. The static data of the gyro were shown in Fig.3.

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Figure 3.

The output of the closed-loop IFOG

In the process of filtering, the SUKF parameter a was equal to 1 and was 2. The gyro’s bias and the change of the standard deviation and Allan variance with various tuning parameter L was separately shown in Fig.4 and Fig.5. From Fig.4 and Fig.5 it can be seen that: with the tuning parameter L increasing, the standard deviation and Allan variance of the gyro’s bias stability decreased gradually, as shown in Table.1, which denoted that the bias stability of the gyro was improved evidently. But there was a drawback that the convergent time of SUKF became long with the parameter L increasing, so the selection of the parameter L had to consider the specifications of response time of the system.

Figure 4.

The result of filter with various tuning parameter L


Figure 5.

Allan variance analysis

Table 1 The Result of SUKF with augmented state

Parameter L Raw Data L=0 L=10 L=20 L=30

)

Bias stability of the gyro Mean

14.6517 14.8708 14.7475 14.7362 14.7359

Standard deviation

0.3086 0.1042 0.0729 0.0635 0.0585

Allan variance

0.008313 0.001996 0.001152 0.000896 0.000794

7. Conclusions

The closed-loop IFOG is a nonlinear response system. The noise in the gyro includes both nonlinear noise and additive noise, which decays the bias stability of the gyro. In this paper, the SUKF with augmented state is employed to filter the data of the gyro. The experiment showed that the algorithm can reduce availably the effect of the noise and enhance obviously the bias stability of the gyro.

Acknowledgment

The author would like to appreciate the help of Professor Jintao Xu.

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