Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l. 10
, No
. 1, Febru
a
r
y
2
020, pp
. 117
~128
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v10
i1
.p
p11
7-1
28
1
17
Jo
urn
a
l
h
o
me
pa
ge
: http://ijece.iaesc
o
re
.c
om/index
.
php/IJ
E
CE
Three-di
mension
a
l st
ruct
ure from
motion recovery of a moving
object with noisy measurement
Z
o
ubaida Mejri
1
, Lilia
Sidho
m
2
, Afef Abdelkrim
3
1,2,3
Research
L
a
b
o
rator
y
L.A.R
.
A
in Autom
a
t
i
c
co
ntrol,
Nation
a
l Engineer
ing Scho
ol of
Tunis (
E
NIT),
University
of
Tu
nis El Manar
,
Tu
nisia
1,3
N
a
tio
n
a
l Engin
eer
ing
School
o
f
Carth
a
ge (ENICarthage)
,
Un
iv
ersity
of C
a
rth
a
ge, Tun
i
sia
1,2
The Nation
a
l
Engineering Sch
ool of B
i
zerte
(ENIB), University of Car
t
hag
e
, Tu
nisia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mei 3, 2019
Rev
i
sed
Au
g 8, 201
9
Accepted Aug 29, 2019
In this paper
,
a Nonlinear Unknown Input Observer (NLUIO) based
approach
is pro
posed for three-dimensional (3
-D) structure fr
om motion
identif
ication
.
Unlike th
e previo
us st
udies that r
e
quire pr
ior knowledge o
f
eith
er the motion parameters
or s
cene geometr
y
, th
e proposed approach
as
s
u
m
e
s
that the object m
o
tion i
s
im
perfectl
y
kn
own and cons
idered as
an
unknown input
to the perspective d
y
nami
cal s
y
stem. The recon
s
truction o
f
the 3-D structur
e of the moving objects can b
e
achieved usin
g just two-
dimensional (2-
D
) images of a m
onocular v
i
sion s
y
stem. The proposed
schem
e
is illu
strated with a
num
erical ex
am
ple in the
presence of
m
eas
urem
ent no
is
e for bo
th s
t
at
ic
and d
y
n
a
m
i
c
s
cenes
.
Thos
e
res
u
lts
ar
e
used
to clearly
d
e
monstrate the
advantag
es
of the proposed
NLUIO.
K
eyw
ords
:
Measurem
ent noise
Monoc
u
lar vision
system
s
N
o
n
lin
ear
u
nkn
own
inpu
t
obs
er
ver
St
ruct
ure fr
om
m
o
ti
on
Copyright ©
202
0 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Zo
ubai
d
a M
e
jr
i
,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
Nat
i
onal
En
gi
n
eeri
n
g
Sc
h
ool
of
C
a
rt
ha
ge (E
NIC
a
rt
ha
ge),
Uni
v
ersi
t
y
o
f
C
a
rt
hage
,
45
R
u
e
des E
n
t
r
e
p
r
e
neu
r
s,
C
h
a
r
g
u
i
a II,
2
0
3
5
.
T
u
ni
s, T
u
ni
si
a.
Em
ail: zoubaida.m
e
jri@enit.utm
.
tn
1.
INTRODUCTION
The p
r
o
g
r
ess i
n
st
ruct
ure a
n
d m
o
t
i
on est
i
m
a
t
i
on (a.k
.a. structu
r
e-
fr
om
-m
oti
on) re
search has been
h
ectic, sti
m
u
l
ated
b
y
r
ecen
t
b
r
eak
t
hr
oug
hs in
co
m
p
u
t
er v
i
sio
n
, th
e ad
v
e
n
t
of
d
i
g
ital p
h
o
t
og
r
a
phy an
d
th
e aug
m
en
ted
reality [1
-6
]. Th
is pro
g
ress
h
a
s th
e
po
ten
tial to
su
bstan
tially in
crease th
e u
s
e of th
e st
ru
cture
fr
om
m
o
t
i
on t
echni
que
f
o
r
a
v
a
ri
et
y
of a
p
pl
i
cat
i
ons,
f
o
r
exa
m
pl
e t
h
e gr
owi
n
g
ap
pl
i
cat
i
on
of
u
n
m
a
nned
a
e
ri
al
vehi
cl
es f
o
r re
m
o
t
e
survey
i
n
g f
o
r a n
u
m
e
rous
of ec
ol
o
g
i
cal
dom
ai
n [7]
.
W
i
de
-reac
hi
n
g
m
a
ri
ne assessm
ent
s
usi
n
g t
h
i
s
t
e
c
h
ni
q
u
e
have
rec
e
nt
l
y
becom
e
pos
si
bl
e i
n
so
m
e
cases l
i
k
e i
n
[
8
,
9]
wi
t
h
dr
one
-
b
ased
ap
pl
i
cat
i
on.
The st
ruct
ure
fr
om
m
o
t
i
on t
echni
que ca
n be use
d
fo
r t
o
po
g
r
ap
hi
c dat
a
col
l
ect
i
on i
n
fi
el
d and l
a
b
o
r
at
ory
studies [10] and as a m
eans of dig
i
t
a
l
p
r
eser
vat
i
on a
n
d f
o
r
doc
um
ent
i
ng a
r
chae
ological excavations
, cul
t
ural
material and architecture [11]. On the
ot
her s
i
de, st
ruct
ure f
r
om
m
o
t
i
on can be a g
o
o
d
l
o
w-c
o
st
al
t
e
rnat
i
v
e t
o
gene
rat
e
hi
g
h
resol
u
t
i
o
n
t
o
po
gra
p
hy
[
12]
,
w
h
ere
l
i
ght
det
e
ct
i
on a
n
d
ra
n
g
i
ng
dat
a
i
s
u
n
a
f
f
o
r
d
a
b
l
e
o
r
sc
arce.
Recently in the area of a
g
ric
u
lture
[13], the use of unm
a
nne
d aerial syst
e
m
s (UAS)
base
d on the s
t
ructure
fr
om
m
o
t
i
on t
echni
que as r
e
m
o
t
e
-sensi
ng
pl
at
fo
rm
s have
m
a
ssi
ve pot
ent
i
a
l
for o
b
t
a
i
n
i
n
g det
a
i
l
e
d
of cr
o
p
feat
ure
s
. The
st
ruct
u
r
e
a
n
d m
o
ti
on fi
el
d of
resea
r
c
h
i
s
w
o
r
r
i
e
d
wi
t
h
t
h
e reco
ve
ry
o
f
3-
D ge
om
et
ry
o
f
the dynam
i
c scene (t
he structure
)
whe
n
obs
e
rve
d
thr
ough
a
m
oving camera (the m
o
tio
n
)
. Basically, stru
cture
fro
m
m
o
tio
n
in
vo
lv
es th
ree
main
step
s: Fi
rst ex
tractin
g feature
s
in images and m
a
tc
hing these fea
t
ures
betwee
n im
ages, then m
odeling the cam
era-object relativ
e m
o
ti
on an
d fi
n
a
l
l
y
recovery
of t
h
e
3-
D st
ru
ct
u
r
e
using t
h
e estimated m
o
tion a
n
d
feature
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
n
t
J
Elec
&
C
o
m
p
Eng
,
Vo
l. 1
0
, N
o
. 1
,
Febru
a
r
y
20
20
:
117
-
1
28
11
8
Keep
i
n
g
i
n
v
i
ew th
e abov
e
literatu
re, sev
e
ral work
s h
a
ve ad
dressed
stru
ct
u
r
e estim
at
io
n
o
b
s
erv
e
r
base
d ap
pr
oac
h
w
h
e
r
e f
u
l
l
v
e
l
o
ci
t
y
pa
ram
e
t
e
rs fee
dbac
k
of t
h
e cal
i
b
rat
e
d cam
era was pr
o
v
i
d
e
d
. S
u
ch a
s
i
n
[
1
4]
whe
r
e
aut
h
ors
desi
g
n
e
d a
no
nl
i
n
ea
r
o
b
se
rve
r
t
o
e
s
t
i
m
a
t
e
an u
n
m
easurabl
e
st
a
t
e cal
l
e
d
dept
h
wi
t
h
kn
o
w
n
dy
n
a
m
i
cs. T
h
at
l
a
st
o
n
e
has
been e
x
peri
m
e
nt
ed
on
a m
o
b
ile ro
bo
t
with
an
on-board cam
era. Authors
i
n
[
15]
have i
n
t
r
o
duce
d
a
n
o
n
l
i
n
ear o
b
se
rve
r
fo
r a
part
i
c
ul
a
r
case
of
feat
u
r
e poi
nt
s o
n
t
h
e
ob
ject
m
ovi
n
g
wi
t
h
co
nstan
t
v
e
l
o
cities an
d
h
a
v
e
ap
pro
v
e
d
in
m
a
n
y
p
r
actical
scen
ari
o
s.
Alth
ou
gh
, i
n
[1
6
]
a
n
o
n
lin
ear ob
serv
er is
defi
ned t
o
rec
ove
r st
r
u
ct
u
r
e
an
d m
o
t
i
on
wi
t
h
l
e
ss
rest
r
i
ct
i
v
e assum
p
t
i
ons
o
n
t
h
e
m
ovi
ng
o
b
j
ect
m
o
ti
on.
A
r
e
du
ced
-
o
r
der
non
lin
ear ob
serv
er
is
p
r
esen
ted
i
n
[
1
7]
to estim
a
t
e the ra
nge
from
a m
oving cam
era to
a feat
ure
poi
nt
on a st
at
i
c
scene. F
u
rt
herm
ore
,
a desi
g
n
of c
o
m
p
l
e
t
e
o
r
de
r o
b
ser
v
e
r
s
based
on
no
n
l
i
n
ear
cont
ract
i
on t
h
eory
a
nd sy
nc
hr
o
n
i
zat
i
on i
s
gi
ve
n i
n
[
1
8]
whe
r
e a
n
g
u
l
a
r
and l
i
n
ea
r v
e
l
o
ci
t
y
m
easurem
ent
s
are also noisy.
The inform
ation of the
camera m
o
tion pa
ram
e
ters
has
been
u
n
a
voi
da
bl
e i
n
t
h
e
pre
cedi
n
g ci
t
e
d
r
e
f
e
r
e
n
ces. V
a
r
i
ou
s stud
ies on
stru
ctur
e fr
om
m
o
tio
n
estimation are also availa
ble
where the cam
era
m
o
tion
i
s
not
k
n
o
w
n.
St
art
i
ng
wi
t
h
[
19]
, sl
i
d
i
ng m
ode
ob
ser
v
ers
were
prese
n
t
e
d
t
o
est
i
m
a
t
e
t
h
e
m
o
t
i
on pa
ra
m
e
t
e
rs
and t
h
e structure of a m
oving object
with the aid of
a ch
an
ge-c
o
upl
e
d
de
vi
ce (
CCD) ca
mera. The a
dvantage
prese
n
t
e
d
by
t
h
e p
r
o
p
o
se
d o
b
ser
v
e
r
s i
s
t
h
at
bot
h
ri
gi
d
and a
ffi
ne m
o
t
i
on pa
ram
e
t
e
r
s
, co
nst
a
nt
o
r
t
i
m
e
-
vary
i
n
g, ca
n b
e
est
i
m
a
t
e
d correct
l
y
. In t
h
e
u
n
i
q
uene
ss co
nt
ext
,
[
20]
i
n
t
r
o
d
u
ced a
de
vel
o
p
e
d n
o
n
l
i
n
ear
re
duce
d
o
r
d
e
r
ob
serv
er wh
ich
on
ly requ
ires
on
e ca
m
e
ra lin
ear
v
e
l
o
city to
esti
m
a
te a sta
tio
n
a
ry
o
b
j
ect
seen
by
a cal
i
b
rat
e
d
cam
e
ra. The m
e
t
hods d
e
scr
i
bed i
n
[
2
1
,
2
2
]
prese
n
t
no
n
l
i
n
ear o
b
ser
v
er
s based o
n
a R
o
b
u
st
Int
e
gral
Si
g
n
e
d
Er
ro
r m
e
t
hod (R
I
S
E) t
o
es
t
i
m
a
t
e
t
h
e un
k
n
o
w
n di
st
ance
bet
w
ee
n t
h
e c
a
m
e
ra and t
h
e
ob
ject
an
d th
e m
o
v
i
ng
cam
era v
e
l
o
cities. Th
is
p
r
o
b
l
em
was al
so
i
n
v
e
stig
ated
in
[23
]
wh
ere
a non
lin
ear red
u
c
ed-
or
der
o
b
ser
v
er
i
s
pro
p
o
se
d t
o
reco
ver t
h
e fe
at
ure
poi
nt de
pth and cam
era
linear ve
l
o
city. Only the camera’s
an
gu
lar v
e
l
o
city is assu
m
e
d
to
b
e
k
nown.
Au
t
h
ors
desc
ri
bed
i
n
[
24]
a
new
ap
p
r
oac
h
base
d
o
n
E
x
t
e
nde
d
Kalm
an Filter to sim
u
ltaneously rec
ove
r c
a
m
e
ra pose
and the
struct
ure
of non-rig
id extensi
b
le
surfaces.
In
o
r
de
r t
o
e
x
t
e
nd t
h
e
pr
o
b
l
e
m
t
o
a
def
o
r
m
abl
e
dom
ai
n,
aut
h
o
r
s
defi
n
e
d t
h
e
o
b
j
ect
’
s
su
rface
m
e
chani
c
s
by
m
eans o
f
Navi
e
r’s
eq
uat
i
ons
.
A rec
e
nt
pa
per
[
2
5]
add
r
esses
t
h
e
c
a
se w
h
ere
a
n
ovel
c
o
m
p
l
e
t
e
-o
rde
r
o
b
s
erv
e
r
is d
e
sig
n
e
d
to
estimate
th
e u
nkno
wn
m
o
tio
n
par
a
m
e
ter
s
an
d
f
eatu
r
e d
e
p
t
h in
th
e p
r
esence o
f
measurem
ent noise. T
h
e
observer is de
rive
d
from
a di
f
f
ere
n
t
i
at
or
based
o
n
t
h
e sl
i
d
i
n
g
-
m
ode t
echni
qu
e.
Th
is
p
a
p
e
r, tack
les th
e prob
lem
o
f
m
o
tio
n
an
d stru
ct
ure re
cove
ry for a cl
ass of system
consisted
on
a
m
ovi
ng ca
m
e
ra
m
ovi
ng
ob
ject
. Nat
u
r
a
l
l
y
,
m
o
ti
ons
are co
nst
r
uct
e
d i
n
co
nt
i
n
u
o
u
s t
i
m
e
sett
i
ngs an
d
t
h
e m
o
t
i
on par
a
m
e
t
e
rs are assum
e
d t
o
be al
l
t
i
m
e
vary
i
n
g
.
The 3
D
posi
t
i
on i
s
est
i
m
a
t
e
d by
usi
ng a s
e
t
of
i
m
ag
e d
a
ta ob
serv
ed
thro
ugh
a d
y
n
a
m
i
c ca
mera with v
a
rying
fo
cal leng
th.
Th
e co
n
t
ribu
tio
n
s
of th
is
p
a
per are
first th
e an
alysis o
f
th
e ex
t
e
n
t
to
wh
ich
a sch
e
m
e
can be de
velope
d that is guaranteed to c
onverge
by obse
rvi
ng a
single
point a
n
d ha
ving
a
n
unknown object
m
o
ti
on. In addition, for a m
o
re accurate t
r
eatment,
th
is p
a
p
e
r ex
t
e
n
s
iv
ely
v
a
lidates th
is appro
ach fo
r
bo
th static an
d dyn
amic o
b
ject
in the
prese
n
ce
of
measurem
ent noise.
The rem
a
i
nder
of t
h
i
s
pa
per
i
s
orga
ni
zed a
s
fol
l
o
ws:
Nec
e
ssary
prel
i
m
inari
e
s an
d st
at
e dy
nam
i
cs
fo
rm
ul
at
i
on ar
e sou
g
h
t
i
n
Se
ct
i
on 2
.
Sect
i
o
n 3
pre
s
ent
s
the d
e
sign
of
th
e N
o
n
lin
ear
Unk
now
n
I
npu
t Ob
serv
er
NLU
I
O t
o
est
i
m
a
t
e
st
ruct
ur
e of a feat
u
r
e
poi
nt
w
h
e
r
e
LM
I-bas
ed f
o
rm
ul
at
i
on i
s
devel
o
pe
d t
o
pro
v
e
asym
pt
ot
i
c
conve
r
g
ence
. I
n
Sect
i
on
4 t
h
e si
m
u
l
a
t
i
on resul
t
s
are
dem
onst
r
at
i
n
g
t
h
e ro
b
u
st
ne
ss o
f
the approac
h
i
n
the
prese
n
ce
of m
easurem
en
t noise
. Fi
n
a
lly,
con
c
lud
i
ng
remark
s
are d
r
awn
in
Sectio
n 5
.
2.
STA
TE DYNA
M
I
C
FOR
M
U
L
A
T
ION
In t
h
i
s
sect
i
on
an o
v
er
vi
ew o
f
t
h
e pers
pect
i
v
e rel
a
t
i
ons
hi
p
s
and ba
si
c ki
nem
a
ti
c i
s
gi
ven m
odel
i
n
g
a cam
era whic
h m
oves
and
obs
erves a
m
oving
object.
Mo
st of
t
h
e co
n
c
ep
ts can be fo
und
,
fo
r ex
am
p
l
e,
i
n
[2
1]
an
d [
2
6]
. C
o
nsi
d
e
r
a
scenari
o
i
n
F
i
gu
re 1
whe
r
e
t
h
e m
o
t
i
on of
a si
ngl
e m
ovi
ng
o
b
ject
i
s
vi
ewe
d
by
a
m
ovi
n
g
c
a
m
e
ra un
der
g
o
i
ng r
o
t
a
t
i
o
n an
d t
r
an
sl
at
i
on.
The equation of a feature
poi
n
t in the object can be
prese
n
ted
in t
h
e refe
re
nce
fra
m
e
as
.
x
xv
(1
)
Whe
r
e, the stat
e vector
3
12
3
()
()
()
()
T
xt
x
t
x
t
x
t
i
s
de
f
i
ned
rat
h
e
r
as
/,
/,
1
/
T
x
XZ
Y
Z
Z
,
W
i
th
:
,
XY
and
Z
ar
e th
e
un
kn
own
Eu
clid
ean
co
ord
i
n
a
tes
of
f
eat
u
r
e po
in
t
in
th
e cam
era’s
inertial fram
e.
3
()
x
t
being
perpe
ndi
cular t
o
the
ca
mera’s im
age plane is
the i
n
verse
of an unmeasura
b
le
focal
distance
.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
E
()
t
ca
m
e
r
b
oun
d
ob
jec
t
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a
of t
h
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n
E
lec & C
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m
p
3-
D st
r
33
re
pres
e
12
(t
)
(t
)
3
0
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vt
Is a
r
r
a v
e
lo
city a
n
d
ed
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n
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i
n
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xt
d
t
w
ith
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ect
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n
d
()
vt
in
to
(
11
2
2
2
33
3
12
c
x
xx
xx
x
vx
y
xx
The e
quat
i
e
m
ovi
n
g
sc
e
n
k
now
n m
o
t
i
o
1
2
(,
)
(,
)
Gu
y
Gu
y
1
2
3
(,
)
(,
)
(,
)
F
xu
F
xu
F
xu
Eng
r
u
c
tu
re fro
m
m
e
nt a
s
k
ew
s
3
)
()
T
t
and
g
32
3
1
21
0
0
r
elative ca
m
e
nd
1
pp
vv
n
uo
usly d
i
ffe
r
d
evel
opm
ent
t
o
a m
ovi
ng
(
1
)
as
fo
llow
s
2
21
2
2
2
21
2
2
1
21
1
2
3
()
T
xx
x
xx
x
xx
x
i
on
s ab
ov
e
ar
e
e
ne.
T
o
re
cu
p
o
n param
e
t
e
rs
12
2
2
2
xx
x
11
3
3
22
3
2
33
2
()
()
(
cc
cc
c
vx
v
x
vx
v
x
vx
x
I
S
m
otion re
cove
r
s
ymmetr
i
c
m
a
g
iv
e
n
b
y
e
ra lin
ear v
e
l
o
23
T
pp
vv
is
r
en
tiab
l
e.
Fi
gu
re 1.
C
ai
m
e
d t
o
des
i
cam
era. The
r
2
31
1
3
32
2
3
2
33
()
(
)
cc
cc
p
vx
v
x
vx
v
vx
e
c
o
m
pos
ed
o
p
e
r
ate the
3
D
of bo
th
came
r
2
12
2
3
12
2
1
3
xx
xx
x
3
11
2
3
)
xx
SSN
:
208
8-8
7
r
y of
a mo
vi
n
g
a
trix
m
a
d
e
f
r
o
city, su
ch
t
h
the feature
p
C
amera objec
t
i
gn
o
f
NLU
I
O
r
elative m
o
ti
o
31
3
1
3
32
3
2
3
)
pp
pp
x
vx
x
v
x
xv
x
x
v
o
f un
m
e
asu
r
a
b
D
structure, t
h
e
ra an
d
ob
jec
t
3
7
08
g
ob
ject
with
n
r
om
the ang
u
h
a
t
cp
vv
v
p
oi
nt
ve
l
o
ci
t
y
t
mo
t
i
o
n
mo
d
e
O
observe
r
s
i
s
o
n
of t
h
e feat
u
3
3
3
x
x
b
l
e
c
o
o
r
di
nat
e
h
e state vect
o
are se
parate
d
n
oi
sy
me
as
ur
e
u
lar m
o
tio
n
o
,
wh
ere
c
c
vv
y
. N
o
te th
at
e
l
s
to
es
t
i
ma
t
e
t
u
re po
in
t can
b
3
x
an
d un
k
n
o
o
r s
h
ould
be
in
th
e
fo
llo
w
i
e
ment (
Z
ou
ba
o
f
t
h
e m
ovi
n
(
12
3
T
c
cc
vv
is
v
e
lo
cities
c
v
a
t
he st
ruct
ure
o
b
e
gi
ve
n by
s
u
(
o
wn m
o
t
i
on i
n
esti
m
a
ted
.
T
h
i
ng
. W
e
d
e
fi
n
(
(
i
d
a Mejri)
11
9
n
g cam
era
(
2)
the linea
r
a
n
d
p
v
are
o
f m
ovi
n
g
u
b
s
t
itu
tin
g
(
3)
n
fo
rm
ation
h
at’s why
n
e
(
3.1)
(
3.2)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
E
12
0
a m
e
a
in
(3
)
Whe
r
cont
a
i
funct
i
angul
ca
m
e
r
1
c
v
v
Whe
r
vecto
r
th
e n
o
suc
h
t
Gi
ve
n
and t
h
form
u
E
lec & C
o
m
p
11
2
2
33
3
p
p
p
Dv
x
Dv
Dv
x
Wi
t
h
2
(
G
a
surable inpu
t
can be
re
wri
t
(,
x
fx
yC
x
e
()
n
xt
i
s
i
n
s
t
h
e poi
n
t
i
on
s.
nq
H
Re
m
a
rk 1
ar v
e
lo
cities
o
r
a and the
11
3
(
pc
v
xv
v
In our cas
e
x
Ax
yC
x
e
(,
)
(
f
xu
f
x
Not
e
t
h
at
r
s
u
ch tha
t
fo
r
(,
)
f
xu
Whe
n
t
h
e
o
nl
i
n
ear
f
unc
t
t
hat
(,
)
f
xu
n
an obse
rvab
l
h
e
ang
u
l
ar
c
a
m
The over
a
u
latio
n
,
the N
L
En
g, V
o
l
.
10
,
31
3
3
2
32
3
3
2
3
p
p
x
xv
x
xx
v
x
(
,)
uy
,
1
(,
F
x
u
t
con
s
titu
ted
b
t
t
e
n i
n
no
nl
i
n
e
)(
,
)
ug
y
u
s
the state of
t
t
v
e
lo
cities,
is the un
kno
w
(
O
b
s
erva
bi
l
i
o
f the camer
a
feature
p
o
32
2
)
pc
p
v
vv
e
,
t
h
e
dy
nam
i
c
(,
)
(
fx
u
g
y
,)
x
uA
x
and
A
t
he no
nl
i
n
e
a
r
th
e sam
e
p
o
s
ˆ
(,
)
fx
u
x
camera is
m
t
ion
(,
)
f
xu
,
i
ˆ
(,
)
(
fx
u
A
x
l
e pe
rs
pectiv
e
m
era v
e
lo
citie
a
ll struct
ure
L
UI
O fo
r
t
h
e
Figu
r
,
No. 1, Febr
u
)
u
,
2
(,
)
F
xu
b
y the angul
a
r
e
ar system
for
()
Hd
t
t
h
e
norm
a
l
i
ze
()
p
yt
t
h
w
n i
n
p
u
t
m
a
t
r
t
y) [27
]
: Th
e
a
are null. T
h
o
in
t of th
e
2
23
3
(
cp
xv
v
c
syste
m
in (
4
,)
y
uH
d
33
A
.
a
r fun
c
tio
n
(
fx
s
itiv
e con
s
tan
t
ˆ
x
x
ovi
ng wi
t
h
f
a
i
t is al
so
a Li
p
ˆ
)(
x
e
syste
m
, the
d
s.
o
f
t
h
e
pr
op
o
s
tru
c
ture est
i
m
r
e 2.
St
r
u
c
t
ur
e
u
ary 20
20
:
1
1
,
3
(,
)
Fx
u
,
d
r
and
th
e lin
e
r
m
as fo
llow
s
d Euclidean
c
h
e out
p
u
t
,
g
r
ix
an
d
p
C
no
n
linear sy
h
at m
eans
(
)
c
vt
m
ovi
ng o
b
3
)0
4
) ca
n
be
re
w
r
,)
x
u
is a Lip
s
t
i
nde
pe
nd
e
a
ster v
e
lo
citi
e
p
sch
itz fun
c
t
i
ˆ
)
xx
d
esi
g
n p
u
r
p
os
e
o
se
d m
e
thod
m
at
io
n
w
ill
b
e
e
o
f
th
e prop
o
1
7 - 12
8
1
()
d
x
,
2
()
dx
e
ar cam
era ve
l
c
oordinat
es,
d
3
(,
)
g
yu
a
p
n
is th
e
ou
tp
stem
in (4) i
s
)
0
and
(
)
t
b
ject follow
r
itten
as follo
w
s
ch
itz fun
c
ti
o
e
nt
of
x
e
s the Lipsch
i
i
on
so
t
h
ere e
x
e
is to
esti
ma
t
is sh
own in
e
achi
e
ve
d i
n
t
se
d s
t
ru
c
t
u
r
e
e
and
3
()
dx
l
o
c
ities. Co
n
s
()
q
d
t
is a
n
a
nd
(,
)
:
f
xu
u
t m
a
trix
.
s
not
o
b
se
rva
b
. In addi
t
th
e sa
m
e
w
s
n [
28]
a
n
d
ˆ
x
t
z consta
n
t
x
ists a
po
si
ti
v
t
e
the coordin
a
Fi
gu
re 2.
A
t
he fo
llo
w
i
n
g
e
st
im
at
i
on m
e
ISS
N
:
2
(
and
u
s
equ
e
n
tly, th
e
(
n
un
know
n
i
n
3
are
b
le if all the
t
io
n, w
h
ere t
h
ray, it
m
e
(
is th
e esti
m
(
is larg
e. A
s
v
e Lip
s
ch
itz
c
(
n
ates
(
)
xt
fr
o
m
A
fter the sta
t
e
sectio
n
e
th
o
d
2
088
-87
08
(
3.3)
T
c
v
is
d
yna
m
i
cs
(
4)
n
pu
t w
h
ich
no
nl
i
n
ea
r
lin
ear and
h
e m
ovi
n
g
e
an
s that
(
5)
m
ated state
(
6)
re
gar
d
s t
o
on
st
an
t
(
7)
m
th
e
lin
ear
e
dy
nam
i
c
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
El
ec &
C
o
m
p
En
g
ISS
N
:
2
0
8
8
-
87
08
3-
D st
r
u
ct
ure
f
r
om
m
o
t
i
o
n re
covery
of
a
mo
vi
ng
o
b
j
ect
w
i
t
h
noi
sy
me
as
ur
ement
(
Z
o
u
b
a
i
d
a
Mej
ri
)
12
1
3.
OBSERV
ER
FORMU
L
A
T
ION
In t
h
i
s
sect
i
o
n
,
an a
s
y
m
pt
ot
ical
l
y
conve
r
g
i
ng
NL
UI
O i
s
con
s
t
r
uct
e
d, t
h
e st
at
e of
w
h
i
c
h f
o
l
l
o
ws
the state
of t
h
e dynam
i
cs syste
m
given in
(5) as
clos
ely
as possi
ble further i
n
t
h
e pre
s
ence of
a
n
unknown
in
pu
t. For th
e rest o
f
th
e
stud
y it is go
ing
to be assu
m
e
d
th
at
th
e fo
llowing
co
nd
itio
ns [29
]
are satisfied
:
H
is assu
m
e
d
to
b
e
co
lu
m
n
rank
m
a
trix
()
(
)
.
ra
nk
CH
ran
k
H
q
Whe
r
e
q
t
h
e
nu
m
b
er of t
h
e
un
kn
o
w
n
i
n
put
.
W
i
t
h
ab
ov
e
con
d
ition
s
, th
e
NLUIO
for system
rep
r
esen
ted
b
y
(5
) can b
e
sh
own
as
fo
llows
ˆ
(,
)
(
,
)
ˆ
zN
z
L
y
M
f
x
u
M
g
y
u
xz
E
y
(8
)
Whe
r
e
ˆ
()
n
xt
is an
esti
m
a
te
o
f
()
x
t
and
()
n
zt
is the state obse
r
ve
r. M
a
trices
nn
N
,
np
L
,
np
E
and
nn
M
are determined to design
the observe
r
such that
()
x
t
ev
entu
ally ten
d
s
to
ˆ
()
x
t
in
th
e
face of unknown
input.
Obse
rver
gain m
a
trices equations can
be expressed in alternate
form
0
()
N
MI
E
C
NM
A
K
C
LK
I
C
E
M
A
E
(9
)
Whe
r
e
K
and
E
are
gai
n
m
a
t
r
i
ces o
f
s
u
i
t
a
bl
e di
m
e
nsi
o
ns s
u
bse
q
u
e
nt
l
y
desi
g
n
e
d
.
The e
r
r
o
r
eq
ua
t
i
on
fo
r sy
st
em
(
5
) a
n
d
NL
UI
O (
8
) i
s
defi
ne
d as
f
o
l
l
o
w
s
ˆ
()
()
(
)
e
t
xt
xt
z
E
y
x
z
M
x
(1
0)
By
su
b
s
titu
t
i
n
g
th
e
syste
m
o
u
t
p
u
t
presen
ted
i
n
(5
) in
to th
e
erro
r equ
a
tio
n
in
(10
)
, t
h
e d
y
n
a
mic erro
r
et
will h
a
v
e
th
e fo
llo
win
g
fo
rm
()
(
)
et
z
I
E
C
x
(1
1)
Th
en
su
b
s
titu
ti
n
g
(5
) an
d (8) in
to
(1
1
)
, th
e
d
y
n
a
mic erro
r can
b
e
ex
p
r
essed
as fo
llo
ws
ˆ
()
(
(
,
)
(
,
)
)
eN
e
N
I
E
C
x
L
C
x
M
f
x
u
f
x
u
M
A
x
M
H
d
ˆ
()
(
(
,
)
(
,
)
)
eN
e
N
M
L
C
M
A
x
M
f
x
u
f
x
u
M
H
d
(12
)
To
ob
tain
matrices, th
e fo
llo
wi
n
g
steps shou
l
d
be
fol
l
o
we
d:
Fi
rs
t
usi
n
g
(9
) t
h
e e
quat
i
o
n
0
NM
LC
M
A
is satisfied
, and
if th
e m
a
trix
E
satisfies (13)
()
0
MD
I
E
C
D
(1
3)
The
n
t
h
e
eq
uat
i
on
o
f
t
h
e
er
ro
r
dy
nam
i
cs i
n
(
1
2
)
y
i
el
ds t
o
ˆ
((
,
)
(
,
)
)
eN
e
M
f
x
u
f
x
u
(1
4)
Th
e con
d
ition
i
n
(13
)
can b
e
written
as
EC
H
H
(1
5)
After th
at, a solu
tio
n
ex
ists
for m
a
trix
E
u
s
ing
gen
e
ralized inv
e
rse as fo
llo
ws
()
(
(
)
(
)
)
q
E
H
CH
Y
I
CH
CH
EF
Y
G
(1
6)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
n
t
J
Elec
&
C
o
m
p
Eng
,
Vo
l. 1
0
, N
o
. 1
,
Febru
a
r
y
20
20
:
117
-
1
28
12
2
Whe
r
e
nq
Y
an
arb
itrary m
a
trix
,
()
F
HC
H
and
2
((
)
(
)
)
G
I
CH
CH
.
Fin
a
lly, b
y
su
bstitu
tin
g
E
i
n
t
o
(
9
) t
h
e
onl
y
un
kn
o
w
ns
are
m
a
t
r
i
ces
K
and
Y
. Th
e fol
l
owi
ng
se
ct
i
o
n
p
r
esen
ts a th
eorem
th
at g
i
v
e
s
a sufficien
t con
d
ition
for ch
oo
sing
t
h
em
.
3.1.
LMI sufficient condition
The
o
rem
:
The
error
()
et
will co
nverg
e asym
p
t
o
t
i
cally to
0
for an
y in
itial v
a
lu
e
(0
)
e
and
th
e NLUIO
in
(8
) is exp
o
n
e
n
tially stab
le su
ch
th
at
0
()
(
)
(
)
et
et
e
x
p
t
, w
h
ere
, if th
ere
ex
ists
P
a p
o
s
itive
sym
m
e
t
ric m
a
trix
0
P
satisfyin
g
t
h
e
fo
llowing
co
nd
itio
n [8
]
22
()
2
0
TT
N
P
PN
PMM
P
I
(1
7)
Pro
o
f
:
Let
’
s
de
fi
ne t
h
e Ly
a
p
u
n
o
v
f
unct
i
o
n c
a
ndi
dat
e
3
:
V
as foll
ows
T
Ve
P
e
(1
8)
Th
is Lyap
uno
v fu
n
c
tion
v
e
ri
fy th
e in
eq
u
a
lit
y b
e
low
22
mi
n
m
a
x
()
()
P
eV
P
e
(1
9)
Whe
r
e
mi
n
and
ma
x
are
t
h
e
m
i
nim
u
m
and t
h
e m
a
xi
m
u
m
Ei
gen v
a
l
u
es o
f
P
. By e
x
p
a
nd
ing
th
e
Lyap
uno
v
can
d
i
d
a
te fun
c
tio
n
of
(18
)
alon
g th
e er
ror
equ
a
tio
n in
(1
4)
t
h
e
f
o
llow
i
ng
ex
pr
essi
o
n
is obtain
e
d
ˆ
()
2
(
(
,
)
(
,
)
)
TT
T
Ve
N
P
P
N
e
e
P
M
f
x
u
f
x
u
ˆˆ
()
2
(
(
,
)
(
,
)
)
2
(
)
TT
T
T
V
e
N
P
PN
e
e
PM
f
x
u
f
x
u
e
P
MA
x
x
ˆ
()
2
2
(
(
,
)
(
,
)
)
TT
T
T
V
e
N
P
PN
e
e
PM
A
e
e
P
M
f
x
u
f
x
u
()
2
2
TT
T
T
V
e
N
P
PN
e
e
PM
e
e
P
M
e
th
en
u
s
i
n
g th
e
b
e
llo
w in
eq
uality, wh
ere
and
.
2
2
2
2
TT
eP
M
e
e
P
M
e
An
d
2
2
2
2
TT
eP
M
e
e
P
M
e
after sim
p
lificatio
n
,
V
m
a
y
be re
con
s
t
r
uct
e
d as
22
()
(
)
2
TT
T
T
T
V
e
N
P
PN
e
e
PMM
P
e
e
e
22
((
)
2
)
TT
T
Ve
N
P
P
N
P
M
M
P
I
e
Defi
n
e
th
e m
a
t
r
ix
0
Q
by
22
()
2
TT
QN
P
P
N
P
M
M
P
I
, hence
t
h
e t
i
m
e
deri
vat
i
v
e
V
is p
r
esen
ted
as
T
Ve
Q
e
(2
0)
Usi
n
g (1
9
)
a
n
d (2
0)
,
t
h
e up
per
b
o
u
n
d
f
o
r
()
Vt
can b
e
written
as
0
()
(
)
(
)
Vt
Vt
e
x
p
t
whe
r
e
max
mi
n
()
()
Q
P
and t
h
e
up
per b
o
u
n
d
fo
r
()
et
i
s
exp
r
esse
d by
0
()
(
)
(
)
et
et
e
x
p
t
wher
e
ma
x
m
i
n
()
/
(
)
PP
.
Th
e co
nd
itio
n
,
0
n
sI
A
D
ra
nk
n
q
s
C
leads to the
fact that
(,
)
M
AC
i
s
obs
er
vabl
e
.
In
co
nse
que
nc
e, t
h
e m
a
t
r
i
x
K
can be obtained suc
h
that
NM
A
K
C
is Hurwitz equ
a
lity.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
El
ec &
C
o
m
p
En
g
ISS
N
:
2
0
8
8
-
87
08
3-
D st
r
u
ct
ure
f
r
om
m
o
t
i
o
n re
covery
of
a
mo
vi
ng
o
b
j
ect
w
i
t
h
noi
sy
me
as
ur
ement
(
Z
o
u
b
a
i
d
a
Mej
ri
)
12
3
No
te th
at th
ere is n
o
systematic way
to
o
b
t
ain
t
h
e ad
ap
tab
l
e NLUIO p
a
ram
e
ters d
i
rectly fro
m
co
nd
itio
n (1
0) an
d th
e exp
r
essio
n
i
n
th
e t
h
eo
rem
g
i
v
e
n
by (17
)
. Th
is al
lo
ws t
o
reformu
l
ate th
em
as
LMIs.
Sub
s
titu
tin
g
N
and
M
fro
m
(9
) i
n
to (1
7) th
e fo
llowing
relatio
n
s
h
i
p
can
t
h
en
b
e
estab
lish
e
d
22
()
()
2
(
)
(
)
(
)
0
T
MA
K
C
P
P
MA
K
C
I
P
I
E
C
I
EC
P
(2
1)
Using
t
h
e so
lu
t
i
o
n
i
n
(16
)
, th
e in
equ
a
lity (21)
b
eco
m
e
s
11
2
2
22
11
2(
)
0
T
TT
T
T
T
T
T
T
A
IF
C
P
P
I
F
C
A
A
C
G
P
P
G
C
A
C
P
P
C
I
P
PFC
P
G
C
P
PFC
P
G
C
(2
2)
Varia
b
les
1
P
PY
and
2
P
PK
are g
i
v
e
n to
m
a
k
e
th
e reso
l
u
tio
n
of th
e non
lin
ear m
a
trix
in
equ
a
lities easier. Ex
pon
en
tial co
nv
erg
e
n
ce t
o
th
e obj
ect coo
r
d
i
n
a
tes is ach
i
ev
ed
.
3.
2.
L
M
I form
ul
a
t
i
o
n
For t
h
e NL
UI
O sy
nt
hesi
s t
h
e fol
l
o
wi
n
g
LM
Is (2
3)
hav
e
feasi
b
l
e
sol
u
t
i
o
ns f
o
r P,
K an
d Y i
n
vo
ki
n
g
th
e in
eq
u
a
lity in
(22
)
tran
sfo
r
med
with
schur’s
co
m
p
le
m
e
n
t
.
0
T
XW
WI
(2
3)
W
h
er
e
1
WP
P
F
C
P
G
C
,
11
2
2
2
T
TT
T
T
T
T
T
XA
I
F
C
P
P
I
F
C
A
A
C
G
P
P
G
C
A
C
P
P
C
I
22
()
.
Using
so
l
u
tio
n o
f
LMI i.e. feasib
le v
a
lu
es of
1
1
YP
P
and
1
2
KP
P
, observe
r
m
a
trices satisfying
th
e requ
isite con
d
ition
s
are
fou
n
d
.
Th
e LMI
feasib
ility can
b
e
so
lv
ed
u
s
ing
stan
d
a
rd LM
I ap
pro
a
ch
[3
0].
4.
RESULTS
A
N
D
DI
SC
US
S
I
ONS
In c
ont
rast
w
i
t
h
pre
v
i
o
us r
e
search t
h
at
assum
e
noi
se-
free m
easure
m
ent
s
and
de
m
a
nd p
r
i
o
r
knowledge of the object a
nd cam
era
mo
tio
n, th
e prop
o
s
ed
m
e
th
o
d
assu
m
e
th
at
th
e ob
j
ect velo
city is
u
nkn
own
.
In
th
e fo
llowing
, th
e
p
e
rfo
r
m
a
n
ce
o
f
th
e NLUIO is v
a
li
d
a
ted th
rou
gh d
i
fferen
t
numerical
sim
u
lations in the prese
n
ce
of m
easurement noise
for bot
h static and dynam
i
c sc
enes.
As the current
sim
u
lation res
u
lts are
rest
ricted to trac
king a si
ngle
p
o
i
n
t
feat
u
r
e.
T
w
o
di
ffe
re
nt
o
b
j
ec
t
m
o
t
i
on m
odel
s
are
considere
d
, and the
proposed NL
UIO
pe
rform
a
nce is evaluated
for
bot
h
cases.
Whe
r
ea
s the usual s
p
e
e
d of
t
h
e m
onocul
a
r
cam
e
ra i
s
30
fram
e
s/
s, t
h
e NLU
I
O i
s
val
i
d f
o
r a c
o
nt
i
n
u
ous
-t
i
m
e sy
st
em
. For t
h
e si
m
u
l
a
t
i
o
n
resu
lts,
SIMULINK is used
with
sam
p
lin
g p
e
ri
o
d
3
10
s
ts
.Th
e
i
n
i
tial Eu
clid
ean
co
ord
i
n
a
tes of th
e ob
j
ect
feature
are
0
()
5
2
1
(
m
)
T
xt
. Sin
ce in
itial targ
et
featu
r
e po
in
t is no
t
kn
own
at th
e
NLUIO, thu
s
th
e system
an
d ob
serv
er start
fro
m
d
i
fferen
t
in
itial co
nd
itio
n
s
.
In
itial con
d
ition
for t
h
e ob
serv
er is tak
e
n as
0
ˆ
()
1
0
.
5
0
.
2
(
m
)
T
xt
.
Matrices
A
,
C
and
H
are gi
ve
n by
01
2
10
1
000
A
10
0
01
0
C
An
d
10
0
T
D
Not
e
,
t
h
e t
h
i
r
d
com
pone
nt
3
x
of the state, whi
c
h is the
unm
e
asura
b
le
distance betwee
n the cam
era
and t
h
e m
ovi
n
g
o
b
j
ect
. C
l
earl
y
t
h
e est
i
m
a
ti
on
of t
h
e t
h
r
ee-di
m
e
nsi
onal
Eucl
i
d
ea
n co
or
di
nat
e
s ca
n
y
i
el
d
the dista
n
ce e
s
tim
a
tion. T
h
e
com
p
aris
on
of
RMS
erro
r v
a
lu
es ob
tain
ed
with
th
e pro
p
o
s
ed
NLUIO with
di
ffe
re
nt
val
u
e
s
o
f
m
easurem
ent
n
o
i
s
e
was
u
s
ed t
o
dem
onst
r
at
e t
h
e
pr
o
p
o
s
ed m
e
t
hod.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
n
t
J
Elec
&
C
o
m
p
Eng
,
Vo
l. 1
0
, N
o
. 1
,
Febru
a
r
y
20
20
:
117
-
1
28
12
4
3.
1.
Sta
t
i
c
sce
n
e
In th
is case, the ob
j
ect
and
t
h
e cam
era v
e
lo
cities p
a
ram
e
ter
s
are cho
s
en
resp
ectiv
ely b
y
1
2(
/
)
0.
2
s
i
n
(
/
4)
c
vm
s
t
,
0
0
/
3
0
(
ra
d
/
s)
T
and
0.
5
0(
/
)
0
p
vm
s
Figure 3 shows the structure estim
a
ti
on of t
h
e o
b
ject
p
o
si
t
i
on i
n
t
h
e
single ca
m
e
ra images. Figure 4
p
r
esen
ts th
e error in
th
e
po
sitio
n
estim
a
tio
n
o
f
th
e st
atic o
b
j
ect. No
tice
th
at
th
e trans
i
ent perform
a
nce of
t
h
e p
r
o
p
o
sed
s
c
hem
e
i
s
si
gni
fi
cant
l
y
l
e
ss t
h
en 4
seco
n
d
.
The R
M
S er
r
o
r val
u
es o
b
t
a
i
n
ed
by
t
h
e
pr
op
ose
d
NLUIO are
given as
follows
e
1
= 0.157
8, e
2
= 0
.
07
89
and
e
3
= 0.077
6.
Fig
u
re
3
.
Tim
e
h
i
stories
o
f
t
h
e static o
b
j
ect
p
o
s
ition
in the
single-ca
m
era im
ag
es; so
lid
lin
e: estimated
and Da
she
d
-dotted line: real s
t
ate
Fig
u
re
4
.
Static ob
j
ect esti
m
a
ti
o
n
error
o
f
th
e
pr
o
pose
d
m
e
t
hod
Next, the m
e
a
s
urem
ents of
c
V
as sho
w
n i
n
Fi
g
u
re 5 i
s
assum
e
d t
o
be co
rr
u
p
t
e
d by
ad
di
n
g
a B
a
nd
Lim
i
t
e
d Whi
t
e
Noi
s
e
(B
L
W
GN
) wi
t
h
5%
of
po
we
r, a c
o
r
r
el
at
i
on t
i
m
e of
0 an
d a c
ova
ri
ance
of i
n
fi
ni
t
y
.
Figure
6 s
h
ows the
struct
ure estim
ation of t
h
e static
object
positi
on i
n
the
single cam
era images in
the presence
of m
easurem
ent noise.
The
erro
r in
t
h
e po
sitio
n
estim
atio
n
o
f
th
e
static o
b
j
ect is d
e
scri
b
e
d
i
n
Fi
gu
re
7.
O
n
l
y
t
h
e t
h
i
r
d c
o
m
pone
nt
of
R
M
S
err
o
r
i
s
cha
n
ge
d e
3
= 0.081
6.
Fig
u
re
5
.
Th
e
measu
r
m
e
n
t
s of th
e cam
er
a v
e
lo
cities in
th
e
presen
ce of
no
ise
0
2
4
6
8
1
01
2
1
41
61
8
2
0
-50
0
50
0
2
4
6
8
1
01
2
1
41
61
8
2
0
-50
0
50
Time(s
)
0
2
4
6
8
1
01
2
1
41
61
8
2
0
-50
0
50
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
El
ec &
C
o
m
p
En
g
ISS
N
:
2
0
8
8
-
87
08
3-
D st
r
u
ct
ure
f
r
om
m
o
t
i
o
n re
covery
of
a
mo
vi
ng
o
b
j
ect
w
i
t
h
noi
sy
me
as
ur
ement
(
Z
o
u
b
a
i
d
a
Mej
ri
)
12
5
Fig
u
re
6
.
Tim
e
h
i
stories
o
f
t
h
e static o
b
j
ect
p
o
s
ition
i
n
the single-cam
era im
ages
in
t
h
e prese
n
ce of noisy
ca
m
e
ra v
e
lo
cit
y
; so
lid
lin
e: esti
m
a
ted
and
Dash
ed
-d
o
tte
d
line: real state
Fi
gu
re
7.
St
at
i
c
o
b
ject
e
s
t
i
m
a
t
i
on
er
ro
r
of
t
h
e
p
r
op
ose
d
m
e
t
hod
i
n
t
h
e prese
n
ce o
f
noi
sy
ca
m
e
ra velocit
y
In
ad
d
ition
to th
e p
r
ev
iou
s
m
easu
r
em
en
t
n
o
i
se
o
f
c
V
, t
h
e pr
op
ose
d
o
b
ser
v
e
r
i
s
val
i
dat
e
d f
o
r
ro
b
u
st
ness
by
t
h
e ad
di
t
i
on o
f
a B
a
nd Li
m
i
t
e
d
W
h
i
t
e
N
o
i
s
e (B
L
W
GN
)
wi
t
h
5%
of
po
we
r t
o
t
h
e
ob
ject
v
e
lo
city. Figu
re 8
shows th
e stru
cture estimatio
n
o
f
t
h
e dy
nam
i
c objec
t
coor
di
nat
e
s wi
t
h
n
o
i
s
y
o
b
j
ect
a
n
d
ca
m
e
ra v
e
lo
cities and
Fi
g
u
re
9 d
e
scrib
e
s t
h
e erro
r i
n
th
e po
si
tio
n
estim
at
io
n o
f
th
e
d
y
n
a
m
i
c. The NLUIO
th
en
y
i
el
ds uni
fo
rm
l
y
asym
pt
ot
i
c
al
l
y
conve
r
g
ent
est
i
m
a
t
e
s of
t
h
e t
h
ree-
di
m
e
nsi
o
nal
E
u
cl
i
d
ean c
o
o
r
di
nat
e
s o
f
the feature
point. In the pre
s
en
ce of no
ise in
th
e
m
o
tio
n
p
a
ram
e
ters,
th
e esti
m
a
ted
state
3
ˆ
x
is corrupted
di
rect
l
y
by
t
h
e sou
r
ce o
f
n
o
i
s
e, t
h
e
r
ef
ore
t
h
e t
h
i
r
d c
o
m
ponent
o
f
R
M
S err
o
r i
n
c
r
eases an
d bec
o
m
e
s
e
3
=0.
2
04
8.
Fig
u
re
8
.
Tim
e
h
i
stories
o
f
t
h
e static o
b
j
ect
p
o
s
ition
i
n
the single-cam
era im
ages
in
t
h
e prese
n
ce of noisy
ca
m
e
ra and
o
b
ject v
e
lo
cities; so
lid
lin
e: estimated
an
d
Dashed-dotted
line: real state
Fi
gu
re
9.
St
at
i
c
o
b
ject
e
s
t
i
m
a
t
i
on
er
ro
r
of
t
h
e
p
r
op
ose
d
m
e
t
hod
i
n
t
h
e prese
n
ce o
f
noi
sy
ca
m
e
ra and
o
b
ject v
e
lo
cities
3.2.
Dynamic
scene
In this case, only the performance
in the pre
s
ence of m
eas
urem
ent
noise for both cam
era and object
v
e
lo
city is studied
. Th
e sam
e
v
a
lu
es
of m
eas
urem
ent noise
are used. T
h
e
ob
ject a
n
d the
ca
m
e
ra velocit
i
es are
chosen res
p
ectively as:
Er
ror 1
Error 2
Error 3
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
0
2
4
6
8
10
12
14
16
18
20
-0.5
0
0.5
Tim
e
(s)
0
2
4
6
8
10
12
14
16
18
20
-1
0
1
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
n
t
J
Elec
&
C
o
m
p
Eng
,
Vo
l. 1
0
, N
o
. 1
,
Febru
a
r
y
20
20
:
117
-
1
28
12
6
1
2(
/
)
0.2
s
in
(
/
4)
c
vm
s
t
,
00
/
3
0
(
r
a
d
/
s
)
T
and
0.2+co
s(
2
t/4)
0(
/
)
0
p
vm
s
.
Fi
gu
re
1
0
pres
ent
s
t
h
e
st
r
u
ct
ure
est
i
m
at
i
on o
f
t
h
e
dy
nam
i
c ob
ject
p
o
si
t
i
on a
n
d
Fi
g
u
r
e
1
1
s
h
o
w
s
th
e erro
r i
n
th
e p
o
s
ition
esti
matio
n
of th
e obj
ect with
no
isy
ca
m
e
ra v
e
lo
city. Th
ese resu
lts d
e
m
o
n
s
trate th
at
t
h
e pr
op
ose
d
NLU
I
O base
d
ob
ject
st
ruct
u
r
e est
i
m
a
ti
on m
e
t
hod can ac
hi
eve sat
i
s
fact
ory
pe
rf
orm
a
nce even
with
cam
era v
e
lo
cities. Th
is o
b
s
erv
e
r
g
i
v
e
s b
e
tter esti
m
a
t
e
s for a sign
ifi
can
t lev
e
l o
f
no
ise ev
en
ch
ang
i
ng
scene. RMS error
values
are
gi
ve
n as
f
o
l
l
o
ws:
e1
=
0
.
15
78
, e2
=
0
.
07
89
and e3
= 0.081
6.
Fig
u
re
10
. Time h
i
stories
o
f
t
h
e
d
y
n
a
m
i
c o
b
j
ect po
sition
in
th
e
presen
ce of
n
o
i
sy cam
e
r
a
v
e
lo
city; so
l
i
d
lin
e:
est
i
m
a
t
e
d and
Dash
ed
-d
o
tted
line: real state
Fi
gu
re
1
1
.
Dy
n
a
m
i
c obj
ect
est
i
m
a
ti
on e
r
r
o
r
o
f
t
h
e
p
r
op
ose
d
m
e
t
hod
i
n
t
h
e prese
n
ce o
f
noi
sy
ca
m
e
ra velocit
y
Fig
u
re
12
shows t
h
e stru
cture estim
at
io
n
o
f
th
e
d
ynam
i
c obj
ect po
sitio
n with
n
o
i
sy
ob
ject and
ca
m
e
ra velocit
i
es and Figure
13 shows the
struct
ure
esti
mation error of the
dynam
i
c
object.
He
re
again,
onl
y
t
h
e t
h
i
r
d
com
pone
nt
of
R
M
S err
o
r a
r
e
cha
n
ged
e
3
=0
.0
992
. Ho
w
e
v
e
r, th
e
p
r
esence of
no
ise
o
n
bo
th
ca
m
e
ra and
obj
ect
v
e
lo
cities
can
si
g
n
i
fican
t
l
y d
e
grad
e th
e p
e
rform
a
n
ce
of NLUIO. Th
erefo
r
e, it can
be seen
th
at th
e
p
r
actical situ
atio
n
do
es req
u
i
re a m
o
re ro
bu
st
n
o
n
linear
o
b
s
erv
e
r
for th
e con
s
id
ered
p
r
ob
lem
.
Fi
gu
re
1
2
. Ti
m
e
hi
st
o
r
i
e
s
of t
h
e
dy
nam
i
c ob
ject
p
o
s
ition
i
n
th
e
p
r
esen
ce
of
n
o
i
sy cam
era an
d
o
b
j
ect
v
e
lo
cities; so
li
d
lin
e: estim
ate
d
an
d Dash
ed
-d
o
tted
line: real state
Fi
gu
re
1
3
.
Dy
n
a
m
i
c obj
ect
est
i
m
a
ti
on e
r
r
o
r
o
f
t
h
e
p
r
op
ose
d
m
e
t
hod
i
n
t
h
e prese
n
ce o
f
noi
sy
ca
m
e
ra and
o
b
ject v
e
lo
cities
Evaluation Warning : The document was created with Spire.PDF for Python.