DSpace at VNU: Errors determination of the MEMS IMU

The demand of navigation and guidance has been urgent for many years.. Nowadays, with the strong growth of Micro-Electro-Mechanical-System MEMS technology, the Inertial Navigation System

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 I M U

L u u M a n h H a 1 , 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 1

1D ep a rtm en t o f Electronics a n d Telecommunications, College o f Technology, V N U

2V ietnam N a tio n a l University, H anoi

A bstract 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),

1 I n t r o d u c t io n

N ow adays, n av ig a tio n and guidance a re very i m p o r t a n t p ro b lem s for m arin e,

a ero n au tics a n d space technology In such system s, I n e r tia l M e a s u r e m e n t U n its (IMUs) a re widely used as th e core of th e I n e r tia l N a v ig atio n S y ste m s (INS) [1] In principle, an IMU consists of gyroscopes a n d acc elero m ete rs which m e a su re a n g u la r velocities a n d a cceleratio n s in th re e d im ensions R ecently, th a n k to the developm ent of MEMS technology, th e IMƯS become sm a lle r, c h e a p e r and more precise However, th e re are still problem s w ith M E M S b a se d th e IM U s which are necessary to be solved The position e rro r of a n IN S in c re a s e s rapidly w ith navigation d u e to th e in te g ra tio n of m e a s u r e m e n t e rr o r s in th e gyroscopes and accelerom eters In o rd er to m ake th e corrections, th e e rro rs a re classified into

determ inistic e rro rs a n d stochastic e rro rs [2],

To e lim in a te th e d e te rm in istic errors, we can specify th e m q u a n tita tiv e ly by calibrating th e device It is, however, more complex in d e te r m in a tio n of the stochastic e rr o r s .An optim al filter such as K a lm a n one is often used In th is case, the p a r a m e te r s of those stochastic e rro rs m u s t n e c e ssa ry to be specified In th is paper, we h a v e d e te rm in e d noise p a ra m e te r s of both d e te r m in is tic an d stochastic errors of M E M S based th e IMƯS For th e d e te rm in istic erro rs, a precise ra te table has been used as a calib ra tio n device For th e sto ch astic e rro rs , we h av e trie d two different m eth o d s PSD a n d A llan variance The PSD is k n o w n a s a classical m ethod

to analyze sig n al, w hile A llan v arian ce is a new m eth o d which can show more

6

E rro rs D e te r m in a tio n o f the M ems IMU 7

in fo rm a tio n t h a n th e PSD C om bining th ese two m ethods will give us a reliable noise model t h a t is a p p lied d irectly to the Noise E lim in a tin g Block (NEB)

2 M e a s u r e m e n t a n d c h a r a c t e r i z a t io n

In th is s tu d y , we u se d th e MICRO-ISƯ BP3010 which consists of th re e ADXRS300 gyros a n d th r e e h e a t com pensated ADXL210E accelerom eters [3] The

m e a s u r e m e n ts a re sy n th e siz e d by IM U’s m icro-controllers a n d t r a n s m i t te d out via RS232 in te rfa c e T h e u n i t t r a n s m its o u tp u t d a ta as a n g u la r in c re m e n ta l a n d velocity in c r e m e n t a l d a ta in se ria l fram es of 16 b ytes a t one of th e user-selectab le freq u en cies of 64 Hz, 32 Hz, 16 Hz or 8 Hz

F ig u re 1: T he M ICRO-ISU BP3010 - A MEM S unit.

2.1 D e te r m in i s ti c erro rs

D e te r m in is tic e r r o r s c o n sist of bias, scale factor, in e r tia l axis m isalig n m en t

t h a t a re c o n sid e re d by th e following error model [2]:

Sab x = a x + a xxa h x + a xya b y + a x a h z

Sah y = a y + a yxa h x + a ^ a ị + a r a z

Sa\ = a z + a a aỊ + a ya b y + a a a\

H = A + A X + + P XA + ( Ạ X + Pxyyab y + Pxyiab 2)a>b y +{Pxzxab x +(ixzyab y +

H = / * , + 4 X + / ? x + 4 X +(Pm a i + p yy y

Scob : = p z + Pzxcoh x + + PIzcob : + (Pzy x + + Ạ X K + ( & x + )(0b y

(1)

w h ere ổũị ,ôũ)ị ( i = X, y, z) a r e accelerom eter a n d gyroscope e rro rs expressed in the

body fram e

ữị - A c c e le ro m e te r b iases [m/s2].

a u - A c ce lero m eter scale factor [unit less].

GCjj - A c c e le ro m e te r i n s t a lla tio n erro r (/ ^ j ) [unit less].

- A c c e le ro m e te r o u t p u t in body fram e coordinates [m/s2]

/? - Gyro b iases [rad/s],

p - Gyro scale factor [unit less].

- Gyro in s ta lla tio n e rro r (i * j ) [unit less].

P ik - Gyro drift dep en d in g on acceleration, flexure e r r o r [m/s2]

ứ / 'G yro o u tp u t in body fram e coordinates [rad/s]

In th is p ap er, all in s ta lla tio n e rro rs a n d flexure e rr o r s will be neglected because th ey are very sm all All of rem a in in g d e te rm in is tic e rr o r s are d e te rm in e d

by th e accelerom eter a n d gyroscope calibrations

a) Accelerometer calibration

u

Ạ z

N

Figure 2: In itia l IMU position for up-dow n c alib ra tio n

In th e c alib ratio n procedure of th e accelerom eters, th e e a r t h g ravity h a s been used In th is m ethod, th e IMƯ is in itially positioned so t h a t th e Z-axis of th e IMƯ aligned w ith the location level fram e s U-axis, th e Y -axis of th e IMU aligned with the N-axis a n d th e X-axis aligned w ith th e E-axis (Fig.2) I t m e a n s t h a t th e gravity

component will affect only th e accelerom eter along Z-axis by a n a m o u n t of +g (g -

9.8 m/s2) If th e IMƯ is th e n ro ta te d 180° a ro u n d th e Y-axis, a new m e a su re m e n t

could be ta k e n w hen the accelero m eter along Z-axis se n se s th e n e g ativ e g rav ity (-g ).

W hen the IM U w ith th e ith accelerom eter a lig n ed w ith th e U-axis in the

navigation fram e, th e o u tp u t of th e accelerom eter is:

R o tatin g th e IMU 180° a ro u n d p e rp e n d ic u la r axis a n d m a k in g a n o th e r

m e a su re m e n t will give th e following o u tp u t of th e accelero m eter:

z2( a ) = a ị - ( a a + l ) g (3)

E rrors D e te r m in a tio n o f the Mems IMU 9

Solving s e t of e q u a tio n s (2) an d (3) above, we can e stim a te of the

a ccelero m eter b ia s a n d scale factor:

a n d , ' ( « * ) - , ’ („,»)

T he collecting d a ta process is perform ed for ab o u t 10 m in u te s for each position, t h e n t h e d a ta is a v e ra g e d to give z '(a * ) a n d z 2(a*) S e t of eq u atio n s (4) is finally u se d to e x tr a c t th e accelero m eter bias a n d scale factor C alib ratio n resu lts showed t h a t th e a cc elero m ete r along Z-axis h a s b ias of 0.1330 m /s2 a n d scale factor

of 0.0041

b) Gyroscope calibration

The m e th o d is a c a lib ra tio n procedure t h a t uses a precise ra te table which contains se q u e n c e of d iffe re n t r a te s for each dim ension h a s been m ade use The IMU is i n itia lly po sitio n ed in c en ter of r a t e table a n d each r a t e is ru n

ap p ro x im ate ly for 10 m in u te s

The e r r o r model e q u a tio n of th e gyro is:

where w is n o m in a l gyro a n g u la r r a te a t tab le a n g u la r r a te Wj [deg/h, rad/s].

Wj a v e r a g e tab le a n g u la r r a t e for d a ta se g m e n t j [deg/h, rad/s]

wex s e n s e d co m p o n e n t of e a r t h ro tatio n r a te [deg/h, rad/s]

p ị - gyro b ia s [deg/h, rad/s].

Pii - gyro scale factor.

From (5), we have:

We can t h e n e s t im a t e gyro bias scale factor b ased on Eq.6 R e su lts showed

t h a t the Z-axis gyro h a s b ia s of 0.3172 °/s a n d scale factor of -0.0070

2.2 S to c h a s tic IM U e rr o rs

Some stochastic errors t h a t affect the Initial Navigation Systems are listed as follows

- Q u a n tiz a tio n noise

Q u a n tiz a tio n noise is m ad e from encoding th e an alo g signal in to digital form This noise is c a u se d by th e sm all difference betw een th e a c tu a l a m p litu d e s of the sam pled sig n a l a n d b i t reso lu tio n of A-D C onverter We can reduce q u a n tiza tio n noises by im p ro v in g encode m ethods, a d ju s tin g sam p le ra te , or in creasin g bit resolution

- W hite noise

W hite noise can be a m ajor source of th e IMƯ e rr o r s a n d it h a s a c o n s ta n t power sp e ctru m over whole frequency axis Angle ran d o m w alk (for gyroscope) a n d velocity ran d o m w alk (for accelerom eter) are caused by th e w h ite noise

- R a n d o m w alk

T his is th e ran d o m process of u n c e rta in origin, possible of a lim itin g case of an exponentially co rre lated noise w ith long co rrelation tim e The gyroscopes a re affected by a n g u la r r a te ran d o m walk, while th e a c c elero m ete rs a re affected by acceleration ran d o m walk

- F licker noise

Flicker noise is low-frequency noise term t h a t show s as b ias flu ctu atio n s in data T his noise is cau sed by the electronics or o t h e r com ponents t h a t are susceptible to ran d o m flickering

In o rd er to an aly ze stochastic IMU errors, we h a v e used th e following methods:

a) Power spectral d en sity an a lysis

The Pow er Sp ectral D ensity A nalysis (PSD) d escrib es how th e power is allotted along th e frequency axis [4] T he o u tp u t d a ta of th e IM U, which is collected during an h o u r, is an aly zed to give th e PSD F ig 3 show s a log'log plot of th e PSD of the X-axis gyro We note t h a t th e re is a b u n ch in g of h ig h frequency a re a It IS difficult to id en tify th e noise te rm s a n d th e p a r a m e te r s a sso c iated w ith th em Thus, the frequency a v e ra g in g tech n iq u e [5] h a s been used to sm ooth th e PSD plot

Figure 3 T h e PSD plot of th e X-axis gyro (a) and th e P S D plot o b ta in e d by the

frequency av erag in g tec h n iq u e (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, 0 and 2 I t m eans th a t the gyro data includes the an g u la r ra te 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 th e a n g u la r ra te flicker noise

E rr o rs D e te r m in a tio n o f the M ems IMU

11

Figure 4 T h e P S D of a cc elero m ete r z obtained by frequency a v erag in g technique

F ig.4 sh o w s th e PSD plot of Z-axis accelerom eter o b tain ed by using the

frequency a v e r a g in g tec h n iq u e The slopes of th e curve com prise -2, -1, 0 1 an d 2

T his curve i n d ic a te s t h a t accelero m eter d a ta includes acceleration ran d o m walk acceleration flick er noise, velocity random walk, and acceleration q u an tiza tio n noise

By u sin g th e c o n v e rtin g form ula in [6], we o b tain from th e PSD plot the

a n g u la r r a t e w h ite noise a n d th e acceleration w hite noise as listed in T a b l

T a b l e 1 E s tim a te d w hite noise in th e in e r tia l sensors

Angular rate white noise °l4 h 0,0560 0,0486 0,0578

Acceleration white noise (m/s)/ 4 h 0.0033 0,0030 0,0028

Analog Device s t a t e s t h a t a n g u la r random w alk h a s valu es from 0 to 6°/4 h for

th e ADXLS300 gyros u se d in th e MIRCO ISU BP3010 ỈMU If we compare it to Table 1, we c a n see t h a t th e w h ite noise level indeed lies w ith in th e lim it of the

m a n u fa c tu re r

b) A lla n va ria n ce a n a ly s is

The A lla n v a r ia n c e is s ta tis tic a l m e a su re to c h a ra c te riz e th e stability of a tim e-freq u en cy s y s te m [7] T h e PSD can only e x tra c t w h ite noise s ta n d a rd deviation In c o n tr a s t , u s in g th e A llan variance, sev eral o th e r e rro r p a ra m e te rs can

be c o m p re h en siv e ly derived

The basic id e a of th e A llan v arian ce is to ta k e a long d a ta sequence a n d divide

it in to s e g m e n ts b a s e d on a n a v e ra g in g t i m e r to process L et give a sequence with

N e le m e n t s y k , k= 0 ,1, , N - l T hen, we define for each n = l , 2 , 3 , , M £ N / 2 a new sequence of a v e r a g e s of s u b seq u e n ce w ith length n:

If t h e s a m p lin g tim e is A / , th e tim e s p a n w ith in an a v e ra g e d se q u en ce of

len g th n is r = n A t T h e A lla n v a ria n c e , for a given su b se q u e n c e len g th n, is

defined as:

'N'

c r l ( T , N ) t ( 1

~N~

- 1 - 2

2 ] ( x J+i ( n ) - X j ( n ) Y

j=0

(8)

T h e ty p ic al slopes of t h e A llan v a ria n c e for th e gyroscope a n d th e

a c c e le ro m e te rs in log-log plo t a re show n in Fig 6 w ith d a t a collected from th e IMU ISƯ BP3010 d u r in g a n h o u r

To d e te r m in e th e noise p a r a m e t e r s , we n e e d to fit th e s t a n d a r d slopes in Fig 5 [8], For e x am p le , if d a t a c o n ta in s w h ite noise, t h e slope -1/2 will a p p e a r in the log- log plot of th e A lla n s t a n d a r d d ev iatio n

Averaging tone (s)

F igure 5 T h e s t a n d a r d slopes of t h e A lla n s t a n d a r d deviation.

T h e log-log plot of th e A llan s ta n d a r d deviation in Fig.6.a indicates th e presence

of a n g u la r r a te q u a n tiz a tio n noise (slope -1), a n g u la r ra te w hite noise (slope -1/2),

a n g u la r ra te ra n d o m w alk (slope 1/2), while a n g u la r r a t e flicker noise (slope 0) is absent T his re s u lt is fully co n sisten t w ith t h a t o b tain ed 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 e v ia tio n for th e

a ccelero m eter T h is show s th e p re s e n c e of a c c e le ro m e te r q u a n tiz a tio n noise (slope - I), a c c e le ro m e te r w h ite noise (slope -1/2), a c c e le ro m e te r flicker noise (slope 0), a n d

a cc elera tio n r a n d o m w a lk (slope 1/2) T h is r e s u l t is g a in well c o n s is te n t w ith t h a t from th e PSD plot In a d d itio n , th is show s t h e p re s e n c e of a c c e le ra tio n t r e 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 sin g t h e PS D plot

E rro rs D e te r m in a tio n o f the M em s IMU 13

Figure 6 T h e A llan s t a n d a r d d e v ia tio n of gyro X (a) a n d of a c c e le ro m e te r z (b) The w h ite n o ise coefficient is o b ta in e d by f ittin g t h e slope lin e a t r = 1 Below

th e table show s t h e e s tim a te d noise coefficients for t h e gyros a n d th e accelerom eters

T a b l e 2 Id e n tif ie d Noise Coefficients, u sin g A lla n v a ria n c e

(Quantization noise)

Q(radI y f s )

(white noise)

B (rad/s) (Flicker noise)

K(rad/s/Vi ) (random walk)

R (rad/s2) (trend)

Accelerometers Qz(m/s) Q(m/s/ yfs ) B(m/S2) K (m/s2/ V T ) R(m/s3)

C h a r a c te r X m e a n s t h a t t h e se n so r lacks th e e rr o r or t h i s one is m u ch sm a lle r

th a n th e o th ers

c) C o m p a riso n between P S D a n d A ll a n variance

T able 3 sh o w s th e c o m p ariso n b e tw e e n th e P S D a n d A lla n v a ria n c e in

e x tra ctin g w hite n o ise coefficients T he r e s u l ts o b ta in e d by t h e two m e th o d s a re much closed w ith e a c h o t h e r w hich confirm ed a s s e r t th e re lia b ility a n d th e accu racy

of the e rr o r m odel a p p lie d to th e p rac tic al I n e r t i a l N a v ig a tio n S y stem s

T a b l e 3 The com parison betw een t h e P S D a n d th e A llan v arian ce.

PSD ([°/VÃ]) Allan([°/V^ ]) PSD{[m/s/Jh ]) Allan [[m/s/yfh ])

3 C o n c lu sio n

T his p a p e r h a s succeeded in specifying t h e p a r a m e t e r s of th e IMU errors, which is a n e ce ssa ry step w hen applying e rro r-p ro c e s sin g a lg o rith m s for th e INS

E stim a tio n of th e stochastic e rro rs is more co m p lica te d t h a n for th e d eterm in istic ones Both of th e two m ethods, PSD and A llan v a ria n c e , h a v e b e en used h ere to estim a te th e sto ch astic e rro rs of th e IMƯ I t is sh o w n t h a t th e A llan v a ria n ce is the more com prehensive m ethod The e x tra cted r e s u l ts will be u sed as th e p a ra m e te rs

in K alm an filter for th e INS-GPS in te g ra te d sy stem

A c k n o w le d g e m e n ts T his work is su p p o rte d by th e V N U p ro g ram QGTD0509

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