INSTRUMENTATION VIEWPOINT Sessions
92
Resul s show ha he me hod can be used o classi y wo ma e ials
wi h simila esponses, as is he case wi h s eel and aluminum. In he
case o he small bea ing-balls, i has been ound ha o he la ge
cylinde s he e is no signi i can di e ence among diame e s, due all
esponses p esen simila cha ac e is ics. In he case o he small cyl-
inde s, i has been ound ha he la ge ball (b1) impac s a e be e , in
o de o obse e di e ence among ma e ials.
Table I. Valida ion es and ANN ou pu . Size=0, 1 means sho , and
long. Type=0, 1 means aluminum, and s eel.
3. Conclusions
We ha e p esen ed a me hod which allows classi ying ma e ials using
impac s and neu al ne wo ks. The main p oblem on ma e ial’s classi-
i ca ion a ises when esponses a e simila , in his case he me hod has
been es ed wi h s eel and aluminum, and he ANN has p o en o be
a obus solu ion o de ec di e ences. Fu he analysis will in ol e
o he ma e ials as well.
4. Acknowledgemen
Au ho s would like o hank i nancial suppo om Minis e io de Edu-
cación y Ciencia de España, unde g an DIP-2003-08637-C03-03.
E. M. M. R. hanks o he i nancial suppo o he Mexican Na ional
Council o Science and Technology (CONACyT).
5. Re e ences
[1] P. Cawley and R. D. Adams, “The mechanics o he coin- ap me hod o
non-des uc i e es ing,” Jou nal o Sound and Vib a ion, 122(2), pp. 299-316,
1988.
[2] A. Miglio i and T. W. Da ling, “Resonan ul asound spec oscopy o ma-
e ials s udies and non-des uc i e es ing,” Ul asonics, Vol 34, pp. 473-476,
1996.
[3] S. Baglio and N. Sa alli, “Fuzzy ap- es ing senso s o ma e ial heal h-s a e
cha ac e iza ion,” IEEE T ans. Ins um. Meas., ol. 55, no. 3, pp. 761-770, Jun.
2006.
[4] H. Wu and M. Siegel, “Co ela ion o Accele ome e and Mic ophone Da a
in he “Coin-Tap Tes ”,” IEEE T ans. Ins um. Meas, Vol 49, No. 3. pp. 493-497,
June 2000.
[5] C. M. Ha is and A. G. Pie sol, “Ha is’ shock and ib a ion handbook”, Mc-
G aw-Hill, 5 h ed., 2002.
Table I. Valida ion es and ANN ou pu . Size=0, 1 means sho , and
long. Type=0, 1 means aluminum, and s eel.
APPLICATION OF A KALMAN FILTER ON MECHANICAL
SYSTEMS TO ANALYZE IMPACTS
Ramón Guzmán, E ik Molino Mine o Re, An oni Manuel Láza o
Escola Uni e si à ia Poli écnica de Vilano a i la Gel ú
Depa men o communica ions and signal heo y
A . Víc o Balague s/n Vilano a i la Gel ú (Ba celona)(Spain)
[email p o ec ed].edu
1. In oduc ion
The pu pose o his pape is o es he easibili y o using a Kalman
i l e and a simple mechanical model o analyze he eloci y and
he accele a ion o an impac gene a ed om he collision be ween
wo igid bodies. To achie e his, he Kalman i l e and a mechanical
model a e compa ed wi h expe imen al signals ha ha e been ob-
ained om eal impac s, be ween a senso ized hamme and a s eel
cylinde .
The cylinde has been modeled as a i s o de dynamic sys em, as
shown in Figu e 1, wi h he impac signal, ( ), applied on i s su ace.
The mechanical model is shown on equa ion (1).
Figu e 1. Me allic cylinde
model.
)(
1
0
10
2
1
2
1 u
x
x
m
k
m
k
x
x
d
d
e
d
e
s⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
=
⎥
⎦
⎤
⎢
⎣
⎡
(1)
Whe e me : e ec i e mass [kg]
kd : damping cons an [N•s/m]
ks : sp ing cons an [N/m]
( )=u( ) : inpu o ce [N]
x1 , x2 : displacemen and speed.
Acco ding o [1], he e ec i e mass is he join masses o he hamme
and he cylinde , and his is gi en by equa ion (2).
whe e mh : hamme mass
mc : cylinde mass
And cons an s kd and ks a e o s eel.
1.1 Kalman Algo i hm
Figu e 2, shows he block diag am o he Kalman es ima o .
M9
INSTRUMENTATION VIEWPOINT
Sessions
93
Figu e 2. Kalman es ima o .
To desc ibe he Kalman algo i hm some equa ions need o be de-
i ned. The disc e e Kalman i l e s a e equa ions a e speci y by (3) and
(4).
(3)
kk
kk wGuBxFx ⋅+⋅+⋅=
+1
(4)
On equa ion (3), yk is he obse able a ime k, and H is he measu e-
men ma ix. The measu emen noise, k, and he p ocess noise, wk,
a e assumed o be addi i e, whi e, and Gaussian, wi h he co a iance
ma ix de i ned by (5) and (6):
R = E{ K T
k} (5)
Q = E{wk wT
k}. (6)
Te ms in (4) a e: xk+1 which is he s a e ec o in k+1; F is he ans-
mission ma ix; xk is he s a e ec o in k; B is he inpu signal ec-
o ; and uk is he inpu signal. The comple e Kalman algo i hm is de-
sc ibed on Figu e 3.
kk
k
xHy +⋅=
Figu e 3. Kalman algo i hm.
2. Resul s and Discussion
Figu e 4 shows he simula ions esul s om he mechanical model,
s a ed on (1), and om he Kalman algo i hm, desc ibed on Figu e 3.
On Figu e 4a, we can obse e he esul s o he Kalman es ima ions
o he sys em, Figu e 4a op, and he sys em simula ion esul s, Fig-
u e 4a bo om. F om hese g aphics a good app oxima ion be ween
bo h can be obse ed, e en i he e is an e o a he beginning o he
es ima ion, shown in Figu e 5, due o he ini ial condi ions de i ned
on ma ix P (co a iance o e o ) and (s a e a iable). A e a ew
cycles, a ound 20, he e o eaches a minimum s able alue. Also,
om Figu e 4a, can be obse ed ano he in e es ing ea u e o he
Kalman es ima o , which is i s capaci y o educe he noise on he
measu ed signal.
On Figu e 4b, we p esen he eloci y and accele a ion o he impac
ob ained by nume ical de i a ion o he displacemen and eloci y
shown on Figu e 4a ( op), om he Kalman es ima o . These signals
a e compa ed wi h he expe imen al da a, shown on Figu e 4c, ob-
ained om a senso ized hamme apping a s eel cylinde . These
g aphics indica e ha he e is a good app oxima ion be ween hem,
and ha Kalman algo i hm can be conside ed as a sui able solu ion
o analyze he model o an impac .
3. Conclusions
On his wo k a me hod o s udy impac s has been p esen ed, using a
Kalman i l e and a mechanical model. Resul s show ha wi h a co -
ec model, he Kalman i l e achie es a good es ima ion.
Figu e 4. a) Simula ion esul s. b) Accele a ion om nume ical de i-
a ion o eloci y. c) Expe imen al signal om a senso ized ham-
me .
Figu e 5. Es ima ion e o
The Kalman i l e is a well explo ed solu ion o many complex p ob-
lems; howe e i equi es a e y good unde s anding o he plan . On
Figu es 4b and 4c, we can see ha he simula ions esul s a e simila
o he expe imen al impac . In u u e wo ks, we will s udy he senso ’s
in l uence by modeling he plan as a second o de dynamic sys em.
4. Re e ences
[1] A. To nambe, Modelling and con olling one-deg ee-o - eedom impac s,
IEEE p oceedings, Roma, 1996.
[2] Jae-Yoo Yoo, Tae-Sik Pa k, Speed Es ima ion o an IM using Kalman Fil e
Algo i hm a Ul a-low Speed, IEEE, Ko ea, 1997.
[3] S. G ewal, P And ews, Kalman i l e ing: Theo y and P ac ice Using Ma lab.,
John Wiley & Sons inc, 2001
M9