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. 252~
258
I
S
SN
: 208
9-4
8
5
6
2
52
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 LQR Optimal Meth
od to Control
the P
o
sition of an Overhead
Crane
J. J
a
f
a
ri
, M.
Ghaz
al
, M.
N
a
z
emi
z
a
deh
*
Departm
e
nt o
f
M
echani
c
s
,
Dam
a
vand Br
anch
, Is
lam
i
c Az
ad Uni
v
ers
i
t
y
,
Dam
a
va
nd, Ir
an
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 15, 2014
Rev
i
sed
Ju
l 6
,
2
014
Accepte
d
J
u
l 26, 2014
In this p
a
per
,
a LQR (
L
in
ear
Quad
ratic Reg
u
lation)
optimal method is
implemented to control
posi
tion
of an ov
erhead
c
a
rne.
To do
this
,
a tr
ack
ing
formulation of
LQR is develo
ped a
nd app
l
i
e
d to the s
y
s
t
em
. Henc
e the
d
y
nam
i
c
m
odel
of the overh
ea
d crane
is
pres
ented
,
the d
y
n
a
m
i
c of the
actu
a
tor m
o
tor o
f
the
troll
e
y is
c
onsider
ed.
As th
e par
a
m
e
ters of
the opt
im
al
controller assign
ed, some simulations ar
e done to show the efficiency
of th
e
proposed metho
d
.
Keyword:
LQR
Op
tim
al co
n
t
rol
Ove
r
head cra
n
e
Po
sition
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
:
M. Nazem
izadeh,
Depa
rt
m
e
nt
of
M
echani
c
s,
Da
m
a
vand
B
r
a
n
c
h
,
Isl
a
m
i
c Azad
Uni
v
ersi
t
y
,
Dam
a
vand,
I
r
a
n
Em
a
il: m
n
.n
aze
m
i
zad
eh
@g
mail.co
m
1.
INTRODUCTION
Cranes are e
x
t
e
nsively applied in
t
r
an
sp
ort
a
t
i
on an
d co
nst
r
uct
i
on
fi
el
ds [
1
, 2]
.
Ove
r
h
e
a
d
cra
n
es are
o
n
e
k
i
n
d
of C
r
an
es u
s
ed
t
o
tran
sp
ort arb
itrary lo
ad
fo
rm a po
sition
to ano
t
h
e
r
o
n
e
.
Th
e
ov
erh
ead
crane
con
s
i
s
t
s
of a ca
rt
or t
r
ol
l
e
y
wh
i
c
h m
oves al
on
g i
t
s
ra
il. More
ove
r, a hoisting m
echanis
m
including a cabl
e
and
a payloa
d is a
ttached to t
h
e
cart.
T
h
e overhead c
r
a
n
es
have exte
nsivel
y used i
n
m
a
ny industries,
because
these syste
m
s
exhi
bit novel features s
u
ch as
low cost
, easy
asse
m
b
ly, and less
m
a
in
tenance [3-5]. The
r
efore,
the overhea
d
c
r
anes
ha
ve attracted a great deal of in
t
e
re
st
s, an
d t
h
e
dy
na
m
i
c
m
odel
i
ng and c
o
nt
r
o
l
of
suc
h
sy
st
em
s are st
udi
ed by
s
o
m
e
r
e
searche
r
s.
Hu
bbel
et
al
. [6]
u
s
ed an
ope
n
-
l
o
op m
e
t
hod t
o
c
ont
rol
t
h
e m
o
t
i
on
o
f
a g
a
n
t
ry crane. In
t
h
is m
e
t
h
od
, th
e inp
u
t
con
t
ro
l pro
f
ile was d
e
term
i
n
ed in su
ch
way th
at
un
wan
t
ed
o
s
cillatio
n
s
and
resi
d
u
a
l
p
e
nd
u
l
ation
s
were av
o
i
d
e
d.
Howev
e
r their app
r
o
ach
was ap
p
l
i
cab
le, bu
t th
e
o
p
e
n-
l
o
o
p
c
ont
rol
s
c
hem
e
i
s
not
ro
bust
t
o
di
st
u
r
b
a
nces a
n
d
para
m
e
t
e
r unce
r
t
a
i
n
t
i
e
s [
7
]
.
M
o
re
ove
r,
a fe
ed
ba
ck P
I
D
ant
i
-
swi
ng c
o
n
t
rol
l
e
r i
s
devel
ope
d i
n
[
8
]
t
o
cont
r
o
l
of a
n
over
h
ea
d cra
n
e. A
h
m
a
d et
al
. [9]
used a
hy
b
r
i
d
i
n
p
u
t
-
s
h
api
ng
m
e
t
hod t
o
c
o
nt
r
o
l
of t
h
e c
a
rne
.
W
a
hy
u
d
i
and Jal
a
ni
[1
0]
em
pl
oy
ed f
u
zzy
l
ogi
c fee
dba
c
k
co
n
t
ro
ller t
o
co
n
t
ro
l an
in
tellig
en
t cran
e. M
o
reov
er, pres
en
ted
an
op
ti
m
a
l co
n
t
ro
l m
e
th
o
d
is u
s
ed
i
n
[1
1
]
t
o
cont
rol the
dyna
m
i
c
m
o
tion of the system
.
Here in, m
i
ni
m
u
m energy of syste
m
and
in
tegrated
ab
so
l
u
te error
of
pay
l
oa
d a
n
gl
e are as
sum
e
d as t
h
ei
r
opt
i
m
i
zat
i
on cri
t
e
ri
on.
Zha
o
a
n
d
Gao
[
12]
st
udi
ed t
h
e c
o
nt
r
o
l
of a
n
ove
rhead cra
n
e. They proposed a fu
zzy meth
od
to
con
t
ro
l th
e in
pu
t delay an
d
actu
a
to
r satu
ratio
n
o
f
th
e
syste
m
. Nazemizadeh et al.
[13] st
udie
d
t
r
acki
n
g control of an under
a
c
tuated ga
ntry
cr
a
n
e. Furthe
rm
ore,
Nazem
izadeh [14]
pre
s
ente
d a
PID tuni
ng m
e
thod
for trac
ki
ng
control of
a
crane
.
In
t
h
is pap
e
r,
a LQR
(Lin
ear Qu
ad
ratic R
e
g
u
l
ation
)
op
timal
m
e
th
o
d
is i
m
p
l
e
m
en
ted
to
co
n
t
rol
p
o
s
ition
of an
o
v
e
rh
ead
carne. To
do
th
is,
a track
ing
fo
rm
u
l
a
tio
n
o
f
LQR is d
e
v
e
l
o
ped
and
app
lied to
th
e
sy
st
em
. Hence
t
h
e dy
nam
i
c
m
odel
of t
h
e
o
v
er
hea
d
cra
n
e
i
s
prese
n
t
e
d
,
t
h
e dy
nam
i
c of t
h
e act
uat
o
r
m
o
t
o
r of
t
h
e t
r
ol
l
e
y
i
s
c
onsi
d
ere
d
.
As
t
h
e
param
e
t
e
rs of t
h
e
opt
i
m
al
cont
rol
l
e
r as
si
gne
d,
som
e
si
m
u
l
a
t
i
ons are
do
ne t
o
sho
w
t
h
e e
ffi
ci
ency
o
f
t
h
e
pr
o
pos
ed
m
e
t
hod.
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:
252 – 258
2
53
In
t
h
is article, t
h
e
p
o
s
ition
co
ntro
l of th
e ov
erh
ead
cran
e is st
u
d
i
ed
b
a
sed on o
p
tim
al co
n
t
rol strateg
y
.
The dy
nam
i
c equat
i
o
ns
of t
h
e
sy
st
em
are deri
ve
d, co
nsi
d
er
i
ng t
h
e m
o
t
o
r
vol
t
a
ge
of a
w
h
eel
of t
h
e t
r
ol
l
e
y
as
the input, and
displacem
ent of the t
r
o
lley
as
the out
put of the
cra
n
e. To
cont
rol t
h
e position of the
cart, an
LQR
m
e
t
hod i
s
use
d
.
T
o
veri
fy
t
h
e
pr
o
p
o
s
e
d
m
e
t
hod,
s
o
m
e
si
m
u
l
a
ti
on
re
sul
t
s
are
d
o
n
e
and
p
r
ese
n
t
e
d.
2.
LQR
OPTIMAL CONT
ROL OF THE
SYSTEM
In t
h
i
s
sect
i
o
n,
t
h
e LQR
c
ont
rol
m
e
t
hod i
s
a
ppl
i
e
d t
o
t
h
e s
y
st
em
. Presum
e t
h
e dy
nam
i
c
equat
i
o
n
o
f
the ove
r
hea
d
c
r
ane
in state-
space from
can
be written as:
CX
y
Bu
AX
X
(1
)
Whe
r
e
X
is t
h
e
state vector
of the system
,
u
is th
e in
pu
t effo
rt,
y
i
s
t
h
e o
u
t
put
, a
n
d
A, B, C
are the c
o
efficien
t
m
a
trices of t
h
e
syste
m
.
Fu
rt
h
e
rm
o
r
e the fin
a
l po
sitio
n o
f
t
h
e cart can
b
e
d
e
fin
e
d
as
r(t)
, and relat
e
d to the
final
state vector
e
X
and
fin
a
l inpu
t of t
h
e system
e
u
b
y
Eq
. (2
)
:
e
e
e
CX
t
r
Bu
AX
)
(
0
(2
)
Thu
s
, co
m
b
in
g Eq
s.
(1) , (2
) resu
lts in
:
X
C
y
u
B
X
A
X
(3
)
Whe
r
e
r
y
y
u
u
u
X
X
X
e
e
,
,
are ass
u
med as re
formatted vect
ors
.
Fu
rt
h
e
rm
o
r
e, to
ap
p
l
y th
e LQR op
ti
m
a
l co
n
t
ro
ller,
an obj
ectiv
e fu
n
c
tion
is con
s
id
ered as
fo
llows:
dt
Ru
u
X
Q
X
J
T
T
0
(4
)
Whe
r
e
Q
and
R
are
weigh
ting
m
a
trices o
f
op
ti
m
a
l co
n
t
ro
ller
wh
ich
is
d
e
fin
e
d
b
y
th
e user.
Using
LQR m
e
th
od
, t
h
e
op
timal feed
b
a
ck
l
a
w is
X
P
B
R
X
K
u
T
1
, whe
r
e can be
ac
hieve
d
from
R
i
ccat
i
’
s equat
i
on
[
15]
:
0
1
Q
P
B
PBR
PA
P
A
T
T
(5
)
Wh
ere is
d
e
fined
as a
p
o
s
itiv
e m
a
trix
.
3.
D
YNA
M
I
C
MOD
ELING
OF THE SYST
EM
In t
h
i
s
sect
i
o
n,
t
h
e dy
nam
i
c
m
odel
i
ng o
f
t
h
e ove
rhea
d ca
r
n
e i
s
pre
s
ent
e
d
.
The
dy
nam
i
c
equat
i
o
ns
of
t
h
e sy
st
em
are deri
ve
d u
s
i
n
g
Lagra
n
ge’s
p
r
i
n
ci
pl
e.
Fi
g
u
re
1 s
h
o
w
s a
n
o
v
e
rhea
d c
r
ane
.
The c
r
ane
i
s
c
onsi
s
t
s
of
a cart
t
r
ans
v
erses i
n
ho
ri
zo
nt
al
di
rect
i
o
n,
whi
l
e
a
pe
nd
ul
um
connect
s
o
n
t
h
e
cart
a
n
d
hoi
st
s t
h
e
pay
l
oad
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
A LQR
Op
timal Meth
o
d
t
o
Con
t
ro
l t
h
e Po
sitio
n o
f
an
Overhea
d
Cran
e (J.
Ja
fa
ri)
25
4
Fi
gu
re 1.
The
ove
r
h
ead
cra
n
e
The pa
ram
e
ters of the syste
m
are:
x
x
,
are the cart position and s
p
ee
d,
,
are t
h
e pe
n
dul
um
angular dis
p
lace
m
e
nt,
l
i
s
t
h
e pe
nd
ul
um
l
e
ngt
h,
M
i
s
th
e
ma
s
s
o
f
th
e
ca
r
t
,
m
is t
h
e
paylo
a
d
m
a
ss,
r
is th
e
radi
us of the wheels of the ca
rt,
e
is th
e DC
m
o
to
r v
o
ltag
e
o
f
th
e cart,
R
i
s
the
m
o
tor armature resistance,
k
is
th
e
m
o
to
r torq
u
e
co
nstan
t
,
p
B
i
s
t
h
e vi
scou
s dam
p
i
ng co
effi
ci
ent
o
f
t
h
e pen
dul
um
axi
s
,
eq
B
is th
e
equi
val
e
nt
vi
sc
ous
d
a
m
p
i
ng c
o
ef
fi
ci
ent
,
a
n
d
g i
s
t
h
e
gra
v
i
t
a
t
i
onal
co
nst
a
nt
of
eart
h
.
To
de
ri
ve t
h
e e
quat
i
o
n
o
f
t
h
e
m
o
ti
on,
t
h
e
ki
net
i
c
ene
r
gy
o
f
t
h
e ca
rt
1
T
an
d t
h
e
ki
net
i
c
e
n
er
gy
o
f
t
h
e
p
e
ndu
lu
m
2
T
are:
2
2
,
1
2
,
1
1
2
1
2
1
x
M
V
V
M
T
y
x
co
s
l
x
sin
l
x
)
l
(
l
x
m
V
V
m
T
y
,
x
,
2
2
2
1
2
1
2
2
2
2
2
2
2
2
(6
)
Furt
herm
ore,
the
pote
n
tial en
ergy
of
the
pay
l
oad is:
cos
mgl
U
2
(7
)
To
deri
ve th
e
d
y
n
am
ic equatio
n
of
the sy
st
em
, the
Lagrangia
n
function is st
ated as:
cos
2
1
cos
2
sin
2
)
(
2
1
2
2
2
2
2
2
1
mgl
x
M
l
x
l
x
l
l
x
m
U
T
T
L
(8
)
An
d t
h
e
dam
p
ing
f
o
rce
o
f
t
h
e
sy
stem
is:
l
B
Q
x
B
Q
p
lost
,
eq
lost
,
x
(9
)
The Lagrange’s prin
ciple is written as:
lost
,
j
j
j
j
Q
Q
q
L
q
L
dt
d
(1
0)
l
m
M
r
f
x
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:
252 – 258
2
55
Whe
r
e
j
q
is the gene
ralized coordinate of the system
,
j
Q
is the ge
neralized force e
x
erte
d to
t
h
e
cor
r
es
po
n
d
in
g
gene
ralized c
o
or
dinate,
an
d
lost
j
Q
,
is the
dam
p
ing
force.
The
n
usin
g La
gra
n
ge’s
p
r
inci
ple, the
n
o
n
lin
ear e
quatio
ns
o
f
the
sy
stem
can
be ac
hieve
d
[
16]
:
0
2
sin
g
l
cos
x
f
sin
ml
cos
ml
x
)
m
M
(
(1
1)
On t
h
e ot
her
hand, the linea
r
force of the ca
rt
f
is origi
n
ated from
the torque
of the
DC
m
o
tor. The
r
efore, the
related equations are:
r
x
e
R
k
R
k
T
f
r
T
2
(1
2)
Whe
r
e
T
is the torque of
the
actuator
,
a
nd
is the angular
velocity of the
m
o
to
r
.
Thu
s
, fr
o
m
Eq
s.
(1
1)
and
(1
2)
, we have:
0
1
2
2
sin
g
l
cos
x
e
R
k
x
Rr
k
r
sin
ml
cos
ml
x
)
m
M
(
(1
3)
To prese
n
t the nonlinear e
quations
of the syste
m
in s
t
ate-space form
,
the state
vector is defi
ned as
x
x
X
, and t
h
e
nonli
n
ear equations
are:
)
cos
(
ml
Ml
Rr
x
k
Rr
ke
cos
sin
g
)
m
M
(
cos
sin
ml
)
(
dt
d
tan
g
)
cos
(
m
M
Rr
x
k
Rr
ke
l
tan
gl
)
m
M
(
sin
ml
x
)
x
(
dt
d
x
2
2
2
2
2
2
2
2
2
1
1
(1
4)
M
o
re
ove
r, to
Use LQR
m
e
tho
d
, the
linea
rization o
f
the
no
nlinear e
q
uation
s
m
u
st
be d
one
. Usi
n
g the
linearization m
e
thod, t
h
e state
-
space
lineariz
ed
Equation of
the syst
em
s are obtained as:
u
D
X
C
y
u
B
X
A
X
(1
5)
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I
J
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9-4
8
5
6
A LQR
Op
timal Meth
od
to
Con
tro
l
th
e Po
sitio
n
o
f
an
Overhea
d
Cran
e (J.
Ja
fa
ri)
25
6
0
0
0
0
1
0
0
0
)
(
0
1
0
0
0
0
0
0
0
1
0
2
2
2
2
D
C
RrMl
k
RrM
k
B
Ml
g
m
M
Ml
Rr
k
M
mg
M
Rr
k
A
4.
SIMULATIONS
AN
D R
E
SU
LTS
In t
h
is section, the L
Q
R cont
rol
of t
h
e
ove
rhead cr
ane is si
m
u
lated. The
param
e
ters of t
h
e system
s
are:
m
l
3302
.
0
,
kg
M
073
.
1
,
kg
m
23
.
0
,
m
r
006
.
0
,
6
.
2
R
e,
rad
Vs
k
/
00767
.
0
,
V
e
12
max
,
and
2
/
81
.
9
s
m
g
[17]
.
As it is m
e
ntioned, the
desired criteri
ons
of
the control
design are: t
h
e tr
olley can set the fi
nal position while
the swaying of the
pe
ndul
um
is dam
p
ed
quic
k
ly, and
t
h
e input voltage
of the
m
o
tor does not
exceed
its
m
a
xim
u
m
value. Fo
r a
r
bitra
r
y
Q
a
nd
R
wei
g
hting m
a
trices,
the LQR feedb
ack controller gains
are s
hown i
n
Table 1.
Table 1.
L
Q
R
cont
roller gai
n
s
Contr
o
ller
gain
R
Q
83742
.
0
97281
.
0
5735
.
1
1
K
1
)
(
diag
1
22028
0
17422
0
57139
0
31623
0
.
.
.
.
K
1
)
.
(
diag
1
0
7238
2
7902
5
3898
4
1623
3
.
.
.
.
K
1
0
.
)
(
diag
1
An
d t
h
e sim
u
lation
results a
r
e
depicted
.
Figure
2. The
displacem
ent of the
cart
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I
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SN:
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-48
56
IJRA Vol. 3, No. 4,
D
ecem
ber 2014:
252 – 258
2
57
Figure
3. The
displacem
ent of the
pe
ndul
um
Figu
re
4.
The
voltage
o
f
t
h
e
cart
As it is seen i
n
the forgoi
ng figures, in
c
r
e
a
sing the wei
g
hting
of the st
ate vector (
Q
)
,
and dec
r
easin
g the
weighting
of t
h
e input (
R
) leads to decreasing the m
a
xim
u
m of
the cart position and increasi
ng
of the
m
a
xim
u
m
values of the in
p
u
t contr
o
ller. T
h
eref
ore
,
o
n
e can choose appropriate va
lues
of
Q an
d R
to obtai
n
desire
d results.
5.
CO
NCL
USI
O
N
In this pa
pe
r, the p
o
sition c
o
ntr
o
l of the
ov
erhe
a
d
cra
n
e has been investi
g
ated using L
Q
R optim
a
l
cont
rol m
e
thod.
At first, t
h
e
no
nlinea
r dy
n
a
m
i
c equati
ons
of t
h
e sy
stem
have
bee
n
der
i
ved
via Lag
r
a
nge
’s
pri
n
ciple, an
d then the
dy
nam
i
c of the DC
m
o
to
r has be
en
applied t
o
the syste
m
.
The voltage of the actuator
of the trolley has been as
sum
e
d as th
e input, and
displacem
ent of t
h
e trol
l
e
y has bee
n
presum
ed as the out
put
of t
h
e system
.
To c
ont
rol t
h
e
position
of t
h
e
cart, th
e LQR m
e
thod has be
en devel
ope
d
a
n
d
som
e
sim
u
lations
have
bee
n
d
o
n
e.
It is concl
ude
d that i
n
cr
easing t
h
e wei
ghting
of t
h
e
state vector
(
Q
) or dec
r
easing
t
h
e
weighting
of t
h
e input (
R
) leads to decreasing the m
a
xim
u
m of
the cart position and increasi
ng
of the
m
a
xim
u
m
values
of
the
in
p
u
t co
ntr
o
ller.
Furt
herm
ore,
si
m
u
lation results properly
de
m
onstrated the
p
o
we
r
and efficiency
of th
e p
r
op
osed
ap
pro
ach.
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I
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9-4
8
5
6
A LQR
Optimal Method t
o
Control t
h
e Pos
ition
of an Overhead
Crane (J.
Jafari)
25
8
REFERE
NC
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