Research on Modeling and Compensating Method of Random Drift of MEMS Gyroscope

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1 01 4th Interntionl Conference on Signl Processing Systems (ICSPS 01) IPCSIT vol. 58 (01) (01) IACSIT Press, Singpore DOI: /IPCSIT.01.V58.9 Reserch on Modeling nd Compensting Method of Rndom Drift of MEMS Gyroscope Yun Li +, Fengyng Dun nd Znping Li Deprtment of Control Engineering, Avition University, Chngchun, Chin Abstrct. Aiming t the performnce of Micro Electro Mech-nicl Systems (MEMS) gyroscope influenced by rndom drift lrgely, its error compensting method is nlyzed. Bsed on the time series nlysis method, the rw mesurement from the MEMS gyroscope is processed nd modeled by the Autoregressive (AR) model. With the AR model, time-vrying fctor dptive Klmn filter (AKF) is designed. The results show tht, fter filtering in the cse of constnt nd chnging ngulr rte, the stndrd devitions re 16% nd 70% of tht before filtering, nd the drift error cn be reduced by the filter effectively. Keywords: MEMS, rndom drift, AR model, AKF. 1. Introduction MEMS gyroscopes, with the dvntges of smll size, light weight nd low power consumption, re more nd more widely pplicted in low cost inertil system. As the precision level of MEMS gyroscopes re not high till now, even in low precision mesurement system, the rndom drift estimtion nd compenstion of gyroscopes re quite necessry. And the min wys of improving the system precision re the model identifiction nd filtering of the MEMS gyroscopes rndom drift [1]. The common methods of modeling of gyroscopes rndom drift lwys get high model order nd lrge mount of computtion, such s neurl networ [] nd wvelet nlysis [3], which pplicted in low cost system cn hrdly ensure rel-time property. Bsed on the time series nlysis theory, n AR model of gyroscope rndom drift is estblished. Then we process the gyroscope rndom drift with time-vrying fctor dptive Klmn filter, nd the method, which is helpful in improving rel-time property of the low cost system, is efficient with low computtion.. Error Model Modeling bsed on the time series nlysis theory requires collecting long-term mesurements, nd ensure tht the signl is zero mens vlue, sttionry nd norml [4]..1. Dt cquisition nd processing In this pper, we te STIM0 gyroscope [5] s reserch object. Under the condition of constnt temperture 5 C,we recorded 1 hours mesurements of gyroscope t uniform intervls of two seconds while it is sttic nd processed the output of Z-Axis. The rw dt contins constnt nd rndom components. So fter the men of it hs been removed, the rest is the signl of gyro rndom drift. During the long-term of smpling, the chnge of electricl prmeters nd outside environment could cuse the signl to generte trend, so it is necessry to te processing to ensure sttionry. + Corresponding uthor. E-mil ddress: @qq.com. 48

2 The liner trend in the signl cn be removed by the first or second order differentil tretment [6], [7]. Fig. 1 shows the signl fter it hs been processed. Fig.1. Processed dt of gyro rndom drift... Model Estblishment After processed, the sttic dt from the MEMS gyroscope turns into rndom error signl. The result of signl testing through reverse order testing method nd its third-order nd forth-order cumulnts show tht the signl hs stisfied modeling of time series nlysis theory requirements. Auto-Regressive nd Moving Averge (ARMA) Model [8] is bsic nd widely used time series model, nd it is defined s X ϕ X ϕ X ϕ X t 1 t 1 t m t m = θ θ θ t 1 t 1 t n t n X t is the time series { X t } t timet, m nd n re the orders of AR model nd Moving Averge (MA) model, ϕ i ( i= 1,,, m) θ j ( j = 1,,, n) is the AR prmeter, is the MA prmeter nd { } t is the stndrd devition of the sensor white noise. θ j = 0( j = 1,,, n) When, ARMA model reduces to AR model s Xt = t + ϕ1xt 1+ ϕxt + + ϕnxt n () ARMA model nd AR model hve different chrcteri-stics in uto-correltion function (ACF) nd prtil uto-correltion function (PACF). In AR model, the ACF is tiled nd the PACF is truncted, while in ARMA model both of ACF nd PACF re tiled. Therefore, the model of signl cn be identified by the chrcteristics of its ACF nd PACF. The ACF nd PACF of gyro rndom drift signl re presented s figure, which shows tht the PACF cuts off wheres the ACF hs jgged exponentil decy. Therefore the AR model is dopted to model the gyro rndom drift signl. After the prmeters of ech low order model hve been estimted by lest-squre method, the order determintion test of the AR model cn be conducted vi Aie Inform-tion Criterion (AIC) nd Byesin Informtion Criterion (BIC), which cn be normlly written s AIC( p) = N lnσ + p (3) BIC( p) = N lnσ + p ln N (4) p denotes the order of model, N denotes the number of Smpling points nd σ denotes is the vrince of residul t. And the test result is presented in Tble 1. As shown, by incresing the order of the AR model, AIC nd BIC vlues don t decrese remrbly. Although those vlues re the lower the better, the mount of computtion rises with the cubic of the order of AR model in the system. To further improve the system reltime property, the AR (1) model is used for the rndom drift error model. After the Model pln hs been finlized, it is necessry to perform the pplicbility test, testing whether { t } is white noise. By clculting, the one-step uto-correltion coeffic-ient of t is ρ,1 = nd the two-step cross-correltion coefficient between t nd X t is ρ X, = Both of the coefficients pproch zero, so { t } is considered to be white noise nd the model is pplicble. (1) 49

3 Fig.. Vlue of ACF nd PACF. Tble 1: Vlue of AIC nd BIC of Models ( 105) Model Criterio n AR(1) AR() AR(3) AR(4) AIC BIC The model is expressed in the form of X = ϕ X + t t 1 t NID σ (5) t ~ (0, ) 3. Filter Design After the drift error model hs been determined, we choose Klmn filer (KF) [8] to compenste the drift error. The liner stte eqution nd mesurement eqution re given s follows X = X +Γ W (6) φ, Z = HX + V (7) X is the system stte vector t time t φ,, 1 t is the stte trnsition mtrix between time point 1 nd t, Γ 1 H is the system noise distribution mtrix, is mesurement observtion mtrix, W 1 nd V re the rndom error of system nd mesurement nd they re per the following error sttistics Q EW [ ] = 0, CovW [, Wj] = Qδ j (8) E[ V] = 0, Cov[ V, Vj] = Rδ j (9) Cov[ W, V j] = 0 (10) nd R re vrinces of W nd V. The regulr KF estblished on this bsis cn effectively reduce the rndom drift error of gyro in the sttic or constnt ngulr rte stte, but in the stte of chnging ngulr rte the KF doesn t wor well. Through the nlysis we lerned tht the filter model chnged while the ngulr rte ws chnging. The more drmtic chnges in ngulr velocity, the pplicbility of the filter model is poorer, cuses the greter system error. But the direct reson for this is tht the old mesurement vlue influences the estimte of the current too much. Incresing the gin of current mesurement vlue in the stte estimtion eqution is n effective men to solve this problem. Bsed on the drift model before, AR(1) model tes the plce of KF stte eqution nd vrince of residul Q tes the plce of σ. H is set by 1. The AKF multiple updte steps my be performed One-step predicted stte estimte Updted stte estimte Filter gin Xˆ = ϕ Xˆ (11) / 1 1 X ˆ = X ˆ + K ( Z X ˆ ) (1) / 1 / 1 K P P R 1 = / 1 ( / 1+ ) (13) 50

4 One-step predicted estimte covrince Updted estimte covrince / 1 = sϕ P 1 + σ P (14) P = ( I K) P/ (15) 1 s s A fctor is dded into recursive eqution of regulr KF. The vlue selection of is not only to be ble to reduce the error under the condition of chnging ngulr rte, but lso to ensure the ccurcy in the constnt ngulr rte stte, nd expressed s follows ω0 s = (16) ω0 Δω Δω ω is the first order differentil of gyroscope mesurements t the current moment, 0 is the reference vlue obtined by experiments. Δω In the condition of constnt ngulr rte, pproches zero nd s pproches 1, the AKF doesn t chnge too much from KF. When the ngulr rte is chnging, s will djust the filter. 4. Simultion nd Experiment 4.1. Constnt ngulr rte experiment In order to verify the effectiveness of the model nd filter, use both regulr KF nd AKF methods to process the sttic dt t first. We set ϕ = nd X ˆ 0 = 0, P 0 [10] nd R [11] re set by 10 times of stndrd devition of rw dt nd one tenth of vrince. Filtering effect is shown in figure 3 nd figure 4, nd the error reduces obviously. The stndrd devition of error before nd fter filtering is shown in tble. The result shows tht the filter nd model estblished in the sttic condition re resonble. Compred with the regulr KF, the ccurcy of AKF doesn t reduce. Under the condition constnt ngulr rte, we get the sme result s it in the sttic condition. After processed by the two inds of filter, the stndrd devition of error is reduced to 16% of the rw dt s. Becuse of the limited spce, the detil description of the experiment is omitted. 4.. Chnging ngulr rte experiment In the sme environment condition, we te out prt of rw chnging ngulr rte dt (figure 5) from the MEMS gyroscope for experiment. Filtering effect is shown in figure 6 nd figure 7. The result shows tht filtering chnging ngulr rte dt with the regulr KF get poor effect nd the error vlue is chnging with the ngulr rte chnging, while the error of AKF isn t influenced by ngulr rte chnging lmost. The stndrd devition of error before nd fter filtering in the tble 3 shows tht AKF cn reduce the rndom drift error under the condition of chnging ngulr rte effectively. From bove, it shows tht bsed on the AR model before, regulr KF just cn reduce the rndom error in the cse of constnt ngulr rte, while AKF cn not only ensure the ccurcy in the constnt ngulr rte condition, but lso improve the filter effect when ngulr rte is chnging. Thus it cn be seen tht this lgorithm wored well. Fig.3. Klmn filter result of sttic experiment. Fig.4. Adptive Klmn filter result of sttic experiment. 51

5 Tble : Stndrd Devition of Error In Sttic Experiment Rw Dt Regulr KF AKF Stndrd devition (deg/s) Fig.5. Chnging ngulr rte dt of gyro. Fig.6. Klmn filter result of chnging ngulr rte experiment. 5. Conclusion Fig.7. Adptive Klmn filter result of chnging ngulr rte experiment. Tble 3: Stndrd Devition of Error In Chnging Angulr Rte Experiment Rw Dt Regulr KF AKF Stndrd devition (deg/s) In this pper, we nlysis the MEMS gyroscope output rw dte, extrct the rndom drift signl bsed on requirements of time series nlysis theory nd estblish n AR(1) model for filter. The results of experiments show tht in the constnt ngulr rte cse, both regulr KF nd AKF hve good performnces, but when the ngulr rte is chnging regulr KF loses efficcy, while AKF still performs well nd the stndrd devition of drift error reduces by 30%. The AKF lgorithm bsed on AR model, which is simple, effective nd high rel-time property, hs some prcticl vlue in MEMS gyroscope-bsed low cost integrted nvigtion system. 6. References [1] H. L. Chng, L. Xue, nd W. Qin, An integrted MEMS gyroscope rry with higher ccurcy output, Sensors, 008(8), pp

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