Noise estimation and filtering method of MEMS gyroscope based on EMMAP

CHEN Guangwu,YU Yue,LI Wenyuan,LIU Hao

(1. Automatic Control Research Institute,Lanzhou Jiaotong University,Lanzhou 730070，China； 2. Gansu Provincial Key Laboratory of Traffic Information Engineering and Control,Lanzhou 730070，China)

Abstract：Aiming at the problems of low measurement accuracy,uncertainty and nonlinearity of random noise of the micro electro mechanical system (MEMS)gyroscope,a gyroscope noise estimation and filtering method is proposed,which combines expectation maximum (EM)with maximum a posterior (MAP)to form an adpative unscented Kalman filter (UKF),called EMMAP-UKF.According to the MAP estimation principle,a suboptimal unbiased MAP noise statistical estimation model is constructed.Then,EM algorithm is introduced to transform the noise estimation problem into the mathematical expectation maximization problem,which can dynamically adjust the variance of the observed noise.Finally,the estimation and filtering of gyroscope random drift error can be realized.The performance of the gyro noise filtering method is evaluated by Allan variance,and the effectiveness of the method is verified by hardware-in-the-loop simulation.

Key words：micro electro mechanical system (MEMS)gyroscope;expectation maximization (EM)algorithm;noise estimation;unscented Kalman filter (UKF)

0 Introduction

In recent years,with the rapid development of micro electro mechanical system (MEMS),MEMS gyroscopes play an increasingly important role in inertial navigation.However,due to the manufacturing process,the usage environment and other factors,the MEMS gyroscope has low precision which limits its application.The study found that deterministic errors and random noise are two important factors affecting the accuracy of MEMS gyroscope[1].Deterministic errors can generally be eliminated by algebraic compensation calculation methods,while the random noise cannot be disposed by simple methods,which is an important factor affecting the accuracy of MEMS gyroscopes and determines the bias stability of gyroscopes[2].Therefore,how to effectively estimate and filter the MEMS gyroscope to improve its measurement accuracy has become an important research direction of MEMS gyroscopes.

Yang et al.[3]proposed a method to eliminate gyro random noise by Kalman filtering aiming at the disadvantage of larger noise when the noise is severely nonlinear.Gao et al.[4]proposed an adaptive neural network filtering method with the advantage of online learning.But it is too computationally complex to implement.Wavelet analysis was applied to gyroscope de-noising,and effective signal and noise were separated by different characteristics of signal and noise in wavelet domain in Ref.[5].However,wavelet analysis lacks adaptive ability and the wavelet function is difficult to implement.Hu et al.[6]used unscented Kalman filter (UKF)to achieve nonlinear filtering,but the inaccurate statistical information would cause filtering divergence because the filtering performance of UKF depends on the prior statistical information of system noise.An adaptive UKF algorithm proposed by Cai et al.[7]is used to compensate the gyroscope error,while the acquisition of the adjustment factors requires some experience.Hu et al.[8]used the residual sequence and the new interest sequence of UKF to estimate the noise characteristics of the system online.Although this method can improve the adaptive ability of UKF,the steady-state estimation error of the covariance matching method limits the filtering accuracy.

Therefore,in the consideration of the unknown time-varying characteristics of MEMS gyro filtering,expectation maximum (EM)algorithm and maximum a posterior (MAP are adopted to form an adaptive UKF algorithm,called EMMAP-UKF,for achieving gyro noise estimation and filtering processing.The greatest advantage of this approach is that it does not require a priori statistics of accurate noise and has strong adaptability compared with the traditional UKF algorithm.Furthermore,the proposed method is easy to implement.Lastly,the effectiveness of the method is verified by a hardware-in-the-loop simulation platform,and its filtering performance is gived by Allan variance.

1 Noise estimation model based on EMMAP

1.1 MAP-based noise estimation model

The maximum value of the conditional probability density function is usually used as the state estimate in the MAP estimation method.When the noise means ofqandrand noise covariances ofQandRare unknown,the MAP estimation can be obtained by maximizing the conditional probability density as

J*=p[q,Q,r,R|Zk],

(1)

whereZk={z1,z2,…,zk}.

The MAP estimator of the noise statistics can be derived from[9]

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

1.2 MAP optimization model

(10)

whereMis the average value of residual vectors recently obtained[11].If the system random drift noise increases,the actual residual variance increases.The theoretical residual variance value is adjusted by the EM algorithm to make it close to the actual value.Similarly,when the system random noise decreases,the actual residual variance decreases.The theoretical residual variance value can be adjusted by introducing the EM algorithm to reduce it to near the actual value.

The EM algorithm can be divided into two steps:the first step is to calculate the expectation.The current estimate of the hidden variable is used to obtain the maximum likelihood result.The second step is to maximize the estimated results of the first step.Finally,the maximized estimate is used for the next expected calculation,so as to implement the iterative operation.

(11)

whereθis the residual variance value to be estimated[12].

Step M:Find the parameter estimations that maximize the objective function.According to the gradient descent method,when the system parameter satisfies Eq.(11),the parameter is the estimation point that minimizes the objective function as

(12)

By calculating the objective function of the partial derivative ofθ,the estimated result that satisfies the condition according to Ref.[13] is

(13)

Here the estimated value of the theoretical residual method calculated by the EM algorithm is obtained.

2 MEMS gyro error compensation method based on EMMAP-UKF

2.1 Mathematical model of MEMS gyroscope

Considering that the order of the random drift error model of MEMS gyroscope is generally low,for the convenience of processing,the ARMA(2,1)in the time series model is used to describe the random drift error of MEMS gyroscope[14]as

(14)

whereXkis the model estimation time series of ARMA(2,1)andakis the white noise sequence.

The state space model of the corresponding filtering equation can be derived as

(15)

The noise statistical characteristics satisfy the following conditions

(16)

From the system point,we can consider that the random drift error of the MEMS gyroscope is the system output with white noise as input.Assume that the system state vector and the process noise are

(17)

The coefficient in Eq.(15)is

(18)

2.2 EMMAP-UKF filtering

In view of the above analysis,the gyro noise estimation filtering method based on EMMAP-UKF is composed of the steps as follows:

① Initialization of system statistical characteristics[15]

(19)

② Sigma point calculation and construction of statistical characteristic coefficients

(21)

③ Measurement update

(22)

Pk,k-1=

(23)

(24)

(25)

(26)

④ Observation update

(27)

(28)

(29)

⑤ Residual variance update by using EM algorithm and estimation criterion

(30)

(31)

(32)

3 Experiment and discussion

In this paper,a low-performance and low-cost MEMS gyroscope 3DM-E10A is taken to implement experiments.The MEMS inertial navigation posture module 3DM-E10A is fixed on the horizontal static two-axis turntable.After the system is stable for 10 min,it will be automatically adjusted to the horizontal state.Then the output signal of the 3DM-E10A is sampled.The sampling frequency is 20 Hz,and the sampling time is 1 050 s.Due to the large error of the low-precision MEMS gyroscope,it is impossible to distinguish the angular rate of the earth rotation.Therefore,the azimuth error of the gyroscope installation is not considered.The signals are processed by the UKF and the EMMAP-UKF respectively.The inertial navigation attitude module and the two-axis turntable are shown in Fig.1.

Fig.1 Inertial navigation attitude module and two-axis turntable

3.1 Hardware-in-loop simulation

According to the above experimental conditions,the collected raw gyroscope data (taking theY-axis as an example)are organized and plotted,as shown in Fig.2.

Fig.2 Diagram of MEMS gyroscope

Then,the data are processed by filtering algorithm in MATLAB.The UKF and the proposed EMMAP-UKF are used respectively.Fig.3 is a signal diagram after UKF processing,and Fig.4 is a signal diagram of gyro after EMMAP-UKF processing.

Fig.3 Comparison between UKF algorithm and original data

Fig.4 Comparison between EMMAP-UKF algorithm and original data

It can be seen from Figs.3 and 4 that after the introduction of the UKF and the EMMAP-UKF method proposed in this paper,the noise of the gyroscope is greatly reduced,and the signal noises are reduced.However,the amplitude of gyro noise processed by EMMAP-UKF is obviously better than that by UKF,and the signal is more stable.In order to further illustrate the scientificity of the method,the above data are separately processed and calculated by statistical methods.The superiority of the method is illustrated by the signal-to-noise ratio and the zero-bias stability.Table 1 shows the standard deviation of the signal and the calculation of the signal-to-noise ratio.

Table 1 Standard deviation and SNR of signal

As seen from Table 1,after the introduction of the EMMAP-UKF,the standard deviation and signal-to-noise ratio are 0.025 °/s and 0.452 dB respectively.However,the standard deviation and signal-to-noise ratio after UKF algorithm are 0.064 °/s and 0.264 dB.Compared with UKF,the signal-to-noise ratio is increased by 71%.Table 2 shows the zero-bias stability comparison of the MEMS gyroscope after denoising.

Table 2 Comparison of MEMS gyroscope bias stability

The zero-bias stability is to measure the dispersion degree of the gyroscope output around its mean value when the input angular velocity is zero.Therefore,the zero-bias stability can indicate the denoising effect of a method for the MEMS gyroscope signal.From Table 2,it can be found that the zero-bias stability of gyroscope processed by EMMAP-UKF method is obviously better than that by the traditional UKF method.

3.2 Noise analysis based on Allan variance

The Allan variance is a method to analyze the stability of the frequency domain in the time domain.It can not only determine the basic random process characteristics of data noise,but also identify the source of noise[16].This paper further analyzes the proposed filtering method by Allan variance.Fig.5 is the Allan variance analysis chart for original noise data of the gyroscope.Fig.6 is the Allan variance analysis chart of gyroscope processed by the UKF.And Fig.7 shows the Allan variance analysis chart of the gyroscope after EMMAP-UKF processing.

Fig.5 Original data Allan standard deviation correlation time curve

Fig.6 Allan standard deviation correlation time curve after UKF treatment

Fig.7 Allan standard deviation correlation time curve after EMMAP-UKF treatment

It can be seen from the above three Allan variance curves that the slope of the curve is approximately 0 in 2 s-5 s.At this point,it is the bias instability noise.The slope of the curve is approximately -1/2 between 5 s-10 s,which is mainly angular random walk noise.During this time,the gyroscope is greatly interfered by the external disturbances.In 10 s-50 s,the slope of the curve is approximately -1,which is mainly quantization noise.Thus,the diversity of MEMS gyroscope errors can be seen.Through the comparison of Figs.5-7,it can be found that the EMMAP-UKF filtering method proposed in this paper is superior to the traditional UKF algorithm in error filtering.Moreover,the method is effective in restraining all kinds of noise.

4 Conclusions

In order to solve the problem of low precision and high noise of low-cost MEMS gyroscope,a UKF gyroscope noise estimation filtering method based on EM algorithm and MAP estimation is proposed.Based on the estimation of UKF noise matrix by MAP algorithm,the theoretical residual variance is adjusted by EM algorithm to realize the unification of actual residual variance and control residual variance.Finally,the method is verified by using the two-axis inertial turntable.The experimental results show that the proposed method is obviously superior to the UKF algorithm in noise estimation filtering.In addition,the zero-bias stability and Allan variance analysis results are given.It will provide some references for noise estimation and filtering research of MEMS gyroscope.