Élcio Jeronimo de Oliveira1,2,*, Waldemar de Castro Leite Filho1, Ijar Milagre da ... 1Instituto de Aeronáutica e Espaço - São José dos Campos/SP –– Brazil.
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Inertial Measurement Unit Calibration Procedure for a 5HGXQGDQW 7HWUDKHGUDO *\UR &RQ¿JXUDWLRQ ZLWK :DYHOHW 'HQRLVLQJ eOFLR-HURQLPRGH2OLYHLUD1,2,*:DOGHPDUGH&DVWUR/HLWH)LOKR1,MDU0LODJUHGD)RQVHFD2 Instituto de Aeronáutica e Espaço - São José dos Campos/SP – Brazil Instituto Nacional de Pesquisa Espacial - São José dos Campos/SP – Brazil
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Abstract: The aim of this paper was to present a calibration procedure applied to an inertial measurement unit FRPSRVHG RI D WULDG RI DFFHOHURPHWHUV DQG IRXU J\URV LQ D WHWUDG FRQ¿JXUDWLRQ 7KH SURFHGXUH KDV WDNHQ into account a technique based on least-square methods and wavelet denoising to perform the best estimate RI WKH VHQVRU D[LV PLVDOLJQPHQWV 7KH ZDYHOHW DQDO\VLV WDNHV SODFH LQ RUGHU WR UHPRYH XQGHVLUDEOH KLJK IUHTXHQF\ FRPSRQHQWV YLD PXOWLUHVROXWLRQ VLJQDO GHFRPSRVLWLRQ DQDO\VLV DSSOLHG J\UR VLJQDOV (TXDWLRQV IRU WKH OHDVWVTXDUH PHWKRGV DQG ZDYHOHWV DQDO\VLV DUH SUHVHQWHG DQG WKH SURFHGXUH LV H[SHULPHQWDOO\ YHUL¿HG Keywords:,QHUWLDOPHDVXUHPHQWVXQLW)LEHURSWLFJ\UR:DYHOHW,08WHWUDG,08FDOLEUDWLRQ
INTRODUCTION
*(20(75,&&21),*85$7,21$1$/ȦxȦyȦz]T is the vector RIDQJXODUUDWHDERXWPDLQD[HV ȦLVWKHDQJXODUUDWHHVWLPDWH vector; and H* is the generalized inverse of H Equation 4 provides the best state estimation in the least VTXDUHVHQVH
3DULW\YHFWRU 7KHVHQVRUHTXDWLRQFRQVLGHUHGLQ(TFDQEHUHZULWWHQ ZLWKDGGLWLRQRIIDXOWVELDVHVDQGQRLVHFRPSRQHQWVDVLQ(T gu o = H~ + dg o + f + hs
Equation 10 leads to a least square estimate of Ȧ,QDFWXDO situations, bias, fault, and random-term vectors are unknown DQGGH¿QHGDV]HUR7KLVHVWLPDWHLVHTXLYDOHQWWR(TDQG FDQEHH[SUHVVHGE\(T ~t LS = (RV )-1
gu o
(12)
7KHPHDQLQJRI(TLVWKDWLIVHQVRUVIDXOWVDQGELDVHV are zero, the resulting product of parity vector with sensor PHDVXUHPHQWVLVDZKLWHQRLVHZLWK]HURPHDQ2WKHUZLVH LIWKHELDVHVDQGRUIDXOWVYDOXHVGLIIHUIURP]HUR(TLVD ³ZHLJKWHG´VHQVRUHUURUVVXPPDWLRQ7KHQU2T is the parity vector (Y) obtained from null space of H
(5) 0,6$/,*10(17(5525(48$7,21
ZKHUHįJo is a constant term (bias) vector; f is the fault vector; and Șs LV D UDQGRP WHUP YHFWRU *DXVVLDQ QRLVH DQG RWKHU UDQGRPGLVWXUEDQFHV Applying the singular value decomposition (SVD) on H, the range and null spaces from this matrix can be obtained (Shim DQGrbH = Hj >~ y(+) -~ y(-)H rc ~ z(+) -~ z(-)
(16)
mia /SFi 6 gvi(+) - gvi(-)@ = 6ra rb rc@ >mib /SFiH + hi mic /SFi
Performing N HYDOXDWLRQV N IRU (T LW FDQ EH REWDLQHG(T mia /SFi >mib /SFi H = (RS R) -1 RS G mic /SFi
where, V R S grvi (1) - grvi (2) W ra (1) rb (1) rc (1) S grvi (3) - grvi (4) W R => h h h H and G = S W h S W ra (k) rb (k) rc (k) S grvi (n - 1) - grvi (h)W X T
and gvi is the mean value of the ith sensor measurement, N is the number of r-vectors, and n is the number of commanded VHTXHQFHVRQWZRD[LVWDEOH ,Q RUGHU WR VROYH (T WKH FRQVWUDLQW LQ (T LV necessary, mia2 + mib2 + mic2 = 1
After estimating the scale factors and misalignments, and UHZULWWHQ(TWKHVHQVRUELDVHVFDQEHREWDLQHGDVLQ(T 1 n u r n R1 (SF (gr v) n - H~ n) = b
(20)
– where, +Ѻ 0+, b is the mean value of bias, and (gv)n is the mean value of sensor output at nth WXUQWDEOH VHTXHQFH 7KHJ\URVXVHGLQWKH,08 )2* KDYHELDVHVZLWKWKHVDPH order of the Earth rotation rate (ȍE 7KXVLWLVDFRPSOLFDWHG task to separate one from another in a skewed sensor FRQ¿JXUDWLRQV,Q(TWKLVSUREOHPLVVROYHGE\SHUIRUPLQJ the average value of the residue, given by (6(Jv)ní+Ȧn), where the ȍELQÀXHQFHLVHOLPLQDWHG7KLVSURFHGXUHZLOOEHFOHDU
J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 2, pp. 163-168, Apr.-Jun., 2012
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Oliveira, E-HWDO
after displaying wavelet multi-resolution decomposition of the VHQVRURXWSXWVDQGDIWHUEHLQJSURYHGE\SDULW\YHFWRUDQDO\VLV :$9(/(775$16)250 After analyzing the spectral content and detecting the presence of long-term components in the sensor outputs, the wavelet analysis was chosen in order to eliminate undesirable high-frequency FRPSRQHQWVDQGWRKLJKOLJKWWKHVHORQJWHUPV GHQRLVLQJ :DYHOHW DQDO\VLV KDV EHHQ XVHG LQ D ZLGH UDQJH RI applications in signal processing techniques, due to its particular properties, such as compressing and denoising with low degradation of the original signal, time-scale representation, application in real-time operations in nonstationary signals, and by exposing hidden aspects of the signals like discontinuities, breakdown points, and trends (O6KHLP\DQG1DVVDU The theoretical foundations that hold the generalization and applicability of wavelet analysis to nonstationary signals FDQEHIRXQGLQ'DXEHFKLHV 0DOODW DQG6WUDQJ DQG1JX\HQ The orthogonality properties of the discrete wavelet transformation make possible the multi-resolution GHFRPSRVLWLRQRIDQ\WLPHVHULHV 'DXEHFKLHV ,QWKH multi-resolution process, a complex function is decomposed at several levels of approximations or resolutions, where ORZ DQG KLJKSDVV ¿OWHUV LQ D ¿OWHU EDQN FRQ¿JXUDWLRQ SHUIRUPWKHGLVFUHWHZDYHOHWWUDQVIRUPDWLRQDWHDFKOHYHO,Q WKLVVFKHPHWKHKLJKSDVV¿OWHUVDUHUHVSRQVLEOHIRUGHWDLOV ZKLOHWKHORZSDVVRQHVDUHUHVSRQVLEOHIRUDSSUR[LPDWLRQV Some details contain noise components of high-frequency and RWKHUV GLVWXUEDQFHV ,Q WKH DSSUR[LPDWLRQV WKHUH DUH ORQJ term components, and thus include frequency components of WKH(DUWKURWDWLRQUDWH$WHDFKOHYHOWKHFXWRIIIUHTXHQF\ fc) RIWKH¿OWHULVRQHKDOIRIWKHPD[LPXPIUHTXHQF\FRPSRQHQWV LQWRWKHSUHYLRXVOHYHO7KHUHIRUHFRQVLGHULQJWKDWWKHRXWSXW of sensor is sampled (fs) at 100 Hz, the maximum frequency component in the signal is 50 Hz (fs/2), and at each level, the cutoff frequency is given by (fc)i Is/2(i+1), where i represents WKHOHYHORIGHFRPSRVLWLRQ The analyses performed here considered eight levels of GHFRPSRVLWLRQZKLFKLQGLFDWHVD¿OWHULQJDERXW+],QWKLV work, the choice of wavelet family was based on properties of continuity, detection of transient singularities, and slow moving DQRPDOLHV,QWKLVFRQWH[WWKH'DXEHFKLHV GE ZDYHOHWIDPLO\ KDVEHHQXVHGLQVHYHUDOZRUNV .LP.LP3DUN DQGLQ WKHVDPHZD\LWZDVDOVRFKRVHQIRURXUSXUSRVH 166
'$7$$&48,6,7,21352&('85( In order to obtain misalignment, scale factor and bias HVWLPDWHVIURP(TWRDVHTXHQFHRISUHYLRXVO\GH¿QHG turntable commands is required to perform the matrix R IRUPHGE\HOHPHQWVFRPSXWHGLQ(T2QFHPDWUL[5KDV rank three, it is not possible to estimate the IMU errors in one step only, forcing the IMU partition into two sub-IMU, and consequent considerations that hold the remarks and the PRGHOUHSUHVHQWHGE\(T$IWHUGHQRLVLQJZLWKZDYHOHW algorithm, the gyro outputs are computed by error model equations, which obey the turntable sequence given in Table 1 7DEOH
6HTXHQFHRIFRPPDQGHGUDWHV w Ȧrt & ȦrtFRV DQG6 ȦrtVLQ
6T n 1 2 2 3 4 5 6 10 11 12 13 14 15 16
5RWD[LV Inner Outer 0 +w 0 -w 0 -w +w -w + w -w +w -w +w -w + w -w +w -w +w -w
X +w -w -w +w -w 0 0 0 0 0 0 0 0 0 0 0 0
Input rate Y 0 0 0 0 0 +w -w +w -w 0 0 0 0 +C -C +C -C
Z 0 0 0 0 0 0 0 0 0 +w -w +w -w +S -S -S +S
Notice that calibration procedures of tetrahedral base ZLWK )2* ZDV SHUIRUPHG RQ WKH WZR'2) URWDWH WDEOH %' ZLWK DWWLWXGH HUURU OHVV WKDQ î í degree, and rate error less than 1×10íGHJUHHVHFRQG7KHUDWH Ȧrt) used for calibration procedure was 10o»s7KH PHWKRG FRXOG EH executed with only six sequences (two rotation directions per axis); however, 16 sequences were employed aiming at minimizing turntable error effects and error covariance matrix ([RT R]í 7KH EORFN GLDJUDP RI WKH DFTXLVLWLRQ V\VWHP LV VKRZQ LQ )LJ 7KH HQVHPEOH LV FRPSRVHG E\
J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 2, pp. 163-168, Apr.-Jun., 2012
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IRXU)2*FRQQHFWHGWRDQHPEHGGHGFRPSXWHU3&YLD bus RS232 asynchronous, which performs the acquisition DQG GDWD VWRUDJH $IWHU WKH H[HFXWLRQ RI FRPPDQGHG VHTXHQFHV LV ¿QLVKHG 7DEOH GDWD DUH WUDQVIHUUHG WR D SHUVRQDOFRPSXWHUYLD)73FRPPXQLFDWLRQDQGSURFHVVHG DFFRUGLQJWRWKHHUURUPRGHO
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7KHUHIRUH (T DSSOLHG LQ WKH VHTXHQFHV IURP Q to Q KROGVWKHHVWLPDWLRQRIWKHPHDQYDOXHVRIWKHELDVHV )LJXUHLOOXVWUDWHVWKHVWDELOLW\RIWKHELDVGHWHUPLQDWLRQ from N 7KH SORW ZDV REWDLQHG IURP WKH SURGXFW RI WKH FRUUHFWHGSDULW\YHFWRU (T ZLWKWKHPHDQYDOXHRIWKH J\URRXWSXWVFRPELQHGWRUHPRYHȦrt (gb DVLQ(T 1 v c gr b = v c 8 S F ((gr v) n + (gr v) n + 1) B 2
(21)
1 v c gr ~ = vc 8 S F ((gr v) n + (gr v) n + 1) B 2
(22)
1 vgr ~ = v 8 S F ((gr v) n + (gr v) n + 1) B 2
(23) (24)
5(68/76 (25) After denoising the output signals, the long-term components are highlighted and represent the earth rotation UDWH )LJ ZKLFK PRGXODWHV WKH J\UR RXWSXWV RI WKH sequences, from N to N (Q , N results from (gv)n=5 - (gv)n=6 and N results from (gv)n=15 - (gv)n=16) related WR RXWHU D[LV URWDWLRQ :LWK WKLV JURXS RI PHDVXUHPHQWV the mean value estimated for ȍE ZDV K ZKLFK UHSUHVHQWV D JRRG DSSUR[LPDWLRQ ZLWK WKH DFWXDO RQH 7KH modulation level at each output is related to the sensor SRVLWLRQ DQG WKH IUHTXHQF\ LV UHODWHG WR WKH WXUQWDEOH UDWH
0.9999895 -0.0019974 0.0041171 M1 = > 0.0004597 0.9999998 -0.0004378H -0.0009892 0.0003843 0.9999994
(26)
)LJXUH $EVROXWH PHDQ YDOXH RI SDULW\ YHFWRU RI VHTXHQFHV k(Ȧ(+)-Ȧ(-)) without bias and ȍE.
)LJXUH 2XWSXWRIWKHJ\UR2ULJLQDO UHG GHQRLVHGZLWKGE DWVFDOH EODFN
Considering that the integration of modulation signal for integer numbers of cycles about the outer axis is zero, the mean value of the gyro outputs in this condition ZLOO FRQWDLQ EDVLFDOO\ WKH FRPPDQGHG UDWH DQG ELDVHV
In addition, for comparison, it was plotted the product RIWKHHVWLPDWHGELDVHV (T ZLWKFRUUHFWHGSDULW\YHFWRU (vcE 7KHUHVXOWVREWDLQHGIRUELDVDQGVFDOHIDFWRUDUHVKRZQ in Table 2, and they are compared with results obtained for WKH VDPH )2* LQ WKH LQGLYLGXDO FDOLEUDWLRQ 6LOYD Therefore, the individual calibration was made in a different VHW ORFDWLRQ DQG WLPH7KH UHVXOWV REWDLQHG IRU WHWUDG DUH TXLWHFRQVLVWHQW)RUWKH)2*XVHGLQWKLVZRUNWKHQRPLQDO YDOXHRIWKHVFDOHIDFWRUZDVP9V7KHPLVDOLJQPHQW matrices (M1 and M2 DUHJLYHQLQ(T
J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 2, pp. 163-168, Apr.-Jun., 2012
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In order to evaluate the improvement of the calibration procedure, and considering that the parity equation should result in a white noise with zero mean, it was plotted in the )LJWKHUHVXOWVRI(TVDQGZKLFKWDNHLQWRDFFRXQW WKHFRPPDQGHGUDWHRQO\ ELDVDQGȦEDUHUHPRYHG ,QWKDW condition, the difference between nominal and corrected SDULW\HTXDWLRQVUHYHDOWKHPLVDOLJQPHQWVLQÀXHQFHRQO\%HDU in mind that, in some situations, not all sensors are excited when the rate is applied on one of the main axes and, in the same way, if the misalignments plane is orthogonal to the LQSXWD[LV FRPPDQGHGUDWH LWZLOOQRWDSSHDU7KHUHIRUH this approach allows us to evaluate the improvement of the PLVDOLJQPHQW HVWLPDWHV LQ DOO GLUHFWLRQV LQ RQH VWHS ,W LV FOHDULQ)LJWKHLPSURYHPHQWRIWKHPHWKRG KRZHYHUD small residual error in the sequences N to N UHPDLQV
individual misalignment residue, but they indicate the amount of error in the overall estimation process in any direction WKDWWKHV\VWHPPD\EHH[FLWHG7KHUHIRUHLWLVDJRRGLQGH[ RITXDOLW\LQWKHFDOLEUDWLRQSURFHVV,QDGGLWLRQWKHSDULW\ equation analysis will give support to fault detect algorithms in terms of threshold determination, consequently, reducing WKHIDOVHDODUPSUREDELOLW\:KHQFRPSDUHGZLWKLQGLYLGXDO calibration, the results obtained by this method are quite FRQVLVWHQWDQGGHQRWHWKHTXDOLW\RIWKHPHWKRGRORJ\XVHG
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