TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4206 ~ 4
2
1
4
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.505
9
4206
Re
cei
v
ed
No
vem
ber 4, 20
13; Re
vised
De
cem
ber 3
0
,
2013; Accep
t
ed Jan
uary 1
9
, 2014
An SLAM Algorithm Based on Square-root Cubature
Particle Filter
Xuefen
g Dai,
Zuguo Ch
en
*, Chao Yan
g
, Laihao Jia
ng, Biao Cai
Coll
eg
e of Co
mputer an
d Co
ntro
l Eng
i
n
eeri
ng, Qiqih
a
r Uni
v
ersit
y
, No.
42
of W
enhua Str
eet, Jianh
an
District, Qiqiha
r, Heilo
ngj
ian
g
,
1610
06, Ch
ina
,
T
e
l (+
86)0452
-273
817
3
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: boss88
8
.coo
l
@
16
3.com
A
b
st
r
a
ct
T
he lack of th
e latest meas
u
r
ement infor
m
ation
a
nd th
e Particle ser
i
ou
s degra
dati
on
cause l
o
w
estimatio
n
pr
ec
ision
in
the tra
d
itio
n p
a
rticle
fi
lter
SLAM (s
imultan
e
o
u
s loc
a
l
i
z
a
t
i
o
n
a
nd
ma
ppi
ng). F
o
r so
l
v
e
this prob
le
m, a SRCPF
-
SLA
M
(square cu
b
a
ture partic
l
e filter si
multa
n
e
ous loc
a
li
z
a
t
i
o
n
and
ma
ppi
n
g
) is
prop
osed
in
thi
s
pa
per. T
h
e
a
l
gorit
hm fuses
the l
a
te
st mea
s
ure
m
e
n
t
infor
m
ati
on
in
the s
t
age
of the
pri
o
r
distrib
u
tion
up
dated
of th
e p
a
r
ticle filter
SLA
M
. It
desig
ns i
m
p
o
rtanc
e d
e
n
s
ity functio
n
by
SRCKF
(Sq
u
a
r
e
-
root Cub
a
ture
kal
m
a
n
filter) that is mor
e
clo
s
e to t
he posterior d
ensity, a
nd it
sprea
d
s the squ
a
re root
of
state covar
i
a
n
c
e. So, th
e a
l
gorith
m
ens
ur
es the
sy
mm
e
t
ry and
the
p
o
sitive
se
mi-
d
efinite
ness
of
the
covari
ance
ma
trix and i
m
prov
es nu
meric
a
l e
s
timati
on pr
eci
s
ion a
nd stabi
li
ty. T
he simulat
i
on res
u
lts sho
w
that the pr
op
os
ed a
l
g
o
rith
m h
a
s hi
gh
er accu
racy of
the stat
e esti
mati
on w
hen c
o
mp
ared
w
i
th the the P
F
-
SLAM (partic
l
e
filter si
multan
eous
loc
a
li
z
a
t
i
on a
n
d
map
p
i
ng) a
l
g
o
rith
m,
EPF
-SLAM (ex
t
end
particl
e fil
t
er
simulta
neo
us local
i
z
a
ti
on a
nd ma
pp
ing)
alg
o
rith
m
a
nd th
e UPF
-
SLAM (u
nsce
nted
particl
e
filter
simulta
neo
us l
o
cali
z
a
tio
n
an
d
map
p
i
ng) al
go
rithm.
Ke
y
w
ords
:
particl
e filter,
squar
e-root cu
bature k
a
l
m
a
n
f
ilter, simulta
neo
us loc
a
l
i
z
a
tion a
nd
map
p
in
g,
m
o
bile robot
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
The mo
bile robot SLAM is that the rob
o
t
does l
o
calization an
d buil
d
s in
creme
n
tal map
on the
ba
si
s
of the
po
sition e
s
timation
and th
e
o
b
se
rvation data of
the se
nor in
the
un
kno
w
n
environ
ment.
It is the ba
si
s that the robo
t finish
e
s
Env
i
ronm
ent det
ection, n
a
vig
a
tion an
d target
tracking. Th
e
SLAM is se
en as the
ke
y that
the robot reali
z
e completely aut
onomy. So, the
mobile robot
has be
en the most po
p
u
lar p
r
oj
e
c
t, the most ab
unda
nt proje
c
t and the
most
pione
erin
g project in the ro
bot resea
r
ch field [1].
The extend kalman filter was first p
r
opo
sed
by smith
and chee
sem
an [2]. This algorith
m
is
widely
used in the SLA
M
. It estimat
e
s the po
sterior probability
density
of the pose and t
h
e
environ
ment feature of ro
b
o
t. The major disadvant
a
g
e
is that this algorith
m
has more amou
n
t
s
of com
putatio
ns. And th
e a
l
gorithm
mu
st assu
re
that t
he inp
u
t noi
se and th
e o
b
s
ervatio
n
noi
se
of system
sh
ould o
bey a
g
aussia
n
di
stri
bution [3]. In
recent yea
r
s,
the pa
rticle fil
t
er was u
s
ed
as
a ne
w m
e
tho
d
to d
eal
with the SLAM
probl
em
by Murp
hy an
d
dou
cent [4].
The al
go
rith
m ca
n
build featu
r
e
map a
nd g
r
id
map a
c
co
rdi
ng to the
nee
ds. And thi
s
algorith
m
can
effectively so
lve
the proble
m
o
f
data
co
rrel
a
tion. But the
choice of
im
po
rtance d
e
n
s
ity function
affects the
pa
rticl
e
filter perfo
rm
ance be
ca
use the pa
rticl
e
need
be ex
tracted f
r
om th
e impo
rtan
ce
den
sity funct
i
on
[4]. However, the state
tran
sition
prio
r di
st
rib
u
t
ion which
doe
s n
o
t contain th
e l
a
test
measurement
data is a
s
the importa
nce
density funct
i
on in the tra
d
itional pa
rticle filter. So, the
algorith
m
introdu
ce
s larg
e
r
weig
ht variance and
it can’t approximate poste
rior
prob
ability very
well [5-6]. Espe
cially, wh
en the me
asurem
ent dat
a app
ear i
n
the tail end
of the tran
sition
probability distribution or
the likelihood function excessive
concentration, compared with
transition probability
di
stribution,
the particle filter may
failure.
For
solving the above probl
ems in
the traditional
particle filter
very well, a lot of
resea
r
ch
schola
r
s h
a
ve done la
rge
amount of wo
rk
in term
s of the ch
oice of importa
nce sa
mpli
ng fun
c
ti
on. The
UKF
has b
een
used to de
sign
the
importa
nce d
ensity fun
c
tio
n
of th
e p
a
rt
icle
by [7], a
nd they
hav
e p
r
op
osed t
he
UPF-SLA
M
algorith
m
. When the ro
bot
builds ma
p, the algo
ri
thm redu
ce
s the n
eede
d parti
cl
e numbe
r. But, it
need
s to ascertain the three un
kno
w
n
para
m
eter
s i
n
scaled u
n
scente
d
tran
sf
ormatio
n
(SUT)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An SLAM Algorithm
Based
on Square-ro
ot
Cubatu
r
e
Particle Filte
r
(Xuefen
g Dai
)
4207
according to
experie
nce. The ch
oice of para
m
eter
di
rectly affects the accu
ra
cy of the SLAM [8-
9].
In ord
e
r to
m
a
ke
up fo
r th
e defe
c
ts
of the SLAM al
g
o
rithm a
nd to
improve
the
pre
c
isi
on
of the mobil
e
robot SLAM
algorith
m
, the
SRPF-S
LAM
algorith
m
is
prop
osed in t
h
is p
ape
r. Th
e
algorith
m
ba
sic id
ea is t
hat the parti
cle filt
ering f
r
ame
w
o
r
k int
r
odu
ce
s the
latest SRCKF
nonlin
ear filtering
algo
rith
m an
d fu
se
s
the late
st
o
b
s
ervatio
n
dat
a. It gen
erate
s
the
imp
o
rta
n
ce
den
sity functi
on of
parti
cle
filter by u
s
ing
the
S
RCKF
algorith
m
. Th
e impo
rtan
ce
den
sity functi
on
is more cl
ose to the
sy
stem posteri
o
r probability dist
ribution ac
cording to giving
consi
deration to
nonlin
earity a
nd non
-ga
u
ssian ch
ara
c
te
ri
stics. And,
the algorith
m
h
a
s spread th
e squ
a
re
root
o
f
state cova
ria
n
ce. So, it ascertai
ns th
e sy
mmetry
and the h
a
lf posit
ive definitivene
ss of
covari
an
ce
matrix. It greatly improv
es th
e pe
rfo
r
man
c
e
of the
standa
rd
parti
cle filte
r
, and
improve
s
the
pre
c
isi
on an
d
the stability
of the mobile robot SLAM algorithm.
2. Square-root Cubature
Particle Filter
2.1. Cuba
tur
e
Rules
To calculate t
he nonlinear tran
sfer probability density
of the Gaussi
an di
stribution is the
most im
porta
nt step th
at it Implement B
a
yesia
n
filt
eri
ng un
de
r ga
u
ssi
an field.
T
hat is to
say, it
will cal
c
ul
ate
the gaussia
n
weig
hted i
n
tegral
of th
e nonlin
ear f
unctio
n
[10]. The gau
ssi
a
n
weig
hted inte
gral of the no
nlinea
r functi
on ca
n be giv
en by:
()
(
;
,
)
R
I
fx
N
x
d
x
(1)
H
e
re, x spec
ifies
n
x
dimen
s
i
onal ve
ctor.
f
(
) spe
c
ifie
s t
he no
nline
a
r
function.
N
（：
）
specifies
the Ga
ussia
n
dist
ribution.
R
spe
c
ifie
s t
he d
o
main
of
integ
r
ation. The analytic value
of
i
n
te
gral
function i
s
o
b
tained
difficultly above t
he integ
r
al f
unctio
n
. Usin
g equ
al
wei
ghts 2
n
x
point
nume
r
ical integral
re
sults
are cl
ose to t
he gau
ssian
weig
hted inte
gral in cubatu
r
e.
2
11
2
1
()
(
[
1
]
)
(
)
22
xx
x
nn
x
Ni
i
i
ii
x
n
If
f
w
f
n
(2)
There, n is the state dimen
s
ion.
i
and
i
w
can
be obtain
ed b
y
:
x
i
i
x
i
n
w
n
2
1
]
1
[
2
2
i=
1
，
…
，
2n
x
(3)
The sq
ua
re-root cub
a
ture
kalma
n
filter doe
s not nee
d to calculate
the Jacobia
n
matrix.
It calculate
s
the con
d
itional tran
sition
prob
ab
ility den
sity by using the
cub
a
ture. So, this
algorith
m
can
be
clo
s
e to
the third o
r
de
r accu
ra
cy
. Th
e a
c
cura
cy of
the al
gorithm
is
highe
r th
a
n
the extend
kalman filter.
The p
ape
r [1
1] gives
det
ai
led the
cal
c
ul
ation ste
p
s of
the squa
re
-root
cub
a
ture
kal
m
an filter. Th
e main cal
c
ul
ation formul
a is as follo
ws.
2.2. Square-r
oot Cu
batu
re Particle Filter
The
parti
cle f
ilter is that th
e integ
r
al
op
erat
ion
tra
n
sl
ates i
n
to the
summ
ation
of sa
mple
points
by usi
ng the Mo
nte
Carl
o metho
d
. Thereby,
it obtain
s
the recu
rsive B
a
yesia
n
e
s
timation
whi
c
h h
a
s th
e State mini
mum vari
ance. The al
gorit
hm ne
eds ob
tain dire
ctly sample p
o
int f
r
om
the po
steri
o
r pro
bability
den
sity of st
ate. But,
the analytic fo
rm of the p
o
sterio
r p
r
o
b
a
b
ility
den
sity can
not be obtai
ned. So, it is difficult to
sampl
e
for p
a
rticle. It req
u
ire
s
u
s
ually
an
importa
nce d
ensity fun
c
tio
n
(0
,
1
)
(
1
,
)
(/
,
)
ii
i
kk
k
xx
y
which i
s
si
milar to t
he posterior
probability
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4206 – 4
214
4208
distrib
u
tion
a
nd it i
s
sampl
ed. The
imp
o
r
tance d
e
n
s
ity function
is
easy to
samp
le. But it doe
s not
u
s
e th
e
la
te
s
t
me
as
ur
e
m
en
ts
da
ta
. O
n
ly a few min
o
r
ities of p
a
rticles
have l
a
rg
er
weig
hts after
several time
s iteration
s
,
and m
o
st
of the p
a
rt
icl
e
s di
e rapidly
.
Thereby, this
exist g
r
e
a
t
differen
c
e
s
b
e
twee
n the
collectio
n of t
he p
a
rticl
e
s
and th
e real
po
sterio
r p
r
obability fun
c
tion
gene
rated
sample
s. In o
r
de
r to ma
ke the im
po
rtance den
sity function m
o
re cl
ose to the
poste
rio
r
pro
bability densit
y, the importance den
sity
function n
eed
s to con
s
id
er
the influen
ce
of
the latest me
asu
r
em
ent value. It makes more parti
cl
e moves into
the higher li
kelih
ood fun
c
tio
n
value area [1
2]. So, the state estimatio
n
inco
rp
orate
s
the late
st measurement
value to de
sign
the importa
nce den
sity function by usi
n
g the S
RCKF
algorithm. A SRCPF al
gori
t
hm is propo
sed
in this pap
er.
The SRCPF
algorithm calcul
ates the
m
ean and
the variance of the importan
c
e
probability density function by
using t
he SRCPF
algorithm,
and produces
a
new posterior
prob
ability density distrib
u
tion. Mean
whi
l
e the
algorit
hm gene
rate
s new p
a
rticl
e
s by the new
poste
rio
r
pro
bability den
si
ty distributio
n and
comp
l
e
tes the
state estimatio
n
. Each pa
rticl
e
sampli
ng p
o
ints have diff
erent G
a
u
ssi
an dist
ri
butio
n. However,
any non
-ga
u
ssi
an di
strib
u
t
ion
can b
e
co
mp
ose
d
of a se
ri
es of differe
nt Gaus
sia
n
distribution com
b
ination
s
. It is re
asonabl
e to
assume
the
importa
nce d
ensity fun
c
tio
n
is Ga
ussia
n
di
stributio
n
[13]. The
n
e
w im
po
rtan
ce
probability density functi
on
can be defined as:
/
0,
1
1
.
/
(/
,
)
(
,
)
i
ii
i
i
kk
kk
k
k
k
x
xy
x
s
(4)
The detail ste
p
of the SRCPF algorithm
is as follo
ws:
Step 1:
Whe
n
k=0, the al
gorith
m
Ch
oo
se th
e
initial p
a
rticl
e
set
0
i
X
from the prior probability
distrib
u
tion P(X
0
). And comp
ute their expe
ctation
s
and
covari
an
ce.
()
0
0
[]
i
i
X
EX
(5)
00
00
0
[(
)(
)
]
ii
ii
i
T
P
E
XX
XX
(6)
Here, i=1, 2,
…, m
Step 2:
Whe
n
K>0, the alg
o
rithm
desi
g
n
s
the
sampling
den
sity function of
the pa
rticle
filter by
usin
g the S
R
CPF alg
o
rith
m. Each
pa
rticle i
s
u
pdat
ed by u
s
in
g
the SRCPF
algorith
m
in t
he
particl
e set at
the importan
c
e pa
rticle
sa
mpling.
Comp
uting cubature point:
()
()
1/
1
1
/
1
ˆ
ii
kk
k
k
XX
(7)
()
()
()
,
1
/1
1
/
1
1
/1
ˆ
ii
i
nk
k
k
k
n
k
k
XS
X
(8)
Whe
r
e n is th
e state dimen
s
ion
Time upd
atin
g:
()
()
,/
1
,
1
/
1
()
ii
nk
k
n
k
k
Xf
X
(9)
()
()
*
/1
,
/
1
1
1
ˆ
m
ii
kk
n
k
k
i
XX
m
(10)
()
()
*
/1
,
/
1
([
])
ii
kk
n
k
k
k
ST
r
i
a
X
Q
(11)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An SLAM Algorithm
Based
on Square-ro
ot
Cubatu
r
e
Particle Filte
r
(Xuefen
g Dai
)
4209
()
*
(
)
*
()
()
*
(
)
(
)
*
()
/1
1
,
/1
/1
2
,
/1
/1
,
/
1
/
1
1
ˆˆ
ˆ
[]
ii
i
i
i
i
i
kk
k
k
kk
k
k
kk
m
k
k
k
k
XX
X
X
X
X
X
m
(12)
Measure upd
ating:
The latest ob
servatio
n info
rmation i
s
ble
nded into the
algorith
m
(
)
()
()
,/
1
/
1
/
1
ˆ
ii
i
nk
k
k
k
i
k
k
XS
X
(13)
()
()
,/
1
,
/
1
()
ii
nk
k
n
k
k
Zh
X
(14)
()
()
/1
,
/
1
1
1
ˆ
m
ii
kk
n
k
k
i
ZZ
m
(15)
()
()
()
()
/1
1
,
/1
/1
2
,
/
1
/1
,
/
1
/
1
1
ˆ
ˆˆ
[]
ii
i
i
kk
k
k
kk
kk
k
k
m
k
k
k
k
ZZ
Z
Z
Z
Z
m
(16)
()
()
,/
1
/
1
{[
]
}
ii
z
z
kk
kk
k
Sq
r
R
(17)
()
()
()
()
()
()
()
/1
1
,
/1
/1
2
,
/1
/1
,
/
1
/
1
1
ˆ
ˆˆ
[]
ii
i
i
i
i
i
kk
k
k
kk
k
k
kk
m
k
k
k
k
XX
X
X
X
X
X
m
(18)
2.1.3. State
Upda
ting
()
()
()
,/
1
/
1
/
1
ii
T
i
x
zk
k
k
k
k
k
PX
(19)
()
()
(
)
(
)
/1
,
/
1
,
/1
,
/
1
()
/
ii
T
i
i
kk
x
z
kk
z
z
k
k
z
z
k
k
KP
s
s
(20)
()
()
()
()
()
//
1
/
1
ˆˆ
()
ii
i
i
i
kk
kk
k
k
kk
XX
K
Z
Z
(21)
()
()
()
,/
1
ii
i
kz
z
k
k
UK
S
(22)
()
()
()
//
1
{,
,
1
}
ii
i
kk
k
k
S
c
hol
updat
e
S
U
(23)
The algo
rith
m rege
nerates the pa
rticle
set ba
sed
on the state estimation
and the
covari
an
ce
matrix of the
SRCKF
alg
o
rithm, an
d
cal
c
ulate
s
th
e impo
rtan
ce
weig
hts of
each
particl
e in the
particle
set.
The impo
rtan
ce weight
s are norm
a
lized.
()
()
()
()
()
1/
ˆ
ˆ
~(
/
,
)
(
,
)
ii
i
i
i
kk
k
k
k
k
k
XX
X
Y
N
X
S
(24)
()
()
()
1
()
()
1
ˆˆ
(/
)
(
/
)
ˆ
(/
,
)
ii
i
i
kk
k
k
k
ii
kk
k
pY
X
P
X
X
XX
Y
(25)
()
()
1
/
m
ii
i
kk
k
j
(26)
Step 3: Particle re-sam
plin
g
Firstly, determine wh
ether it
needs sa
mples a
c
co
rd
ing to the effective parti
cl
e numbe
r.
If it needs sa
mples, th
e al
gorithm
re
-sa
m
pled
parti
cl
es
by usi
ng t
he pa
rticl
e
co
mbination
me
thod
[14]. The re-sampled p
a
rti
c
les are
1
,
m
j
sj
j
xn
. Otherwise
，
the alg
o
rithm do
state estimation.
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ISSN: 23
02-4
046
TELKOM
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KA
Vol. 12, No. 6, June 20
14: 4206 – 4
214
4210
Comp
uting State estimatio
n
:
()
11
1
mm
j
j
j
kj
k
s
jj
n
X
nX
X
NN
(27)
//
/
1
1
()
(
)
N
j
jT
kk
k
k
k
k
kk
j
PX
X
X
X
N
(28
)
Accordi
ng to t
he new pa
rticle state estim
a
tion, vari
ance
matrix
and t
he
above probability
model, re
peat
the steps a
b
o
ve for the ne
xt filter.
3. An SLAM
Algorithm Based on Squ
a
re-r
oot
Cub
a
ture Par
t
icl
e
Filter
In ord
e
r to
overcome
problem
s
whi
c
h the SLAM
algo
rithm b
a
se
d o
n
con
v
entional
particl
e filter
have large
r
calcul
ated am
ount an
d
pa
rticle d
egradati
on, the thou
g
h
t of the SRCKF
algorith
m
lea
d
in the pa
rticle filter to i
m
pr
ove the
sampli
ng p
r
o
c
e
ss. It enri
c
hes th
e pa
rticle
sampl
e
s in t
he process
of parti
cl
e filter re
sam
p
li
ng. The alg
o
rithm integ
r
ates the lat
e
st
observation
data at the prio
r distri
but
ion
update
stage and de
sign
s the im
portan
c
e d
e
n
s
ity
function
by the SRCKF al
gorithm.
It make
s
state e
s
timation clo
s
er to the p
o
sterior
pro
babi
lity
den
sity of the sy
stem
state.
The alg
o
rithm also
has
spre
ad
t
he
squa
re
root of the
st
ate
covari
an
ce. It can insure t
he symmetry
and the
positive semi-def
initene
ss of the cova
rian
ce
matrix. Thus,
it can improve the numerical accu
racy and the stabilit
y of the SLAM algorithm.
The state ve
ctor
k
X
sho
u
ld i
n
clu
de the p
o
sition info
rm
ation
s
G
and the
environ
menta
l
information
N
P
of
the mobile ro
bot in the mobile rob
o
t SLAM algorithm
.
T
N
s
k
P
G
X
,
(29)
Her
e
,
,,
s
kk
k
Gx
y
spe
c
ifie
s the po
sition
coordinate
a
nd the angl
e coo
r
din
a
te of the
mobile
un
der global coo
r
di
nate.
N
P
is N env
ironm
ent co
ordinate set.
The b
a
si
c th
o
ught of th
e S
L
AM alg
o
rith
m ba
sed
on
square-root
cu
bature
pa
rticl
e
filter i
s
to estimate the mobile
ro
bot trajecto
ry
and the
po
sterio
r pro
babil
i
ty density of
the environ
m
ent
map on the b
a
si
s of the perceptual info
rmation
of the robot in the
environme
n
t and the upd
ate
informatio
n of the robot po
se. The sp
ecifi
c
step
s of the
algorithm
s are as follo
ws:
1)
The alg
o
rith
m pro
d
u
c
e
randomly the
parti
cle
set
whi
c
h i
s
con
s
tituted by the N
particl
e and t
he parti
cle weight is
0
W
. Then, this particle i
s
initialized b
y
using (5
)-(6
) equatio
ns.
2)
Particle
s Pre
d
icted: The a
l
gorithm
cal
c
ul
ates the inf
o
rmatio
n vector and matri
x
of
the particl
es
according to
control input
k
u
and the ro
bot
position di
stri
bution at K time.
3)
Observation
and Data
Correlatio
n
:
T
he algo
rithm cal
c
ul
ates the p
a
rticle
corre
s
p
ondin
g
feature
poi
nt coo
r
din
a
te
s an
d ma
ke
s it to asso
ci
ation with a
landma
r
k in the
environ
ment l
andma
r
k set. Then, it ma
ke
s the
ob
se
rva
t
ion ne
w info
rmation to
associate
with th
e
estimation
m
ap at th
e K-1 time by
u
s
ing
the
dat
a a
s
soci
ation
method
of t
he mini
mizati
on
observation probability function.
4)
Particle
s
Up
d
a
ted: When
the
robot
obt
ai
ns the
ne
w ob
servatio
n
feature
point
s o
r
the ro
bot p
r
e
d
icted
po
se
at K-1 time
compa
r
i
ng
wit
h
one
at K ti
me have
gre
a
t ch
ange
s, t
h
e
SRCPF al
gorithm update
s
particl
e by u
s
ing
(7)-(1
3
)
equatio
ns. It will cal
c
ul
ate
the informati
on
vector a
nd m
a
trix and the
importan
c
e
weig
ht of
the each p
a
rti
c
le at K time
and the pa
rti
c
le
importance weights
will be
norm
a
lized.
5)
The al
gorith
m
re
sam
p
le
parti
cle in
the pa
rticle
set a
c
cordin
g to the
pa
rticle
importa
nce weights
j
k
and
removes the
small wei
ghts
particl
e in th
e
parti
cle
set a
nd reserve
s
the big wei
g
h
t
s parti
cle in the parti
cle se
t.
6)
The al
gorith
m
upd
ates fe
ature i
n
form
a
t
i
on in the
m
ap by u
s
in
g (27)-(2
8)
equ
a
t
ions
and ma
ke
s th
e co
rrelation
failure o
b
serv
ation in
format
ion a
s
the ne
w feature info
rmation
adde
d
to the map.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An SLAM Algorithm
Based
on Square-ro
ot
Cubatu
r
e
Particle Filte
r
(Xuefen
g Dai
)
4211
4. The Experimental Con
c
lusion and
Analy
s
is
4.1. Experiment Mod
e
ling
Before th
e S
L
AM sim
u
lati
on exp
e
rime
nt, we
need
to build
a
system mo
del
for the
mobile
rob
o
t. The e
s
tabli
s
he
d mo
del
s mainly in
clu
de sy
stem
model, robot
locatio
n
mo
del,
control co
m
m
and mod
e
l, environme
n
t map model,
robot motion
model, sen
s
or mea
s
u
r
em
ent
model an
d sy
stem noi
se m
odel. In this p
aper, the Bail
ey SLAM model is u
s
ed [1
5].
(1) T
he motio
n
model can
be obtain
ed b
y
:
,1
,
1
,
1
,1
,
1
,
1
,1
,1
1
1
,
co
s
(
)
=
=
c
o
s
(
)
sin
Vx
k
V
x
k
k
V
k
k
Vy
k
V
y
k
k
V
k
k
Vk
Vk
k
k
Vk
xx
T
V
x
xx
T
V
x
xT
V
x
B
x
(30)
Input:
,
Vk
x
spe
c
ifies th
e p
o
se
o
f
the robot
at
time k.
T
spe
c
i
f
ies the
samp
ling time
of
the dead
re
ckoni
ng sen
s
o
r
s.
k
V
specifie
s
the velocity of the robot.
k
is rudd
er an
gle
.
B is tw
o
interaxial wheelbas
es
. Output:
,1
Vk
x
spe
c
ifies
the pose of t
he robot at time k+1.
(2) O
b
servati
on model
ca
n
be obtaine
d by [16]:
22
,,
,
,
,
()
(
)
=
=
ar
ct
an
iV
x
k
iV
y
k
i
k
iV
y
k
Vk
iV
x
k
xx
y
x
r
yx
i
x
xx
z
(31)
Input:
,
ii
x
y
sp
ecifi
e
s th
e p
o
sitio
n
coordinate
s
of dete
c
ted
the
i
th landmark features
.
Output:
i
r
and
i
respe
c
tively spe
c
ifie
s the
rang
e of the
i
th landma
r
k
feature
relate
d to the
robot an
d an
gle of the
i
th
landma
r
k feature related to
the robot direction.
4.2. Experimental Analy
s
es
In this pape
r, it adopts resp
ectively the
PF algori
t
hm, the EPF algorithm,
the UPF
algorith
m
an
d the SRCPF
algorithm in
the SL
AM experim
ent, an
d comp
ares t
he experi
m
e
n
ta
l
results of fou
r
kin
d
s of alg
o
rithm. It co
mpares
m
a
inl
y
the deviation of the state
estimation
with
the re
al path
value, the
sta
t
e estimatio
n
covari
an
ce, t
he po
ste
r
ior
prob
ability distribution
and t
he
runni
ng tim
e
and
so
on. I
n
orde
r to m
a
ke
the
ex
pe
rimental
re
su
lts have
mo
re unive
rsality, it
take
s the av
erag
e re
sult
of the 20 time rep
eat
ed th
e experim
ent
s final re
sult
were compa
r
i
n
g
analyzed.
Figure 1. Re
sults of the PF-SLAM Algori
t
hm, the EPF
-SLAM Algori
t
hm, the UPF-SLAM
Algorithm an
d the SRCPF
-
SLAM Algori
t
hm unde
r the Gau
ssi
an Noise
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
Ti
m
e
F
i
lte
r
e
s
tim
a
te
s
vs
T
r
u
e
s
t
a
t
e
l
a
ndm
ar
k
T
r
ue x
P
F
e
s
ti
m
a
te
E
K
F
-
PF
e
s
ti
m
a
te
U
K
F
-
PF
e
s
ti
m
a
te
S
R
C
K
F
-
PF
e
s
ti
m
a
te
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4206 – 4
214
4212
Figure 1 i
s
th
e chart
betwe
en
state e
s
timation a
nd th
e re
al p
a
th va
lue deviatio
n
that the
PF algo
rithm,
the EPF al
g
o
rithm, the
UPF algo
rithm
and th
e SRCPF algo
rithm
are
used i
n
t
h
e
mobile
ro
bot
SLAM. Fro
m
Figu
re
1
sho
w
n, th
e d
e
viation of th
e PF
algo
rith
m an
d the
UPF
algorith
m
bet
wee
n
the
state estim
a
tion
and the
re
al va
lue i
s
g
r
eate
r
du
ring th
e fi
rst h
a
lf pe
riod
of
the mobile ro
bot SLAM. The deviation
of the PF
algorithm bet
we
en the state
estimation a
n
d
the
real valu
e is the gre
a
test
. The deviati
on of
the E
P
F algorith
m
and the S
R
CPF alg
o
rith
m
betwe
en the state estimati
on and the re
al value is
rel
a
tively less. Among then, the deviation of
the SRCPF a
l
gorithm
bet
ween th
e
state
estimatio
n
a
nd the
re
al v
a
lue i
s
the
mi
nimum. So, t
h
e
estimation
a
c
curacy
of the
SRCPF
algo
ri
thm is th
e hi
g
hest. T
he e
s
ti
mation a
c
cu
racy of th
e EPF
algorith
m
i
s
slightly poo
rer
than
that of
the
SRCPF al
gorithm, but
t
he e
s
timation
accu
ra
cy of t
h
e
EPF algo
rith
m is
high
er t
han th
at of the
UPF alg
o
rithm and
the
PF algo
rith
m. The
estim
a
tion
accuracy
of the
UKF alg
o
rithm is
nea
r t
o
that of
the
EPF algo
rith
m, but the
estimation a
c
cu
racy
of the UKF
al
gorithm i
s
hi
g
her th
an that
of the
PF alg
o
rithm. Th
e e
s
timation
accura
cy of the
PF
algorith
m
is the wo
rst.
The d
e
viation
of the fo
ur
ki
nds of alg
o
rit
h
ms
between
the
state e
s
t
i
mation a
nd t
he real
value sh
ows
a trend of de
cre
a
si
ng du
ri
ng the se
co
n
d
half peri
o
d
of the mobile rob
o
t SLAM.
Among th
en,
the d
e
crea
sing rate of t
he d
e
viation
of the EPF
algo
rithm
b
e
twee
n the
state
estimation a
n
d
the real value is larger.
It can
be see
n
from Figure 1, the deviation of the PF
algorith
m
bet
wee
n
the stat
e estimation
and the re
al
value is big
g
e
r than that o
f
the other three
kind
s of algo
rithms. The de
viation of the SRCP
F alg
o
ri
thm betwee
n
the state esti
mation and th
e
real val
ue i
s
t
he minim
u
m.
The d
e
viation
of t
he
UPF a
l
gorithm
between the
state
estimation
an
d
the real value
is relatively smaller tha
n
that of the EPF
algorith
m
s.
Hen
c
e, the e
s
timation a
ccura
cy of the SRCP
F al
gorithm is the hi
ghe
st. The e
s
timation
accuracy of t
he UPF al
go
rithm is
sligh
t
ly poor
er th
an that of the SRCPF
al
gorithm, b
u
t the
estimation a
c
curacy of the
UPF algorith
m
is hi
ghe
r than that of the EPF algorithm and the
PF
algorith
m
. The estimation
accuracy of the EPF al
gori
t
hm is poo
r, but the estim
a
tion accu
ra
cy of
the EPF algorithm is high
er than that of the PF
algorithm. The e
s
timation a
c
cura
cy of the
PF
algorith
m
is t
he worst. Th
e deviation
of the SR
CPF
algorith
m
bet
wee
n
the
state estimatio
n
and
the real valu
e is always
the minimum
during
all the pro
c
e
s
se
s of the mo
bile rob
o
t SLAM.
Accordi
ng on this, It may obtai
ns that
the state estimation
and the stabilit
y of the SRCPF
algorith
m
is b
e
tter than tha
t
of the other three ki
nd
s of algorithm
s.
Figure 2. The
State Estimates Cova
rian
ce of t
he SRCPF Algorithm
on the X, Y Coordi
nate
s
of
the Mobile Robot Position
with the State Estima
tes Covarian
ce of the Late
s
t UPF Algorithm o
n
the X, Y Coordinate
s
of the Mobile
Ro
bo
t Position und
er the Ga
ussi
an Noi
s
e
It compa
r
e
s
the state
e
s
timates
cova
riance of the
SRCPF
alg
o
rithm
on th
e X, Y
coo
r
din
a
tes o
f
the mobile robot po
sition
with t
he stat
e estimate
s covarian
ce of
the latest UP
F
algorith
m
on
the X, Y coordi
nate
s
of
the mobile
robot p
o
sitio
n
in Figu
re
2. As Figu
re
2
experim
ental
re
sults, it
shows the
foll
owin
g an
alysis results. From the al
go
rithm estimati
on
pre
c
isi
on an
alysis, the state estimate
s cova
ri
an
ce
of the SRCPF algo
rith
m for the X,
Y
coo
r
din
a
tes o
f
the mobile robot po
sition
is relati
vely smaller tha
n
that of the UPF algorithm. T
he
state e
s
timat
e
s
cova
ria
n
ce of th
e S
R
CPF algo
rithm
for th
e X
co
ordin
a
tes of t
he m
obile
ro
bot
positio
n is 0.
5m aro
und
smaller tha
n
that of
the UPF algorithm.
The state e
s
ti
mates
covari
ance
of the SRCP
F algo
rithm f
o
r the
Y coo
r
dinate
s
of the mo
bile
ro
bot po
sition i
s
0.1m
arou
nd
smalle
r than t
hat of the UP
F algo
rithm. The e
s
ti
matio
n
accu
ra
cy of the
SRCPF algorith
m
for the
10
0
10
1
10
2
0
0.
5
1
1.
5
2
ti
m
e
va
r
(
x)
c
o
v
a
r
i
a
n
c
e
o
f
th
e
X
-
a
x
i
s
s
t
a
t
e
e
s
ti
m
a
te
s
S
RCP
F
UP
F
10
0
10
1
10
2
0.
5
1
1.
5
2
ti
m
e
va
r(
y)
c
o
v
a
r
i
a
n
c
e
of
t
h
e
Y
-
a
x
i
s
st
at
e
est
i
m
a
t
e
s
S
RCP
F
UP
F
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An SLAM Algorithm
Based
on Square-ro
ot
Cubatu
r
e
Particle Filte
r
(Xuefen
g Dai
)
4213
coo
r
din
a
tes o
f
the mobile robot po
sition is highe
r
than
that of
the UPF algorithm.
The estimati
on
accuracy
of the SRCPF
al
gorithm
for th
e X coordinat
es
of the m
o
bile robot
po
sition i
s
hi
gh
er
than the esti
mation a
c
curacy of the SRCPF al
go
ri
thm for the Y coo
r
din
a
tes o
f
the mobile robot
positio
n.
From the alg
o
rithm e
s
tima
tion stability analysi
s
, the stability of th
e SRCPF alg
o
rithm is
better than that of the
UPF
algorithm. Fi
gure 3 i
s
the
posterior probability dist
ribution map of t
h
e
SRCPF al
go
rithm and the
UPF alg
o
rith
m. It can be
see
n
from
Fi
gure
3
, the po
sterio
r p
r
ob
a
b
ility
distrib
u
tion
map of the
SRCPF al
gorithm is gentl
e
r than that
of
the UPF algorith
m
. Th
is is
equivalent to
increa
sin
g
the pa
rticl
e
filter di
stributio
n ra
nge i
n
th
e sa
mplin
g p
r
ocess. So, t
h
e
gene
rating
p
a
rticle
sampl
e
by u
s
ing t
he SRCP
F
algorith
m
is
clo
s
er to the
true
po
steri
o
r
prob
ability de
nsity dist
ribut
ion than th
at by usin
g th
e UPF al
gori
t
hm duri
ng t
he mobil
e
ro
bot
SLAM.
Figure 3. The Posterior Probab
ility Distri
bution Map of
the
SRCPF
Algorithm and the UPF
Algorithm un
der the G
aussian
Noi
s
e
The data
con
t
rast
s of the system n
o
ise
and
the ob
servation noi
se unde
r the Gau
ssi
an
white noi
se condition is
sh
own in the Ta
ble 1. From the run
n
ing time cont
ra
st
，
the runni
ng time
of the UPF algorith
m
is the longe
st. The ru
nnin
g
time of the
SRCPF alg
o
rithm is relati
vely
sho
r
ter th
an
that of the UPF algorith
m
. The ru
nnin
g
time of the
EPF algorit
hm is
relativ
e
ly
sho
r
ter th
an t
hat of the SRCPF alg
o
rith
m. The ru
nni
ng time of th
e PF algo
rith
m is the
sho
r
test.
Becau
s
e th
e SRCPF al
gorithm, the UPF algorith
m
a
nd the EPF a
l
gorithm joi
n
respe
c
tively the
SRCKF alg
o
r
ithm, the UKF algorithm
and the EKF
algorithm i
n
the pro
c
e
s
s of gene
rati
ng
importa
nt de
nsity functio
n
.
Duri
ng the
pro
c
e
s
s of
th
e
estim
a
tion error co
ntra
st,
the
error of the
PF algo
rithm
is the
big
g
e
s
t. The
erro
r of
the S
RCPF
algorith
m
i
s
smaller than
that of th
e
UPF
algorith
m
. The error of the UPF algo
ri
thm is sm
all
e
r than that of the EKF a
l
gorithm. So, the
validity of the
SRCPF
-
SLA
M
algorithm i
s
verified.
Table 1. The
Data Contra
st of the Syste
m
No
ise and
the Observati
on Noi
s
e u
n
d
e
r the Ga
ussi
an
White Noise Condition
SLAM Run
time
precision
Gm/t
PF
1.3125s
First-Order Accur
a
te
5.2356
EPF
2.0741s
First-Order Accur
a
te
4.2651
UPF
4.8643s
Second-Order
A
ccurate
3.6952
RCPF
3.5271s
Second-Order
A
ccurate
3.1256
Here, Gm/t is
the root mean s
q
uare er
ro
r of the map estimation
re
spe
c
tively.
5. Conclusio
n
In this
pap
e
r
, the S
RCP
F-SLAM
alg
o
rithm i
s
propo
sed.
This algo
rithm
whi
c
h i
s
estimating
th
e robot
po
sition a
nd
upd
ating the
la
n
d
m
ark info
rma
t
ion ha
s th
e
more
effect
s
than
the PF al
gorit
hm, the EPF
algorith
m
a
n
d
the
UPF
al
g
o
rithm. T
he i
m
porta
nce d
ensity fun
c
tio
n
is
0
5
10
0
10
20
0
0.
5
1
P
a
r
t
i
c
l
e
F
i
l
t
er
(
U
K
F
pr
opo
sal
)
y
t
Ti
m
e
(
t
)
p(
y
t
|y
t-
1
)
0
5
10
0
10
20
0
0.
5
1
x
t
Ti
m
e
(
t
)
p(
x
t
|x
t-
1
)
0
5
10
0
10
20
0
0.
5
1
P
a
r
t
i
c
l
e
F
i
l
t
er
(
S
R
C
K
F
p
r
o
posal
)
y
t
Ti
m
e
(
t
)
p(
y
t
|y
t-
1
)
0
5
10
0
10
20
0
0.
5
1
x
t
Ti
m
e
(
t
)
p(
x
t
|x
t-
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4206 – 4
214
4214
the impo
rtan
t factor
of the pa
rticl
e
filter
s re
sam
p
l
e
.
Wh
ether can
de
sign
rea
s
on
ably
t
h
e
importa
nt de
nsity fun
c
tion
, it impact
s
di
rectly o
n
the
perfo
rman
ce
of parti
cle filt
er. Th
e alg
o
ri
thm
fuse th
e late
st mea
s
u
r
em
ent informati
on in th
e
sta
ge of the
pri
o
r di
strib
u
tio
n
upd
ated
of the
particl
e filter
SLAM. It designs im
po
rta
n
ce
den
si
ty func
tion by SRCKF
(Squ
a
r
e-root Cub
a
ture
Kalman Filter) that is mo
re
close to the
poste
rio
r
den
sity, and it sp
read
s t
he
squ
a
re root of st
ate
covari
an
ce.
So, the algo
ri
thm en
sures
the symme
t
r
y and the
po
si
tive semi-defi
n
itene
ss
of the
covari
an
ce
matrix and improve
s
nu
meri
cal e
s
ti
mation preci
s
ion a
nd sta
b
ility. And th
at the
SRCKF al
gorithm decrea
s
es the mo
bil
e
rob
o
t
SLAM algorith
m
runni
ng time
whe
n
co
mpa
r
ed
with the UPF
algorith
m
. So, the algorith
m
has
more
real-time
com
parin
g with th
e UPF alg
o
rit
h
m.
The paper
consi
ders t
he operational
efficiency, the accu
racy
and the
st
ability of these
algorith
m
s. T
he SRCPF a
l
gorithm i
s
a
better way
that can imp
r
ove pa
rticle
filter algorit
hm
related to the
PF algorithm,
the EPF algorithm and the
UPF algo
rith
m.
Ackn
o
w
l
e
dg
ements
This
wo
rk i
s
sup
porte
d by
the Postg
r
ad
uat
e technol
o
g
y innovation
proje
c
t Fu
nd
s of the
Qiqiha
r University, Heilon
g
jiang P
r
ovin
ce G
o
ve
rnme
nt, China, u
n
der
Gra
n
t YJSCX201
3-0
3
0
X.
The
autho
rs also
gratef
ully ackn
owl
edge
the
he
lpful comme
nts a
n
d
sug
gestio
n
s of
the
reviewers, wh
ich have im
p
r
oved the pre
s
entation.
Referen
ces
[1]
Bailey
T
,
Durrant-Why
te
HF. Si
multaneous localiz
ation and mapping (SLAM):
Part II, Sta
t
e of the art
.
IEEE Robotics
and Auto
matio
n
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z
i
ne.
2
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3): 108
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h
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YF
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D, et al. Sim
u
lta
neo
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w
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p
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p
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c
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h
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e
g
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M
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on to
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h
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a
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a
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