J. Aerosp. Technol. Manag. vol.4 número2

Inertial Measurement Unit Calibration Procedure for a

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eOFLR-HURQLPRGH2OLYHLUD1,2,*:DOGHPDUGH&DVWUR/HLWH)LOKR1,MDU0LODJUHGD)RQVHFD2 1_{Instituto de Aeronáutica e Espaço - São José dos Campos/SP – Brazil}

2_{Instituto Nacional de Pesquisa Espacial - São José dos Campos/SP – Brazil}

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

The main error sources in the inertial navigation computation are associated with the sensor biases and scale factors, as well as the overall misalignments of the sensor FRQ¿JXUDWLRQV7KHUHIRUHDPHWKRGDQGDQHUURUPRGHOIRU LQHUWLDOPHDVXUHPHQWXQLW,08FDOLEUDWLRQDUHUHTXLUHG,Q order to design a proper method and an error model, spectral analysis and wavelet denoising were performed to highlight the long-term component and to remove high-frequency GLVWXUEDQFHV *HQHUDOO\ WKH SHUIRUPDQFH RI WKH PHWKRG LV YHUL¿HG E\ FRPSDULVRQ EHWZHHQ D UHIHUHQFH FRPPDQG and sensor computations, after error compensation (Cho DQG 3DUN ,Q WKLV ZRUN LW ZDV XVHG D SDULW\ YHFWRU DQDO\VLVWRYHULI\WKHRYHUDOOSHUIRUPDQFHRIWKHFDOLEUDWLRQ Such analysis was possible due to a redundant sensor that KROGV WKH H[LVWHQFH RI WKH SDULW\ YHFWRU 7KH SHUIRUPDQFH analysis, based on parity vector, allows us to verify the DPRXQW RI DOLJQPHQW HUURU LQ DQ\ GLUHFWLRQ RI URWDWLRQ ,Q the sequence, this paper has developed the geometric, parity vector, and error model formulations, the wavelet application, WKH FDOLEUDWLRQ WHFKQLTXH DQG WKH H[SHULPHQWDO UHVXOWV _____________________

Received: 23/12/11 Accepted: 16/04/12 DXWKRUIRUFRUUHVSRQGHQFHHOFLRHMR#LDHFWDEU

3o0DO(GXDUGR*RPHV&(3±6mR-RVp dos Campos/SP – Brazil

*(20(75,&&21),*85$7,21$1$/<6,6

Geometry

The geometrical arrangement used in this work considers four gyros mounted on the faces (black hexagon) of a tetrahedral structure (tetrad), and three accelerometers in a triad FRQ¿JXUDWLRQEOXHEORFNLQWHUQDOO\¿[HGLQWKHWHWUDKHGUDO The analysis performed here takes into account the gyros only, DQGWKHH[WHQVLRQIRUDFFHOHURPHWHUVFDQEHPDGHHDVLO\7KH WHWUDGFRQ¿JXUDWLRQLVVKRZQLQ)LJ

The mathematical representation of the gyro arrangement is given in terms of measurement matrix, where each line represents the direct cosines vector of the sensor axis with UHVSHFW WR DQDO\WLFDO WULRUWKRJRQDO D[HV DQDO\WLFDO WULDG 7KHUHIRUHFRQVLGHULQJWKHVFKHPDWLFUHSUHVHQWHGLQ)LJ GRLMDWP

)LJXUH7HWUDKHGUDOEDVH

X

Y

Z(North) g4

3 2

1 g g

where Į žWKHPHDVXUHPHQWPDWUL[LVJLYHQE\(T

H

1/ 3 1/ 3 1/ 3 1

0 2 /2

2 /2 0

6 /3 6 /6 6 /6 0 =

-J

L K K K KK

N

P O O O OO

(1)

The matrix H _{relates the sensor measurements (g}

i) with the DQJXODUUDWHLQWKHPDLQD[HVLQWKHIRUPRIWKHHT

go=H~ (2)

The estimate of the angular rate components in the main D[HVFDQEHREWDLQHGIURP(TDVLQ(TVDQG

H* go

=

~t ₍₃₎

H*₌(H HT )-1HT

(4)

where, the superscript ( T _{) indicates the transpose; g}

o= [g1 g2 g3 g4] T is the vector of the gyro outputs; Ȧ >ȦxȦyȦz]

T _{is the vector} RIDQJXODUUDWHDERXWPDLQD[HVȦLVWKHDQJXODUUDWHHVWLPDWH 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

guo=H~+dgo+ +f hs (5)

ZKHUHįJo is a constant term (bias) vector; f is the fault vector; and Ș_sLVDUDQGRPWHUPYHFWRU*DXVVLDQQRLVHDQGRWKHU UDQGRPGLVWXUEDQFHV

Applying the singular value decomposition (SVD) on H_{, the}

range and null spaces from this matrix can be obtained (Shim DQG<DQJ,QDGGLWLRQLWFDQDOVRFRPSXWHWKHELDVHV LQÀXHQFHRQWKHDUUDQJHPHQW'HFRPSRVLQJHDV(TVDQG

U HV

0

T ∑

K

= =e o (6)

H=U V_K S

where, U_, ȁ_{, and} VT_=V=I

n are matrices obtained from SVD of H The matrix Ȉ is a diagonal one, whose elements are eigenvalues of H

$SSO\LQJ(TLQWRDQGPXOWLSO\LQJERWKVLGHVE\UT_{, it} FDQEHREWDLQHGWKHUHODWLRQVKLSDVLQ(T

Partitioning UDV(T

U=6U U1h 2@

where U 1אR

4×3 _{and U} 2אR

4×1DQGDSSO\LQJLWLQWR(TWKH UHVXOWLQJHTXDWLRQVFDQEHZULWWHQDV(TVDQG

U g1Tuo=(RV)~+UT1(dgo+ +f hs) (10)

(11)

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

( V) g

LS 1 o

~_t = R - _u

(12)

7KHPHDQLQJRI(TLVWKDWLIVHQVRUVIDXOWVDQGELDVHV are zero, the resulting product of parity vector with sensor PHDVXUHPHQWVLVDZKLWHQRLVHZLWK]HURPHDQ2WKHUZLVH LIWKHELDVHVDQGRUIDXOWVYDOXHVGLIIHUIURP]HUR(TLVD ³ZHLJKWHG´VHQVRUHUURUVVXPPDWLRQ7KHQU

2

T _{is the parity}

vector (Y) obtained from null space of H

0,6$/,*10(17(5525(48$7,21

The estimation of the sensor axis misalignments takes an LPSRUWDQWUROHRQWKH,08FRQVWUXFWLRQ7KH\DUHGLVDJUHHV between predicted (or nominal) sensor axis angles and actual RQHVDIWHUEXLOW&RQVHTXHQWO\WKHVHDOLJQPHQWHUURUVPXVWEH HVWLPDWHGDQGLQFRUSRUDWHGWRSHUIRUPDQHZVHQVRUPDWUL[ In addition, the scale factor is another important element to be HVWLPDWHGDQGLWFDQEHSHUIRUPHGLQWKHVDPHH[SHULPHQW

In this work, a method based on least square technique was used (Cho and Park, 2005) to estimate misalignments and scale factors of the sensors, and the experimental procedures to obtain the proper sensor outputs for calibration were executed RQGHJUHHRIIUHHGRP'2)WXUQWDEOH

,QHUWLDO0HDVXUHPHQW8QLW&DOLEUDWLRQ3URFHGXUHIRUD5HGXQGDQW7HWUDKHGUDO*\UR&RQ¿JXUDWLRQZLWK:DYHOHW'HQRLVLQJ

LVGLYLGHGLQWRWZRVXEV\VWHPV7KH¿UVWRQHLVFRPSRVHGRI

gyros g₁, g₂ and g₃LQDWULRUWKRJRQDOFRQ¿JXUDWLRQDQGWKH second one is composed of gyro g₄, whose measurement

D[LVLVFROOLQHDUWRWKHYHUWLFDO;D[LV7KHUHIRUHWKHHUURU PRGHOLQJFDQEHPDWKHPDWLFDOO\H[SUHVVHGDV(T

S g

_{F v}

=

MH

_~

+

_b

+

_h

(13)

where, 6₎ is a diagonal matrix whose elements are the sensor scale factors in (o»

s)/mV, gv is the sensor outputs in mV, M is

the misalignment matrix, and b is the bias vector;

The misalignment matrix (M_{) can be seen as a rotation}

IURPQRPLQDOVHQVRUD[HVSRVLWLRQWRWKHUHDORQH6HSDUDWLQJ (TLQWRWZRVXEV\VWHPVWKHUHLV(T 0 S g S g S g g 0 S M M H H b b b b 1 3 3×3 F v F v F v F v × x y z 1 1 2 2 3 3 4 4 1 2 1 2 1 2 3 4 ~ ~ ~ h = + + R T S S S S S R T S S S S S

= =

>

V X W W W W W V X W W W W W

G G

H

(14)

where, ; ; ; M m m m m m m m m m

M m m m H h h h h h h h h h H I

x y z

x x x y y y z z z 1 11 21 31 12 22 32 13 23 33

2 4 4 4 1

1 2 3 1 2 3 1 2 3 2 3

=

>

H

=6 @ =

>

H

=

$WWKLVSRLQWVRPHUHPDUNVDUHQHHGHGWRFODULI\(T ‡ WKH PLVDOLJQPHQW RI J4 is estimated independently of

the other sensors, and it is considered, without loss of generality, as a part of triorthogonal arrangement (pseudo-arrangement) coincident with the main axes;

‡ LQVSLWHRIPDWUL[H _{has 4×3 dimension, it is extended}

to accommodate the axes y and z in order to permit the estimate of misalignment between g₄ and those two axes;

‡ DWD¿UVWJODQFHWKHPDWUL[M _{could be 4×4 dimensional,}

given H _{has 4×3 dimension; however, this consideration}

ZRXOGOHDGWRVLQJXODULW\LQWKHPHWKRG

(TXDWLRQLVWKH(TUHZULWWHQLQDFRPSDFWIRUP

S gFi vi=6m m mia ib ic@Hj~+bi+hi (15)

where,

i = 1, 2, 3, 4 – is the sensor number;

j = 1,2 – is the sensor subsystem; for , ,

, ,

j a b c

j a x b y c z

1 1 2 3

2 " " = = = = = = = = ) 'H¿QLQJWKHGLIIHUHQFHEHWZHHQWZRVXFFHVVLYHVHTXHQFHV

by Ȧl

(+)íȦ

l

(-) _{, where} O [\] _and Ȧí _{is the positive/}

negative commanded turntable rate, the following vectors can

EHFRQVWUXFWHG(T

r

r

r

H

( ) ( ) ( ) ( ) ( ) ( ) a b c j x y z x y z

~

~

~

~

~

~

=

-+ + +

->

H

>

H

(16)

8VLQJ(TVDQGDIWHUVRPHPDQLSXODWLRQVLWFDQEH UHDFKHGWRDUHODWLRQVKLSH[SUHVVHGLQ(T / / /

g g r r r

m S m S m S

( ) ( )

vi vi a b c

ia Fi ib Fi ic Fi i h - = + +

-6 @ 6 @

>

H

Performing N HYDOXDWLRQV N• IRU (T LW FDQ EH

REWDLQHG(T / / / ( ) m S m S m S

R R R G

ia Fi

ib Fi

ic Fi

1

= S - S

>

H

where, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R r r k r r k r r k andG g g g g

g n g

1 1 1 1 2

3 4 1 a a b b c c vi vi vi vi vi vi

h h h

h h = = -- -r r r r r r R T S S S S S

>

V X W W W W W

H

and g_vi 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,

mia mib mic 1

2 2 2

+ + =

After estimating the scale factors and misalignments, and

UHZULWWHQ(TWKHVHQVRUELDVHVFDQEHREWDLQHGDVLQ(T

( ( ) )

n S g H b

1 n

F v n n

1 ~

R r -u =r ₍₂₀₎

where, +Ѻ 0+, b– _{is the mean value of bias, and (g}

v)n is

the mean value of sensor output at nthWXUQWDEOHVHTXHQFH 7KHJ\URVXVHGLQWKH,08)2*KDYHELDVHVZLWKWKHVDPH

order of the Earth rotation rate (ȍ_E7KXVLWLVDFRPSOLFDWHG task to separate one from another in a skewed sensor

FRQ¿JXUDWLRQV,Q(TWKLVSUREOHPLVVROYHGE\SHUIRUPLQJ

after displaying wavelet multi-resolution decomposition of the

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:$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

FRPSRQHQWVDQGWRKLJKOLJKWWKHVHORQJWHUPVGHQRLVLQJ :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'DXEHFKLHV0DOODWDQG6WUDQJ 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

RWKHUVGLVWXUEDQFHV,QWKHDSSUR[LPDWLRQVWKHUHDUHORQJ

term components, and thus include frequency components of

WKH(DUWKURWDWLRQUDWH$WHDFKOHYHOWKHFXWRIIIUHTXHQF\fc)

RIWKH¿OWHULVRQHKDOIRIWKHPD[LPXPIUHTXHQF\FRPSRQHQWV LQWRWKHSUHYLRXVOHYHO7KHUHIRUHFRQVLGHULQJWKDWWKHRXWSXW

of sensor is sampled (f_s) at 100 Hz, the maximum frequency component in the signal is 50 Hz (f_s/2), and at each level, the cutoff frequency is given by (f_c)_i I_s/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

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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 6HTXHQFHRIFRPPDQGHGUDWHVw Ȧ_rt& Ȧ_rtFRVž DQG6 ȦrtVLQž

6T 5RWD[LV Input rate

n Inner Outer X Y Z

1 +w 0ž +w 0 0

2 -w 0ž -w 0 0

2 -w 0ž -w 0 0

3 +w ž +w 0 0

4 -w ž -w 0 0

5 ž + _{w 0 +w 0}

6 ž - w 0 -w 0

ž + w 0 +w 0

ž - w 0 -w 0

ž + w 0 0 +w

10 ž - w 0 0 -w

11 ž + _{w 0 0 +w}

12 ž -w 0 0 -w

13 ž +w 0 +C +S

14 ž -w 0 -C -S

15 ž +w 0 +C -S

16 ž - w 0 -C +S

Notice that calibration procedures of tetrahedral base

ZLWK )2* ZDV SHUIRUPHG RQ WKH WZR'2) URWDWH WDEOH

%'ZLWKDWWLWXGHHUURUOHVVWKDQîí _{degree, and}

rate error less than 1×10íGHJUHHVHFRQG7KHUDWHȦ_rt) used for calibration procedure was 10o»

s7KHPHWKRGFRXOGEH

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}

IRXU)2*FRQQHFWHGWRDQHPEHGGHGFRPSXWHU3&YLD bus RS232 asynchronous, which performs the acquisition DQG GDWD VWRUDJH $IWHU WKH H[HFXWLRQ RI FRPPDQGHG VHTXHQFHVLV¿QLVKHG7DEOHGDWDDUHWUDQVIHUUHGWRD SHUVRQDOFRPSXWHUYLD)73FRPPXQLFDWLRQDQGSURFHVVHG DFFRUGLQJWRWKHHUURUPRGHO

)LJXUH%ORFNGLDJUDPRIWKHFDOLEUDWLRQVHWGDWDDFTXLVLWLRQ

5(68/76

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 ȍ_E ZDV žK ZKLFK

UHSUHVHQWV D JRRG DSSUR[LPDWLRQ ZLWK WKH DFWXDO RQH7KH modulation level at each output is related to the sensor SRVLWLRQDQGWKHIUHTXHQF\LVUHODWHGWRWKHWXUQWDEOHUDWH

)LJXUH 2XWSXWRIWKHJ\UR2ULJLQDOUHGGHQRLVHGZLWKGE DWVFDOHEODFN

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

7KHUHIRUH (T DSSOLHG LQ WKH VHTXHQFHV IURPQ to

Q KROGVWKHHVWLPDWLRQRIWKHPHDQYDOXHVRIWKHELDVHV

)LJXUHLOOXVWUDWHVWKHVWDELOLW\RIWKHELDVGHWHUPLQDWLRQ

from N 7KH SORW ZDV REWDLQHG IURP WKH SURGXFW RI WKH

FRUUHFWHGSDULW\YHFWRU(TZLWKWKHPHDQYDOXHRIWKH

J\URRXWSXWVFRPELQHGWRUHPRYHȦrt (gbDVLQ(T

(( ) ( ) )

S

2 1

v gcrb=vc8 F grv n+ grv n 1+ B (21)

(( ) ( ) )

S

2 1

v gcr~=vc8 F grv n+ grv n 1+ B (22)

(( ) ( ) )

S

2 1

vgr~=v8 F grv n+ grv n 1+ B (23)

(24)

(25)

. . .

. . .

. . .

M

0 9999895 0 0004597 0 0009892

0 0019974 0 9999998 0 0003843

0 0041171 0 0004378 0 9999994 1=

->

H

(26)

)LJXUH $EVROXWH PHDQ YDOXH RI SDULW\ YHFWRU RI VHTXHQFHV k(Ȧ(+)_-Ȧ(-)_{) without bias and} ȍ

E.

In addition, for comparison, it was plotted the product RIWKHHVWLPDWHGELDVHV(TZLWKFRUUHFWHGSDULW\YHFWRU

(v_cE7KHUHVXOWVREWDLQHGIRUELDVDQGVFDOHIDFWRUDUHVKRZQ

in Table 2, and they are compared with results obtained for WKHVDPH)2*LQWKHLQGLYLGXDOFDOLEUDWLRQ6LOYD Therefore, the individual calibration was made in a different VHWORFDWLRQDQGWLPH7KHUHVXOWVREWDLQHGIRUWHWUDGDUH TXLWHFRQVLVWHQW)RUWKH)2*XVHGLQWKLVZRUNWKHQRPLQDO YDOXHRIWKHVFDOHIDFWRUZDVP9žV7KHPLVDOLJQPHQW

matrices (M₁ and M₂DUHJLYHQLQ(T

7DEOH5HVXOWVIURPLQGLYLGXDOVHQVRUDQGWHWUDGFDOLEUDWLRQV *\UR ,QGLYLGXDOFDOLEUDWLRQžV 7HWUDGFDOLEUDWLRQžV

%LDVžV S₎P9žV %LDVžV S₎P9žV

1

2

3

4

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[LVFRPPDQGHGUDWHLWZLOOQRWDSSHDU7KHUHIRUH this approach allows us to evaluate the improvement of the PLVDOLJQPHQW HVWLPDWHV LQ DOO GLUHFWLRQV LQ RQH VWHS ,W LV FOHDULQ)LJWKHLPSURYHPHQWRIWKHPHWKRGKRZHYHUD small residual error in the sequences N to N UHPDLQV

)LJXUH 0HDQ YDOXH RI SDULW\ YHFWRU RI VHTXHQFHV NȦ(+)_-Ȧ(-)₎ without Ȧ_rt

&21&/86,216

In this paper, a method based on least square parameter estimation was used to estimate the misalignments of an ,08 LQ D WHWUDG FRQ¿JXUDWLRQ DQG SDULW\ HTXDWLRQV ZHUH DOVR XVHG WR HYDOXDWH WKH TXDOLW\ RI WKH FDOLEUDWLRQ ,Q this arrangement, the parity equations do not provide the

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

5()(5(1&(6

&KR6<3DUN&*³$FDOLEUDWLRQWHFKQLTXHIRUD redundant imucontaining low-grade inertial sensors”, ETRI -RXUQDO9RO1RSS

'DXEHFKLHV,³7HQOHFWXUHVRQZDYHOHWV´3KLODGHOSKLD 6,$0

(O6KHLP\ 1 1DVVDU 6 ³:DYHOHW GHQRLVLQJ IRU ,08DOLJQPHQW´,((($ (6\VWHPV0DJD]LQHSS

.LP 6 .LP < 3DUN & ³)DLOXUH GLDJQRVLV RI VNHZFRQ¿JXUHG DLUFUDIW LQHUWLDO VHQVRUV XVLQJ ZDYHOHW decomposition”, IET Control Theory and Applications, 9ROSS

0DOODW 6 * ³$ WKHRU\ RI PXOWLUHVROXWLRQ VLJQDO decomposition: the wavelet representation”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9ROSS

6KLP'6<DQJ&.³*HRPHWULF)',EDVHGRQ SVD for redundant inertial sensor systems”, 5th Asian Control &RQIHUHQFH9ROSS

6LOYD - )) ³3URFHGLPHQWRV SDUD HQVDLRV GH FDUDFWHUL]DomR GH JLUR GH ¿EUD ySWLFD´ 7HFKQLFDO 5HSRUW ASE-MT-001-2010, Instituto de Aeronáutica e Espaço, IAE, %UDVLO