TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 9, September
2014, pp. 67
5
8
~ 676
3
DOI: 10.115
9
1
/telkomni
ka.
v
12i9.506
1
6758
Re
cei
v
ed
No
vem
ber 7, 20
13; Re
vised
Apr 29, 201
4; Accept
ed Ju
ne 2, 2014
Forward Position Solution of 3-RPS in-Parallel
Manipulator Based on Particle Swarm Optimization
Zhang Hongl
i*, Ren Tianti
an
Coll
eg
e of Elec
trical Eng
i
ne
eri
ng, Xinj
ian
g
Un
iversit
y
, Ur
umq
i
830
04
7, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: 1606
82
927
4
@
qq.com
A
b
st
r
a
ct
Particle sw
ar
m opti
m
i
z
at
io
n is
introd
uce
d
to
solv
e th
e pr
ob
l
e
m in t
h
is p
a
p
e
r. Instead
of
solvin
g
a
grou
p
of n
on-l
i
near
e
quati
ons
, forw
ard ki
ne
matics
is
so
lv
e
d
by
co
mputin
g the
extre
n
u
m
of a
functi
on.
An
d
accurate so
luti
ons can b
e
obt
ain
ed by the gl
oba
l and
l
o
cal
search
ing a
b
il
ities of adva
n
ce
d particl
e sw
ar
m
opti
m
i
z
at
ion. I
t
overco
mes the s
hortag
e
t
hat pr
ec
isi
on
is gre
a
tly i
n
fl
uenc
ed
by i
n
i
t
ial va
lu
es w
i
t
h
conventional numerical m
e
thods.
Calc
ulation results sh
ow that this new method
is sim
p
le, convenient, and
w
i
th genera
lity for solvin
g the para
lle
l mani
pu
lator forw
ard ki
ne
matics pr
obl
ems.
Ke
y
w
ords
: par
ticle sw
arm o
p
timi
z
a
tio
n
, para
l
lel
ma
nip
u
lat
o
r, forw
ard kine
matics
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
Comp
ared
wi
th se
rial me
chani
sm, the p
a
rallel
manip
u
lator i
s
wi
de
ly used i
n
ap
plicatio
n
fields, whi
c
h has
several
a
d
vantage
s su
ch as
hig
h
st
if
f
ness,
st
r
ong
load
-c
ar
ry
ing
ca
pa
cit
y
,
sm
all
self-weig
h
t/load ratio, fast
re
spon
se
speed,
ni
ce d
y
namic
p
e
rfo
r
man
c
e and so
o
n
.
Forward
kinem
atics is
to determi
ne
the po
sition a
nd o
r
ientat
ion
of the platform with give
n
limbs le
ngth
s
,
while
inverse
kin
e
mati
cs i
s
to
dete
r
min
e
the
limb
s
l
ength
s
fo
r gi
ven po
sition
and
orie
ntation of
the platform.
Contra
ry to serial me
ch
anism,
inve
rse ki
nemati
c
s of parall
e
l
mecha
n
ism
is
relatively ea
sy to achieve
but the forward ki
nema
t
ics i
s
more
compli
cate
d. The an
alysi
s
o
f
forwa
r
d
kine
matics i
s
n
o
t only one of t
he ba
sic
pr
o
b
lems
of the
theory of pa
rallel me
cha
n
i
s
m
s
,
but also the
foundation f
o
r an
alysis
and synth
e
si
s of mechan
ism, sol
u
tion
of velocity and
accele
ration,
dynamics an
alysis a
nd error analy
s
is
. Rese
arche
r
s h
a
ve ca
rrie
d
o
u
t many studi
es
on nume
r
i
c
al
solutio
n
and
analytical
sol
u
tion, and ha
ve made a se
ries of p
r
og
re
sses [1
-3].
The e
s
sen
c
e
of forward
kin
e
matics i
s
to
sovle hi
ghly n
online
a
r e
qua
tions, an
d the
main
method
s in
clude an
alytic method a
nd
homotopy m
e
thod. Analytic metho
d
is to redu
ce the
unkno
wn nu
mbers of me
cha
n
ism
co
n
s
train
ed e
q
u
a
tions
by elimination, thu
s
the in
put-o
utput
equatio
n turn
s into a hi
ghe
r equ
ation
co
ntaini
ng o
n
ly one un
kn
own
numbe
r. The
advantage
s
of
this m
e
thod
are
no
nee
d
of initial val
ue, getting
a
ll the p
o
ssibl
e
solution
s
and
having
no
limitations of
som
e
spe
c
i
a
l me
chani
sm co
nfigur
ations.
Ho
weve
r, the elimin
ation p
r
o
c
e
s
s is
usu
a
lly diverse and
compli
cated, an
d so
lving sp
e
ed is lowe
r. Mean
while, ho
mot
opy method h
a
s
its advantag
e
s
such
as
no
need of elimi
nation an
d in
i
t
ial value, getting all po
ssi
b
l
e solutio
n
s, b
u
t
its solving
sp
eed is l
o
w.
Mean
while
some sch
o
lars do studi
es o
n
neu
ral net
work meth
od
for
forwa
r
d kine
matics
[4]. The nonlin
ea
r mappin
g
fro
m
the joint-variabl
e-spa
c
e
to the opera
t
ion-
variable
-
spa
c
e for th
e platf
o
rm i
s
a
c
com
p
lish
ed
with
neural n
e
two
r
k
after traini
ng an
d le
arni
ng
so
as to get
sol
u
tion
s ea
sier by avoi
d
i
ng
compli
cat
ed formula
d
e
rivation
and
pro
g
rammin
g
cal
c
ulatio
n. But the method
need
s furthe
r study to sol
v
e multiple so
lutions p
r
obl
e
m
.
PSO algorith
m
is a n
on-n
u
meri
c pa
rall
el algo
rithm
based on
po
pulation. Rel
a
tive to
traditional
evolutiona
ry alg
o
rithm
s
, PSO has fe
wer
a
d
justme
nt pa
ramete
rs
and
no com
p
licated
operation
s
of
auto re
gulati
ng, and it ha
s better
gl
ob
al sea
r
chin
g optimizatio
n
cap
a
city. In this
pape
r, PSO algorith
m
is use
d
for forward kin
e
mat
i
cs of 3RPS
parall
e
l mani
pulator, an
d the
result is
s
a
tis
f
ac
tory.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Forward Posi
tion Solution of 3-RPS in-P
arallel M
anip
u
lator Ba
sed
on… (Z
han
g Hon
g
li)
6759
2. For
w
a
rd
Kinematics o
f
3RPS Parallel Mechanis
ms
3RPS paralle
l mecha
n
ism, which wa
s p
r
opo
se
d
by Hunt in 19
83
, is a typical parall
e
l
mech
ani
sm
with fe
w d
e
g
r
ees of freed
o
m
. As Fi
gu
re
1
sho
w
s th
a
t
the lo
we
r pl
atform ABC (the
fixed platform) and the up
per platform abc (th
e
moving platform
) are both eq
ui
lateral trian
g
l
e
s
with their
circumra
diu
s
of R and
r re
sp
ectively
. The uppe
r platform is co
nne
ct
ed with cylin
der
linka
ge
s by spheri
c
al joi
n
ts, and the lower on
e is con
necte
d with t
he bottom by revolute joint
s
of
whi
c
h th
e a
x
es a
r
e
pe
rpendi
cula
r to
the axi
s
of
cylind
e
r li
nkage
s. The
u
pper platfo
rm is
prom
oted by cylinde
r linka
ges (revol
ute joints) m
o
vin
g
.
Figure 1. 3RPS Parallel Manipul
ator
O-XYZ is
the referenc
e
c
o
or
din
a
te system on th
e fixed platform, while
p-x
y
z is the
moving coordinate sy
ste
m
on the mo
ving platfo
rm
. The co
ordi
nates of the
uppe
r a
c
me
s are
r
e
pr
es
e
n
t
ed
a
s
: A
)
0
,
0
,
(
R
, B
)
0
,
2
3
,
2
(
R
R
, C
)
0
,
2
3
,
2
(
R
R
; the coordin
a
tes of the lowe
r
acme
s a
r
e re
pre
s
ente
d
as:
a
)
0
,
0
,
(
r
, b
)
0
,
2
3
,
2
(
r
r
, c
)
0
,
2
3
,
2
(
r
r
.
r
is
a vecto
r
of the movi
ng coordinat
e sy
ste
m
, an
d it can
be t
r
an
sform
ed a
s
R
by
coo
r
din
a
te tra
n
sformation
matrix T to the referen
c
e coordi
nate sy
stem.
c
o
s
.
c
o
s
.
co
s
s
i
n
.
s
i
n
co
s
.
co
s
.
s
i
n
s
i
n
.
c
o
s
co
s
.
s
i
n
s
i
n
.
c
o
s
.
c
o
s
c
os
.
s
i
n
s
i
n
.
cos
.
s
i
n
c
os
.
c
os
s
i
n
.
s
i
n
s
i
n
.
c
o
s
s
in
.
s
in
c
o
s
T
R=
T
r+p
Whe
r
e p
)
,
,
(
p
p
p
Z
Y
X
is th
e po
sition
ve
ctor of the
m
o
ving
coo
r
din
a
tes
origi
n
in
referen
c
e
co
o
r
dinate
system; Eule
r angl
es
,
,
re
pre
s
ent movi
ng platform
attitude. Coo
r
dinate
s
of t
he upp
er
platform a
c
m
e
s in th
e refe
ren
c
e
co
ordi
n
a
te syste
m
a
r
e functio
n
s
of
p
p
p
Z
Y
X
,
,
,
,
, only three
of
whi
c
h a
r
e in
depe
ndent p
a
ram
e
ters fo
r 3RPS p
a
ra
llel mechani
sms
[5]
. Selec
t
,
,
p
Z
as the
indep
ende
nt
output po
sture param
ete
r
s, and the oth
e
r three
w
ill
be re
present
ed by them.
The
3RPS p
a
rall
el
mechani
sm
s ch
ara
c
te
rs show that th
e
moving pl
atform
can
not rotate ab
out
z-a
x
is
of the m
o
vin
g
platfo
rm, so we
can
ge
t
0
, that is
. As
a
res
u
lt, the formulas
f
o
r
cal
c
ulatin
g limbs len
g
ths
are represent
ed as follo
ws:
2
1
22
(c
o
s
1
)
[c
o
s
2
(
c
o
s
1
)
2
](
s
i
n
c
o
s
)
p
r
Lr
RZ
r
(
1
)
2
2
22
2
(c
os
1
)
(
3
si
n
2
cos
2
)
(
1
c
o
s
)
[
44
si
n
(
co
s
3
sin
)
][
]
22
(
1
co
s
)(3
si
n
2
3
c
os
2
)
[
4
3(
1
c
o
s
)
]
42
p
rr
L
Rr
Z
r
rR
(
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 67
58 – 676
3
6760
2
3
22
2
(
1
cos
)
(
3
si
n
2
cos
2
)
(
1
c
os
)
[
44
si
n
(
co
s
3
sin
)
][
]
22
(
1
cos
)(3
si
n
2
3
c
o
s
2
)
3
[]
42
p
rr
L
Rr
Z
rR
(
3
)
Forward kin
e
m
atics i
s
to d
e
termin
e
,
,
p
Z
for given
3
2
1
,
,
L
L
L
, and its esse
nce is
to solve
highly no
nlin
ear
equ
ation
s
me
ntione
d
above.
Ne
wton-Rap
h
son method
i
s
a usu
a
l
nu
meri
cal
solutio
n
. Accordin
g to the
stru
ctural
chara
c
te
risti
c
s of 3RPS and the identity condition
s
for
mech
ani
sm
s motion
s, L
i
Shujun fro
m
No
rthea
st
ern Unive
r
si
ty
extends the
su
cce
ssi
ve
approximatio
n method to the forwa
r
d kinematics so
l
u
tions of the spatial pa
rall
el mech
ani
sms,
whi
c
h i
s
a
n
e
w a
p
p
r
oa
ch
to the po
siti
on an
alysi
s
o
f
parall
e
l me
cha
n
ism. So
me othe
r
sol
u
tion
method
s are
propo
se
d by schola
r
s
[7], but generally spe
a
ki
n
g
, some pro
b
lems a
r
e n
o
t
compl
e
tely so
lved su
ch a
s
initial value se
ns
itivity, solution sp
eed a
n
d
conve
r
ge
nce probl
em.
3. Particles
S
w
a
r
m Opti
mization
Particle
s
swarm
optimi
z
ation al
gorit
hm, ha
s
b
e
com
e
an i
m
porta
nt b
r
anch of
Evolutionary Algorithm
[8], which wa
s
origin
ally pro
posed
by Ameri
c
an
so
ci
al psycholo
g
i
s
t
Jame
s K
enn
edy an
d el
ectrical
engi
nee
r Russel
Eb
e
r
ha
rt in
199
5
and
in
spired
by the
so
ci
al
behavio
rs
of
animal
s
such
as
bird flo
c
ki
ng an
d fish
scho
oling. PS
O algo
rithm
has
bee
n wi
d
e
ly
applie
d in
m
any field
su
ch a
s
fu
nction
optimi
z
ation,
neu
ral
net
work d
e
si
gn, I
C
d
e
si
gn, p
o
w
er
netwo
rk
plan
ning an
d so on be
cau
s
e
of its simple
con
c
e
p
t, easi
l
y implement, high spee
d an
d
better glob
al sea
r
ching a
b
i
lity for large scal
e mathe
m
atical optim
ization p
r
obl
e
m
s and
high
er
conve
r
ge
nce spe
ed than g
enetic al
gorit
hm.
The ba
sic p
r
i
n
cipl
e of PSO is describe
d
that
in the
n-dim
e
n
s
iona
l space, m particle
s
with thei
r co
ordin
a
tes
as
)
,...
,
(
2
1
in
i
i
i
x
x
x
X
re
spe
c
tivel
y
and fitness co
rrel
a
ted t
o
optimi
z
atio
n
obje
c
t functio
n
(u
sually o
b
j
e
ctive fun
c
tio
n
is u
s
ed
as f
i
tness directly
), have their
resp
ective flig
h
t
s
p
ee
ds
r
e
p
r
es
e
n
t
ed
as
)
,...
,
(
2
1
in
i
i
i
v
v
v
V
. For the i
-
th p
a
rticle, it
s be
st po
sition n
a
med a
s
Pb
est
c
a
n
be
r
e
pr
es
e
n
t
ed
a
s
)
,...
,
(
2
1
in
i
i
i
p
p
p
P
, a
nd all parti
cle
s
’ be
st positio
n named a
s
Gbe
s
t can b
e
r
e
pr
es
e
n
t
ed
a
s
)
,...
,
(
2
1
n
g
g
g
g
P
. For the i-th parti
cle of
the t-th generation, its j-t
h
dimen
s
ion
a
l
spe
ed
and
p
o
sition
of the
(t+1
)-th ge
n
e
ration
can
be
solved
according
to th
e follo
wing t
w
o
equatio
ns:
11
22
(1
)
(
)
(
(
)
(
)
)
()
(
(
)
(
)
)
ij
ij
j
i
j
i
j
jg
j
i
j
vt
v
c
r
t
pt
xt
cr
t
p
t
x
t
(
4
)
))
1
(
)
(
)
1
(
t
v
t
x
t
x
ij
ij
ij
(5)
Whe
r
e i
=
1,2,
… , m. m is the numb
e
r
of particl
es.
2
1
r
r
、
are
ran
dom
numbe
rs in t
he
rang
e (0,1
).
2
1
c
c
、
are the a
c
cel
e
ration
wei
g
h
t
s. In addition
, the particl
e spe
ed is limit
ed by the
maximum sp
eed. The first part of Equation (4),
rep
r
e
s
entin
g
the current
spee
d, prov
ides
necessa
ry momentum for
particl
es’ flyin
g
in the
searching sp
ace. The se
con
d
pa
rt, the cognitive
portion, repre
s
ent
s pa
rticle
s’ thinki
ng an
d impels
p
a
rt
icle
s to fly to
the person
a
l best po
sitio
n
Pbest. Th
en t
he thi
r
d
part i
s
the
soci
al p
o
rtion,
rep
r
e
s
ents th
e m
u
tu
al coop
eratio
n an
d influ
e
n
c
e
betwe
en p
a
rti
c
le
s an
d imp
e
ls p
a
rticl
e
s fly to the
be
st positio
n Lb
est in the neig
h
borh
ood i
n
itia
lly
to find the
gl
obal
optimal
solutio
n
a
s
m
u
ch
a
s
p
o
ssi
b
le a
nd fin
a
ll
y fly to the gl
obal
be
st po
sition
Gbes
t. The 1
st
portion
of th
e sp
eed Evol
ution Equatio
n gua
rante
e
s
the glob
al se
arching
ca
pa
city
and the
othe
r two p
o
rtion
s
gua
rante
e
th
e local
sea
r
ching
cap
a
city
. A modified
PSO is u
s
u
a
l
l
y
applie
d
[9], and the co
rrespondi
ng spee
d evolut
ion e
quation i
s
rep
r
esented a
s
:
11
22
(1
)
(
)
(
)
(
(
)
(
)
)
()
(
(
)
(
)
)
ij
ij
j
i
j
i
j
jg
j
i
j
vt
w
t
v
c
r
t
p
t
x
t
cr
t
p
t
x
t
(6)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Forward Posi
tion Solution of 3-RPS in-P
arallel M
anip
u
lator Ba
sed
on… (Z
han
g Hon
g
li)
6761
Whe
r
e
w
i
s
inertia
wei
g
ht. Similar t
o
the tem
p
e
r
ature pa
ram
e
ter in
sim
u
l
a
ted an
neali
n
g
algorith
m
, larger
w
me
an
s better glo
b
a
l sea
r
ching
ability and smaller
w
m
e
ans b
e
tter lo
ca
l
sea
r
ching a
b
i
lity. As the ite
r
ation time
s i
n
crea
se
s,
w
decrea
s
e
s
g
r
adually an
d the algo
rithm
gets
better gl
obal
co
nverg
e
n
c
e ability initi
a
lly and
better lo
cal
con
v
ergen
ce
abi
lity later. Sel
e
ct
max
5
.
0
9
.
0
k
k
w
, where
k
i
s
iteration time
s,
and
max
k
is cut-off iteration time
s, so the
actu
al
variation rang
e of
w
is 0.9~0.4.
In usu
a
l ap
pli
c
ation
s
, a
co
nstrai
ned
con
d
ition
mu
st b
e
take
n into
a
c
count that th
e ce
nter
point of the
uppe
r platfo
rm is requi
red
to run
a
c
co
rding to the
p
r
econ
ceived t
r
ack. Ba
sed
on
3RPS pa
ralle
l mechani
sm
s st
ru
ctural
chara
c
te
risti
c
s, points a
, b
,
c of the u
p
per
platform
are
limited in th
re
e plan
es:
,
0
Y
X
Y
3
，
X
Y
3
,
and a
s
a re
sult the up
per
platform
cent
er p
o
int
is co
nst
r
aine
d by the following e
quatio
ns:
2
)
cos
1
(
2
sin
,
2
)
1
(cos
2
cos
r
Y
r
X
P
P
If the c
enter point is
required to
follow circle locus with radius
0
r
, then:
0
2
0
2
2
r
Y
X
P
p
That is:
0
)
2
)
cos
1
(
2
sin
(
)
2
)
1
(cos
2
cos
(
2
0
2
2
r
r
r
(
7
)
The
nonlin
ea
r e
quatio
ns
made
up
of
Equation
(1
),
(2
), (3) an
d
(7
)
are
mo
dified to
homog
ene
ou
s equ
ation
s
a
s
follows:
0
)
,
,
(
2
1
1
L
Z
f
p
(8)
0
)
,
,
(
2
2
2
L
Z
f
p
(9)
0
)
,
,
(
2
3
3
L
Z
f
p
(10)
0
)
,
,
(
4
p
Z
f
(11)
Then a ne
w functio
n
ca
n b
e
con
s
tru
c
ted
by the four equation
s
ab
o
v
e:
22
2
2
11
2
2
22
2
33
4
(,
,
)
(
)
(
)
()
p
f
Zf
L
f
L
fL
f
(
1
2
)
So the mini
mum value
of the un
con
s
train
ed fun
c
tion
)
,
,
(
p
Z
f
is the
solutio
n
of trajecto
ry-
boun
ded
nonl
inear eq
uatio
ns. In PSO,
)
,
,
(
p
Z
f
is u
s
e
d
a
s
th
e fitness fu
nction to eval
ua
te the
positio
n of pa
rticle
s, and it is the fitnes
s value that guid
e
s the evoluti
on pro
c
e
s
s.
The cal
c
ul
ation flow of PSO is as follo
ws:
1)
Initialize the p
a
rticle
swarm
(i
nclu
ding
ra
ndom po
sitio
n
and spee
d);
2)
Cal
c
ulate fitn
ess value of every parti
cle
,
assu
me the
initial position
as the histo
r
i
c
al
best po
sition
n
i
P
i
,...
2
,
1
,
, and obtain th
e global b
e
st
positio
n
g
P
;
3)
Comp
are the fitness valu
e of every par
ti
cle with its hi
stori
c
al be
st positio
n
i
P
, and
let it be the current be
st position
if the fitness value is better;
4)
Comp
are the fitness valu
e of every
parti
cle with the gl
obal be
st po
sition
g
P
, and let it
be the histo
r
i
c
al be
st po
sition if the fitness valu
e is be
tter;
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 67
58 – 676
3
6762
5)
Evolve the speed an
d po
sition of particl
e
s
acco
rdin
g to formula
s
(5
) and (6), and
obtain a ne
w
gene
ration of
particl
e swa
r
m;
6)
If the termination co
ndition
(usually
goo
d
enoug
h fitness valu
e or p
r
esu
ppo
se
maximum alg
ebrai
c multipli
city) is not satisf
ied, retu
rn
2; otherwi
se f
i
nish the
whol
e
p
r
oc
es
s
.
4. Example of For
w
a
rd
Kinematic
s
of Parallel Manipulator
Based on PS
O
As Fig
u
re
1
shows th
at the
radii
of the
u
pper an
d lo
wer pl
atform
s
are
40
cm a
n
d
30
cm,
and the limb
s
lengths a
r
e
betwe
en 50
cm and 100
cm.
Assum
e
the si
ze of pa
rticle
swa
r
m
n=3
0
,
dimen
s
ion
d
=
3,and m
a
ximum iteratio
n t
i
mes i
s
100.
The
cente
r
p
o
int of up
per platform
ru
n
s
followin
g
the circl
e
with radiu
s
4.4cm.
Choo
se
arbi
trarily 5 grou
ps of limbs l
ength
s
, and the
cal
c
ulatio
n re
sults of Ta
ble
1 sho
w
s that
calculation p
r
eci
s
io
n alrea
d
y have rea
c
hed
4
10
.
Table 1. Re
sult of PSO
Num
b
er
Rod
Len
g
t
h
/m
Real Euler
A
n
gl
es/rad
PSO
Cac
u
late
d A
ngle
s
/rad
Error×10
-4
/rad
Fi
rs
t
L
1
0.5494
0.5236
0.523789
1.89
L
2
0.7022
0.7854
0.785446
0.46
L
3
0.9035
0.7000
0.699997
0.03
Second
L
1
0.5554
0.6283
0.628370
0.07
L
2
0.6803
0.7854
0.785381
0.19
L
3
0.9165
0.7000
0.700002
0.02
Thi
r
d
L
1
0.5685
0.7854
0.785203
1.97
L
2
0.6487
0.7854
0.785470
0.70
L
3
0.9313
0.700000
0.700015
0.15
Fouth
L
1
0.6023
1.047200
1.047198
0.02
L
2
0.6023
0.785400
0.785342
0.58
L
3
0.9411
0.700000
0.699997
0.03
Fifth
L
1
0.7022
1.570800
1.570606
1.94
L
2
0.5494
0.785400
0.785446
0.46
L
3
0.90
35
0.700000
0.699997
0.03
(a) fitne
ss val
ue ch
angi
ng
curve
(b) p
a
rticl
e
s d
i
stributio
n of the 1st ge
nera
t
ion
(c
) Particl
e
s d
i
stributio
n
of the 15th ge
ne
ration
(d) p
a
rticl
e
s d
i
stributio
n of the 100th
gene
ration
Figure 2. Evolution of PSO Algorithm
0
10
20
30
40
50
60
70
80
90
10
0
-0
.
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
0
5
10
15
20
25
30
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
5
10
15
20
25
30
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
5
10
15
20
25
30
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Forward Posi
tion Solution of 3-RPS in-P
arallel M
anip
u
lator Ba
sed
on… (Z
han
g Hon
g
li)
6763
Takin
g
the 2
nd
group data
as an exam
pl
e, the changi
ng cu
rve of particle
swarm
fitness
value is
sh
owed a
s
Figu
re
2(a
)
. Obviou
sly the av
erag
e fitness valu
e of every ge
neratio
n pa
rti
c
le
swarm
de
cre
a
se
s
co
nsta
n
t
ly and fina
lly conve
r
ge
s
to
zero.
Ta
kin
g
L
1
of the
2
nd
g
r
oup
data
a
s
an
example
and
sh
owi
ng
as
Figure 2
(
b
)
,
2(c), 2
(
d
)
, p
a
r
ticle
swa
r
m
unde
r th
e g
u
i
dan
ce
of fitn
ess
value co
nverge gra
dually from initial r
a
n
dom dist
ributi
on to actual v
a
lue.
5. Conclusio
n
The gl
obal
o
p
timal solutio
n
can b
e
o
b
tained
better
by the mo
dified PSO
and i
t
s pa
rallel
sea
r
ching
ab
ility. The method overco
mes th
e pro
b
lem that tra
d
itional
soluti
on of no
nlin
ear
equatio
ns i
s
sen
s
itive to the initial poi
nt, and
avoi
ds of formul
a derivatio
n
and
compli
ca
ted
cal
c
ulatio
n for forward
kin
e
matics, and
need
s no speci
a
l form o
f
equation
s
. Comp
ared wi
th
other evoluti
onary alg
o
rit
h
m, it is easy
to understand and p
r
ogra
m
need
s fewer empi
rical
para
m
eters. The re
sults sho
w
th
at
PSO
algo
rith
m is
a n
e
w effective m
e
thod fo
r fo
rward
kinem
atics.
Ackn
o
w
l
e
dg
ements
This wo
rk
is supp
orted
by Xinjiang
Uygur A
u
to
nomou
s
Reg
i
on Nature
Scien
c
e
Found
ation u
nder G
r
a
n
t (2
0122
11A00
3)
Referen
ces
[1]
W
en F
uan, L
i
ang
Cho
n
g
g
a
o
.
Displac
eme
n
t ana
l
y
sis of St
e
w
art p
l
atform
mecha
n
isms.
MMT
. 1994;
29(4): 54
7-5
5
7
.
[2]
Mcaree
PR, D
ani
el
RW
. A fa
st robust s
o
luti
on to
the St
e
w
art pl
atform for
w
a
r
d k
i
n
e
matic
s
. Rob
o
S
y
st.
199
6; 13(7): 40
7-42
7.
[3]
Li W
e
ij
ia. A stu
d
y
on the
direc
t
kinematic so
l
u
tion
of ste
w
a
r
t
platforms.
Jou
r
nal of H
u
a
z
h
o
ng Un
iversity
of Science a
n
d
T
e
chnol
ogy
. 1
997; 25(
4): 38-
40.
[4]
Lou YF
, Brun
n P.
A hybrid artificial
neur
al network inv
e
rs
e kinem
atic
solution
for accurate robot
pathco
n
trol.
Proc Instn Mech Engrs. 199
9; 2
13(1): 23-
32.
[5]
F
ang Yu
efa H
uan
g Z
hen. Ins
t
antane
ous mo
tion an
al
ys
is of a three d
egre
e
of freedom
3
R
PS para
lle
l
robot man
i
p
u
la
tor.
Mechanic
a
l
Science a
nd T
e
chn
o
lo
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199
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934.
(in Chi
nese).
[6]
Li S
huj
un, W
a
ng
yu
e, W
a
n
g
Xi
ao
gua
ng. F
o
r
w
a
r
d P
o
siti
on
Anal
ys
is of
3-
RPS in-P
ara
lle
l Man
i
p
u
lat
o
r
Using S
e
lf-mo
difie
d
Success
ive Ap
pro
x
im
ation Met
hod.
J
ourn
a
l of N
o
rtheaster
n
Un
iv
ersity (Natura
l
Scienc
e)
. 200
1
;
22(6): 285-2
8
7
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[7]
Han KY, C
h
u
ng W
Y
, You
m
Y. Ne
w
r
e
soluti
on
sch
e
m
e of the for
w
a
r
d ki
nem
ati
cs of paral
le
l
mani
pul
ators u
s
ing e
x
tras
ens
or.
T
r
ansaction
s of the ASME
J of Mech Desi
gn.
199
6; 118(
2): 214-2
19.
[8]
Kenn
ed
y J Eb
erhart RC.
Par
t
icle sw
arm op
timi
z
a
t
i
o
n
. Proceedings of IEEE In
ternational Conferenc
e
on Ne
ural N
e
tw
o
r
ks. Perth, W
A
, Australia. 199
5: 194
2 -19
48.
[9]
Shi Y, E
ber
har
t RC.
A M
odifi
ed P
a
rticle
Sw
arm Opti
mi
z
a
t
i
on.
Pro
c
e
e
d
i
ngs o
f
th
e 19
99
C
o
ng
re
ss
on
Evoluti
onar
y
C
o
mputati
on. IEEE Pre
ss, Piscata
w
a
y
,
NJ. 19
98; 69-7
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.