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IJ
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2089
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4856
Op
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2.
RE
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ip
u
lato
r
tr
aj
ec
to
r
y
p
lan
n
i
n
g
an
d
o
p
ti
m
izatio
n
.
N
in
g
et
al.
[
1
]
an
al
y
ze
d
th
e
d
y
n
a
m
ic
m
o
v
e
m
e
n
t
o
f
p
r
i
m
it
i
v
es
a
n
d
p
r
o
p
o
s
ed
a
n
o
v
el
s
c
h
e
m
e
f
o
r
g
en
er
ati
n
g
tr
aj
ec
to
r
y
.
T
h
ey
co
m
p
ar
ed
th
e
p
o
s
itio
n
co
o
r
d
in
ates
an
d
v
elo
cit
y
at
s
tar
t
en
d
p
o
in
ts
o
f
a
tr
aj
ec
to
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y
o
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tain
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d
f
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o
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th
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e
th
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to
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h
at
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f
t
h
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m
ea
s
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r
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v
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lu
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n
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f
o
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n
d
it
to
b
e
v
er
y
p
r
ec
is
e
an
d
ac
cu
r
ate.
Gasp
ar
etto
et
al.
[
2
]
g
av
e
s
tr
es
s
u
p
o
n
p
lan
n
in
g
a
s
m
o
o
th
t
r
aj
ec
to
r
y
f
o
r
r
o
b
o
t
m
an
ip
u
lato
r
s
.
T
h
e
y
m
o
d
eled
an
o
b
j
ec
tiv
e
f
u
n
ct
io
n
i.e
.
i
m
p
l
icitl
y
d
ep
e
n
d
e
n
t
u
p
o
n
th
e
i
n
te
g
r
al
tak
e
n
o
v
er
t
h
e
s
q
u
ar
ed
j
er
k
as
w
el
l
as
to
tal
ex
ec
u
t
io
n
t
i
m
e.
C
h
i
u
[
3
]
d
ev
e
lo
p
ed
A
s
ad
a‟
s
i
n
er
tia
el
lip
s
o
i
d
an
d
Yo
s
h
ik
a
w
a‟
s
m
an
ip
u
lab
ilit
y
ellip
s
o
id
.
T
h
ese
to
o
ls
to
g
eth
er
r
es
u
lt
a
p
er
f
o
r
m
a
n
ce
p
ar
a
m
eter
o
f
v
elo
cit
y
as
w
el
l
as
s
ta
tic
fo
r
ce
.
B
o
r
b
o
w
[
4
,
5
]
tr
aj
ec
to
r
y
p
la
n
n
in
g
s
t
u
d
y
w
as
b
as
ed
o
n
o
p
ti
m
izi
n
g
ti
m
e
an
d
later
o
n
h
e
a
ls
o
d
em
o
n
s
tr
ated
co
n
tr
o
l o
f
h
is
o
p
ti
m
ized
p
ath
-
p
lan
n
i
n
g
r
esu
l
t.
E
ld
er
s
h
a
w
et
al.
[
6
]
u
s
ed
p
o
l
y
n
o
m
ial
i
n
ter
p
o
latio
n
alo
n
g
w
it
h
g
en
e
tic
al
g
o
r
ith
m
w
h
ich
is
b
ased
u
p
o
n
t
h
e
n
at
u
r
al
s
elec
tio
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p
r
o
ce
d
u
r
e
in
o
r
d
er
to
tack
le
th
e
tr
a
j
ec
to
r
y
p
lan
n
i
n
g
p
r
o
b
le
m
.
T
ian
et
al
[
7
]
as
w
ell
a
s
Yu
n
a
n
d
Xi
[
8
]
also
p
er
f
o
r
m
ed
th
e
s
a
m
e
tas
k
b
y
e
m
p
lo
y
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n
g
g
e
n
etic
alg
o
r
it
h
m
.
T
h
is
m
eth
o
d
w
as
f
u
r
th
er
d
ev
elo
p
ed
b
y
Z
h
a
[
9
,
1
0
]
w
h
o
co
m
p
ar
ed
th
e
tr
aj
e
cto
r
y
to
a
r
u
led
s
u
r
f
ac
e
a
n
d
in
co
r
p
o
r
ated
in
ter
p
o
latio
n
u
s
i
n
g
B
ez
ier
cu
r
v
es
b
et
w
ee
n
d
i
f
f
er
e
n
t
p
o
s
es.
A
n
o
v
e
l
s
c
h
e
m
e
o
f
t
r
aj
ec
to
r
y
p
lan
n
i
n
g
w
as
s
u
g
g
e
s
ted
b
y
Olab
i
et
al.
[
1
1
]
w
h
ic
h
co
n
s
id
er
ed
co
n
ti
n
u
o
u
s
m
ac
h
i
n
i
n
g
.
A
p
ar
a
m
et
r
ic
s
p
ee
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in
ter
p
o
lato
r
ca
m
e
in
to
p
ict
u
r
e
w
h
ic
h
r
esu
lt
s
in
s
m
o
o
th
tr
aj
ec
to
r
ies.
I
n
o
r
d
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r
to
ac
h
iev
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m
o
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ac
cu
r
ac
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p
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ig
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r
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ly
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o
m
ia
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w
as
i
n
co
r
p
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r
ated
b
y
B
o
r
y
g
a
et
al.
[
1
2
]
.
Mu
l
ti
d
eg
r
ee
Sp
li
n
es
w
as
in
tr
o
d
u
ce
d
to
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y
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o
f
p
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n
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n
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m
et
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o
d
s
b
y
L
iu
et
al
.
[
1
3
]
.
Go
u
as
m
i
et
al.
[
1
4
]
im
p
le
m
en
ted
d
u
al
q
u
a
ter
n
io
n
m
e
th
o
d
f
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k
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m
ati
c
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al
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is
o
f
r
o
b
o
t m
a
n
ip
u
lato
r
.
Sh
a
h
et
al.
[
1
5
]
h
av
e
tak
e
n
f
ee
d
f
o
r
w
ar
d
ANN
an
d
tr
ai
n
ed
th
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ata
o
b
tain
ed
f
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r
a
3
-
d
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f
m
a
n
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lato
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MT
AL
A
B
to
o
lb
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to
s
h
o
w
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at
A
NN
is
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est
m
et
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o
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to
f
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n
d
in
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er
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k
i
n
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m
atic
s
o
lu
tio
n
.
J
h
a
et
al.
[
1
6
]
h
a
v
e
p
r
o
p
o
s
ed
a
s
tr
u
ctu
r
ed
ar
tific
ial
n
e
u
r
al
n
et
w
o
r
k
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p
r
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a
ch
i.e
.
m
u
l
ti
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l
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y
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p
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rc
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p
tro
n
n
e
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ra
l
n
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tw
o
rk
(
M
L
PN
N)
to
s
o
lv
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in
v
er
s
e
k
i
n
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m
a
tics
p
r
o
b
le
m
b
y
co
n
s
id
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in
g
4
-
d
o
f
S
C
AR
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m
an
ip
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lato
r
.
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h
en
t
h
e
m
o
tio
n
ac
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a
ll
y
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s
p
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f
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th
r
o
u
g
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i
n
ter
p
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latin
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eq
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f
v
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p
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in
t
s
,
it
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s
n
o
t
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s
il
y
p
r
ed
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le.
T
h
e
n
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li
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it
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f
t
h
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k
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m
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m
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d
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e
v
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in
t
m
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tio
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s
p
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o
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d
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to
ac
h
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a
s
m
o
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th
m
o
tio
n
in
C
ar
tes
ian
s
p
ac
e,
w
h
en
d
at
a
o
b
tain
ed
f
r
o
m
i
n
ter
p
o
latio
n
s
ch
e
m
e
ar
e
m
ap
p
ed
in
to
j
o
in
t
s
p
ac
e.
Su
b
s
eq
u
en
tl
y
,
a
n
u
n
e
v
en
an
d
i
n
ac
cu
r
ate
m
o
tio
n
i
n
t
h
e
C
ar
tesi
a
n
s
p
ac
e
co
u
l
d
b
e
ac
q
u
ir
ed
.
R
o
d
n
a
y
et
al.
[
1
7
]
an
al
y
ze
d
th
e
d
y
n
a
m
i
c
ch
ar
ac
ter
is
tic
s
o
f
2
DOFs
r
o
b
o
t
in
3
D
s
p
ac
e.
T
h
e
y
s
i
g
n
i
f
ied
tr
aj
ec
to
r
ies
b
y
all
o
w
i
n
g
th
e
m
a
n
ip
u
la
to
r
to
p
er
f
o
r
m
f
r
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m
o
v
e
m
en
t
b
y
i
n
tr
o
d
u
ci
n
g
g
eo
d
esics
o
n
t
h
e
d
y
n
a
m
ic
s
u
r
f
ac
e.
Z
ef
r
a
n
e
t
al.
[
1
8
]
to
o
k
a
L
ie
g
r
o
u
p
ap
p
r
o
ac
h
to
g
en
er
ate
tr
aj
ec
to
r
ies
r
esu
ltin
g
f
r
o
m
n
o
n
lin
ea
r
m
o
tio
n
.
T
h
ey
c
h
o
s
e
L
ie
g
r
o
u
p
an
d
co
n
s
eq
u
e
n
tl
y
d
e
f
i
n
ed
a
lef
t
in
v
ar
ia
n
t
R
ie
m
an
n
ia
n
m
etr
ic
o
n
it
to
g
e
n
er
ate
s
m
o
o
t
h
tr
a
j
ec
to
r
ies.
Selig
et
al.
[
1
9
]
s
teer
ed
th
e
r
o
b
o
t m
an
ip
u
lato
r
s
alo
n
g
h
elica
l
tr
aj
ec
to
r
ies
b
y
r
o
b
o
tic
co
n
tr
o
l
an
d
tr
aj
ec
to
r
y
p
la
n
n
i
n
g
.
T
h
e
y
u
s
ed
a
L
ie
g
r
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u
p
m
et
h
o
d
w
h
ich
m
ad
e
th
e
p
lan
n
i
n
g
co
m
p
le
x
,
ab
s
tr
ac
t,
a
n
d
n
o
t
ea
s
y
to
p
u
t
i
n
t
o
r
ea
l
p
r
ac
tice.
C
h
e
n
et
al.
[
20
]
h
a
v
e
d
e
m
o
n
s
tr
ated
t
h
e
ef
f
ec
tiv
e
n
e
s
s
o
f
g
eo
d
esic
p
la
n
n
i
n
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b
y
co
n
d
u
cti
n
g
s
i
m
u
lat
io
n
ex
p
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m
en
t
s
.
T
h
e
y
s
p
ec
i
f
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a
R
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m
a
n
n
ian
m
etr
ic
f
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r
ea
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f
th
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p
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s
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s
p
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an
d
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tatio
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s
p
ac
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ar
ately
to
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m
p
lis
h
th
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e
o
d
esic
m
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tio
n
.
T
h
e
g
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d
esic
eq
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alitie
s
ar
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s
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lv
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n
u
m
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d
t
h
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r
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lts
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u
s
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to
m
an
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late
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r
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b
o
t
.
Sin
ce
th
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li
n
k
s
o
f
th
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r
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b
o
t
m
an
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lato
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ar
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in
te
r
lin
k
ed
,
w
e
ca
n
‟
t
p
lan
tr
aj
ec
to
r
ies
s
ep
ar
atel
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f
o
r
ea
ch
s
p
ac
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I
n
o
r
d
er
to
g
et
a
s
h
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test
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d
ac
c
u
r
ate
p
at
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w
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p
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t
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k
in
e
tic
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n
er
g
y
in
v
ar
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n
t,
Z
h
a
n
g
et
al.
[
2
1
]
ch
o
s
e
th
e
p
ar
a
m
eter
s
as
ar
c
le
n
g
th
an
d
k
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n
etic
e
n
er
g
y
an
d
co
n
s
tr
u
cted
R
ie
m
a
n
n
i
an
m
etr
ic
ac
co
r
d
in
g
l
y
.
B
u
t
h
i
s
m
eth
o
d
h
ad
s
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m
e
s
h
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r
tco
m
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s
w
h
ich
d
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r
ea
s
e
d
th
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cr
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ib
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o
f
th
a
t
m
et
h
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d
.
T
h
e
au
th
o
r
s
d
id
n
o
t
tak
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m
o
r
e
t
h
a
n
3
DOFs
r
o
b
o
t to
p
lan
tr
aj
ec
to
r
ies an
d
also
n
o
t ta
ck
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t
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tatio
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p
r
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b
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s
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.
P
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f
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4856
IJ
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3
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Sep
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1
6
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1
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192
j
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n
a
s
in
g
le
ar
m
.
T
h
er
ef
o
r
e,
th
e
tr
aj
ec
to
r
ies
n
ee
d
to
b
e
d
eter
m
in
ed
b
y
co
n
s
id
er
i
n
g
t
h
e
co
m
b
i
n
atio
n
o
f
p
o
s
itio
n
an
d
o
r
ien
tatio
n
j
o
in
t
s
in
an
i
n
te
g
r
ated
m
a
n
n
er
.
I
n
o
r
d
er
to
o
u
tr
u
n
t
h
e
ab
o
v
e
m
e
n
tio
n
ed
d
r
a
w
b
ac
k
s
,
th
is
p
a
p
er
p
r
esen
ts
a
g
eo
d
esic
tr
aj
ec
to
r
y
p
lan
n
i
n
g
m
eth
o
d
b
y
co
m
b
i
n
in
g
t
h
e
p
o
s
itio
n
a
n
d
o
r
ien
tatio
n
o
f
j
o
in
t.
Geo
d
esic
is
in
tr
i
n
s
ic
i
n
n
at
u
r
e
an
d
h
as
n
o
r
elatio
n
w
i
th
t
h
e
co
o
r
d
in
ates.
Un
li
k
e
th
e
p
o
l
y
n
o
m
ial
m
et
h
o
d
w
h
ic
h
is
a
m
er
e
ap
p
r
o
x
i
m
ati
n
g
in
ter
p
o
latio
n
s
ch
e
m
e
,
th
is
g
eo
d
esic
m
et
h
o
d
p
r
o
v
id
es
an
ex
ac
t
an
d
ac
cu
r
ate
s
o
lu
tio
n
.
T
h
e
w
o
r
k
v
o
lu
m
e
o
f
en
d
-
e
f
f
ec
to
r
i.e
.
p
o
s
itio
n
an
d
o
r
ien
tatio
n
s
p
ac
e
ar
e
co
m
b
in
ed
to
g
eth
er
to
d
ef
i
n
e
th
e
R
ie
m
an
n
ia
n
m
etr
ic
to
attai
n
g
eo
d
esic
m
o
tio
n
s
.
Fir
s
t,
j
o
in
t
v
ar
iab
les
ar
e
co
n
s
id
er
ed
as
l
o
ca
l
co
o
r
d
in
ates
o
f
p
o
s
itio
n
an
d
o
r
ien
tatio
n
s
p
ac
e.
T
h
en
g
eo
d
esics
ar
e
o
b
tain
ed
f
r
o
m
m
ath
e
m
atica
l
f
o
r
m
u
latio
n
f
o
llo
w
ed
b
y
o
b
tain
in
g
j
o
in
t
tr
aj
ec
to
r
ies
a
n
d
ch
ar
ac
ter
izin
g
C
ar
te
s
ian
t
r
aj
ec
to
r
ies
b
y
j
o
in
t
tr
aj
ec
to
r
i
es.
T
h
e
n
atu
r
e
o
f
g
eo
d
esic
i
m
p
licitl
y
m
a
k
e
s
b
o
th
eq
u
i
v
ale
n
t
tr
aj
ec
to
r
ies
(
C
ar
tesi
a
n
a
n
d
J
o
in
t)
s
m
o
o
t
h
an
d
r
elat
iv
el
y
less
er
r
o
n
eo
u
s
.
T
h
is
m
et
h
o
d
i
m
p
li
citl
y
f
ilter
s
m
u
ltip
le
s
o
l
u
tio
n
s
r
esu
lti
n
g
f
r
o
m
i
n
v
er
s
e
k
i
n
e
m
a
tics
an
d
r
esu
lts
t
h
e
o
p
tim
a
l o
n
e.
3.
M
AT
H
E
M
AT
I
CAL M
O
DE
L
L
I
N
G
USI
NG
G
E
O
DE
SI
C
F
O
R
SH
O
RT
E
S
T
P
AT
H
T
h
e
s
h
o
r
test
p
ath
co
n
n
ec
ti
n
g
an
y
t
w
o
p
o
in
t
s
o
n
a
R
ie
m
a
n
n
i
an
m
a
n
i
f
o
ld
alo
n
g
i
ts
el
f
is
ca
l
led
as
th
e
g
eo
d
esic.
I
t
h
as
a
n
o
th
er
p
r
o
p
e
r
t
y
th
at
v
elo
cit
y
alo
n
g
th
is
g
e
o
d
esic
cu
r
v
e
r
e
m
ai
n
s
i
n
v
ar
ian
t.
T
h
e
b
ac
k
g
r
o
u
n
d
o
f
g
eo
d
esic h
a
s
b
ee
n
d
is
c
u
s
s
e
d
in
b
r
ief
in
t
h
e
later
s
ec
tio
n
.
3
.
1
.
R
iema
n
n
ia
n
Ma
n
ifo
ld
A
m
a
n
i
f
o
ld
„
M
n
‟
is
d
escr
ib
e
d
as
a
Ha
u
s
d
r
o
f
f
to
p
o
lo
g
ical
s
p
ac
e
f
o
r
w
h
ic
h
an
y
p
o
in
t
„
p
‟
h
as
a
n
eig
h
b
o
r
h
o
o
d
U
⊂
M
n
h
o
m
o
m
o
r
p
h
ic
to
an
o
p
en
s
u
b
s
et
o
f
t
h
e
E
u
clid
ian
s
p
ac
e
„
R
n
‟
.
I
f
w
e
d
e
f
i
n
e
a
f
u
n
ct
io
n
:
Ф:
U
→
Ф
(
U)
⊂
R
n
(
1
)
A
R
ie
m
a
n
n
ia
n
m
a
n
i
f
o
ld
is
d
ef
in
ed
b
y
(
M
n
,
g
)
,
w
h
er
e
„
M
n
‟
i
s
a
n
n
-
d
i
m
e
n
s
io
n
al
d
i
f
f
er
en
tiab
le
m
an
if
o
ld
an
d
„
g
‟
is
a
R
ie
m
an
n
ia
n
m
etr
ic.
E
v
er
y
R
ie
m
a
n
n
ia
n
m
etr
ic
h
a
s
a
u
n
iq
u
e
p
r
o
p
er
ty
th
at
it
is
s
y
m
m
etr
ic,
p
o
s
iti
v
e
d
e
f
in
ite
q
u
ad
r
atic
f
o
r
m
.
B
asicall
y
d
i
s
ta
n
ce
alo
n
g
th
e
m
a
n
i
f
o
ld
i
s
r
eg
ar
d
ed
as
th
e
m
etr
ic.
I
n
an
y
n
ei
g
h
b
o
r
h
o
o
d
U
o
f
a
p
o
in
t
in
m
a
n
i
f
o
ld
w
e
d
ef
in
e
l
o
ca
l
co
-
o
r
d
in
ates
(
ϴ
1
,
ϴ
2,
ϴ
3,
….
.
,
ϴ
n
)
,
th
e
n
t
h
e
m
etr
ic
ca
n
b
e
w
r
itte
n
as:
∑
(
2
)
w
h
er
e,
I
f
w
e
tak
e
a
cu
r
v
e
o
n
a
R
ie
m
a
n
n
ia
n
m
an
if
o
ld
:
(
3
)
th
en
,
it
s
ta
n
g
e
n
t
v
ec
to
r
ca
n
b
e
d
ef
in
ed
b
y
:
∑
(
4
)
Geo
d
esic is d
escr
ib
ed
as
th
e
s
h
o
r
test
p
at
h
alo
n
g
t
h
e
R
ie
m
a
n
n
ian
m
an
i
f
o
ld
co
n
n
ec
ti
n
g
an
y
t
w
o
p
o
in
ts
b
elo
n
g
i
n
g
to
it.
Hen
ce
,
i
f
a
r
c
len
g
t
h
is
co
n
s
id
er
ed
as
a
v
ar
iab
le
an
d
tak
en
as
t
h
e
co
v
ar
ian
t
d
er
iv
ati
v
e
o
f
eq
u
atio
n
(
4
)
an
d
m
ak
e
s
it z
er
o
th
en
eq
u
a
tio
n
(
4
)
m
ak
e
s
o
u
t t
h
e
g
eo
d
esic e
q
u
at
io
n
(
5
)
,
i.e
.
(
5
)
w
h
er
e,
is
th
e
C
h
r
i
s
to
f
f
el
s
y
m
b
o
l a
n
d
is
d
ef
in
ed
b
y
:
(
6
)
W
h
er
e
,
is
a
g
en
er
al
ele
m
en
t
o
f
in
v
er
s
e
m
atr
ix
o
f
R
ie
m
a
n
n
i
an
m
etr
ic
co
e
f
f
icien
t
m
atr
i
x
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
Op
tima
l Tr
a
jecto
r
y
P
la
n
n
in
g
o
f I
n
d
u
s
tr
ia
l R
o
b
o
ts
u
s
in
g
Ge
o
d
esic
(
P
r
a
d
ip
K
u
ma
r
S
a
h
u
)
193
4.
G
E
O
D
E
S
I
C
AP
P
RO
ACH
F
O
R
T
RAJ
E
C
T
O
RY
P
L
AN
NIN
G
T
h
e
lin
k
f
r
a
m
e
s
o
f
r
o
b
o
t
m
a
n
ip
u
lato
r
s
h
a
v
e
b
ee
n
a
ttach
ed
an
d
t
h
e
co
r
r
esp
o
n
d
in
g
li
n
k
p
ar
am
eter
s
h
av
e
b
ee
n
f
o
u
n
d
o
u
t
f
r
o
m
t
h
e
D
-
H
r
ep
r
esen
tatio
n
o
f
th
e
li
n
k
f
r
a
m
e
s
.
T
h
e
g
e
n
er
al
tr
a
n
s
f
o
r
m
atio
n
m
atr
ix
i
-
1
T
i
f
o
r
a
s
in
g
le
li
n
k
ca
n
b
e
d
ef
in
e
d
as f
o
llo
w
s
:
i
-
1
T
i
=
[
]
(
7
)
W
h
er
e
,
s
θ
i
=
s
in
θ
i
, c
θ
i
=
co
s
θ
i,
θ
i
is
th
e
i
th
j
o
in
t r
o
tatio
n
an
g
le,
sα
i
=
s
i
n
α
i,
cα
i
=
co
s
α
i,
α
i
is
t
w
i
s
t
an
g
le,
a
i
is
len
g
th
o
f
lin
k
,
d
i
is
o
f
f
s
et
d
is
tan
ce
at
j
o
in
t
i.
T
h
e
f
o
r
w
ar
d
k
i
n
e
m
atics
o
f
th
e
e
n
d
-
e
f
f
ec
to
r
w
it
h
r
e
s
p
ec
t
to
t
h
e
b
ase
f
r
a
m
e
is
d
eter
m
i
n
ed
b
y
m
u
ltip
l
y
in
g
all
o
f
t
h
e
i
-
1
T
i
m
at
r
ices o
f
lin
k
f
r
a
m
e
s
.
base
T
end
-
effector
=
0
T
1
*
1
T
2
*
2
T
3
……
n
-
1
T
n
(
8
)
Ass
u
m
in
g
t
h
e
f
i
n
al
tr
a
n
s
f
o
r
m
a
tio
n
,
th
e
T
m
atr
ix
o
f
a
r
o
b
o
t e
n
d
-
e
f
f
ec
to
r
as:
T
=
(
)
=
(
)
(
9
)
W
h
er
e
,
R
an
d
P
r
ep
r
esen
t th
e
o
r
ien
tatio
n
(
n
,
o
,
a)
an
d
p
o
s
itio
n
(
)
o
f
th
e
en
d
-
ef
f
ec
to
r
r
esp
ec
tiv
el
y
.
B
y
co
m
b
in
in
g
to
g
et
h
er
,
R
a
n
d
P
as
o
r
ien
tatio
n
s
p
ac
e
a
n
d
p
o
s
itio
n
s
p
ac
e
r
esp
ec
ti
v
el
y
,
a
R
i
e
m
an
n
ia
n
m
etr
ic
f
o
r
t
h
e
w
o
r
k
s
p
ac
e
ca
n
b
e
co
n
s
tr
u
cted
f
o
r
th
e
T
m
atr
i
x
.
4
.
1
.
Select
io
n o
f
L
o
ca
l C
o
o
rdina
t
es
I
n
o
r
d
er
to
o
b
tain
a
s
o
lu
tio
n
t
o
th
e
in
v
er
s
e
k
in
e
m
atic
p
r
o
b
le
m
o
f
th
e
r
o
b
o
t
m
a
n
ip
u
la
to
r
,
an
ex
p
licit
s
ch
e
m
e
s
h
o
u
ld
b
e
m
o
d
eled
to
in
ter
-
r
ela
te
j
o
in
t sp
ac
e
a
n
d
C
a
r
tesi
an
s
p
ac
e.
I
f
a
ll j
o
in
t
v
ar
ia
b
les ar
e
ch
o
s
e
n
a
s
a
co
o
r
d
in
ate
s
y
s
te
m
o
f
C
ar
te
s
ia
n
s
p
ac
e,
th
en
b
o
th
t
h
e
s
p
ac
es
ca
n
b
e
ea
s
il
y
i
n
ter
-
r
elate
d
.
Ho
w
e
v
er
th
i
s
p
r
o
ce
s
s
h
as
s
o
m
e
s
h
o
r
tco
m
in
g
s
.
I
f
t
h
e
j
o
in
t
s
p
ac
e
is
to
b
e
m
ap
p
ed
to
C
ar
tesi
a
n
s
p
ac
e,
t
h
en
f
o
r
w
ar
d
k
in
e
m
atic
s
h
as
to
b
e
u
s
ed
.
T
h
e
in
ten
tio
n
is
to
m
ap
C
ar
tesi
an
s
p
ac
e
to
j
o
in
t
s
p
ac
e.
Mu
ltip
le
s
o
l
u
tio
n
s
m
a
y
b
e
o
b
tain
ed
in
s
tead
o
f
a
s
in
g
le
s
o
lu
tio
n
.
He
n
ce
,
j
o
in
t
v
ar
iab
les
s
h
o
u
ld
b
e
tr
ea
ted
as
lo
ca
l
co
o
r
d
in
ates
in
s
tead
o
f
g
en
er
al
co
o
r
d
in
ates
to
tack
le
esp
ec
iall
y
th
e
i
n
v
er
s
e
k
in
e
m
atic
s
p
r
o
b
lem
s
.
4
.
2
.
Appl
ica
t
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n o
f
G
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des
ic
f
o
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t
ra
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t
o
r
y
pla
nn
ing
C
o
n
s
id
er
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g
a
n
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u
c
lid
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n
s
p
ac
e
w
it
h
d
is
ta
n
ce
m
etr
ic,
th
e
g
eo
d
esic
b
ec
o
m
e
s
a
s
tr
ai
g
h
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lin
e.
A
s
E
u
clid
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n
s
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ac
e
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s
ta
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e
th
e
p
o
s
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tio
n
s
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ac
e,
h
e
n
ce
a
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a
n
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ia
n
m
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ic
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a
s
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n
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a
s
d
is
ta
n
c
e
m
etr
ic.
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h
e
R
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a
n
n
ia
n
m
etr
i
c
is
g
i
v
en
b
y
:
(
1
0
)
w
h
er
e,
is
th
e
d
er
iv
ativ
e
o
f
th
e
p
o
s
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n
v
ec
to
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an
d
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d
ar
e
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e
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er
iv
ativ
es
o
f
th
e
o
r
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tatio
n
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to
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,
,
,
r
esp
ec
tiv
el
y
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A
f
lo
w
c
h
ar
t
o
f
th
e
r
o
b
o
tic
m
an
ip
u
lato
r
tr
aj
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to
r
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lan
n
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n
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u
s
in
g
g
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d
esic
m
et
h
o
d
is
r
ep
r
esen
ted
in
Fig
u
r
e
1.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
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3
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9
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194
4
.
1
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Rie
m
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du
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n
tr
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h
e
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ic
s
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ld
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e
s
y
m
m
etr
ic,
p
o
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itiv
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e
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i
n
ite
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ad
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atic
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o
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o
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n
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s
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1
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icie
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=
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.
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ter
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h
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at
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h
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d
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ar
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tain
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in
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d
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lt
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lis
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ed
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o
r
t
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i
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t tr
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o
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4
.
1
.
M
a
t
he
m
a
t
ica
l F
o
r
m
u
la
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io
n a
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Nu
m
er
ica
l C
o
m
p
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RS0
6
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Ro
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t
Ka
w
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R
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ate
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h
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k
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2
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ith
t
h
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f
o
r
m
as
eq
u
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n
(
9
)
,
w
h
er
e
,
[
]
[
]
[
]
[
]
Fig
u
r
e
1
.
Flo
w
c
h
ar
t s
h
o
w
in
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s
eq
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tial p
r
o
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s
s
o
f
m
a
n
i
p
u
lato
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tr
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to
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lan
n
i
n
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b
y
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eo
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et
h
o
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
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N:
2089
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4856
Op
tima
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1
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6
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0
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6
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
IJ
RA
Vo
l.
5
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No
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3
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Sep
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h
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3
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4
.
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e
tr
a
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o
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th
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g
5
.
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f
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6
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.
0
0
4
5
7
,
-
0
.
2
3
4
1
8
,
-
0
.
9
7
2
1
8
)
(
0
.
1
0
4
3
7
,
-
0
.
3
0
8
9
7
,
-
0
.
9
4
5
3
2
)
T
ab
le
3
.
C
o
o
r
d
in
ates o
f
en
d
p
o
in
ts
i
n
j
o
in
t sp
ac
e
θ
s
I
n
itial st
a
te
(
t=0
s
ec
)
in
r
ad
Fin
al
s
tate
(
t=5
s
ec
)
in
r
ad
θ
1
-
1
.
7
3
1
7
3
-
1
.
3
3
1
2
2
θ
2
-
0
.
4
3
8
1
6
-
0
.
3
8
4
8
8
θ
3
-
1
.
9
3
8
3
8
-
1
.
8
9
7
4
3
θ
4
0
.
0
6
5
1
4
0
.
0
5
5
6
7
θ
5
-
0
.
5
3
1
8
3
-
0
.
5
2
8
8
9
θ
6
-
0
.
6
5
8
2
5
-
0
.
5
5
2
7
9
5.
RE
SU
L
T
S O
F
T
H
E
S
I
M
UL
AT
I
O
N
T
h
e
m
et
h
o
d
p
r
o
p
o
s
ed
in
th
is
p
ap
er
h
as
b
ee
n
v
er
if
ied
in
s
i
m
u
latio
n
s
f
o
r
th
e
Ka
w
a
s
ak
i
R
S
0
6
L
r
o
b
o
t.
T
h
e
s
i
m
u
latio
n
r
es
u
lt
s
co
n
f
ir
m
th
e
ac
c
u
r
ac
y
,
s
m
o
o
th
n
es
s
an
d
th
e
o
p
ti
m
alit
y
o
f
t
h
e
e
n
d
-
e
f
f
ec
to
r
m
o
tio
n
,
w
h
ic
h
ar
e
r
ep
r
esen
ted
in
Fi
g
u
r
e
3
an
d
Fig
u
r
e
4
Si
m
ilar
l
y
,
th
e
ac
cu
r
ac
y
a
s
w
ell
a
s
s
m
o
o
th
n
ess
o
f
e
n
d
-
e
f
f
ec
to
r
o
r
ien
tatio
n
v
ec
to
r
i
s
r
ep
r
esen
ted
in
Fi
g
u
r
e
5
T
h
e
j
o
in
t
tr
aj
ec
to
r
ies
an
d
th
eir
d
er
i
v
ati
v
e
s
ar
e
f
o
u
n
d
to
b
e
s
m
o
o
th
,
w
h
ic
h
is
p
r
esen
ted
in
Fig
u
r
e
6
.
Hen
ce
th
e
m
et
h
o
d
ca
n
b
e
im
p
le
m
e
n
ted
f
o
r
s
m
o
o
t
h
an
d
ac
cu
r
ate
tr
aj
ec
to
r
y
p
lan
n
i
n
g
f
o
r
en
d
-
ef
f
ec
to
r
s
an
d
th
e
r
elev
a
n
t j
o
in
t tr
aj
ec
to
r
ies o
f
th
e
r
o
b
o
t m
an
ip
u
l
a
to
r
s
.
6.
CO
NCLU
SI
O
N
AND
F
U
T
U
RE
WO
RK
An
o
p
ti
m
al
tr
aj
ec
to
r
y
p
lan
n
in
g
w
it
h
s
m
o
o
t
h
a
n
d
ac
c
u
r
ate
m
o
v
e
m
e
n
t
f
o
r
r
o
b
o
t
m
a
n
ip
u
lato
r
s
co
m
b
i
n
i
n
g
b
o
th
p
o
s
it
io
n
a
n
d
o
r
ien
tatio
n
s
p
ac
e
b
y
i
m
p
le
m
e
n
ti
n
g
g
eo
d
esic
m
et
h
o
d
is
p
r
es
en
ted
i
n
t
h
is
p
ap
er
.
T
h
e
k
e
y
in
ten
t
is
to
as
s
ig
n
a
n
ap
p
r
o
p
r
iate
m
etr
ic
to
ac
q
u
i
r
e
t
h
e
n
ec
es
s
ar
y
g
eo
d
esic
m
o
tio
n
.
T
h
e
g
eo
d
esic
eq
u
atio
n
s
ar
e
s
o
lv
ed
n
u
m
er
ic
all
y
b
y
s
i
m
u
lta
n
eo
u
s
R
u
n
g
e
-
Ku
tta
4
t
h
s
tep
m
et
h
o
d
an
d
t
h
e
r
esu
lt
s
ar
e
u
s
ed
to
co
n
tr
o
l
th
e
r
o
b
o
t.
Geo
d
esic
s
im
u
latio
n
s
o
f
t
h
e
r
es
u
lts
u
s
in
g
th
e
Ka
w
a
s
ak
i
R
S0
6
L
r
o
b
o
t
ar
e
p
r
esen
ted
.
T
h
e
p
r
e
s
en
ce
o
f
o
b
s
tacle
in
g
eo
d
esic
f
o
r
m
u
lat
io
n
ex
ten
d
s
t
h
e
p
r
o
ce
s
s
o
f
f
i
n
d
in
g
g
eo
d
esic
f
r
o
m
i
n
itia
l
p
o
in
t
to
o
b
s
tacle
an
d
ag
ai
n
f
r
o
m
o
b
s
tacle
to
f
in
a
l
tar
g
et
p
o
in
t.
T
h
e
p
r
o
p
o
s
ed
m
et
h
o
d
w
ill
h
av
e
a
h
ig
h
i
m
p
ac
t
o
n
t
h
e
r
o
b
o
tic
w
eld
i
n
g
a
n
d
m
ac
h
i
n
i
n
g
ap
p
licatio
n
,
w
h
er
e
th
e
en
d
-
ef
f
ec
to
r
h
as
to
p
er
f
o
r
m
a
s
m
o
o
t
h
an
d
ac
cu
r
at
e
m
o
tio
n
in
lea
s
t
d
u
r
atio
n
co
n
s
id
er
in
g
t
h
e
tr
u
e
n
ess
o
f
g
eo
d
esic
m
eth
o
d
to
o
p
tim
izatio
n
,
s
m
o
o
th
n
e
s
s
an
d
ac
cu
r
ac
y
.
T
h
e
m
et
h
o
d
co
n
ce
r
n
s
to
r
o
b
o
ts
w
it
h
le
s
s
th
a
n
o
r
eq
u
a
l to
s
ix
DOFs
an
d
f
o
c
u
s
es
ex
cl
u
s
iv
e
l
y
o
n
li
n
ea
r
m
o
tio
n
s
.
F
u
t
u
r
e
w
o
r
k
w
ill
b
e
f
o
cu
s
in
g
o
n
tr
aj
ec
to
r
y
p
la
n
n
i
n
g
o
f
co
m
p
le
x
r
o
b
o
ts
h
a
v
in
g
m
o
r
e
th
an
s
i
x
DOF
s
w
it
h
d
i
f
f
er
e
n
t t
y
p
e
s
o
f
n
o
n
-
lin
ea
r
m
o
tio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
SS
N:
2089
-
4856
Op
tima
l Tr
a
jecto
r
y
P
la
n
n
in
g
o
f I
n
d
u
s
tr
ia
l R
o
b
o
ts
u
s
in
g
Ge
o
d
esic
(
P
r
a
d
ip
K
u
ma
r
S
a
h
u
)
197
RE
F
E
R
NCE
S
[1
]
K.
Nin
g
,
e
t
a
l.
,
“
Ac
c
u
ra
te
p
o
siti
o
n
a
n
d
v
e
lo
c
it
y
c
o
n
tro
l
f
o
r
traje
c
to
ries
b
a
se
d
o
n
d
y
n
a
m
ic
m
o
v
e
m
e
n
t
p
rim
it
iv
e
s”
,
IEE
E
In
tern
a
ti
o
n
a
l
Co
n
f
e
re
n
c
e
o
n
Ro
b
o
ti
c
s
a
n
d
A
u
to
m
a
ti
o
n
2
0
1
1
,
ICCS
,
Ch
in
a
,
M
a
y
0
9
-
1
3
,
2
0
1
1
,
p
p
.
5
0
0
6
–
5
0
1
1
,
2
0
1
1
[2
]
A
.
G
a
sp
a
re
tt
o
,
e
t
a
l.
,
“
A
n
e
w
m
e
th
o
d
f
o
r
sm
o
o
t
h
traje
c
to
ry
p
l
a
n
n
in
g
o
f
ro
b
o
t
m
a
n
ip
u
lat
o
rs”
,
M
e
c
h
a
n
ism
a
n
d
M
a
c
h
in
e
T
h
e
o
ry
,
4
2
(4
)
,
p
p
.
4
5
5
–
4
7
1
,
2
0
0
7
.
[3
]
S
.
L
.
Ch
iu
,
“
T
a
sk
c
o
m
p
a
ti
b
il
it
y
o
f
m
a
n
ip
u
lato
r
p
o
stu
re
s”
,
T
h
e
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
R
o
b
o
ti
c
s
Res
e
a
rc
h
,
7
(
5
),
p
p
.
13
–
2
1
,
1
9
8
8
.
[4
]
J.
Bo
b
ro
w
,
“
Op
ti
m
a
l
ro
b
o
t
p
a
th
p
lan
n
i
n
g
u
sin
g
th
e
m
in
i
m
u
m
-
ti
m
e
c
rit
e
rio
n
”
,
IEE
E
J
o
u
rn
a
l
o
f
Ro
b
o
ti
c
s
a
n
d
Au
to
m
a
ti
o
n
,
4
(4
),
p
p
.
4
4
3
–
4
4
9
,
1
9
8
8
.
[5
]
J.
Bo
b
r
o
w
,
e
t
a
l.
,
“
T
ime
o
p
ti
m
a
l
c
o
n
tro
l
o
f
ro
b
o
ti
c
m
a
n
ip
u
lato
rs
a
lo
n
g
sp
e
c
if
ied
p
a
th
s”
,
T
h
e
I
n
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
R
o
b
o
ti
c
s R
e
se
a
rc
h
,
4
(3
),
p
p
.
3
–
1
7
,
1
9
8
5
.
[6
]
C.
El
d
e
rsh
a
w
,
e
t
a
l.
,
“
Us
in
g
g
e
n
e
ti
c
a
lg
o
rit
h
m
s
to
so
lv
e
th
e
m
o
ti
o
n
p
la
n
n
i
n
g
p
ro
b
lem
”
,
J
o
u
rn
a
l
o
f
Un
ive
rs
a
l
Co
mp
u
ter
S
c
ien
c
e
,
6
(
4
),
p
p
.
4
2
2
–
4
3
2
,
2
0
0
0
.
[7
]
L
.
F
.
T
ian
,
e
t
a
l.
,
“
A
n
e
ff
e
c
ti
v
e
r
o
b
o
t
traje
c
to
ry
p
lan
n
in
g
m
e
th
o
d
u
sin
g
a
g
e
n
e
ti
c
a
lg
o
rit
h
m
”
,
M
e
c
h
a
tro
n
ics
,
1
4
(5
)
,
p
p
.
4
5
5
–
4
7
0
,
2
0
0
3
.
[8
]
W
.
M
.
Yu
n
,
e
t
a
l.
,
“
Op
ti
m
u
m
m
o
ti
o
n
p
lan
n
in
g
in
jo
i
n
t
sp
a
c
e
f
o
r
ro
b
o
ts
u
si
n
g
g
e
n
e
ti
c
a
lg
o
rit
h
m
s
”
,
Ro
b
o
ti
c
s
a
n
d
Au
to
n
o
m
o
u
s
S
y
ste
ms
,
3
(4
)
,
p
p
.
3
7
3
–
3
9
3
,
1
9
9
6
.
[9
]
X
.
F
.
Z
h
a
,
“
Op
ti
m
a
l
p
o
se
traje
c
to
ry
p
lan
n
in
g
f
o
r
ro
b
o
t
m
a
n
ip
u
lat
o
rs”
,
M
e
c
h
a
n
ism
a
n
d
M
a
c
h
in
e
T
h
e
o
ry
,
3
7
(
1
),
p
p
.
1
0
6
3
–
1
0
8
6
,
2
0
0
2
.
[1
0
]
X
.
F
.
Z
h
a
,
e
t
a
l.
,
“
T
ra
jec
to
ry
c
o
-
o
rd
i
n
a
ti
o
n
p
lan
n
in
g
a
n
d
c
o
n
tr
o
l
f
o
r
ro
b
o
t
m
a
n
ip
u
lato
rs
in
a
u
t
o
m
a
ted
m
a
t
e
rial
h
a
n
d
l
in
g
a
n
d
p
r
o
c
e
ss
in
g
”
,
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Ad
v
a
n
c
e
d
M
a
n
u
f
a
c
tu
re
T
e
c
h
n
o
lo
g
y
,
2
3
(
1
1
/
1
2
)
,
p
p
.
8
3
1
–
8
4
5
,
2
0
0
4
.
[1
1
]
A
.
Ola
b
i,
e
t
a
l
.
,
“
F
e
e
d
ra
te
p
lan
n
in
g
f
o
r
m
a
c
h
in
in
g
w
it
h
i
n
d
u
strial
six
-
a
x
is
ro
b
o
ts”
,
Co
n
tro
l
E
n
g
i
n
e
e
rin
g
Pra
c
ti
c
e
,
1
8
(
5
),
p
p
.
4
7
1
–
4
8
1
,
2
0
1
0
.
[1
2
]
M
.
Bo
ry
g
a
,
e
t
a
l.
,
“
P
lan
n
i
n
g
o
f
m
a
n
ip
u
lat
o
r
m
o
ti
o
n
traje
c
to
ry
w
i
th
h
ig
h
e
r
-
d
e
g
re
e
p
o
ly
n
o
m
i
a
ls
u
s
e
”
,
M
e
c
h
a
n
ism
a
n
d
M
a
c
h
i
n
e
T
h
e
o
ry
,
4
4
(7
),
p
p
.
1
4
0
0
–
1
4
1
9
,
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0
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9
.
[1
3
]
H.
L
iu
,
e
t
a
l.
,
“
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ime
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ti
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a
l
a
n
d
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rk
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ra
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a
n
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p
u
lato
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e
m
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ti
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n
stra
in
ts”
,
Ro
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o
t
ics
a
n
d
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m
p
u
ter
-
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teg
ra
ted
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,
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9
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,
p
p
.
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0
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–
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3
.
[1
4
]
M
.
G
o
u
a
s
m
i
,
e
t
a
l.
,
“
Ro
b
o
t
K
in
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m
a
ti
c
s
Us
in
g
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l
Qu
a
tern
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s”
,
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ter
n
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ti
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l
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o
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a
l
o
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o
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s
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n
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to
m
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ti
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n
,
1
(1
),
p
p
.
1
3
-
3
0
,
2
0
1
2
.
[1
5
]
J.
S
h
a
h
,
e
t
a
l
.
,
“
Kin
e
m
a
ti
c
A
n
a
l
y
sis
o
f
a
P
lan
e
r
Ro
b
o
t
Us
i
n
g
A
rti
ficia
l
Ne
u
ra
l
Ne
tw
o
rk
”
,
In
ter
n
a
ti
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a
l
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p
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5
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1
,
2
0
1
2
.
[1
6
]
P
.
Jh
a
,
e
t
a
l.
,
“
A
Ne
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ra
l
Ne
t
wo
rk
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p
p
ro
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v
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rse
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e
m
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ti
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R
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n
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ter
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ti
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l
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rn
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3
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p
p
.
5
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1
,
2
0
1
4
.
[1
7
]
G
.
Ro
d
n
a
y
,
e
t
a
l.
,
“
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m
e
tri
c
v
isu
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li
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ti
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ig
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ra
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s
p
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w
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re
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s
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of
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re
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o
m
m
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h
a
n
is
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s”
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M
e
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h
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n
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n
d
M
a
c
h
in
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T
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e
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ry
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3
6
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p
.
5
2
3
–
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4
5
,
2
0
0
1
.
[1
8
]
M
.
Zef
ra
n
,
e
t
a
l.
,
“
On
t
h
e
g
e
n
e
ra
ti
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n
o
f
s
m
o
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h
re
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l
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o
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m
o
ti
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n
s”
,
IE
EE
T
ra
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sa
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ti
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s
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n
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d
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t
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p
.
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6
–
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9
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9
9
8
.
[1
9
]
Y.
Ch
e
n
,
a
t
a
l.
,
“
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m
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n
d
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c
c
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ra
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ra
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to
ry
P
lan
n
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n
g
fo
r
In
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u
strial
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b
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,
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v
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n
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in
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h
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rti
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le ID
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1
3
7
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1
4
.
[2
0
]
J.M
.
S
e
li
g
,
e
t
a
l.
,
“
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a
n
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p
u
lati
n
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r
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b
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ts
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lo
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e
li
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l
traje
c
to
ries
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o
ti
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,
1
4
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p
.
2
6
1
–
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6
7
,
1
9
9
6
.
[2
1
]
L
.
Zh
a
n
g
,
e
t
a
l.
,
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a
n
ip
u
lato
r
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ra
jec
to
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p
lan
n
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n
g
u
sin
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g
e
o
d
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ic
m
e
th
o
d
”
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o
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we
l
d
in
g
,
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n
telli
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n
c
e
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n
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to
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ti
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6
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e
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tu
re
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n
Co
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tro
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p
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7
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RAP
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ra
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ip
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p
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m
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l
s.
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