A MODIFIED ADAPTIVE KALMAN FILTER FOR FIBER OPTIC GYROSCOPE

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1 Électronique et transmission de l information A MODIFIED ADAPTIVE KALMAN FILTER FOR FIBER OPTIC GYROSCOPE VOLKAN Y. SENYUREK, ULVI BASPINAR, HUSEYIN S. VAROL Key words: Fiber optic gyroscope, Adaptive Kalman filter, Allan variance, Noise covariance matrix, Angle random wal (ARW), Rate random wal (RRW). In this article, an adaptive Kalman filter was designed in order to filter angle random wal (ARW) and rate random wal (RRW) noises observed in the output of the fiber optic gyroscope (FOG). Especially in case of rapid maneuvers, a new innovation approach, which is based on adaptive mechanism, has been introduced to overcome the shortcomings of conventional and innovation-based adaptive Kalman filters. Process and measurement noise covariances Q and R were estimated successfully and by using these parameters comparisons of conventional and adaptive Kalman filters were made on both real and artificial signals. The results of this study shows that the designed filter reduces ARW, RRW and flicer noises successfully.. INTRODUCTION The noise and drifts in FOG output should be eliminated in order to increase the measurement accuracy of FOG. Kalman filter (KF) is the most widely used effective technique [ 5]. To design Kalman filter, statistical model of random processes in FOG output should be nown correctly. Random processes in FOG output are modeled by being examined with analysis techniques such as Allan variance and power spectral density (PSD). In accordance with the statistical information obtained as a result of the analysis, measurement and process noise covariance matrices R and Q are estimated for KF. Conventional KF using constant noise variance provides quite good performance when FOG is quiescent. However in many practical applications, the performance of the Kalman filters reduces because the value of R and Q matrices are not nown correctly. Especially in case of rapid changes such as maneuver, filtering becomes fully inefficient. Therefore, adaptive filtering techniques should be used. There are two basic approaches for adaptive Kalman filtering: multiple model based adaptive estimation (MMAE) and innovation based adaptive estimation (IAE). MMAE system includes more than one KF and calculates the weighted state variables in parallel way [6]. MMAE University of Marmara, Turey, Department of Electronics and Computer, Faculty of Technical Education, istanbul; ubaspinar@marmara.edu.tr Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 59, 2, p , Bucarest, 204

2 54 Volan Y. Senyure, Ulvi Baspinar, Huseyın S. Varol 2 hasn t been widely used because of its complicated calculation [7]. IAE method predicts R and Q matrices by using residuals obtained from the results of prior states [8, 9]. IAE method taes as basis the fact that whether the innovation series is white noise or not [6]. This approach becomes inefficient for the non-white noises such as rate random wal (RRW) in FOG output. Dynamic sliding window is used in some studies in order to increase the performance of adaptive Kalman filters in maneuvering state. In these studies window size of Kalman filters is changed by using maneuver detection algorithms [7]. The disadvantages of this method are use of excessive memory and process load. In this article, an adaptive Kalman filter was designed for stochastic model for ARW and RRW noises in FOG output. The proposed method filtered successfully the signals in FOG output during both quiescent position and maneuver. Also, this method did not require additional memory use. 2. STOCHASTIC MODEL OF FIBER OPTIC GYROSCOPE In FOG signals, there are errors including unpredicted stochastic processes. These processes are seen as noise in the output or as slow parameter changes within time. Since the contribution of these errors in a certain moment cannot be nown definitely, this affects the performance of the gyroscope negatively. In the analysis of stochastic processes, power spectral sensity and Allan variance analysis are main methods. According to IEEE STD 952, Allan variance approach is a powerful method which is preferred for stochastic error identification [0,]. In data analysis with Allan variance, it is assumed that the uncertainty in the data is produced from noise sources having a specific character. The covariance value of each noise source can be predicted from the data [2]. In IEEE standards, Allan variance σ Ω 2 (τ) for Ω angular rate is defined as () for finite number of samples. K 2 2 Ω M [ + M M ] 2( K ) = σ ( ) ω ( ) ω ( ), () where M is the number of the sample in a cluster and K is the number of the cluster. ω (M ) is the mean of the cluster -th. Gyroscope output is affected by several types of noises. The main five of them are angle random wal (ARW), rate random wal (RRW), flicer noise which is main reason of the bias instability (BI) in the FOG, quantization noise (QN) and rate ramp (RR) [3]. After obtaining Allan variance for different correlation times, the noise parameters are calculated by using least squares method or log-log axis graph of σ(τ) [4, 5]. The simplified stochastic model for FOG is shown in Fig.. Flicer and quantization noises were not shown in the simplified model.

3 3 Modified adaptive Kalman filter for fiber optic gyroscope 55 Fig. Simplified stochastic model. A proper state-space model of FOG should be formed in order to use KF. After stochastic noises are determined with Allan variance model, state-space model is formed. State-space model comprises two equations. The first one of them is the process model (2) which determines how the states within the system changes with control signal or noise, and the other one is measurement model (3) where measurements are obtained from state variables. Discrete time indication of state space model is as follows: x ' [ w ] E w j ' [ v ] E v j = Φx + Γw +, (2) z + + = Hx v, (3) = Q δ = R δ j j Q = 0 R = 0 = j, j (4) = j j (5) where x is system state vector; Φ is system state matrix; Γ is noise matrix; z is observation vector; H is observation matrix; w is white noise with zero mean and it is called as process noise. v is called as measurement noise and it is white noise with zero mean. Q and R are process and measurement noise covariance matrices respectively. Among the random noise sources of FOG, only ARW is white noise. The other noise sources are non-white and the ones which have private spectrum. While ARW noise can be expressed directly in state-space model, this is not possible for other noises. The modeling of the non-white noises is provided by using shaping filters. Shaping filter is a linear system which converts the white noise to the desired spectral function [6]. Expanded model is formed by using proper shaping filters for obtaining state-space model. In this article, ARW and RRW noises were evaluated as white and non-white measurement noises respectively and the following model was formed.

4 56 Volan Y. Senyure, Ulvi Baspinar, Huseyın S. Varol 4 x x [ + ] RRW [ + ] Φ = 0 Φ x x 0 [ ] RRW RRW [ ] Γ + 0 Γ 0 RRW 0 v RRW [ ], (6) x z = H H + v x [ ] [ ] [ ] + RRW ARW RRW[ ]. (7) 3. KALMAN AND ADAPTIVE KALMAN FILTER Kalman filter (KF) is a discrete estimator which predicts the state space variables of a dynamic system. It is commonly used for signal filtering in navigation systems [7]. KF eliminates the noises and errors by using the state space model of the system and the uncertainties within the system. These uncertainties are called as measurement and process noise. Kalman filter comprises two equation groups: time updating (prediction) and measurement updating (correction) equations [8]. Time updating equations are as follows: x, (8) = Φˆ x P ˆ, (9) T = ΦP Φ + Q where xˆ denotes the estimated state vector, x is the predicted state vector for the next epoch, Pˆ is the estimated state covariance matrix and P is predicted state covariance matrix. Measurement updating equations are as follows: T ( HP H + R ) T = P H K, (0) ( z Hx ) xˆ = x + K, () ( I K H ) P P ˆ =, (2) where Q is process noise covariance. K is Kalman gain and determines the updating weight between measurement and state prediction [9]. Q and R should be nown completely for the effective use of KF. These parameters are estimated with the statistical information obtained as a result of the analysis and are used in KF by being accepted as constant. However in many real applications, the performance of KF reduces or the filter may become divergent within time since Q and R covariance values cannot be nown completely or they may vary within time. Various adaptive methods were examined in order to reduce the effect of indefinite parameters. In the MMAE approach, many Kalman filters

5 5 Modified adaptive Kalman filter for fiber optic gyroscope 57 are operated in a parallel way. Each filter has a different model and each filter maes its own state prediction and then the system calculates the weighted state variables [6, 20, 2]. In the references [6, 8, 9], IAE method was examined. Innovation based adaptive estimation method predicts R and Q matrices by using residuals obtained from the results of prior states. 4. INNOVATION BASED MODIFIED ADAPTIVE ESTIMATION Following the procedures proposed by (Mehra, 970, 97; Mohamed and Schwarz, 999), in IAE approach, v is the difference between the actually observed value and the predicted value. In the adaptation of R, it is assumed that Q is nown and it is constant. In this case, R is adapted as follows: v = z Hx, (3) ˆ T Cv HP H, (4) R = ˆ where Ĉ v is predicted variance-covariance matrix and it is calculated by using the differences between the measurement and the predicted value. Ĉ v is obtained by calculating the mean within a movable window with m size [9]. m T Cˆ v = v iv i. (5) m i= In the event that ARW measurement noise in FOG output increases, R should also increase proportionally for maing the filtering optimum. In the mentioned adaptive structure, variance-covariance value increases since the difference between the measurement and prediction values will increase. Thereby the adaptation of R is provided. However when FOG shows rapid changes, this adaptive filter structure remains slow even compared to conventional KF in respect to following the signal. If Q is assumed as constant, the value of R should be decreased in order to provide quic follow-up of Kalman filter in case of maneuver. In case of quic maneuver, the difference between measurement value and the predicted output will increase. In this adaptive approach, R value also increases during passing moment since this situation is considered as the increase of measurement noise. In this case performance reduces significantly because the error in filtering will increase. This negative situation can be prevented by maing the following addition to the adaptation formula in (6). The basic approach here is the fact that mean value of the noise should be zero. In the optimum state, the mean of the differences between the observed data and the predicted data is approximately zero. Diverging of innovation mean from zero indicates that the adaptation is not sufficient. The disadvantages of KF are reduced by increasing the adaptation rate with the

6 58 Volan Y. Senyure, Ulvi Baspinar, Huseyın S. Varol 6 insertion of innovation mean to the formula. In Fig. 2, the comparison of conventional KF, adaptive Kalman filter (AKF) and modified Kalman filter (MAKF) on an artificial signal is shown. An ARW noise whose measurement noise variance is approximately 3 was added to a signal in the form of a step function whose angular rate is 0 degree/hour. Q was ept constant as 0-5 for each of three Kalman filters. The size of window was taen as 00 samples in adaptive filters. 2 ˆ ˆ T = + m R Cv HP H vi. (6) = m i Fig. 2 Comparison of filtering methods on step function. The presented method gives more correct results compared to the others in the region where signal changes rapidly (Fig. 2). When the constant region of the signal is examined, it is seen that AKF and the presented MAKF have the same performance. It is assumed that R is nown completely the Q is approximately as follows in the stable state: Qˆ = K Cˆ K. (7) The adaptation of Q taes the innovation information as basis as it is in R. Q is estimated depending on the difference between the observation data and the predicted data. Filtering performance remains insufficient during the rapid changes in course of maneuver. In addition, a correct adaptation is not able to be provided with the increase of RRW noise in FOG. In order to eliminate these problems, the innovation mean which is applied in the adaptation of R was also taen here and (7) was rearranged as follows: v T m T Qˆ = KCˆ vk vi. (8) m In Fig. 3, the comparison results of KF, AKF(Q) and MAKF(Q) on gyroscope and artificial signal are shown. ARW value in the signal is deg/h /2 and RRW value is.55 deg/h 3/2. R value was taen as 0.03 for each of three Kalman filters. Q for KF was taen as In the presented AKF, the i=

7 7 Modified adaptive Kalman filter for fiber optic gyroscope 59 dynamic model presented in (7) and (8) was used as model. The window width was taen as 00 samples in AKF. When Fig. 3 is examined, it is seen that KF and AKF do not provide a quic follow-up during sharp edge changes in angular rate. The presented method provided much closer results to the ideal value. Fig. 3 Comparison of KF, AKF(Q) and MAKF(Q) methods. Table Comparison of Kalman filtering methods for constant angular rate Standard variance (deg/h) KF AKF(Q) MAKF(Q) In Table, the variance values of the data filtered for the region of gyroscope which has constant rate are shown. The variance of the signal filtered with the presented modified Kalman filter (MAKF) is less compared to other methods. 5. EXPERIMENTAL STUDY The method which is explained above and tested with artificial signals was carried out with the signals received from a specific type of gyroscope. In order to form stochastic model, the output signal was recorded for 0 hours in the rate of 250 samples/s by placing FOG on a still horizontal table. The Allan variance analysis of the recorded data was made by using (). The performance of FOG was obtained by determining the noise parameters in FOG output with the use of asymptotic approach (Fig. 4). In these figure, there are four straight lines which have different slope. The standart deviation of these straight lines at different time points are used for calculation of noise parameters [2]. Table 2 summarizes the obtained noise parameters of the FOG. KF model was formed by using ARW and RRW coefficients determined at the end of Allan variance analysis and by using the state-space model given in (6) and (7). The method which is presented by applying a damped oscillation to test platform was compared with KF and AKF.

8 60 Volan Y. Senyure, Ulvi Baspinar, Huseyın S. Varol 8 In Fig. 5, the section where the angular rate is high and the forms filtered with KF and AKF(Q) are shown. In this comparison, R (30) value was ept as constant. In KF, Q was taen as The window width was,000 samples in the AKF and MAKF. Fig. 4 Allan standard deviation analysis of fiber optic gyroscope. Table 2 Noise parameters of fiber optic gyroscope ARW BI RRW RR deg/h /2 0.2 deg/h.55 deg/h 3/ deg/h 2 It was seen that AKF(Q) gives a correct output in a delayed way for a certain time. MAKF(Q) followed FOG information quicly in this region. The quiescent state and filtered forms of FOG are shown in Fig. 5b. KF and AKF(Q) produced results which are very close to each other. MAKF(Q) provided better filtering compared to other methods. Fig. 5 a) FOG signal filtered with KF, AKF and MAKF; b) filtering of quiescent part of FOG signal. Comparison of filtering methods in respect of Allan variance analysis is shown in Fig. 6 and Table 3. Allan variance analysis was made in the () as stated after ten-hour of gyroscope output is filtered. KF AKF(Q) provided a slight improvement on ARW value. It did not provide any improvement on other noise parameters. The presented MAKF(Q) reduced significantly ARW, BI and RRW

9 9 Modified adaptive Kalman filter for fiber optic gyroscope 6 values. RR remained same. The measurement covariance is related to measurement noise so the adaptation of the R parameter reduces only the ARW noise. Fig. 6 Allan standard deviation analysis of three filtering methods adaptation of Q. Table 3 Comparison of filtering methods in respect of Allan variance analysis Raw data KF AKF(Q) MAKF(Q) AKF(R) MAKF(R) ARW (deg/h /2 ) BI (deg/h) RRW (deg/h 3/2 ) RR (deg/h 2 ) CONCLUSIONS In this article, a modified adaptive Kalman filter, which reduces the drifts in FOG output, was presented for two situations: a) angular rate eeps constant b) angular rate changes rapidly. The presented modified adaptive Kalman filter was developed by taing the innovation principle as basis. The filter was tested with both real and artificial signals, and it was compared with conventional KF and AKF. The presented method eliminated successfully ARW, RRW and flicer noises in FOG output. The ARW and flicer noise parameters were improved in the proportion of 70%. RRW was reduced in the proportion of 43%, therefore the sudden angular rate changes in FOG signal was transferred successfully to the filter output, so the filter performance in the maneuvering case is greatly improved. ACKNOWLEDGMENTS This study was supported by Marmara University Scientific Research Projects Support Fund (FEN-C-DRP ) and the doctorate scholarship program of The Scientific and Technological Research Council of Turey (TÜBİTAK). Received on 9 April 203

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