Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 2
,
A
p
r
il
201
6, p
p
.
90
1
~
90
8
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
2.9
594
9
01
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Robust Backstepping Tracking Co
ntrol of Mobile Robot Based
on Nonlinear Disturbance Observer
Mahm
ood Ali
Moqbel
Ob
aid*
,
**, Abdul
Ras
h
id
H
u
s
a
in*,
Ali Abd
o
Mohammed Al-kub
ati
*
*
* Faculty
of
Electrical En
g
i
neer
ing, Universiti Te
knologi Malay
s
ia, Malay
s
ia
** Faculty
of
Co
mputer Scien
c
e
and Eng
i
nee
r
ing
,
Hodeid
ah Univ
ersit
y
,
Yem
e
n
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 27, 2015
Rev
i
sed
D
ec 24
, 20
15
Accepte
d
Ja
n 16, 2015
This paper pres
ents a robust b
ackstepp
i
ng con
t
rol (BC) meth
odbased on
nonlinear disturbance observer (NDOB
) for trajectory
tracking of the
nonholonomic
wheeled mobile robot (WMR
) in the pr
esence of extern
al
disturbances an
d parameters uncertain
ties. At first, a
bound
ed Fuzzy
logic
based backstepp
i
ng controller (
B
FLBC)
is designed to con
t
rol the WMR
without consid
ering th
e effects of
the external disturb
a
nces and the
parameters un
certainties.
Ty
pically
,
the conv
entional BC con
t
roller dep
e
nds
upon the state
tr
acking
errors an
aly
s
is, wher
e un
bounded velo
city
sign
al is
produced for
th
e applications that ha
v
e
huge tracking errors. Therefor
e, a
fuzzy
logic con
t
roller (FLC
) is introduced
in this resear
ch in
order to
normalize
the state
track
ing err
o
rs, so that th
e
input errors to
the BC ar
e
bounded to a fin
ite interval. Finally
,
th
e designed
BFLBC is integrated with
the nonlinear
disturbance observer in
order
to attenu
ate t
h
e extern
al
disturbances an
d model uncertainties
. Th
e si
m
u
lation results
show the
effectiven
ess of the proposed co
ntroller
to gen
e
r
a
te
a bounded velocity
sign
al
as well as to st
ab
iliz
e th
e tra
c
king
errors
to z
e
ro. I
n
addition
,
th
e re
sults prove
that the proposed controller provide an
excellen
t disturbance attenuation as
well
as robustness against
th
e parameters un
certainties
.
Keyword:
B
o
u
n
d
ed
bac
k
s
t
eppi
n
g
c
o
nt
r
o
l
Fuzzy logic c
o
ntrol
No
nl
i
n
ea
r di
st
ur
ba
nce obse
r
v
e
r
Tracki
n
g control
Wheele
d
m
obile robot
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
M
a
hm
ood Al
i
M
o
q
b
el
Obai
d,
Facu
lty of Electri
cal Engineering,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia,
Em
ail: eng_m
ahka
h192@yahoo.com
1.
INTRODUCTION
Th
e r
e
sear
ch
on
tr
aj
ect
o
r
y tr
ack
i
ng
pr
ob
lem o
f
wh
eeled
mo
b
ile ro
bo
t (WMR) g
a
in
s a
great in
terest
in the recent years due
to its prom
ising appl
ications
in m
a
n
y
fields such a
s
fact
ory aut
o
mation, trans
p
ortation,
room
cleaning , security and s
p
ace exploration. The m
a
in
m
o
tivations be
hind these considera
b
le interests are
t
h
e une
x
p
ect
ed gr
o
w
t
h
i
n
t
h
e
area
s of
wi
rel
e
ss
c
o
m
m
uni
cat
i
on,
c
o
m
put
i
n
g
m
a
chi
n
e
r
y
and s
e
ns
or
s
technology. The purpose of the path
track
i
ng
con
t
ro
ller is to
force th
e
WMR to
ach
iev
e
a d
e
sired
p
a
th su
ch
that the tracki
ng e
r
ror is stabilized
to
zero. Howev
e
r, the track
ing
erro
r is m
o
stly
u
n
a
vo
id
ab
le since th
e
per
f
o
r
m
a
nce o
f
t
h
e
W
M
R
ca
n
be a
ffect
e
d
by
di
ffe
rent
t
y
pes
of
unce
r
t
a
i
n
t
i
e
s suc
h
as s
e
ns
ors
an
d act
uat
o
rs
faul
t
s
, sl
i
p
page
, fri
ct
i
on a
n
d u
n
m
odel
e
d dy
n
a
m
i
cs. T
hus,
d
e
si
gn
of r
o
bust
pat
h
t
r
acki
ng
cont
rol
l
e
r f
o
r WM
R
is still an
op
en
issu
e in rob
o
tics co
mm
u
n
ity [1
].
The pat
h
tracking problem
of W
M
R curre
nt
ly r
eceives a lot of attention from
both academ
ics and
aut
o
m
obi
l
e
i
ndust
r
i
a
l
resear
ches. T
h
e a
v
a
i
l
a
bl
e W
M
R
t
r
acki
ng a
p
p
r
o
aches ha
ve
b
een re
po
rt
ed i
n
t
h
e
literatu
re can
be classified
in
to
b
a
ck
steep
i
n
g [1
-7
], s
lid
ing
m
o
d
e
[8
-9
], lin
earizatio
n
[1
0
]
, n
e
u
r
al n
e
t
w
ork
[11-
12]
a
n
d
f
u
zzy
l
ogi
c c
ont
rol
[
13]
.
Al
t
h
ou
g
h
t
h
e ext
e
nsi
v
e
r
e
search
ha
s be
en
do
ne
o
n
W
M
R
t
r
acki
n
g c
ont
rol
,
but
m
o
st
of
t
h
e
pre
v
i
o
us w
o
rk [1
-
10]
has assum
e
d
t
h
at
t
h
e W
M
R
has a
pu
re no
n
-
h
o
l
on
om
i
c
const
r
ai
nt
.
Ho
we
ver
,
t
h
e pu
re r
o
l
l
i
ng a
nd
no
n-
sl
i
ppi
n
g
assum
p
t
i
ons
cann
o
t
be gu
arant
e
e
d
d
u
e t
o
a set
of fact
ors t
h
at
i
n
fl
ue
nce t
h
e p
e
rf
orm
a
nce of
t
h
e
W
M
R
suc
h
as shar
p t
u
rn
i
ng m
o
t
i
on at
hi
g
h
spee
d [
1
3
]
. To ove
rc
om
e t
h
ese
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
90
1 – 9
0
8
90
2
l
i
m
i
t
a
t
i
ons, so
m
e
researc
h
ers
ha
ve
pr
op
ose
d
a t
r
acki
n
g
contr
o
ller
f
o
r
th
e
W
M
R w
ith non
id
eal
no
nho
lo
no
m
i
c
co
nstr
ain
t
s [13-
14
]. Fo
r
i
n
st
ance, a
fuzzy logic controller
i
s
pr
o
pose
d
i
n
[
13]
f
o
r t
r
acki
n
g co
nt
r
o
l
o
f
t
h
e
W
M
R
i
n
t
h
e prese
n
ce
of ki
nem
a
ti
c
and
dy
nam
i
c uncert
a
i
n
t
i
e
s,
w
h
i
c
h t
h
e ki
nem
a
t
i
c
di
st
urba
nc
e i
s
assum
e
d to be a
fun
c
tion
o
f
bo
th
lin
ear as
well
as an
gu
lar
v
e
l
o
cities.
Am
ong t
h
e
p
r
evi
o
us c
ont
rol
t
echni
ques
,
t
h
e ba
ckst
e
ppi
ng c
o
nt
r
o
l
l
e
r
(B
C
)
i
s
t
h
e
m
o
st
pop
ul
ar
m
e
t
hod f
o
r
no
nh
ol
o
n
o
m
i
c WM
R
t
r
acki
ng c
ont
rol
[
1
-
6
]
.
In
part
i
c
ul
ar, t
h
e
ki
nem
a
t
i
c
based B
C
appr
oac
h
for
a
no
n
hol
on
om
i
c
W
M
R
was fi
rst
sug
g
est
e
d by
Kanay
a
m
a
et
al
. (199
0
)
, whi
c
h i
t
has been assum
e
d t
h
at
t
h
e
WMR h
a
s a pure non
ho
lono
mic co
n
s
t
r
ain
t
.
Nev
e
rth
e
less, a n
e
w ch
alleng
i
n
g
will b
e
im
p
o
s
ed
in
th
e contro
ller
d
e
sign
b
y
v
i
olatin
g
th
e id
eal n
o
nho
lono
m
i
c co
n
s
train
t
s in
wh
ich
th
e co
n
t
ro
ller sh
ou
l
d
in
corp
orates th
e
di
st
ur
ba
nce re
m
oval
t
o
achi
e
ve hi
g
h
t
r
ac
k
i
ng acc
uracy
.
There
f
ore, i
n
t
h
i
s
pa
per, a
n
on l
i
n
ea
r di
st
u
r
ba
nce
obs
er
ver
(N
D
O
B
)
i
s
c
o
m
b
i
n
ed wi
t
h
t
h
e B
C
cont
r
o
l
l
e
r i
n
or
der
t
o
i
m
prove t
h
e
di
st
ur
b
a
nces at
t
e
n
u
at
i
ons
an
d
th
e robu
stn
e
ss
ag
ain
s
t th
e p
a
ra
m
e
ters u
n
cert
a
in
ties. Essen
tially, th
e NDOB h
a
s b
een
app
lied
in
th
e literatu
re
fo
r m
a
ny
no
nl
i
n
ear
sy
st
em
s, such
as m
i
ssi
l
e
s [
1
5
-
1
7
]
,
ro
b
o
t
m
a
ni
pul
at
o
r
[1
8-
2
0
]
an
d
pen
d
u
b
o
t
sy
st
em
[2
1]
.
Thi
s
resea
r
ch
pr
o
pose
d
a r
o
bust
B
C
f
o
r
u
n
cert
a
i
n
ki
ne
m
a
t
i
c
m
odel
of t
h
e n
o
n
h
o
l
o
nom
i
c
W
M
R
base
d o
n
N
D
O
B
.
Th
e pr
o
pos
ed c
ont
rol
l
er i
n
cl
u
d
es t
w
o
part
s.
Fi
r
s
t
,
a bo
un
de
d
Fuzzy
l
ogi
c
based
back
st
ep
pi
n
g
cont
rol
l
e
r (B
FL
B
C
)
i
s
desi
g
n
e
d t
o
co
nt
r
o
l
t
h
e
W
M
R
wi
t
h
out
c
onsi
d
eri
n
g t
h
e ef
fect
s o
f
t
h
e
ex
tern
al d
i
stu
r
b
a
n
c
es and
the p
a
ram
e
ters u
n
c
ertain
ties.
Mean
wh
ile, the fu
zzy lo
g
i
c co
n
t
ro
ller (FLC) is
in
teg
r
ated
with th
e BC to
ov
erco
m
e
th
e un
realistic larg
e v
e
lo
city sig
n
a
l that p
r
o
d
u
c
ed
b
y
th
e BC cau
sed
by
the huge state tracking errors
. There
f
ore
,
it
can ach
iev
e
a sm
o
o
t
h
m
o
tio
n
of th
e
W
M
R with
ou
t an
y sh
arp
spee
d jum
p
s even
for the syste
m
s th
at have large state tracki
ng e
r
rors. Finally, the designe
d BFLB
C is
in
teg
r
ated
with th
e
NDOB i
n
o
r
d
e
r to
tack
le
th
e ex
tern
al
d
i
stu
r
b
a
n
ces and
m
o
d
e
l u
n
c
ertain
ties.
The
pape
r is
outlined as
follows: In section
2,
t
h
e m
e
thods
are
prese
n
ted, whic
h incl
ude
the m
odel
o
f
th
e non
ho
l
o
n
o
m
ic W
M
R with
k
i
n
e
m
a
tic
u
n
c
ertain
ties, th
e co
n
t
ro
ller desig
n
, and
th
e stab
ility an
alys
is. Th
e
sim
u
l
a
t
i
on set
u
p i
s
gi
v
e
n
i
n
se
ct
i
on
3.
T
h
e
re
sul
t
s
o
f
t
h
e
pr
o
pos
ed
co
nt
r
o
l
l
e
r as
com
p
are
d
t
o
t
h
e c
o
nve
nt
i
onal
BC are
d
i
scu
s
sed
in section
4
.
Fin
a
lly, th
e con
c
lu
si
o
n
is presen
ted in
secti
o
n 5.
2.
R
E
SEARC
H M
ETHOD
2.
1. N
o
nh
ol
o
n
omi
c
W
M
R Mo
del
w
i
th K
i
nemati
c Unce
rtai
n
t
i
e
s
The
nonhol
onom
ic W
M
R model in a
2-D
Cartes
ian workspace is shown in Fi
gure
1,
whe
r
e
{X
, O,
Y}
and
{D
, C,
L
}
are t
h
e
g
l
obal and
lo
cal Cartesian
coo
r
d
i
nate syste
m
, resp
ectiv
ely.
D
is th
e d
r
iv
i
n
g d
i
rectio
n
(l
o
ngi
t
u
di
nal
d
i
rect
i
on)
an
d
L
is th
e lateral d
i
rectio
n (lat
itu
d
i
n
a
l
d
i
rectio
n).
A
p
o
stu
r
e o
f
a non
ho
lon
o
m
ic
WMR in
the glo
b
a
l Cartesian coo
r
d
i
n
a
te syste
m
{X, O,
Y}
can be represe
n
ted by
a vector
Ѳ
,
w
h
er
e
C
is th
e cen
ter
o
f
t
h
e
robo
t,
(
,
)
is th
e
W
M
R
coord
i
n
a
te an
d
Ѳ
is t
h
e orien
t
ation
an
g
l
e of th
e
WM
R.
D
L
X
Y
c
x
c
y
c
c
o
{
r
v
δ
w
δ
s
V
Fig
u
re 1
.
W
M
R
m
o
d
e
l
with
k
i
n
e
m
a
tics
u
n
c
ertain
ties
Th
e
k
i
n
e
m
a
tics
m
o
d
e
l o
f
th
e
no
nho
lon
o
m
ic WMR can b
e
written
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ro
bust
Backst
e
ppi
ng
Tr
acki
n
g
C
o
nt
rol
of
M
o
bi
l
e
Ro
b
o
t
B
a
s
e
d
on
N
o
nl
i
n
e
a
r Di
st
ur
b
a
n
ce
…
(
M
.A.M. O
b
aid)
90
3
Ѳ
cos
Ѳ
rsin
Ѳ
sin
Ѳ
r
cos
Ѳ
01
0
(1
)
whe
r
e
,
and
rep
r
ese
n
t
t
h
e
f
o
r
w
ar
d
vel
o
ci
t
y
,an
gul
a
r
ve
l
o
city and the
di
stance from
the confi
g
uration
center of the
WMR to t
h
e
wheels
ce
nter, res
p
ectively.
More
ove
r,
a
n
d
re
pre
s
ent
i
ng
t
h
e
ki
nem
a
t
i
c
s
u
n
c
ertain
ties in
th
e
D
a
nd
L
coo
r
di
nat
e
s, r
e
spect
i
v
el
y
.
B
a
sed o
n
st
u
d
y
[1
3]
, t
h
e eq
uat
i
ons
of t
h
e
ki
n
e
m
a
t
i
c
u
n
c
ertain
ties are g
i
v
e
n
as fo
llows:
Ѳ
Ѳ
(2
)
Ѳ
Ѳ
(3
)
whe
r
e,
,
0
.
2
t
a
n
h
(4
)
2.
2. B
o
un
ded
Fuz
z
y
Logic B
a
sed B
a
cks
t
ep
ping
Contr
o
ll
er Desi
gn
The
pu
r
pose
o
f
W
M
R
pat
h
t
r
acki
n
g c
ont
r
o
l
l
er i
s
t
o
fi
n
d
a
cont
rol
l
o
w i
n
put
q
q
v
w
th
at
force th
e
ro
bot to
fo
llo
w
refe
rence t
r
aject
ory with
position
Ѳ
and
vel
o
ci
t
y
, suc
h
th
at th
e
p
o
sture error
Ѳ
converges
t
o
ze
ro as
the
tim
e approaches
i
n
finite. T
h
e
con
v
e
n
t
i
onal
B
C
m
e
t
hod
f
o
r
t
r
acki
n
g
co
nt
r
o
l
o
f
t
h
e
W
M
R
k
i
nem
a
t
i
c
m
odel
i
s
gi
ve
n
by
[
1
]
cos
Ѳ
sin
Ѳ
(5
)
Th
e BC ap
pro
a
ch
is li
mited
fo
r th
e ap
p
lication
s
th
at
have s
m
all tracking e
r
rors since the
cont
roller is
d
i
rectly related
to th
e state
track
ing
erro
rs. The FL
C com
b
in
ed
with
t
h
e BC ap
pro
a
ch
to ov
erco
m
e
th
is
li
mitatio
n
o
f
th
e BC m
e
th
o
d
. Firstly, th
e FLC is u
tilized
to
norm
a
l
i
ze t
h
e track
i
ng
erro
r in
t
h
e long
itu
d
i
nal
di
rect
i
o
n
.
Th
en
, th
e BC app
r
o
ach is app
lied to con
t
ro
l
t
h
e m
o
ti
on o
f
t
h
e WM
R
base
d o
n
t
h
e n
o
rm
al
i
z
e
d
t
r
acki
n
g e
r
r
o
rs
, w
h
i
c
h a
r
e
obt
ai
ned
f
r
om
t
h
e
FLC
.
T
h
e
B
F
L
B
C
fo
r t
h
e
ki
n
e
m
a
t
i
c
m
odel
of
t
h
e
W
M
R
w
i
t
hout
consideri
n
g the
effects
of
th
e
ex
tern
al d
i
st
u
r
b
a
n
c
es an
d th
e
p
a
ram
e
ters un
certain
ties is g
i
ven
as fo
llo
ws:
1
Ѳ
Ѳ
Ѳ
(6
)
whe
r
e
is th
e desired
forward v
e
lo
city,
th
e d
e
sired
an
gu
lar v
e
l
o
city,
is the norm
a
lized error i
n
th
e lon
g
itud
i
n
a
l d
i
rection
wh
i
c
h
is
ob
tain
ed
b
y
th
e
FLC and
T
is th
e ti
m
e
co
nstan
t
.
2.3. Nonlinear
Disturbance Obser
v
er Design
A NDOB is dev
e
lop
e
d in
this sectio
n
to
esti
m
a
te
th
e
WMR k
i
n
e
m
a
tic
u
n
c
ertain
ties. Th
en
, the
NDOB is in
tegrated
with
th
e
p
r
op
o
s
ed
BFLBC to
con
s
tru
c
t th
e con
t
ro
llero
f t
h
e
W
M
R.
Th
e
k
i
n
e
m
a
tic
s m
o
d
e
l
o
f
th
e
no
nho
lon
o
m
ic W
M
R can
b
e
rewritten as:
(7
)
whe
r
e
,
cos
Ѳ
rs
in
Ѳ
sin
Ѳ
rcos
Ѳ
,
,
Thr
o
ug
h
out
t
h
i
s
pa
per
,
t
h
e r
o
bot
ki
nem
a
t
i
c
unce
r
t
a
i
n
t
i
e
s a
r
e ass
u
m
e
d t
o
be sl
owl
y
t
i
m
e va
ry
i
n
g
.
M
o
t
i
vat
e
d
b
y
th
e
w
o
r
k
pr
opo
sed
i
n
[15-
20
], th
e
f
o
ll
ow
ing
ob
ser
v
er is d
e
sign
ed
to
estim
a
t
e th
e u
nkn
own
k
i
ne
m
a
tic
u
n
c
ertain
ties
, g
i
ven
by
(8
)
(9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
90
1 – 9
0
8
90
4
whe
r
e
,
and
rep
r
ese
n
t
t
h
e
est
i
m
a
t
e
d di
st
ur
ba
nce, t
h
e
o
b
se
rve
r
internal state, and t
h
e
obse
r
ver
gain,
respectively. T
h
e
observer gai
n
is g
i
v
e
n as
fo
llo
ws:
(1
0)
whe
r
e
2.
4. B
a
cks
t
ep
ping
Contr
o
ll
er Desi
gn B
a
s
e
d on
Nonlinear
Distur
banc
e Obser
v
er
In
t
h
is sectio
n, th
e co
m
p
o
s
it
e co
n
t
ro
ller law is d
e
si
g
n
e
d for th
e
u
n
c
ertain
k
i
n
e
m
a
tic
m
o
d
e
l
W
M
R
.
Th
e
pr
o
pose
d
c
o
nt
r
o
l
l
e
r i
s
c
o
m
p
o
s
ed
of
t
h
e
B
F
LB
C
as gi
ve
n
i
n
eq
uat
i
o
n
(6
)
an
d t
h
e
ND
O
B
as de
si
g
n
ed
i
n
t
h
e
pre
v
i
o
us
sect
i
o
n.T
h
e c
o
m
posi
t
e
cont
rol
l
o
w
i
s
gi
ve
n as
f
o
l
l
o
ws:
(1
1)
whe
r
e
is t
h
e
feed
b
a
ck BFLB
C fo
r th
e
WM
R withou
t co
nsid
ering th
e effects of th
e un
certain
ties,
is
t
h
e di
st
ur
ban
c
e
com
p
ensat
i
o
n
gai
n
a
n
d
i
s
t
h
e
est
i
m
a
t
e
d di
st
ur
ba
nce
usi
n
g t
h
e
pr
o
pose
d
N
DOB
.
1
Ѳ
Ѳ
Ѳ
(1
2)
The di
st
u
r
ba
nc
e
com
p
ensat
i
o
n gai
n
is
g
i
v
e
n as fo
llows:
C
cos
Ѳ
rs
in
Ѳ
sin
Ѳ
rcos
Ѳ
,c
c
0
0c
(1
3)
whe
r
e
c
and
c
are
t
h
e c
ont
rol
l
e
r
gai
n
t
o
be
desi
gn
.
2
.
5
.
Sta
b
ility Ana
l
y
s
is
Theore
m 1:
If t
h
e c
o
nt
r
o
l
l
e
r (
6
) i
s
a
ppl
i
e
d t
o
t
h
e
WM
R
ki
nem
a
t
i
c
m
odel
gi
ven i
n
(
1
)
,
0
is a sta
b
le equ
ilib
ri
u
m
poi
nt
.
Proof
:
A l
y
ap
un
o
v
fu
nct
i
o
n
can
di
dat
e
V
is cho
s
en
as fo
llo
ws;
1
2
1
1
Ѳ
(1
4)
Clearly,
0
and
0
if
0
,
0
and
Ѳ
0
sin
Ѳ
Ѳ
(1
5)
cos
Ѳ
sin
Ѳ
sin
Ѳ
(1
6)
1
cos
Ѳ
cos
Ѳ
sin
Ѳ
sin
Ѳ
(1
7)
1
sin
Ѳ
sin
Ѳ
Ѳ
sin
Ѳ
(1
8)
1
Ѳ
sin
Ѳ
0
(1
9)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ro
bust
Backst
e
ppi
ng
Tr
acki
n
g
C
o
nt
rol
of
M
o
bi
l
e
Ro
b
o
t
B
a
s
e
d
on
N
o
nl
i
n
e
a
r Di
st
ur
b
a
n
ce
…
(
M
.A.M. O
b
aid)
90
5
There
f
ore
is
neg
a
tiv
e sem
i
-defin
ite and
t
h
e
resu
lting
system
is asy
m
p
t
o
tically stab
le.
3.
SIMULATION SET
U
P
Th
e
pr
opo
sed con
t
ro
ller
a
nd th
e
k
i
n
e
m
a
ti
c m
o
d
e
l of
t
h
e
r
obo
t as def
i
n
e
d
in eq
uatio
n
(
1
)
are
si
m
u
lated
in
MATLAB
-
SIMULINK. Th
e si
m
u
latio
n
in
terv
al is fro
m
0
to
4
0
second
,
an
d
th
e sam
p
lin
g
tim
e
t
is 0
.
01
secon
d
. Th
e inp
u
t
p
a
ra
m
e
ters to
th
e
syste
m
are as
fo
llows:
1
,
4
,
2
.
5
,
1
,
1
,
1
,
3
,
1
,
Ѳ
1
.
5
,
1
.
3
,
1
.
3
an
d T
=
1
.
Moreover,
the NDOB
gai
n
is selected a
s
fo
llow:
28
0
0
2
8
(2
0)
4.
R
E
SU
LTS AN
D ANA
LY
SIS
Th
is sectio
n
p
r
esen
ts two
case stu
d
i
es th
at
will illu
strate t
h
e ro
bu
st
n
e
ss
an
d
t
h
e effecti
v
en
ess of th
e
p
r
op
o
s
ed
con
t
ro
ller to
tack
l
e
th
e ex
tern
al
d
i
stu
r
b
a
n
c
es an
d
m
o
d
e
l u
n
certain
ties. For th
e first trajecto
r
y
track
ing
case,
th
e propo
sed
co
n
t
ro
ller is app
lied
to
trac
k
a si
m
p
le straig
h
t
p
a
th
with
a co
nstan
t
lin
ear and
an
gu
lar
v
e
lo
city. Th
en, a circu
l
ar p
a
t
h
is set to
b
e
a refere
nce traject
ory. Finally
, t
h
e di
st
urba
nce re
je
ct
i
o
n
ab
ility o
f
th
e propo
sed con
t
roller is co
m
p
ared
with
th
e conv
en
tion
a
l BC ap
pro
ach.
4.
1. Str
a
ight P
a
th
Trac
king
In
itially, a si
mp
le case stud
y
to
track a strai
g
h
t
p
a
th
is
prop
o
s
ed. Th
e d
e
sired
strai
g
h
t
p
a
th
is d
e
fin
e
d
as
1
0
and
1
0
i
n
t
h
e Ca
rtesian c
o
ordinates worksp
ace. The
initial postur
e of
the
WMR is
(0,0,0),
wh
ile th
e d
e
si
red
i
n
itial p
o
s
ture is (1
0,10
,0).
Th
u
s
th
e in
itial trac
k
i
ng
error is (1
0,10
,0). Figures2
-
3
sho
w
t
h
e
per
f
o
rm
ances o
f
s
t
rai
ght
pat
h
t
r
aject
o
r
y
usi
n
g
t
h
e
pr
o
pose
d
cont
rol
l
e
r.
It
c
a
n
be
obs
er
ve
d f
r
o
m
Fi
gu
re
2 t
h
at
t
h
e p
r
op
ose
d
c
ont
rol
l
e
r
ge
ner
a
t
e
s cont
i
n
u
o
u
s
, b
o
u
n
d
e
d
an
d sm
oot
h c
ont
rol
si
g
n
al
s
wi
t
h
ze
r
o
val
u
e at
t
h
e st
art
i
ng t
i
m
e. The post
u
re er
ro
r
of t
h
e st
rai
ght
pat
h
t
r
a
j
ect
o
r
y
i
s
show
n i
n
Fi
gu
re 3
.
As i
t
can be
seen, the
proposed c
o
nt
ro
ller stab
ilizes th
e track
i
ng
erro
r to
zero.
4.
2. Ci
rcul
a
r
Pat
h
T
r
acki
n
g
In
th
is sectio
n, a circu
l
ar p
a
th is set
to
b
e
a r
e
feren
ce traj
ecto
ry. Th
e sim
u
latio
n
in
terv
al is fro
m
0
to
4
0
second
, an
d th
e sam
p
lin
g
ti
m
e
t
is 0
.
01
secon
d
. The initial p
o
s
tu
re
of th
e
W
M
R is (0
,
0
.
6
,
0
)
, wh
ile th
e
d
e
sired
in
itial p
o
s
t
u
re is
(1
5, 1
3
, 0).
Th
erefo
r
e, th
e in
itial
track
ing
erro
r is
(15
,
1
3
, 0
)
. Th
e d
e
sired
ci
rcu
l
ar
pat
h
i
n
t
h
e
C
a
r
t
esi
a
n co
or
di
na
t
e
s i
s
de
fi
ne
d a
s
f
o
l
l
o
ws:
15
2
cos
2
,
15
2
sin
2
(2
1)
Fi
gu
res
4
-
5
sh
ow
t
h
e
ci
rcul
a
r
pat
h
t
r
acki
n
g
per
f
o
r
m
a
nce o
f
t
h
e
p
r
op
os
ed c
ont
rol
l
e
r
a
n
d
t
h
e
st
an
dar
d
B
C
m
e
t
hod,
res
p
e
c
t
i
v
el
y
.
In c
o
m
p
ari
s
on wi
t
h
t
h
e co
n
v
ent
i
o
nal
B
C
ap
pr
oa
ch, t
h
e
si
m
u
l
a
ti
on
resul
t
s
pr
o
v
e t
h
at
t
h
e t
r
acki
n
g pe
rf
orm
a
nce can be im
pro
v
e
d
si
gni
fi
cant
l
y
usi
ng t
h
e
pr
op
ose
d
co
nt
rol
l
e
r
,
w
h
ere i
t
i
s
capabl
e
t
o
track a circula
r
path acc
urate
l
y even
in the prese
n
ce of external distur
ba
nces an
d pa
ra
m
e
t
e
rs uncert
a
i
n
t
i
e
s.
Fig
u
res
6
-
7
sho
w
t
h
e v
e
l
o
cities resp
on
se
o
f
th
e propo
sed
co
n
t
ro
ller and
th
e conv
en
ti
o
n
a
l BC, resp
ectively. It
can
b
e
seen
for
m
Fig
u
r
e
6
that th
e pr
opo
sed
co
n
t
ro
ller
is
cap
ab
le
of
prod
u
c
i
n
g a
b
ound
ed as
w
e
ll as
sm
o
o
t
h
v
e
lo
city sign
al with
zero
v
a
lu
e at th
e starti
n
g
ti
m
e
.
H
o
w
e
ver
,
as s
h
ow
n
i
n
Fi
gu
re
7 t
h
e f
o
r
w
ar
d
vel
o
ci
t
y
of
th
e con
v
e
n
tional BC j
u
m
p
s to
m
o
re th
an
70
m
/
s at
th
e in
itial t
i
m
e, wh
ich
is d
i
fficu
lt to b
e
im
p
l
e
m
en
t
a
b
l
e in
an act
ual
r
o
bot
i
c
sy
st
em
. The post
u
re e
r
r
o
r o
f
t
h
e
pr
o
pose
d
cont
rol
l
e
r a
n
d t
h
e t
h
e c
o
nve
nt
i
onal
B
C
are
s
h
o
w
n
i
n
Fi
gu
res 8 a
n
d 9,
respect
i
v
e
l
y
.
It
can be ob
serve
d
t
h
at
t
h
e
no
rm
ali
z
i
ng err
o
r
usi
n
g FLC
i
s
conve
rg
ed t
o
t
h
e
n
a
tural erro
r as th
e ti
me ap
p
r
oach
in
fi
n
ity. In ad
d
itio
n,
th
e propo
sed
co
n
t
roller is stab
ilize
d
th
e track
i
n
g
erro
rs
t
o
zero
.
H
o
we
ver
,
as sh
ow
n
i
n
Fi
gu
re 9, the co
nv
en
tio
n
a
l
BC is failed
t
o
stab
ilize th
e track
ing
error to
zero
.
Fi
gu
res
1
0
-
1
1
sho
w
t
h
e est
i
m
at
i
on
of
t
h
e
di
st
ur
bance
s
a
n
d
pa
ram
e
t
e
rs un
cert
a
i
n
t
i
e
s usi
n
g t
h
e
N
D
O
B
.
I
t
can
be o
b
se
rve
d
t
h
at
t
h
e pr
op
ose
d
co
nt
r
o
l
l
e
r ha
s an excel
l
e
nt
di
st
ur
ba
nce at
t
e
nuat
i
o
n as
we
l
l
as st
ron
g
r
o
b
u
st
nes
s
ag
ain
s
t t
h
e
p
a
ra
m
e
ters u
n
c
ert
a
in
ties.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
90
1 – 9
0
8
90
6
Fi
gu
re
2.
C
o
nt
r
o
l
si
g
n
al
s
Fi
g
ure
3. Posture erro
r
Fi
gu
re
4.
Tra
j
e
c
t
o
ry
of ci
rcl
e
t
r
acki
n
g
f
o
r
t
h
e p
r
op
ose
d
c
ont
rol
l
e
r
Fi
gu
re
5.
Tra
j
e
c
t
o
ry
of ci
rcl
e
t
r
acki
n
g
f
o
r
st
anda
rd
bac
k
s
t
eppi
n
g
c
o
nt
r
o
l
l
e
r
Fi
gu
re
6.
C
o
nt
r
o
l
si
g
n
al
s
fo
r t
h
e
pr
o
pose
d
c
o
nt
r
o
l
l
e
r
Fi
gu
re
7.
C
o
nt
r
o
l
si
g
n
al
s
fo
r t
h
e
st
anda
rd
bac
k
s
t
eppi
n
g
c
o
nt
r
o
l
l
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ro
bust
Backst
e
ppi
ng
Tr
acki
n
g
C
o
nt
rol
of
M
o
bi
l
e
Ro
b
o
t
B
a
s
e
d
on
N
o
nl
i
n
e
a
r Di
st
ur
b
a
n
ce
…
(
M
.A.M. O
b
aid)
90
7
5.
CO
NCL
USI
O
N
A ro
b
u
st
B
C
cont
rol
l
e
r b
a
se
d o
n
n
onl
i
n
ea
r
di
st
ur
bance
o
b
ser
v
e
r
i
s
pres
ent
e
d i
n
t
h
i
s
r
e
search
fo
r
t
r
acki
n
g co
nt
r
o
l
of t
h
e ki
nem
a
t
i
c
m
odel
of the n
o
n
h
o
l
o
no
m
i
c W
M
R
i
n
the pre
s
ence
of
ext
e
rnal
di
st
u
r
bance
s
an
d p
a
r
a
m
e
ter
s
un
cer
tain
ties. Th
e pr
opo
sed
c
on
tro
ller
is
cap
ab
leo
f
p
r
odu
cing
a
b
ound
ed v
e
l
o
city sig
n
a
ls
rega
rdl
e
ss
of t
h
e am
ount
of t
h
e st
at
e t
r
acki
n
g err
o
rs. T
h
ere
f
o
r
e, i
t
can ach
i
e
ve a sm
oot
h
m
o
ti
on f
o
r t
h
e
WM
R
ev
en fo
r t
h
e ap
p
lication
s
th
at h
a
v
e
hug
e st
ate track
ing
errors.
In
add
ition
,
t
h
e sim
u
lati
o
n
resu
lts h
a
s
sh
own
t
h
at
t
h
e pr
op
os
ed co
nt
r
o
l
l
e
r p
r
o
v
i
d
e a
n
exce
l
l
e
nt
di
st
urba
n
ce at
t
e
nuat
i
on
as wel
l
as st
rong r
o
bu
st
ness
agai
nst
th
e p
a
ram
e
ters u
n
c
ertain
ties.
REFERE
NC
ES
[1]
S.X. Yang, A.
Zhu, G. Yuan, an
d M.
Q.H. Meng
, (2012). A
bioin
s
pired neurod
y
n
amics-based app
r
oach to
tr
ack
in
g
control of mobile robots.
IEEE Trans. Ind.
Electr
on
., 59(8), 3211
–3220.
[2]
Y. Kanay
a
ma,
Y. Kimura, F.
Miy
a
zak
i, T. No
guchi, (1990)
. A
stable
track
ing
control method f
o
r an autonomo
u
s
mobile robot.
In
Proc.
I
E
EE
In
t. Conf. Robot.
Au
tomat
, 384–389
.
Figu
re
8.
P
o
stu
r
e er
r
o
r
f
o
r t
h
e
p
ro
p
o
s
ed
con
t
ro
lle
r
Figu
re
9.
P
o
stu
r
e er
r
o
r
f
o
r t
h
e
standa
rd
back
st
ep
pi
n
g
c
ont
rol
l
e
r
Fi
gu
re
1
1
.
Di
st
ur
ba
nce est
i
m
at
i
on i
n
t
h
e
L
coo
r
di
nat
e
usi
n
g ND
OB
.
Fi
gu
re 1
0
. Di
st
ur
ba
nce
est
i
m
at
i
on
i
n
t
h
e D
co
o
r
di
na
t
e
usi
n
g
N
D
O
B
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
90
1 – 9
0
8
90
8
[3]
I.M.H. Sanhour
y
,
S.H.M.
Amin, A.R. Husain
, (2011).Tr
a
jecto
r
y
tr
ack
ing of steer
ing s
y
stem mobile robot.
4th
International Co
nference
on
Mechatronics (
I
COM)
, 1-5.
[4]
R. Fierro
, F.L.
Lewis, (1997)
.
Contro
l of a no
nholomic mobile robot: b
ack
stepping kinematics into d
y
namics
.
Journal of Robotic S
y
stems,
149–
163.
[5]
Chwa, D. (2010
). Tr
ack
ing con
t
rol of d
i
ffer
e
ntial-drive
wheeled
mobile robots us
ing a backstepping-like feedb
ack
line
a
riz
a
tion
.
IEEE Transactions
on Syst
ems, Man and Cybernetics
, Part A: Sy
stems and Hu
man
s
, 40(6), 1285-
1295.
[6]
Obaid, M.A
.
M.
and Husain,
A.R. (2015). Time var
y
ing
ba
ckstepping
control f
o
r trajec
tor
y
tracking of
mobile
robot.
In
t. J. Co
mputational Vision and Robo
tics
, Vol. X, No. Y,
pp. 000–000
.
[7]
Qiao, W.Y. (201
3). Backstepping
Adaptiv
e Fuzzy Scheme for SCARA GRB400
Robot.
TELKOMNIKA Indonesian
Journal of Electr
ical Engineerin
g
, 11(8)
, 4229-4
237.
[8]
D. Chwa, (2004)
.Sliding-mode tr
acking
control o
f
nonhol
onomic wheeled mobile
robot
s in polar
coordinates.
IEEE
Trans. Control S
y
st
.
Technol., 12
(4), 637–644
.
[9]
G.G. Rigatos,
C.S. Tzafes
tas,
and S.G. Tzaf
estas, (2000). M
obile
robo
t motion contro
l in
partially
unkno
wn
environments us
ing a sliding-
mo
de fuzzy
logic controller.
Robot. Auton
.
S
y
st
., 33
(1), 1–11
.
[10]
D.H. Kim and
J.H. Oh, (19
99). Tr
acking
control
of
a
two-wheeled m
obile robo
t using input–outpu
t
line
a
riz
a
tion
.
Co
ntrol Eng
.
Practi
ce
, 7(3), 369–37
3.
[11]
Mohareri, O., D
h
aouadi, R., &
Rad, A.B
.
(201
2). Indi
r
ect adaptive
track
ing contro
l of
a non
holonomic mobile
robot via n
e
ural
networks.
N
e
urocomputing
, 88, 5
4
-66.
[12]
Yan-dong, L., L
i
ng, Z., & Ming, S. (2013). Adaptiv
e RBF
NN
Form
ation Control of Multi-m
obile Robots wit
h
Actuator
D
y
n
a
m
i
cs
.
TELKOMNI
KA Indonesian
Journal
of Electrical Engineering
, 11(4), 1797-180
6.
[13]
Chwa, D. (2012). Fuzzy
adaptiv
e
track
ing contr
o
l of wheel
ed
mobile robots with state-d
e
pend
ent kinematic and
d
y
namic disturb
a
nces.
IEEE Transactions on
Fuzzy Systems
, 20(3
)
, 587-593
.
[14]
Taher
i
Kalan
i
, J., & Khosrowjerd
i
, M.J. (2014)
. Adaptiv
e tr
ajector
y
tr
ack
ing contr
o
l of wheeled mobile robots with
disturbance observer.
Internation
a
l Journal
of Ad
aptive Control a
nd Signal Processing
, 28(1), 14-
27.
[15]
Chen, W.H. (2
004). Disturban
ce observ
e
r b
a
sed con
t
rol
for
nonlinear s
y
s
t
ems.
IEEE/ASM
E
T
r
ansactions on
Mechatronics
, 9
(
4), 706-710
.
[16]
Chen, W.H. (2003). Nonlinear
disturbance observer-e
nh
anced d
y
namic inversi
on control of missiles.
Journal of
Guidance, Contr
o
l, and
Dynamics
, 26(1), 161-16
6.
[17]
Yang, J., Ch
en, W.H., & Li, S. (2011). Non-linear dist
urb
a
nce observer-based
robust control
for s
y
stems with
m
i
s
m
atched dis
t
urbances
/un
cer
ta
inties
.
IET
contr
o
l th
eory
&
appl
ications
, 5(18)
,
2053-2062.
[18]
Chen, W.H., B
a
llan
ce, D.J., Gawthrop,
P.J.,
&
O'
Reilly
, J. (200
0). A non
lin
ear
disturbance observer for
robotic
manipulators.
IEEE Transactions
on I
ndustrial Electronics
, 47(4)
, 932-938.
[19]
Mohammadi, A., Tav
a
koli, M.,
Marquez,
H
.
J.,
&Hashemzadeh
, F. (2013). Nonl
inear d
i
sturbance observer desig
n
for robotic manipulators.
Contro
l Eng
i
neering
Pr
actice
, 21(3)
, 25
3-267.
[20]
Nikoobin, A.,
& Haghighi, R
.
(2009).
Ly
apu
nov-based nonlin
ear distu
r
bance observer for s
e
rial n-link rob
o
t
manipulators.
Jo
urnal of In
tellig
ent and
Robotic S
y
stems
, 55(2-3)
,
135-153.
[21]
Eom, M., & Ch
wa, D. (2015).
Robust Swing-Up and B
a
lan
c
ing
Control Using
a Nonlin
ear
Disturbance Observer
for the Pendubot S
y
stem with D
ynamic Friction
.
I
EEE Transactio
ns on Robotics
, 31(2),
331-343
.
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