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Ro
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Aut
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(
I
J
RA
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Vo
l.
5
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No
.
4
,
Dec
em
b
er
2
0
1
6
,
p
p
.
27
7
~
28
3
I
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N:
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4856
277
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M
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M
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De
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NIT
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6
In
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C
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Natio
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1.
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b
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r
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[
1
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.
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[
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]
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[
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IJ
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Dec
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27
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28
3
278
2.
DYNA
M
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CS O
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WO
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M
ANIPULA
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O
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Her
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o
t
m
a
n
i
p
u
lato
r
[
9
]
:
(
̈
̈
)
(
̈
̈
)
(
)
̈
̇
̇
̇
(
)
̇
(
6
)
(
̈
̈
)
(
̈
)
̈
̇
(
7
)
w
h
er
e
c
1
=
co
s
(
q
1
)
,
s
1
=sin
(
q
1
)
,
s
12
=
s
in
(
q
1
+q
2
)
,
c
2
=
co
s
(
q
2
)
,
s
2
=si
n
(
q
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
Dyn
a
mic
A
n
a
lysi
s
o
f Tw
o
Lin
k
R
o
b
o
t Ma
n
i
p
u
la
to
r
fo
r
C
o
n
tr
o
l D
esig
n
Usi
n
g
P
I
D
C
o
mp
u
te
d
(
Jo
lly
A
tit S
h
a
h
)
279
W
h
ich
ca
n
b
e
r
e
w
r
itte
n
as
[
1
0
]
,
(
)
̈
(
̇
)
̇
(
)
(
8
)
T
h
e
p
ar
am
eter
s
v
i
an
d
k
i
ar
e
v
i
s
co
u
s
a
n
d
C
o
u
lo
m
b
f
r
ictio
n
co
ef
f
icien
ts
,
r
esp
ec
ti
v
el
y
.
T
h
e
eq
u
atio
n
ca
n
b
e
w
r
it
ten
i
n
th
e
f
o
r
m
o
f
̈
̇
(
9
)
w
h
er
e
[
]
(
)
(
)
[
̇
̇
̇
̇
]
,
w
it
h
(
)
[
]
[
̇
̇
]
[
]
[
(
)
(
)
(
)
(
)
]
*
+
[
(
̇
)
(
̇
)
]
3.
DE
S
I
G
N
ST
E
P
S F
O
R
P
I
D
CO
M
P
UT
E
D
T
O
RQ
UE
CO
NT
RO
L
O
F
RO
B
O
T
M
ANIPULAT
O
R
S
An
y
m
ec
h
an
ical
s
y
s
te
m
h
av
i
n
g
tr
an
s
latio
n
m
o
tio
n
i
s
d
escr
ib
ed
as b
elo
w
s
h
o
w
n
i
n
F
ig
u
r
e
2
.
Fig
u
r
e
2.
Me
ch
an
ical
s
y
s
te
m
w
it
h
tr
a
n
s
latio
n
al
m
o
t
io
n
Op
en
lo
o
p
d
y
n
a
m
ic
eq
u
a
tio
n
f
o
r
s
u
ch
t
y
p
e
o
f
s
y
s
te
m
w
ill b
e
g
iv
e
n
b
y
f
o
llo
w
i
n
g
eq
u
a
tio
n
,
(
1
0
)
ste
p
1
Ass
u
m
in
g
t
h
at
w
e
d
o
n
'
t k
n
o
w
v
alu
e
o
f
m
,
b
,
k
.
I
n
m
o
d
el
b
ased
co
n
tr
o
l
m
eth
o
d
m
,
b
,
k
ar
e
ca
lcu
la
ted
u
s
i
n
g
r
ea
l
ti
m
e
ca
lc
u
lat
io
n
s
.
T
h
u
s
s
y
s
te
m
w
i
ll
b
e
s
i
m
p
li
f
ied
to
m
a
s
s
o
n
l
y
s
y
s
te
m
.
A
p
p
l
y
i
n
g
m
o
d
el
b
ased
co
n
tr
o
l
la
w
to
eq
u
atio
n
(
1
0
)
b
ec
o
m
e,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
5
,
No
.
4
,
Dec
em
b
er
2
0
1
6
:
27
7
–
28
3
280
w
h
er
e
,
w
h
ic
h
is
n
e
w
in
p
u
t to
s
y
s
te
m
a
s
s
h
o
w
n
i
n
F
ig
u
r
e
3
o
n
n
e
x
t p
ag
e
.
Fig
u
r
e
3
.
Stru
ct
u
r
e
o
f
co
m
p
u
t
er
to
r
q
u
e
co
n
tr
o
l
Ste
p 2
A
p
p
l
y
a
n
y
cla
s
s
ical
co
n
tr
o
l la
w
li
k
e
P
D,
P
I
,
P
I
D.
I
f
w
e
ap
p
l
y
s
er
v
o
co
n
tr
o
l la
w
t
o
ab
o
v
e
eq
u
atio
n
,
it
w
i
ll g
et
c
o
n
v
er
ted
in
to
co
m
p
u
ted
to
r
q
u
e
co
n
tr
o
l s
y
s
te
m
.
T
o
im
p
r
o
v
e
t
h
e
ac
c
u
r
ac
y
w
e
w
il
l
u
s
e
f
ee
d
b
ac
k
co
n
tr
o
l.
C
o
m
p
u
ted
to
r
q
u
e
co
n
tr
o
l
alo
n
g
w
it
h
P
I
D
f
ee
d
b
ac
k
co
n
tr
o
l
is
k
n
o
w
n
a
s
P
I
D
co
m
p
u
ted
to
r
q
u
e
co
n
tr
o
l
m
et
h
o
d
.
T
o
tal
s
y
s
te
m
b
lo
ck
d
iag
r
a
m
is
s
h
o
w
n
b
elo
w
in
Fig
u
r
e
4
.
Fig
u
r
e
4
.
P
I
D
f
ee
d
b
ac
k
lo
o
p
c
o
m
p
u
ted
to
r
q
u
e
co
n
tr
o
ller
th
u
s
to
tal
co
n
tr
o
l la
w
is
d
e
f
in
e
d
as,
(
)
(
̈
̇
)
(
̇
)
(
)
w
h
er
e
(
)
is
i
n
teg
r
al
o
f
tr
ac
k
i
n
g
e
r
r
o
r
e(
t
)
an
d
k
d
, k
p
, k
i
ar
e
co
n
tr
o
l g
ain
.
4.
SI
M
UL
AT
I
O
N
A
ND
RE
SU
L
T
S
Fo
r
s
i
m
u
latio
n
p
u
r
p
o
s
e
f
o
llo
win
g
v
al
u
es a
r
e
tak
e
n
i
n
M
A
T
L
A
B
P
r
o
g
r
a
m
m
in
g
.
T
o
tal
Sim
u
latio
n
T
im
e
=
T
1
=
1
0
s
ec
an
d
T
2
=
1
2
0
s
ec
;
Sa
m
p
li
n
g
T
i
m
e
=
T
=
0
.
0
0
2
s
e
c
;
m
as
s
=
m
=
[
2
;2
]
u
n
it;
l1
=
1
u
n
it;
l2
=
1
u
n
it;
g
r
av
=
9
.
8
u
n
it
;
Gain
s
:
Kd
=
[
1
0
0
0
0
;0
1
0
0
0
]
;
Kp
=
[
1
0
0
0
;0
1
0
0
]
;
Ki
=
[
1
0
0
;
0
1
0
]
;
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
Dyn
a
mic
A
n
a
lysi
s
o
f Tw
o
Lin
k
R
o
b
o
t Ma
n
i
p
u
la
to
r
fo
r
C
o
n
tr
o
l D
esig
n
Usi
n
g
P
I
D
C
o
mp
u
te
d
(
Jo
lly
A
tit S
h
a
h
)
281
S
i
m
u
latio
n
is
d
o
n
e
to
co
n
f
ir
m
th
e
r
esu
lt
s
o
f
th
e
p
r
o
p
o
s
ed
c
o
n
tr
o
l
s
ch
e
m
e.
Si
m
u
la
tio
n
is
d
o
n
e
f
o
r
f
ir
s
t
1
0
s
ec
an
d
th
e
n
f
o
r
s
ec
o
n
d
ti
m
e
1
2
0
s
ec
,
an
d
at
ev
er
y
0
.
0
0
2
s
ec
th
e
s
a
m
p
le
r
ea
d
in
g
w
a
s
tak
e
n
to
p
lo
t th
e
g
r
ap
h
s
.
Fig
u
r
e
5
s
h
o
w
s
t
h
e
p
o
s
itio
n
er
r
o
r
s
f
o
r
jo
in
t 1
an
d
j
o
in
t 2
f
o
r
1
0
s
ec
an
d
F
ig
.
6
s
h
o
w
s
t
h
e
v
el
o
cit
y
er
r
o
r
s
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
1
0
s
ec
.
Fig
u
r
e
5
.
P
o
s
itio
n
er
r
o
r
s
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
1
0
s
ec
Fig
u
r
e
6
.
V
elo
cit
y
er
r
o
r
s
f
o
r
jo
in
t 1
an
d
j
o
in
t 2
f
o
r
1
0
s
ec
Fig
u
r
e
7
an
d
F
i
g
u
r
e
8
s
h
o
ws
th
e
co
m
p
ar
i
s
o
n
f
o
r
d
er
iv
e
d
an
d
p
r
ed
icted
p
o
s
itio
n
f
o
r
lin
k
1
an
d
lin
k
2
r
esp
ec
tiv
el
y
f
o
r
1
0
s
ec
.
F
ig
u
r
e
9
s
h
o
w
s
t
h
e
to
r
q
u
e
r
eq
u
ir
ed
f
o
r
j
o
in
t
1
a
n
d
j
o
in
t
2
f
o
r
co
n
tr
o
l
p
u
r
p
o
s
e
f
o
r
1
0
s
ec
.
Fig
u
r
e
9
.
T
o
r
q
u
e
r
eq
u
ir
ed
f
o
r
jo
in
t 1
an
d
j
o
in
t 2
f
o
r
1
0
s
ec
.
Fig
u
r
e
10
s
h
o
w
s
t
h
e
p
o
s
itio
n
er
r
o
r
s
f
o
r
j
o
in
t
1
an
d
jo
in
t
2
f
o
r
1
2
0
s
ec
an
d
F
ig
u
r
e
11
s
h
o
w
s
th
e
v
elo
cit
y
er
r
o
r
s
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
1
2
0
s
ec
.
Fig
u
r
e
10
.
P
o
s
itio
n
er
r
o
r
s
f
o
r
jo
in
t 1
an
d
j
o
in
t 2
f
o
r
1
2
0
s
ec
Fig
u
r
e
11
.
V
elo
city
er
r
o
r
s
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
1
2
0
s
ec
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
5
,
No
.
4
,
Dec
em
b
er
2
0
1
6
:
27
7
–
28
3
282
Fig
u
r
e
1
2
an
d
F
i
g
u
r
e
13
s
h
o
ws
th
e
co
m
p
ar
is
o
n
f
o
r
d
er
i
v
ed
an
d
p
r
e
d
icted
p
o
s
itio
n
f
o
r
lin
k
1
an
d
lin
k
2
r
esp
ec
tiv
el
y
f
o
r
1
2
0
s
ec
.
Fig
u
r
e
12
.
C
o
m
p
ar
is
o
n
o
f
d
er
i
v
ed
an
d
p
r
ed
icted
p
o
s
itio
n
f
o
r
lin
k
1
f
o
r
1
2
0
s
ec
.
Fig
u
r
e
13
.
C
o
m
p
ar
is
o
n
o
f
d
er
i
v
ed
an
d
p
r
ed
icted
p
o
s
itio
n
f
o
r
lin
k
2
f
o
r
1
2
0
s
ec
.
F
ig
u
r
e
14
s
h
o
w
s
t
h
e
to
r
q
u
e
r
eq
u
ir
ed
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
co
n
tr
o
l p
u
r
p
o
s
e
f
o
r
1
2
0
s
ec
.
Fig
u
r
e
14
.
T
o
r
q
u
e
r
eq
u
ir
ed
f
o
r
j
o
in
t 1
an
d
j
o
in
t 2
f
o
r
1
2
0
s
ec
.
5.
CO
NCLU
SI
O
N
I
n
th
i
s
p
ap
er
w
e
ai
m
ed
to
in
tr
o
d
u
ce
a
n
e
w
co
n
tr
o
ller
s
y
s
t
e
m
w
it
h
th
e
h
elp
o
f
t
w
o
co
n
v
en
t
io
n
al
co
n
tr
o
ller
P
I
D
an
d
C
o
m
p
u
ter
t
o
r
q
u
e
co
n
tr
o
l.
T
h
e
s
i
m
u
latio
n
r
esu
lt
s
s
h
o
w
th
e
v
a
lid
it
y
o
f
t
h
e
p
r
o
p
o
s
ed
m
et
h
o
d
an
d
g
i
v
e
th
e
p
o
s
s
ib
ili
t
y
o
f
a
P
I
D
co
m
p
u
ted
to
r
q
u
e
co
n
t
r
o
l
f
o
r
2
DOF
r
o
b
o
t
m
an
ip
u
lato
r
.
P
o
s
itio
n
er
r
o
r
s
an
d
Velo
cit
y
er
r
o
r
s
ar
e
co
m
in
g
to
b
e
less
th
an
1
%.
A
n
d
ev
e
n
a
f
ter
f
e
w
s
ec
o
n
d
s
th
o
s
e
ar
e
g
et
tin
g
al
m
o
s
t
ar
o
u
n
d
ze
r
o
as
s
h
o
w
n
i
n
s
i
m
u
latio
n
g
r
ap
h
s
.
Si
m
u
latio
n
s
g
r
ap
h
s
als
o
s
h
o
w
s
th
a
t
er
r
o
r
b
et
w
ee
n
d
e
r
iv
ed
an
d
p
r
ed
icted
p
o
s
itio
n
f
o
r
lin
k
1
an
d
li
n
k
2
i
s
al
m
o
s
t a
r
o
u
n
d
ze
r
o
.
RE
F
E
R
E
NC
E
[
1
]
Am
o
l
A
.
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