TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5342 ~ 53
4
9
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.501
2
5342
Re
cei
v
ed O
c
t
ober 3
1
, 201
3; Revi
se
d March 23, 201
4
;
Accepte
d
April 6, 2014
Dynamic Error Analysis of CMM Based on Variance
Analysis
and Improved PLSR
Zhang Mei*
1
,
Cheng Fan
g
2
, Li Guihua
3
1
,3
Coll
eg
e of Electrical E
ngi
ne
erin
g and A
u
to
mati
on, Anh
u
i
Univers
i
t
y
, H
e
fei, 230
60
1, Chi
n
a
2
School of Mec
han
ical & Aer
o
space En
gi
nee
ring, Na
n
y
an
g
T
e
chnolog
ica
l
Univers
i
t
y
, 6
3
9
798, Sin
g
a
pore
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: hfren@1
26.c
o
m
1
, chengf@
a
rtc.a-star.edu.
sg
2
, guihu
ali
1
@sina.c
o
m
3
A
b
st
r
a
ct
It is difficult to build
an
a
ccu
ra
te
mo
de
l
to p
r
edict the
dyna
mic
error of C
MM by
ana
l
y
zi
ng
err
o
r
sources. A
n
in
novativ
e
mo
del
ing
metho
d
b
a
s
ed o
n
Va
ri
ance
An
aly
sis
an
d Improve
d
P
a
rtial L
east-sq
u
a
re
regressi
on
(IPLSR)
is pr
op
o
s
ed to
av
oid
a
naly
z
i
n
g
th
e
i
n
teraction
of
err
o
r so
urces
an
d to
overc
o
me
th
e
mu
lti-col
lin
ear
ity of Ordi
nary
Le
as
t-squ
a
re
regr
essio
n
(
O
LSR). A
m
on
g
ma
ny
impac
t factors the
mo
s
t
influ
entia
l par
a
m
eters
are se
l
e
cted as
the
in
dep
en
dents of
the mod
e
l, by
me
ans of var
i
a
n
ce an
alys
is.T
he
prop
osed
mod
e
lin
g
metho
d
IPLSR can n
o
t only avo
i
d
the analys
is
of the error sources an
d
the
interacti
ons, b
u
t can als
o
so
lve the pro
b
l
e
m of mult
i-c
o
ll
i
near
ity in OLS
R
. F
r
om exp
e
r
i
menta
l
data t
h
e
expos
itory ca
p
abil
i
ty of th
is IP
LSR
mo
de
l ca
n b
e
ca
lc
ul
ated
as 8
5
.62
4
per
cent, an
d th
e
me
an
squ
a
re
e
rror
is 0.94
μ
m. As comparis
on, th
e mean s
quar
e valu
es
of co
nventi
o
n
a
l PL
SR and OLS
R
are 1.04
μ
m
an
d
1.39
μ
m, resp
e
c
tively. So IPLSR has hi
gh
er pred
ic
ting pr
eci
s
ion a
nd b
e
tter expository ca
p
abil
i
ty.
Ke
y
w
ords
: dy
na
mic err
o
r, pa
rtial le
ast-squa
res regress
i
on
(PLSR), varia
n
c
e ana
lysis, multi-col
lin
ear
ity
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
In the fiel
d of
dynami
c
error m
odeli
ng f
o
r
Co
ordi
nate
Mea
s
u
r
ing
Machi
n
e
(CM
M
), ma
ny
efforts
have
been
mad
e
on the
an
alysis of e
r
ror
sou
r
ces [1-3]. It has nota
b
le the
o
reti
cal
signifi
can
c
e
esp
e
ci
ally for designi
ng a
measu
r
ing
machi
ne, but
for practi
cal
measu
r
em
e
n
t its
appli
c
ation i
s
limited beca
u
se the m
o
d
e
ling a
c
cura
cy is notably influen
ced by
the intera
ctions
betwe
en diffe
rent e
r
ror
sou
r
ce
s.
Wee
k
e
r
s an
d Schelle
ken
s
[4] u
s
ed
8 po
sition
se
nso
r
s to dete
c
t
the defo
r
mati
on a
nd
accel
e
ration
chan
g
i
ng of
8 m
a
in
arti
culating
p
o
ints
on
a
bri
dge type
CM
M.
With the
s
e
p
a
ram
e
ters
a
dynamic e
rro
r mo
del
of
th
e
p
r
obe
wa
s establi
s
h
ed. Don
g
[5]
di
re
ctly
measured th
e an
gula
r
e
r
ror of th
e mai
n
conne
ct
ing
mech
ani
sm d
u
ring
the m
o
vement to m
odel
the pro
b
ing e
rro
rs. Zh
ang
Yi and Liu Ji
zhu[6], Wei Ji
nwe
n
and
Ch
en Yanling [7
] focuse
d on t
he
deform
a
tion
of the cro
s
sbeam. By a
nalyzin
g th
e
crossb
eam deform
a
tion with
a
c
cele
ra
tion,
c
o
ns
tant s
peed
and dec
e
lera
tion with
ANSYS, a common mo
del of beam
deformation
was
establi
s
h
ed
unde
r any lo
ad. But this
method i
s
o
n
ly useful fo
r erro
r comp
ensation in
one
dire
ction.
It’s kno
w
n that the influence of each error
source will be fi
nally reflecte
d in the
measured va
lues, na
mely
, (x, y, z) coordi
nat
e
s
. On the othe
r hand, the
Dire
ct Comp
uter
Control
（）
DCC
para
m
eters,
which are ef
fective durin
g
the whole p
r
oce
s
s of mea
s
ureme
n
t, are
easy fo
r con
t
rolling a
nd
sampli
ng. So
if these
pa
ramete
rs are
see
n
a
s
th
e inde
pen
de
nt
variable
s
, th
e
co
mpli
cated
sou
r
ce
analy
s
is for dyna
mic
errors ca
n be
avoid
e
d
.
So it’s a
no
ve
l
resea
r
ch to st
udy the influe
ntial po
wer of
each
ind
epe
ndent va
riabl
e as
well
as the elimin
atio
n of
the multi-colli
nearity in bet
wee
n
[8-9].
Partial Lea
st
-sq
u
a
r
e
s
(P
LS) Re
gression [
10-13] has attracte
d many re
searche
r
s’
intere
sts the
s
e years. Recently some i
m
prov
e
d
PLS algorithm
s [14-15] a
r
e pro
posed and h
a
ve
some
su
cce
ssful a
pplica
t
ion in different fi
elds. Similar with
PCR (P
rin
c
i
pal Com
pon
ent
Reg
r
e
ssi
on
), PLS i
s
al
so effective
for
red
u
ci
ng
the dim
e
n
s
io
ns
and
elimi
nating th
e m
u
lti-
collin
earity. The method of
PCR, however, ha
s not
a
c
ceptabl
e fitti
ng for dep
en
dents, be
ca
u
s
e it
only co
ncentrates o
n
the
prin
ciple
co
m
pone
nts of
th
e inde
pend
e
n
ts, but it’s i
r
relative with t
h
e
depe
ndent
s. As com
p
a
r
iso
n
, PLS starts
from the
dep
ende
nts to find a linear
co
mbination of t
h
e
indep
ende
nts which h
a
ve
the most influential po
wers. The
r
ef
ore it has b
e
tter predi
cti
v
e
cap
ability than PCR. Be
si
des, In the
case
s that th
e
sampl
e
si
ze
is sm
aller th
a
n
the qua
ntity of
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Erro
r Analysi
s
of CMM Ba
sed
on Varia
n
ce Analysis a
nd
Im
proved PL
SR (Zh
ang M
e
i)
5343
variable
s
, it
still has
acce
p
t
able expo
sit
o
ry c
apa
bility [16-1
9
]. Yan
g
Hongtao
a
nd Liu
Yong[
8
]
use
d
the hybrid mod
e
ling
method ab
ou
t PLS regre
s
sion a
nd the sup
port vecto
r
machine, a
nd
Zhang M
e
i[9] applied 3B spline-orth
ogo
nal proj
ectio
n
PLS regre
ssion model to
analyze
CM
M
dynamic erro
r. The
re
sult
s sh
ow th
at th
e PLS c
an
b
e
u
s
ed to
buil
d
the hi
gh
-preci
sion
mod
e
l of
CMM’
s dyna
mic erro
rs. T
he autho
rs’ g
r
oup h
a
s
em
ployed imp
r
o
v
ed PCR in t
he field of CMM
dynamic e
r
ro
rs. By analyzing the
rela
tionshi
p between (x, y, z) coo
r
dinate
s
and the DCC
para
m
eters, an accu
rate
model with
cl
ear phy
sical meanin
g
s
ca
n be esta
blished.
2. Rese
arch
Metho
d
Assu
ming th
ere i
s
a
sin
g
l
e
dep
end
ent
Y
, a set of in
depe
ndent
s
1
p
x
x
, and n
sampl
e
points a
r
e acq
u
ire
d
. Wi
th the n-di
mensi
onal d
epen
dent vector an
d the p-dime
nsi
onal
indep
ende
nt vector, an n×p
ob
se
rving
matrix can
b
e
configu
r
ed
as:
1
[]
pn
p
Xx
x
. Then the
PLS algorith
m
[10] can be
descri
bed a
s
follow.
In the ob
servi
ng matrix
X
a
comp
one
nt
1
t
, a linea
r comb
ination of
1
()
p
x
x
, is extracted,
whi
c
h
sho
u
ld
to the large
s
t extent incl
ude the m
u
tation informa
t
ion and
ha
s most
correla
t
ion
with
Y
[11].
So
1
t
include
s most inform
a
t
ion of
X
and has a goo
d exposito
r
y ca
pability for
Y
.
Then PLS re
gre
ssi
on of
X
on
1
t
and that of
Y
on
1
t
can be work
ed out, res
p
ec
tively. If
the
reg
r
e
ssi
on e
quation ha
s
rea
c
he
d the requi
re
d a
ccura
cy the op
eration
stop
s; otherwi
se the
resi
dual
info
rmation i
n
X
sho
u
ld
be
e
x
tracted
for the n
e
xt ope
ration. Thi
s
it
erative
pro
c
e
s
s
sho
u
ld be repeate
d
until the require
d ac
curacy is achi
eved. Finally if
k
comp
one
nts
are
extracted fro
m
X
:
1
k
tt
，
the regre
s
sive op
eration
s
of
Y
on
1
k
tt
sho
u
ld b
e
done. Th
e
n
the
reg
r
e
ssi
on m
odel can be e
x
presse
d in form of
1
()
p
Yf
x
x
.
2.1. Modeling Process
Acco
rdi
ng to
the refere
n
c
e
s
[10-1
1
], the modelin
g
process ca
n be sum
m
a
r
ize
d
as
belo
w
:
Step 1: Standardi
zation
The observin
g
matrix
X
is stan
dardize
d a
s
00
1
0
()
pn
p
EE
E
;
The single dep
end
ent
vec
t
or
Y
is st
anda
rdi
z
ed a
s
00
1
1
()
n
FF
.
Step 2: Comp
onent
s extraction.
k
co
mpon
ent
s ca
n be extracted a
s
Equ
a
tion (1
):
1
kk
k
tE
w
(1)
Whe
r
e
1
0
1
2
10
,,
k
T
k
k
kk
k
k
k
k
k
k
Et
EF
wE
E
t
p
p
EF
t
, and,
1
k
EE
are the
resid
ual e
r
ror
matrixes afte
r the standa
rdi
z
ation of ind
e
pend
ents.
For th
e
k
th
compon
ents,
the
coeffici
ent
s of
the fittin
g
eq
uation
can b
e
d
e
termined
by
iterative ope
ration, expre
s
sed by:
1
2
T
kk
k
k
F
t
r
t
(2)
Whe
r
e
11
,
kk
k
k
k
F
Ft
r
F
F
a
r
e t
he resi
dual
e
rro
r ve
ctors
a
fter the
stan
d
a
rdi
z
ation
of
the depe
nde
n
t
.
Then the
k
th fitting equatio
n can b
e
expressed a
s
:
11
2
2
*
kk
yr
t
r
t
r
t
(3)
Step 3: Numb
er of com
pon
ents.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5342 – 53
49
5344
The b
e
st
co
mpromi
se
sh
ould b
e
ma
d
e
to dete
r
min
e
the nu
mbe
r
of compo
n
e
n
ts. The
extracted co
mpone
nts sh
ould
have e
noug
h
expo
si
tory cap
abili
ty to the system, but the
modelin
g reliability can
not
be de
crea
sed by ov
e
r
fitting. Curren
tly the method called
Cross
Validation (CV) is wid
e
ly used to dete
r
mine the
nu
mber of com
pone
nts for L
S
. The pro
c
e
ss of
CV can b
e
de
scribe
d as foll
ow:
The
i
th
sam
p
le i
s
remov
ed from
the
s
a
mple data
s
e
t. With the res
t
s
a
mple data, a
reg
r
e
ssi
on eq
uation is
wo
rked o
u
t on th
e
k
PLS com
pone
nts. Wh
en the
i
th sample is taken into
this reg
r
e
ssi
o
n
equation, the fit value o
f
the
i
th sample can b
e
worked out, na
mely,
()
ˆ
ki
y
. For
any
i
= 1,
2,
…,
n
, the above operation is
repeated.
Then
the P
R
ESS (Predi
cti
on Residual Error
Sum of Squares)
can b
e
ca
lculate
d
:
2
()
1
ˆ
()
(
)
n
ik
i
i
PRESS
k
y
y
(4)
With all the sample data, a
nother
reg
r
e
s
sion e
quation
on k com
pon
ents can be d
e
rived.
A
ssu
ming
ˆ
ki
y
is the
cal
c
ulat
ed fro
m
all t
hese
sampl
e
data,
with t
he
same
op
eration
in th
e
above pa
rag
r
aph, the sum
of squa
re
s ca
n be define
d
as:
2
1
ˆ
()
(
)
n
ik
i
i
SS
k
y
y
(5)
CV can b
e
de
fined as:
2
()
1
(1
)
k
P
RE
S
S
k
Q
SS
k
(6)
Only if
Q
2
k
≥
0.0975, the
model
quality
can
be i
m
prov
ed by in
creasi
ng the
n
u
mbe
r
of
PLS c
o
mponents
,
k
t
.
Step 4: This
reg
r
e
ssi
on e
quation
with the
optimized
comp
one
nts is then de
d
u
cted to
that with o
r
ig
inal vari
able
s
to analy
z
e t
he
di
re
ction and extent,
to
which
the depe
ndent
s are
influen
ced by
the indepe
nd
ents [10
-
14].
2.2. Impro
v
e
m
ent of PLS Model (IPLS)
In the pro
c
e
s
s of PLS re
gression, t
he fu
ndame
n
tal p
r
i
n
cipl
e of extracting
k
t
is to
make
the covari
an
ce of th
e d
epe
ndent
Y
to
a
c
qui
re
the
m
a
ximal value.
The
covari
a
n
ce
of
Y
can be
defined a
s
,
k
Cov
t
Y
. Normally
k
t
ha
s the be
st exposito
r
y cap
a
b
ility to the d
epen
dent
s. But
there’
s
still a problem: I
t
’s seen from
equation
(7) that a bi
g value of
,
k
Co
v
t
Y
will not
necessa
rily result in
a bi
g
value of
,
k
tY
, the co
rrelation
coeffici
ent of
k
t
and
Y
. In
some
ca
se
s, therefore, a
big
val
ue of
k
Va
r
t
, the va
rian
ce
of
k
t
, may ca
use
wro
ng
sele
ction
of the
comp
one
nt
k
t
.
22
,,
kk
k
Co
v
t
Y
t
Y
V
a
r
t
V
ar
Y
(7)
To solve this pro
b
lem, Ch
eng a
nd
Wu
[15]
pro
p
o
s
ed an
improved alg
o
rithm
of PLS:
Firstly the
ort
hogo
nal m
a
trix of
Y
is
w
o
r
k
ed
ou
t a
n
d
n
a
m
ed
as
B
, whi
c
h i
s
co
mposed
by the
eigenve
c
tors
11
,
p
bb
corre
sp
on
ding to the zero ei
gen
values of
TT
XY
Y
X
. Then the
eigenvalu
e
s
and eige
nvectors of
TT
B
XX
B
are ca
lculate
d
and
named a
s
11
,
p
and
11
,
p
,
respe
c
tively. Among the
eigenvalu
e
s the large
s
t
s
values
1
,
s
are extra
c
te
d and the
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Erro
r Analysi
s
of CMM Ba
sed
on Varia
n
ce Analysis a
nd
Im
proved PL
SR (Zh
ang M
e
i)
5345
corre
s
p
ondin
g
eige
nvecto
rs
are
sele
cted to form a
matrix
A
. T
he dete
r
min
a
t
ion of
s
s
hou
ld
make the val
ue of
1
11
p
s
ii
ii
close to 99%.
A new o
r
thog
onal m
a
trix o
f
Y
is config
u
r
ed as
UX
B
A
. So the p
r
oje
c
tion
of
X
in
the
dire
ction that is ortho
gon
al to
U
can b
e
e
x
presse
d by:
11
TT
TT
pU
p
I
P
X
X
U
U
U
UX
X
I
B
A
U
U
UX
X
D
(8)
Whe
r
e
I
P
is an identity matrix,
P
U
X
is
a
proje
c
tion
wh
ich i
s
th
e
X
on
th
e
U
. Th
e
p
r
oc
es
s
of proj
ectio
n
help
s
to elimi
nate the i
n
formation
with u
nobviou
s
rela
tivity of
Y
. The ope
ration
o
f
PLS with
XD
sho
w
s the improvem
ent of convention
a
l
PL
S modeling, which i
s
na
med as IPLS in
this pap
er:
=
I
PLS
P
LS
I
P
LS
YX
D
X
(9)
This IPLS mo
del ha
s the b
e
st expo
sitory
capability to the depe
nd
ents an
d ca
n
improve
the predi
cting
accura
cy.
2.3. Acquisition of Exper
i
mental Da
ta
Yang etc [1-3], [8] proposed the experi
m
ental sc
he
me by drivin
g the prob
e runnin
g
in
the mea
s
u
r
in
g sp
ace freel
y, without tou
c
hin
g
or
pra
c
tical mea
s
u
r
e
m
ent. So onl
y the position
i
ng
errors of different po
sitio
n
s are sam
p
led. In
this study the pra
c
tical e
r
rors are a
c
qui
red
by
touchi
ng th
e
sp
eci
m
en
a
t
different
p
o
sition
s
and
with
differe
nt DCC p
a
rameters,
su
ch a
s
positio
ning v
e
locity v
1
, to
uchi
ng velo
ci
ty v
2
and ap
proa
chi
ng di
stance a. Thi
s
pro
c
e
ss of the
experim
ent correspon
ds to the defin
ition of dynamic errors and in
clud
es the co
nsid
eratio
n of all
t
he main
er
r
o
r
sou
r
c
e
s,
su
ch a
s
me
cha
n
ic
al
st
ru
cture, guild way,
environ
ment,
and
most
importa
nt, the probing
errors. T
he
co
mposite
er
ro
rs, therefore,
are
acquired
in the p
r
op
o
s
ed
experim
ental
pro
c
e
ss. Fig
u
r
e 1 is the p
r
i
n
cipl
e
diag
ra
m of dynamic erro
r collecti
on experi
m
en
t.
Figure 1. Prin
ciple
Diag
ram
of Dy
namic
Erro
r Coll
ecti
on Experime
n
t
A moving bri
dge CM
M MC85
0 (eq
u
ipp
ed with the p
r
obe
Reni
sh
a
w
TP20, stylu
s
length:
20mm, tip bal
l diameter: 4
mm) is u
s
ed t
o
test
ify the propo
se
d mod
e
ling metho
d
. The experim
ent
for error
sam
p
ling is a
r
ran
ged a
s
followi
ng.
Table 1. Data
Colle
ction Plan of Dynami
c
Mea
s
u
r
ing
Erro
r
Independents
Values
Number of
group
s
Number of
data
x
/(mm)
0,150,300,45
0,6
00,750
72
3456
y
/(mm)
150,300,450,
550
z
/(m
m
)
-581,-4
73,-32
4
zv
1
/(mm/s)
20,60,100
48
a
/(mm)
1,2,5,8
v
2
/(mm/s)
2,4,6,8
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5346
There are 6
indepen
den
ts: spatial coordi
nate
s
x(mm), y(mm
), z(mm), an
d DCC
para
m
eters:
positio
ning v
e
locity v
1
(mm
/
s), app
ro
achi
ng dista
n
ce a
(
mm), a
nd co
ntacting velo
city
v
2
(mm/s). Th
e depe
ndent i
s
the com
p
o
s
ite spatial dy
namic e
r
ror e
(
μ
m). Differe
nt values of e
a
ch
indep
ende
nt
are
u
s
ed
an
d
in
every in
di
vidual exp
e
ri
ment o
n
ly on
e vari
able
is
cha
nge
d. Ta
ble 1
sho
w
s the
combinatio
ns
of the in
dep
ende
nts.
Th
e
r
e a
r
e
totally 345
6
combi
nation
s
of th
e
variable
s
.
In order to te
stify the prop
ose
d
method,
only 5% of
these 3456 g
r
oup
s data, about 173
points, we
re randomly se
l
e
cted. Fo
r e
a
ch point th
e m
easure
m
ent
s
were repe
ate
d
five times
a
n
d
then the m
e
a
n
value
s
a
r
e
worke
d
o
u
t. So147
data
we
re u
s
e
d
for th
e mod
e
l eval
uation, while t
he
rest 26
data
were used for modeling.
3. Results a
nd Analy
s
is
3.1. Analy
s
is of Experim
e
ntal Data
In 1923 R.A
.
Fishe
r
pro
p
o
se
d Analysi
s
of Varian
ce (ANOVA
), which is u
s
ed to
determi
ne th
e influential
factors
of a
ce
rtain va
ri
able
and th
e inte
rsection
s
amon
g the
s
e fa
ctors.
This meth
od
is widely used in biology
and agri
c
ul
t
u
re, but still seld
om used
in the field
of
mech
ani
cal
e
ngine
erin
g. In o
r
de
r to
de
termine
the
i
n
fluen
cing
va
riable
s
of dy
namic e
r
rors
and
their interacti
ons (expressed in form of
produ
cts
in f
o
llowin
g
se
cti
ons), the method of varian
ce
analysi
s
is e
m
ployed in this study. The
experim
ent i
s
rep
eated b
y
5 times because it’s nee
ded
to distin
gui
sh
the inte
ra
ction of th
e infl
uential
fa
ctors a
gain
s
t the
ran
dom
erro
rs. T
he
analy
s
is
c
an be done by the s
o
ftware SPSS16.0. T
he res
u
lts
are rec
o
rded in Table 2.
Table 2. Vari
ance Analysi
s
the Im
pa
ct Facto
r
s of Me
asu
r
ing Erro
r
Fac
t
or
T
y
pe III Sum
of SquaresDegree of Freedom
Mean SquareF V
a
lueP
V
a
lue
C
290.49
1
290.49
61.91
0.00
x
335.75
1
335.75
71.55
0.00
y
100.92
1
100.92
21.51
0.00
z
530.50
1
530.50
113.06
0.00
v
1
13.86
1
13.86
2.95
0.08
a
28.77
2
14.39
3.07
0.05
v
2
22.89
2
11.44
2.44
0.09
v
1
*a
30.37
2
15.19
3.24
0.04
a*v
2
101.81
4
25.45
5.42
0.00
x*a
40.20
2
20.10
4.28
0.00
y*a
4.58
2
2.29
0.49
0.62
z*
a
26.22
2
13.11
2.79
0.08
v
1
*v
2
1.78 2
0.89
0.19
0.83
x*v
1
132.40
1
132.40
28.22
0.00
y*v
1
14.69
1
14.69
3.13
0.07
z*
v
1
13.16
1
13.16
2.81
0.09
x*v
2
27.37
2
13.68
2.92
0.06
y*v
2
31.90
2
15.95
3.40
0.04
z*
v
2
35.20
2
17.60
3.75
0.03
x*y
135.16
1
135.16
28.80
0.00
x*
z
15.05
1
15.05
3.21
0.07
y*
z
35.07
1
35.07
7.47
0.01
e
525.53
112
4.69
Sum 2493.65
147
Note: The p
r
o
duct of two variabl
es exp
r
es
sed the int
e
ra
ction bet
ween the varia
b
les
The data
in
Table 2
sh
o
w
that except
the intera
cti
on bet
wee
n
coo
r
din
a
te va
lue y and
approa
chin
g distan
ce a
(e
xpresse
d
by y*a),
and
the intera
ction be
tween po
sitio
n
ing
velo
city and
conta
c
ting
ve
locity (expre
ssed
by v
1
*v
2
), all the
othe
r facto
r
s have
sig
n
ifica
n
t in
fluence o
n
th
e
dynamic
erro
rs at the level
of 10%. Beca
use
all t
he fa
ctors affe
ct the dynami
c
e
r
rors t
o
differe
nt
extents, they are all use
d
as
the ind
epen
dent
s for the model
of dynamic
error p
r
edi
cti
on.
Beside
s, it
sh
ould b
e
con
s
i
dere
d
that th
e coor
dinate
s
(x, y, z),
posi
t
ioning velo
cit
y
, appro
a
chin
g
distan
ce
an
d
co
ntactin
g
velocity may
affect
th
e
measurement
erro
rs in f
o
rm
of vari
a
b
le
nonlin
earity.
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Dynam
ic Erro
r Analysi
s
of CMM Ba
sed
on Varia
n
ce Analysis a
nd
Im
proved PL
SR (Zh
ang M
e
i)
5347
3.2. IPLS M
odeling for
D
y
namic Err
o
rs
It’s kno
w
n from expe
rien
ce that the
meas
urement
errors of a
CMM h
a
ve a
nonline
a
r
relation
shi
p
with the
sel
e
cted
inde
p
ende
nts. In
pra
c
tice
the
influen
ce
o
f
errors
can
be
synthe
sized i
n
form of sum
:
22
2
2
2
2
0
1
2
3
4
5
1
6
2
7
8
9
10
11
1
1
2
2
13
14
15
16
1
1
7
2
1
8
19
1
2
0
2
21
22
1
2
3
2
24
1
2
5
2
ex
y
z
a
v
v
x
y
z
a
v
v
x
y
x
z
xa
x
v
xv
yz
yv
yv
a
z
z
v
z
v
a
v
a
v
(10
)
Whe
r
e
i
（～
）
i=0
2
5
i
s the
para
m
eter th
at needs e
s
t
i
mation and
is the item
o
f
rand
om erro
r. The pro
d
u
c
tions of varia
b
l
e
s re
sp
ect th
e intera
ction
s
in betwee
n
.
With O
r
di
nary
Lea
st Sq
uare (OLS), th
e l
a
rge
s
t
val
ue
of VIF (Va
r
ia
nce
Inflation
Facto
r
) i
s
as high a
s
625.47, whi
c
h mean
s se
rious multi-
col
linearity exist
s
. To overco
me the above
limitation, IPLS is employ
ed to elimina
t
e the
dynamic error exp
r
essed
by eq
uation (1
0). The
whol
e process can be
di
vided into two
step
s:
①
T
he orth
ogo
na
l proje
c
tion
o
f
indepen
den
ts
matrix is wo
rked o
u
t by MATLAB.
②
PLS regressio
n
is wo
rked o
u
t by SICAM-P. The result
is
sho
w
n in Ta
b
l
e 3.
T
abl
e 3. Para
meters of the model Fitting
Ef
fect
Component
R
2
R
2
(c
um)
Q
2
Q
2
(c
um
)
e/
μ
m
Comp1
0.64473
0.64473
0.59605
0.59605
Comp2
0.21151
0.85624
0.20168
0.79773
Comp3
0.07101
0.92725
0.07116
0.86889
The a
nalysi
s
in sectio
n 2.1
sh
ows that
only if Q
2
≥
0.
0975
can the
modeli
ng
qu
ality be
improve
d
by
increa
sing t
he compo
n
e
n
ts of PLS.
Whe
n
two
compon
ents
a
r
e extra
c
ted,
the
exposito
r
y
capability of t
he m
odel
is R
2
=0.856
24
and
the
CV of the
dy
namic e
r
rors is
Q
2
=0.79
773,
whi
c
h mea
n
s
the model ha
s goo
d pre
c
i
s
ion.
Figure 2. V
a
ri
able Impo
rtan
ce
It’s see
n
fro
m
Figu
re 2
(Variabl
e Imp
o
rtan
ce
(VIP)) that amo
n
g
the inde
pen
dents th
e
coo
r
din
a
te x and z
have t
he mo
st influ
ence upo
n th
e dynami
c
errors; the influ
ences of v
1
, y,
a
are
weaker
and v
2
i
s
th
e wea
k
e
s
t. This ph
enom
enon
can
b
e
explai
ned
by the p
r
a
c
ti
cal
con
d
ition
s
: for a bri
dge
-type machine th
at is dr
iven
o
n
one
side, a
closer p
o
siti
on to the driv
ing
side
will
cau
s
e la
rge
r
e
r
rors
(x axis). A
highe
r p
o
si
tion
w
ill a
l
s
o
c
a
u
s
e lar
g
er
er
ro
r
s
(z a
x
is)
.
Bu
t
along th
e dri
v
ing sid
e
(y
axis) the
po
sition ha
s
le
ss influe
nce. All t
he other factors have
no
notable influ
e
n
ce o
n
the dynamic e
r
rors.
Figure 3 li
sts
the re
gre
s
sio
n
co
efficient
s of
the re
gression
equ
ation
for the
stand
ardi
zed
data, whi
c
h h
a
ve no items
of con
s
t
ants.
It’s seen th
at the items z, v
2
, z
2
, v
2
2
,
z*a,
z*v
1
, z
*
v
2
and
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5348
y*z have negative effects,
whi
c
h means the small
e
r t
hey are, the
bigger
error they will cause.
The oth
e
r ite
m
s, ho
weve
r,
have the
po
sitive effect
s.
This
co
ncl
u
si
on p
r
ovide
s
t
he in
stru
ction
for
optimizin
g the para
m
eters combin
ation.
Figure 3. Coe
f
ficient Plot
3.3. Anal
y
s
is
of IPLS Predicting Effect
26 sets
of da
ta amon
g all
173 a
r
e
be
selecte
d
an
d t
a
ke
n into th
e
fitting functi
on, the
predi
cted
me
an
squ
a
re
error (MSE) of
IPLS re
g
r
e
s
sion
eq
uation
,
whi
c
h
ca
n
be eval
uate t
he
predi
cting
effects, i
s
cal
c
u
l
ated to be
0
.
94
μ
m.As co
mpari
s
o
n
, the predi
cted
MSE of OLS
and
PLS are calculated to be
1.39
μ
m and
1.04
μ
m, respectively. Figure 4
sho
w
s the predi
cti
o
n
accuracy of IPLS and PLS. This re
sults
sho
w
that IPLS has the bet
ter pre
d
ictin
g
effect.
Figure 4. IPLS & PLS Observed vs. Pre
d
icted Plot
4. Conclusio
n
The error sou
r
ce
s
of CMM
are
very com
p
licate
d
a
nd
have u
n
certai
n inte
ra
ction
s
. So it i
s
difficult to establish an accurate mo
del to predi
ct
the dynamic e
r
rors by analyzi
n
g error sou
r
ces.
In this
pape
r an im
prove
d
mod
e
ling
method
ba
se
d on P
L
S re
gre
ssi
on i
s
p
r
opo
se
d to
a
v
oid
analyzi
ng th
e intera
ction
of error
so
urces
and
t
o
overcom
e
the multi-collinea
rity of OLS
reg
r
e
ssi
on. The re
sult
s sho
w
that the prop
osed
method IPLS has better
perfo
rman
ce
of
predi
cting a
n
d
better expli
c
ability, comp
ared
with OL
S and PLS.
Ackn
o
w
l
e
dg
ements
This
work
wa
s su
ppo
rt
ed by the
Young
Ta
lents
Fun
d
of Anhui Province
(NO.2012S
Q
R
L0
12),
the
Key Unive
r
sit
i
es
Natu
ral
S
c
ien
c
e
Resea
r
ch
Proje
c
t of
Anhui
Provin
ce
(NO. KJ2010
A036, NO.KJ2011A0
1
2
)
, the Re
sea
r
ch
Fund for the
Docto
r
of Anhui Unive
r
si
ty,
and the Fun
d
i
ng for traini
ng
young
teach
e
rs of Anh
u
i University.
Referen
ces
[1]
F
e
i Yetai, Z
h
a
o
Jing, W
a
n
g
Hon
g
tao, Ma
Xi
us
h
u
i. A Re
vie
w
of Res
e
a
r
ch on D
y
n
a
m
i
c Errors of
Coor
din
a
te Me
asuri
ng Mach
in
es.
Chin
ese Jo
urna
l of Scienti
f
ic Instrume
nt
. 200
4; 25(4): 77
3-77
6.
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TELKOM
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046
Dynam
ic Erro
r Analysi
s
of CMM Ba
sed
on Varia
n
ce Analysis a
nd
Im
proved PL
SR (Zh
ang M
e
i)
5349
[2]
Yang H
o
n
g
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