Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
V
o
l.
3, N
o
. 3
,
Sep
t
em
b
e
r
2014
, pp
. 18
4
~
19
0
I
S
SN
: 208
9-4
8
5
6
1
84
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
A New Meth
od for Time-Jerk Op
timal Trajectory Planning
Under Kino-dynamic Constraint
of Robot Manipulators in
Pick-and
-Place Operation
s
Bendali
Nadir*,
Ouali Mohammed*, Ki
fouche Abdess
alam**
* S
t
ructur
al
M
e
c
h
anics
R
e
s
ear
ch
Labora
t
or
y,
Dep
a
rtm
e
nt of
M
ech
anic
al
Engin
eeri
ng, Univers
i
t
y
B
lida
,
Alg
e
ria
** IBISC, Univ
ersity
Val d’Essonne,
Evr
y
, France
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 10, 2014
Rev
i
sed
May 10
, 20
14
Accepte
d
J
u
n 3, 2014
A new
method for time-jer
k
optim
al planning under Kino-d
y
namic
constrain
t
s of ro
bot manipulators
in p
i
ck-
a
nd-place oper
a
tions
is d
e
scribed
in
this paper
.
In or
der to
ensure th
at the r
e
sulting
tr
ajector
y
is sm
ooth enough, a
cost function
co
ntaining
a term proportional to the integr
al of the squared
jerk (def
ined as
the deriv
a
tive
of th
e acceler
a
tion) along the
tr
ajector
y
is
considered
. Moreover, a second
term
, proportio
nal to the total execution
time, is added
to the expr
ession of
the cost function. A C
ubic Spline
functions ar
e th
en used to compose overa
ll
t
r
a
j
ec
tory
.
T
h
i
s
me
t
h
od ma
ke
s i
t
possible to deal
with the kine
m
a
tic constra
i
nt
s as well as the d
y
n
a
m
i
c
constrain
t
s impo
sed on the robot manipul
ator. Th
e algorithm has
been tested
in simulation
y
i
elding good
results.
Keyword:
Cubic s
p
lines
Fi
ft
h key
w
or
d
K
i
no
-d
yn
am
ic co
nstr
ain
t
s
M
i
nim
i
zati
on of
je
rk
R
o
b
o
t
m
a
ni
pul
at
ors
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
:
B
e
ndal
i
Nadi
r,
Struct
ural Mec
h
anics
Researc
h
La
boratory,
Depa
rt
m
e
nt
of
M
echani
cal
E
n
gi
nee
r
i
n
g,
U
n
i
v
ersi
t
y
Saa
d
D
a
hl
ab B
l
i
d
a
,
B
P
2
7
0
r
o
ut
e
de
soum
aa, Al
geri
a.
Em
a
il: n
a
d
i
r_b1
102
@yah
oo
.co
m
1.
INTRODUCTION
Th
e
d
e
term
in
atio
n
o
f
t
h
e time-j
e
rk
o
p
timal traj
ectory plan
n
i
ng
fo
r m
a
n
i
pu
lato
rs is
an
im
p
o
r
tan
t
pr
o
b
l
e
m
i
n
ro
b
o
t
t
r
aject
ory
pl
anni
ng
. Li
m
i
t
i
ng t
h
e je
r
k
i
s
very
i
m
port
a
nt
, beca
use
hi
g
h
jer
k
val
u
es ca
n
wear
out
t
h
e r
o
bot
st
ruct
u
r
e, a
nd
heavi
l
y
exci
t
e
i
t
s
resona
nce
freq
u
e
n
ci
es;
vi
b
r
at
i
ons i
n
d
u
ced
by
no
n
-
s
m
oot
h
tr
aj
ector
ies can
d
a
m
a
g
e
th
e
r
obo
t actu
a
tor
s
, and in
trod
u
c
e lar
g
e erro
r
s
w
h
ile t
h
e
r
obot is p
e
r
f
o
r
m
i
n
g
task
s
suc
h
as traject
ory trac
king
more
over low-je
rk tra
j
ectories
can be e
x
ec
ut
ed m
o
re rapi
dl
y
and acc
urat
el
y
.
Al
so
,
d
ecreased th
e ex
ecu
tion
ti
me o
f
th
e task
is
v
e
ry
imp
o
rtan
t t
o
i
n
crease t
h
e
p
r
od
u
c
tiv
ity
o
f
th
e
robo
t
m
a
ni
pul
at
o
r
s;
t
h
i
s
can be t
h
e case of t
h
e
han
d
l
i
ng
of
ob
ject
s, t
h
e p
o
i
n
t
-
t
o
-
poi
nt
wel
d
i
ng
or t
h
e i
n
st
al
l
a
t
i
on
o
f
t
h
e el
ect
ro
ni
c c
o
m
pone
nt
s.
So
m
e
of t
h
ese a
p
pl
i
cat
i
ons
re
qu
i
r
e t
h
e
use
o
f
a
t
r
aject
ory
pl
an
ner
t
h
at
y
i
el
ds,
fo
r
a
g
i
v
e
n
p
e
rform
a
n
ce criterion
,
op
ti
m
a
l o
r
n
ear-o
p
tim
al so
lu
tio
n
s
wh
ile con
s
id
eri
n
g fu
ll d
yna
m
i
cs [1
].
M
a
ny
w
o
r
k
i
n
t
h
e fi
el
d
o
f
r
o
b
o
t
i
c
s has
been
dev
o
t
e
d
t
o
t
h
e
st
udy
of
t
h
e
p
r
obl
em
of m
o
t
i
o
n
pl
a
nni
n
g
,
we ci
t
e
i
n
t
h
i
s
cont
e
x
t
t
h
e
w
o
rk
o
f
[
2
]
T
h
e a
u
t
h
ors
ha
ve t
r
e
a
t
e
d t
h
e
p
r
o
b
l
e
m
of t
r
aject
ory
pl
an
ni
n
g
o
f
r
o
bot
m
a
ni
pul
at
o
r
i
n
im
posed t
a
sk
s by
con
s
i
d
e
r
i
ng t
h
e ki
nem
a
t
i
c
const
r
ai
nt
s
,
and t
o
o
p
t
i
m
i
ze t
h
e cost
f
u
nct
i
o
n
wh
ich
represents a wei
g
h
ting
b
e
tween
th
e execu
tio
n tim
e o
f
th
e task
and
t
h
e in
terv
al squared j
e
rk th
ey
were
use
d
t
h
e
se
que
nt
i
a
l
q
u
ad
rat
i
c
pr
o
g
ram
m
i
ng
f
unct
i
o
n.
I
n
[
3
]
a ne
w a
p
proac
h
called interval analysis is
used t
o
devel
o
p an al
gorithm
that minimizes
th
e
m
a
x
i
m
u
m
ab
so
lu
te
v
a
lu
e
of j
e
rk along t
h
e trajectory, the cubic
spl
i
n
es
were
u
s
ed t
o
re
pre
s
e
n
t
t
h
e t
r
a
j
ect
or
y
im
posed t
a
s
k
s;
t
h
i
s
pr
o
b
l
e
m
i
s
sol
v
e
d
w
i
t
hout
c
o
nsi
d
e
r
ed t
h
e
dy
nam
i
cs of t
h
e ro
bot
.
In
[4]
t
h
e aut
h
o
r
s p
r
o
pos
ed a m
e
t
hod base
d
on P
S
O
t
o
o
p
t
i
m
i
z
e the cost
f
u
nct
i
o
n use
d
in
[3
] cu
b
i
c
splin
es were
u
s
ed
to in
terpo
l
ate b
e
t
w
een
the
nodes
of t
h
e trajectory in
an
im
posed
t
a
sk
of
t
h
e
robo
t. In
[5
] the au
thors
u
s
ed
th
e prin
ci
p
l
e of Pon
t
ryag
in
t
o
o
p
tim
ize th
e co
st fun
c
tio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
3, N
o
. 3,
Se
pt
em
ber 20
1
4
:
18
4 – 19
0
1
85
Th
is wo
rk
is stru
ctured
as
follo
ws: Sectio
n
2
pres
en
ts th
e ap
pro
ach
tak
e
n
in
[1
] to
refo
rm
u
l
ate th
e
vari
ous c
o
nst
r
ai
nt
s. Sect
i
o
n 3 p
r
ese
n
t
s
t
h
e
ref
o
rm
ul
at
i
on
of t
h
e c
o
st
f
u
n
c
t
i
on. Sect
i
o
n
4 co
nsi
d
e
r
s t
h
e
C
ubi
c
Spl
i
n
e
fu
nct
i
o
n
s
use
d
t
o
m
ode
l
t
h
e di
ffe
re
nt
segm
ent
s
. Sect
i
on
5
prese
n
t
s
t
h
e t
ech
ni
q
u
e o
f
ge
net
i
c
al
go
ri
t
h
m
s
.
The S
ect
i
o
n
6
prese
n
t
s
t
h
e re
capi
t
u
l
a
t
i
o
n
o
f
t
h
i
s
m
e
t
hod.
The
di
f
f
ere
n
t
r
e
sul
t
s
o
b
t
a
i
n
e
d
an
d t
h
e
co
ncl
u
si
o
n
s
are
prese
n
t
e
d
i
n
Sect
i
o
n
7 a
n
d
8
usi
n
g a
pl
a
n
ar
r
o
b
o
t
m
a
ni
pul
at
o
r
.
2.
TREAT
M
ENT OF THE
KINO
-D
YN
A
M
I
C
CO
NSTR
A
I
NTS
To
so
lv
e th
e
p
r
ob
lem
o
f
trajectory planni
ng i
n
the free
tasks
for a
n
optim
al
trajectory Q(T
)
,
we
carried out a st
anda
rdization
of t
h
e tim
e sca
l
e that trans
f
o
r
m
s
t
h
e pr
o
b
l
e
m
of a researc
h
on a
n
i
n
t
e
r
v
a
l
of a
n
i
nde
fi
ni
t
e
t
e
rm
i
n
al
[0,
T
]
t
o
w
a
rds a
not
her
b
e
i
ng e
qui
val
e
n
t
who
s
e t
e
rm
inal
of
researc
h
i
s
kn
o
w
n
[0
,
1
]
and
who
will b
e
easier to
so
l
v
e.
qt
Q
t
O
r
:
tt
T
wit
h
:
0,
1
(1
)
We ca
n b
r
ea
k
up t
h
i
s
pr
ofi
l
e
of t
r
aject
ory
i
n
t
o
a way
P
an
d
m
ovem
e
nt
on t
h
i
s
way
, w
h
ich
will b
e
form
u
l
ated
as
fo
llowi
n
g
:
QP
(2
)
Ap
pl
y
i
ng t
h
e n
o
rm
al
i
zati
on (
1
)
of t
i
m
e s
cale for a give
n
generalized tra
j
ectory
qt
, the generalized
v
e
lo
cities
qt
, generalized
accele
r
ation
qt
and jerk
qt
of
th
e
trajecto
r
y can
b
e
written
as
fo
llo
ws:
23
11
1
qt
Q
,
qt
Q
a
n
d
q
t
Q
TT
T
(3
)
2.
1.
T
r
e
a
t
men
t of Ki
nem
a
ti
c
C
o
ns
trai
n
t
s
From
t
h
e e
quat
i
on
(
3
) t
h
e
ki
ne
m
a
t
i
c
const
r
ai
n
t
s can
be
fo
rm
ul
at
ed as
f
o
l
l
o
wi
n
g
:
Velocit
y
.
i
max
ii
ma
x
i
1
,
...,
n
0
,
1
i
Q
t0
,
T
;
q
t
q
T
m
a
x
m
a
x
q
O
r
:
V
TT
(4
)
Acceleration.
1
2
i
max
ii
ma
x
i
1
,
...,
n
0
,
1
i
Q
t0
,
T
;
q
t
q
T
m
a
x
m
a
x
q
Or:
A
TT
(5
)
Jerk.
1
2
i
max
ii
ma
x
i
1
,
...,
n
0
,
1
i
Q
t0
,
T
;
q
t
q
T
m
a
x
m
a
x
q
O
r
:
J
TT
(6
)
For
a
give
n tra
j
ectory
pr
ofile
Q
, the optim
al tim
e
Q
T
m
u
st satisfy
the
kinem
a
tic const
r
aints is:
*
Q
TT
Wi
t
h
:
11
23
ii
i
*
ma
x
m
a
x
ma
x
i1
,
.
.
.
,
n
0
,
1
ii
i
QQ
Q
Tm
a
x
m
a
x
,
,
qq
q
(7
)
2.
2. T
r
e
a
t
men
t
of
the
D
y
n
a
mi
c C
o
nst
r
ai
n
t
s
The equation of the dy
nam
i
c
m
odel of
robot is written:
n
ii
j
j
i
j
i
1
t
M
qt
q
t
C
q
t
,
qt
G
q
t
(8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN:
208
9-4
8
5
6
A N
e
w
Met
h
o
d
f
o
r Ti
me-Je
r
k
Opt
i
m
al
T
r
aj
ec
t
o
ry Pl
a
n
n
i
n
g
Un
der K
i
no
-
d
y
n
a
m
i
c
…
(
B
e
n
dal
i
N
adi
r)
18
6
Whe
r
e
ij
M
is
the inertia m
a
trix,
i
C
is the vector
of
C
o
rio
lis and centrifugal forces,
i
G
is the vector
of potential forces and
i
is the
vector
of actuat
o
r efforts.
By using the e
quation (1) a
nd the equations
of the
velocity and the accele
r
ations (3) in the equations
of
the
dy
nam
i
c m
odel (8
)
we
obtain:
ii
i
2
1
hG
T
Wi
t
h
:
ii
i
ii
i
ma
x
m
a
x
ma
x
ii
i
hG
;h
;
G
(9
)
The s
o
lutio
n
of
the eq
uatio
n (
9
) c
o
m
p
ared t
o
Tfo
r
a
give
n
v
a
lue
gives a
n
acceptable inte
rval for T
of lower li
m
i
t
i
L
T
and
higher limit
i
R
T
or
ii
LR
TT
,
T
, and the intersection
of the i
n
terval
s
ii
LR
T,
T
for
i
1
,
.
..,
n
along the
trajectory gi
ve ac
ceptable inte
rval
for the tim
e of
displacem
ent respecting dy
nam
i
c
constraints
due to the torques:
gd
TT
,
T
Wit
h
:
ii
gL
d
R
i
1
,...,
n
0
,
1
i
1
,
.
..,
n
0
,
1
T
m
ax
max
T
;
T
mi
n
m
i
n
T
(1
0)
If we de
note
ad
I
the interval durations T
which satis
fy all kino-dynam
ic constraints:
*i
n
f
s
u
p
a
d
g
d
ad
ad
IT
,
T
,
T
I
,
I
(1
1)
3.
REFO
RM
UL
ATIO
N O
F
T
H
E C
O
ST F
U
NCTI
O
N
The c
o
st function i
n
our case represe
n
ts a
weigh
ting
between the tim
e
tran
sfer and t
h
e Je
rk, its
form
ula is written:
f
T
2
n
fi
i1
0
JT
1
q
t
d
t
(1
2)
Wi
t
h
is
a wei
ght coe
fficient change
bet
w
ee
n 0
a
n
d
1 acc
o
r
di
ng
to t
h
e
us
er
need
s ca
n
fa
vo
r eithe
r
the execution t
i
m
e
of the task
is the
j
e
rk. Using equation
(3
)
,
the e
q
uatio
n (
1
2
)
bec
o
m
e
s:
2
1
n
fi
3
i1
f
0
1
JT
1
Q
d
T
(1
3)
Wi
t
h
:
2
32
2
3
i
32
2
3
d
Q
d
d
Qd
d
d
Qd
Q3
dd
d
dd
d
d
The tim
e of
displacem
ent
m
T
whi
c
h m
i
nim
i
zes the c
o
st f
u
nctio
n
fo
r the
p
r
ofile
Q
is:
2
7
m
1
S
T6
S
Wi
t
h
:
2
1
n
12
i
i1
0
S;
S
(
1
)
Q
d
(1
4)
4.
MODELING THE
FUNC
TIONS OF THE WAY
AN
D THE
MO
VE
MENT
We chose a m
odel by Cubic Spline functions,
these
functi
ons a
r
e com
posed
by pieces of
polynom
i
als of three degree,
and whic
h
will be written
usi
n
g the standardi
zation of t
h
e ti
m
e
of equation (1)
as followi
ng:
23
i
0
i
1
i
i
1
2
ii
1
3
ii
1
Qa
a
a
a
F
o
r:
i1
i
(1
5)
And t
h
e
deri
vatives of the
j
o
i
n
t variation
i
qt
are:
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I
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56
IJR
A
V
o
l. 3, N
o
. 3,
Se
ptem
ber 20
1
4
:
18
4 – 19
0
1
87
2
i
1
i
2
ii
1
3
ii
1
i2
i
3
i
i
1
2
i3
i
3
1
qt
a
2
a
3
a
T
1
q
t
2a
6a
T
1
qt
6
a
T
4.
1. Func
ti
on
of
the
W
a
y
This
function
is m
odeled by
Cubic Splines ca
lled Natural
(Figure 1),
the boundary
conditions
i
m
posed on t
h
e joint
positions
of
th
e ro
bo
t ma
nipulator are:
in
i
f
i
n
P0
q
a
n
d
P
1
q
(1
6)
Fig
u
r
e
1
.
Represen
tatio
n on
t
h
e
profile
of the path with
C
N
2
p
o
i
n
ts of
c
o
ntr
o
l
4.
2. Func
ti
on
of
the
M
o
vem
e
nt
We ha
ve ad
op
ted to rep
r
ese
n
t the pr
ofiles
of this fu
ncti
on
by
C
ubic S
p
lines called C
l
am
ped this
m
odel is well adopted to take
the bound
ary
conditions of ve
locities (Figure 2).
01
0
(1
7)
In addition, thi
s
profile of m
ovem
e
nt is composed by
poin
t
s
of cont
rol
placed i
n
a standardized
plan
so t
h
at the
first and the
last point a
r
e
fixe
d acc
ording
t
o
(18), while the interi
or points
are placed
freely
according t
o
the conditions
(17) and
(19):
00
a
n
d
1
1
(1
8)
0
(1
9)
Figu
re
2.
R
e
p
r
esentation
o
n
t
h
e
pr
ofile
of
th
e m
ovem
e
nt w
ith
C
N2
poi
nts of
c
o
ntr
o
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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S
SN:
208
9-4
8
5
6
A N
e
w
Met
h
o
d
f
o
r Ti
me-Je
r
k
Opt
i
m
al
T
r
aj
ec
t
o
ry Pl
a
n
n
i
n
g
Un
der K
i
no
-
d
y
n
a
m
i
c
…
(
B
e
n
dal
i
N
adi
r)
18
8
5.
OPTIMIZ
A
T
I
ON US
ING ALGO
RITH
MS
Th
e
u
s
e of
a
gen
e
tic algo
r
ithm
star
ts with
th
e cr
eatio
n
o
f
an
i
n
itial p
opulatio
n
o
r
chr
o
m
o
so
m
e
in
genetics,
this c
h
r
o
m
o
som
e
is com
posed
by
gene
s
or t
h
eir
num
ber is
de
fined
acc
or
din
g
to the
n
u
m
b
er
of
th
e
poi
nts o
f
c
ont
r
o
l use
d
t
o
ge
n
e
rate the
fu
nct
i
on
way
an
d at
the sam
e
tim
e
the f
u
nction
m
ovem
e
nt. Th
e steps
use
d
in our calculation
code a
r
e as
followi
ng:
1) B
e
gi
nni
n
g
.
Selected chrom
o
som
e
num
ber, chrom
o
som
e
size, the probability of
crossover
and the
probability of
m
u
ta
tion.
2
)
Initi
a
liza
t
io
n.
Ge
ne
rate a ra
nd
om
po
pulati
o
n
o
f
n
c
h
r
o
mo
s
o
me
s
.
3)
C
o
st
f
uncti
on.
Calcu
l
ate
Q
JT
ofeachc
h
rom
o
som
e
.
4)
New
gener
a
ti
on
.
C
r
eate a
new
p
o
pulatio
n
by
re
peatin
g
the f
o
llo
win
g
g
e
netic o
p
erat
or
s:
Selection.
Chrom
o
som
e
s wit
h
best cost functi
on
have the
ability to select m
o
re.
Cros
sing
.
C
r
e
a
te two c
h
ild
re
n
by
m
a
king a
m
i
xture
of c
h
r
o
m
o
som
e
s fro
m
both pa
re
nts.
Mu
ta
tion
.
T
h
i
s
o
p
erat
or
allows
the
em
ergence
o
f
ne
w
ge
nes
by
e
x
p
l
orin
g a
r
eas
o
f
the
searc
h
space t
h
at coul
d
not
be
visited by a
sim
p
le
application
of the
crossing
opera
t
or.
5)
T
e
st.
If the initial cond
itions are
satisfied,
stop go t
o
st
ep
6, i
f
not
go to step 3.
6)
G
oal
.
Obtain the m
i
ni
m
a
l value
of
QQ
JT
.
6.
R
E
CA
P
I
TU
LA
TI
ON
OF
TH
E M
ETHOD
OF
R
E
S
O
LU
TI
ON
To see
k
the optim
a
l traj
ectory, we m
u
st generate by c
h
a
n
ce
accordi
ng
to the ge
netic technique
of
optim
izat
ion
of the al
gorithm
s
a prof
ile
of way and a
profile of m
oveme
nt which
will give
us thereafter a
profile
of traj
ectory (Figure
3), candi
date the latter will be
evaluated therea
fter and com
p
ared
with other,
this
operation is
repeated for all the intr
o
duce
d
ch
rom
o
som
e
s, an
d t
h
e
best
re
sult, it is t
h
at which satisfi
es the
give
n c
r
iterio
n
co
nve
r
g
ence
.
It s
h
o
u
ld
be
note
d
that
a
n
y
pr
o
f
ile o
f
w
a
y
whic
h
wo
ul
d
violate o
n
e
of t
h
e
geom
etrical constraints, as any profile
of tra
j
ectory wh
ich
wo
uld
vi
olate
one
o
f
t
h
e c
o
n
s
traints
kinem
a
tics or
dynam
i
cs will be autom
a
tical
l
y
rej
ected.
Figu
re
3.
Flo
w
ch
art of Resolution
7.
RESULTS OF SIMUL
A
TION
We will consi
d
er a planar robot 3R, we ask him
to carry out a displace
m
e
nt between the initial
configurations
T
in
it
q0
,
3
,
1
0
to the final confi
g
urations
T
fi
n
q,
0
,
0
.We will fix the rate or the
probability of
crossi
ng equa
l
to 65% and the probability of m
u
tation
equal 4%. The col
l
ect
ed results and the
optim
al aspects f
o
r t
h
e m
ove
m
e
nt of
a
plan
ar r
o
bot
3R
ar
e r
e
p
r
esen
ted r
e
sp
ectiv
ely in
Fig
u
r
e
4
an
d Figu
r
e
5.
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I
S
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-48
56
IJR
A
V
o
l. 3, N
o
. 3,
Se
ptem
ber 20
1
4
:
18
4 – 19
0
1
89
Table
1. T
h
e
P
a
ram
e
ters of Pl
anar
R
o
bot
3R
Seg
m
ent i
σ
i
L
i
α
i
d
i
θ
i
m
i
[K
g
]
x
gi
[m
]
I
zi
[K
g
.
m
2
]
ݍ
௫
[r
a
d
]
ݍ
ሶ
௫
[
r
ad/s
]
ݍ
ሷ
௫
[
r
ad/s
2
]
ݍ
ഺ
௫
[
r
ad/s
3
]
߬
௫
[N
.
m
]
1 0
0.
7
0
0
q
1
7
0.
35
0.
8
π
3
8
10
30
2 0
0.
5
0
0
q
2
5
0.
5
0.
5
3
π
/4 3
8
15
25
3 0
0.
5
0
0
q
3
5
0.
5
0.
5
3
π
/4 3
8
20
25
Figure
4. Resul
t
s Positions, Velocities, Acce
lerations
, Jerk,
Torques, and successi
ve
configurations
for a
trajectory
opti
m
ized of Plana
r
Robot 3R
Figu
re
5.
O
p
tim
al aspects f
o
r
a m
ovem
e
nt of
planar
r
o
bot
3R
8.
CO
NCL
USI
O
N
Through this
work, we could show
the
possibility of this approach
of giving us
the sub-optim
al
results f
o
r t
h
e
pr
oblem
s of optim
al
trajectory
pla
nni
ng
of the
ro
b
o
t m
a
nipulato
r
s
by
m
i
nim
i
zing
a cost
-1
0
1
-1
0
1
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
X
Y
Z
Ro
b
o
t3
R
x
y
z
t0
-1
0
1
-1
0
1
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
X
Y
Z
Ro
b
o
t3
R
x
y
z
Q
1
t.
T
2
-1
0
1
-1
0
1
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
X
Y
Z
Ro
b
o
t3
R
x
y
z
Q
tT
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I
J
RA I
S
SN:
208
9-4
8
5
6
A N
e
w
Met
h
o
d
f
o
r Ti
me-Je
r
k
Opt
i
m
al
T
r
aj
ec
t
o
ry Pl
a
n
n
i
n
g
Un
der K
i
no
-
d
y
n
a
m
i
c
…
(
B
e
n
dal
i
N
adi
r)
19
0
fu
nctio
n
whic
h
prese
n
ts Ti
m
e
/Jerk a
n
d
with im
posed
Kin
o
-
dy
nam
i
c co
nstrai
nts.
We c
oul
d
dea
l
with a
problem
of trajectory pla
nni
ng in
Pi
ck a
n
d
Place operations
by usi
ng C
ubic Splines
functions who allow
to
guarantee the s
m
oothing of the traj
ect
ory and at the sam
e
ti
me
the c
ontinuity of the
velocities, the
accelerations, and Je
rk for a Planar
Robot
m
a
nipulator
3R, noting that
the tim
e
execution
of the
code
com
puter re
q
u
i
r
es
34
sec
o
n
d
s
by
usi
n
g
10
0 c
h
r
o
m
o
som
e
s in ge
netic alg
o
rithm
s
and
o
n
a
P
C
of
2
G
hz.
ACKNOWLE
DGE
M
ENTS
We
tha
nk Pr
of
H.E. Lehtihet fo
r
his usef
ul
s
u
g
g
es
tio
ns a
n
d
his assistance
and
his a
dvice
du
rin
g
thi
s
work.
REFERE
NC
ES
[1]
M. Haddad,
et a
l
.,
“Trajector
y
Planning of Unicy
c
le Mobile R
obots With a T
r
apezo
idal-V
elo
c
ity
Constrain
t
”,
IEEE Transactio
ns On Robotics,
vol. 26
, no
5, Oct 2010.
[2]
A. Gasparetto
and V. Zanatto
,
“A New Method for Smooth Trajector
y
Planning of Robot Manipulato
r
s”,
Mechanism and
Machine Theory 42
,
p455-471 , 2007.
[3]
A. Piazzi
and A. Visioli, “
G
lobal
m
i
nim
u
m
-
jerk
traje
c
tor
y
pl
anni
ng of robot m
a
n
i
pulators
”
,
IEEE Transactions on
Industrial Electronics 47 (
1
)
, 200
0, 140-149
.
[4]
R.H. Lin, and
Y. Liu
,
“Minimum-
Jerk Robot Joint
Trajecto
r
y
Using Par
t
icle Swarm Optimization
”
,
First
International Co
nference on
Ro
b
o
t, Vision and
Signal
Processing,
2011.
[5]
K.J. K
y
riakopou
los a
nd G.N. Sar
i
dis, “
Minimum jerk path gen
e
ration
”, Proceed
ing
s
of th
e 1988 IEEE International
Conference on
R
obotics
and Automa
tion, Philad
elphia, 1988
, pp
.
364–369.
Evaluation Warning : The document was created with Spire.PDF for Python.