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In this paper, an integrated MEMS gyroscope array method composed of two levels of optimal filtering was designed to improve the accuracy of gyroscopes. In the firstlevel filtering, several identical gyroscopes were combined through Kalman filtering into a single effective device, whose performance could surpass that of any individual sensor. The key of the performance improving lies in the optimal estimation of the random noise sources such as rate random walk and angular random walk for compensating the measurement values. Especially, the cross correlation between the noises from different gyroscopes of the same type was used to establish the system noise covariance matrix and the measurement noise covariance matrix for Kalman filtering to improve the performance further. Secondly, an integrated Kalman filter with six states was designed to further improve the accuracy with the aid of external sensors such as magnetometers and accelerometers in attitude determination. Experiments showed that three gyroscopes with a bias drift of 35 degree per hour could be combined into a virtual gyroscope with a drift of 1.07 degree per hour through the firstlevel filter, and the bias drift was reduced to 0.53 degree per hour after the secondlevel filtering. It proved that the proposed integrated MEMS gyroscope array is capable of improving the accuracy of the MEMS gyroscopes, which provides the possibility of using these low cost MEMS sensors in highaccuracy application areas.
Angular rate sensors have wide applications in the automotive, aerospace and consumer electronics sectors. Gyroscopes fabricated by microelectromechanical system (MEMS) technology offer revolutionary improvements in cost, size, and ruggedness relative to fiberoptic and spinning mass technologies. At least twenty angular rate sensors of this kind with different structures and principles have been presented by various groups in the past twenty years. However, the current stateoftheart MEMS gyroscopes have lowgrade performance and can not compete with the established sensors in highaccuracy application areas such as guidance and navigation, where the bias drift of the gyroscope is the most popular term used to define performance. At the end of 2007 the best published accuracy for a MEMS gyroscope was about one degree drift per hour [
The advantages of using multiple sensors over a single sensor to improve the accuracy of acquired information about an object have been recognized and employed by many engineering disciplines ranging from applications such as a medical decisionmaking aid system to a combined navigation system [
Through analyzing the current various multisensor fusion methods, we find that these approaches could be improved further in several ways for better accuracy improvements. In the virtual gyroscope, Bayard established the covariance matrices of process noises and measurement noises separately for the Kalman filtering. The parameters of the noise covariance matrices were stationary. However, the work conditions of the sensor especially the gyroscope are subject to changes due to many other factors such as in the case of high maneuverability. In such case, adopting the values of stationary noise sources could result in large estimation errors and low filtering performance. In order to well satisfy the requirement of high rate maneuvering operating conditions, Lam proposed a high order gyroscope model which contained scale factor and misalignment errors in addition to the usual noises such as angular random walk (ARW) and rate random walk (RRW). And a fifteenstate Kalman calibration filter was designed to enhance the performance of estimation [
Therefore, in this paper we will combine both homogeneous and heterogeneous sensor data fusion to improve the MEMS gyroscope accuracy. The proposed method consists of two levels of robust and optimal estimators. In the first level, several gyroscopes will be combined into a single effective device through minimum variance estimation approach. In the second level, the output of first level will be integrated with external aiding sensors such as magnetometers and accelerometers to improve the gyroscope accuracy to a better degree.
In the proposed integrated MEMS gyroscope array method, the correlation between the elements in the array is the theoretical basis of improving the accuracy. Bayard had given the accuracy improvement relationship with the correlation factor between the sensors as shown in
However, the separate MEMS gyroscopes are independent of each other. Therefore there should be no correlation between the separate sensors. But if we could fabricate these MEMS gyroscopes simultaneously on a single silicon chip within a very narrow area of about several square millimeters through various micromachining process as shown in
The proposed MEMS integrated gyroscope array method was composed of two levels, as shown in
In most applications, the gyroscopes will not be used alone but combined with other sensors such as accelerometers or magnetometers in an attitude determination system. Therefore in the integrated gyroscope array, the output of first level was undertaken by the second level filtering with aiding of the external sensors. With the external signals, the random errors of MEMS gyroscope are set as the state vectors of integrated filter. Then the output of the firstlevel Kalman filter could be compensated by the optimal estimation of random errors through the integrated filter, therefore the accuracy will be improved to a larger degree.
The Kalman filtering approach has the advantage of being a systematic method to ensure minimum variance rate estimation. Furthermore, it is suitable for dealing with dynamic data and has high realtime performance in comparison with other minimum variance estimation approach. So the Kalman filtering was used here to implement the integrated gyroscope array technology.
The systematic errors of gyroscope due to biases, scale factors and misalignments could be compensated for via an onboard Kalman filtering approach [
Where
Using the above gyroscope model the units for each term were unified through the scaling as shown in
Computation of the Allan variance is a powerful method for estimating the gyroscope random noise sources [
After the random noise modeling, the filter design which provides the optimal estimation is the key to success of the proposed accuracy improving method. As Kalman filtering was widely used in the combined navigation system and was used by the previous researchers, herein the algorithm will be also used as the foundation for the filter design.
In this paper we took three gyroscopes to form the sensor array. In order to improve the accuracy we treated the true angular rate
In
After discretion, the
By this the firstlevel filter for the integrated gyroscope array was established (
In the various application areas of MEMS gyroscopes, there are more external sensors such as the magnetometers and accelerometers for combination with gyroscope signal. The information from external sensors is usually to determine the attitude of the aircraft, but also can be used to improve the accuracy of gyroscope by heterogeneous sensors data fusion. Herein in the secondlevel filter, the gravitational field and earth magnetic field information obtained from the accelerometers and magnetometers were used to be integrated with output of firstlevel filter to enhance the accuracy of gyroscope. In order to well describe the establishment of the secondlevel integrated Kalman filter, it needs to make some definitions as shown in
Based on Euler's rotation theorem, the error quaternion
After differentiation and simplification,
In
The integrated Kalman filter used the information of earth magnetic and gravity field as external calibration from which the random errors of MEMS gyroscope were deduced. According to the attitude quaternion, the error of Direction Cosine Matrix was written as:
Direction Cosine Matrix from
In
We defined
Then the measurement residual of magnetometer in
If the similar deduction was carried out for the measurement residual of accelerometer, then the following linear measurement equation was obtained:
So the measurement vector was established as:
The error quaternion and estimation error of rate random walk were defined as the state vectors. According to the state and measurement equations established above, the Kalman filter was established as following by using
The Kalman filter coefficient matrix
In the measurement noise vector
We took three separate MEMS gyroscopes with identical specifications to form a gyroscope array because the gyroscope array as shown in the
In the secondlevel integrated Kalman filter, the outputs of magnetometer and accelerometer were used to feed into the Kalman filter to correct the drifts of gyroscope which were the outputs of firstlevel Kalman filter. The ARW noise matrix
Three gyroscopes with same specifications were used for the firstlevel optimal filtering. When the correlation factor is set about 0.5, the bias drift of three MEMS gyroscopes with lower accuracy of 35 degree per hour was improved to 1.07 degree per hour by the firstlevel self compensation in the gyroscope array (
The direct comparison with virtual gyroscope method would be useful to verify the proposed method in addition to the experiment. However such direct comparison is not possible because of the difficulty to attain the same group of drift data to compare both methods. Therefore the verification in the paper was only implemented through such experiment.
Furthermore, the drift was improved to a better degree of about 0.53 degree per hour (
The correlation factor plays an important role in this integrated MEMS gyroscope array method. As shown in
In this paper an integrated MEMS gyroscope array method to improve the accuracy of MEMS sensors was presented. Contrasted with conventional methods for improving the accuracy of MEMS gyroscopes, which focused on the improvements on interface circuitry or mechanical sensing scheme, the method proposed here enhanced the accuracy from the view of algorithm design approach. Through experiment results, it was proven that the proposed method is very effective at improving the accuracy of MEMS sensors. The firstlevel optimal filter is very effective for improving the accuracy through homogeneous sensors data fusion within the identical sensors array. The accuracy improvement largely depends on the correlation between the sensors. How to obtain the real correlation factor in the sensor array is a main problem in the implementation of this method. The secondlevel integral optimal filter can improve the accuracy of the gyroscope further through heterogeneous sensors data fusion techniques.
In the future fabrication of the MEMS gyroscope array on a single chip and the achievement of online correlation factor and filtering would provide a real smart high accuracy angular rate sensor. On the other hand, the measurement random noises of external sensors can be set as a part of the estimator states to get higher accuracy.
The authors gratefully acknowledge Chinese National Science Foundation's financial support (Contract No. 50505038) and Chinese HiTech Research and Development Program's financial support (Contract No.2006AA04Z306).
Normalized drift of a four component virtual gyroscope versus different correlation factor.
Schematic of singlechip gyroscope array.
Structure of the proposed integrated gyroscope array method.
Random error model with unit scaling for the MEMS gyroscope.
Allan variance plot of three gyroscopes' bias drift.
Principle block diagram of firstlevel Kalman filter.
Improvements of the bias drift through firstlevel filtering of gyroscope array.
Improvement of the bias drift through secondlevel integrated filtering.
Plot of drift reduction versus different correlation factor
Error terms obtained through the Allan variance analysis.
Error terms  Allan variance  Unit  Slope 

ARW 


1/2 
Bias Instability 

deg/ 
0 
RRW 

deg/ 
+1/2 
Definitions of symbols for the integrated filter design.
Symbols  Description 

Body frame of aircraft  
Navigation frame  
Attitude quaternion  
Attitude error quaternion  

Direction Cosine Matrix (DCM) 
Measurement value of Earth magnetic  
Measurement value of gravity field  
Estimation of rate random walk 