Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
Vol.
3, No. 4, Decem
ber
2014, pp. 245~
251
I
S
SN
: 208
9-4
8
5
6
2
45
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJRA
Res
e
arch and ap
plicati
o
n on f
r
act
i
onal-
o
rder Darwinian P
S
O
based ad
aptive extended k
a
lman fil
t
erin
g
al
gorithm
Zh
u
Q
i
-
g
ua
ng (
朱奇
光
)
a) b)
†
, Yu
an
Mei
(
袁
梅
)
a) b)
, Liu Yao-long (
刘要
龙
)
a) b)
, Chen
Wei-d
o
n
g
(
陈卫
东
)
a) b)
, Chen
Yin
g
(
陈
颖
)
c)
,
and Wan
g
Hong-r
ui
(
王洪
瑞
)
c)
a) School of Info
rmation Science
and Eng
i
neerin
g
,
Yanshan
University
, Qinhu
angd
ao 066004, Chin
a
b) The Key
Lab
o
rator
y
for
Special Fib
e
r
and Fiber Se
nsor of Heb
e
i Province, Ya
nshan University
,
Qinhuangdao
066004, Chin
a
c) School of
Electrical En
g
i
neer
ing, Yanshan
Univer
sity
, Qinhuan
gdao 066004, C
h
ina
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 25, 2014
Rev
i
sed
Ju
l 27
,
20
14
Accepted Aug 20, 2014
To resolve th
e d
i
fficu
lt
y in est
a
b
lishing ac
cura
te
priori noise m
odel for the
extend
ed Kalm
an filtering algori
t
hm
,
propose the fraction
a
l-order
Darwinian
parti
c
le swarm
optim
iza
tion (
PSO)
algorithm has been pro
posed and
introduced in
to the fuzzy
ad
aptiv
e
ex
tended
Kalman filtering
alg
o
rithm. Th
e
natural selection
method has been adopted
to im
prove the standard particle
swarm
optim
izat
ion algor
ithm
,
w
h
ich enh
a
nced the diversity
of p
a
rticles
and
avoided
the pr
emature. In
addition, th
e
fra
ction
a
l c
a
lcu
l
us has
been used
to
improve the evo
l
ution speed of
partic
les.
The P
S
O algorithm
after im
prove
d
has
been
appl
ied
to tr
ain
fuz
z
y
a
d
aptiv
e ex
tend
e
d
Kalm
an fi
lter
and a
c
hiev
e
the sim
u
lt
aneou
s
loca
liz
ation
a
nd m
a
pping.
T
h
e sim
u
lat
i
on r
e
sults hav
e
s
hown that com
p
ared with the g
ees
e part
icl
e
s
w
arm
optim
izatio
n training of
fuzz
y
adap
tive
e
x
tended
Kalm
an
filt
er
localization and mapping
algor
ithm,
has been
greatly
improved in
ter
m
s of localizatio
n and mapp
ing.
Keyword:
Darwin
ian
Fractional calculus
Fuzzy a
d
apti
ve
extende
d
Kalm
an
filterin
g
M
a
ppi
ng
Particle swarm op
ti
m
i
zatio
n
Si
m
u
ltan
e
ou
s Lo
cation
Copyright ©
201
4 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
:
Zh
u Qi
-g
ua
ng
(
朱奇光
),
Sch
ool
o
f
I
n
fo
r
m
at
i
on Sci
e
nce
an
d E
n
gi
neeri
n
g
,
Yans
ha
n
U
n
i
v
e
r
si
t
y
, Qi
n
h
u
an
g
d
ao
0
6
6
0
0
4
, C
h
i
n
a
The
Key
La
bo
r
a
t
o
ry
f
o
r S
p
eci
al
Fi
ber
an
d
Fi
ber
Se
nso
r
of
Hebei
Pr
o
v
i
n
c
e
, Ya
ns
ha
n
Un
i
v
ersi
t
y
,
Q
i
nh
u
a
ng
d
a
o
06
600
4,
C
h
in
a
Em
a
il: zh
u
7
8
8
0
@
ysu
.
ed
u.cn
PACS
: 05.40.-a, 45.10.Hj, 07.05.Mh
1.
INTRODUCTION
Sim
u
l
t
a
neo
u
s
l
o
cat
i
on a
n
d
m
a
ppi
n
g
(
S
L
A
M
)
m
eans t
h
at
w
h
e
n
a
r
o
b
o
t
m
oves i
n
a
n
u
n
k
n
o
w
n
envi
ronm
ent, during the m
ovem
e
nt, the location can
be achieve
d according to the position esti
m
a
ti
on a
n
d
sens
or
obs
er
va
t
i
on an
d t
h
e s
u
rr
ou
n
d
i
n
g en
vi
ro
nm
ent
m
a
p can be est
a
bl
i
s
h
e
d si
m
u
l
t
a
neou
sl
y
,
whi
c
h m
a
kes up
th
e
preco
n
d
ition
o
f
au
t
o
no
m
o
u
s
op
eration
fo
r
m
o
b
ile
robots. Th
erefore, t
h
e SLA
M
prob
lem
re
m
a
in
s o
n
e
of
r
e
sear
ch
ho
tspo
ts o
f
t
h
e r
obo
t
f
i
eld
.
So fa
r
，
th
e
g
e
n
e
ral algo
rithm
s
so
lv
in
g
SLAM p
r
ob
lem
s
in
clu
d
e
ex
tended
Kalm
an
filt
ering
,
p
a
rticle
filtering, unsce
n
ted
Kalm
an filtering, a
nd
s
o
on
[1,
2].
Am
ong those algorithm
s
, extende
d
Kalm
an filtering
Pro
j
ect
s
u
p
p
o
rt
e
d
by
t
h
e
Fo
un
dat
i
o
n f
o
r Key
P
r
o
g
r
a
m
of M
i
ni
st
ry
of E
d
ucat
i
o
n
of C
h
i
n
a
(
G
r
a
n
t
N
o
s.
2110
23
)
,
t
h
e N
a
tu
r
a
l
Scien
ce Fo
und
atio
n o
f
Heb
e
i
Pr
ov
in
ce, Ch
in
a (G
r
a
n
t
N
o
s. F201
2203
169
),
an
d
th
e Ch
in
a
Postdoct
oral Science
Fo
und
at
io
n(
Gr
an
t No
s.
2012M
76
5
)
Co
rr
espon
d
i
n
g
au
thor
. E-
m
a
il
:zh
u
7
880
@ysu.edu
.cn
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
245 – 251
24
6
an
d
so
m
e
i
m
p
r
ov
ed
ex
tend
ed
Kalm
an
filte
r are u
s
ed
m
o
re co
mm
o
n
l
y. Ex
tend
ed
Kal
m
an
filter h
a
s g
o
o
d
n
o
n
lin
ear an
d
math
e
m
atica
l
rig
o
r,
h
o
wev
e
r it n
eed
s to
ob
tain
th
e priori in
fo
rm
atio
n
,
su
ch as th
e
syste
m
n
o
i
ses
and the
obse
rvation
noises, i
n
whic
h t
h
e sy
ste
m
noises
ca
n
be
obtained t
h
rough e
x
peri
ments, while
because
of t
h
e
great
i
n
fl
l
u
e
n
ces
f
o
r
en
vi
ro
nm
ent
a
l
fact
ors
,
t
h
e
obs
er
vat
i
o
n
n
o
i
s
es are
ha
r
d
t
o
get
i
n
a
d
vance
.
In
su
fficien
c
y an
d in
accu
r
acy
o
f
prio
ri i
n
fo
rmatio
n
m
a
y
le
ad
to
l
o
wer
filtering
accu
r
acy
and
ev
en
d
i
v
e
rg
en
ce.
Th
erefo
r
e,
fu
zzy lo
g
i
c h
a
s
been
u
s
ed
t
o
adj
u
st th
e ob
servatio
n
no
ise
o
f
ex
tend
ed
Kal
m
an
filter alg
o
rith
m
in
refe
rence [3], but the fuzzy rules ar
e selected through expe
ri
ence, whic
h may cause a ce
rtain de
viation
in the
appl
i
cat
i
o
n
of
dy
nam
i
c envi
r
onm
ent
.
I
n
re
f
e
rence
[
4
]
,
PS
O al
g
o
r
i
t
h
m
has bee
n
use
d
t
o
t
r
ai
n
fuzzy
r
u
l
e
s t
o
o
b
t
ain th
e
p
r
op
er
p
a
ram
e
ters
o
f
m
e
m
b
ersh
i
p
fun
c
tio
ns,
b
u
t it is easy to
fall in
to
prem
at
u
r
e con
v
e
rg
ence and
l
o
cal
o
p
t
i
m
al
pro
b
l
e
m
.
For
t
h
e di
sa
dva
nt
age
s
o
f
P
S
O
al
g
o
r
i
t
h
m
,
a l
o
t
of
i
m
prove
d P
S
O
al
go
ri
t
h
m
s
hav
e
be
e
n
put
forward, s
u
ch as the
gee
s
e pa
rticle swa
r
m
algorithm
[5
], ch
ao
tic p
a
rticle swarm
al
g
o
rith
m
[6
], d
i
screte
p
a
rticle swarm
alg
o
r
ith
m
[7
]. Th
ese im
p
r
ov
ed
al
g
o
rith
ms in
crease th
e d
i
v
e
rsity o
f
p
a
rticles to
a certai
n
ext
e
nt
, a
n
d t
h
e
pr
o
b
l
e
m
of p
r
e
m
at
ure co
n
v
er
gence
an
d l
o
ca
l
opt
i
m
al
has b
een i
m
pro
v
ed
.
In t
h
i
s
pa
per
,
t
h
e fract
i
onal
-
o
r
de
r
Dar
w
i
n
i
a
n pa
rt
i
c
l
e
swarm
al
gori
t
h
m
has bee
n
i
n
t
r
od
uce
d
t
o
o
p
tim
ize
th
e key p
a
ram
e
ters
o
f
fu
zzy log
i
c. Th
e n
a
t
u
ral selectio
n
m
e
th
od
h
a
s
b
e
en
ado
p
t
ed
to
im
p
r
ov
e th
e
st
anda
rd
part
i
c
l
e
swarm
opt
i
m
i
zat
i
on al
gor
i
t
h
m
,
whi
c
h e
nha
nce
d
t
h
e di
versi
t
y
of
part
i
c
l
e
s and av
oi
d
e
d t
h
e
p
r
em
atu
r
e.
In ad
d
ition
,
t
h
e
fractio
n
a
l calcu
l
u
s h
a
s b
e
en
u
s
ed
to im
p
r
o
v
e
th
e ev
o
l
u
tio
n sp
eed of
p
a
rticl
e
s. Th
e
PSO al
g
o
rith
m after im
p
r
o
v
e
d
h
a
s
b
e
en
app
lied
to
trai
n
fu
zzy ad
ap
tiv
e
ex
tend
ed
Kal
m
an
filter an
d
ach
ieve
t
h
e si
m
u
l
t
a
neous l
o
cal
i
zat
i
on
and
m
a
ppi
ng
.
2.
FR
AC
TIONAL-
O
RD
ER DA
RWINIAN
PA
R
T
IC
LE SWAR
M A
L
GOR
I
THM
2.1.
Darwinian P
a
rticle Sw
arm
Algorithm
Part
i
c
l
e
swarm
opt
i
m
i
zati
on a
l
go
ri
t
h
m
ori
g
i
n
at
es f
r
om
t
h
e researc
h
on t
h
e gr
o
up m
ovi
n
g
be
ha
vi
o
r
s
of bi
rd fl
ock
s
and fi
s
h
sch
o
o
l
s
, and
has bee
n
wi
del
y
u
s
ed
to
so
lv
e th
e com
p
lex
o
p
t
i
m
iz
atio
n
prob
lem
s
. Jason
Tillett
an
d
T.M. Rao
p
u
t
fo
rward
th
at Darwin
ian
n
a
tural
selectio
n
in
b
i
o
l
og
y can
b
e
used
to
i
m
p
r
ov
e p
a
rticle
swarm
algorithm
[8]. The basic idea of Darwi
n
ian pa
r
ticle swarm alg
o
r
ith
m
is:
at
e
ach m
o
m
e
nt
there are
sev
e
ral
p
a
rticle swarm
s
search
ing
sim
u
ltan
e
o
u
sly. Each
particle swarm
can
run in
acc
or
da
nce
wi
t
h
s
t
anda
r
d
p
a
rticle swarm alg
o
rith
m
,
an
d so
m
e
ru
l
e
s are
adde
d t
o
si
m
u
l
a
t
e
nat
u
ral
sel
ect
i
on
du
ri
n
g
t
h
e r
u
nni
ng
, i
n
whi
c
h
t
h
e ap
p
r
o
p
ri
at
e
swa
r
m
woul
d
be c
h
o
s
en
in
t
h
e con
tin
uou
s sw
arm
com
b
ination.
In the sea
r
chi
n
g process
,
whe
n
on
e o
f
th
e swarm
s
ten
d
s
to fall in
to
th
e lo
cal o
p
t
i
m
u
m
,
t
h
e search
ing
i
n
t
h
i
s
area w
oul
d be
di
scar
ded i
m
m
e
di
at
el
y
and t
u
r
n
t
o
t
h
e searc
h
i
n
g i
n
ot
he
r areas
. The pa
rt
i
c
l
e
swarm
s
with
b
e
tter
fitness will g
e
t reward
s,
o
t
h
e
rwise, tho
s
e tend
to
stagn
a
te w
ill b
e
pu
n
i
sh
ed
.
Re
m
o
v
i
ng
a particle
(or a
p
a
rticle swarm
)
n
e
ed
s t
o
co
m
p
ly with
th
e fo
llowing
ru
les:
(1
)
Wh
en
t
h
e
part
i
c
l
e
num
ber
of
t
h
e
swa
r
m
i
s
l
e
ss t
h
an
t
h
e speci
fi
ed m
i
ni
m
u
m
num
ber
of
pa
rt
i
c
l
e
s,
it will b
e
rem
o
v
e
d.
(2)
If th
e op
timal fitn
ess
o
f
a
p
a
rticle swarm
h
a
s no
t
been
im
p
r
ov
ed
with
in th
e
pred
eterm
i
n
e
d
search
i
n
g
nu
mb
er, th
e p
a
rticle with
th
e
worst fitn
ess w
ill b
e
rem
o
v
e
d
.
In
add
itio
n, th
e search
i
n
g
will n
o
t
b
e
reset to
0, bu
t
reset to
a v
a
l
u
e clo
s
e to
th
e
max
i
m
u
m
searching num
b
er.
It can be
desc
ribed as the
followi
ng
fo
rm
ula:
1
1
1
max
kill
c
kill
c
N
SC
N
SC
(1
)
The
ge
nerat
i
o
n
of a
ne
w
p
a
rt
i
c
l
e
swarm
m
u
st
m
e
e
t
with
two
con
d
itio
n
s
:
No
p
a
rticle h
a
s
b
een
rem
oved
from
this pa
rticle swarm
and
pa
rticle swarm
num
b
er has
not
reached the m
a
xim
u
m
.
Even i
f
these
two
co
nd
ition
s
are m
e
t, th
e e
m
erg
i
ng
pro
b
ab
ility o
f
a n
e
w p
a
rticle swarm
is
NS
f
p
，
f
i
s
a r
a
nd
om
num
ber
bet
w
e
e
n 0 a
n
d 1
,
N
S
i
s
part
i
c
l
e
s
w
arm
num
ber.
The f
u
nct
i
o
n
of t
h
i
s
co
ndi
t
i
on i
s
t
o
rest
ri
ct
t
h
e
g
e
n
e
ration
of new p
a
rticle.
Wh
en
a
n
e
w p
a
rticle swarm
is
gene
rat
i
n
g, t
h
e
part
i
c
l
e
s o
f
pa
rent
s
w
arm
have n
o
t
got
a
ffect
e
d
.
I
n
c
h
i
l
d
re
n s
w
a
r
m
s
, hal
f
of
pa
rt
i
c
l
e
s are
ran
d
o
m
l
y
sel
ect
ed fr
om
parent
s
w
arm
s
, t
h
e
ot
h
e
r
hal
f
are ran
d
o
m
ly
selected
p
a
rticles. If th
e
n
u
m
b
er o
f
a p
a
rticle swarm c
a
n
no
t reach
th
e in
itial n
u
m
b
e
r
of
p
a
rticles, th
en
th
e swarm
’
s o
t
h
e
r
p
a
rticles co
u
l
d
b
e
i
n
itiali
zed
rand
o
m
ly
an
d
add
e
d
t
o
a n
e
w
p
a
rticle swarm
.
Whe
n
a
particle swa
r
m
reaches a
ne
w
opti
m
al fitness
va
lue while
the particle
num
b
er of t
h
is s
w
arm
has
not
reache
d
t
h
e m
a
xim
u
m
,
a new
particle swa
r
m em
erges [9].
Li
ke t
h
e
st
an
da
rd
PS
O al
g
o
r
i
t
h
m
,
m
a
ny
para
m
e
t
e
rs of
Darwin
ian PSO al
so
n
e
ed
to
set reason
ab
ly to
m
a
ke t
h
e al
go
r
i
t
h
m
bet
t
e
r pe
r
f
o
r
m
a
nce, i
n
cl
udi
ng:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Researc
h
a
n
d
ap
pl
i
c
at
i
o
n o
n
f
r
act
i
o
n
a
l
-
or
de
r
D
a
rw
i
n
i
a
n
P
S
O
b
a
se
d a
d
a
p
t
i
ve
ext
ende
d …
(
Z
hu
Qi
-
g
ua
ng)
24
7
(1) Particle num
b
e
r o
f
t
h
e i
n
itializat
io
n
p
a
rti
c
le swarm
s
.
(2) Th
e m
i
n
i
mu
m
an
d
m
a
x
i
mu
m
p
a
rticle n
u
m
b
er th
at a swarm
allo
w to
ex
ist.
(3) Th
e
nu
m
b
er
o
f
i
n
itializati
o
n p
a
rticle swarm
s
.
(4) Th
e m
i
n
i
mu
m
an
d
m
a
x
i
mu
m
n
u
m
b
e
r of
th
e p
a
rticle swarm
th
at allo
wed
to ex
ist.
(5) T
h
e t
h
res
h
old t
h
at the
sea
r
chi
n
g ends.
2.
2. I
m
pr
ove
d
D
a
rw
i
n
i
a
n
P
a
rti
c
l
e
Sw
arm
Al
g
o
ri
t
h
m u
s
i
n
g Fr
acti
on
al
C
a
l
c
ul
us
The t
i
m
e dom
ai
n f
r
act
i
o
n
a
l
-
o
r
der
cal
cul
u
s
e
q
uat
i
o
n
[
1
0]
de
f
i
ned
by
Gr
u
n
w
a
l
d
-Let
ni
k
o
v
i
s
0
0
1
1
1
1
1
lim
k
k
h
k
k
kh
t
x
h
t
x
D
(2
)
Fro
m
th
e Equ (2), it h
a
s been
sh
own
that in
teg
e
r-ord
e
r d
e
riv
a
tiv
e co
n
t
ains fi
n
ite series,
wh
il
e
fract
i
o
nal
-
or
de
r de
ri
vat
i
v
e c
o
nt
ai
ns i
n
fi
ni
t
e
seri
es. T
h
e m
o
st
si
gni
fi
ca
nt
d
i
ffere
nce
bet
w
een f
r
act
i
o
n
a
l
-
or
der
calcu
lu
s and
in
teg
e
r-ord
e
r
calcu
lu
s is t
h
at fraction
a
l-ord
e
r calcu
l
u
s
is related
to
all th
e po
in
ts' p
a
st
i
n
f
o
rm
at
i
on, i
s
a gl
o
b
al
ope
rat
o
r
.
Form
ul
a o
f
fra
ct
i
onal
-
or
der
d
e
ri
vat
i
v
e
use
d
i
n
di
scret
e
t
i
m
e
i
s
ap
pr
oxi
m
a
t
e
as
r
k
k
k
k
kT
t
x
T
t
x
D
0
1
1
1
1
1
(3
)
Whe
r
e,
T
rep
r
e
s
ents the
sam
p
ling
pe
rio
d
,
r
re
prese
n
ts t
h
e st
op
o
r
der.
Usi
n
g t
h
e fr
act
i
onal
-
or
de
r cal
cul
u
s
,
t
h
e u
p
d
a
t
i
ng vel
o
ci
t
y
of Da
r
w
i
n
i
a
n
PSO ca
n be i
m
prove
d, a
n
d
t
h
e vel
o
ci
t
y
u
p
d
at
i
n
g
f
o
rm
ul
a o
f
D
a
r
w
i
n
PS
O ca
n
be
rear
ra
nge
d a
s
[
1
1]
t
id
t
d
i
t
id
t
id
t
id
t
id
x
p
r
c
x
p
r
c
v
v
1
2
2
1
1
1
(4
)
Su
pp
ose t
h
at
1
,
th
e left sid
e
of th
e equ
a
tio
n is a d
e
riv
a
tive in
d
i
screte fo
rm
, o
r
d
e
r num
b
e
r
1
(su
p
pose
1
T
),
the
n
the
fo
rm
ula ab
ove
tu
rns
to
t
id
t
d
i
t
id
t
id
t
x
p
r
c
x
p
r
c
v
D
1
2
2
1
1
1
(5
)
Usi
n
g f
r
act
i
o
n
a
l
-
or
de
r cal
cul
u
s i
d
ea,
vel
o
ci
t
y
deri
vat
i
v
e'
s or
der ca
n be e
x
t
e
n
d
ed t
o
real
num
ber i
n
th
e li
m
its o
f
1
0
，
wh
ich
will cau
se th
e ch
an
ges m
o
re stab
le an
d
t
h
e m
e
m
o
ry effect m
o
re
lo
ng
er.
In
or
der t
o
di
sc
us
s t
h
e effect
of
'
s
val
u
e on t
h
e
per
f
o
r
m
a
nce of i
m
pro
v
ed
P
S
O al
g
o
ri
t
h
m
,
m
a
ke
change
s
fr
om
0 t
o
1,
st
ep l
e
n
g
t
h
1
.
0
, to
calcu
late th
e op
ti
m
a
l so
lu
tio
n o
f
so
m
e
fun
c
tio
n
s
. Accord
i
n
g
to
t
h
e
expe
rim
e
nts, the pe
rform
a
nce can achieve t
h
e best whe
n
6
.
0
.As th
e fractio
nal-o
r
d
e
r calcu
lu
s is in
fin
i
t
e
d
i
m
e
n
s
io
n
a
l, its "in
f
in
ite me
m
o
ry" ch
aracteristics lead
to
th
e d
i
fficu
lties o
f
d
i
g
ital realizatio
n
.
Existin
g
sim
u
l
a
t
i
on t
o
ol
s can
’t
deal
wi
t
h
no
n i
n
t
e
ger
-
o
rde
r
cal
cul
u
s
directly, so
wh
en
fraction
a
l-o
r
d
e
r calcu
lu
s is
u
s
ed
,
it is n
ecessary
to
approx
im
at
e it wirh
t
h
e fi
n
ite d
i
m
e
n
s
io
nal fun
c
tio
n [1
2]. If
we co
n
s
ume
4
r
, in
wh
ich
o
n
l
y th
e first fo
ur term
s are
bein
g
con
s
id
ered
, t
h
e eq
u
a
ti
o
n
abo
v
e
can turns in
to
t
id
t
d
i
t
id
t
id
t
t
t
t
t
x
p
r
c
x
p
r
c
v
v
v
v
v
1
2
2
1
1
3
2
1
1
2
1
24
1
1
6
1
2
1
,
(6
)
That is
t
id
t
d
i
t
id
t
id
t
t
t
t
t
t
x
p
r
c
x
p
r
c
v
v
v
v
v
v
1
2
2
1
1
3
2
1
1
2
1
24
1
1
6
1
2
1
(7
)
After m
a
n
y
si
m
u
la
tio
n
s
we can
con
c
lud
e
that th
e in
crease
o
f
r
d
o
little to
i
m
p
r
o
v
e
th
e perform
a
n
ce
o
f
th
e al
g
o
rithm
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
245 – 251
24
8
3.
FUZ
Z
Y
ADAPTIVE E
X
TE
NDE
D KAL
M
AN
FILTER SL
AM AL
GORITHM
SLAM
al
go
ri
t
h
m
based
on
EKF m
a
i
n
l
y
cont
ai
n
s
t
w
o st
eps:
p
r
e
d
i
c
t
i
o
n
pr
ocess
an
d
obs
er
vat
i
o
n
u
p
d
a
te
p
r
o
cess. First, estab
lish
th
e system
m
o
d
e
l. Th
e sp
atial en
v
i
ron
m
en
t is exp
r
essed as
T
m
v
k
x
k
x
k
x
,
(8
)
Whe
r
e,
k
x
v
rep
r
es
ent
s
r
o
bot
’s
po
se at
t
i
m
e
k,
k
x
m
re
prese
n
t
s
a
feat
ure
l
o
cat
i
o
n
of
t
h
e m
a
p.
Pred
ictio
n
process:
k
u
i
s
defi
n
e
d as ro
b
o
t
cont
rol
vect
or
at
any
gi
ven
t
i
m
e
k.
k
z
is th
e
obs
er
vat
i
o
n
va
l
u
e.
k
Q
and
k
R
are c
o
vari
a
n
ce m
a
t
r
ices of
sy
st
em
at
i
c
noi
se a
nd
obs
er
vat
i
o
n
n
o
i
se. The
p
r
ed
ictio
n equatio
n
s
are
k
u
k
k
x
k
F
k
k
x
1
(9
)
k
k
x
k
H
k
k
z
1
1
(1
0)
k
Q
k
F
k
k
p
k
F
k
k
p
T
1
(1
1)
In t
h
ese e
quat
i
ons
,
k
F
and
k
H
are the syste
m
state transition m
a
trix
and
th
e observ
a
tion
m
a
tri
x
at ti
me
k.
Obse
r
v
at
i
o
n
u
pdat
e
p
r
oces
s:
obs
er
vat
i
o
n
va
l
u
e
1
k
z
and predic
tion
value at ti
me k+1 a
r
e
us
ed t
o
u
p
d
a
te th
e sp
atial en
v
i
ron
m
en
t.
k
k
z
k
z
k
v
1
1
1
(1
2)
1
1
1
k
R
k
H
k
k
p
k
H
k
s
T
(1
3)
1
1
1
1
1
k
v
k
K
k
k
x
k
k
x
(1
4)
1
1
1
1
1
1
k
K
k
s
k
K
k
k
p
k
k
p
T
(1
5)
1
1
1
1
k
s
k
H
k
k
p
k
K
T
(1
6)
Whe
r
e,
1
k
v
is th
e i
n
nov
atio
n at time k
+
1
;
1
k
s
is th
e th
eoretical cov
a
rian
ce m
a
tri
x
o
f
inn
o
v
a
tion.
The
differe
n
ce
of actual c
o
varia
n
ce m
a
trix
of i
n
n
o
v
a
tion
an
d th
eo
reti
cal co
v
a
rian
ce m
a
trix
of
in
no
v
a
tion is
nk
c
ln
, use
diagonal ele
m
ents of
nk
c
ln
to a
d
just t
h
e
diagonal elem
en
ts in
co
v
a
rian
ce
matrix
o
f
ob
serv
ation
n
o
i
se , t
h
at is t
h
e
fu
zzy ad
ap
tiv
e ex
tend
ed kal
m
an
filter.
k
s
k
v
k
v
c
T
nk
ln
(1
7)
Use fraction
a
l-o
r
d
e
r
Darwin
ian
PSO to
trai
n fu
zzy
syste
m
,
there are there
m
e
m
b
ersh
ip
fu
n
c
tion
s
in
fuzzy
sy
st
e
m
, so t
h
e vari
a
b
l
e
o
f
m
e
m
b
ershi
p
f
u
nct
i
ons
has
9 di
m
e
nsi
ons
, w
h
i
c
h i
s
T
w
w
w
b
b
b
a
a
a
x
3
2
1
3
2
1
3
2
1
. Th
e
p
u
rpo
s
e
is to
ob
tain th
e min
i
m
u
m
difference
of act
ual cova
riance
and the
o
retical covaria
n
ce.
T
h
ere
f
ore, the
object
function i
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Researc
h
a
n
d
ap
pl
i
c
at
i
o
n o
n
f
r
act
i
o
n
a
l
-
or
de
r
D
a
rw
i
n
i
a
n
P
S
O
b
a
se
d a
d
a
p
t
i
ve
ext
ende
d …
(
Z
hu
Qi
-
g
ua
ng)
24
9
obs
N
n
cnobs
j
j
nk
N
j
c
f
obs
obs
cnobs
1
2
ln
(1
8)
Whe
r
e,
obs
N
is th
e to
tal nu
m
b
er of
o
b
s
erv
a
tio
n in
iteratio
n
,
cnobs
j
is dia
g
onal elem
ents
＇
nu
m
b
er
of
nk
c
ln
.
4.
AN
ALY
S
IS
O
F
E
X
PE
RI
M
E
NT RES
U
L
T
S
We can ass
u
me that the m
obile robot
＇ｓ
，
simu
latio
n
env
i
ronmen
t is a p
l
ane
rectangula
r
area
in this
envi
ronm
ent there are 28 feat
ure poi
n
ts and
14
p
a
th
po
in
ts, wh
ich
resp
ecti
v
ely rep
r
esen
ted
b
y
〝〞
〝
〞
*a
n
do.
Fig
u
re
1
is th
e m
a
p
th
at g
e
ts b
y
th
e
u
s
e
o
f
fu
zzy ad
ap
tiv
e
ex
tend
ed Kalman
filter
SLAM alg
o
rith
m
b
a
sed
on
th
e g
eese,
while fig
u
re 2 is th
e m
a
p
th
at g
e
ts b
y
u
s
ing
fu
zzy ad
ap
ti
v
e
ex
tend
ed
Kalm
an
filter SLAM
al
go
ri
t
h
m
based o
n
t
h
e
fra
ct
i
onal
-
or
der
Dar
w
i
n
i
a
n PS
O. R
e
d
〝〞
+
i
n the
picture
are the feat
ure poi
nt
s
＇
po
sitio
n
s
bu
i
lt b
y
th
e
robo
t.
Fi
gu
re
1.
SL
A
M
base
d
on
t
h
e
geese
PS
O
Fi
gu
re
2.
SL
A
M
base
d
on
t
h
e
fract
i
o
nal
-
o
r
d
e
r
Darwin
ian
PSO
As is
sho
w
n fro
m
th
e two
fig
u
res m
e
n
tio
n
e
d abov
e, the fu
zzy ad
ap
tiv
e ex
tend
ed
Kalm
an
filter
SLAM
al
g
o
ri
t
h
m
based on t
h
e fr
act
i
onal
-
o
r
de
r Da
rwi
n
i
a
n PS
O pe
rf
or
m
s
m
u
ch bet
t
e
r t
h
an
fuzzy
a
d
apt
i
v
e
ex
tend
ed Kalman
filter SLAM alg
o
rith
m
b
a
sed
on
th
e g
e
ese in
feature
po
in
ts estim
atio
n
an
d th
e ro
bo
t
’
s
p
o
se.
In
fi
gu
re 1,
t
h
e
l
o
cat
i
o
n of
est
i
m
a
t
e
d
feat
ure
poi
nt
s has a l
a
rge de
vi
at
i
on t
o
t
h
e act
ual
feature p
o
i
n
t
s
, and
t
h
ere
are m
a
ny points are
m
i
staken as featur
e
poi
n
t
s. In fi
gu
re 2,
t
h
e l
o
cat
i
on
of
est
i
m
a
t
e
d feat
ure p
o
i
n
t
s
an
d act
ual
feature
points
are m
u
ch im
prove
d i
n
c
o
m
p
ared with fi
gure
1, the
num
b
er
of erro
r poi
nts decrease
s
a
l
o
t.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
245 – 251
25
0
Fi
gu
re
3.
Locat
i
on e
r
r
o
rs
of
S
L
AM
t
h
e gee
s
e P
S
O
i
n
X
di
rect
i
o
n
base
d
on
Error
s
of X ax
is
Fi
gu
re 4.
Locat
i
on
e
r
r
o
rs of
S
L
AM
base
d on
t
h
e
fraction
a
l-o
r
d
e
r Darwin
ian PSO i
n
X
d
i
rectio
n
Fi
gu
re 5.
Locat
i
on
e
r
r
o
rs of
S
L
AM
base
d on
t
h
e
geese PSO
in Y direction
0
1000
2000
3000
-1
-0
.5
0
0.5
1
Ob
se
rva
t
i
on ti
me
s
Error
s
of Y
axis
y
Fi
gu
re 6.
Locat
i
on
e
r
r
o
rs of
S
L
AM
base
d on
t
h
e
fraction
a
l-o
r
d
e
r Darwin
ian PSO i
n
X
d
i
rectio
n
Fi
gu
re 7.
Locat
i
on
e
r
r
o
rs of
S
L
AM
base
d on
t
h
e
geese
PSO in a
ngle
Fi
gu
re 8.
Locat
i
on
e
r
r
o
rs of
S
L
AM
base
d on
t
h
e
fraction
a
l-o
r
d
e
r Darwin
ian PSO i
n
an
g
l
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Researc
h
a
n
d
ap
pl
i
c
at
i
o
n o
n
f
r
act
i
o
n
a
l
-
or
de
r
D
a
rw
i
n
i
a
n
P
S
O
b
a
se
d a
d
a
p
t
i
ve
ext
ende
d …
(
Z
hu
Qi
-
g
ua
ng)
25
1
As i
s
sh
o
w
n i
n
Fi
gu
re 3 t
o
Fi
gu
re 8
,
com
p
ar
i
ng t
h
e
pose (or angle) error e
s
tim
a
ted by fuzzy adaptive
ex
tend
ed Kal
m
an
filter SLAM algo
rith
m
b
a
sed
on
th
e fraction
a
l-o
r
d
e
r
Darwi
n
ian
PSO
an
d
fu
zzy
ad
ap
tiv
e
ex
tend
ed
Kalman
filter SLAM alg
o
rith
m
b
a
sed
on
th
e ge
ese algo
rith
m
,
th
e m
a
x
i
m
u
m
o
f
ro
bo
t l
o
catio
n errors
i
n
X di
r
ect
i
on
has dec
r
ease
d
f
r
om
2.5 t
o
1.
5,
whi
l
e
m
a
xim
u
m
val
u
e of e
r
r
o
rs i
n
Y di
rect
i
on
has d
r
op
pe
d
from
1.
5 t
o
1.
0, an
d
t
h
e
m
a
xim
u
m
angl
e er
ro
r has
decrease
d
fr
o
m
0.07 t
o
0
.
0
6
,
t
h
e per
f
o
r
m
a
nce of
fuzzy
a
d
apt
i
v
e
ex
tend
ed Kalm
an
filter SLAM alg
o
rith
m
h
a
s b
e
en
im
p
r
oved
g
r
eatly.
5.
CO
NCL
USI
O
N
To
ov
er
co
m
e
t
h
e shor
tco
m
in
g
of
the inac
c
u
racy for the
pri
o
r c
h
ar
acteristics of system
noise and
observation
noise is inaccura
te in extended Kalm
an
filter
SLAM,
using fuzzy ad
a
p
tive extended
Ka
l
m
an
filter SLAM
alg
o
rith
m
,
th
e no
ises co
u
l
d b
e
fu
zzily
reg
u
l
ated on
line b
y
d
e
tecting
th
e d
i
fferences of
cova
ri
ance m
a
t
r
i
x
o
f
p
r
edi
c
t
i
on a
n
d act
ual
val
u
e
di
ffe
re
nc
e and ide
a
l covaria
n
ce m
a
trix, the
r
efore t
h
e effect
th
at ti
m
e
-v
aryin
g
n
o
i
ses
po
sed
on
filter stab
i
lity can
b
e
restrain
ed
. However, fu
zzy lo
g
i
c
n
o
t
o
n
l
y h
a
s a
lo
wer
accuracy,
but also lacks syst
e
m
atic para
m
e
ter design m
e
thods. For this
defect,
na
tural
selection in
biology
and
fractional calculus are
used to
im
p
r
o
v
e p
a
rticle o
p
t
i
m
izatio
n
,
an
d
fraction
a
l-o
r
d
e
r Darwin
ian
particle
swarm
alg
o
rith
m
is u
s
ed
t
o
o
p
tim
ize th
e k
e
y p
a
ram
e
ters
of
f
u
zzy
l
o
gi
c,
t
hus
re
g
u
l
a
t
i
on acc
uracy
of
fuzzy
lo
g
i
c co
u
l
d
b
e
i
m
p
r
ov
ed, as
well as th
e
p
e
rform
a
n
ce
o
f
Kal
m
an
filter SLAM.
Accord
in
g
t
o
th
e sim
u
latio
n
s
,
th
e p
e
rfo
r
m
a
n
ce o
f
fu
zzy adap
tiv
e ex
tend
ed
Kalm
an
filter SLAM algo
rith
m
b
a
sed
o
n
th
e fractio
n
a
l
-
ord
e
r
Dar
w
i
n
i
a
n PS
O has
bee
n
i
m
pr
o
v
ed
ob
vi
o
u
s
l
y
, and t
h
i
s
i
m
prove
d st
rat
e
gy
can b
e
ap
pl
i
e
d t
o
t
h
e o
p
t
i
m
i
zat
i
o
n
o
f
o
t
h
e
r
swarm in
tellig
en
ce al
g
o
rith
m
s
.
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Evaluation Warning : The document was created with Spire.PDF for Python.