Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
Vol.
3, No. 4, Decem
ber
2014, pp. 221~
233
I
S
SN
: 208
9-4
8
5
6
2
21
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJRA
Dynami
c Behavi
or of a SCARA Ro
bot by using N-E Method for
a Straight Line and Simulation
of
Moti
on by using Solidworks
and Verification by Matlab/Simulink
Brahim Fer
n
ini, Mustapha
Temmar
Departem
ent
of
M
echani
cal
Eng
i
neering
,
Alg
e
ria
Univers
i
t
y
of
Bli
d
a 1,
Alger
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
J
u
n 17, 2013
R
e
vi
sed Oct
6,
2
0
1
3
Accepte
d Apr 2, 2014
SCARA (Selec
tive Com
p
lian
t
Assem
b
ly
Ro
bot Arm
)
robot of seria
l
archi
t
ec
ture
is
widel
y
us
ed
in
as
s
e
mbly
op
erations and op
erations "pick-
plac
e", it has been shown that use
of robots
im
proves the accura
c
y
o
f
a
sse
mbly
,
a
nd sa
ve
s a
sse
mbly
ti
me
a
nd c
o
st
a
s
we
l
l
.
T
h
e
most i
m
port
a
nt
condition for th
e choice of this kind of robot is the d
y
namic beh
a
vior for
a
given p
a
th, no
closed solution
f
o
r the d
y
namics
of th
is important robot has
been reported. This paper presents th
e stud
y
of the kinematics (forward and
inverse) b
y
usin
g D-H notation and the d
y
namics of SC
ARA rob
o
t b
y
using
N-E methods. A computer cod
e
is develop
e
d
for trajector
y
g
e
neration b
y
using inverse kinematics, a
nd calcu
l
ates the var
i
ations of the to
rques of the
links for a straight line (path rest to
rest) between
two positions fo
r operation
"pick-place". SC
ARA robot is construc
ted to achieve “pick-place»
operation
using Solid Works software.
And ve
rific
a
tio
n b
y
Mat
l
ab/Si
m
u
link. Th
e
results of sim
u
lations wer
e
di
scussed. An ag
reem
ent b
e
twee
n the tw
o
softwares is
certain
ly
obtained
h
e
rein
.
Keyword:
Dy
nam
i
c B
e
havi
o
r
Matlab
/
Si
m
u
li
n
k
pat
h
pl
an
ni
ng
Ro
bo
tics
SC
AR
A
R
o
bot
Si
m
u
latio
n
So
lid
Work
s
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Brah
im
Fern
in
i
,
Lab
o
rato
ry
of Struct
ures
, De
partm
e
nt of M
echanical E
n
gineeri
n
g,
Un
i
v
ersity of
Blid
a 1
,
Al
g
e
ri
a
e-m
a
i
l: fern
in
i.b
r
ah
im
1
2
@
g
m
ail.co
m
Nom
e
ncla
ture
:
1
i
i
A
D-H tran
sformatio
n
m
a
trix
f
o
r a
d
jace
nt
fram
e
s,
i
and
1
i
i
C
Co
sin
e
i
ijk
C
Co
sin
e
ijk
= cosi
ne {(
i
+
j
) +
k
}
i
d
Distance
from
the origi
n
of t
h
e (
1
i
) th
e coo
r
d
i
nate fram
e
to
the in
tersection
of th
e
1
i
Z
ax
is with th
e
i
X
axis along
1
i
Z
axi
s
i
e
is th
e
p
o
s
ition
v
ector
o
f
t
h
e C
O
M of link
i
with
resp
ect to fra
m
e
i
i
F
I
npu
t fo
r
c
e fo
r
i
th
jo
in
t
i
f
F
o
rce
exe
r
ted
on link
i
by
l
i
nk
1
i
at the c
o
ordinate fram
e (
1
i
X
,
1
i
Y
,
1
i
Z
)
to
su
ppo
rt link
i
and
th
e lin
ks ab
ov
e it
i
In
p
u
t tor
q
ue
fo
r
i
th
jo
in
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
221 – 233
22
2
i
I
In
ertia m
a
trix
o
f
lin
k
i
abou
t its cen
ter of m
a
ss with referen
c
e to
th
e coord
i
n
a
te system
(
0
X
,
0
Y
,
0
Z
)
i
J
In
ertia m
a
trix
o
f
lin
k i abou
t
its cen
ter
o
f
mass referred to
its o
w
n
link
coo
r
d
i
n
a
te system
(
i
X
,
i
Y
,
i
Z
)
i
l
The
s
h
ortest distance betwee
n
1
i
Z
and
i
Z
axes
ef
f
m
Effectiv
e m
a
ss
.
i
m
Mass of the
i
th
lin
k
i
n
M
o
m
e
nt exe
r
ted
on link-
i
by lin
k
1
i
at the coordinate
fram
e
(
1
i
X
,
1
i
Y
,
1
i
Z
)
*
i
p
i
s
t
h
e
di
spl
ace
m
e
nt
fr
om
t
h
e ori
g
i
n
of
f
r
am
e
i
−
1
to
fram
e
i
with respect
t
o
fram
e
i
i
Th
e
jo
in
t an
g
l
e fro
m
1
i
X
ax
is to
t
h
e
i
X
ax
is abou
t the
1
i
Z
ax
is (u
sing
t
h
e
r
i
gh
t h
a
nd
ru
le)
1
i
R
A 3 ×
3
ro
tatio
n m
a
trix
wh
ich
tran
sform
s
an
y
vect
or
wi
t
h
refe
rence
t
o
c
o
or
di
nat
e
fram
e
(
i
X
,
i
Y
,
i
Z
)
t
o
t
h
e c
o
or
di
na
t
e
sy
st
em
(
1
i
X
,
1
i
Y
,
1
i
Z
)
i
S
Sine
i
ijk
S
Sine
ijk
= sine {(
i
+
j
) +
k
}
i
V
Li
near
vel
o
ci
t
y
o
f
t
h
e
co
or
di
n
a
t
e
sy
st
em
(
i
X
,
i
Y
,
i
Z
) wi
t
h
respect
t
o
base c
o
or
di
nat
e
sy
st
em
(
0
X
,
0
Y
,
0
Z
)
i
An
g
u
l
a
r
vel
o
ci
t
y
of t
h
e c
o
o
r
di
nat
e
sy
st
em
(
i
X
,
i
Y
,
i
Z
)
wi
t
h
res
p
ect
t
o
base
co
or
di
nat
e
syste
m
(
0
X
,
0
Y
,
0
Z
)
1.
INTRODUCTION
Pick
And Plac
e cycle is the ti
me
, i
n
sec
o
n
d
s
,
t
o
exec
ut
e t
h
e
f
o
l
l
o
wi
ng
m
o
t
i
on
seq
u
e
n
ce:
M
ove
d
o
w
n
one
i
n
c
h
,
g
r
as
p a
rat
e
d
pay
l
oad;
m
ove u
p
o
n
e i
n
ch;
m
ove ac
ro
ss t
w
el
ve i
n
ches;
m
ove
d
o
w
n
o
n
e
i
n
ch;
u
ngrasp
;
m
o
v
e
up
o
n
e
in
ch
; an
d ret
u
rn
to start lo
cation
.
Th
e SC
ARA
Selectiv
e Co
m
p
lian
t
Assem
b
ly
Ro
bo
t Arm
o
r
Selectiv
e Com
p
l
i
an
t Articulated
Ro
bo
t
Arm
i
s
wi
del
y
use
d
f
o
r
ope
rat
i
ons “
p
i
c
k
-
pl
ac
e”. The
robo
t was called
Sel
ectiv
e Co
m
p
lia
n
ce Assem
b
ly
Ro
bo
t
Arm
,
SC
AR
A
.
It
s arm
was ri
gi
d i
n
t
h
e Z
-
ax
i
s
and
pl
i
a
bl
e i
n
the
XY-a
xes, whic
h a
llowed
it to
ad
ap
t to h
o
l
es
in
th
e
XY-ax
e
s.
By v
i
rtu
e
o
f
t
h
e SCAR
A's
p
a
rallel-ax
is jo
in
t layo
u
t
, the arm is sl
ig
htly co
m
p
lian
t
in
th
e X-Y
di
rect
i
o
n
but
ri
gi
d i
n
t
h
e
‘Z
’
di
rect
i
o
n,
he
nc
e t
h
e t
e
rm
:
Sel
ect
i
v
e C
o
m
p
l
i
ant
.
T
h
i
s
i
s
a
d
vant
a
g
eo
us
f
o
r
m
a
ny
types of assem
b
ly operations:
pick-place, ins
e
r
ting
a
round pin
i
n
a round hole without bi
ndi
ng.
Th
e seco
nd
att
r
ibu
t
e o
f
t
h
e SCARA is th
e
jo
in
ted
t
w
o-link
arm
layo
u
t
si
m
i
lar to
ou
r
hu
m
a
n
arm
s
,
hence t
h
e often-use
d
term
,
Articulated. T
h
is feature a
llo
ws t
h
e arm
t
o
ex
ten
d
i
n
to
confine
d
area
s
and the
n
ret
r
act
or “
f
ol
d
up”
out
of t
h
e
way
.
Thi
s
i
s
a
dva
nt
age
o
us f
o
r t
r
ans
f
er
ri
n
g
p
a
rt
s fr
om
one cel
l
t
o
anot
he
r o
r
f
o
r
l
o
adi
n
g/
unl
oa
di
n
g
pr
ocess
st
at
i
ons t
h
at
are
encl
ose
d
.
The
SCAR
A robots a
r
e
generally faster and
cleaner
t
h
an com
p
arable Cartesian system
s. Their
singl
e
pede
st
al
m
oun
t
req
u
i
r
es
a sm
al
l
fo
ot
pri
n
t
an
d
pr
ovi
des a
n
easy
,
u
nhi
n
d
er
ed f
o
rm
of m
ount
i
n
g.
O
n
t
h
e
ot
he
r
h
a
nd
, SC
ARA's can
b
e
m
o
re ex
p
e
n
s
iv
e t
h
an
co
m
p
arab
le Cartesian
syste
m
s an
d
th
e co
n
t
ro
lling
software
requ
ires inv
e
rse k
i
n
e
m
a
tics fo
r li
n
ear i
n
terp
o
l
ated m
o
v
e
s. Th
is software typ
i
cally co
mes with
th
e
SCARA
th
ou
gh
and
is
usu
a
lly tran
sp
aren
t to
t
h
e end
-
u
s
er.
In this
work, 4 axes«
R
-
R-P-R
»
robot syst
e
m
s for
ope
rat
i
on
pic
k
a
n
d place will be
de
signe
d a
nd
devel
ope
d
usi
n
g S
o
l
i
d
wo
rk
s
pr
o
g
ram
as sh
ow
n i
n
fi
gu
re
1, a
n
d m
odel
e
d
by
M
a
t
l
a
b/
Sim
u
l
i
nk as s
h
o
w
n
i
n
figu
re 2
.
Sim
u
l
a
tio
n
b
y
u
s
ing
MATLAB/Simu
lin
k
so
ftware
will b
e
carried ou
t. Th
e Re
sults o
f
bo
th sofwares
will b
e
presented
and
d
i
scu
s
sed
.
In
t
h
e
p
a
p
e
r, th
e e
q
u
a
ti
o
n
s
of k
i
n
e
m
a
tics fo
r «
R
-
R-P-R
»
robo
t wi
th
th
e
robot dy
nam
i
c
s
for
each joint
we
re
devel
ope
d with D-H form
ula
tion.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Dyn
a
m
i
c
Be
ha
vi
or
of
a
SC
AR
A Ro
b
o
t
by
u
s
i
n
g
N
-
E
met
h
o
d
f
o
r
a
St
rai
ght
Li
ne …
(
B
r
ahi
m Fer
n
i
n
i
)
22
3
The
pape
r i
s
or
gani
ze
d as
f
o
l
l
o
ws:
Fi
rst
,
a
n
i
n
t
r
od
uct
i
o
n t
o
SC
AR
A
r
o
b
o
t
,
ki
nem
a
t
i
c
s i
s
present
e
d i
n
sect
i
on
2. I
n
s
ect
i
on 3
,
t
h
e
d
y
n
am
i
c
behavi
or
.I
n sect
i
o
n
4,
t
h
e ap
pl
i
cat
i
o
n
.
Sect
i
o
n
s
5,
6
and
7, t
h
e dy
n
a
m
i
cs
sim
u
l
a
t
i
on,
di
s
c
ussi
o
n
a
n
d c
o
ncl
u
si
on
res
p
e
c
t
i
v
el
y
and
f
o
l
l
o
we
d
by
t
h
e
t
h
e refe
re
nces.
Fi
gu
re
1.
SC
A
R
A r
o
bot
m
o
d
e
l
e
d by
Sol
i
d
Wo
r
k
s
Fi
gu
re
2.
SC
A
R
A r
o
bot
m
o
d
e
l
e
d by
M
a
t
l
a
b
/
Sim
u
l
i
nk
Previous W
o
r
k
The pre
v
i
o
us wo
rk
[
1
] [2
]
st
udi
e
d
t
h
e
dy
na
m
i
c of t
h
i
s
r
o
b
o
t
by
usi
ng
N
-
L m
e
t
hod
, b
u
t
t
h
i
s
m
e
t
hod i
s
not com
m
only used for real
tim
e
c
ontrol as its need large am
ount of
c
o
m
putation tim
e and space,
and the
st
udy
of
t
h
e
dy
nam
i
c behavi
o
r
i
s
do
ne
fo
r
p
a
t
h
creat
e
d
by
t
h
e j
o
i
n
t
s
p
ace
,
t
h
i
s
l
a
st
d
o
es
not
gi
ve
t
h
e
de
si
re
d
traj
ectories like (strai
g
h
t
lin
e, circle,..).
Present W
o
rk
The p
r
esent
an
al
y
s
i
s
of t
h
i
s
rob
o
t
i
s
carri
ed
out
t
o
st
udy
t
h
e dy
nam
i
cs behavi
or f
o
r a st
rai
ght
l
i
n
e
(rest
t
o
rest
pat
h
)
by
usi
n
g
N-
E m
e
t
hod. T
h
e
si
gni
fi
cance
o
f
t
h
i
s
st
udy
l
i
e
s
i
n
t
h
e
fact
t
h
at
i
t
gi
ves i
n
si
g
h
t
i
n
t
o
t
h
e dy
nam
i
c behavi
or
o
f
t
h
i
s
r
o
b
o
t
.
The
di
rect
ki
n
e
m
a
t
i
c
s al
l
o
ws us t
o
fi
nd
t
h
e
rel
a
t
i
ons
hi
p
b
e
t
w
een t
h
e a
n
gul
a
r
di
s
p
l
ace
m
e
nt
and
t
h
e
p
o
s
ition
of th
e en
d-effector, t
h
e inv
e
rse
k
i
ne
m
a
tics al
lo
w
u
s
to
co
nn
ect b
e
tween
two
po
sitio
n
s
b
y
a straig
ht
lin
e (rest to
rest p
a
th).
Sol
i
d
Wo
r
k
s and M
a
t
l
a
b Si
m
u
li
nk so
ft
wa
res are use
d
t
o
m
odel
an
d
check t
h
e r
o
bot
m
o
t
i
on
si
m
u
latio
n
.
2.
ROBOT KINEMATICS
2.1 Direct Kinematics
The
De
navi
t
-
H
a
rt
en
ber
g
(D
-H
)
param
e
t
e
rs fo
r SC
AR
A
r
o
b
o
t
sh
ow
n i
n
Fi
g
1
are
de
fi
ne
d i
n
Tabl
e
1.
Table 1. D-
H p
a
ram
e
ters
of
S
C
ARA
R
o
b
o
t.
L
i
nk
i
a
i
i
d
i
1
1
l
0 0
*
1
2
2
l
0 0
*
2
3 0
0
*
3
d
0
4 0
0
*
4
d
*
4
*
:
jo
in
t
v
a
riab
l
e
s
The e
x
pressi
on
f
o
r t
h
e e
n
d
ef
f
ect
or
fram
e
rel
a
t
i
v
e t
o
t
h
e
bas
e
fram
e
i
s
gi
ve
n
by
t
h
e a
r
m
m
a
t
r
i
x
(
0
4
T
) as:
00
1
2
3
41
2
3
4
TT
T
T
T
, w
h
e
r
e:
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I
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:
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56
IJRA Vol. 3, No. 4,
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ecem
ber 2014:
221 – 233
22
4
11
1
1
11
1
1
0
1
0
0
00
1
0
00
0
1
cs
l
c
s
cl
s
T
,
22
2
2
22
2
2
1
2
0
0
00
1
0
00
0
1
cs
l
c
s
cl
s
T
,
2
3
3
10
0
0
01
0
0
00
1
000
1
T
d
,
44
44
3
4
4
00
00
00
1
00
0
1
cs
sc
T
d
After m
u
ltip
lic
atio
n
an
d use
of ad
d
ition
m
a
trices,
on
e
g
e
ts t
h
e
h
o
m
o
g
e
n
e
ou
s tran
sform
a
ti
o
n
m
a
trix
;
124
1
24
1
1
2
1
2
12
4
1
24
1
1
2
1
2
0
4
34
0
0
00
1
00
0
1
cs
l
c
l
c
s
cl
s
l
s
T
dd
.
2.2 Inverse
Ki
nematics
2.
2.
1 In
verse
Sol
u
ti
on
f
o
r P
o
si
ti
o
n
s
Desi
re
d l
o
cat
i
o
n
of
R
o
bot
:
00
0
1
XX
X
X
yy
y
y
R
H
z
zz
z
no
a
p
no
a
p
T
no
a
p
The
fi
nal
e
q
uat
i
on
re
prese
n
t
i
n
g t
h
e
r
o
bot
i
s
[
3
]
:
0
4
R
H
TT
We ge
t:
11
2
1
2
X
p
lc
l
c
,
11
2
1
2
y
pl
s
l
s
.
B
y
usi
n
g Kram
er
m
e
t
hods we fi
n
d
[
4
]
;
Equation
of el
bow u
p
:
2
2
2
ta
n
s
a
c
,
22
1
2
2
1
12
2
2
2
()
tan
()
(
)
xy
xy
p
ls
p
l
l
c
a
p
ll
c
p
l
s
Equation
of el
bow d
o
wn:
2
2
2
ta
n
s
a
c
,
22
1
2
2
1
12
2
2
2
()
tan
()
(
)
xy
xy
p
ls
p
l
l
c
a
p
ll
c
p
l
s
Inverse
sol
u
ti
on for velocity:
12
12
1
12
Xy
Pc
P
s
ls
11
2
1
2
1
1
2
1
2
2
12
2
()
(
)
Xy
Pl
c
l
c
P
l
s
l
s
ll
s
Inverse
sol
u
ti
on for accelerati
on:
12
12
12
12
12
1
2
1
2
1
12
()
(
)
Xy
X
y
P
s
P
c
Pc
Ps
l
c
ls
11
1
1
2
1
2
2
2
12
2
()
(
)
yx
y
x
Ps
P
c
l
P
s
P
c
l
ll
s
2
1
1
1
1
12
12
2
1
2
1
2
2
2
12
2
()
(
)
yx
X
y
Pc
P
s
l
P
s
P
c
l
l
l
c
ll
s
3.
ROBOT DYNAMICS
We
fi
n
d
t
h
e
dy
nam
i
cs equat
i
o
ns
of
m
o
t
i
on o
f
r
o
b
o
t
s
by
t
w
o
m
e
t
hods:
Newt
on
-E
ul
er
and
Lag
r
a
nge
.
The
Newt
on
-E
ul
er m
e
t
hod i
s
m
o
re fu
ndam
e
nt
al
an
d fi
nds
t
h
e dy
nam
i
c
eq
u
a
tion
s
t
o
determin
e th
e requ
ired
act
u
a
tors’ fo
rce and
torq
u
e
t
o
m
o
v
e
the robo
t, as
well as th
e jo
in
t
forces.
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I
J
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9-4
8
5
6
Dyn
a
m
i
c
Be
ha
vi
or
of
a
SC
AR
A Ro
b
o
t
by
u
s
i
n
g
N
-
E
met
h
o
d
f
o
r
a
St
rai
ght
Li
ne …
(
B
r
ahi
m Fer
n
i
n
i
)
22
5
Lagra
n
ge m
e
tho
d
p
r
o
v
i
d
e
s
onl
y
t
h
e
req
u
i
r
ed
di
ffe
rent
i
a
l
eq
uat
i
o
ns t
h
at
det
e
rm
i
n
e t
h
e
act
uat
o
rs’
f
o
r
ce an
d
t
o
r
que
. [
5
]
The N-E
m
e
t
hod
i
s
base
d on t
w
o recu
rsi
o
ns fo
rwa
r
d
an
d ba
ckwa
r
d
rec
u
r
s
i
v
e
eq
uat
i
o
ns.
The
f
o
rwa
r
d
recursive
e
q
uation
is used for the
kinem
a
tics
inform
ation
such as
velocities and accelerat
ions at the ce
nt
er of
mass of
each link. T
h
e
bac
k
ward rec
u
rsi
v
e e
quation is
use
d
for t
h
e
forces
and m
o
m
e
nts exerte
d
on each link
from
the end effector to t
h
e
ba
se of the
robot.
Th
e
ro
tation
m
a
trices are as fo
llo
ws:
11
0
11
1
0
0
00
1
CS
RS
C
,
22
1
22
2
0
0
00
1
CS
RS
C
,
44
3
44
4
0
0
00
1
CS
RS
C
,
2
3
100
01
0
00
1
R
12
12
0
21
2
1
2
0
0
00
1
CS
RS
C
,
00
32
RR
,
124
12
4
0
4
1
24
124
0
0
00
1
CS
RS
C
,
11
1
01
1
0
0
00
1
CS
RS
C
22
2
12
2
0
0
00
1
CS
RS
C
,
12
1
2
2
01
2
1
2
0
0
00
1
CS
RS
C
32
00
RR
,
*
11
1
1
1
,,
0
T
Pl
C
l
S
,
*
21
1
2
1
1
2
,,
0
T
Pl
C
l
S
,
*
33
0,
0,
T
Pd
*
4
0,
0,
0
T
P
,
00
0
0
V
,
0
0,
0,
T
Vg
Forwar
d recursive:
11
01
0
0
0
1
1
00
1
T
RR
Z
22
1
02
1
0
1
0
2
1
2
00
1
T
RR
R
Z
33
2
03
2
0
2
1
2
00
1
T
RR
R
44
3
04
3
0
3
0
4
12
4
00
1
T
RR
R
Z
11
01
0
0
0
1
0
0
1
1
00
1
T
RR
Z
Z
22
1
1
02
1
0
1
0
2
0
1
0
2
12
00
1
T
RR
R
Z
R
Z
33
2
03
2
0
2
1
2
00
1
T
RR
R
44
3
3
04
3
0
3
0
4
0
3
0
4
12
4
00
1
T
RR
R
Z
R
Z
11
1
*
1
0
1
01
0
1
01
11
*
1
1
2
01
0
1
0
0
0
1
1
1
1
,,
T
RV
R
R
p
R
R
Rp
R
R
V
l
l
g
22
2
*
2
02
0
2
0
2
0
2
22
*
2
1
02
0
2
1
0
1
2
2
11
2
1
1
2
2
1
2
2
21
2
1
1
2
1
1
2
,
,
T
RV
R
R
p
R
RR
p
R
R
V
lS
l
C
l
ll
S
l
C
g
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:
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IJRA Vol. 3, No. 4,
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221 – 233
22
6
33
2
3
3
*
03
2
0
3
0
2
0
3
0
3
33
03
2
0
3
33
3
*
02
0
3
0
3
32
3
3
*
20
2
0
3
0
3
33
3
*
03
03
0
3
2
2
11
2
1
1
2
2
1
2
2
11
2
1
1
2
2
1
2
2
,
,
T
RV
R
Z
R
V
R
R
p
RR
Z
RR
R
p
RR
V
R
R
p
RR
R
p
lS
l
C
l
lC
l
S
l
g
44
4
*
04
0
4
0
4
44
4
*
04
04
0
4
43
30
3
2
2
1
1
24
1
1
24
2
1
2
4
1
2
4
2
2
11
2
4
11
2
4
2
1
2
4
1
2
4
,
,
T
RV
R
R
p
RR
R
p
RR
V
lS
l
C
l
C
S
lC
l
S
l
S
C
g
The position
of center
of mass:
11
1
1
1
/2
,
/
2
,
0
T
el
C
l
S
22
1
2
2
1
2
/2
,
/
2
,
0
T
el
C
l
S
33
0,
0,
/
2
T
ed
4
0,
0
,
0
T
e
11
1
1
1
1
0
1
01
0
1
01
0
1
0
1
11
2
01
0
1
1
1
1
1
/2
,
/
2
,
T
R
aR
R
e
R
R
R
e
RV
R
V
l
l
g
22
2
2
2
2
0
2
02
0
2
02
02
0
2
22
02
02
2
2
11
2
1
1
2
2
1
2
2
11
2
1
1
2
2
1
2
/2
,
/2
,
T
R
a
R
Re
R
R
Re
RV
RV
lS
l
C
l
lC
l
S
l
g
33
3
3
3
3
0
3
03
0
3
03
0
3
0
3
33
03
0
3
2
2
11
2
1
1
2
2
1
2
2
11
2
1
1
2
2
1
2
,
,
T
R
a
R
Re
R
R
Re
RV
R
V
lS
l
C
l
lC
l
S
l
g
44
4
04
0
4
0
4
44
4
04
0
4
0
4
44
04
0
4
2
2
11
2
4
11
2
4
2
1
2
4
1
2
4
2
2
1
1
2
4
1
1
2
4
212
4
1
2
4
,
,
T
Ra
R
R
e
RR
R
e
RV
RV
lS
l
C
l
C
S
lC
l
S
l
S
C
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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SN
:
208
9-4
8
5
6
Dyn
a
m
i
c
Be
ha
vi
or
of
a
SC
AR
A Ro
b
o
t
by
u
s
i
n
g
N
-
E
met
h
o
d
f
o
r
a
St
rai
ght
Li
ne …
(
B
r
ahi
m Fer
n
i
n
i
)
22
7
Backwar
d
rec
ursive:
We have:
55
0
fn
44
5
4
4
04
5
0
5
4
0
4
40
4
2
2
11
2
4
11
2
4
2
1
2
4
1
2
4
4
2
2
11
2
4
11
2
4
2
1
2
4
1
2
4
,
,
T
Rf
R
R
f
m
R
a
m
R
a
lS
l
C
l
C
S
m
lC
l
S
l
S
C
g
33
4
3
03
4
0
4
3
0
3
2
2
11
2
1
1
2
2
1
2
34
2
11
2
1
1
2
2
1
2
,
,
T
Rf
R
R
f
m
R
a
lS
l
C
l
mm
lC
l
S
l
g
22
3
2
02
3
0
3
2
0
2
2
2
11
2
1
1
2
2
1
2
2
11
2
1
1
2
2
1
2
,
,
T
Rf
R
R
f
m
R
a
xl
S
l
C
y
l
x
l
C
l
S
yl
xg
11
2
1
01
2
0
2
1
0
1
2
2
11
1
2
1
2
2
12
2
11
1
2
1
21
2
2
1
2
2
/2
,
/2
,
T
Rf
R
R
f
m
R
a
lx
m
y
l
C
S
lx
m
x
mg
yl
S
C
23
4
x
mm
m
23
4
/2
ym
m
m
The
moments exerte
d
on the
links:
44
5
5
*
5
04
5
0
5
0
4
0
5
4*
4
4
4
04
0
4
4
0
4
4
0
4
44
04
4
0
4
Rn
R
R
n
R
p
R
f
Rp
R
e
m
R
a
J
R
RJ
R
0
0
i
ii
i
JR
I
R
1,
2
,
3
,
4
i
Gene
rally
, the
m
a
ss and t
h
e l
e
ngt
h
of lin
k
(
4
) a
r
e
very
sm
all in com
p
arison to
othe
r links; the i
n
ertia
of link
(4
) is e
v
aluate
d to
be
zer
o.
4
04
0
Rn
33
4
4
*
4
03
4
0
4
0
3
0
4
3*
3
3
3
03
0
3
3
0
3
3
0
3
32
4
33
03
3
0
3
33
2
11
2
1
1
2
2
1
2
2
2
12
2
1
1
2
2
1
2
/2
,
,0
T
Rn
R
R
n
R
p
R
f
Rp
R
e
m
R
a
J
R
dm
m
RJ
R
ml
lC
l
S
l
lS
l
C
l
Evaluation Warning : The document was created with Spire.PDF for Python.
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56
IJRA Vol. 3, No. 4,
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221 – 233
22
8
22
3
3
*
3
02
3
0
3
0
2
0
3
2*
2
2
2
02
0
2
2
0
2
2
0
2
22
02
2
0
2
2
11
2
1
1
2
2
1
2
2
2
12
2
1
1
2
2
1
2
2
22
21
1
2
1
1
2
2
1
2
,,
T
Rn
R
R
n
R
p
R
f
Rp
R
e
m
R
a
J
R
RJ
R
lC
l
S
l
lS
l
C
l
l
g
y
ly
l
C
l
S
l
23
4
3
3
4
3
3
/2
dm
m
d
m
m
m
l
23
4
/3
mm
m
11
2
2
*
2
10
0
2
0
1
0
2
1*
1
1
01
0
1
10
1
11
1
10
1
0
1
1
0
1
11
2
1
2
2
22
2
21
2
2
2
2
11
2
1
2
2
2
1
2
2
22
1
22
11
2
1
2
,
/2
,
/3
2
Rn
R
R
n
R
p
R
f
Rp
R
e
m
R
a
JR
R
J
R
ll
C
lg
y
S
lS
ll
S
l
C
lg
x
y
C
m
lx
m
l
l
l
21
2
21
2
2
2
1
2
2
2
1
2
2
Cy
ll
l
C
y
l
l
S
y
The jo
int to
rque of
link (1
):
2
11
11
10
1
0
0
1
2
21
2
2
2
21
2
2
2
1
2
2
2
1
2
/3
2
2
T
lx
m
Rn
R
Z
ll
l
C
y
ll
l
C
y
l
l
S
y
The jo
int to
rque of
link (2
):
22
2
2
2
22
1
0
1
2
2
2
1
2
2
1
2
2
1
T
R
n
R
Z
l
l
yC
l
l
l
l
yS
The force e
x
e
r
ted
on the
link (3):
33
30
3
2
0
3
4
T
eff
FR
f
R
Z
m
m
g
m
g
4.
APPLI
CATI
O
N
:
C
onsi
d
er a res
t
-to-
rest C
a
rtesian path f
r
o
m
poi
nt
(1
.5
, 1
)
to poi
nt (1
.5
,-
1)
on strai
ght
line x=1
.
5
du
rin
g
1s
with
12
1
ll
.A cubic polynom
ial can sat
i
sfy the
positio
n and vel
o
city constr
aints at initial and
final poi
nts.
0
(0
)
1
yy
0
(0
)
0
yy
(1
)
1
f
yy
(1
)
0
f
yy
Th
e co
eff
i
cien
ts of
th
e po
lynomial ar
e:
0
1
a
1
0
a
2
6
a
3
4
a
;
The Cartesian
path is :
23
()
1
6
4
yt
t
t
1.
5
x
The traj
ect
ory
si
m
u
lation:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN:
208
9-4
8
5
6
Dynamic Beh
a
vio
r
o
f
a S
C
ARA Robo
t b
y
usin
g N-
E
met
h
o
d
for
a
Strai
ght
Line …
(
B
r
ahi
m Fer
n
ini)
22
9
For
the
traje
c
to
ry
sim
u
lation
we
use el
bo
w
do
w
n
a
n
d
elb
o
w
up
.
The
figure shows the sim
u
lati
on bl
ock to
sim
u
la
te th
e trajectory
by
in
ver
s
e ki
nem
a
tics of
SC
AR
A
ro
b
o
t.
Figu
re
3.
The
traject
ory
ge
ner
a
tion
of
a SC
A
R
A r
o
bot
with
M
a
tlab/Sim
u
link
by
usi
n
g
in
verse
ki
nem
a
tics
Trajec
tor
y
si
mulati
on:
Elbow down
Figu
re 4.
M
a
tlab/Sim
u
link
Figu
re
5.
S
o
lidw
o
r
k
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
089
-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
221 – 233
23
0
Elbow up
Figu
re 6.
M
a
tlab/Sim
u
link
Figu
re 7.
S
o
lid
wo
rk
s
The Tra
j
ect
ory
obtaine
d
whe
t
her by
usin
g
Solid
Wo
rk
s o
r
by
M
A
TL
A
B
/Sim
ulink is exactly
the
sam
e
(a straight line), so t
h
e
position constr
ai
nt is
verified at
initial and final poi
nts.
The jo
int v
e
locit
y
o
f
the ro
bo
t by
M
a
t
l
a
b
/Simlink:
Figure
8. The joint
velocity(1)
Figure
9. The joint
velocity(2)
The angul
a
r
velocity of
the l
i
nks of r
o
b
o
t
by
Solidwor
k
s
:
Elbow down
Fig
u
r
e
10
.
Link
(
1
)
Fig
u
r
e
11
.
Link
(
2
)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-2
.
5
-2
-1
.
5
-1
-0
.
5
0
0.
5
ti
m
e
(
s
e
c
)
j
o
i
n
t
ve
l
o
ci
t
y
(
r
a
d
/
se
c)
el
b
o
w
do
w
n
el
b
o
w
up
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
ti
m
e
(
s
e
c
)
jo
in
t
v
e
lo
c
i
t
y
(
r
a
d
/
s
e
c
)
el
bo
w
d
o
wn
el
bo
w
u
p
0.
00
0.
10
0.
20
0
.
30
0.
40
0.
50
0
.
60
0.
7
0
0.
80
0.
90
1.
00
ti
m
e
(
s
e
c
)
0
33
66
99
131
angu
l
a
r
v
e
l
o
c
i
t
y
(
deg/
s
e
c
)
0.
00
0.
10
0.
20
0.
30
0.
40
0.
50
0.
60
0.
70
0.
80
0.
90
1.
00
t
i
m
e
(
s
ec
)
0
33
66
99
131
a
ngul
ar
vel
o
ci
t
y
(
d
eg/
sec)
Evaluation Warning : The document was created with Spire.PDF for Python.