TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 82
9
2
~ 830
2
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.61
36
8292
Re
cei
v
ed Ap
ril 24, 2013; Revi
sed Septe
m
ber
12, 201
3; Acce
pted
Octob
e
r 2, 20
13
Pose Error Analysis Model Based on Binocular Vision
for Rigid-Body
Niu Fenglian
Schoo
l of Mechan
ical a
nd El
ectrical En
gin
e
e
rin
g
, Ning
bo
Dah
ong
yi
n
g
U
n
iversit
y
,
N
i
n
gb
o
31
51
75, P
.
R
.
C
h
i
n
a
email: nfl
197
9
@
12
6.com
A
b
st
r
a
ct
In order to s
a
tisfy the orie
ntat
ion
measur
in
g req
u
ire
m
ent
s of rigi
d-bo
dy
such as w
o
rk
piec
e,
cutting to
ol i
n
i
ndustry a
nd
medic
a
l i
n
stru
me
nts for inv
a
sive
surgery, th
is p
aper
pres
ents
a bi
noc
ular v
i
si
o
n
detectio
n
techn
i
qu
e base
d
on
spatia
l positi
o
n
informatio
n
of mark
ers to extract rigid-b
ody
pose i
n
for
m
ati
o
n
and
ana
ly
z
e
s t
he p
o
se acc
u
r
a
cy of rigi
d-b
o
d
y usin
g the
pr
incip
a
l c
o
mpo
n
ent an
alysis (P
CA) and th
e l
e
ast
squar
e metho
d
(LSM) w
hen spatia
l pos
ition err
o
r of
mark
ers exi
s
t. T
he simul
a
tion ex
per
i
m
en
t
de
mo
nstrates
t
he maxi
mu
m ang
le erro
r
of
orie
ntation
is
abo
ut 0.5
9
de
gree
w
hen
the
pos
ition
err
o
r
of
mark
ers satisfy the Gaussia
n
distrib
u
tion w
i
th the m
ean is
z
e
r
o
an
d the stand
ard
d
e
viati
on is 0~
3
m
m. T
h
e
exper
imenta
l
results verify thi
s
meth
od ca
n robustly so
lve
the orie
ntatio
n of rigi
d b
ody u
s
ing the p
o
siti
o
n
infor
m
ati
on of
mark
ers w
i
th positio
n error
s
, and it
prov
i
des a the
o
reti
cal an
d exp
e
ri
me
ntal b
a
sis f
o
r
orie
ntation me
asure
m
ent of rigid b
ody.
Ke
y
w
ords
:
error a
nalys
is, orie
ntatio
n d
e
tection,
bin
o
c
ular
v
i
sio
n
, camera c
a
li
br
ation, th
e pri
n
cip
a
l
compo
nent a
n
a
lysis, the le
as
t square
meth
o
d
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
E
n
g
i
n
eerin
g and
Scien
ce.
All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
With the developme
n
t of
machi
ne visio
n
tec
hnolo
g
y
,
orientaion d
e
tection of work pie
c
e
and mde
d
ical
instrum
ents
for invasive
surge
r
y is on t
he increa
se
based on visual inspe
c
tio
n
in
indu
strial technolo
g
y and medical diag
nosi
s
[1-4]. Compute
r
vision and photog
rammet
r
y have
become
a core issue,
po
se
estimatio
n
a
s
a
re
sea
r
ch
hot in the
field of compute
r
visio
n
resea
r
ch
has be
en carried on for m
any years [5]
.
The issue of study here is a
dynamic po
se e
s
timation
of
the rigid
body
. For rigi
d-b
o
d
y pose esti
mation the
r
e
are a l
a
rg
e n
u
mbe
r
of stu
d
ies [6
-7]
in
cl
uding
the method about the minimum su
m of squa
re
s
and sing
ular v
a
lue de
comp
osition which
is
robu
st an
d fa
st cal
c
ul
ation.
Some researche
s
h
a
ve st
udied
po
sitio
n
error of the
marke
r
s
how to
af
fect attitude
solving,
su
ch as
W
o
lt
ring
et al [8] pro
posed a
n
e
s
timation meth
od of maxim
u
m
error for solvi
ng a given tracki
ng error of the
probe wh
en markers is symme
trically distributed on
the pro
be, M
o
rri
s a
nd
Don
a
th el at [9] e
x
tended
the
W
o
ltrin
g
’
s
stu
d
y and q
uanti
f
y the cumul
a
tive
impact
of mul
t
iple so
urce
s
of error i
n
clu
d
ing dy
n
a
mic deform
a
tion
error al
go
rith
m target
s an
d so
on. However
,
no quantitative analysis a
bout the
influence of these met
hod
s position e
rro
r how
to influent th
e pose of rigid body
.This
article mainly
focuse
s on pose estimati
on of rigid body
based on bi
nocular visi
o
n
includi
ng calibratio
n
of
int
r
insi
c an
d
ex
t
r
insic p
a
r
amet
e
r
s of
t
he
came
ra, the
solving
abo
ut theori
entatio
n rel
a
ti
on of
two
came
ra
.
Accordi
ng t
o
the bi
no
cul
a
r
came
ra pa
ra
meters, imag
e position inf
o
rmatio
n
of markers in two im
age coordination sy
stems
are used to derive the sp
atial position of the ma
rkers in the worl
d coor
dinate
system. In pose
solutio
n
of rig
i
d body
, the l
east
squ
a
re
s
method
com
b
ined with
pri
n
cipal com
pon
ent
analy
s
is a
r
e
use
d
to di
scu
ss
ho
w to u
s
e ina
c
curate l
o
catio
n
information of ma
rke
r
s to an
alyze the
orie
nta
t
ion
error an
d use
simulation a
nalysi
s
to
describ
e the entire resea
r
ch proce
s
s.
2.
Setting up o
f
V
i
sion Dete
ction Sy
stem
The po
se d
e
tection
syste
m
con
s
i
s
ts of
two
T
o
shib
a indu
strial ca
mera
s, two P
C
I frame
grab
be
rs whi
c
h type is Maxtor general a
nd three
infra
r
ed LED lingh
ts.
The using of infrared LE
D
is mainly on accou
n
t of th
e ambient light af
fect
ing little to
the acq
u
ired ima
ge in the experiment
and dedu
ce the segm
enta
t
ion dif
f
iculty
of the or
dinary image. Stereo vision system is used to
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Pose Error A
nalysi
s
Model
Based on Bi
nocular
Vi
sio
n
for Rigid
-
Bo
dy (Niu Fe
ngl
ian)
8293
detect and track three IR
markers and
calcul
ate
the
orientation a
nd positio
n of the rigid body
relative to the worl
d coo
r
dinate frame
by merg
in
g vir
t
u
a
l r
e
c
o
n
s
tr
uc
tio
n
s
y
s
t
e
m
a
s
F
i
gu
r
e
1
sho
w
s.
Figure 1(a
)
. Binocular visi
o
n
system mo
del
Figui
re 1(b
)
. V
i
tual pose
rec
o
ns
truc
tion s
y
tem
2.1. Camer
a
Cali
bration
T
o
obtain
spa
t
ial positon of
marke
r
s in the wo
rld co
ordinate syste
m
, the first proce
d
ue
r
is to calib
rati
on two came
ras
and o
b
ta
in the rela
tiv
e
orientatio
n. Intr
insi
c parameters of the
came
ra i
s
used to a
quire
ef
fective len
s
focal l
engt
h, optical ce
nter
,
and
le
ns distortion.
Extrin
sic
calib
ration i
s
need
ed to de
termine the o
r
ientation
re
la
iton of the stereo
cam
e
ra
s with respe
c
t to
one an
othe
r
.
Becau
s
e th
e accu
ra
cy a
bout came
ra
calib
ration
p
a
ram
e
ters directly af
fects
the
spatial lo
catio
n
accura
cy of marke
r
s, and there
are some method
s of the camera calibration to
solve this
problem.
T
a
ki
n
g
into acco
u
n
t the
need
s of system
desi
g
n, Zha
n
g
ZY’
s
came
ra
calib
ration al
gorithm [10] i
s
ado
pted, which d
o
e
s
not
requi
r
e the 3
D
sp
atial po
si
tion informati
o
n
of marke
r
s a
nd only need
s to know 2
D
plane po
si
tion informati
o
n of marke
r
s in two cam
e
ra
coo
r
din
a
te systems, so it is very con
v
enient
to o
perate an
d has high ro
b
u
stne
ss. Before
calib
ration
e
x
perime
n
t, a high-quality
printe
r are
employee
d
to print bl
ack an
d wh
ite
che
c
kerboa
rd
squares 19
×21 corn
er p
o
ints on
A0 pape
r
,
and p
a
ste the template on a flat
woo
den su
rfa
c
e, whe
r
e the
length and width of eac
h grid are equ
al and the side length is 20cm
and pixle
accura
cy is
0.1
mm as Figu
re 3
sho
w
s.
During
the cali
bration,
t
w
o cameras wo
ul
d
be
put in front of calibration
template, transl
a
te dista
n
ce a
n
d rota
tion angle d
onn’t exce
ed
the
scope of the came
ra view
angle, t
he ca
mera auto
m
a
t
ically detect the co
rne
r
in the boa
rd usi
n
g
Susan
corne
r
detection alg
o
rithm
[1
1]
, Figure 2 is the left camera first image co
rner dete
c
ted.
In
orde
r to g
e
t the came
ra m
a
trix and
dist
ortion
param
eters,
we
mu
st get at le
ast three o
r
mo
re
dif
f
erent po
se
images.
Figure 2. Acq
u
ired im
age
s
on the left ca
mera
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8292 – 83
02
8294
The cali
ration
process can
be de
scrible
d
as follo
ws
, firs
tly c
a
libration template is
placed
in front of bin
o
cul
a
r
came
ra, and a
c
qui
re image p
o
in
ts by detectin
g
che
s
sbo
a
rd
corner
so th
at
image
co
rne
r
point an
d
co
rre
sp
ond
en
ce
co
rne
r
s
on
-b
oard
are built
. which
shoul
d be
explaine
d
that the first corne
r
of the lowe
r left corn
er on t
he cali
bration plate i
s
use
d
as a calibratio
n
plate
coo
r
din
a
te sy
stem ori
g
in p
o
int, while the
image co
or
dinate system
origin p
o
int is in the upper l
e
ft
corne
r
of the
image. In
order to
solve
the intri
n
si
c a
nd extrin
si
c p
a
ram
e
ter
of camera, thre
e
or
more dif
f
e
r
en
t pose ima
g
e
s
are nee
ded
, so that the came
ra m
a
tri
x
and disto
r
tion pa
ramete
rs
prop
osed by
Zhang Z
hen
g
y
u are obtai
n
ed. Whe
n
the
four co
rne
r
p
o
ints an
d the
corre
s
po
nding
image of
the
pun
ctuation board cor
ners corre
s
po
nd
ing relation
s
are
esta
blish
ed, Intern
al a
n
d
external p
a
ra
meters of the
equation
are
solved. In
the calib
ration,
the relative
positio
ns of t
he
came
ra
s a
r
e
fixed, right an
d left came
ra
image
s ar
e
available, an
d so the inte
rnal and exte
rnal
para
m
eters o
f
two came
ra
s are o
b
taine
d
at the same
time.
For the devel
opment of cal
i
brat
ion
software, we m
a
ke
use of the Intel's Ope
n
CV
library
to implement the calibratio
n
for our research.
The image re
sol
u
tion is 768 pixel×5
76 pixel.
The
model pla
n
e
contain
s
a
pattern of 8x
1
1
che
c
ker
boards, a
nd
the size of every sq
uare
is
20cmx2
0
cm. It is printed with a high-q
uality printe
r
.
In theory
,
three image
s p
o
sition
ed in three
dif
f
erent di
re
ction
s
can
sa
tisfy the requ
ireme
n
t, in
order to g
e
t be
tter solutio
n
of intrinsi
c a
nd
extrinsi
c pa
ra
meter
,
6 im
a
ges
are
u
s
ed
to calib
rate t
he left and
ri
ght ca
meras.
Figure 3
sho
w
s
the image
s a
nd its co
mers acqui
red by the left came
ra.
Figure 3. Corner poi
nt extration of calib
ration image
Usi
ng Zhan
g
'
s nonlin
ea
r optimizatio
n techni
que
ba
sed on the maximum likelihoo
d
crite
r
ion; we can g
e
t the two came
ra p
a
ram
e
ters.
T
a
ble 1 de
scrib
e
s the left an
d right pa
ram
e
ter
.
Table 1. The
Intrinsi
c Para
meter of the Left and Rig
h
t
Camera
Left c
a
mera(m
m)
Ri
ght c
a
mera(m
m)
Focal length
1002.406 10
34.
403
996.238 10
28.6
67
Principal point
358.465 32
1.97
0
376.992
315.856
2.2.
Relativ
e
Orienta
t
ion Solution of T
w
o
Cameras
Whe
n
intrinsi
c and extrin
si
c paramete
r
s of
matrix are obtained, the following m
e
thod is
to obtain the relation
shi
p
of position and
orientat
ion a
bout two cam
e
ra
s.
The ori
entation rel
a
tion
for two came
ras can be d
e
rives as follo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Pose Error A
nalysi
s
Model
Based on Bi
nocular
Vi
sio
n
for Rigid
-
Bo
dy (Niu Fe
ngl
ian)
8295
Figure 4. Location of
feature point ba
sed
on bino
cula
r vision
Suppo
sing th
ere i
s
a featu
r
e poi
nt
P
in sp
ace
as
sho
w
n in Figu
re 4,
the relatio
n
about
the feature p
o
int in left and worl
d co
or
d
i
nate system
can b
e
expre
s
sed a
s
:
LL
W
W
PL
P
(2)
Whe
r
e
L
P
is the spaci
a
l coo
r
dinate betwe
en the left camera coo
r
di
nate system,
W
P
is the
spa
c
ial
co
ord
i
nate in the
worl
d coo
r
di
nate sy
stem,
L
w
L
is the
relat
i
ve orientaito
n abo
u
t the
worl
d co
ordin
a
te system a
nd left came
ra coo
r
din
a
te system.
Similarly
,
the relation a
bout
the feature p
o
in
t betwe
en
right and
wo
rl
d coo
r
din
a
te system
can b
e
expre
s
sed a
s
:
RR
W
W
PL
P
(3)
By (2) and (3) the equation
can b
e
getten
,
11
LL
R
R
TT
LP
L
P
That is,
1
LL
R
R
TT
PL
L
P
Therefore, th
e rotation an
d
translatio
n
m
a
trix about the came
ra
s ca
n be expre
ssed as:
1
RR
L
LW
W
TL
L
(4)
T
abl
e 2. Orie
ntation matrix
about two ca
mera
s for fou
r
calib
ratio
n
e
x
perime
n
ts
0.89578
0.014684
-0.44426
551.51
-0.005952
4
0.99976
0.021043
3.5124
0.44446
-0.016205
0.89565
151.8
0 0
0
1
0.89698
0.013141
-0.44187
549.48
-0.002142
2
0.99968
0.025379
-3.7665
0.44206
-0.021819
0.89672
149.72
0 0
0
1
0.89849
0.017266
-0.43866
544.3
-0.008750
4
0.99973
0.021427
2.6137
0.43891
-0.015413
0.8984
146.4
0 0
0
1
0.89594
0.017502
-0.44384
551.13
-0.005705
6
0.99959
0.027899
-5.9424
0.44415
-0.022463
0.89567
150.26
0 0
0
1
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8292 – 83
02
8296
Con
s
id
erin
g t
he o
r
ientai
on
relatio
n
b
e
twee
n two
ca
mera
s
didn’t
ch
ang
ed
du
ring th
e
came
ra
calib
ration process, thus
R
L
T
shoul
d be con
s
tant matrix, but experim
ents
re
sults sho
w
ed
that the orient
ation matrix is not entirely
con
s
i
s
t
ent by four cali
bratio
n ex
perim
ent
s. In theory, th
e
two
came
ra
s
relative o
r
ient
aiton remain
e
d
un
cha
nge
d
after
compl
e
tion of the
pa
ckage, T
able
2
is the orie
ntai
on matrix for four caliba
r
ion
expeirme
n
ts.
From the calibration mat
r
ix we coul
d co
nclu
de
that calibratio
n
mo
del of binocular vision
is limited by the experi
m
en
tal. The limitaion wa
s ca
u
s
ed by the detetion error of
corno
r
point a
n
d
the vibration
of the cam
e
ra, which
a
ffects t
he small cha
nge
s of calib
rati
on pa
ramete
r. To
improve
cali
b
a
tion a
c
cura
cy, the least-squares
me
th
od can o
p
timize
calib
ration
matrix to obt
ain
the optimal p
a
ram
e
ters, the l
east-sq
ua
res can expre
s
sed a
s
:
2
1
4
2
1
mi
n
mi
n
K
R
Lk
o
p
t
i
m
a
l
k
K
R
op
ti
m
a
l
L
k
k
tt
R
R
(5)
The re
ctified
orientatio
n m
a
trix is,
1
0
0
0
54
.
149
89661
.
0
018975
.
0
44239
.
0
8957
.
0
023937
.
0
99969
.
0
0056376
.
0
11
.
549
44216
.
0
015648
.
0
8968
.
0
optimal
T
3.
Marker T
r
ac
king and Lo
cation
In order to realize real time and rob
u
s
t tracking of
marke
r
s, ea
ch cam
e
ra should be
pro
c
e
s
sed at least more than 25HZ a
nd image re
solutio
n
sho
u
ld be less than one pixel.
Although u
s
i
ng ste
r
eo vi
sion co
ntrib
u
tes to hig
h
a
nd ro
bu
st tra
cki
ng, but th
ese
req
u
ire
m
ents
need a comp
utational co
st.
T
o
overcom
e
this probl
e
m
, firstly we
predi
ct the region of interest.
The procedu
re of tracking i
s
emple
m
ent
ed as follo
ws:
a)
Grab th
e fra
m
e
b)
Thre
sh
old the
Frame into bi
nary imag
e
c)
Filtering a
nd l
abelin
g
d)
Find the ce
nter of marke
r
Con
n
e
c
ted compon
ent
la
beling algo
rithm
is
used t
o
extra
c
t the
blob
regio
n
a
nd canny
edge
dete
c
tio
n
is e
m
ploye
ed to get
con
t
our line
and
extract bl
ob
aera. F
o
r
eve
r
y blob
we
can
give two th
re
shol
ds
abo
ut area t
o
di
sca
r
d the
blob
s
with le
ss th
a
n
one
pixel a
nd mo
re tha
n
fifty
pixels. If the
numbe
r of every image is more than
three, we use the round
ne
ss
of the blob to
get
the best thre
e blob
s.
After that, geom
etric
cente
r
a
s
criteria i
s
u
s
ed to di
sco
v
er every blo
b's
cente
r
, Figure 5 sho
w
s two image
s of markers a
c
q
u
ired by the left camera a
nd right ca
m
e
ra.
Whe
n
ko
wni
n
g the relative
orientaiton
matrix
about
two ca
meras
and imag
e p
o
int of marke
r
s in
the two imag
e coo
r
din
a
tes, the next question i
s
ho
w to extract the spa
c
ial p
o
i
n
t in the cam
e
ra
coo
r
din
a
te sy
stem from the
image co
ordi
nate syste
m
.
Figure 5. Maker imag
e in the left and righ
t cameras
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TELKOM
NIKA
ISSN:
2302-4
046
Pose Error A
nalysi
s
Model
Based on Bi
nocular Vi
sio
n
for Rigid
-
Bo
dy (Niu Fe
ngl
ian)
8297
In previo
us
studie
s
, the
comm
only t
r
iang
ulation
method i
s
u
s
ed to
calcu
l
ate 3D
coo
r
din
a
te of
markers
acco
rding
to the
correspon
ding
left image
po
ints coo
r
din
a
tes
(,
)
ll
uv
and
right ima
ge p
o
ints
coo
r
din
a
tes
(,
)
rr
uv
. For
si
mplified calculation in th
e
solutio
n
p
r
o
c
e
ss, two
came
ra
coo
r
dinate sy
ste
m
s no
rmali
z
ed into the
same coordin
a
te system i
s
ne
ce
ssary
.
The
intersectio
n
sapci
a
l poi
nt o
f
two straight
lines
i
n
the
same
coordin
a
te
syste
m
can be
obtain
ed.
The imag
e po
int of left camera coo
r
din
a
te system in t
he right coo
r
d
i
nate system i
s
as follo
ws:
1
(1
)
T
R
LR
L
L
l
l
VT
A
u
v
(6)
Whe
r
e,
1
1
T
Ll
l
Au
v
re
pre
s
ent
s image
point of marke
r
s in the left image coordi
nate
transfo
rmatin
g into the left camera co
ordin
a
te,
LR
V
indicate
s that image poi
nts in the left
came
ra
coo
r
dinate are tanrsfo
r
matin
g
into the
right
came
ra coordinate sy
ste
m
. So the whole
probl
em i
s
converted i
n
to
solving i
n
te
rse
c
ting
poin
t
of the two
strai
ght line
s
in the
sa
me
coo
r
din
a
te sy
stem.
T
a
kin
g
into accou
n
t the ac
tual m
easure
m
ent, came
ra
calib
ration erro
r a
n
d
extraction
errors
of the image point
s will
result in
no intersection
about two
straight spacial lines,
the publi
c
ve
rtial line of t
w
o
spa
c
ial li
ne are sovle
d
to define t
he interse
c
tin
g
point
whi
c
h is
defined by th
e the cente
r
o
f
public vertical line.
4.
Solv
ing the Pose of Rigi
d-body
The above discusse
s how
to solve the p
r
oble
m
of spacial location
of marke
r
, next we’ll
discu
ss
ho
w to use ma
rker spatial l
o
catio
n
info
rmation to
solve rigid
-
b
o
d
y positio
n
an
d
orientatio
n ch
angin
g
. It is
not robu
st using the
dire
ct linear sol
u
tion method to
sovle becau
se
markers
posi
t
ion errors
woul
d influe
nt the po
se
accu
ra
cy
. In practi
cal
appli
c
ation
s
,
it is
necessa
ry to
find a m
o
re
rob
u
st
pose
optimization
algo
rithm fo
r the
solutio
n
usi
ng
optical
tracking ma
rker location informatio
n.
As for the lo
catio
n
information
with errors, the optimizatio
n
method of pri
n
cip
a
l com
p
o
nent analy
s
is
will re
d
u
ce the impact on p
o
sition a
nd orientation.
For the probl
em of pose estimation, ma
rke
r
s can be elabo
rated a
s
follows: For a given
three
-
dime
nsi
onal d
a
ta po
ints
1
i
y
and
2
i
x
, where
1
,
2
,
.....,
im
, they respectively re
p
r
esent
coo
r
din
a
te value
s
of on
e data poi
nt in two
co
o
r
dinate
syste
m
s, the rota
tion matrix and
transl
a
tion m
a
trix expre
ssi
ng tran
sform
a
tion relatio
n
from
2
i
y
to
1
i
x
, res
p
ec
tively is
1
R
and
t
and
therefo
r
e:
21
1
ii
yR
x
t
1
,
2
,
.....,
im
(1)
For this equa
tion, in previous stu
d
ie
s,
ther
e are several method
s for solving such a
s
linear lea
s
t square metho
d
, singula
r
value analysi
s
, quaternion, e
t
c.
All
these method
s sha
r
e
a
comm
on solu
tion, i.e. the
solutio
n
is di
vided into
two step
s, the first ste
p
is to
use a give
n two
pairs of p
o
int
s
seri
es to
solve rotatin
g
matr
ix, and t
hen
solve th
e tran
slatio
n
matrix. Some
of
orientatio
n estimation alg
o
rithm
s
are
used su
ch
as least-sq
uare
s
metho
d
and prin
ci
pal
comp
one
nt a
nalysi
s
[6]. Here, lea
s
t squ
a
re
s o
p
timization m
e
thod combi
n
e
d
with pri
n
ci
pal
comp
one
nt a
nalysi
s
are e
m
ployeed to
solve the
ri
gi
d-bo
dy po
se,
the wh
ole
sol
u
tion process can
be expre
s
sed
as the followi
ng equ
ation:
2
1
(,
)
K
kk
k
k
R
tw
R
t
yx
(
7)
Whe
r
e,
k
is the numbe
r of marke
r
s,
k
y
is coordi
nate value of No. k marke
r
in the global
coo
r
din
a
te sy
stem (ri
ght camera coo
r
di
nate system
),
k
x
is coordi
nat
e value of No. k marke
r
in
the local coo
r
dinate
sy
ste
m
(ri
gid-body
coo
r
din
a
te
system),
k
w
is th
e wei
ghting f
a
ctor for the
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8292 – 83
02
8298
rigid
-
bo
dy
, all the weight coef
ficient
s are
s
e
t to 1. Error matrix can
be written as
:
2
''
1
(,)
K
kk
k
R
tR
t
yx
(8)
Whe
r
e,
kk
xx
x
,
kk
yy
y
,
y
and
x
is resp
ect
i
vely center
point of all markers in the
global
coo
r
di
nate syste
m
and lo
cal coo
r
dinate
syste
m
.
Acco
rdi
ng to equatio
n (8
), erro
r mat
r
ix can be expressed a
s
:
''
'
1
(,)
(
)
(
)
K
kk
kk
k
R
t
Rt
Rt
yx
yx
And it can be
simplified a
s
:
()
(
2
)
1
K
TT
T
T
RR
kk
k
k
k
k
k
yy
x
x
y
x
(9)
Minimization of
function
()
R
is equivalent to maximizati
on minimi
zati
on of functio
n
()
f
R
which ca
n be
expresse
d a
s
:
1
()
K
TT
kk
k
f
Ry
R
x
(10)
The above e
q
uation can be
solved by sin
gul
ar valu
e d
e
com
p
o
s
ition
and be written as:
1
()
(
)
(
)
K
TT
kk
k
f
R
T
ra
ce
y
R
x
T
ra
ce
R
H
(1
1)
Whe
r
e,
1
K
T
kk
k
H
xy
, if
H
can be broke
n
into
T
A
A
, in which
()
Trac
e
R
H
is maximum
value of the function.
usin
g the si
ngula
r
value
decom
po
sition
H
can expre
s
seda
s a
s
,
T
HU
V
. where,
U
and
V
are for o
r
tho
gonal matrix.
Γ
is a non-n
e
g
a
tive diagona
l matrix
expre
s
sed a
s
:
00
1
00
2
00
3
00
0
0
11
00
0
0
22
00
00
33
T
CC
Suppo
sing
T
X
VU
, then:
TT
T
T
X
HV
U
U
V
V
C
C
V
So
A
VC
,
T
X
HA
A
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Pose Error A
nalysi
s
Model
Based on Bi
nocular Vi
sio
n
for Rigid
-
Bo
dy (Niu Fe
ngl
ian)
8299
Whe
r
e,
XH
is a positive sym
m
etric m
a
trix, when
T
X
VU
,
(,
)
R
t
is
the s
m
alles
t, so the
solutio
n
ha
s been
a
rot
a
tion mat
r
ix
R
, as
for
T
VU
the orthog
onal
matrix,
so
R
can be
expre
s
sed a
s
:
10
0
01
0
00
d
e
t
(
)
T
T
RV
U
VU
(12)
So the transl
a
tion matrix is
tR
=y
-
x
5. Pose
Error
Analy
s
is
For po
se e
s
timation, it is the key probl
em about ho
w to determi
ne the pose estimatio
n
error be
ca
use of spa
c
ial p
o
sition e
r
ror
of markers.
Here we ca
n e
x
press the o
r
i
entation erro
r as
a micro
-
chan
ge for the real positio
n and orie
ntation. Given a real rotatio
n
matrix
R
an
d
transl
a
tion m
a
trix
T
, the micro
-
vari
able
rotation
R
and
transl
a
tion m
a
trix
T
, the rel
a
tion
betwe
en re
al matrix and mi
cro
-
vari
ble m
a
trix can be d
e
scrbil
ed a
s
:
Re
r
r
R
R
Te
r
r
T
T
Bec
a
us
e
T
R
VDU
,
,
UV
is the matrix of
the SVD so
lution. In order to determine
R
and
T
, we mu
st
kno
w
m
a
rke
r
positio
n e
r
ro
r ho
w to
af
fect matrix
H
, the error
matrix
V
and
U
is
res
p
ec
tively applied to the matrix
,
UV
, the
chan
ge of matrix
,
UV
can exp
r
essed a
s
:
VV
V
W
e
defin
ed two matri
c
e
s
A
and
B
in orde
r to expre
ss thi
s
tran
sform
a
tion
:
VA
V
,
UB
U
Given that these pa
ram
e
ters
,,
VU
D
are f
r
o
m
marke
r
s
a
nd rel
a
ted in
formation to
determi
ne the
erro
r
,
and
err
R
ca
n be expre
ssed as:
()
(
)
()
(
)
err
T
RR
R
IA
R
I
B
VV
D
U
U
T
a
king into a
c
count the m
a
trix exponen
tial
can be carri
ed out u
s
ing
T
a
ylo
r
expan
sio
n
approximatio
n:
23
11
1,
26
A
eA
A
A
The first
-
orde
r app
roximati
on ca
n be wri
tten as:
()
(
)
e
e
e
e
AB
err
IA
I
B
AB
A
B
T
R
Re
R
eR
eR
I
e
R
R
R
(13)
UU
U
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8292 – 83
02
8300
Therefo
r
e, th
e rotation e
r
ror
R
can be exp
r
essed a
s
”
e
AB
T
Re
R
R
This
R
ef
fectively descri
b
le t
he imp
a
ct a
b
out
spatial
positio
n e
rro
r of ma
rkers to
orientaito
n errors for rigi
d-body
.
6.
Simulation analy
s
is
This re
se
arch
makes u
s
e of matlab software to simu
late position error of markers h
o
w
to af
fect orie
n
t
ation of rigid
-
body
, an
d em
ployee
s the l
east-sq
ua
re
s
method
to fit the entire dat
a
.
Gau
ssi
an noi
se i
s
add
ed t
o
the marke
r
point in
the lo
cal
coo
r
din
a
te system, a
n
d
the varia
n
ce
is used to de
termine the
e
rro
r characte
ri
stics of markers.
Th
e ra
n
g
e of varian
ce
is 0-3mm
and the mea
n
is zero.
Assumptio
n
err
R
,
err
T
is the expected
error matrix
of local coordinate
s
according to the above stat
ement, avera
ge error matri
x
E
in t
h
e t
r
acki
ng coo
r
din
a
t
e
sy
st
em can
be expre
s
sed
as:
1
1
K
k
e
rr
k
e
rr
k
E
yR
x
T
K
Figure 6(a
)
. Position e
r
ror o
f
single ma
rker
in X direction
Figure 6(b
)
. Position e
r
ror o
f
single ma
rker in Y
dire
ction
Figure 6(c): P
o
sition e
r
ror o
f
single ma
rker in Z dire
cti
on
Whe
n
the ran
ge of the variance is 0-3m
m,
we simul
a
te 1000 time
s and u
s
e the a
v
erag
e
error to analyze and predi
ct the relationship bet
wee
n
pose error an
d gaussian n
o
ise. Simulat
e
d
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Pose Error A
nalysi
s
Model
Based on Bi
nocular
Vi
sio
n
for Rigid
-
Bo
dy (Niu Fe
ngl
ian)
8301
prob
e ha
s three marke
r
po
ints, the cu
rve relation
shi
p
betwee
n
po
sition erro
r of single ma
rker
and varible
is shown in Figure 2.
As
expected, wh
en the varian
ce
increa
se
s, position erro
r
is also on the
increa
se. Error di
stributio
n
of the ma
rke
r
is dif
f
ere
n
t for a given p
o
i
nt in dif
f
eren
t
dire
ction
s
wh
en error vari
ance
is chan
g
i
ng whi
c
h re
spectively is 0~1.8mm in the x axis, 0 ~
0.61mm in the y axis and 0.27 ~ 0mm in the z axis.
In order to describ
e better
R
chan
ging,
quaternion m
e
thod is use
d
to repre
s
e
n
t
pose error as in the a
b
ove studie
s
er
ror
RR
R
,
therefo
r
e
-1
e
rror
RR
R
, so the correspondi
ng quat
ernio
n
can
be expre
s
se
d as follows,
-1
error
qq
q
, which is a
n
g
le error a
s
well as in
clu
d
i
ng rotation a
n
gle erro
r of the rotation a
x
is.
Figure 7 sho
w
s the relati
onship betwe
en angle e
rror of rigid-b
o
d
y and the detection po
sit
i
o
n
errors of three marke
r
points, the fluctuation
rang
e of the erro
r is also g
r
o
w
ing, wh
en the
locatio
n
error of the point
mark sat
s
fied
with Gaussi
an noise vari
ance are cha
nged from 0
~
3
mm, its pose
cha
nge
s is in
the
angle ran
ge -0.59
- 0.4
1
degree.
Figure 7. Rel
a
tionship bet
wee
n
po
sition
variance and
pose
cha
nge
s
7. Conclu
sion
In this pap
e
r
, the po
sition
and p
o
se
rel
a
tion abo
u
t two
came
ra
s
are d
e
rived
b
y
came
ra
calib
ration ex
perim
ents u
s
i
ng the intrin
si
c and extr
in
si
c paramete
r
s of the two cameras. Due
to
the dete
c
ting
error of im
ag
e co
rno
r
, the
orientatio
n rel
a
tionship o
b
tained
by ea
ch image
grou
p is
not con
s
i
s
ten
t, the optimized ori
entatio
n param
ete
r
s are a
c
qui
ed
by optimizin
g the obtain
ed
orientatio
n matrix of multi group imag
es. and us
e
the same co
ordin
a
te syst
em to solve
the
probl
em of sp
acial p
o
ints.
The lea
s
t sq
u
r
es
met
hod
combine
d
with
prin
cipal
co
mpone
nt anal
ysis
method a
r
e
e
m
ployeed to
analyze wo
rk-pie
ce o
r
ient
a
t
ion and
set u
p
the mathe
m
atical m
o
de
l of
orientatio
n error p
r
op
agati
on. Simulation analysi
s
establi
s
he
d the relation ab
out the position
error of markers and po
se erro
r of rigid-bo
dy and ve
rified that it is feas
ible and ef
fective, it
provide
s
a th
eoreti
c
al a
nal
ysis a
nd exp
e
rime
ntal
ba
sis ab
out the
pose dete
ctio
n of rigid
bo
dy
su
ch worke pi
ece a
nd medi
cal in
strum
e
n
t
.
Ackn
o
w
l
e
dg
ements
This p
r
oje
c
t is su
ppo
rted b
y
Natural Sci
ence Foun
dat
ion of Ning
bo
City (2013A6
1
004
8
)
Referen
ces
[1]
Agustin N
a
var
r
o, Edgar Vill
arrag
a
, Joan
Arand
a.
Relati
ve Pose Esti
mation of Sur
g
ical T
ools i
n
Assisted Minim
a
lly
Invas
i
ve S
u
rger
y
.
Pattern
Recog
n
itio
n a
nd Image A
nal
ysis
. 2007; 4
4
7
8
: 428-4
35.
[2] Vinas
FC
，
Z
a
moran
o
L, e
l
at. Applic
atio
n
accurac
y
st
u
d
y
of a s
e
mi
p
e
rman
ent fid
u
c
ial s
y
stem f
o
r
frameless ster
eota
x
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mp
uter Aide
d Sur
gery
.
1997
;
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: 257-26
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[3]
F
abrice C, Lav
alle
e S.
Experi
m
e
n
tal Protoc
o
l
for Accuracy Evalu
a
tion of 6
-
d Local
i
z
e
r
s for Co
mput
er-
Integrated S
u
r
gery:
Appl
icati
on to F
our O
p
ti
cal Loc
ali
z
e
r
s.
Lecture N
o
tes
In Comput
er Scienc
e. 199
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[4]
Micha H
e
rsch,
Aude Bi
llar
d
, Sven Bergm
a
nn. Itera
tive E
s
timation of R
i
gid-B
o
d
y
T
r
ansformations.
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