Pre-Filtering Low-Cost Inertial Measuring Unit using a Wavelet Thresholding De-noising for IMU/GPS Integration

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1 ISG & ISPRS 11, Sept. 7-9, 11 Shh Alm, MALAYSIA Pre-Filtering Low-Cost Inertil Mesuring Unit using Wvelet Thresholding De-noising for IMU/GPS Integrtion Elkhidir T. Y. 1 & Shuhimi M. 1, Mus T. A., A. Stti 3 1 UTM Aeronutic Lb, Fculty of Mechnicl Engineering, Universiti Teknologi Mlysi, 8131 Skudi, Johor, Mlysi. Emil: khidirty@yhoo.com GNSS & Geodynmics Reserch Group, Fculty of Geoinformtion & Rel Estte, Universiti Teknologi Mlysi, 8131 Skudi, Johor, Mlysi. 3 Electricl nd Computer Engineering Deprtment, Engineering College, Krry University, Omdurmn, Sudn. ABSTRACT Low-cost micro-electromechnicl system (MEMS) inertil sensors now re vilble which llow the development of inertil nvigtion system (INS)/Globl Positioning System (GPS) integrtion. Unfortuntely, MEMS-bsed INS performnce is ffected by stochstic complex errors. This study proposes wvelet multi-resolution nlysis (WMRA) for pre - filtering MEMS - IMU. Two threshold de - noising functions re proposes to rise bove the limittions of hrd nd soft threshold de-noising. Simultion experiment uses MEMS - IMU output dt. The result show tht the proposed threshold de - noising functions compred with soft nd hrd thresholding de - noising methods mkes the de - noising better, nd enhnces the signl to noise rtio. Keywords: MEMS - IMU; wvelet trnsform; thresholding de-noising. 1. Introduction Improving signl to noise rtio () is importnt objective in ctul pplictions in prticulr, nvigtion system; this cn be chieved by filtering out the noise from signl of noisy inertil sensors. The present de-noising method in engineering pplictions is Fourier Trnsform (FT). Wvelet nlysis hs dvntges over FT such s: mplifiction, zoom out, trnsltion nd other functions to overcome the shortcomings tht window size does not chnge with frequency, reserches the chrcteristics of signl by inspecting the chnge under different mgnifiction (Qi & Yi 9). Therefore, using wvelet nlysis for signl processing, give more signl chrcteristics thn Fourier nlysis. From the mid 198s, Wvelet Trnsform (WT) nd wvelet nlysis hve received gret del of ttention in engineering fields, beginning from seismology (Mtti 8). The wvelet trnsform is time-frequency nlysis method which combine with time nd frequency, Donoho (1995) proposed wvelet soft nd hrd thresholding de-noising method bsed on WT, this method esy nd vluble, to be wide rnge of ttention nd reserch. Noise usully mnifests highfrequency signls to threshold processing of high-frequency coefficients of wvelet decomposition (WD). Hrd-threshold hs drwbcks such s, its function is not continuous nd the constnt devition between estimted wvelet coefficients nd tht of noisy signl (Qi & Yi 9). In the softthresholding cse, there re devitions between wvelet coefficient nd estimted wvelet coefficient, which ffect the ccurcy of the signl when reconstructed. 1

2 . Problem Sttement Driven by their low cost nd smll size, MEMS inertil sensors hve been used to produce low cost INS tht cn be widely dopted in severl nvigtion pplictions. However, Klmn Filter (KF) bsed INS/GPS integrtion techniques my not be suitble for MEMS-bsed nvigtion systems, this is for: KF suppresses the inertil sensors noise effect using GPS updtes within smll frequency bnd. If KF technique pplied to MEMS-bsed INS, this technique my be risked the ccurcy becuse MEMS-bsed inertil sensor errors re mixed with the motion dynmics. KF needs certin predefined nd ccurte stochstic model for ech inertil sensor (gyroscopes nd ccelerometers), epiclly in cse of long-term behviour of the corresponding sensor error. In fct, this is extremely difficult to chieve if the MEMS-bsed inertil sensors re involved in the integrted INS/GPS system due to their reltively high mesurement noise. Therefore, the noise level t the output of MEMS-bsed inertil sensors must be reduced nd the sensor errors seprte from motion dynmics prior to processing their mesurements by the KF module. The most pre-filtering techniques used to overcome this problem re WT. There re two mjor problems with WT when pplied to IMU; first problem is tht, the IMU dt hs two frequency levels in sttic nd kinemtic so there re need to multi-frequency filtering. The second problem is tht, the soft-threshold function nd hrd-threshold function hve limittions such s the degree of smoothing is inversely proportionl to the conservtive rel signl informtion. This study proposes WMRA for pre-filtering IMU dt, nd use two modified threshold to overcome the problems of thresholding. 3. Discrete Wvelet Trnsform (DWT) Since the inertil sensor signls re discrete, the DWT is used s n lterntive of the Continuous Wvelet Trnsform (CWT). The DWT for discrete time sequence x (n) cn be stted s (Slm & Ahmed 9): Where is the scle function, designed to rejects high frequency components of the signl nd is the wvelet function nd (, nd ) re the scled nd shifted versions of nd, respectively, bsed on the vlues of S (scling coefficient) nd k (shifting coefficient). The S nd k coefficients tke integer vlues for different scling nd shifting versions of, nd,, respectively. x(n) cn be generted from the corresponding wvelet function using: The difference between the wvelet function nd FT or short-time FT (STFT) is tht the former function is not limited to exponentil functions. The only limittion on is tht it must be short nd oscilltory. This limittion ensures tht the summtion in the DWT eqution is finite. 3.1 Wvelet Multi-Resolution Anlysis (WMRA): WMRA is method used to crry out DWT. It performs different resolution levels to decompose signls.

3 WMRA is proposed tool to develop MEMS-bsed inertil sensors performnce, nd implemented to enhnce the sensors, remove sensor errors tht re mixed with motion dynmics, nd provide more relible dt to the KF module for integrtion MEMS-IMU/GPS. Applying WMRA to MEMS inertil signl comprises two min steps. The first involves eliminting the high frequency sensor noise using wvelet de-noising methods. The second step seprted the motion dynmics from the short or/long term errors nd other disturbnces without deformtion the bsic signl. This step chieve by specifying correct threshold. Disturbnces cn be clssified minly into two types, the first noise type is sensor noise which is normlly Gussin, nd secondry dynmics being coloured or hving sinusoidl signture. The second noise type usully rises in kinemtic mode due to vehicle motion but it cnnot be considered s prt of the motion of interest. Therefore it must be seprte these sinusoidl noise components nd remove them from the true motion signl; for this purpose we must identify their frequency bnd nd ccording to this select the suitble (proper) filter. To this, simply nlyze the power spectrum of inertil sensor output signls in both sttic nd kinemtic modes. It is very difficult to nlyze non sttionry signl in rel time, becuse nlyzing non sttionry signl demnd determintion of the trnsient moment. 3. Wvelet decomposition Wvelet multi-level decomposition implemented to ech IMU signl to estblish MEMS-IMU error used in modelling INS solution error. WD mthemticl procedure cn be performed s following steps (Slm & Ahmed 9) 1. Clculte pproximtion coefficient t S th resolution level (RL) for IMU outputs:. Determine pproximtion using the pproximtion coefficient using: 3. Clculte the detils coefficient t Sth RL: (6). Clculte detil from detils coefficient: 5. Return to step one nd continue the wvelet decomposition process until pproprite level of decomposition (LOD) is reched. 6. Apply threshold function to de-noise the detils of the IMU signls.. Thresholding Algorithm in Wvelet Coefficients Aprt from the Gussin noise, sensor signl is often infected with sinusoidl interference of single frequency nd its hrmonics for some resons such s indequte shielding of the sensor electronics or vibrtions nd secondry dynmics in kinemtic systems. Since sinusoidl interference 3

4 frequency is lwys within the frequency bnd of the motion dynmics, specil tretment is required for its seprtion without degrding the signl of interest (motion dynmics) (Wled 5). For tht reson, it must be successfully removing this sinusoidl interference; this depends on the threshold estimtion nd seprtes the noisy component from the originl signl. To void the limittion of the liner filtering, thresholding is implemented on ll wvelet coefficients. The bsic principle of thresholding here depends on the WT (Goswmi & Chn 1999). By compring threshold vlue with the empiricl wvelet coefficient we cn estimte the empiricl wvelet coefficients. Cutting off some of the noise in the error signl nd improving its cn be chieved by the thresholding procedure. There re two thresholding opertors, the soft nd the hrd opertors, re proposed by Donoho et l. (199, 1995). In the cse of hrd-thresholding, ny wvelet coefficient with n bsolute vlue below the threshold is replced by zero. Coefficients with n bsolute vlue bove the threshold re kept the sme. In the cse of soft thresholding, coefficients with mgnitude bove the threshold re replced (nd hence reduced in vlue) by the threshold vlue. There is lso more sophisticted shrinking function, which utilizes more complex lgorithms to estimte the shrink vlue. The hrd nd soft thresholds re given s: The hrd-thresholding formul: The soft-threshold formul: Where : is the threshold. The soft-thresholding de-noising method cn obtin compression signl which is continuous dmped, nd the compression of the wvelet coefficients is not continuous by the hrd-thresholding de-noising method. Therefore, the soft-thresholding de-noising will usully mke signl smooth, but will lose some of the fetures; while the hrd threshold will retin the chrcteristics of the signl, but lcks the smooth (Qi & Yi 9). There re potentil deficiencies in hrd threshold function nd soft threshold function. In the soft threshold noise reduction method though the whole estimted vlue of the continuity of is better, but when, nd there is lwys constnt bis which directly ffects the ner degree of the signl fter the noise reduction nd the rel signl nd will inevitbly ffect the signl noise reduction effect. In the hrd threshold noise reduction method, the estimted vlue of is not continuous somewhere t nd. At this time the use of estimtes vlue of to reconstruct the signl fter noise reduction my produce some oscilltion (Li 1). In order to overcome the different defects of soft threshold function nd the hrd threshold function. Herein propose to use two modified threshold function, these functions re not new but their ppliction to IMU inertil sensor is novel. First modified threshold function clled comprise is proposed by Song Guoxing nd Zho Ruizhen (1) nd stte tht the threshold formul is given by:

5 Especilly when tking nd 1, the formul is the estimtion methods of hrd threshold nd soft threshold. For generl 1, the method estimtes the size of dt processed between soft nd hrd methods, so clled soft nd hrd threshold compromise lw. The threshold function hs infinite continuous derivtive which is fcilitte to del with vrious mthemticl tretment, nd t the sme time it significntly reduced the constnt bis tht produced from soft threshold method nd improved the reconstruction precision. It cn be seen tht this threshold is better nd more flexible choice reltive to hrd nd soft threshold function. As long s you djust the size of between nd l ppropritely then you cn get better noise reduction effect. Second modified thresholding method clled modulus ws proposed by M liyun nd et l (1) nd stte tht the threshold formul is given by: 5. The simultion bsed on Mt lb In order to show the effect of the proposed thresholding functions upon IMU signls. IMU dt re obtined from internet. The experiment results re showed s figure 1 nd tble 1. The tble 1 shows (Signl-to-Noise-Rtio) nd S (Root Men Error Squre) for the soft, hrd nd the proposed functions for IMU X-xis signl. The results for other IMU signls re present on ppendix A. Tble 1: nd S for ccelerometer X-xis Threshold-function -vlue S Soft Hrd Comprise modulus Tble 1 shows vlues of nd S for ccelerometer X-xis, the following figure 1 shows nd S for this signl with different vlues of (for comprise, nd for modulus ). 5

6 vrition of with vrition of in comprise threshold vrition of with vrition of in comprise threshold vrition of with vrition of in modulus threshold vrition of with vrition of in modulus threshold Figure1: nd S for different vlues of Figure below shows the noisy signl nd de-noised signl for the four thresholding functions De-noising of Accelerometer X-xis Rw Dt with Hrd thresolding method Accel X-xis Rw dt Accel X-xis De-noised dt 8 X-xis Accelertion De-noising of Accelerometer X-xis Rw Dt with Soft thresolding method 81 8 Accel X-xis Rw dt Accel X-xis De-noised dt X-xis Accelertion

7 8 81 De-noising of Accelerometer X-xis Rw Dt with Modulus thresolding method Accel X-xis Rw dt Accel X-xis De-noised dt 8 X-xis Accelertion De-noising of Accelerometer X-xis Rw Dt with Comprise thresolding method Accel X-xis Rw dt Accel X-xis De-noised dt 8 X-xis Accelertion Figure : noisy nd de-noised ccelerometer X-xis signls. 6. Discussion From figure 1 we cn stte tht: for comprise function, when = ; t mximum nd t minimum, this cse is similr to hrd thresholding. Also we cn see tht when chnge in the intervl ( to.1) decrese fst, fter tht (.1 to 1) decrese slowly; while increse grdully. When vlue reches 1 the cse will be similr to soft thresholding. On other hnd for modulus function, when = 1; t minimum nd t mximum, nd this cse is similr to soft thresholding. Also we cn see tht when chnges in the intervl (1 to ) increse fst nd decrese fst, fter tht the increse of nd the decrese of is slowly. Therefore vlue for comprise function must be between.1 nd.9. vlue for modulus function must be chosen greter thn. 7. Conclusion Since the problems exist in soft nd hrd thresholding de-noising function, this study proposed two thresholding functions, compred through simultion experiment, results show tht we cn get very good de-noising effects, nd the hd further improve. The comprise method of the hrd- nd soft-thresholding is clerly better thn the hrd- or soft-thresholding method. The modulus function gives out new threshold function which improves the effect with wvelet threshold. Through comprison with other functions, this function is more flexible nd better in threshold de-noising. 8. References Dvid L. Donoho (1995): De-Noising by Soft-Thresholding. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 1, pp

8 Goswmi J. C. nd Chn A. K, (11): Fundmentls of Wvelets Theory, Algorithms, nd Applictions. nd Edition, John Wiley & Sons, Inc. Li Xipn, Zhng Anbing nd Hu Jing, (1): The Study on Noise Reduction Model of GPS Dynmic Deformtion Dt Bsed on Wvelet Anlysis. IEEE, Vol3, pp M liyun, Dun yonggng, Li yongjun, Wng tinhui (1): Improved Algorithm for De-noising Bsed on Wvelet Threshold nd Performnce Anlysis. First Interntionl Conference on Pervsive Computing, Signl Processing nd Applictions, IEEE, pp Mtti De Agostino, (8): A multi-frequency filtering procedure for inertil nvigtion. IEEE, Qi Zho nd Yi Liu, (9): Genetic Optimized Algorithms in Wvelet Thresholding De-noising. IEEE, pp Slm A. Ismeel nd Mr. Ahmed M. Hssn (11): GPS/INS System Integrtion Bsed on Neuro- Wvelet Techniques. citeseerx.ist.psu.edu/viewdoc/summry?doi= Song Guoxing nd Zho Ruizhen, (1): Three Novel Models of Threshold Estimtor for Wvelet Coefficients. Springer Verlg Berlin Heidlberg, pp Wlid Abdel-Hmid, (5): Accurcy Enhncement of Integrted MEMS-IMU/GPS Systems for Lnd Vehiculr Nvigtion Applictions. PhD thesis, UNIVERSITY OF CALGARY. 8

9 Appendices Appendix (A) results of IMU signls De-noising of Accelerometer Y-xis Rw Dt with Hrd thresolding method Accel Y-xis Rw dt Accel Y-xis De-noised dt 86 Y-xis Accelertion De-noising of Accelerometer Y-xis Rw Dt with Soft thresolding method Accel Y-xis Rw dt Accel Y-xis De-noised dt 86 Y-xis Accelertion De-noising of Accelerometer Y-xis Rw Dt with Modulus thresolding method Accel Y-xis Rw dt Accel Y-xis De-noised dt 86 Y-xis Accelertion De-noising of Accelerometer Y-xis Rw Dt with Comprise thresolding method Accel Y-xis Rw dt Accel Y-xis De-noised dt 86 Y-xis Accelertion Figure A.1 noisy nd de-noised ccelerometer Y-xis signls. 9

10 vrition of with vrition of in comprise threshold vrition of with vrition of in comprise threshold vrition of with vrition of in modulus threshold vrition of with vrition of in modulus threshold Figure A.: nd S for different vlues of De-noising of Accelerometer Z-xis Rw Dt with Hrd thresolding method Accel Z-xis Rw dt Accel Z-xis De-noised dt 7 Z-xis Accelertion De-noising of Accelerometer Z-xis Rw Dt with Soft thresolding method Accel Z-xis Rw dt Accel Z-xis De-noised dt 7 Z-xis Accelertion De-noising of Accelerometer Z-xis Rw Dt with Moduls thresolding method Accel Z-xis Rw dt Accel Z-xis De-noised dt 7 Z-xis Accelertion

11 76 75 De-noising of Accelerometer Z-xis Rw Dt with Comprise thresolding method Accel Z-xis Rw dt Accel Z-xis De-noised dt 7 Z-xis Accelertion Figure A.3 noisy nd de-noised ccelerometer Z-xis signls. vrition of with vrition of in comprise threshold vrition of with vrition of in comprise threshold vrition of with vrition of in modulus threshold vrition of with vrition of in modulus threshold Figure A.: nd S for different vlues of De-noising of Gyro X-xis Rw Dt with Hrd thresolding method Gyro X-xis Rw dt Gyro X-xis De-noised dt X-xis Rte ngle De-noising of Gyro X-xis Rw Dt with Soft thresolding method Gyro X-xis Rw dt Gyro X-xis De-noised dt X-xis Rte ngle

12 -1-15 De-noising of Gyro X-xis Rw Dt with Modulus thresolding method Gyro X-xis Rw dt Gyro X-xis De-noised dt -13 X-xis Rte ngle De-noising of Gyro X-xis Rw Dt with comprise thresolding method Gyro X-xis Rw dt Gyro X-xis Denoised dt -13 X-xis Rte ngle Figure A.5 noisy nd de-noised gyroscope X-xis signls. vrition of with vrition of in comprise threshold vrition of with vrition of in comprise threshold vrition of with vrition of in modulus threshold vrition of with vrition of in modulus threshold Figure A.6: nd S for different vlues of 95 De-noising of Gyro Y-xis Rw Dt with Hrd thresolding method Gyro Y-xis Rw dt Gyro Y-xis De-noised dt 9 X-xis Rte ngle

13 95 De-noising of Gyro Y-xis Rw Dt with Soft thresolding method Gyro Y-xis Rw dt Gyro Y-xis De-noised dt 9 X-xis Rte ngle De-noising of Gyro X-xis Rw Dt with Modulus thresolding method Gyro Y-xis Rw dt Gyr Y-xis De-noised dt 9 Y-xis Rte ngle De-noising of Gyro X-xis Rw Dt with comprise thresolding method Gyro Y-xis Rw dt Gyro Y-xis De-noised dt 9 Y-xis Rte ngle Figure A.7 noisy nd de-noised gyroscope Y-xis signls. vrition of with vrition of in comprise threshold vrition of with vrition of in comprise threshold vrition of with vrition of in modulus threshold vrition of with vrition of in modulus threshold Figure A.8: nd S for different vlues of 13