TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.4, April 201
4, pp. 2613 ~ 2
6
2
1
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i4.4342
2613
Re
cei
v
ed Se
ptem
ber 6, 2013; Re
vi
sed
Octob
e
r 14, 2
013; Accepte
d
Octob
e
r 31,
2013
Robust Adaptive Sliding Mode Control Based on Fuzzy
Compensation for Robot
Yufeng LI
1
*, Kui
w
u
LI
2
, Yutian PAN
1
, Kelei LI
1
1
School of Co
mputer an
d Co
ntrol Eng
i
n
eeri
ng, Nort
h Un
iv
ersit
y
of C
h
in
a,
T
a
i
y
u
an 0
3
0
0
51, Chi
n
a
2
North
w
est Institute of Mecha
n
ical & El
ectr
ic
al Eng
i
ne
eri
ng,
Xia
n
y
a
ng 7
1
2
099, Ch
in
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: 1133
46
71@
q
q
.com
A
b
st
r
a
ct
Aimed at the
prob
le
ms of l
o
w
control ac
curacy an
d w
eak rob
u
stnes
s influe
nce
d
b
y
externa
l
disturb
ance, fri
c
tion, lo
ad c
h
a
nges,
mo
de
lin
g errors
an
d o
t
her issu
es i
n
ammuniti
on
au
to-loa
din
g
ro
b
o
t
control system, a new robus
t
adaptive fu
z
z
y
s
l
iding
mo
de
controller based on
fu
z
z
y
compensation is
proposed. The
control
arc
h
itec
ture em
ploys fu
z
z
y
system
s t
o
com
p
ensate
adaptively for plant unc
ertainties
to disti
ngu
ish
different
distur
banc
e co
mpe
n
s
ation
ter
m
s a
nd
appr
oxi
m
at
e e
a
ch
of the
m
r
e
spectiv
e
ly.
T
h
e
stability of the r
obust ad
aptiv
e fu
zz
y
sli
d
i
ng
mode co
ntrol (S
MC) and the c
onver
genc
e of the tracking
error
s
are e
n
sure
d b
y
usin
g the L
y
apu
nov th
eor
y. By analy
z
i
n
g an
d co
mp
ar
ing th
e si
mu
la
tion res
u
lts, it is
obviously
shown that
the c
o
ntrol system
c
an lighten t
he
effect
on the c
ontrol syst
em
caused by
diff
erent
disturbance fac
t
ors and elim
inate the
system chattering
inst
ead
of traditional SMC. As
a r
e
sult, the contr
o
l
system
has gr
eat
dy
nam
i
c
fe
atures
and robust stabilit
y and
m
eets the r
e
quir
e
ment
that
the actual
motion
of amm
u
nition
auto-
loading robot quick
ly tracks the scheduled trajectory.
Ke
y
w
ords
: fu
z
z
y
co
mp
ens
ati
on, slid
ing
mo
d
e
contro
l, no
nl
i
near syste
m
, u
n
certai
nty distu
r
banc
e
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
It is wid
e
ly reco
gni
zed th
at rob
o
tic
ro
bot
have to
face m
any u
n
ce
rtaintie
s i
n
their
dynamics,in particula
r structur
ed un
ce
rtainty,su
ch as payloa
d
para
m
eter, a
nd
u
n
structu
r
ed
one,such a
s
f
r
iction
a
nd
di
sturb
a
n
c
e
[1]. It is
di
fficult t
o
obtai
n the
desi
r
ed
control pe
rforman
c
e
whe
n
the
co
ntrol alg
o
rith
m is only b
a
se
d on
th
e
robot dyn
a
m
ic mo
del.
To overcom
e
this
probl
em,
a
n
adaptive co
ntrol of
robot robot
u
s
in
g
fu
zzy co
mpe
n
sator wa
s de
si
gned
in
[1], a
nd
the adaptive
control sch
e
m
es utili
zed
an fuzzy l
ogi
c syst
em (FL
S
) as a
com
pen
sator fo
r
any
uncertainty. I
n
[2] a
fuzzy
co
mpe
n
satio
n
ba
se
d o
n
comp
uted
torque
co
ntrolle
r i
s
d
e
si
gne
d
to
handl
e
in
evitable un
certai
nties,an
d
the
fuzzy co
mp
e
n
sative
co
ntroller i
s
u
s
ed
for
app
roxima
ting
lumped
un
ce
rtainty. A practical a
nd
effective co
mpen
sato
r b
a
se
d o
n
a
d
a
p
tive fuzzy l
ogi
c
system
s is e
m
ployed to compen
sate t
he joint fricti
on in [3-6]. The basi
c
ide
a
of the adaptive
fuzzy logi
c co
ntrol ari
s
e
s
from the fact a wide
cla
s
s of nonlinea
r sy
stem
ca
n be
approximated
to
arbitrary cl
osene
ss
by a fuzzy lo
gic
syst
em. Adaptive
fuzzy e
s
tima
tor provide
s
a
tool for ma
ki
ng
use
of the fuzzy info
rmati
on in a
syst
ematic a
nd e
fficient mann
er.Hsu
Chu
n
-
Fei et al [7,
8]
sug
g
e
s
ted th
e method
s of
robu
st wavel
e
t-ba
s
ed ada
ptive
neural
controlle
r de
si
gn with a fu
zzy
comp
en
sato
r. But the
structure
of ne
ural
network
and
the ad
aptive l
a
ws h
a
ve to
be fou
nd
by the
trial-a
nd-erro
r method. To
overcome th
ese difficu
ltie
s, in [9] a ro
bust ad
aptive
fuzzy control
o
f
robot
s ba
sed
on fuzzy com
pen
sation
wa
s pro
p
o
s
ed.
In the re
cent
decade,
slidi
ng mod
e
con
t
rol (SM
C
)
which
has
goo
d co
ntrol p
e
rf
orma
nce
for nonlin
ea
r system
s, wa
s used in ro
botic robot
control in [10,
11], and the most sig
n
ificant
prop
erty of S
M
C i
s
it
s rob
u
stne
ss.In fa
ct, a p
u
re
S
M
C suffers
f
r
om some
di
sadvantag
es. First,
there is the
probl
em of chattering, which is
the hi
gh-frequency oscilla
tions
of the controller
output, bro
u
g
h
t about by the high
spe
e
d
swit
chin
g
n
e
ce
ssary for
the establi
s
h
m
ent of a sli
d
ing
mode. Se
con
d
, an SM
C is extremely vu
lnera
b
le to m
easure
noi
se
sin
c
e th
e in
put dep
end
s
on
the sig
n
of
a mea
s
u
r
ed
variable
that
is very
clo
s
e to zero. T
h
ird, the SM
C may e
m
pl
oy
unne
ce
ssarily
larg
e contro
l sign
als to
overcome th
e pa
ramet
r
ic uncertai
n
tie
s
. To
attenu
ate
these
difficult
ies,
seve
ral
method
s
we
re p
r
opo
se
d i
n
[12
-
14].To
overcome
th
ese
difficultie
s, in
this pap
er
we pro
p
o
s
e th
e rob
u
st ad
a
p
tive fu
zzy sl
iding mo
de control
schem
es which utili
ze
fuzzy lo
gic
sy
stem
s as
co
mpen
sato
rs f
o
r any u
n
cert
ainty to alleviate the chattering
and
red
u
ce
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2613 – 2
621
2614
the tra
ckin
g
errors, and
re
strain f
r
iction,dist
urban
ce, loa
d
variations
and
other n
onlin
ea
r
influen
cing fa
ctors.
This p
ape
r is organi
zed
as follows:
Section 2
provide
s
the
dynamic m
odel of
ammunition auto-loading robot.In
section 3
classi
cal sliding mode contro
ller f
o
r robot and its
stability are g
i
ven. Section 4 pre
s
ent
s ro
bust ad
apt
ive
fuzzy slidi
n
g
mode co
ntro
llers
with fuzzy
comp
en
satio
n
are give
n. Finally, experiment re
sults
and con
c
lu
si
ons a
r
e give
n
in se
ction 5
and
6 r
e
spec
tively.
2. D
y
namic
Model of Am
m
unition Au
to-loa
ding Robot
The ammunition auto-loading robot is
s
h
ow
n in Fig.1.Its
dynamic
model
may be
expre
s
sed in
the followin
g
Lagrang
e form:
)
,
,
(
)
(
)
,
(
)
(
q
q
q
F
q
G
q
q
q
C
q
q
D
(1)
Whe
r
e
]
,
,
[
T
n
1
q
q
q
is an
1
n
vector of join
t position,
T
n
1
]
,...,
[
q
q
q
is an
1
n
vec
t
or of joint veloc
i
ty,
]
,
,
[
1
n
q
q
q
is an
1
n
vector of j
o
int accel
e
rat
i
on,
)
(
q
D
is an
n
n
inertia matrix,
)
,
(
q
q
C
is an
n
n
matrix resulting from
Cori
o
lis a
nd centrifugal forces,
)
(
q
G
is an
1
n
gravity vecto
r
,
is the
contro
l input.
)
,
,
(
q
q
q
F
is the
uncertainty
g
enerated
by f
r
iction
r
F
,
load chan
ge
s and the distu
r
ban
ce
d
addi
ng
on the
.
Figure 1. Ammunition Auto
-loadi
ng Rob
o
t
3. Classical
Sliding Mode Con
t
roller for Ro
bot
In the design
of sliding mode controll
e
r
for ammunit
i
on auto-l
oadi
ng rob
o
t, the control
obje
c
tive is t
o
drive th
e jo
int positio
n
q
to the d
e
si
red
positio
n
d
q
. So by definin
g the tra
c
king
error to
be
in
the follo
wing
form:
q
q
e
d
.The
sl
iding su
rface can
be written
a
s
:
e
e
s
,where
,
]
,
,
,
,.
diag[
1
n
i
in whi
c
h
i
is a
po
sitive co
nsta
nt. T
he control ob
jective can
now be
achie
v
ed by
choo
sing the
co
ntrol input
so
th
at the
sliding
surfa
c
e
satisf
ies th
e follo
wi
ng
s
u
ffic
i
ent c
o
ndition:
i
i
i
s
s
dt
d
2
1
,where
i
is a
po
sitive co
nsta
nt,which i
ndi
cate
s that the
energy of
s
sho
u
ld de
cay a
s
long a
s
s
is
not zero.
To set up
the co
ntro
l
, define the
referen
c
e
states to b
e
:
e
q
s
q
q
d
r
,and
e
q
s
q
q
d
r
.Now th
e cont
rol inp
u
t
can b
e
c
h
os
en to be in the following form:
s
K
sgn
ˆ
,
ˆˆ
ˆ
ˆ
rr
Dq
C
q
G
A
s
(
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Adapt
ive Sliding M
ode Control B
a
se
d
on Fu
zzy Com
pen
sati
on for Ro
bot (Yufeng LI)
2615
Whe
r
e
]
,
,
,
,.
diag[
11
nn
ii
k
k
k
K
is a
di
agon
al po
siti
ve definite m
a
trix in
whi
c
h
s
k
ii
'
are
po
sitive con
s
tant
s an
]
,
,
,
,.
diag[
1
n
i
a
a
a
A
is a
diag
onal
po
sitive defi
n
ite matrix in
whi
c
h
s
a
i
'
are also positive con
s
tants. So we
have:
s
K
f
s
A
C
s
D
sgn
)
(
(3)
Whe
r
e
G
G
G
C
C
C
D
D
D
G
q
C
q
D
f
r
r
ˆ
,
ˆ
,
ˆ
,
. It c
an be
proved
that b
y
cho
o
si
ng
K
su
ch t
h
at
bound
ii
i
f
k
,where
bound
i
f
is th
e b
ound
ary of
i
f
, the
overall sy
ste
m
is asympto
t
ically stable.
A Lyapun
ov functio
n
is
a
scala
r
fun
c
tion
(x)
L
defined on
a
regio
n
D
that is contin
uou
s
,
pos
itive definite, (
(x)
L
>0
f
o
r
a
l
l
0
y
), and
ha
s
contin
uou
s fi
rst-ord
e
r pa
rtial de
rivatives at
every point o
f
D
. The deriva
t
ive of
L
with res
p
ec
t to the
s
y
s
t
em
(x
)
x
f
, written as
(x)
L
is
defined
a
s
th
e dot
produ
ct
. The
existe
n
c
e
of a
Lyap
unov fun
c
tion
for
whi
c
h
(x)
0
L
on som
e
regio
n
D
contain
i
ng
the origi
n
,
guarantee
s the
stability o
f
the ze
ro
sol
u
tion of
(x
)
x
f
, while
the existen
c
e
of a Lya
pun
ov function
for
whi
c
h
(x)
L
is
negative d
e
finite on
som
e
regi
on
D
contai
ning the origin guarantees the asy
m
pt
otical stability of the zero sol
u
tion of
(x
)
x
f
.
Con
s
id
er a L
y
apunov function candi
dat
e
Ds
s
L
T
2
1
,s
inc
e
D
is sy
mmetric and
positive
definite, then for
0
,
0
L
s
. It can be proved that:
0
)))
(
sgn
(
(
)
)
(
sgn
)
(
(
2
1
1
As
s
As
s
s
K
f
s
Cs
s
K
f
s
A
C
s
s
D
s
s
D
s
L
T
T
i
ii
i
i
n
i
T
T
T
(4)
Thus, Equ
a
tion (4
) gua
ra
ntees the d
e
c
ay of the e
nergy of
s
as l
ong a
s
0
s
.
T
he
s
u
ffic
i
ent c
o
ndition is
thus
s
a
tis
f
ied.
4. Sliding M
ode Con
t
roll
er Bas
e
d on
Fuzz
y
Compensa
tion
Select the L
y
apunov fun
c
tion a
s
:
)
~
~
(
2
1
1
n
i
i
i
T
i
T
Ds
s
L
, where
i
i
i
*
~
，
*
i
is the desi
r
e
d
para
m
eter,
0
i
.We have:
r
r
q
D
F
G
q
C
q
D
q
D
s
D
(5)
)
~
~
2
1
1
n
i
i
i
T
i
T
T
s
D
s
s
D
s
L
n
i
i
i
T
i
r
r
T
F
G
q
C
q
D
s
1
~
~
)
(
(6)
Whe
r
e
)
,
,
(
q
q
q
F
is un
known no
nline
a
r fun
c
tion.
)
|
,
,
(
ˆ
q
q
q
F
based
on the m
u
lti input-
multi output (MIMO) fu
zzy
system
i
s
ad
opted to ap
proximate to
)
,
,
(
q
q
q
F
. In orde
r to elim
inate
the influence
of approximation error and keep
st
abili
zation, robust
adaptive fuzzy sliding mode
controlle
r is d
e
sig
ned a
s
:
)
sgn(
)
|
,
,
(
ˆ
)
(
)
,
(
)
(
s
W
s
K
q
q
q
F
q
G
q
q
q
C
q
q
D
D
r
r
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2613 – 2
621
2616
Whe
r
e
)
(
i
D
K
diag
K
,
,
2
,
1
,
0
n
i
K
i
n
i
w
w
w
diag
W
i
Mi
M
M
n
,
,
2
,
1
,
],
,
,
[
1
1
, and:
)
,
,
(
)
,
,
(
)
,
,
(
)
|
,
,
(
ˆ
)
|
,
,
(
ˆ
)
|
,
,
(
ˆ
)
|
,
,
(
ˆ
T
n
T
2
T
1
2
2
1
1
q
q
q
q
q
q
q
q
q
q
q
q
F
q
q
q
F
q
q
q
F
q
q
q
F
n
n
(8)
Fuzzy app
rox
i
mating erro
r is
)
|
,
,
(
ˆ
)
,
,
(
F
*
1
q
q
q
F
q
q
q
, and we h
a
v
e:
n
i
i
i
T
i
T
s
W
s
q
q
q
F
q
q
q
s
L
1
D
1
~
~
))
sgn(
K
)
|
,
,
(
ˆ
)
,
,
(
F
(
))
,
,
(
~
~
~
(
)
K
(
1
D
q
q
q
s
s
W
s
s
T
i
i
n
i
i
i
T
i
T
(9)
Whe
r
e
)
,
,
(
,
~
*
q
q
q
is the fuzzy sy
sterm
.
The adaptive rule is:
n
i
q
q
q
s
i
i
,
2
,
1
),
,
,
(
~
1
(10)
Therefore, we have:
0
K
K
D
D
s
s
s
W
s
s
s
L
T
T
T
(11)
Becau
s
e th
e
unce
r
tainty of ammunitio
n
auto-l
oadin
g
rob
o
t, inclu
d
ing fri
c
tion, external
disturban
ce, and
l
oad ch
a
nge exist
si
m
u
ltaneo
usly, we
ca
n
con
s
i
der
th
e ca
se of
rob
u
st ada
ptive
fuz
z
y
c
o
mpens
a
tion
with res
p
ec
t to fric
t
i
on, ex
tern
al
disturban
ce
a
nd lo
ad
cha
n
ge. Becau
s
e
the
load
chan
ge i
s
rel
a
tive to velocity, the fu
zzy
sy
stem which app
roxi
mate
extern
al
disturban
ce can
b
e
w
r
itte
n
as
)
|
,
,
(
ˆ
1
q
q
q
F
. In orde
r to
decrea
s
e
the
num
ber of f
u
zzy rule
s, th
e un
ce
rtaintie
s
are
de
com
p
o
s
ed
an
d the
method
ba
se
d on
the t
r
ad
itional fu
zzy
comp
en
satio
n
is ad
opted
to
desi
gn the
controlle
r. The
r
efor
e, the d
y
namic e
quat
ion for am
m
unition auto
-
l
oadin
g
co
uld
be
descri
bed a:
d
)
(
)
t
,
,
,
(
)
(
)
,
(
)
(
q
F
q
q
q
e
q
G
q
q
q
C
q
q
D
r
(12)
Whe
r
e,
(
,
)
(
,
,
)
;
()
(
,
)
;
(
,
,
,
t
)
[
()]
[
(
,
)
]
[
()
]
nn
D
C
G
C
q
q
C
m
q
q
G
q
G
m
q
e
q
q
q
e
D
qq
e
C
qqq
e
G
q
(
,
)
(
,
)
;
(
,,
)
(
,,
)
;
(
,
)
(
,
)
Dn
c
n
C
n
c
n
G
n
c
n
e
D
m
q
q
D
mq
q
e
C
m
q
q
q
C
mq
q
q
e
G
m
q
G
m
q
; and
n
m
is known
nom
inal value an
d
nc
m
is a
c
tual val
ue. The un
ce
rtain pa
rts ca
n be expre
ssed as:
d
)
(
)
t
,
,
,
(
)
,
,
(
q
F
q
q
q
e
q
q
q
F
r
(13)
The above
formula
can b
e
di
vided into:
)
,
(
)
,
(
)
,
,
(
2
1
q
q
F
q
q
F
q
q
q
F
.
Whe
r
e,
d
C
1
)
(
)]
(
[
e
]
)
,
(
[
)
,
(
q
F
q
G
q
q
q
C
e
q
q
F
r
C
,
]
)
(
[
)
,
(
2
q
q
D
e
q
q
F
D
. The rob
u
st
adaptive fuzzy adaptive sli
d
ing mode
controller is designed as:
)
sgn(
)
|
,
(
ˆ
)
|
,
(
ˆ
)
(
)
,
(
)
(
2
2
1
1
s
W
s
K
q
q
F
q
q
F
q
G
q
q
q
C
q
q
D
D
r
r
(14)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Adapt
ive Sliding M
ode Control B
a
se
d
on Fu
zzy Com
pen
sati
on for Ro
bot (Yufeng LI)
2617
Whe
r
e
.
,
,
2
,
1
,
],
,
,
[
2
1
1
1
n
i
w
w
w
diag
W
i
i
m
m
m
i
n
The a
daptive rule i
s
:
n
i
q
q
s
q
q
s
i
i
i
i
i
i
,
2
,
1
),
,
(
),
,
(
2
1
2
2
1
1
1
1
(15)
Select the Lyapun
ov functi
on as:
)
~
~
)
~
~
(
2
1
1
2
2
2
1
1
1
1
n
i
i
i
T
i
n
i
i
i
T
i
T
Ds
s
L
(16)
Therefore:
n
i
i
i
T
i
n
i
i
i
T
i
r
T
F
G
q
C
q
D
s
L
1
2
2
2
1
1
1
1
~
~
~
~
)
(
(17)
Fuzzy app
rox
i
mating erro
r is
)
|
,
(
ˆ
)
,
(
*
1
1
1
1
q
q
F
q
q
F
,
)
|
,
(
ˆ
)
,
(
*
2
2
2
2
q
q
F
q
q
F
Therefore,
)
)
,
(
~
~
~
(
))
,
(
~
~
~
(
)
(
K
1
2
2
2
2
2
1
1
1
1
1
1
2
1
D
n
i
T
i
i
i
i
T
i
T
i
n
i
i
i
i
T
i
T
T
q
q
s
q
q
s
w
s
s
s
L
)
(
K
2
1
D
w
s
s
s
T
T
(18)
The FLS is
compo
s
ed
of four mai
n
co
mpone
nts: a
fuzzifie
r
, a fuzzy rule ba
se
, a fuzzy
inferen
c
e e
n
g
i
ne and a d
e
fuzzfier a
s
sh
own in Fig
u
re
2.
Figure 2. Block
Diag
ram o
f
Fuzzy Lo
gic Systems
And the fuzzy system is:
)
|
,
(
ˆ
)
|
,
(
ˆ
)
|
,
(
ˆ
)
|
,
(
ˆ
)
|
,
(
ˆ
)
|
,
(
ˆ
)
|
,
,
(
ˆ
2
2
1
2
2
2
2
1
2
2
2
1
2
1
1
1
1
n
n
n
n
q
q
F
q
q
F
q
q
F
q
q
F
q
q
F
q
q
F
q
q
q
F
(19)
5. Experiments fo
r Amm
unition Au
to
-loading Rob
o
t
The kin
e
tics equatio
n of ammunition a
u
t
o-loadi
ng rob
o
t is:
2
1
2
1
2
2
1
1
2
1
2
12
1
2
12
2
2
12
2
1
2
22
2
12
2
21
2
11
)
,
,
(
)
(
)
(
0
)
)(
(
C
-
)
(
C
)
(
C
-
)
(
)
(
)
(
)
(
q
q
q
F
g
q
q
g
g
q
q
g
q
q
q
q
q
q
q
q
q
q
D
q
D
q
D
q
D
(20
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2613 – 2
621
2618
),
cos(
2
)
(
)
(
2
2
1
2
2
2
2
2
1
2
1
2
11
q
r
r
m
r
m
r
m
m
q
D
),
cos(
)
(
)
(
2
2
1
2
2
2
2
2
21
2
12
q
r
r
m
r
m
q
D
q
D
,
)
(
2
2
2
2
22
r
m
q
D
).
sin(
)
(
2
2
1
2
2
12
q
r
r
m
q
C
Whe
r
e
1
m
and
2
m
are
the mass of link
1
and lin
k
2
, and
1
r
and
2
r
are the le
ngths of lin
k1
and link2. Le
t
T
T
T
q
q
q
q
x
q
q
y
]
,
,
,
[
,
]
,
[
,
]
,
[
2
2
1
1
2
1
2
1
.
The paramet
ers:
kg
m
kg
m
m
r
m
r
5
.
1
,
1
,
8
.
0
,
1
2
1
2
1
. The
co
n
t
rol obj
ect
s
t
o
ma
ke
the outp
u
ts
2
1
,
q
q
track
the des
i
red trajec
tories
t
y
d
sin
3
.
0
1
and
t
y
d
sin
3
.
0
2
r
e
spec
tively.
The memb
ership fun
c
tion
as sho
w
n in
Figure 3 is de
fined as:
)
)
24
/
(
exp(
)
(
2
l
i
i
i
l
i
x
x
x
A
(21)
Whe
r
e
l
i
x
are
12
/
,
0
,
12
/
,
6
/
and
6
/
, res
p
ec
tively,
5
,
4
,
3
,
2
,
1
i
,
l
i
A
is the
fuzzy set incl
uding
NB,NS,ZO,PS,PB belong to the fuzzy rule.
Figure 3.
Membershi
p
Fun
c
tion
The control b
a
se
d on fri
c
ti
on, external
dist
urban
ce
a
nd loa
d
ch
an
ges
com
pen
sation is
use
d
for th
e
ca
se
with f
r
iction,
external di
sturb
a
n
c
e a
nd lo
ad
cha
nge
s, a
nd the
cont
roller
para
m
eters a
r
e:
.
0001
.
0
,
10
,
10
2
1
2
1
I
K
D
The initial
states are:
.
0
)
0
(
)
0
(
)
0
(
)
0
(
2
1
2
1
q
q
q
q
The fric
tion is
].
2
,
2
[
,
)
20
sin(
1
.
0
)
20
sin(
05
.
0
,
)
sgn(
6
15
)
sgn(
6
15
)
(
2
2
1
1
dia
W
t
t
q
q
q
q
q
F
d
The fu
zzy slidi
ng
mode
controlle
r is
given in Eq
uation
(14
)
, and the
ada
ptive rule i
s
given in E
quation
(15
)
.
T
he
simulatio
n
re
sults
by matla
b
are
sh
own i
n
Figu
re 4
-
Fi
gure
8, and t
he sli
d
ing m
o
de control
(S
MC)
without fu
zzy compe
n
sation an
d with a
daptive fuzzy
comp
en
satio
n
are
appli
e
d
respe
c
tively for
tracking
control of am
muni
tion auto
-
loa
d
i
ng robot i
n
t
he ca
se with friction,
exte
rnal
di
sturban
ce
and loa
d
ch
a
nge
s.
It is seen in Figure 4 and 5 that tracki
n
g
ac
curacy with adaptive fuzzy comp
en
sation is
more hi
ghe
r than that in S
M
C with
out fuzzy co
m
pen
sation, an
d tracking
traje
c
tory and d
e
si
red
trajec
tory almos
t
c
o
inc
i
de completely in the former.
-0
.
8
-0
.
6
-0
.
4
-0
.
2
0
0.
2
0.
4
0.
6
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
x
M
e
m
b
er
s
h
i
p
f
u
nc
t
i
o
n
de
gr
ee
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Adapt
ive Sliding M
ode Control B
a
se
d
on Fu
zzy Com
pen
sati
on for Ro
bot (Yufeng LI)
2619
(a)
(b)
Figure 4. Position Tra
cki
ng
of Joints 1 a
nd 2
(a) SM
C with
out comp
en
sation and (b)
SMC with ad
aptive fuzzy compen
satio
n
(a)
(b)
Figure 5. Position Tra
cki
ng
Error of Joint
s
1 and 2
(a) SM
C with
out comp
en
sation and (b)
SMC with ad
aptive fuzzy compen
satio
n
(a)
(b)
Figure 6. Speed Tra
c
king o
f
Links 1 a
nd
2
(a) SM
C with
out comp
en
sation and (b)
SMC with ad
aptive fuzzy compen
satio
n
In Figure 6 th
e sp
eed
traje
c
tory
without
fu
zzy com
p
e
n
satio
n
ha
s seriou
s ch
attering
an
d
great e
r
ror,
but there i
s
a slig
ht error at
the begi
nning
of the
spe
ed tra
c
ki
ng in SM
C
with
adaptive fuzzy compensati
on,
then tracking effect i
s
better.Re
ferring to Fi
gure 7, the
contro
l
torque
inp
u
ts ch
attering
i
s
effe
ctively eliminated
by
usi
ng the
a
daptive fu
zzy
com
pen
sati
on
whi
c
h
ca
n
well re
du
ce th
e influe
nce o
f
friction,exte
rnal
distu
r
b
a
n
ce
an
d lo
ad
ch
ang
e o
n
t
h
e
0
1
2
3
4
5
6
7
8
9
10
-0
.
4
-0
.
2
0
0.
2
0.
4
ti
m
e
(
s
)
P
o
s
i
t
i
on t
r
ac
k
i
n
g
1
I
dea
l
pos
i
t
i
on s
i
g
nal
pos
i
t
i
on t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0
.
4
-0
.
2
0
0.
2
0.
4
ti
m
e
(
s
)
P
o
s
i
t
i
on t
r
ac
k
i
ng 2
I
dea
l
pos
i
t
i
on s
i
g
nal
pos
i
t
i
on t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0.
4
-0.
2
0
0.
2
0.
4
ti
m
e
(
s
)
P
o
s
i
t
i
on
t
r
ac
k
i
ng 1
I
d
e
a
l
po
s
i
t
i
on s
i
gnal
pos
i
t
i
o
n
t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0.
4
-0.
2
0
0.
2
0.
4
ti
m
e
(
s
)
P
o
s
i
t
i
on t
r
ac
k
i
n
g
2
I
d
e
a
l
po
s
i
t
i
on s
i
gnal
pos
i
t
i
o
n
t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0
.
0
5
0
0.
05
0.
1
0.
15
tim
e
(
s
)
P
o
s
i
t
i
on
t
r
ac
k
i
ng
er
r
o
r
of
j
o
i
n
t
1
0
1
2
3
4
5
6
7
8
9
10
-0
.
0
1
0
0.
01
0.
02
0.
03
tim
e
(
s
)
P
o
s
i
t
i
on t
r
ac
k
i
ng er
r
o
r
o
f
j
o
i
n
t
2
0
1
2
3
4
5
6
7
8
9
10
-0
.
0
2
0
0.
02
0.
04
ti
m
e
(
s
)
P
o
s
i
t
i
on
t
r
ac
k
i
ng
e
r
r
o
r
o
f
j
o
i
n
t
1
0
1
2
3
4
5
6
7
8
9
10
-0
.
0
2
-0
.
0
1
0
0.
01
0.
02
ti
m
e
(
s
)
P
o
s
i
t
i
on
t
r
ac
k
i
ng
e
r
r
o
r
o
f
j
o
i
n
t
2
0
1
2
3
4
5
6
7
8
9
10
-0
.
4
-0
.
2
0
0.
2
0.
4
ti
m
e
(
s
)
S
p
eed t
r
ac
k
i
ng
f
o
r
Li
nk
1
I
d
ea
l
S
p
e
e
d s
i
gna
l
S
p
ee
d
t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0
.
4
-0
.
2
0
0.
2
0.
4
ti
m
e
(
s
)
S
p
eed t
r
a
c
k
i
ng
f
o
r
Li
n
k
2
I
d
ea
l
S
p
e
e
d s
i
gna
l
S
p
ee
d
t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0.
4
-0.
2
0
0.
2
0.
4
ti
m
e
(
s
)
S
p
e
ed t
r
a
c
k
i
ng
f
o
r
Li
n
k
1
I
deal
S
p
e
ed s
i
gnal
S
p
e
ed t
r
ac
k
i
ng
0
1
2
3
4
5
6
7
8
9
10
-0.
4
-0.
2
0
0.
2
0.
4
ti
m
e
(
s
)
S
peed t
r
ac
k
i
n
g
f
o
r
L
i
nk
2
I
deal
S
p
e
ed s
i
gnal
S
p
e
ed t
r
ac
k
i
ng
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2613 – 2
621
2620
system.F
ricti
on,external d
i
sturb
a
n
c
e a
nd load
cha
nge fuzzy compen
satio
n
F
p1
and F
p2
are
sho
w
n in Fi
g.8, and th
e adaptive
fuzzy
comp
e
n
satio
n
ca
n
comp
en
sat
e
the effect
o
f
friction,extern
al disturban
ce and loa
d
ch
ange
s in am
munition auto
-
loadi
ng robot
.
(a)
(b)
Figure 7. Con
t
rol Torque In
puts of links 1
and 2
(a) SM
C with
out comp
en
sation and (b)
SMC with ad
aptive fuzzy compen
satio
n
Figure 8. Friction, External Distu
r
ba
nce and Lo
ad Ch
ange
Fuzzy Com
p
ensation F
p
1
a
nd F
p
2
Table
1 su
mmari
ze
s a
nume
r
ical
comp
ari
s
o
n
of
the
two
control sche
mes, with
maximum of
absolute val
ue an
d root-mean
-squa
re
(rm
s) val
u
e
of the tra
cki
ng e
rro
rs. F
r
om
these
com
p
a
r
ative simulati
on re
sult
s, it is foun
d
that the propo
se
d
control sch
e
me is
sup
e
ri
o
r
to
the traditiona
l SMC witho
u
t compe
n
sa
tion. C
onseq
uently, we have found
that the SMC with
adaptive fu
zzy comp
en
sati
on
sch
eme f
o
r a
mmunitio
n
auto
-
loa
d
in
g ro
bot i
s
fea
s
ible
and
ro
b
u
st
to the friction
s, external di
sturb
a
n
c
e
s
a
nd load
cha
n
ges throug
h the simul
a
tion
s.
Table 1. Co
m
pari
s
on of the
Trackin
g
Errors
T
y
pe of
Cont
roller
Trac
k
i
ng Erro
r (
m
ax)
Trac
k
i
ng Erro
r (r
ms
)
Joint1
Joint 2
Joint 1
Joint 2
SMC w
i
thout
Fuz
z
y
Compensatio
n
0.0521
0.0252
0.0315
0.0123
SMC
w
i
th Adapti
v
e Fuzzy
Compe
n
sation
0.0375
0.0181
0.0156
0.0052
0
1
2
3
4
5
6
7
8
9
10
-1
0
0
10
20
30
ti
m
e
(
s
)
C
o
n
t
r
o
l in
p
u
t
o
f
lin
k
1
0
1
2
3
4
5
6
7
8
9
10
-5
0
5
10
15
ti
m
e
(
s
)
C
ont
r
o
l
i
n
p
u
t
of
l
i
n
k
2
0
1
2
3
4
5
6
7
8
9
10
0
5
10
15
20
ti
m
e
(
s
)
C
ont
r
o
l
i
n
p
u
t
of
l
i
nk
1
0
1
2
3
4
5
6
7
8
9
10
-5
0
5
10
ti
m
e
(
s
)
C
o
nt
r
o
l
i
nput
o
f
l
i
n
k
2
0
1
2
3
4
5
6
7
8
9
10
-2
0
2
4
6
ti
m
e
(
s
)
F
1
and
F
p
1
Pr
a
c
ti
c
a
l
F
Es
ti
m
a
ti
o
n
o
f
F
0
1
2
3
4
5
6
7
8
9
10
-2
-1
0
1
2
ti
m
e
(
s
)
F
2
and
F
p
2
Pr
a
c
ti
c
a
l
F
Es
ti
m
a
ti
o
n
o
f
F
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TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Adapt
ive Sliding M
ode Control B
a
se
d
on Fu
zzy Com
pen
sati
on for Ro
bot (Yufeng LI)
2621
6. Conclusio
n
In this pa
pe
r, a
slidi
n
g
mode
cotrol
with
ro
bu
st ada
ptive fu
zzy
co
mpe
n
s
ation
is
pre
s
ente
d
to compe
n
sate
the effect of fr
iction, external di
stu
r
b
ance and lo
ad ch
ange
s
on
ammunitio
n
auto-lo
adin
g
robot. The control ar
chite
c
ture e
m
ploy
s adaptive fu
zzy sy
stems to
comp
en
sate
adaptively for plant
un
ce
rtainties to
di
st
ingui
sh different
di
sturban
ce co
mpe
n
sa
tion
terms and
a
pproxim
ate e
a
ch
of them
respe
c
ti
vely.
Re
sults are comp
ared wi
th
SMC with
out
f
u
zzy
comp
e
n
sat
i
o
n
,
whi
c
h sh
ow
s t
h
a
t
f
u
zzy
com
pen
sation i
s
essential fo
r obtaini
ng l
o
w
trajec
tory track
i
ng errors
. It is
s
h
own that the
control
system ca
n lig
hten the effect on the control
system
cau
s
ed by differe
nt dist
urban
ce facto
r
s a
n
d
eliminate t
he sy
stem chattering th
at the
traditional SM
C witho
u
t fuzzy com
pen
sa
tion can n
o
t accompli
sh.
Ackn
o
w
l
e
dg
ements
This proje
c
t
is
sup
p
o
r
ted
by the
National
Natu
re
Scien
c
e
Fou
ndation
of
China
(No.
5127
5489
)
Referen
ces
[1]
Yoo BK, H
a
m W
C
. Adapt
ive C
ontrol
of
Rob
o
t Man
i
p
u
lator
Usin
g
F
u
zz
y
Comp
e
n
sator.
IEEE
Tra
n
s
a
c
ti
on
s on
Fu
z
z
y System
s
. 20
00; 8(2):
186-1
99.
[2]
Z
uoshi S
o
n
g
, Jian
qi
ang
Yi,
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g
b
i
n Z
h
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. A comp
ute
d
torqu
e
c
ont
roller
for u
n
c
e
rtain r
o
b
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u
zzy
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u
zzy Sets and Systems
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[3] Chen
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he
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u
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ode
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uncerta
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a
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n
-F
ei Hs
u, Chih-M
in
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h
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a
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y
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a
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F
u
z
z
y
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eur
al C
o
n
t
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e
si
g
n
w
i
t
h
a Co
nstru
c
tive Lear
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u
zzy Systems
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11; 13(3): 1
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ong
D
ong.
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u
st
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d
aptiv
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F
u
zz
y
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l of
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nip
u
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u
zz
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h
i
dbor
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e
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F
u
zz
y
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l
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d
in
g
mo
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o
l f
o
r a
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o
t
ma
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u
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h
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u
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y
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y
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y
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e Sl
idi
n
g Mo
de
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l.
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