Multi-scale prediction of MEMS gyroscope random drift based on EMD-SVR

HE Jia-ning, ZHONG Ying, LI Xing-fei

(State Key Laboratory of Precision Measurement Technology and Instruments, Tianjin University, Tianjin 300072，China)

Abstract：To improve the prediction accuracy of micro-electromechanical systems(MEMS)gyroscope random drift series, a multi-scale prediction model based on empirical mode decomposition(EMD)and support vector regression(SVR)is proposed.Firstly, EMD is employed to decompose the raw drift series into a finite number of intrinsic mode functions(IMFs)with the frequency descending successively.Secondly, according to the time-frequency characteristic of each IMF, the corresponding SVR prediction model is established based on phase space reconstruction.Finally, the prediction results are obtained by adding up the prediction results of all IMFs with equal weight.The experimental results demonstrate the validity of the proposed model in random drift prediction of MEMS gyroscope.Compared with a single SVR model, the proposed model has higher prediction precision, which can provide the basis for drift error compensation of MEMS gyroscope.

Key words：random drift; MEMS gyroscope; empirical mode decomposition(EMD); support vector regression(SVR); phase space reconstruction; multi-scale prediction

0 Introduction

With the development of micro-electromechanical systems(MEMS)technology, MEMS gyroscope has been widely used in the field of inertial navigation due to its advantages of low cost, small size, light weight and low power consumption.However, limited to the structural defects and processing technology, the performance of MEMS gyroscope is severely affected by its drift error, especially the random drift.Therefore, the accurate prediction and compensation of random drift are of vital importance for improving the measurement precision of MEMS gyroscope.

The random drift series of MEMS gyroscope have the characteristics of non-stationarity, high nonlinearity and slow time-varying.In recent years, extensive research on the prediction methods of MEMS gyroscope random drift has been performed.A commonly used method is based on the theory of traditional time series analysis represented by auto regressive and moving average(ARMA)[1-4].Traditional time series models are mainly suitable for stationary and linear time series, which have low prediction accuracy about random drift.With the development of artificial intelligence, nonlinear models represented by neutral networks have achieved better results in random drift prediction[5-11].However, considering the complicated mechanism and multiple time scales of random drift series, the prediction accuracy of single models has encountered a bottleneck.To solve this problem, wavelet analysis has been gradually used in random drift prediction of MEMS gyroscope[12-15].Through the time-frequency analysis of drift series on different time scales, the prediction capability of nonlinear models can be fully excavated.Nevertheless, the wavelet basis and decomposition level are difficult to select, which restricts the application of wavelet analysis.Empirical mode decomposition(EMD)algorithm is a signal decomposition algorithm proposed by Huang in 1998.With the EMD, the original time series can be decomposed into a finite number of intrinsic mode functions(IMFs)with different time scales gradually.Compared with the wavelet transform, EMD is adaptive without selecting the basis function and decomposition level in advance.Multi-scale prediction models combining EMD algorithm and nonlinear models have been successfully applied to fiber optic gyroscope and proved to be more effective and accurate than single models[16-18].

In this paper, a multi-scale prediction model based on EMD and support vector regression(SVR)is firstly applied to the random drift prediction of MEMS gyroscope.SVR is a nonlinear model which overcomes the defects of neural networks such as over fitting and easily falling into local minima.Moreover, compared with many other nonlinear models, SVR is more suitable for real-time prediction due to its sparse solution.After the EMD, SVR models are established for each IMF separately to reveal the law of random drift on different time scales.The feasibility and effectiveness of the proposed model are proven by experiments.

1 Basic principle

1.1 EMD algorithm

EMD algorithm is a signal decomposition algorithm proposed by Huang in 1998.It can adaptively decompose the complex original signal into a finite number of IMFs based on the characteristic time scales of the signal itself to realize the separation of different modes.

IMF needs to satisfy two conditions[19]: 1)in the whole data set, the number of extremas and the number of zero crossings must either equal or differ at most by one; and 2)at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

The decomposition steps of EMD are as follows.

1)After finding all the extremas of the original signalx(t), the upper envelopeeu(t)and lower envelopeed(t)are generated by fitting the local maxima sequence and local minima sequence based on piecewise cubic spline interpolation principle, respectively.

2)The mean value of the upper and lower envelope is computed by

(1)

3)The mean valuem(t)is subtracted from the original signalx(t)to get a new signal as

h(t)=x(t)-m(t).

(2)

4)Steps 1-3 are repeated untilh(t)satisfies the IMF conditions.At this time, the first IMF is obtained, which is denoted asc1(t)=h(t).

5)By regarding the residual termr1(t)=x(t)-c1(t)as the signal to be decomposed, the above steps are repeated until the termination condition is satisfied.After a series of decomposition, the original signalx(t)can finally be decomposed into a finite number of IMFs and a residual term as

(3)

ThesemIMFs represent the fluctuations of the original signal on different time scales and their frequency decreases successively.The residual term is monotonous and represents the trend item.The essence of EMD algorithm is to reduce the data complexity by decomposing the original signal into the superposition of several stationary signals.

1.2 SVR model

SVR model is a nonlinear prediction model based on statistical learning theory, which can be viewed as the expansion of SVM model on regression problem.The principle of SVR is as follows[20].

For a given data set {(xn,yn)}(n=1,2,…,N), wherexn∈Rdis the input vector andyn∈Ris the output value, the aim of regression problem is to obtain the optimal decision function betweenxnandyn.

eε(yn,h(xn))=

(4)

Taking the sum of error term and regularizer as the objective function, the primal problem of SVR is expressed as

(5)

g(x)=(w*)TФ(x)b*,

(6)

wherew*is the optimal weight vector, andb*is the optimal offset.

For complex feature transformation, the dimension of feature spaceZis usually large or even infinite, which makes it difficult to solve the primal problem of SVR directly.In this case, the primal problem needs to be transformed into dual problem by Lagrange multiplier method.The dual problem of SVR is still a quadratic programming problem and is expressed as

(7)

K(xn,xm)=exp(-γ‖xn-xm‖2),

(8)

whereγdetermines the width of RBF.By solving the dual problem, the optimal decision function can be obtained as

(9)

whereα∧*andα∨*are the optimal Lagrange multiplier vectors; andb*is the optimal offset which can be derived according to Karush-Kuhn-Trcker(KKT)condition.

SVR model includes three parameters: the error insensitivityε, kernel parameterγand penalty factorC.These three parameters need to be selected in advance and have a significant influence on the performance of SVR model.εdetermines the error sensitivity,γandCdetermine the generalization ability.Improper selection of(C,γ)may lead to over fitting or under fitting.

1.3 Phase space reconstruction

Phase space reconstruction is an important step of nonlinear time series analysis.As a kind of time series with chaotic characteristic, gyroscope drift needs to be embedded in an auxiliary phase space to restore the original dynamic system.The principle is as follows.

For a given time series {x(n)}(n=1,2,…,N), after selecting the embedding dimensionand time delay, the original time series can be reconstructed intoMvectors inm-dimensional phase space as

P(i)=(x(i),x(i+τ),…,x(i+(m-1)τ)),

i=1,2,…,M,

(10)

where the number of vectors in-dimensional phase space isM=N-(m-1)τ.

To ensure the equivalence betweenP(i)and the original time series {x(n)}, C-C method is applied to selectmandτrationally.C-C method, proposed by Kim in 1999[21], is a joint design algorithm ofmandτwith little computation, simple operation and good performance.

The firstM-1 vectorsP(i)(i=1,2,…,M-1)are taken as the input vectors, and the last sampling points of time series {x(n)} are taken as the corresponding output values.Then a data set that containsM-1 samples can be constructed as

Y=[x(2+(m-1)τ)x(3+(m-1)τ) …x(N)].

(11)

All of theseM-1 samples are divided into the training set and testing set for SVR.

2 Multi-scale prediction model based on EMD-SVR for random drift of MEMS gyroscope

The prediction steps of the proposed multi-scale prediction model are as follows.

1)With the EMD, the random drift of MEMS gyroscope is decomposed into several IMFs.

2)According to the time-frequency characteristic of each IMF, SVR prediction models are established based on phase space reconstruction, respectively.

3)The final prediction results are the superimposition of the prediction results of each IMF with equal weight.

The flow chart of the proposed model is shown in Fig.1.

Fig.1 Flow chart of multi-scale prediction model based on EMD-SVR

3 Experiments and results

3.1 Experimental procedures

In order to verify the validity of multi-scale prediction model based on EMD-SVR, the static drift data of MEMS gyroscope CRM100 were collected.The sampling period was 0.01 s and the sampling duration was 80 min.A drift series of length 5 000 was extracted as the experimental data.By eliminating the singularity points and using wavelet threshold denoising method, the following steps were performed.

1)EMD

With the EMD, the drift series of gyroscope was decomposed into nine IMFs and a residual term.Fig.2 shows the raw drift series and decomposition results.Since the residual term representes the trend term of drift series which belonges to deterministic error, the random drift of gyroscope is obtained by eliminating the residual term, as shown in Fig.3.

Fig.2 Raw drift series and decomposition results

Fig.3 Random drift of MEMS gyroscope

2)Phase space reconstruction

The next step was to reconstruct each IMF separately.C-C method was used to determine the embedding dimensionmand time delayτ.

For all IMFs, the last 1 000 samples reconstructed by phase space reconstruction were divided into testing set, and the rest were for training.

3)Sample normalization

To accelerate the convergence speed and improve the prediction accuracy of SVR model, sample normalization is often required before training.Z-score standardization method is widely used for sample normalization and its formula is

(12)

Z-score standardization was performed on the output values and each feature dimension of training set separately.To ensure the consistency of training and testing, each feature dimension of testing set was transformed in the same way as training set.

4)Parameter selection for SVR model

The error insensitivityεwas 0.05, and the model parameters(C,γ)were selected by 4-fold cross validation.

5)Training and testing

Based on the selected model parameters, training and testing were conducted for each IMF.Since the data were normalized before training, anti-normalization was needed to restore the desired prediction results.The formula is

(13)

6)Synthesis of prediction results

The final prediction results of the last 1 000 sampling points were obtained by adding up the prediction results of all IMFs with equal weight.

7)Evaluation of prediction accuracy

To quantitatively evaluate the prediction accuracy of each IMF and the overall, the root mean square errorERMSEand correlation coefficientρwere used as evaluation indices.The formulas ofERMSEandρa(bǔ)re

(14)

(15)

3.2 Experimental results and discussion

In order to further validate the superiority of the proposed model in prediction accuracy, the single SVR model was used for comparison based on the same experimental data.The prediction results of the last 1000 sampling points using different models are shown in Figs.4 and 5, respectively.The parameter selection and prediction accuracy of the two models are listed in Tables 1 and 2, respectively.

Fig.4 Prediction results of multi-scale prediction model based on EMD-SVR

Fig.5 Prediction results of single SVR model

Table 1 Parameter selection and prediction accuracy of multi-scale prediction model based on EMD-SVR

Table 2 Parameter selection and prediction accuracy of single SVR model

As shown in Figs.4 and 5, it is evident that the prediction values of the proposed model are closer to the actual random drift than those of SVR model.According toERMSEandρof the two models given in Tables 1 and 2, the prediction accuracy of the proposed model is higher than that of SVR model.It is proved that the multi-scale prediction model based on EMD-SVR can reduce the data complexity by decomposing the original random drift series into a series of stationary time series so as to achieve better performance than the single SVR model.

4 Conclusions

In this paper, a multi-scale prediction model based on EMD-SVR is proposed for random drift prediction of MEMS gyroscope.The main conclusions are as follows.

1)The random drift of MEMS gyroscope has the characteristic of multiple time scales, which makes it difficult to achieve high prediction accuracy with the single SVR model.

2)With the EMD, the gyroscope random drift is decomposed into several stationary IMFs with the frequency descending successively.These IMFs represent the fluctuations of random drift series on different time scales.Compared with the original random drift series, IMF is much simpler and easier to model.

3)The proposed model reduces the data complexity by using EMD so as to achieve better performance than the single SVR model.

4)Experimental results prove the validity of the proposed model in random drift prediction of MEMS gyroscope, and the prediction accuracy of the proposed model is higher than that of the single SVR model.