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al
(
li
n
ea
r
)
d
is
p
lace
m
en
t.
T
h
e
li
n
k
s
o
f
t
h
e
m
a
n
ip
u
lato
r
ca
n
b
e
co
n
s
id
er
ed
to
f
o
r
m
a
k
i
n
e
m
atic
ch
ai
n
[1
]
,
[
9
]
.
T
h
e
en
d
o
f
th
e
k
i
n
e
m
atic
ch
ai
n
o
f
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h
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m
a
n
ip
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lato
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is
ca
lled
th
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en
d
e
f
f
ec
to
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it
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alo
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m
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d
.
T
h
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en
d
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f
f
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t
o
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ca
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esig
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er
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ir
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task
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ch
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ip
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,
s
p
in
n
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tc.
,
d
ep
en
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g
o
n
th
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ap
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licatio
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s
u
c
h
a
s
w
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ld
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p
lace
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t
d
u
r
i
n
g
ass
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m
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l
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r
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f
f
r
ee
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w
n
i
n
Fi
g
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r
e
1
.
Fig
u
r
e
1
.
R
o
b
itic
ar
m
w
it
h
s
ev
en
d
eg
r
ee
s
o
f
f
r
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d
o
m
[
1
]
R
o
b
o
t
ar
m
s
ar
e
ca
teg
o
r
ized
b
y
t
h
e
n
u
m
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er
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f
co
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tr
o
lled
d
eg
r
ee
o
f
f
r
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m
(
DOF)
t
h
e
y
c
an
ex
ec
u
te.
T
h
is
n
u
m
b
er
is
eq
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al
to
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s
u
m
o
f
th
e
DOF
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ea
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f
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r
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b
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m
‟
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Gen
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w
il
l
b
e
eith
er
Hi
n
g
e
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ts
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P
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o
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h
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m
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ep
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at
w
o
u
ld
h
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v
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to
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if
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in
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er
to
lo
ca
te
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i
n
w
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h
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r
o
b
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ar
m
ca
n
m
o
v
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[
8
]
.
A
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o
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m
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DO
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itch
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y
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w
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itc
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h
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w
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m
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io
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2
-
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ac
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li
k
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to
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r
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m
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cl
u
d
e
d
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p
lace
m
e
n
t
alo
n
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th
e
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Y
ax
es,
p
lu
s
r
o
tatio
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a
T
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Di
m
en
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io
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a
l
(
3
-
D)
s
p
ac
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ix
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o
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f
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m
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n
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is
t
o
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n
t
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er
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icu
lar
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s
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X,
Y,
a
n
d
Z
)
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r
o
tatio
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o
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s
a
m
e
ax
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s
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ea
v
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: M
o
v
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p
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d
o
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Su
rg
e
: M
o
v
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f
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Sw
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R
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u
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bo
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I
t
'
s
a
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f
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m
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
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R
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Mo
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a
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d
p
lace
ap
p
licatio
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o
f
s
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a
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m
b
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ee
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Fig
u
r
e
2
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A
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[
1
5
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A
r
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t
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co
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ec
tio
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w
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r
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e
[
1
5
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,
[
2
0
]
.
P
a
ra
llel
Ro
bo
t
:
Used
f
o
r
m
o
b
ile
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latf
o
r
m
h
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t p
r
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3.
O
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Op
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cie
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m
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ai
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n
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ai
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a
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d
o
b
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tiv
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f
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/
f
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s
.
Desig
n
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les
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e
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ep
en
d
en
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v
ar
iab
les
o
f
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Ser
ia
l
an
d
P
ar
allel
R
o
b
o
t
Ma
n
ip
u
lato
r
s
–
Ki
n
e
m
at
ics,
D
y
n
a
m
ic
s
,
C
o
n
tr
o
l
a
n
d
Op
ti
m
izatio
n
o
b
j
ec
tiv
e
f
u
n
ctio
n
a
n
d
ca
n
ta
k
e
co
n
t
in
u
o
u
s
o
r
d
i
s
cr
ete
v
al
u
e
s
.
T
h
e
r
an
g
e
s
o
f
th
e
s
e
v
ar
iab
les
ar
e
g
iv
e
n
f
o
r
t
h
e
p
r
o
b
le
m
s
.
C
o
n
s
tr
ain
ts
ar
e
th
e
f
u
n
ctio
n
s
o
f
d
e
s
ig
n
v
ar
iab
les
a
n
d
li
m
it
t
h
e
s
ea
r
c
h
s
p
ac
e.
T
h
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
is
t
h
e
m
ain
f
u
n
ctio
n
d
ep
en
d
en
t
o
n
t
h
e
d
e
s
ig
n
v
ar
iab
les.
I
f
t
h
er
e
is
m
o
r
e
th
a
n
o
n
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
,
th
e
p
r
o
b
le
m
is
ca
lled
m
u
lti
-
o
b
j
ec
tiv
e
o
p
ti
m
izatio
n
p
r
o
b
le
m
.
4.
K
I
NE
M
AT
I
CS
I
t
(
f
r
o
m
Gr
ee
k
κιν
εῖν
,
k
in
ei
n
,
to
m
o
v
e)
is
t
h
e
b
r
an
ch
o
f
class
ical
m
ec
h
an
ic
s
t
h
at
d
esc
r
ib
es
th
e
m
o
tio
n
o
f
b
o
d
ies
(
o
b
j
ec
ts
)
an
d
s
y
s
te
m
s
(
g
r
o
u
p
s
o
f
o
b
j
ec
ts
)
w
it
h
o
u
t
co
n
s
id
er
atio
n
o
f
t
h
e
f
o
r
ce
s
th
at
ca
u
s
e
t
h
e
m
o
tio
n
.
K
in
e
m
atic
s
i
s
n
o
t
to
b
e
co
n
f
u
s
ed
w
i
th
an
o
t
h
er
b
r
an
ch
o
f
c
lass
ical
m
ec
h
a
n
ic
s
:
a
n
al
y
tical
d
y
n
a
m
ic
s
(
th
e
s
tu
d
y
o
f
th
e
r
elatio
n
s
h
ip
b
et
w
ee
n
t
h
e
m
o
tio
n
o
f
o
b
j
ec
ts
an
d
its
ca
u
s
es),
s
o
m
eti
m
es
s
u
b
d
iv
id
ed
in
to
k
in
e
tics
(
th
e
s
tu
d
y
o
f
th
e
r
elat
io
n
b
et
w
ee
n
ex
ter
n
al
f
o
r
ce
s
a
n
d
m
o
tio
n
)
an
d
s
tatic
s
(
th
e
s
t
u
d
y
o
f
t
h
e
r
elatio
n
s
in
a
s
y
s
te
m
at
eq
u
ilib
r
i
u
m
)
.
Ki
n
e
m
a
tics
al
s
o
d
if
f
er
s
f
r
o
m
d
y
n
a
m
ics a
s
u
s
ed
in
m
o
d
er
n
-
d
a
y
p
h
y
s
ics to
d
escr
ib
e
ti
m
e
-
e
v
o
lu
tio
n
o
f
a
s
y
s
te
m
[
1
]
.
Kin
e
m
a
tics
i
s
t
h
e
p
r
o
ce
s
s
o
f
ca
lc
u
lati
n
g
th
e
p
o
s
itio
n
i
n
s
p
ac
e
o
f
th
e
e
n
d
o
f
a
lin
k
ed
s
tr
u
ct
u
r
e,
g
i
v
e
n
th
e
a
n
g
les o
f
all
t
h
e
j
o
in
ts
.
I
t is ea
s
y
,
an
d
th
er
e
is
o
n
l
y
o
n
e
s
o
lu
tio
n
.
I
n
v
er
s
e
Ki
n
e
m
atics
d
o
es
th
e
r
ev
er
s
e.
Giv
e
n
t
h
e
en
d
p
o
in
t
o
f
th
e
s
tr
u
ct
u
r
e,
w
h
at
an
g
le
s
d
o
th
e
j
o
in
ts
n
ee
d
to
b
e
in
th
e
ac
h
iev
e
t
h
a
t
en
d
p
o
in
t.
I
t
ca
n
b
e
d
if
f
icu
l
t,
an
d
th
er
e
ar
e
u
s
u
all
y
m
an
y
o
r
in
f
in
ite
l
y
m
an
y
s
o
lu
tio
n
s
.
T
h
is
p
r
o
ce
s
s
ca
n
b
e
ex
tr
e
m
el
y
u
s
e
f
u
l
i
n
r
o
b
o
tics
.
Yo
u
m
a
y
h
av
e
a
r
o
b
o
tic
ar
m
w
h
ic
h
n
ee
d
s
to
g
r
ab
an
o
b
j
ec
t
as
s
h
o
w
n
i
n
Fig
u
r
e
3
.
I
f
th
e
s
o
f
t
w
ar
e
k
n
o
w
s
w
h
er
e
th
e
o
b
j
ec
t
is
in
r
elatio
n
to
th
e
s
h
o
u
ld
er
,
it
s
i
m
p
l
y
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
6
,
No
.
1
,
Ma
r
ch
2
0
1
7
:
1
–
14
4
n
ee
d
s
to
ca
lcu
late
t
h
e
an
g
les
o
f
th
e
j
o
in
ts
to
r
ea
ch
it.
T
h
e
s
i
m
p
le
s
t
ap
p
licatio
n
o
f
k
i
n
e
m
atics
is
f
o
r
p
ar
ticle
m
o
tio
n
,
tr
a
n
s
latio
n
al
o
r
r
o
tatio
n
al
[
7
]
.
T
h
e
n
ex
t
le
v
el
o
f
c
o
m
p
le
x
it
y
co
m
e
s
f
r
o
m
t
h
e
i
n
tr
o
d
u
ctio
n
o
f
r
i
g
id
b
o
d
ies,
w
h
ic
h
ar
e
co
llectio
n
s
o
f
p
ar
ticl
es
h
a
v
i
n
g
ti
m
e
in
v
ar
ian
t
d
i
s
ta
n
ce
s
b
et
w
ee
n
th
e
m
s
elv
es.
R
i
g
id
b
o
d
ies
m
i
g
h
t
u
n
d
er
g
o
tr
an
s
latio
n
an
d
r
o
tatio
n
o
r
a
co
m
b
i
n
atio
n
o
f
b
o
th
.
A
m
o
r
e
co
m
p
licated
ca
s
e
is
t
h
e
k
i
n
e
m
atic
s
o
f
a
s
y
s
te
m
o
f
r
ig
id
b
o
d
ies,
w
h
ich
m
a
y
b
e
lin
k
ed
to
g
et
h
er
b
y
m
ec
h
a
n
ical
j
o
in
ts
.
F
o
rwa
rd
K
ine
m
a
t
ics
:
T
h
e
f
o
r
w
ar
d
k
in
e
m
atic
i
s
th
a
t
th
e
p
o
s
itio
n
s
o
f
p
ar
ticu
lar
p
ar
ts
o
f
th
e
m
o
d
el
at
a
s
p
ec
if
ied
ti
m
e
ar
e
ca
lcu
lated
f
r
o
m
t
h
e
p
o
s
itio
n
an
d
o
r
ien
tati
o
n
o
f
th
e
o
b
j
ec
t,
to
g
eth
er
w
i
th
an
y
in
f
o
r
m
a
tio
n
o
n
th
e
j
o
in
ts
o
f
an
ar
tic
u
lated
m
o
d
el.
I
f
th
e
o
b
j
ec
t
to
b
e
an
im
ated
is
an
ar
m
w
it
h
t
h
e
s
h
o
u
l
d
er
r
em
ai
n
i
n
g
at
a
f
i
x
ed
lo
ca
tio
n
,
th
e
lo
ca
tio
n
o
f
th
e
tip
o
f
t
h
e
th
u
m
b
w
o
u
ld
b
e
ca
lcu
lated
f
r
o
m
th
e
a
n
g
l
es
o
f
th
e
s
h
o
u
ld
er
,
elb
o
w
,
w
r
is
t,
t
h
u
m
b
a
n
d
k
n
u
c
k
le
j
o
in
t
s
.
T
h
r
ee
o
f
th
e
s
e
j
o
in
t
s
(
th
e
s
h
o
u
ld
er
,
w
r
i
s
t
a
n
d
t
h
e
b
ase
o
f
th
e
t
h
u
m
b
)
h
av
e
m
o
r
e
th
a
n
o
n
e
d
eg
r
ee
o
f
f
r
ee
d
o
m
,
all
o
f
w
h
ich
m
u
s
t
b
e
tak
en
i
n
to
ac
co
u
n
t.
I
f
t
h
e
m
o
d
el
w
er
e
a
n
en
t
ir
e
h
u
m
a
n
f
i
g
u
r
e,
th
en
t
h
e
lo
ca
tio
n
o
f
th
e
s
h
o
u
ld
er
w
o
u
ld
also
h
av
e
to
b
e
ca
lcu
lated
f
r
o
m
o
th
er
p
r
o
p
e
r
ties
o
f
th
e
m
o
d
el
[
3
]
,
[
5
]
.
I
nv
er
s
e
K
i
ne
m
a
t
ics
:
I
t
w
i
ll
e
n
ab
le
u
s
to
ca
lc
u
late
w
h
at
ea
c
h
j
o
in
t
v
ar
iab
le
m
u
s
t
b
e
i
f
w
e
d
esire
th
at
t
h
e
h
an
d
b
e
lo
ca
ted
at
p
ar
ticu
lar
p
o
in
t
an
d
h
av
e
a
p
ar
ticu
lar
p
o
s
itio
n
.
T
h
e
p
o
s
itio
n
an
d
o
r
ien
tatio
n
o
f
th
e
en
d
ef
f
ec
to
r
r
elativ
e
to
th
e
b
a
s
e
f
r
a
m
e
co
m
p
u
te
all
p
o
s
s
ib
le
s
e
ts
o
f
j
o
in
t a
n
g
le
s
a
n
d
lin
k
g
eo
m
etr
ies
w
h
i
ch
co
u
ld
b
e
u
s
ed
to
attain
t
h
e
g
i
v
e
n
p
o
s
itio
n
a
n
d
o
r
ien
tatio
n
o
f
th
e
e
n
d
ef
f
ec
to
r
[
1
]
,
[
4
]
.
Fig
u
r
e
3
.
T
w
o
lin
k
r
o
b
o
tic
ar
m
[
4
]
Fo
r
w
ar
d
k
i
n
e
m
atics
(1
-
2)
x
=
l
1
co
s
θ
1
+
l
2
co
s
(θ
1
+θ
2
)
(
1
)
y
=
l
1
s
i
n
θ
1
+
l
2
s
i
n
(
θ
1
+
θ
2
)
(
2
)
I
n
v
er
s
e
k
i
n
e
m
a
tics
(
3
-
4)
x
2
+y
2
=l
1
2
co
s
2
θ
1
+l
2
2
co
s
2
(
θ
1
+
θ
2
)
+2
l
1
l
2
co
s
θ
1
co
s
2
(
θ
1
+
θ
2
)
+l
1
2
s
in
2
θ
1
+l
2
2
si
n
2
(
θ
1
+
θ
2
)
+2
l
1
l
2
s
in
θ
1
s
in
2
(
θ
1
+
θ
2
)
(
3
)
=l
1
2
+l
2
2
+2
l
1
l
2
co
s
θ
1
co
s
(
θ
1
+
θ
2
)
+sin
θ
1
s
i
n
2
(
θ
1
+
θ
2
)
(
4
)
Nex
t
w
e
u
s
e
t
h
e
f
o
llo
w
i
n
g
eq
u
alities
(5
-
6)
s
in
(
x
±
y
)
=si
n
x
co
s
y
±
co
s
x
s
i
n
y
(
5
)
co
s
(
x
±
y
)
=c
o
s
x
co
s
y
±
s
i
n
x
s
i
n
y
(
6
)
T
h
er
ef
o
r
e
(7
-
8)
x
2
+
y
2
=l
1
2
+l
2
2
+2
l
1
l
2
[
co
s
θ
1
co
s
θ
2
-
s
i
n
θ
1
s
i
n
θ
2
+
s
in
θ
1
(
co
s
θ
2
s
i
n
θ
1
+c
o
s
θ
1
s
in
θ
2
)]
(
7
)
=l
1
2
+l
2
2
+2
l
1
l
2
[
co
s
2
θ
1
co
s
θ
2
+si
n
2
θ
2
co
s
θ
2
]
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
R
o
b
o
tic
A
r
m
Mo
ve
men
t O
p
timiz
a
tio
n
u
s
in
g
S
o
ft C
o
mp
u
tin
g
(
S
u
r
en
d
er K
u
ma
r
)
5
=l
1
2
+l
2
2
+2
l
1
l
2
co
s
θ
2
(
8
)
An
d
(
9
)
co
s
θ
2
=
(
9
)
Her
e,
th
e
an
g
le
d
ir
ec
tl
y
u
s
i
n
g
th
e
ar
c
co
s
f
u
n
ctio
n
b
u
t
t
h
is
f
u
n
ctio
n
is
v
er
y
i
n
ac
cu
r
ate
f
o
r
s
m
all
an
g
le
s
,
.
th
e
t
y
p
ical
w
a
y
to
a
v
o
id
th
is
ac
c
u
r
ac
y
is
to
co
n
v
er
t f
u
r
th
er
u
n
t
il
w
e
ca
n
u
s
e
th
e
a
ta
n
2
f
u
n
ctio
n
:
co
s
2
θ
2
+sin
2
θ
2
=1
an
d
s
in
θ
2
=±
√
th
e
t
w
o
s
o
l
u
tio
n
s
co
r
r
esp
o
n
d
in
g
to
th
e
„
elb
o
w
u
p
‟
an
d
„
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
0
8
9
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4856
IJ
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IJ
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2089
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4856
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.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
6
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No
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s
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AL
5
B
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ass
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s
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h
o
w
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i
n
Fi
g
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r
e
9
.
T
h
e
y
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estab
lis
h
ed
u
s
i
n
g
th
e
p
r
in
cip
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o
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t
h
e
De
n
av
i
t
-
Har
ten
b
ar
g
(D
-
H)
co
n
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en
tio
n
.
Fo
r
th
e
k
in
e
m
atic
m
o
d
el
o
f
5
d
o
f
r
o
b
o
tic
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m
f
ir
s
t
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v
e
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r
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e
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ch
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o
m
b
ase
(
f
r
a
m
e
0
)
to
en
d
-
e
f
f
ec
to
r
(
f
r
a
m
e
5
)
.
Fig
u
r
e
9
.
C
o
o
r
d
in
ate
f
r
a
m
e
as
s
ig
n
m
en
t [
1
9
]
4
.
3
.
M
a
t
he
m
a
t
ica
l a
nd
K
ine
m
a
t
i
c
M
o
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g
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f
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DO
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Ro
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t
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c
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m
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o
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ip
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a
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h
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ip
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m
h
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s
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h
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h
m
ea
n
th
e
r
o
b
o
t
h
a
s
5
DOF.
T
h
e
k
in
e
m
at
ics
r
o
b
o
t
m
an
ip
u
lato
r
is
d
er
iv
ed
b
y
u
s
in
g
De
n
av
it
-
Har
ter
b
er
g
(
DH)
r
ep
r
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tatio
n
.
I
n
t
h
is
co
n
v
e
n
tio
n
,
e
ac
h
h
o
m
o
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tr
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A
i
i
s
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as a
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o
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f
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r
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asic tr
a
n
s
f
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r
m
atio
n
s
as
s
h
o
w
n
i
n
(
2
8
-
2
9
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
R
o
b
o
tic
A
r
m
Mo
ve
men
t O
p
timiz
a
tio
n
u
s
in
g
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o
ft C
o
mp
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tin
g
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S
u
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en
d
er K
u
ma
r
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9
T
e
=
R
Z
(Ɵ
i
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D
Z
(d
i
)
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x
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i
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x
(α
i
)
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8
)
[
]
[
]
[
]
[
]
=
[
]
(
29)
w
h
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e
R
X
a
n
d
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r
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tat
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h
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o
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atic
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th
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it
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ase
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r
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e
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n
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(
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)
:
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[
]
(
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h
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r
kj
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r
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atr
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itio
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itio
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r
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th
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it
h
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to
th
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ase
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g
i
v
en
b
y
(
31
-
3
2
)
0
T
5
=
0
T
1
1
T
2
2
T
3
3
T
4
4
T
5
(
3
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
6
,
No
.
1
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Ma
r
ch
2
0
1
7
:
1
–
14
10
[
]
(
31
)
o
T
5
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e
W
h
er
e,
r
11
=C
12
C
34
5
r
12
=S
12
r
13
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S
345
r
21
=S
12
C
345
r
22
=
-
C
12
r
23
=S
12
S
345
r
31
=S
345
r
32
=0
r
33
=
-
C
34
5
p
x
=S
12
d
5
+C
12
a
4
C
34
+C
12
a
3
C
3
p
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=
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C
12
d
5
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a
4
C
34
+S
12
a
3
C
3
p
z
=a
4
S
34
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5
S
5
+d
1
I
NVE
R
S
E
KINEM
A
T
I
C
S
A
N
AL
Y
SIS
T
h
e
f
o
llo
w
i
n
g
eq
u
a
tio
n
s
w
ill
b
e
u
s
ed
to
o
b
tain
th
e
s
o
lu
tio
n
f
o
r
th
e
in
v
er
s
e
Ki
n
e
m
atic
s
p
r
o
b
le
m
.
0
T
5
=
0
T
1
1
T
2
2
T
3
3
T
4
4
T
5
=
T
e
I
n
v
er
s
e
k
in
e
m
atic
s
s
o
lu
tio
n
f
o
r
th
e
f
ir
s
t
j
o
in
t
as
a
f
u
n
ctio
n
o
f
th
e
k
n
o
w
n
ele
m
e
n
t
s
o
f
T
e
,
th
e
lin
k
tr
an
s
f
o
r
m
atio
n
in
v
er
s
e
s
ar
e
p
r
e
m
u
ltip
lied
as f
o
llo
w
s
:
X
1
=[
(
0
T
1
)
-
1
]*
0
T
5
=
1
T
2
*
2
T
3
*
3
T
4
*
4
T
5
=
1
T
Si
m
i
lar
l
y
,
to
f
i
n
d
th
e
o
t
h
er
v
ar
iab
les,
th
e
f
o
llo
w
i
n
g
eq
u
atio
n
s
ar
e
o
b
tain
ed
in
s
i
m
i
lar
m
a
n
n
e
r
.
X
2
=
[(
0
T
1
)(
1
T
2
)]
-
1
*
0
T
5
=
2
T
3
*
3
T
4
*
4
T
5
=
2
T
5
X
3
=
[(
0
T
1
)(
1
T
2
)
(
2
T
3
)]
-
1
*
0
T
5
=
3
T
4
*
4
T
5
=
3
T
5
X
4
=
[(
0
T
1
)(
1
T
2
)(
2
T
3
)
(
3
T
4
)]
-
1
*
0
T
5
=
4
T
5
=
4
T
5
B
y
s
o
lv
i
n
g
th
e
s
e
eq
u
atio
n
s
,
we
ca
n
ca
lcu
late
t
h
e
v
alu
e
s
o
f
Ɵ
1
, Ɵ
2
, Ɵ
3
, Ɵ
4
,
an
d
Ɵ
5.
So
,
th
e
v
al
u
es
w
i
ll b
e
(
3
2
-
36)
Evaluation Warning : The document was created with Spire.PDF for Python.