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i
n
[
17]
,
[
18]
.
F
o
r
th
e
d
if
f
e
r
e
n
tia
l e
q
u
a
tio
n
o
f
ti
m
e
a
n
a
l
y
s
is
,
t
h
e
lite
r
a
tu
r
e
p
r
o
p
o
s
e
d
f
in
ite
ti
m
e
s
ta
b
ili
t
y
.
S
o
,
to
s
t
u
d
y
a cl
a
s
s
o
f
t
h
e s
y
s
t
e
m
s
t
a
b
i
l
i
t
y
f
o
r
a l
i
m
i
t
ed
t
i
m
e b
eco
m
es
r
eas
o
n
ab
l
e [
1
9
]
-
[
2
2
]
.
T
h
e
p
ap
er
i
s
o
r
g
an
i
zed
as
f
o
l
l
o
w
s
.
S
ect
i
o
n
I
I
p
r
es
en
t
s
th
e
m
o
b
ile
r
o
b
o
t d
y
n
a
m
ic
m
o
d
e
lin
g
.
S
e
c
tio
n
I
I
I
r
e
s
u
m
e
s
R
B
F
n
e
u
r
a
l
n
e
t
w
o
r
k
.
T
he
R
B
F
-
G
F
S
M
c
o
n
tr
o
lle
r
is
p
r
e
s
e
n
te
d
in
s
e
c
tio
n
I
V
.
T
h
e
s
im
u
la
tio
n
a
n
d
a
n
a
l
y
s
i
s
o
f
th
e
i
m
p
r
o
v
e
d
a
lg
o
r
ith
m
ar
e p
r
es
en
t
ed
i
n
S
ect
i
o
n
V
.
F
i
n
al
l
y
,
co
n
cl
u
s
i
o
n
s
ar
e d
r
a
w
n
i
n
S
ect
i
o
n
V
I
.
2.
M
O
B
ILE R
O
B
O
T M
O
D
ELIN
G
T
h
e
m
obi
l
e
r
obot
m
ode
l
i
ng
c
on
s
i
s
t
s
i
n
t
w
o
m
ode
l
s
:
ki
n
e
m
a
t
i
c
a
n
d d
y
n
a
m
i
c
m
ode
l
s
.
T
he
r
o
bo
t
'
s
ki
ne
m
a
t
i
c
s
i
s
d
e
f
i
ne
d
b
y
(1
)
:
c
o
s
0
si
n
0
01
x
py
q
θ
θ
=
=
θ
(1
)
w
h
er
e q
r
ep
r
es
en
t
s
t
h
e co
n
t
r
o
l
v
ect
o
r
(
v
,
ω
)T
.
G
en
er
al
l
y
d
y
n
a
m
i
c
m
o
d
el
i
n
g
i
s
t
he
s
ys
t
e
m
m
o
t
i
o
n
s
t
ud
y i
n
w
h
i
c
h
f
o
r
c
e
s
a
r
e
m
o
d
e
l
e
d
a
nd
i
t
c
a
n
i
n
cl
u
d
e en
er
g
i
e
s
an
d
t
h
e s
p
eed
s
as
s
o
ci
at
ed
w
i
t
h
t
h
e
m
o
t
i
o
n
s
.
T
h
e g
en
er
al
d
y
n
a
m
i
c
m
o
d
el
o
f
m
o
b
i
l
e r
o
b
o
t
can
be
de
s
c
r
i
be
d
by
t
h
e
f
ol
l
o
w
i
ng
e
qu
a
t
i
on
(2
)
:
()
(
,
)
()
()
()
d
M
qq
V
qqq
F
q
G
q
B
q
ττ
+
+
+
+
=
()
T
A
q
λ
−
(2
)
w
h
er
e
M
(
q
)
is
t
h
e
s
y
m
m
e
tr
ic
p
o
s
itiv
e
d
e
f
in
ite
i
n
e
r
tia
m
a
tr
i
x
,
(
)
V
q,
q
is
t
h
e
c
e
n
tr
ip
e
ta
l a
n
d
c
o
r
io
lis
m
a
tr
ix
,
(
)
Fq
is
th
e
s
u
r
f
a
c
e
f
r
ic
t
io
n
m
a
t
r
ix
,
G
(
q
)
is
th
e
g
r
a
v
ita
tio
n
a
l
v
e
c
to
r
,
τ
d
d
e
n
o
te
d
b
o
u
n
d
e
d
u
n
k
n
o
w
n
d
is
tu
r
b
a
n
c
e
s
in
c
lu
d
i
n
g
u
n
s
tr
u
c
tu
r
e
d
n
o
t
m
o
d
e
le
d
d
y
n
a
m
ic
s
,
B
(
q
)
is
th
e
in
p
u
t tr
a
n
s
f
o
r
m
a
t
io
n
m
a
tr
i
x
,
τ
i
s
t
h
e
i
np
ut
ve
c
t
o
r
,
A
T
(
q
)
is
th
e
m
a
t
r
ix
a
s
s
o
c
ia
te
d
w
i
th
t
h
e
co
n
s
t
r
ai
n
t
s
,
λ
i
s
t
h
e
co
n
s
t
r
ai
n
t
f
o
r
ces
v
ect
o
r
.
T
he
a
b
o
ve
s
y
s
t
e
m
c
a
n
be
t
r
a
n
s
f
or
m
e
d i
n
t
o a
m
or
e
s
ui
t
a
bl
e
r
e
pr
e
s
e
n
t
a
t
i
on
f
or
c
on
t
r
ol
a
n
d s
i
m
u
l
a
t
i
on
pu
r
pos
e
s
.
T
h
e
t
w
o
f
o
llo
w
in
g
m
a
tr
ic
e
s
a
r
e
d
e
f
i
n
e
d
to
d
o
th
is
tr
a
n
s
f
o
r
m
a
tio
n
a
s
s
ho
w
n i
n (
3
-
4)
:
v
ϑ
ω
=
(3
)
(
)
0
0
01
c
o
s
S
q
si
n
θ
θ
=
(4
)
T
h
e
m
a
tr
i
x
S
(
q
)
h
a
s
t
h
e
f
o
llo
w
i
n
g
r
e
la
tio
n
w
it
h
m
a
tr
ix
A
(
q
)
(5
)
:
(
)
(
)
0
TT
S
qA
q
=
(5
)
T
h
e eq
u
at
i
o
n
(
2
)
can
b
e r
ew
r
i
t
t
en
as
s
h
o
w
n
i
n
(
6
-
10)
:
()
(
,
)
()
h
M
hq
V
hq
q
B
hq
ϑ
ϑτ
τ
+
+
=
(6
)
wi
t
h
:
(
)
(
)
(
)
(
)
T
Mh
q
S
q
M
q
S
q
=
(7
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA
I
SSN
:
2089
-
4856
D
y
nam
i
c
C
ont
r
ol
of
M
obi
l
e
R
obot
U
s
i
ng R
B
F
G
l
o
bal
F
as
t
Sl
i
di
ng m
od
e
(
A
li M
a
lle
m
)
161
(
)
T
hd
Sq
ττ
=
(8
)
(
)
(
)
(
)
T
Bh
q
S
q
B
q
=
(9
)
(
)
(
)
(
)
(
)
(
)
,
T
TT
V
h
S
q
M
qS
q
S
q
V
q
qS
=
+
(
10)
E
q
ua
t
i
o
n (
6
)
i
s
t
he
e
q
ua
t
i
o
n w
h
i
c
h i
s
us
e
d
f
o
r
t
he
c
o
nt
r
o
l
a
nd
s
i
m
ul
a
t
i
o
n a
na
l
ys
i
s
o
f
t
he
r
o
bo
t
.
T
he
dy
n
a
m
i
c
m
ode
l
i
ng
of
t
h
e
r
obo
t
i
s
pr
e
s
e
n
t
e
d i
n
[
23]
.
3.
RB
F
NE
URAL
N
E
T
W
O
RK
R
B
F
ne
t
w
o
r
ks
a
r
e
a
d
a
p
t
i
ve
l
y us
e
d
t
o
a
p
p
r
o
xi
m
a
t
e
t
he
u
nc
e
r
t
a
i
n no
nl
i
ne
a
r
f
u
nc
t
i
o
n.
T
he
a
l
go
r
i
t
h
m
o
f
a r
ad
i
al
b
as
i
s
f
u
n
ct
i
o
n
(
R
B
F
)
n
et
w
o
r
k
s
i
s
d
ef
i
n
ed
i
n
[
2
4
]
as
s
h
o
w
n i
n
(
11)
:
2
2
(
)/
)
j
ij
j
h
gx
c
b
=
−
(
11)
(
)
t
f
W
hx
ε
=
+
w
h
er
e
x
i
s
t
he
i
np
ut
s
t
a
t
e
o
f
t
h
e
ne
t
w
o
r
k,
i
i
s
t
h
e
i
n
put
num
b
e
r
of
t
h
e
n
e
t
w
or
k
,
j
i
s
t
h
e
num
be
r
of
h
i
dde
n
l
a
y
e
r
n
ode
s
.
I
n
t
he
ne
t
w
o
r
k,
h
=
[
h
1
h
2
…
…
h
n
]
T
i
s
t
he
o
ut
p
ut
o
f
G
a
us
s
i
a
n
f
u
nc
t
i
o
n.
W
i
s
t
h
e
ne
ur
a
l
ne
t
w
o
r
k
w
e
i
g
ht
s
,
a
nd
t
he
pr
opa
g
a
t
i
on e
r
r
or
i
s
ε
≤
ε
N.
R
B
F
n
e
t
w
or
k a
ppr
ox
i
m
a
t
i
on
f
i
s
us
e
d.
I
n
F
i
gur
e
1
i
s
r
ep
r
es
en
t
s
a
n R
B
F
ne
t
w
o
r
k
.
T
he
o
ut
p
ut
o
f
R
B
F
ne
t
w
o
r
k i
s
(
12)
:
(
)
ˆ
ˆ
()
T
f
x
W
hx
=
(
12)
F
i
g
ur
e
1
.
R
B
F
ne
ur
a
l
ne
t
w
o
r
k
.
T
he
G
a
us
s
i
a
n
f
u
nc
t
i
o
n c
a
n d
e
f
i
ne
a
s
s
ho
w
n
i
n
(
13)
:
(
)
2
2
e
x
p
(
)
2
r
hx
σ
=
−
(
13)
4.
RB
F
-
GF
S
M
-
C
O
N
TR
O
LLE
R
I
n t
hi
s
w
o
r
k t
w
o
c
a
s
e
s
o
f
c
o
nt
r
o
l
a
r
e
p
r
op
o
s
e
d
,
t
he
f
i
r
s
t
w
i
t
ho
ut
d
i
s
t
ur
b
a
nc
e
s
,
t
he
s
e
c
o
nd
i
n t
he
p
r
es
en
ce o
f
d
i
s
t
u
r
b
an
ces
.
I
n F
i
g
ur
e
2
b
e
lo
w
r
e
s
um
e
s
t
h
e
c
on
t
r
ol
s
t
r
a
t
e
gy
pr
opos
e
d i
n
t
h
i
s
w
or
k
.
T
he
f
i
g
ur
e
b
e
lo
w
r
e
s
um
e
s
t
h
e
c
ont
r
ol
s
t
r
a
t
e
gy
pr
opos
e
d i
n
t
h
i
s
w
or
k
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
2
08
9
-
4856
I
J
RA
,
V
o
l.
7
, N
o
.
3
,
S
e
pt
e
m
be
r
201
8
:
15
9
–
1
68
162
Fi
g
ur
e
2
.
C
o
n
tr
o
l s
tr
a
te
g
y
4
.1
.
C
o
nt
r
o
l
w
i
t
ho
ut
di
s
t
ur
b
a
nc
e
s
I
n
t
h
e cas
e o
f
d
i
s
t
u
r
b
a
n
ces
ab
s
en
ce,
t
h
e eq
u
at
i
o
n
(
6
)
b
eco
m
e
s
(
14)
:
()
(
,
)
()
M
hq
V
hq
q
B
hq
ϑ
ϑτ
+=
(
14)
T
h
e
v
e
lo
c
itie
s
er
r
o
r
s
ar
e d
ef
i
n
i
t
e as
s
h
o
w
n i
n
(
15)
:
v
r
r
r
e
vv
e
e
ϑ
ω
ϑϑ
ωω
−
=
−=
=
−
(
15)
T
h
e d
er
i
v
at
i
v
e o
f
(
1
5
)
i
s
o
b
t
ai
n
ed
as
s
ho
w
n i
n (
1
6
)
:
r
e
ϑ
ϑϑ
=
−
(
16)
A
cco
r
d
i
n
g
t
o
t
h
e eq
u
at
i
o
n
(
1
6
)
,
t
h
e eq
u
at
i
o
n
(
1
4
)
can
b
e r
ew
r
i
t
t
en
as
s
ho
w
n
i
n (
1
7
)
:
(
)(
)
(
,
)(
)
(
)
rr
M
hq
e
V
hq
q
e
B
hq
ϑϑ
ϑ
ϑτ
−+
−
=
(
)
(
)
(,
)
(,
)
rr
M
hq
M
hqe
V
hq
q
V
hq
qe
ϑϑ
ϑϑ
−+
−
()
Bh
q
τ
=
(
17)
P
u
ttin
g
(
18)
:
(
)
(
)
(
)
,
rr
f
x
M
hq
V
hq
q
ϑϑ
=
+
(
18)
wi
t
h
[]
T
r
T
TT
r
x
ϑ
ϑ
ϑ
=
R
e
pl
a
c
i
ng
(
18)
i
n
(
17)
a
nd
a
s
s
ho
w
n
i
n (
1
9
)
:
(
)
(
)
(
)
,
()
M
hqe
V
hq
qe
f
x
B
h
q
ϑϑ
τ
−
−
+=
(
19)
T
he
s
l
i
d
i
ng
m
o
d
e
c
o
n
tr
o
lle
r
p
r
o
p
o
s
e
d
in
th
is
w
o
r
k
is
b
a
s
e
d
o
n
g
lo
b
a
l f
a
s
t s
lid
in
g
m
o
d
e
c
o
n
tr
o
l,
th
is
c
o
n
tr
o
l
c
a
n
m
a
ke
t
he
s
ys
t
e
m
s
t
a
t
e
s
c
o
nve
r
ge
t
o
z
e
r
o
i
n a
f
i
ni
t
e
t
i
m
e
.
A
ki
nd
o
f
f
a
s
t
t
e
r
m
i
na
l
s
l
i
d
i
ng
s
ur
f
a
c
e
i
s
pr
op
os
e
d a
s
s
ho
w
n i
n (
2
0
)
:
/
0
qp
sx
x
x
α
β
=
++
=
(
20)
W
h
er
e
x
∈
R
i
s
t
h
e
s
t
a
t
e
a
n
d
α
>
0
.
T
h
e r
each
i
n
g
t
i
m
e o
f
t
h
e s
l
i
d
i
n
g
s
u
r
f
ace t
o
zer
o
i
s
d
ef
i
n
ed
a
s
s
h
o
w
n i
n
(
21)
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA
I
SSN
:
2089
-
4856
D
y
nam
i
c
C
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r
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of
M
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R
obot
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B
F
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l
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t
Sl
i
di
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(
A
li M
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m
)
163
(
0)
ln
()
pq
p
s
px
t
pq
α
β
αβ
−
+
=
−
(
21)
T
h
e g
l
o
b
al
f
as
t
s
l
i
d
i
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g
s
u
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f
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s
s
el
ect
ed
as
s
ho
w
n i
n (
2
2
)
:
/
00
0
qp
ss
s
s
αβ
=
++
(
22)
W
h
er
e
β
>0
an
d
q,
p
(
q
<p
)
a
r
e
pos
i
t
i
v
e
s
odd n
um
be
r
s
.
T
h
e
n
,
to
o
b
ta
in
th
e
c
o
n
tr
o
l la
w
,
a
s
lid
i
n
g
s
u
r
f
a
c
e
is
c
h
o
o
s
i
n
g
(
23)
:
se
ϑ
=
(2
3
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wi
t
h
:
1
2
s
s
s
=
A
c
c
o
r
d
i
n
g t
o
t
he
e
q
ua
t
i
o
ns
(
2
3
)
a
nd
(
2
2)
,
o
ne
c
a
n ha
ve
(
24)
:
(
)
/
1
qp
e
ee
ϑ
ϑϑ
α
β
=
−+
−
(
24)
R
e
pl
a
c
i
ng
(
24)
i
n
(
19)
a
nd
a
s
s
ho
w
n
i
n (
2
5
)
:
(
)
(
)
(
)
(
)
/
1,
qp
M
hq
e
e
V
hq
qe
ϑϑ
ϑ
αβ
++
−
+
(
)
()
f
x
Bh
q
τ
=
(
25)
T
h
e
c
o
n
tr
o
l la
w
is
o
b
ta
in
e
d
a
s
s
ho
w
n
i
n (
2
6
)
:
(
)
(
)
1
/
()
qp
v
B
h
C
e
f
x
M
hqe
ϑϑ
τβ
−
=
++
(
26)
S
uc
h t
ha
t
(
27)
:
(
)
(
)
(
)
1,
v
C
M
hq
V
hq
q
α
=
+−
(
27)
4
.2
.
C
o
nt
r
o
l
i
n
p
res
en
ce o
f
d
i
s
t
u
rb
a
n
ces
I
n
t
h
i
s
cas
e,
t
h
e d
i
s
t
u
r
b
an
ces
ar
e co
n
s
i
d
er
ed
;
t
h
er
ef
o
r
e t
h
e
n
eu
r
al
n
et
w
o
r
k
co
n
t
r
o
l
l
er
i
s
i
n
t
r
o
d
u
ced
.
T
h
e eq
u
at
i
o
n
(
1
9
)
i
s
d
ef
i
n
ed
as
s
h
o
w
n i
n
(
28)
:
(
)
(
)
(
)
,
()
d
M
hqe
V
hq
qe
f
x
B
h
q
ϑϑ
ττ
−
−
+
−=
(
28)
T
h
e
c
on
t
r
ol
l
a
w
de
s
i
gn
e
d i
n
e
qu
a
t
i
on
(
26
)
can
b
e r
ew
r
i
t
t
en
as
s
h
o
w
n i
n
(
29)
:
(
)
(
)
1
/
ˆ
()
qp
v
B
h
C
e
f
x
M
hqe
ϑϑ
τβ
−
=
++
(
29)
w
h
e
n
ˆ
()
f
x
i
s
t
he
o
ut
p
ut
o
f
R
B
F
ne
t
w
o
r
k.
ˆ
()
f
x
a
ppr
ox
i
m
a
t
e
s
()
f
x
R
B
F
n
e
t
w
or
k
c
a
n
be
a
dopt
e
d t
o a
ppr
ox
i
m
a
t
e
f
(x
).
T
h
e d
es
i
r
ed
al
g
o
r
i
t
h
m
o
f
R
B
F
n
et
w
o
r
k
i
s
(3
0
-
32)
:
2
2
(
)/
i
ii
gx
c
b
ϕ
=
−
i
=
1
,2
,……..n
(
30)
(
)
*
T
yW
x
ϕ
=
(
31)
(
)
*
()
T
f
x
W
x
ϕε
=
+
(
32)
x
i
s
t
he
i
np
ut
s
t
a
t
e
o
f
ne
t
w
o
r
k,
φ
(x
)
=
[
φ
1
φ
2
…
.
φ
n
]
T
,
ε
i
s
t
h
e
a
p
p
r
o
x
i
m
a
t
i
o
n
e
r
r
o
r
o
f
ne
t
w
or
k
.
W*
is
th
e
w
ei
g
h
t
v
ect
o
r
o
f
d
es
i
r
ed
R
B
F
n
et
w
o
r
k
.
R
e
pl
a
c
i
ng
(
29)
i
n
(
2
8)
a
nd
a
s
s
ho
w
n i
n (
3
3
)
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
2
08
9
-
4856
I
J
RA
,
V
o
l.
7
, N
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3
,
S
e
pt
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m
be
r
201
8
:
15
9
–
1
68
164
(
)
(
)
(
)
(
)
/
3
,
qp
v
M
hqe
V
hq
q
C
e
M
hqe
ϑ
ϑϑ
β
=
−
+−
+
0
µ
(
33)
w
h
e
n:
(
)
0
d
f
x
µτ
=
−
,
a
nd
(
)
(
)
ˆ
()
f
x
f
x
f
x
=
−
T
he
o
ut
p
ut
o
f
t
he
ne
t
w
o
r
k i
s
g
i
vi
ng a
s
s
ho
w
n i
n (
3
4
)
:
ˆ
()
()
T
f
x
W
x
ϕ
=
(
34)
S
e
le
c
tin
g
:
*
ˆˆ
WW
W
=
−
,
*
m
a
x
WW
≤
T
h
er
ef
o
r
e
(
35
)
:
(
)
(
)
0
ˆ
T
dd
f
x
w
x
µ
τ
ϕ
ετ
=
−
=
+
−
(
35)
T
h
e
c
o
n
tr
o
l la
w
d
e
s
ig
n
e
d
in
e
q
u
a
tio
n
(
2
9
)
c
a
n
r
e
w
r
ite
a
s
s
ho
w
n
i
n
(
36)
:
11
ˆ
()
(
()
)
q
p
v
f
Bh
Bh
C
e
M
h
q
x
e
ϑϑ
ττ
β
−
−
=
++
=
−
ξ
(
36)
w
h
er
e
ξ
i
s
t
h
e
r
o
b
u
s
t
el
e
m
e
n
t
i
n
t
r
o
d
u
ced
t
o
el
i
m
i
n
at
e
t
h
e
n
e
t
w
o
r
k
ap
p
r
o
x
i
m
at
i
o
n
er
r
o
r
ε
a
n
d
t
h
e
d
i
s
t
u
r
b
an
ces
τ
d
R
e
pl
a
c
i
ng
t
h
e
e
q
u
a
t
i
on
(
35)
i
n
(
33)
a
nd
s
ho
w
n i
n (
3
7
)
:
(
)
(
)
(
)
(
)
/
,
qp
v
M
hqe
V
hq
q
C
e
M
hqe
ϑ
ϑ
ϑ
β
−
+−
=
+
(
)
ˆ
ξ
d
T
W
x
ϕ
ετ
+
−
+
(
37)
P
u
ttin
g
(
38)
:
(
)
1
ˆ
ξ
t
d
Wx
µ
ϕ
ετ
=
+
−
+
(
38)
R
e
pl
a
c
i
ng
(
36)
i
n
(
35)
a
nd
a
s
s
ho
w
n
i
n (
3
9
)
:
(
)
(
)
(
)
(
)
/
,
qp
v
M
hqe
V
hq
q
C
e
M
hqe
ϑ
ϑ
ϑ
β
=
−
+−
+
1
µ
(
39)
w
it
h
:
(
)
,
v
Vh
q
q
C
+
=
(
)
(
)
1
Mh
q
α
+
T
h
e
r
o
b
u
s
t
el
e
m
en
t
ξ
i
s
d
es
i
g
n
ed
as
s
ho
w
n i
n (
4
0
)
:
(
)
Nd
ξ
ε
b
s
i
gn
(
)
e
ϑ
=
−
+
(
40)
W
h
er
e:
N
εε
≤
,
dd
b
τ
≤
T
h
e can
d
i
d
at
e f
u
n
ct
i
o
n
o
f
l
y
ap
u
n
o
v
i
s
s
e
l
ect
ed
as
s
ho
w
n i
n (
4
1
)
:
(
)
1
w
1
1
ˆ
(F
)
2
2
ˆ
TT
e
M
hqe
t
r
W
W
ϑϑ
ρ
−
=
+
(
41)
T
h
e d
er
i
v
at
i
v
e o
f
t
h
e l
y
ap
u
n
o
v
f
u
n
ct
i
o
n
i
s
d
ef
i
n
ed
as
s
h
o
w
n i
n
(
42)
:
(
)
(
)
1
1
(F
)
2
TT
T
w
M
e
M
hqe
e
qe
t
r
h
WW
ϑϑ
ϑϑ
ρ
−
=
+
+
(
42)
F
r
o
m
e
qu
a
t
i
on
(
37)
t
o be
(
43)
:
(
)
(
)
(
)
/
1
T
T
qp
e
M
hq
e
e
M
hqe
ϑ
ϑϑ
ϑ
ρ
α
β
=
−
+−
+
(
)
(
)
1
F(
ξ
)
T
TT
wd
t
rW
W
x
e
e
ϑ
ϑ
ϕ
ετ
−
+
+
−+
(4
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA
I
SSN
:
2089
-
4856
D
y
nam
i
c
C
ont
r
ol
of
M
obi
l
e
R
obot
U
s
i
ng R
B
F
G
l
o
bal
F
as
t
Sl
i
di
ng m
od
e
(
A
li M
a
lle
m
)
165
S
e
le
c
t:
(
)
w
F
T
W
xe
ϑ
ϕ
=
T
h
e ad
ap
t
i
v
e r
u
l
e o
f
n
et
w
o
r
k
is
(
44)
:
ˆ
()
T
w
W
F
xe
ϑ
ϕ
=
(4
4
)
T
h
er
ef
o
r
e
(
45
)
:
(
)
(
)
(
)
/
1
T
T
qp
e
M
hq
e
e
M
hqe
ϑ
ϑϑ
ϑ
ρ
α
β
=
−
+−
+
(
ξ
)
T
d
e
ϑ
ετ
−+
(4
5
)
C
o
ns
i
d
e
r
i
ng t
he
t
e
r
m
(
46)
:
(
)
(
)
ξ
T
TT
dd
e
ee
ϑ
ϑϑ
ετ
ετ
ξ
−
+=
−
+
(
)
T
d
e
e
ϑϑ
ετ
=
−−
(
)0
Nd
b
ε
+
≤
(4
6
)
S
uc
h
t
ha
t
m
a
t
r
i
x
Mh
d
e
f
in
e
p
o
s
itiv
e
,
α
a
n
d
β
a
r
e
p
o
s
itiv
e
s
,
p
a
nd
q
a
r
e
p
o
s
itiv
e
s
o
d
d
in
te
g
e
r
s
(0
<
q
/p
<
1
)
;
t
h
er
ef
o
r
e:
0
ρ
≤
5.
Si
m
ul
a
t
i
o
ns
a
n
d R
e
s
u
l
t
s
I
n
th
i
s
s
e
c
tio
n
t
h
e
s
i
m
u
la
tio
n
u
s
i
n
g
M
a
tla
b
/S
i
m
u
li
n
k
is
a
p
p
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o
n
th
e
d
y
n
a
m
ic
m
o
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ile
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b
o
t s
y
s
te
m
.
F
ir
s
tl
y
,
t
h
e d
i
s
t
u
r
b
an
ce
s
ar
e ex
cl
u
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ed
,
s
eco
n
d
l
y
t
h
e d
i
s
t
u
r
b
an
ces
ar
e i
n
j
ect
ed
,
an
d
f
in
a
lly
th
e
R
B
F
n
e
u
r
a
l
n
et
w
o
r
k
i
s
i
n
t
r
o
d
u
ced
t
o
es
t
i
m
at
e t
h
e s
y
s
t
e
m
n
o
n
l
i
n
ear
f
u
n
ct
i
o
n
an
d
d
eal
t
h
e d
i
s
t
u
r
b
an
ces
.
L
e
t
u
s
c
ons
i
de
r
:
v
r
=
1
m/
s
,
ω
r
= 1
r
ad
/
s
.
T
h
e
d
i
s
t
u
r
b
an
ces
τ
d
=
[0
.
1
.
s
i
n
(t
) 0
.
1
.
c
o
s
(t
)].
α
=
1
, β
=
2
,
p
=7
,
q
=
5
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N
=
0
.2
, b
d
=
0.
1.
T
he
ne
ur
a
l
ne
t
w
o
r
k i
s
c
ho
s
e
n
w
i
t
h s
e
ve
n
hi
d
d
e
n,
t
he
i
ni
t
i
a
l
w
e
i
ght
m
a
t
r
i
x i
s
s
e
l
e
c
t
e
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a
s
0
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1
.
b
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T
h
e m
at
r
i
ces
v
al
u
es
o
f
t
h
e d
y
n
a
m
i
c
m
o
d
el
ar
e t
ak
en
f
r
o
m
[
2
5
]
.
F
i
gur
e
s
3
a
n
d
4
s
h
o
w
t
h
at
act
u
al
f
o
r
w
ar
d
an
d
a
n
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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SSN
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4856
D
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s
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R
EF
ER
EN
C
ES
[
1]
C
.
S
a
m
s
on,
V
e
l
oc
i
t
y
a
nd t
or
qu
e
f
e
e
dba
c
k
c
ont
r
ol
of
a
nonhol
onom
i
c
c
a
r
t
,
i
n L
e
c
t
ur
e
N
ot
e
s
i
n C
on
t
r
ol
a
n
d
I
n
f
o
r
m
at
i
o
n
S
ci
en
ce,
C
.
C
an
u
d
as
d
e W
i
t
,
E
d
.
B
er
l
i
n
,
G
er
m
an
y
:
S
p
r
i
n
g
er
-
V
e
r
l
a
g,
1
99
1,
pp
.
1
25
–
15
1.
[
2]
C
.
C
a
nu
da
s
de
W
i
t
,
H
.
K
he
n
no
u
f
,
C
.
S
a
m
s
on,
a
nd
O
.
J
.
S
or
da
l
e
n,
N
o
nl
i
ne
a
r
c
ont
r
ol
de
s
i
g
n f
or
m
obi
l
e
r
obo
t
s
,
i
n
R
e
c
e
nt
T
r
e
nds
i
n M
ob
i
l
e
R
o
bo
t
s
,
Y
.
F
.
Z
he
n
g,
E
d.
Si
ng
ap
or
e
:
W
or
l
d Sc
i
e
nt
i
f
i
c
,
19
93,
p
p.
12
1
–
15
6.
[
3]
A
.
M
.
B
l
oc
h,
M
.
R
e
y
ha
nog
l
u,
a
nd N
.
H
.
M
c
C
l
a
m
r
oc
h,
C
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ol
a
n
d s
t
a
bi
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t
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of
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m
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s
y
s
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m
s
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I
E
E
E
T
r
a
n
s
. A
u
t
o
m
a
t
.
C
o
n
t
r
., v
o
l
. 3
7
, p
p
.
1
7
4
6
–
17
57
,
1
99
2.
[
4]
Y
.
K
a
na
y
a
m
a
,
Y
.
K
i
m
ur
a
,
F
.
M
i
y
a
z
a
k
i
,
a
nd T
.
N
og
uc
hi
,
A
s
t
a
bl
e
t
r
a
c
k
i
ng
c
ont
r
ol
m
e
t
hod f
or
a
n a
ut
o
nom
ous
m
obi
l
e
r
ob
ot
,
i
n
P
ro
c
.
I
E
E
E
I
nt
.
C
o
nf
.
R
obo
t
.
A
ut
om
at
.
,
1
99
0,
pp.
3
84
–
389
.
[
5]
F
uk
a
o,
T
.
,
N
a
k
a
g
a
w
a
,
H
.
a
nd A
da
c
hi
,
N
,
A
da
pt
i
v
e
t
r
a
c
k
i
ng
c
ont
r
ol
of
a
non
hol
on
om
i
c
m
obi
l
e
r
obot
.
I
E
E
E
T
r
ans
ac
t
i
ons
on
R
o
bo
t
i
c
s
an
d A
ut
om
at
i
on
.
20
00
1
6 (
5)
,
60
9
–
61
5
.
[
6]
H
w
a
ng
,
C
.
L
.
(
20
04)
.
A
nov
e
l
T
a
ka
g
i
-
S
ug
e
no
-
ba
s
e
d r
ob
us
t
a
da
pt
i
v
e
f
uz
z
y
s
l
i
di
ng
-
m
ode
c
ont
r
o
l
l
e
r
.
I
E
E
E
T
r
ans
ac
t
i
ons
on
F
uz
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)
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7.
[
7]
M
o
h
ar
er
i
,
O
.
,
D
ha
o
ua
di
,
R
.
,
S
h
i
r
azi
,
M
.
M
.
I
n
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l
l
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g
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ur
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l
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w
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k
ba
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ont
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ol
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r
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p.
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-
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.
[
8]
Ko
z
l
o
w
sk
i
,
K.
a
nd
M
aj
ch
r
zak
,
J
.
A
ba
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ppi
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ppr
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h t
o c
o
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r
ol
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oc
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gs
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vol
.
4
.
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p
p.
39
72
-
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77.
[
9]
N
ga
n
ga
-
Ko
u
y
a
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an
d
O
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.
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.
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p
p.
85
-
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[
1
0]
S
ong
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.
T
.
,
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.
H
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T
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.
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.
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ont
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ack
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01
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3
05
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18.
[
1
1]
Y
a
ng
,
J
.
M
.
a
nd K
i
m
,
J
.
H
.
S
l
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m
ode
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or
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s
.
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E
T
r
a
ns
ac
t
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ons
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R
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bot
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ut
om
at
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on
.
19
99
1
5 (
3)
,
57
8
–
587
.
[
1
2]
R.
F
i
e
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a
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d
F
.
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.
l
e
w
i
s
.
C
ont
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ol
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nom
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c
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obo
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ur
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ans
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ur
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9
(4
).
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89
-
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00(
19
98)
.
[
1
3]
J
i
n O
h J
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.
A
da
pt
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ur
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m
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t
(
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11)
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,
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67
-
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[
1
4]
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k
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S
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,
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oo S
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ode
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17,
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o.
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07
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14,
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an 20
09
.
[
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O
u
. M
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–
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[
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6]
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u
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–
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[
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7]
D
u
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i
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.J
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r
. 5
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11
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717
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[
1
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hoo
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,
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L
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nd M
a
n,
Z
.
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obus
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Y
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., K
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(1
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61
-
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77
.
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2
1]
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o
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(
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)
,
21
06
–
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10
.
[
2
2]
K
h
o
o
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., Y
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., M
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u
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to
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,
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13)
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9
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5
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140
3
-
14
10.
[
2
3]
R
. F
i
e
r
r
o
a
n
d
F
. L
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e
w
i
s
,
C
ont
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of
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N
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nom
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obi
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ur
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on N
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v
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.
9,
pp.
5
89
-
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0,
J
ul
y
19
98
.
[
2
4]
P
a
r
k
J
,
S
a
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be
r
g
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ni
v
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s
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91,
3:
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7
.
[
2
5]
D
uc
D
o,
K
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,
Z
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ong
-
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J.
,
P
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n
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l
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t
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t
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obot
s
,
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E
E
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r
a
n
s
. A
u
t
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m
a
t
.
C
o
n
t
r
., V
3
0
, N
3
,
p
p
. 5
8
9
–
59
4,
06
/
0
4
.
Evaluation Warning : The document was created with Spire.PDF for Python.