TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4778 ~ 4
7
8
6
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.552
2
4778
Re
cei
v
ed
De
cem
ber 3
0
, 2013; Re
vi
sed
March 11, 20
14; Accepted
March 26, 20
14
Optimal Asymmetric S-shape Acceleration/Deceler
ation
for Multi-axial Motion Systems
Chan
g-
y
a
n Chou
1
, Sh
y
h
-Leh
Chen
2
Dep
a
rtment of Mecha
n
ica
l
En
gin
eeri
ng a
nd
Adv
anc
ed Insti
t
ute of Manufa
c
turing
w
i
t
h
Hi
gh-tech
Innovati
ons, N
a
tion
al Ch
un
g Che
ng Un
ivers
i
t
y
, T
a
i
w
a
n
, R
O
C
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: shermie
013
1
@
hotmai
l
.com
1
, imeslc@ccu.e
du.t
w
2
A
b
st
r
a
ct
In this study, an opti
m
i
z
at
io
n alg
o
rith
m is p
r
opos
ed for
asymmetric
s-shape
acceleration/deceler
ation to ac
hiev
e better contour ac
c
u
racy
for biaxial system
s.
The
optim
i
z
at
ion is bas
ed
on the
m
e
thod of genetic algori
thm
inc
o
rporated with the cons
traints
m
a
de by the m
o
tion system
.
Nu
meric
a
l s
i
mulati
ons
of an
XY tabl
e dr
ive
n
by
lin
ear
motors foll
ow
ing
a
si
mpl
e
cor
neri
ng p
a
th v
e
rify t
h
e
effectiveness
o
f
the propos
ed
alg
o
rith
m.
Ke
y
w
ords
: ge
netic al
gorith
m
,
contouri
ng err
o
r, inte
rpo
l
atio
n, s-shape acc
e
ler
a
tion/d
e
ce
l
e
ratio
n
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
Accel
e
ration/
deceleration
(a
cc/d
e
c)
algor
ith
m
pl
ays an i
m
p
o
rtant role
in CNC
machi
n
ing.
There are
several met
hod
s, su
ch
as linea
r
acc/de
c,
expone
ntial acc/de
c,
trigono
metri
c
function a
c
c/de
c,
and
s-sh
ape
acc/d
e
c, etc. [1
]
, [2
]
. Among them, the lin
ear
acc/de
c is th
e most
com
m
only used
one be
ca
use
of easy calculation. Ho
wever, the sud
den
cha
nge
of a
c
cele
ration
ca
n cau
s
e th
e
shockin
g
vi
bra
t
ion an
d lo
w
accuracy i
n
t
he motio
n
co
ntrol,
whi
c
h i
s
not
suitable fo
r hig
h
-spee
d an
d
high-prec
i
s
io
n co
ntou
ring.
On t
he
co
ntra
ry, the s-shap
e
acc/de
c i
s
m
o
re
and
mo
re
re
cog
n
ized
b
e
ca
use of
it
s
smooth
velo
ci
ty profile that
can
re
du
ce th
e
jerk limitation
Error!
Refer
e
nce source not found.
. Shi
et al.
imple
m
ent the effe
ct of the
s-sh
ape
acc/de
c [1] and Ha
o
et al.
transfo
rm th
e linear a
c
c/d
e
c into the s-sha
pe form [
2
]. Furtherm
o
re
,
Cao
et al.
integrate the lo
ok-ahe
ad st
rat
egy with the s-sha
pe a
c
c/de
c
Error! Refere
nce
s
o
urce
not fou
nd.
. Al
l of these recent studie
s
p
o
int out t
he developm
ent
of acc/dec
control
al
gorit
hm
and the adva
n
tage
s of
s-shape a
c
c/de
c.
In this arti
cle,
the
asymm
e
tric s-shap
e a
cc/
d
e
c is introdu
ced
in
se
ction
2. It ha
s a m
o
re
flexible adju
s
tment of the a
cc/d
e
c
i
n
terv
als. The
r
e
are four p
a
ram
e
ters i
n
a a
cc/dec p
r
o
c
e
ss
with
the constrai
nt of the
adm
issi
ble
set.
We i
n
vest
igate the
conditi
ons of the practi
cability
and
analyze the boun
ds of th
ese a
c
c/de
c para
m
eter
s a
c
cordi
ng to the path information and t
he
limitation of this motion
system. In
secti
on 3, the gen
etic algo
rithm
Error!
Refer
e
nce source not
fou
nd.
i
s
ad
opted to fin
d
the optim
al set of
acc/de
c pa
ram
e
ters i
n
the
same tracking
perfo
rman
ce
of the motion
system. The
prop
agatio
n
is also discussed in
det
ail. A numeri
c
al
simulatio
n
of
an XY tabl
e
driven
by lin
ear
motors i
s
de
signe
d fo
r validation
of the p
r
op
ose
d
geneti
c
algo
rithm by the corne
r
ing
pat
h, and a
blo
ck
ch
ang
e criterio
n “exa
ct stop fine”,
is
integrate
d
to
deal
with the
corne
r
erro
r. The set
up a
n
d
re
sults
are
given in the
section
4. Fina
lly,
con
c
lu
sio
n
s a
r
e drawn in section 5.
2. As
y
mmetric S-shape
Acc/De
c
This stu
d
y propo
se
s a
n
o
p
timization
al
gorit
hm
for a
s
ymmetri
c
s-sha
pe
acc/de
c u
s
in
g
the method o
f
genetic alg
o
r
ithm with the
con
s
tr
aint
s
made by the motion syste
m
. The algori
t
hm
is verifie
d
n
u
m
eri
c
ally in t
he
simulatio
n
s
of
an XY t
able
driven
b
y
linear moto
rs foll
owi
ng
a
simple
corne
r
ing path with
the block ch
ange criteri
o
n “exact stop
fine” to deal with the corner
errors. Th
e
si
mulation
re
su
lts sho
w
that
the
cont
o
u
rin
g
erro
r a
nd
co
rner
erro
r can
be redu
ce
d
b
y
meta-g
eneti
c
cycl
e a
nd th
e effect
of a
symmetric
S-shape
a
cc/d
e
c is
su
peri
o
r to the
symme
tric
spe
c
ification. We can obtai
n the opt
imal para
m
eters o
f
s-shape a
cc/dec.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Asymm
e
tric S-sh
ape Accele
rat
i
on/De
cel
e
rati
on for Multi-a
x
ial… (Ch
ang
-ya
n
Ch
ou
)
4779
There a
r
e
two field
s
in
the
co
ntou
ring
p
r
oble
m
whi
c
h
incl
ude
interpolation
an
d t
r
aje
c
tory
tracking
cont
rol. Th
e inte
rpolatio
n i
s
t
o
ge
nerate seri
al comm
and
s
le
adin
g
the cont
roll
ed
comp
one
nts to follow the
desi
r
ed p
a
th. The pro
c
e
ss of interpolat
e in a block
(a ro
w of co
de)
contai
ns
acceleratio
n
, co
nstant
spe
e
d
, and fi
nal
ly decel
erati
on. Thi
s
is the so-call
e
d
accele
ration/
deceleration (acc/de
c
)
.
T
h
e
r
e a
r
e
s
e
ver
a
l
s
h
ap
es
of
acc/de
c, an
d it can
be
an
option a
c
cording to the d
i
fferent appli
c
ations.
Th
ere
are several
sha
p
e
s
of acc/de
c, and th
e
most u
n
iversal shap
e of
acc/de
c i
s
th
e trap
ez
oidal
velocity p
r
of
ile. Accordin
g to the
defi
ned
(acce
p
table
)
axis-a
ccele
r
ation, trap
e
z
oid
a
l
acc/d
e
c a
c
cele
rat
e
s to feed
-rate an
d finally
decelerates t
o
ze
ro
-velo
c
ity at
the end
-point. The
ad
vantage of
trape
zoid
al is t
i
me-o
ptimize
d
;
the path
will
be fini
she
d
in the
sho
r
test time. But, the effici
ency al
so
comes with t
he
discontin
uou
s of the accel
e
ration
profil
e that cau
s
e
the infinity jerk whi
c
h
sho
c
k the sy
stem
on
the contrary.
This impact
will
wear the system and reduce th
e accuracy in
the contouri
ng
pro
c
e
ss.
Ano
t
her
sh
ape
o
f
acc/de
c i
s
calle
d
“s-sha
pe a
c
c/de
c”
whi
c
h
po
sse
s
ses a t
r
ian
g
u
lar
acceleration
profile
as shown i
n
Figure
1. In ot
her
words, the jerk is
pi
ecewise
constant
s. It will
lead to the
smooth, s-sha
ped velo
city profile
sh
o
w
n
in Figu
re 2.
Comp
ared
wi
th the co
mm
only
use
d
trape
zoi
dal acc/de
c, it can gene
rat
e
more
smoo
th comma
nds, resulting in
better accu
ra
cy.
Ho
wever, the
s-shap
e a
c
c/dec involve
s
more
pa
ram
e
ters, ma
kin
g
the tuning
pro
c
ess mo
re
difficult. By appli
c
ation
o
f
acc/d
e
c be
fore inte
rpol
a
t
ion, there
a
r
e 4
pa
rame
ters
of s-sha
p
e
acc/de
c:
F
to
E
from
Interval
Time
:
E
to
D
from
Interval
Time
:
C
to
B
from
Interval
Time
:
B
to
A
from
Interval
Time
:
2
1
2
1
e
e
s
s
T
T
T
T
In a curve. If we
consider t
he case of a
corner
path, t
here
will be 8 parameters t
o
optimize the
conto
u
rin
g
a
nd co
rne
r
errors
at the sa
me time
(the
curve b
e
fore the co
rne
r
and after). Most
appli
c
ation
s
assume the
s-sha
pe a
cc/
dec to be
sy
mmetric, i.e., the accelera
tion time interval
(from A to
C) e
qual
s d
e
cel
e
ratio
n
o
ne (from D
to F). In this work,
asy
mmetric sh
a
pe
is
con
s
id
ere
d
to keep the p
a
rameter tuni
ng
more flexible
.
Figure 1. Acceleratio
n
Prof
ile
Figure 2. Velocity Profile
2
1
1
2
1
2
1
1
1
,
)
(
0
,
)
(
s
s
s
s
s
s
s
s
s
T
T
t
T
t
T
T
J
J
T
t
t
J
t
a
(1)
Whe
r
e
1
s
J
is the jerk of the interval from A to B, and
2
s
J
is the jerk of the interval from
B
to C. Due to the re
stri
ction
of
mathemati
c
al represent
ation, we
onl
y derive the equatio
ns of t
he
accele
ration
zon
e
(from
A to C). The
deceleration
zon
e
is
sim
ilar to the a
c
celeration p
a
rt.
(Re
p
la
ce
1
s
T
and
2
s
T
with
2
e
T
and
1
e
T
, resp
ectively). From Equati
on (1
), we ca
n get the velocity
of ac
celeration interval.
M
V
D
C
E
F
t
B
A
v
(
t
)
D
C
E
F
t
B
A
a
(
t
)
M
A
M
A
M
A
M
A
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: 4778 – 4
786
4780
2
1
1
2
1
2
1
2
1
1
2
1
2
2
2
1
1
2
1
2
1
,
)
(
)
(
0
,
)
(
s
s
s
s
s
s
s
s
s
s
s
s
T
T
t
T
T
J
J
t
T
J
J
t
J
T
t
t
J
t
v
And the total displ
a
cement
from A to C.
3
2
2
6
1
2
1
2
1
1
2
1
3
1
1
6
1
0
)
(
)
(
2
1
s
s
s
s
s
s
s
s
s
T
T
s
T
J
T
T
T
T
J
T
J
dt
t
v
L
s
s
(2)
For the
four
para
m
eters
(time interval
s) to
b
e
ad
ju
s
t
e
d
in
an
ac
c/d
e
c
pr
oc
ess
,
s
o
me
con
s
trai
nt co
ndition
s mu
st be impo
se
d. A
ssu
me that
the total len
g
th of path i
s
L
, maximum
feed-rate is
M
V
, maximum acceleration is
M
A
, and maximum jerk is
M
J
. These con
s
traint
s
rest
rict the up
per an
d lower bound
s of pa
ramete
rs a
s
f
o
llow:
1) The total movement of
acceleration
and de
cele
ra
tion time interval (from A to C and
form D to F) should b
e
less than
L
.
2) The velo
cit
y
must arrive
to
M
V
in the c
o
ns
tant veloc
i
ty time interval (C to D).
3) Accele
ratio
n
and de
cel
e
ration ca
nnot
be larg
er tha
n
M
A
.
4) Je
rk ca
n’t be larg
er tha
n
M
J
.
Her
e
L
,
M
V
,
M
A
, and
M
J
are give
n
con
s
tant
s. According
to th
e Figu
re
1 a
nd 2
)
, we
have:
M
s
s
s
s
s
s
s
s
V
T
T
A
T
J
T
J
A
)
(
,
2
1
2
1
2
2
1
1
(3)
Hen
c
e,
M
M
s
s
M
s
s
M
s
A
V
T
T
A
T
T
V
A
2
2
2
1
2
1
(4)
By constraint 3). Com
b
inin
g Equation (3
) with
Equatio
n (4), we obta
i
n the jerk e
q
uation:
)
(
2
,
)
(
2
2
1
2
2
2
1
1
1
s
s
s
M
s
s
s
s
M
s
T
T
T
V
J
T
T
T
V
J
(5)
Furthe
rmo
r
e,
M
M
s
s
s
M
M
s
s
s
J
V
T
T
T
J
V
T
T
T
2
)
(
,
2
)
(
2
1
2
2
1
1
(6)
Is available b
y
const
r
aint 4). Addition o
f
the two equ
ation in Equat
ion (5
), we ha
ve:
M
M
s
s
M
s
s
M
s
s
J
V
T
T
J
T
T
V
J
J
2
1
2
1
2
1
2
2
(7)
Finally, we ob
tain the total displ
a
cement
of accel
e
rati
on interval.
l
)
2
(
2
1
3
1
s
s
M
s
T
T
V
L
(8)
By substitutin
g
Equation (5
) into Equatio
n (2).
In the sam
e
way, the restri
ction
s
of
par
am
eters in deceleration interval
are al
so
available.
M
M
e
e
A
V
T
T
2
2
1
(9)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Asymm
e
tric S-sh
ape Accele
rat
i
on/De
cel
e
rati
on for Multi-a
x
ial… (Ch
ang
-ya
n
Ch
ou
)
4781
M
M
e
e
J
V
T
T
2
1
(10)
)
2
(
2
1
3
1
e
e
M
e
T
T
V
L
(11)
By constraint 1),
L
L
L
e
s
.
M
e
e
s
s
V
L
T
T
T
T
3
2
2
2
1
2
1
(12)
It also mean
s:
M
s
s
V
L
T
T
3
2
2
1
(13)
As a
re
sult, accele
ration
para
m
eters
)
,
(
2
1
s
s
T
T
must
satisfy the eq
uation
s
(4),
(7
), an
d
(13
)
. In othe
r
words, th
e int
e
rsectio
n
of
Equation
(4
), (7), a
nd
(13
)
sho
u
ld b
e
the
non
-empty
set
in the two
-
di
mensi
onal
sp
ace
co
mpo
s
e
d
of
)
,
(
2
1
s
s
T
T
.Then, we ca
n ap
ply the s-sha
pe a
cc/d
e
c i
n
this
c
a
s
e
and find the optimal s
o
lution.
By the si
mple
cal
c
ul
ation
of
alge
braic ge
om
etry, we
can obtain t
he necessa
ry co
ndition
s
of the s-shap
e acc/de
c are
:
2
3
9
8
L
J
V
M
M
and
M
M
M
M
J
V
L
L
A
V
3
2
2
8
9
9
8
(14)
Equation (14) sho
w
s that
the admi
ssi
ble
set won’t exi
s
t if the feed-rate,
M
V
, is too large
or the limit of
L
,
M
A
, and
M
J
are t
oo
small. Th
e
s
e
re
sults are
rea
s
o
nabl
e.
Und
e
r th
e
co
ndition
of Equation (14), there are ex
actly two intersectio
n
points
cro
s
sed by the
boun
dary of the
Equation (7)
and Equatio
n
(13)
. Th
ese two poi
nts are
:
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
J
V
L
V
V
L
J
M
L
V
V
L
Q
J
V
L
V
V
L
J
V
L
V
V
L
Q
3
2
3
2
2
3
2
3
2
1
8
9
4
1
4
3
,
8
9
2
1
2
3
:
8
9
4
1
4
3
,
8
9
2
1
2
3
:
If the bounda
ry of Equation (4) p
a
ss through
1
Q
, this
res
u
lts
:
M
M
M
M
M
M
J
V
L
V
V
L
A
V
3
2
8
9
4
1
4
9
2
Or if the boun
dary of Equat
ion (4
) pa
ss t
h
rou
gh
2
Q
, this
res
u
lt
s
:
M
M
M
M
M
M
J
V
L
V
V
L
A
V
3
2
8
9
4
1
4
9
2
We have the
simila
r re
sults in decel
erati
on part
)
,
(
2
1
e
e
T
T
.
Synthesizi
n
g
these
an
alyse
s
of a
c
c/d
e
c p
a
ramete
rs ab
ove, the
r
e a
r
e th
ree
possibl
e
ca
se
s of the admissibl
e se
t:
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: 4778 – 4
786
4782
(i) Ca
se
1:
M
M
M
M
J
V
A
V
4
2
. As sho
w
n in
Figure 3, the
admi
ssibl
e set is surrou
n
ded by
the boun
da
ri
es of Equati
on (7
) an
d Equation
(13
)
. Then, the u
pper
and lo
wer bo
und of
the
acc/de
c para
m
eters are li
mited by the points,
)
,
(
2
1
Q
Q
. That
is
:
M
M
M
M
e
s
M
M
M
M
J
V
L
V
V
L
T
or
T
J
V
L
V
V
L
3
2
2
1
3
2
8
9
2
1
2
3
)
(
8
9
2
1
2
3
(15)
M
M
M
M
e
s
M
M
M
M
J
V
L
V
V
L
T
or
T
J
V
L
V
V
L
3
2
1
2
3
2
8
9
4
1
4
3
)
(
8
9
4
1
4
3
(16)
(ii)
Case
2:
M
M
M
M
M
M
M
M
J
V
L
V
V
L
A
V
J
V
3
2
8
9
4
1
4
9
2
4
. As
sh
o
w
n i
n
Fi
gu
re 4, th
e
admissibl
e se
t is the same
regio
n
as
(i).
(iii) Case 3:
M
M
M
M
M
M
M
M
M
M
J
V
L
V
V
L
A
V
J
V
L
V
V
L
3
2
3
2
8
9
4
1
4
9
2
8
9
4
1
4
9
. As
sho
w
n i
n
Figure 5,
there is a
n
intersectio
n
poi
nt.
M
M
M
M
M
M
A
V
V
L
V
L
A
V
Q
2
3
,
3
4
:
3
Cro
s
sed by t
he bou
nda
ry of eq(4)
an
d eq(13)
. Thi
s
situatio
n causes the u
p
per a
nd
lowe
r bou
nd
become:
)
(
3
4
2
1
e
s
M
M
M
T
or
T
V
L
A
V
M
M
M
M
J
V
L
V
V
L
3
2
8
9
2
1
2
3
(17)
)
(
8
9
4
1
4
3
1
2
3
2
e
s
M
M
M
M
T
or
T
J
V
L
V
V
L
M
M
M
A
V
V
L
2
3
(18)
By synthesizi
ng the discu
s
sion
s,
we h
a
ve the con
c
lu
si
ons b
e
lo
w:
1. Given
the
data,
M
M
A
V
L
,
,
and
M
J
, we can
ch
eck
wheth
e
r the
e
q
(14
)
i
s
sati
sfied o
r
not. If not,
the setting of the path or feed
-rate sh
ould b
e
modified.
2. In order to
meet the co
nstrai
nt 1 to 4,
these a
cc/
dec p
a
ra
met
e
rs
sho
u
ld satisfy th
e
Equation (4), (7), (8), (1
2)
and (1
3).
3. Acco
rdin
g to the given informatio
n like
M
M
A
V
L
,
,
and
M
J
, rou
ghly analysi
s
the
uppe
r and lo
wer b
oun
d of the acc/de
c p
a
ram
e
ters in
admissibl
e se
t by the intersection
set of the
Equation (4
), (7), and
(1
3). Ho
wever,
it’
s
just
th
e
be
gin
n
ing if
we
wa
nt to ra
ndo
ml
y cho
o
se a
set
of parameters
satisfying t
he uppe
r and lower
bound. We
still need to check if
the set of dat
a
,
)
,
,
,
(
2
1
2
1
e
e
s
s
T
T
T
T
, loc
a
te in the admiss
i
ble
set or not.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Asymm
e
tric S-sh
ape Accele
rat
i
on/De
cel
e
rati
on for Multi-a
x
ial… (Ch
ang
-ya
n
Ch
ou
)
4783
Figure 3. Ca
se 1 of the Admissi
ble Set
Figure 4. Ca
se 2 of the Admissi
ble Set
Figure 5. Ca
se 3 of the Admissi
ble Set
3. Optimizati
on B
y
Genetic Algorithm
This stu
d
y propo
se
s a
n
o
p
timization
al
gorit
hm
for a
s
ymmetri
c
s-sha
pe
acc/de
c u
s
in
g
the method o
f
genetic alg
o
r
ithm with the
con
s
tr
aint
s
made by the motion syste
m
. The algori
t
hm
is verifie
d
n
u
m
eri
c
ally in t
he
simulatio
n
s
of
an XY t
able
driven
b
y
linear moto
rs foll
owi
ng
a
simple
corne
r
ing path with
the block ch
ange criteri
o
n “exact stop
fine” to deal with the corner
errors. Th
e
si
mulation
re
su
lts sho
w
that
the
cont
o
u
rin
g
erro
r a
nd
co
rner
erro
r can
be redu
ce
d
b
y
meta-g
eneti
c
cycl
e an
d th
e effect of
asymmetric
s-shape acc/de
c
is sup
e
rio
r
t
o
the symme
tric
spe
c
ification. We can obtai
n the opt
imal para
m
eters o
f
s-shape a
cc/dec.
The o
r
igin
o
f
Geneti
c
al
gorithm
co
m
e
s fr
om the
Da
rwi
n
’s
“Natural Sel
e
cti
on”
and
“Survival of
the Fitte
st”. In
196
0, John
Ho
lla
nd
appli
ed the
s
e
con
c
ept
s, in
cludi
ng p
r
op
agati
on,
mutation, and s
e
lec
t
ion repetitively, to s
earc
h
t
he o
p
timal sol
u
tion
by mathemati
c
al calculatio
n.
Re
cently, it has be
en broa
dly applied to
s
earch all
kin
d
s of optimal
probl
em
s.
The procedu
re of the genet
ic algo
rithm is listed as follo
w:
Step 1) Ra
nd
om initializati
on of popul
ation
Step 2) The fi
ttest of chrom
o
som
e
in fitness functio
n
Step 3) Natural selection
Step 4) Crossover
Step 5) Mutat
i
on
Rep
eating th
e step
2) to
5), the
sup
e
ri
or offsprin
g
may app
ear
and it’s
po
ssi
ble to get
the best on
e. In section 2,
the uppe
r a
nd lowe
r bo
u
nd of the acc/dec pa
ram
e
ters
have bee
n
approximatel
y estimated. We want to sea
r
ch the
o
p
timal sol
u
tio
n
in the admi
ssi
ble set of the
spa
c
e co
nstructed by
1
2
1
,
,
e
s
s
T
T
T
, an
d
2
e
T
. Bec
a
us
e
a s
e
t of parameters
)
,
,
,
(
2
1
2
1
e
e
s
s
T
T
T
T
whic
h
satisfy the bo
und are not necessa
ry sa
tisfying the admissible
se
t
,
the genetic algorith
m
whi
c
h
we ap
ply to find the optimal solutio
n
of s-
shap
e a
cc/d
e
c
sho
u
l
d
be modifie
d
to adapt the
constrai
nts.
We will go into detail as foll
ows.
Step 1) Ran
dom initiali
za
tion: Accordi
ng to th
e d
a
t
a,
M
M
A
V
L
,
,
and
M
J
, ra
ndomly
gene
rate 4
chrom
o
some
s
)
,
,
,
(
2
1
2
1
e
e
s
s
T
T
T
T
of the initial populatio
ns i
n
the interval
with the lo
wer
and
upp
er b
ound
an
alyze
d
in
se
ction
2, until
we
h
a
ve
8 set
s
o
f
acc/de
c parameters, whi
c
h
1
s
T
2
S
T
1
Q
Eq
(7)
Eq(13)
2
Q
Eq
(4)
3
Q
1
s
T
2
S
T
1
Q
Eq
(7)
Eq(13)
2
Q
Eq
(4)
2
S
T
1
Q
Eq
(7)
Eq(13)
2
Q
Eq
(4)
1
s
T
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: 4778 – 4
786
4784
belon
g to
the
admi
s
sible
set (omit
the
non-coi
n
ci
de
nt one
). T
h
e
n
, use th
e bi
nary
en
codin
g
fo
r
these
initial
p
opulatio
ns in
10
bits
(g
en
e)
with 0
000
0000
00
co
rre
s
po
ndin
g
to t
he lo
we
r b
o
u
n
d
and 11
111
11
111 corre
s
po
nding to the u
pper b
oun
d.
Step 2) Survi
v
al of the Fittest: De
co
de thes
e ch
rom
o
some
s of p
o
pulation a
nd
simulate
the ca
se of
s-shap
e acc/de
c.
Cal
c
ul
ate the cont
ourin
g errors and co
rne
r
erro
r form
the
simulatio
n
re
sults. Th
en, compa
r
e
the fitness of ea
ch
popul
ation by
:
)
*
2
.
0
*
8
.
0
(
*
10000
erro
r
corne
r
erro
r
contouring
F
itness
Where 10000 is a multiplier to increase t
he se
nsitivity of the fitness
function.
Step 3) Natu
ral sele
ction:
We reserve 4
supe
rio
r
pop
ulation
s
(sele
c
tion rate=50
%
) to be
the su
rvival base on the
fitness,
a
nd
weig
ht these
four po
pulat
ion to gen
erate the offsp
r
ing
possibly by the weig
hting functio
n
.
keep
N
n
keep
n
n
n
N
P
1
1
Whe
r
e
keep
N
is the
popul
ation n
u
mbe
r
which
we reserve, a
nd
n
P
is the probability of the
th
n
individual cho
s
en to p
r
opa
g
a
te.
Step 4) Crossover: We ra
ndomly gen
e
r
ate two
nu
m
bers from the
interval [0, 1] twice.
Acco
rdi
ng to
the wei
ghting
pro
bability, we h
a
ve
two
pairs to p
r
op
agate 4 filial
gene
ration
s.
The
method
of p
r
opag
ation i
s
singl
e poi
nt crossove
r,
a
n
d
the
cross
p
o
int is ra
ndo
m. So, we
h
a
ve
totally 8 samples (4 pop
ul
ations a
nd 4 filial gene
ratio
n
s) afte
r prop
agation.
Step 5) Muta
tion: Except the optimal g
enerat
ion, the mutation p
o
ssibly occu
rs to all
gene of ea
ch
individual. T
he num
bers
of muta
tion is co
ntroll
ed
by the mutation rate,
%
5
(us
ually
). The
n
,
bits
pop
N
N
numbers
Mutation
)
1
(
Whe
r
e
gene
chromosome
bits
N
N
N
is th
e
total bits
of th
e individu
al.
We
ran
domly
mutate the
g
ene
(1
become
s
to 0
or contra
rily) di
re
ctly and repeat to the
mutation
num
bers. Finally, we de
co
de th
e
new mutative
generation a
nd che
c
k if the set of a
cc/d
e
c pa
ramete
rs locate in the admissibl
e set
or not. (If not, we thro
w it o
u
t.) In this wa
y, we
execut
e step 2
)
to step 5) repetiti
v
ely to find th
e
optimal sol
u
tion.
4. Simulation Verificatio
n
Con
s
id
erin
g
a co
mplete
motion control syste
m
, it inclu
d
e
s
co
mmand
co
de
, acc/d
e
c,
interpol
ation, and plant wit
h
servo
loo
p
control,
a
s
sh
own
in
Figu
re
6. Th
e pl
ant
is the
XY tabl
e
driven by line
a
r motors whi
c
h refe
r to the manual
of
Siemen
s. We
choo
se the
motor type “1
FN3
”
and cite
the
excogitative para
m
eters. After
the se
rv
o loop
of the
PPI control,
the ba
ndwi
d
th of
the positio
n is about 100
Hz.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Asymm
e
tric S-sh
ape Accele
rat
i
on/De
cel
e
rati
on for Multi-a
x
ial… (Ch
ang
-ya
n
Ch
ou
)
4785
Figure 6. The
Servo Loop
of the XY Table
We are plan
n
i
ng to conto
u
r
a turnin
g line by th
is XY
table. It starts at the origin
al point
(0, 0), an
d tra
ck
a straight l
i
ne with the l
ength is
L
meter to the corn
er poi
nt alon
g the dire
ctio
n
of
45
upper ri
gh
t. Then, turn to the upper left side a
nd track the
other straig
ht line to th
e
endp
oint. We
adopt the
cri
t
erion of exa
c
t stop fine
to deal with
the probl
em
of corne
r
a
c
cura
cy.
Whe
n
both
o
f
the toleran
c
e
s
of X an
d Y-axis
are
less than
5
10
meter, the a
c
tion of bl
ock
cha
nge
is
execute
d
to int
e
rpol
ate the
se
con
d
strai
ght line. T
h
e
s
e t
w
o lin
es
are
sp
ecifie
d
to
accele
ration
and de
cele
rat
i
on by the S-sha
pe acc/
de
c spe
c
ificatio
n (analy
z
ed i
n
sectio
n 2), and
the pa
ramete
rs
of acc/de
c time,
1
s
T
,
2
s
T
,
1
e
T
, and
2
e
T
are
cal
c
ul
a
t
ed optimally
by the ge
ne
tic
algorith
m
discu
s
sed in section 3.
In this sim
u
la
tion, we set the length
1
.
0
L
met
e
r, and th
e fe
ed-rate
min
/
6000
max
mm
V
(
sec
/
1
.
0
meter
). The limitation of accel
e
ration,
max
A
, is
2
/
1
s
m
and the jerk limitation,
max
J
, is
3
/
10
s
m
. There
are 4
paramete
r
s,
1
s
T
,
2
s
T
,
1
e
T
, and
2
e
T
of th
e first
strai
g
h
t
line, and
an
other
4
para
m
eters,
12
s
T
,
22
s
T
,
12
e
T
, and
22
e
T
of the
se
con
d
on
e.
So, there
are
totally 8 ch
ro
moso
me
s in t
h
is
ca
se. The up
per an
d lower bound
s are d
e
fined by the con
d
ition
s
introdu
ce
d in the se
ction 3.
The se
ction
of the simula
tion about ge
netic
algo
rith
m follow the optimizatio
n method
discu
s
sed i
n
se
ction
3 a
s
a mo
del. Afte
r the
p
r
opa
ga
tions
of two h
undred
s g
e
n
e
ration
s pa
ssed
throug
h, the
optimal pa
ra
meters of the S-s
hape a
cc/d
e
c are o
b
tained.
Th
e
s
e solution
s are
3191
.
0
1
s
T
,
1397
.
0
2
s
T
,
0228
.
0
1
e
T
,
0217
.
1
2
e
T
,
9328
.
2
12
s
T
,
4348
.
0
22
s
T
,
7037
.
0
12
e
T
,
and
5424
.
0
22
e
T
, and
the
co
rrespon
di
ng
conto
u
rin
g
e
rro
r i
s
m
7
10
0348
.
4
a
n
d
c
o
r
ner
e
rro
r i
s
m
7
10
9319
.
3
. Figure 7 i
s
the evoluti
on of the fitness fun
c
tio
n
, and Figu
re 8 sho
w
s the
conto
u
rin
g
pe
rforma
nce.
Figure 7. The
Evolution of the Fitness Fu
ncti
on
Figure 8. Optimal Perform
a
nce: (a
) the
evolution of the avera
ge contourerro
r; (b) the
evolution of the co
rne
r
5. Conclusio
n
This stu
d
y propo
se
s a
n
o
p
timization
al
gorit
hm
for a
s
ymmetri
c
s-sha
pe
acc/de
c u
s
in
g
the method o
f
genetic alg
o
r
ithm with the
con
s
tr
aint
s
made by the motion syste
m
. The algori
t
hm
is verifie
d
n
u
m
eri
c
ally in t
he
simulatio
n
s
of
an XY t
able
driven
b
y
linear moto
rs foll
owi
ng
a
simple
corne
r
ing path with
the block ch
ange criteri
o
n “exact stop
fine” to deal with the corner
errors. Th
e
si
mulation
re
su
lts sho
w
that
the
cont
o
u
rin
g
erro
r a
nd
co
rner
erro
r can
be redu
ce
d
b
y
meta-g
eneti
c
cycl
e an
d th
e effect of
asymmetric
s-shape acc/de
c
is sup
e
rio
r
t
o
the symme
tric
spe
c
ification. We can obtai
n the opt
imal para
m
eters o
f
s-shape a
cc/dec.
0
20
40
60
80
10
0
12
0
14
0
16
0
18
0
20
0
4
4.
5
5
5.
5
6
6.
5
7
x 1
0
-3
G
e
ner
at
i
o
n
Se
l
e
c
t
i
o
n
Er
r
o
r
0
20
40
60
80
10
0
120
140
160
180
200
4
4.
2
4.
4
4.
6
4.
8
5
x 1
0
-7
G
ene
r
a
t
i
on
A
v
ar
ag
e C
ont
our
i
ng E
r
r
o
r
(
m
)
0
20
40
60
80
10
0
120
140
160
180
200
0
0.
5
1
1.
5
2
x 1
0
-6
G
ene
r
a
t
i
on
C
o
rn
e
r
E
rro
r (m
)
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: 4778 – 4
786
4786
Ackn
o
w
l
e
dg
ement
This work
was
su
ppo
rted
in
part
by t
he Mi
ni
stry
o
f
Econ
omic
Affairs, Tai
w
an, ROC,
unde
r G
r
ant
101-E
C
-17
-
A-05
-S1-189,
and by the
Na
tion
al Sci
ence Coun
cil
,
Taiwa
n
, ROC,
unde
r Grant NSC 99
-2
221
-E-19
4
-0
42 -MY3.
Referen
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ong
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i
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/
De
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Al
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Evaluation Warning : The document was created with Spire.PDF for Python.