VNU JOURNAL OF SCIENCE, Mathematics - Physics, T XXII, N04 - 20 06 E R R O R S D E T E R M I N A T I O N O F T H E M E M S IM U L u u M a n h H a , T r a n D u e T a n 1, N g u y e n T h a n g L o n g 1, N g u y e n D i n h D u e 2, N g u y e n P h u T h u y 1D epartm ent o f Electronics an d Telecommunications, College o f Technology, V N U 2Vietnam National University, H anoi Abstract The demand of navigation and guidance has been urgent for many years In fact, INS is daily used in controlling flight dynamics Nowadays, with the strong growth of Micro-Electro-Mechanical-System (MEMS) technology, the Inertial Navigation Systems (INS) is applied widely However, there are existing errors in the accelerometer and gyroscope signals that cause unacceptable drifts There are two kinds of noise in the INS: deterministic and stochastic errors The deterministic noises are usually eliminated by the carefully calibration process but the stochastic noises are always difficult to treat In this paper, we have determined successfully the characteristics of the MEMS sensors’ noise by analyzing the Power Spectrum Density (PSD) and the Allan variance of the experiment data Combining these two methods will give us a reliable noise model that is applied directly to the Noise Eliminating Block (NEB), In tr o d u c tio n Now adays, navigation and guidance a re very im p o r ta n t problem s for m arine, aero nautics a n d space technology In such system s, I n e rtia l M e a s u re m e n t U nits (IMUs) are widely used as the core of the In e rtia l N avigation S y stem s (INS) [1] In principle, an IMU consists of gyroscopes and accelerom eters which m easure a n g u la r velocities and accelerations in th ree dim ensions Recently, th a n k to the developm ent of MEMS technology, the IMƯS become sm aller, c heaper and more precise However, th ere are still problem s w ith M EM S b ased th e IM U s which are necessary to be solved The position error of an IN S in cre ases rapidly with navigation d ue to the in te g ratio n of m e a su re m e n t e rro rs in the gyroscopes and accelerom eters In order to m ake the corrections, th e e rro rs a re classified into deterministic e rro rs a n d stochastic e rro rs [2], To e lim in a te the determ inistic errors, we can specify th em q u a n titativ e ly by calibrating th e device It is, however, more complex in d e te rm in a tio n of the stochastic e rro rs An optim al filter such as K alm an one is often used In this case, the p a m e te rs of those stochastic erro rs m u st n e cessary to be specified In this paper, we h a v e d e te rm in e d noise p a m e te rs of both d e te rm in istic and stochastic errors of M EM S based the IMƯS For the d e te rm in istic errors, a precise te table has been used as a calibration device For the stochastic erro rs, we have trie d two different m ethods PSD and Allan variance The PSD is know n as a classical method to analyze signal, while A llan variance is a new m ethod which can show more E rrors D e te r m in a tio n o f the M ems IMU in fo rm atio n t h a n th e PSD Com bining these two m ethods will give us a reliable noise model t h a t is app lied directly to the Noise E lim in atin g Block (NEB) M e a s u r e m e n t a n d c h a r a c te r iz a tio n In th is s tu d y , we used the MICRO-ISƯ BP3010 which consists of th ree ADXRS300 gyros a n d th re e h e a t compensated ADXL210E accelerom eters [3] The m e a s u re m e n ts a re sy n th e size d by IMU’s micro-controllers and tr a n s m itte d out via RS232 in terface T he u n it tr a n s m its o u u t d a ta as a n g u la r in cre m e n tal and velocity in c r e m e n ta l d a ta in serial fram es of 16 bytes a t one of the user-selectable frequencies of 64 Hz, 32 Hz, 16 Hz or Hz F igure 1: T he MICRO-ISU BP3010 - A MEMS unit 2.1 D e te r m in is tic errors D e te rm in istic e rro rs consist of bias, scale factor, in e rtia l axis m isalignm ent t h a t a re c o n sidered by the following error model [2]: Sabx = a x + a xxa xh + a xya yb + a x.a hz Sah y = a y + a yxa xh + a ^ a ị + a r a z Sa\ = a z + a a aỊ + a ya yb + a a a\ H =A +A X + H = / * , + X + / ? x + X +(Pm a i + p yy y Scob : = p z + Pzxcoh x+ +PXA + ( Ạ X + Pxyyab y + Pxyiab 2)a>b y +{Pxzxab x +(ixzyaby + + PIzcob : + (Pzy x + +Ạ X K +(& x + )(0b y (1) w here ổũị ,ôũ)ị ( i = X, y, z) a re accelerom eter and gyroscope e rro rs expressed in the body fram e ữị - A ccelerom eter b iases [m/s2] a u - A ccelero m eter scale factor [unit less] GCjj - A ccelerom eter in s ta lla tio n error (/ ^ j ) [unit less] - A c celero m eter o u u t in body fram e coordinates [m/s2] Luu M anh Ha, Tran D ue Tan, Nguyen Thang Long /? - Gyro biases [rad/s], p - Gyro scale factor [unit less] - Gyro in stallatio n error (i * j ) [unit less] P ik - Gyro drift depending on acceleration, flexure e rro r [m/s2] ứ / 'G yro o u u t in body fram e coordinates [rad/s] In th is paper, all in stallatio n errors and flexure e rro rs will be neglected because they are very sm all All of rem ainin g d e te rm in istic e rro rs are d eterm ined by the accelerom eter a n d gyroscope calibrations a) Accelerometer calibration u Ạ z N Figure 2: In itial IMU position for up-down calibration In the calibration procedure of the accelerom eters, th e e a r t h gravity h as been used In this method, th e IMƯ is initially positioned so t h a t th e Z-axis of the IMƯ aligned w ith the location level fram es U-axis, the Y-axis of th e IMU aligned with the N-axis and the X-axis aligned w ith the E-axis (Fig.2) It m e a n s th a t th e gravity component will affect only the accelerom eter along Z-axis by a n a m o u n t of +g (g 9.8 m/s2) If the IMƯ is th e n ro ta te d 180° around th e Y-axis, a new m ea su re m e n t could be ta k e n when the accelerom eter along Z-axis sen ses the negative gravity (-g ) W hen the IMU w ith the ith accelerom eter aligned w ith th e U-axis in the navigation frame, the o u u t of th e accelerom eter is: z \ a Ị ) = a { + ( a u + \)g (2) R otating th e IMU 180° a roun d p erp e n d icu lar axis a n d m ak in g a n oth er m easurem ent will give the following o u u t of the accelerom eter: z2( a ) = a ị - ( a a + l)g (3) Errors D e te rm in a tio n o f the Mems IMU Solving se t of e q u a tio n s (2) and accelerom eter b ia s a n d scale factor: (3) above, and we can e stim ate of the , ' ( « * ) - , ’ („,») 2g " The collecting d a ta process is perform ed for about 10 m in utes for each position, th e n th e d a ta is a v erag e d to give z '(a * ) a n d z 2(a*) S et of equations (4) is finally used to e x tra c t th e accelerom eter bias an d scale factor Calibration results showed t h a t th e accelerom eter along Z-axis h a s bias of 0.1330 m/s2 and scale factor of 0.0041 b) Gyroscope calibration The m e th o d is a c alib ratio n procedure t h a t uses a precise te table which contains se q u en c e of d ifferent te s for each dim ension h a s been m ade use The IMU is in itia lly positioned in center of te table and each te is ru n approx im ately for 10 m in u te s The e rr o r model e q u a tio n of the gyro is: w*i = fit + (A i + ! ) ( Wy + ) (5) where w is n o m inal gyro a n g u la r te a t table a n g u la r te Wj [deg/h, rad/s] Wj a v e g e table a n g u la r te for d ata segm ent j [deg/h, rad/s] wex se n se d com ponent of e a r th rotation te [deg/h, rad/s] p ị - gyro b ias [deg/h, rad/s] Pii - gyro scale factor From (5), we have: _ K - w ) - ( „ ) (w, - w2) and _ _ + (6) We can th e n e s tim a te gyro bias scale factor based on Eq.6 R esu lts showed th a t the Z-axis gyro h a s b ias of 0.3172 °/s and scale factor of -0.0070 2.2 S to c h a s tic IM U errors Some stochastic errors th a t affect the Initial Navigation Systems are listed as follows - Q u a n tiza tio n noise Q u a n tiz a tio n noise is m ade from encoding the analog signal into digital form This noise is c au se d by th e sm all difference betw een th e a c tu al a m plitud es of the sam pled sig n a l a n d b it resolu tion of A-D Converter We can reduce quantization noises by im p ro ving encode m ethods, adju stin g sam ple rate , or increasing bit resolution - W hite noise Luu M anh Ha, Tran Due Tan , Nguyen Thang Long 10 W hite noise can be a major source of the IMƯ e rro rs an d it h a s a c o n stan t power spectrum over whole frequency axis Angle random w alk (for gyroscope) and velocity random walk (for accelerom eter) are caused by th e w hite noise - R a n d o m walk This is the random process of u n certain origin, possible of a lim iting case of an exponentially correlated noise with long correlation tim e The gyroscopes a re affected by a n g u la r te random walk, while th e accelerom eters a re affected by acceleration random walk - Flicker noise Flicker noise is low-frequency noise term th a t shows as bias fluctuations in data This noise is caused by the electronics or o th e r com ponents t h a t are susceptible to random flickering In o rder to analyze stochastic IMU errors, we have used the following methods: a) Power spectral density analysis The Pow er Spectral D ensity Analysis (PSD) describes how the power is allotted along th e frequency axis [4] The o u u t d a ta of th e IMU, which is collected during an hour, is analyzed to give th e PSD Fig.3 shows a log'log plot of th e PSD of the X-axis gyro We note t h a t th ere is a bunching of high frequency area It IS difficult to identify the noise term s and the p a m e te rs asso ciated w ith them Thus, the frequency a v erag in g technique [5] h as been used to sm ooth the PSD plot a b Figure 3.T he PSD plot of the X-axis gyro (a) and th e PSD plot ob tain ed by the frequency averaging technique (b) Fig.3.b show s th e PSD plot of the X-axis gyro obtained by the frequency averaging technique The slopes of the curve comprise -2, and It m eans th a t the gyro data includes the angular rate random walk, th e angle random walk and the quantization noise The PSD of the X-axis gyro (see Fig 3b) doesn’t have the inclination o f —1, which m eans th a t the Z-axis gyro lacks the a n g u la r rate flicker noise E rrors D e te rm in a tio n o f the Mems IMU 11 Figure T he PSD of accelerom eter z obtained by frequency averaging technique Fig.4 show s th e PSD plot of Z-axis accelerom eter obtained by using the frequency a v e g in g tech niq ue The slopes of the curve comprise -2, -1, and This curve in d ic a te s t h a t accelerom eter d a ta includes acceleration random walk acceleration flicker noise, velocity random walk, and acceleration quantization noise By u sin g th e conv ertin g form ula in [6], we obtain from the PSD plot the a n g u la r te w h ite noise a n d th e acceleration w hite noise as listed in T ab l T a b l e E s tim a te d white noise in th e in e rtia l sensors Noise term X Y z Angular rate white noise °l4h 0,0560 0,0486 0,0578 Acceleration white noise (m/s)/ h 0.0033 0,0030 0,0028 Analog Device s ta te s t h a t a n g u la r random walk h a s values from to 6°/4 h for the ADXLS300 gyros used in th e MIRCO ISU BP3010 ỈMU If we compare it to Table 1, we can see t h a t the w hite noise level indeed lies w ithin th e limit of the m an u fa ctu rer b) A lla n variance an a lysis The A llan v a ria n c e is sta tistic a l m easu re to characterize th e stability of a tim e-frequency sy ste m [7] T he PSD can only e x tra ct w hite noise sta n d ard deviation In c o n tr a s t, u sin g th e Allan variance, several o th e r e rro r p a m eters can be com prehensively derived The basic id e a of th e A llan variance is to tak e a long d a ta sequence and divide it into se g m en ts b a se d on a n a v erag in g t i m e r to process Let give a sequence with N e le m e n ts y k , k= 0,1, , N - l Then, we define for each n = l ,2 ,3 , ,M £ N /2 a new sequence of a v e g e s of subsequence with length n: Luu M anh Ha, T ran D ue Tan, N guyen T hang Long 12 If th e sa m p lin g tim e is A / , th e tim e sp a n w ith in an a v erag e d sequence of len g th n is r = n A t T h e A llan v a ria n ce , for a given su bsequ ence length n, is defined as: 'N'1 - - c r l ( T ,N ) t ( ~N~ -1 \ n / T he typical slopes of th e A llan ] ( 8) (x J+i ( n ) - X j ( n ) Y j=0 v a ria n c e for th e gyroscope and the accelerom eters in log-log plot a re show n in Fig w ith d a ta collected from the IMU ISƯ BP3010 d u rin g a n hour To d e te rm in e th e noise p a m e te r s , we n e ed to fit th e s t a n d a r d slopes in Fig [8], For exam ple, if d a ta c o n ta in s w hite noise, th e slope -1/2 will a p p ea r in the loglog plot of th e A llan s t a n d a r d deviation Averaging tone (s) Figure T h e s t a n d a r d slopes of th e A llan s ta n d a r d deviation T he log-log plot of th e Allan s ta n d a rd deviation in Fig.6.a indicates th e presence of a n g u la r te q u a n tiza tio n noise (slope -1), a n g u la r te w hite noise (slope -1/2), an g u la r te ran dom w alk (slope 1/2), while a n g u la r te flicker noise (slope 0) is absent This resu lt is fully sistent w ith t h a t o btained by the PSD plot F ig u re 6b show s th e log-log plot of th e A llan s t a n d a r d d eviation for the accelerom eter T h is show s th e presence of a c c elero m ete r q u a n tiz a tio n noise (slope I), a cc elero m ete r w h ite noise (slope -1/2), a c c elero m ete r flicker noise (slope 0), and acceleration ran d o m w a lk (slope 1/2) T his r e s u l t is gain well c o n siste n t w ith t h a t from th e PSD plot In a d d itio n , th is shows th e p resen c e of a cceleratio n tre n d (slope 1) t h a t is u n a b le to be in d ic a te d by only u sing t h e PSD plot E rrors D e term in a tio n o f the Mems IMU a 13 b Figure T he A llan s t a n d a r d deviation of gyro X (a) a n d of acc elero m ete r z (b) The w hite noise coefficient is o b tain ed by fittin g th e slope lin e a t r = Below the table shows th e e s tim a te d noise coefficients for th e gyros and the accelerom eters T a b l e Id e n tifie d Noise Coefficients, u sin g A llan v a ria n ce Gyros Qz(rad) (Quantization noise) X K(rad/s/Vi ) (white noise) B (rad/s) (Flicker noise) 1,504*10-® 1.368M0-5 X 5,617*1 O'7 X Y 1,655*106 1,517*1 o*5 5,315*10* X X z 1,668*10'6 1,535*10-5 5,556'IQ-6 4,892*10'7 X K (m/s2/ V T ) R(m/s3) Accelerometers Qz(m/s) Q(radI y f s ) Q(m/s/ yfs ) B(m/S2) (random walk) R (rad/s2) (trend) X 1,352*10-5 4,734*10'5 4,155*10'5 1.161*105 X y 1,400*105 5,169*10'5 4,713*10'5 7,588’ K)-6 5.0685*107 z 1,339*10 5,688‘ 10'5 4,025*10'5 9,197*10-« 7.4025*1 O'7 C h a c te r X m e a n s t h a t th e sen sor lacks th e e rro r or th is one is m uch sm a ller th an the others c) Com parison between P S D a n d A lla n variance Table show s th e com parison b etw een th e PSD a n d A llan v a ria n ce in extracting w hite noise coefficients The r e s u lts o b ta in e d by th e two m eth o d s a re much closed w ith each o th e r w hich confirm ed a s s e r t th e relia b ility a n d th e accuracy of the e rro r model a p p lied to th e practical I n e r tia l N a v ig a tio n S ystem s Luu Manh Ha, T ran Due Tan , N guyen T h an g Long 14 T a b l e The com parison betw een th e PSD a n d th e A llan variance Accelerometers Gyros PSD ([°/VÃ]) Allan([°/V^ ]) PSD{[m/s/Jh ]) Allan [[m/s/yfh ]) X 0,0560 0.0470 0,0033 0,0028 Y 0,0486 0.0522 0,0030 0,0031 z 0,0578 0,0528 0,0028 0.0026 C o n clu sio n This p a p er h a s succeeded in specifying th e p a m e te r s of th e IMU errors, which is a necessary step w hen applying erro r-p ro c essin g a lg o rith m s for the INS E stim ation of th e stochastic errors is more com plicated t h a n for th e determ inistic ones Both of the two m ethods, PSD and A llan v a ria n ce , hav e been used here to estim ate the stochastic erro rs of the IMƯ I t is show n t h a t th e A llan variance is the more com prehensive m ethod The extracted r e s u lts will be used as the p aram eters in K alm an filter for the INS-GPS in te g rate d system A ck n o w led g em en ts This work is su ppo rted by th e VNU p ro g ram QGTD0509 R eferen ces Vikas K u m a r N, Integration o f In ertial N a v ig a tio n S y ste m a n d Global Positioning S ystem Using K a lm a n F iltering , M Tech D issertatio n , Indian In s titu te of Technology, Bombay, Ju ly 2004 Oleg S Salychev, A p p lied Inertial N avig ation : P roblem s a n d Solutions, BMSTU Press, Moscow R ussia, 2004 Georey J.B u lm er, In M IC R O -IS Ư B P A n O E M M in ia tu re Hybrid Degrees-Of-Freedom Inertial Sensor Unit Gyro S y m p o s iu m , S t u t t g a r t 16th-17th Septem ber, 2003 P e ter s Maybeck, Stochastic models, estim a tio n , a n d control, Academic Press, Vol 1, 1994 IE EE Std 1293-1998, Ieee s ta n d a rd specification fo rm at guide and test procedure for single - a x i s in terferom etric fiber optic gyros Wang, J., Lee, H.K., Rizos, c , G P S / I N S In tegration : Ả S ensitivity A na lysis, U n iversity of New S o u th W ales, Sydney Performance Haiying Hou, M odeling inertial sensors errors u sin g A lla n variance, ƯCEGE reports n u m b er 20201, M aster's thesis, U n iv e rs ity of C algary, Septem ber 2004 Gyro, Accelerom eter P a n e l of the IE E E A erospace, and Electronic Systems Society, D raft recom m ended practice for in e r tia l sen so r te s t equipm ent in stru m e n ta tio n , d a ta acquisition and a n aly sis In IE E E S t d W o rkiig Drafi P 1554/D 14 ... technique The slopes of the curve comprise -2, and It m eans th a t the gyro data includes the angular rate random walk, th e angle random walk and the quantization noise The PSD of the X-axis gyro... ) W hen the IMU w ith the ith accelerom eter aligned w ith th e U-axis in the navigation frame, the o u u t of th e accelerom eter is: z a Ị ) = a { + ( a u + )g (2) R otating th e IMU 180°... the location level fram es U-axis, the Y-axis of th e IMU aligned with the N-axis and the X-axis aligned w ith the E-axis (Fig.2) It m e a n s th a t th e gravity component will affect only the