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Ma
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ch
20
2
6
,
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33
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42
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33
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[
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Evaluation Warning : The document was created with Spire.PDF for Python.
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task
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w
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o
ller
to
en
h
an
ce
t
h
e
p
er
f
o
r
m
an
ce
o
f
a
g
en
etic
alg
o
r
ith
m
(
GA)
o
p
tim
i
ze
d
n
o
n
lin
ea
r
PID
co
n
tr
o
ller
p
ar
am
eter
s
.
An
o
th
e
r
ap
p
r
o
ac
h
u
s
in
g
a
f
u
zz
y
l
o
g
ic
b
ased
s
lid
in
g
m
o
d
e
co
n
tr
o
l
tec
h
n
iq
u
e
h
as
b
ee
n
p
r
o
p
o
s
ed
i
n
[
2
4
]
.
T
h
e
p
r
o
b
lem
o
f
tr
ajec
to
r
y
tr
ac
k
in
g
f
o
r
a
ca
r
-
lik
e
Au
to
n
o
m
o
u
s
Veh
icle
h
as
b
ee
n
th
e
f
o
cu
s
o
f
th
e
wo
r
k
p
r
esen
ted
in
[
2
5
]
.
I
n
th
is
s
tu
d
y
,
th
e
co
n
tr
o
l
s
tr
ateg
y
was
b
ased
o
n
a
“lin
ea
r
p
a
r
a
m
eter
v
ar
y
in
g
”
(
L
PV)
s
y
s
tem
f
o
r
wh
ich
t
h
e
p
ar
a
m
eter
s
o
f
a
m
o
d
el
p
r
ed
ictiv
e
co
n
tr
o
ller
(
MPC
)
wer
e
o
p
ti
m
ized
th
r
o
u
g
h
a
p
ar
ticle
s
war
m
o
p
tim
izatio
n
(
PS
O)
te
ch
n
iq
u
e.
T
r
ajec
to
r
y
tr
ac
k
in
g
in
3
D
s
p
ac
e
is
also
o
n
e
o
f
th
e
m
o
s
t
attr
ac
tin
g
to
p
ics
in
th
e
liter
atu
r
e,
a
m
o
n
g
th
e
m
an
y
p
r
o
p
o
s
ed
co
n
tr
o
llin
g
a
p
p
r
o
ac
h
es
th
e
au
th
o
r
s
o
f
[
2
6
]
h
av
e
d
esig
n
ed
a
n
eu
r
al
f
u
zz
y
b
ased
PID
c
o
n
tr
o
ller
to
m
a
k
e
an
u
n
m
an
n
ed
ae
r
ial
v
eh
icle
(
U
AV)
f
o
llo
w
a
p
r
ed
ef
in
e
d
tr
aj
ec
to
r
y
.
Aim
in
g
to
s
o
lv
e
th
e
s
am
e
p
r
o
b
lem
,
th
e
au
th
o
r
s
in
[
2
7
]
h
av
e
ad
o
p
ted
a
“Su
p
er
twis
tin
g
s
lid
in
g
m
o
d
e
co
n
tr
o
ller
”
co
m
b
in
ed
with
a
f
u
zz
y
PID
s
u
r
f
ac
e
to
ad
ju
s
t
th
e
g
ain
p
ar
am
eter
s
an
d
th
er
eb
y
r
e
d
u
cin
g
th
e
ch
atter
in
g
ef
f
ec
t
in
d
u
ce
d
b
y
th
is
k
in
d
o
f
co
n
tr
o
l
tech
n
iq
u
e
.
As
p
r
e
v
io
u
s
ly
m
en
t
io
n
ed
,
th
e
m
ain
f
o
cu
s
o
f
th
e
p
r
esen
t
wo
r
k
is
to
d
esig
n
a
co
n
t
r
o
ller
e
n
ab
lin
g
th
e
m
o
b
ile
r
o
b
o
t
to
f
o
llo
w
a
p
r
e
s
cr
ib
ed
tr
ajec
to
r
y
,
in
an
o
b
s
t
ac
le
-
f
r
ee
en
v
ir
o
n
m
e
n
t.
T
h
e
r
e
f
o
r
e,
th
is
is
o
n
l
y
a
p
ar
tial
s
o
lu
tio
n
to
th
e
m
o
r
e
g
lo
b
al
p
r
o
b
lem
o
f
m
o
tio
n
p
lan
n
in
g
,
wh
ich
in
clu
d
es
p
ath
p
lan
n
in
g
,
to
g
e
n
er
ate
a
s
u
cc
ess
io
n
o
f
way
p
o
in
ts
th
at
th
e
r
o
b
o
t
s
h
o
u
ld
g
o
th
r
o
u
g
h
,
in
clu
d
in
g
th
e
tim
e
s
am
p
les
at
wh
ich
th
e
r
o
b
o
t
is
s
u
p
p
o
s
ed
t
o
r
ea
ch
th
ese
way
p
o
in
ts
.
Path
p
lan
n
in
g
is
th
e
r
ef
o
r
e
an
o
th
er
im
p
o
r
tan
t
p
r
o
b
lem
th
at
h
as
b
ee
n
tak
e
n
ca
r
e
o
f
i
n
m
an
y
r
esear
c
h
wo
r
k
s
.
T
h
is
was
th
e
ca
s
e
in
[
2
8
]
,
w
h
er
e
th
e
a
u
th
o
r
s
h
av
e
s
u
g
g
ested
a
“Bi
d
ir
ec
tio
n
al
r
ap
id
ly
-
e
x
p
lo
r
in
g
r
an
d
o
m
tr
ee
s
tar
”,
in
teg
r
ated
with
a
“Dy
n
am
ic
win
d
o
w
ap
p
r
o
ac
h
”
an
d
a
n
“Ad
ap
tiv
e
m
o
n
te
C
ar
lo
lo
ca
lizatio
n
”
tech
n
iq
u
e
.
I
n
th
is
p
ap
er
,
u
s
in
g
s
im
p
le
g
eo
m
etr
ic
h
o
m
o
g
en
e
o
u
s
tr
an
s
f
o
r
m
atio
n
p
r
o
p
e
r
ties
,
an
er
r
o
r
d
y
n
am
ic
m
o
d
el
will
b
e
d
er
iv
ed
an
d
u
s
ed
to
o
b
tain
a
two
-
co
m
p
o
n
e
n
t
n
o
n
lin
ea
r
co
n
tr
o
l
s
ch
em
e
f
o
r
b
o
th
th
e
lin
ea
r
an
d
an
g
u
lar
s
p
ee
d
s
o
f
th
e
m
o
b
ile
r
o
b
o
t.
As
f
o
r
an
y
o
th
er
co
n
tr
o
ller
,
th
e
en
d
g
o
al
o
f
its
d
esig
n
is
to
im
p
lem
en
t
a
cu
s
to
m
ized
v
e
r
s
io
n
o
f
it
o
n
a
r
ea
l
h
ar
d
war
e
p
latf
o
r
m
.
I
n
th
e
p
ar
tic
u
lar
ca
s
e
o
f
a
m
o
b
ile
r
o
b
o
t,
t
h
is
co
u
l
d
p
r
o
v
e
to
b
e
v
er
y
tim
e
-
co
n
s
u
m
in
g
,
esp
ec
ially
wh
en
tr
y
in
g
to
tu
n
e
th
e
u
n
k
n
o
wn
p
ar
am
eter
s
.
T
h
is
is
ca
u
s
ed
b
y
th
e
f
ac
t th
at
we
n
ee
d
t
o
u
p
d
ate
th
e
f
ir
m
war
e
o
f
th
e
o
n
b
o
a
r
d
s
y
s
tem
ev
er
y
tim
e
a
c
h
an
g
e
o
f
t
h
ese
p
ar
am
eter
s
is
r
eq
u
ir
ed
.
On
e
o
f
th
e
k
ey
f
ea
tu
r
es
o
f
th
e
p
r
esen
t
wo
r
k
is
to
a
d
d
r
ess
th
is
p
r
o
b
lem
u
s
in
g
a
h
a
r
d
war
e
-
in
-
t
h
e
-
lo
o
p
ar
ch
itectu
r
e,
allo
win
g
to
d
eb
u
g
th
e
co
n
t
r
o
llin
g
p
ar
t
o
n
a
s
tatio
n
ar
y
PC
,
wh
ile
leav
in
g
u
n
ch
an
g
ed
th
e
f
ir
m
war
e
r
u
n
n
in
g
o
n
t
h
e
m
o
v
i
n
g
r
o
b
o
t.
W
h
at
m
ak
es th
is
d
ep
lo
y
m
en
t
ev
en
f
aster
is
th
e
ab
ilit
y
to
ca
r
r
y
o
u
t th
e
c
o
n
tr
o
ller
tu
n
in
g
p
h
ase
u
s
in
g
a
r
o
b
o
t
d
y
n
am
ic
m
o
d
el
as
p
ar
t
o
f
a
co
m
p
lete
s
im
u
latio
n
p
r
o
g
r
am
,
b
ef
o
r
e
r
e
p
lacin
g
th
is
m
o
d
el
with
th
e
c
o
n
n
ec
ted
h
ar
d
war
e
p
latf
o
r
m
.
T
h
e
co
n
ten
t
o
f
th
is
p
ap
e
r
ca
n
b
e
s
u
m
m
ar
ize
d
as
f
o
llo
ws;
I
n
s
ec
tio
n
2
,
a
c
o
m
p
r
e
h
en
s
iv
e
a
n
d
d
etailed
d
escr
ip
tio
n
o
f
th
e
k
in
em
atic
m
o
d
el
is
g
iv
e
n
.
T
h
e
r
esu
ltin
g
,
n
o
n
lin
ea
r
s
tate
s
p
ac
e
m
o
d
el
i
s
th
en
u
s
ed
t
o
d
esig
n
a
co
n
tr
o
ller
f
o
r
s
teer
in
g
t
h
e
r
o
b
o
t
to
war
d
s
th
e
d
esire
d
p
ath
.
T
h
e
r
esu
lts
o
f
s
o
m
e
illu
s
tr
ativ
e
ex
am
p
les,
u
s
ed
t
o
test
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
p
r
o
p
o
s
ed
co
n
tr
o
ller
,
will b
e
d
is
c
u
s
s
ed
in
s
ec
tio
n
3
.
T
h
e
m
o
s
t i
m
p
o
r
tan
t f
ea
tu
r
es o
f
th
e
p
r
o
p
o
s
ed
co
n
tr
o
l sy
s
tem
w
ill b
e
s
u
m
m
ar
ized
in
s
ec
tio
n
4
.
2.
M
O
DE
L
I
NG
AN
D
CO
NT
R
O
L
2
.
1
.
P
a
t
h f
o
llo
wing
s
t
ra
t
e
g
y
On
e
o
f
th
e
m
o
s
t
p
o
p
u
lar
m
eth
o
d
s
u
s
ed
to
m
ak
e
a
r
o
b
o
t
tr
ac
k
a
p
r
e
d
ef
in
ed
tr
ajec
to
r
y
,
c
o
n
s
is
ts
of
d
esig
n
in
g
a
co
n
tr
o
ller
ca
p
ab
le
o
f
d
r
iv
in
g
th
is
r
o
b
o
t
to
war
d
s
an
o
th
er
v
ir
tu
al
r
o
b
o
t
th
at
is
p
er
f
ec
tly
f
o
llo
win
g
th
e
tar
g
et
o
r
d
esire
d
tr
ajec
to
r
y
.
T
h
is
ap
p
r
o
ac
h
is
illu
s
tr
ated
i
n
Fig
u
r
e
1
,
wh
er
e,
in
th
is
ca
s
e
th
e
v
ir
t
u
al
r
o
b
o
t
is
d
esig
n
ated
b
y
th
e
letter
B
,
wh
er
ea
s
th
e
co
n
t
r
o
lled
r
o
b
o
t
is
lab
eled
r
o
b
o
t
A
.
T
o
ac
h
iev
e
th
is
g
o
al,
we
f
ir
s
t
n
ee
d
to
ex
p
r
ess
th
e
p
o
s
tu
r
e
(
p
o
s
itio
n
an
d
o
r
ien
tatio
n
)
o
f
r
o
b
o
t
B
r
elativ
e
to
th
e
f
r
am
e
attac
h
ed
to
r
o
b
o
t
A
.
I
n
t
h
e
f
o
llo
win
g
,
we
will
u
s
e
th
e
n
o
tatio
n
to
d
ef
in
e
a
tr
a
n
s
f
o
r
m
at
io
n
o
f
a
m
o
v
in
g
(
M)
f
r
a
m
e
w
ith
r
esp
ec
t
to
a
f
ix
ed
(
F)
f
r
am
e.
C
o
n
s
id
er
in
g
th
at
th
e
p
o
s
tu
r
es
o
f
r
o
b
o
t
A
an
d
B
ca
n
b
e
r
ep
r
esen
ted
b
y
th
eir
r
esp
ec
tiv
e
tr
an
s
f
o
r
m
atio
n
s
in
t
h
e
wo
r
ld
o
r
f
ix
ed
f
r
am
e
(
0
)
we
h
av
e:
0
=
[
c
os
1
−
s
in
1
1
s
in
1
c
os
1
1
0
0
1
]
(
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
E
xp
erimen
ta
l v
a
lid
a
tio
n
o
f a
t
r
a
jecto
r
y
tr
a
ck
in
g
co
n
tr
o
ller
fo
r
a
tw
o
-
w
h
ee
led
mo
b
ile
r
o
b
o
t
(
B
o
u
a
lem
K
a
z
ed
)
35
0
=
[
c
os
2
−
s
in
2
2
s
in
2
c
os
2
2
0
0
1
]
(
2
)
Fig
u
r
e
1
.
Sam
p
le
p
o
s
tu
r
es o
f
th
e
ac
tu
al
(
g
r
ey
)
a
n
d
v
i
r
tu
al
r
o
b
o
t (
g
r
ee
n
)
T
h
e
tr
an
s
f
o
r
m
atio
n
m
atr
ix
0
m
o
v
es
an
d
r
o
tates
th
e
f
r
a
m
e
(
0
)
f
r
o
m
(
0
,
0
,
0
)
to
th
e
p
o
s
tu
r
e
(
2
,
2
,
2
)
.
T
h
e
f
ix
e
d
f
r
am
e’
s
tr
a
n
s
f
o
r
m
atio
n
m
atr
i
x
ca
n
b
e
wr
itten
as
0
0
=
,
with
b
ein
g
th
e
id
e
n
tity
m
atr
ix
.
R
ewr
itin
g
0
as
0
=
0
0
0
ex
p
r
ess
es
th
e
f
ac
t
th
at
0
ca
n
b
e
v
iewe
d
as
an
o
p
er
at
o
r
ap
p
lied
o
n
th
e
f
r
am
e
(
0
)
to
m
o
v
e
it
an
d
p
lac
es
it
o
n
f
r
am
e
(
2
)
,
attac
h
ed
t
o
r
o
b
o
t
B
.
I
f
we
ch
an
g
e
th
e
s
tar
tin
g
p
o
s
tu
r
e
an
d
m
ak
e
it
o
n
f
r
am
e
(
1
)
wh
ic
h
is
attac
h
ed
to
r
o
b
o
t
A
,
0
0
will
b
e
r
ep
lace
d
b
y
0
,
th
er
ef
o
r
e
0
ca
n
also
b
e
ex
p
r
ess
ed
as
0
=
0
,
if
we
p
r
e
-
m
u
ltip
ly
b
o
th
s
id
es b
y
(
0
)
−
1
we
g
et:
=
(
0
)
−
1
0
=
[
c
os
1
s
in
1
−
1
c
os
1
−
1
s
in
1
−
s
in
1
c
os
1
1
s
in
1
−
1
c
os
1
0
0
1
]
[
c
os
2
−
s
in
2
2
s
in
2
c
os
2
2
0
0
1
]
=
[
c
os
(
2
−
1
)
−
s
in
(
2
−
1
)
(
2
−
1
)
c
os
1
+
(
2
−
1
)
s
in
1
s
in
(
2
−
1
)
c
os
(
2
−
1
)
(
2
−
1
)
c
os
1
−
(
2
−
1
)
s
in
1
0
0
1
]
(
3
)
W
e
ca
n
ea
s
ily
n
o
tice
th
at
th
e
m
atr
ix
o
b
tain
e
d
in
(
3
)
r
e
p
r
esen
ts
a
h
o
m
o
g
e
n
eo
u
s
tr
a
n
s
f
o
r
m
a
tio
n
with
an
an
g
u
lar
r
o
tatio
n
o
f
=
(
2
−
1
)
ar
o
u
n
d
th
e
z
ax
is
,
p
er
p
en
d
ic
u
lar
an
d
p
o
in
t
in
g
o
u
t
o
f
th
e
p
ag
e,
an
d
two
tr
an
s
latio
n
s
:
=
(
2
−
1
)
c
os
1
+
(
2
−
1
)
s
in
1
an
d
=
(
2
−
1
)
c
os
1
−
(
2
−
1
)
s
in
1
,
alo
n
g
th
e
an
d
ax
is
r
esp
ec
tiv
ely
.
T
a
k
in
g
th
e
d
er
iv
ativ
es o
f
th
ese
lin
ea
r
an
d
an
g
u
lar
p
o
s
itio
n
s
,
we
g
et:
̇
=
(
̇
2
−
̇
1
)
c
os
1
−
(
2
−
1
)
̇
1
s
in
1
+
(
̇
2
−
̇
1
)
s
in
1
+
(
2
−
1
)
̇
c
os
1
̇
=
−
(
̇
2
−
̇
1
)
s
in
1
−
(
2
−
1
)
̇
1
c
os
1
+
(
̇
2
−
̇
1
)
c
os
1
−
(
2
−
1
)
̇
s
in
1
̇
=
̇
2
−
̇
1
(
4
)
I
n
m
atr
ix
f
o
r
m
eq
u
atio
n
(
4
)
ca
n
b
e
wr
itten
as (
5
)
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
33
-
42
36
[
̇
̇
̇
]
=
[
−
̇
1
s
in
1
̇
1
c
os
1
0
−
̇
1
c
os
1
−
̇
1
s
in
1
0
0
0
0
]
[
2
−
1
2
−
1
2
−
1
]
+
[
c
os
1
s
in
1
0
−
s
in
1
c
os
1
0
0
0
1
]
[
̇
2
−
̇
1
̇
2
−
̇
1
̇
2
−
̇
1
]
(
5
)
w
h
ich
ca
n
f
u
r
th
er
b
e
ex
p
r
ess
ed
as (
6
)
.
[
̇
̇
̇
]
=
[
−
̇
1
s
in
1
̇
1
cos
1
0
−
̇
1
cos
1
−
̇
1
s
in
1
0
0
0
0
]
[
]
+
[
cos
1
s
in
1
0
−
s
in
1
cos
1
0
0
0
1
]
[
̇
2
̇
2
̇
2
]
−
[
cos
1
s
in
1
0
−
s
in
1
cos
1
0
0
0
1
]
[
̇
1
̇
1
̇
1
]
(
6
)
Fro
m
Fig
u
r
e
1
,
̇
1
an
d
̇
1
ca
n
b
e
r
e
p
lace
d
as
f
o
llo
ws
̇
1
=
1
c
os
1
an
d
̇
1
=
1
s
in
1
,
th
e
a
n
g
u
lar
s
p
ee
d
o
f
th
e
r
o
b
o
t a
r
o
u
n
d
t
h
e
v
er
tical
a
x
is
z
,
̇
1
will b
e
d
en
o
ted
as
Ω
1
.
Usi
n
g
th
ese
ex
p
r
ess
io
n
s
th
e
last
ter
m
o
f
(
6
)
b
ec
o
m
es
(
7
)
.
[
cos
1
s
in
1
0
−
s
in
1
cos
1
0
0
0
1
]
[
̇
1
̇
1
̇
1
]
=
[
̇
1
cos
1
+
̇
1
s
in
1
−
̇
1
s
in
1
+
̇
1
cos
1
̇
1
]
=
[
1
cos
1
cos
1
+
1
s
in
1
s
in
1
−
1
cos
1
s
in
1
+
1
s
in
1
cos
1
̇
1
]
=
[
1
0
Ω
1
]
(
7
)
Similar
ly
n
o
tin
g
th
at
̇
2
=
2
c
os
2
an
d
̇
2
=
2
s
in
2
eq
u
atio
n
(
6
)
ca
n
b
e
r
ed
u
ce
d
to
:
[
̇
̇
̇
]
=
−
̇
1
[
s
in
1
−
c
os
1
0
c
os
1
s
in
1
0
0
0
0
]
[
]
−
[
1
0
Ω
1
]
+
[
c
os
1
s
in
1
0
−
s
in
1
c
os
1
0
0
0
1
]
[
2
c
os
2
2
s
in
2
Ω
2
]
(
8
)
W
h
ich
ca
n
b
e
r
ewr
itten
as:
[
̇
̇
̇
]
=
−
̇
1
[
s
in
1
−
c
os
1
0
c
os
1
s
in
1
0
0
0
0
]
[
]
−
[
1
0
Ω
1
]
+
[
−
2
c
os
1
c
os
2
+
2
s
in
1
s
in
2
2
s
in
1
c
os
2
+
2
c
os
1
s
in
2
Ω
2
]
An
d
th
en
:
[
̇
̇
̇
]
=
−
̇
1
[
s
in
1
−
c
os
1
0
c
os
1
s
in
1
0
0
0
0
]
[
]
−
[
1
0
Ω
1
]
+
[
2
c
os
(
2
−
1
)
2
s
in
(
2
−
1
)
Ω
2
]
Fro
m
th
e
th
ir
d
co
lu
m
n
o
f
th
e
t
r
an
s
f
o
r
m
atio
n
m
atr
ix
(
3
)
,
we
h
av
e:
=
(
2
−
1
)
c
os
1
+
(
2
−
1
)
s
in
1
=
c
os
1
+
s
in
1
=
(
2
−
1
)
c
os
1
−
(
2
−
1
)
s
in
1
=
c
os
1
−
s
in
1
Fin
aly
th
e
er
r
o
r
d
y
n
am
ics o
f
r
o
b
o
t
A
r
elativ
e
to
r
o
b
o
t
B
ca
n
ex
p
r
ess
ed
as:
[
̇
̇
̇
]
=
−
Ω
1
[
−
0
]
−
[
1
0
Ω
1
]
+
[
2
c
os
2
s
in
Ω
2
]
=
[
Ω
1
−
1
+
2
c
os
−
Ω
1
+
2
s
in
Ω
2
−
Ω
1
]
(
9
)
Fo
r
m
o
r
e
co
n
v
en
ie
n
ce
let
u
s
d
ef
in
e
th
e
s
tate
v
ec
to
r
=
[
1
2
3
]
=
[
]
.
T
h
is
g
iv
es u
s
th
e
f
o
llo
win
g
s
tate
s
p
ac
e
s
y
s
tem
in
(
1
0
)
.
{
̇
1
=
Ω
1
2
−
1
+
2
c
os
3
̇
2
=
−
Ω
1
1
+
2
s
in
3
̇
3
=
Ω
2
−
Ω
1
(
1
0
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
E
xp
erimen
ta
l v
a
lid
a
tio
n
o
f a
t
r
a
jecto
r
y
tr
a
ck
in
g
co
n
tr
o
ller
fo
r
a
tw
o
-
w
h
ee
led
mo
b
ile
r
o
b
o
t
(
B
o
u
a
lem
K
a
z
ed
)
37
2
.
2
.
Co
ntr
o
l
d
esig
n
T
o
m
a
k
e
r
o
b
o
t
A
t
r
a
c
k
t
h
e
d
e
s
ir
e
d
t
r
a
j
e
c
t
o
r
y
,
t
h
e
l
i
n
ea
r
a
n
d
a
n
g
u
l
a
r
s
p
e
e
d
s
1
a
n
d
Ω
1
,
n
e
e
d
t
o
b
e
p
r
o
p
e
r
l
y
e
v
a
l
u
a
t
e
d
.
T
o
t
h
is
e
n
d
,
l
e
t
u
s
d
e
f
i
n
e
t
h
e
p
o
s
it
i
v
e
d
e
f
i
n
it
e
f
u
n
c
ti
o
n
(
1
1
)
a
s
a
L
y
a
p
u
n
o
v
f
u
n
c
t
i
o
n
c
a
n
d
i
d
a
t
e
.
=
1
2
(
1
2
+
2
2
)
+
1
−
c
os
3
(
1
1
)
T
h
e
ter
m
(
1
−
c
os
3
)
in
s
u
r
es
t
h
at
is
alwa
y
s
p
o
s
itiv
e
d
ef
in
ite
with
r
esp
ec
t
t
o
th
e
o
r
ien
tatio
n
3
;
(
1
,
2
,
3
)
>
0
∀
(
1
,
2
,
3
)
≠
(
0
,
0
,
0
)
(
0
,
0
,
0
)
=
0
,
th
e
d
er
iv
ativ
e
o
f
is
o
b
tain
ed
as
(
1
2
)
.
̇
=
1
̇
1
+
2
̇
2
+
̇
3
s
in
3
̇
=
1
(
Ω
1
2
−
1
+
2
c
os
3
)
+
2
(
−
Ω
1
1
+
2
s
in
3
)
+
(
Ω
2
−
Ω
1
)
s
in
3
̇
=
1
(
2
c
os
3
−
1
)
+
(
2
2
+
Ω
2
−
Ω
1
)
s
in
3
(
1
2
)
I
f
we
ch
o
o
s
e
(
2
c
os
3
−
1
)
=
−
1
1
,
an
d
(
2
2
+
Ω
2
−
Ω
1
)
=
−
2
s
in
3
,
with
1
,
2
>
0
s
o
m
e
tu
n
in
g
p
ar
am
eter
s
,
we
g
et
co
n
tr
o
l la
ws
in
(
1
3
)
.
{
1
=
1
1
+
2
c
os
3
Ω
1
=
2
2
+
Ω
2
+
2
s
in
3
(
1
3
)
W
ith
th
ese
ch
o
ices (
1
2
)
r
e
d
u
c
es to
(
1
4
)
.
̇
=
−
1
1
2
−
2
s
in
2
3
(
1
4
)
Fro
m
(
1
4
)
we
ca
n
o
b
s
er
v
e
t
h
a
t
̇
(
0
,
2
,
0
)
=
0
wh
ich
m
ea
n
s
th
at
̇
is
n
eg
ativ
e
s
em
i
-
d
ef
in
ite.
I
n
t
h
is
p
ar
ticu
lar
ca
s
e,
co
n
s
id
er
i
n
g
t
h
e
f
ac
t
th
e
s
ec
o
n
d
ter
m
s
o
f
th
e
s
y
s
tem
d
escr
ib
ed
b
y
th
e
d
if
f
er
en
tial
eq
u
atio
n
s
(
1
0
)
ar
e
tim
e
-
v
ar
y
in
g
,
we
o
n
ly
n
ee
d
to
p
r
o
v
e
th
at
th
is
s
y
s
tem
is
m
ar
g
i
n
ally
o
r
l
o
ca
lly
s
tab
le
.
T
h
is
c
o
m
es
f
r
o
m
th
e
f
ac
t
th
at
th
e
m
o
s
t
im
p
o
r
ta
n
t
r
o
le
o
f
th
e
co
n
tr
o
ller
is
to
d
r
iv
e
th
e
ac
tu
al
r
o
b
o
t
t
o
war
d
s
t
h
e
tar
g
et
tr
ajec
to
r
y
,
k
ee
p
in
g
it
v
e
r
y
clo
s
e
to
th
e
v
ir
tu
al
r
o
b
o
t.
T
h
e
r
ef
o
r
e
,
if
w
e
co
u
ld
p
r
o
v
e
th
at
t
h
e
c
o
n
tr
o
ller
s
(
1
3
)
p
r
o
v
id
e
a
lo
ca
lly
s
tab
le
s
o
lu
tio
n
f
o
r
t
h
e
s
y
s
tem
(
1
0
)
th
is
p
r
o
b
lem
wo
u
l
d
th
en
b
e
s
o
lv
ed
.
Usi
n
g
th
e
L
y
ap
u
n
o
v
th
eo
r
y
a
n
d
th
e
B
ar
b
alat’
s
lem
m
a
[
2
9
]
a
s
u
f
f
icien
t
co
n
d
itio
n
is
to
p
r
o
v
e
th
at
th
e
s
ec
o
n
d
d
er
iv
ati
v
e
o
f
th
e
p
r
o
p
o
s
ed
L
y
ap
u
n
o
v
f
u
n
ctio
n
is
b
o
u
n
d
e
d
.
Fro
m
(
1
4
)
we
h
av
e:
̈
=
−
2
1
1
̇
1
−
2
2
̇
3
c
os
3
R
ep
lacin
g
̇
1
an
d
̇
3
f
r
o
m
(
1
0
)
an
d
u
s
in
g
1
an
d
Ω
1
d
ef
in
ed
in
(
1
3
)
we
g
et
(
1
5
)
.
̈
=
−
2
1
1
(
(
2
2
+
Ω
2
+
2
s
in
3
)
2
−
1
1
)
−
2
2
(
2
2
−
2
s
in
3
)
c
os
3
(
1
5
)
E
x
ce
p
t
f
r
o
m
th
e
c
o
n
s
tan
ts
1
an
d
2
all
elem
en
ts
o
f
(
1
5
)
d
e
p
en
d
o
n
th
e
s
tate
v
ar
iab
les
1
,
2
an
d
3
.
As
alr
ea
d
y
m
en
tio
n
e
d
,
̇
≤
0
th
is
m
ea
n
s
th
at
t
h
e
L
y
ap
u
n
o
v
f
u
n
ctio
n
d
ef
in
e
d
i
n
(
1
1
)
is
n
o
n
in
cr
ea
s
in
g
o
r
b
o
u
n
d
ed
,
th
er
ef
o
r
e
1
,
2
an
d
3
ar
e
b
o
u
n
d
e
d
,
wh
ich
i
n
tu
r
n
p
r
o
v
es t
h
at
̈
is
also
b
o
u
n
d
ed
.
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
NS
3
.
1
.
I
m
ple
m
ent
a
t
io
n det
a
ils
T
h
e
o
v
er
all
ar
c
h
itectu
r
e
o
f
t
h
e
ad
o
p
ted
c
o
n
tr
o
l
s
y
s
tem
is
s
u
m
m
ar
ized
in
Fig
u
r
e
2
.
Fro
m
th
is
b
lo
c
d
iag
r
am
we
ca
n
s
ee
th
at,
in
ad
d
itio
n
to
th
e
p
r
o
p
o
s
ed
m
ai
n
co
n
tr
o
ller
,
we
also
h
av
e
t
wo
PID
co
n
tr
o
ller
s
s
p
ec
if
ically
d
esig
n
ed
to
m
ak
e
th
e
le
f
t
an
d
r
ig
h
t
r
o
b
o
t
w
h
ee
ls
r
o
tate
with
th
e
a
n
g
u
lar
s
p
ee
d
s
an
d
,
r
eq
u
ir
ed
b
y
th
e
i
n
v
er
s
e
k
i
n
em
atics
b
lo
c.
I
t
s
h
o
u
ld
b
e
n
o
ted
t
h
at
th
ese
PID
co
n
tr
o
ller
s
h
a
v
e
b
ee
n
t
u
n
ed
tak
in
g
in
to
ac
co
u
n
t
th
e
ac
tu
al
r
o
b
o
t
d
y
n
am
ics,
in
clu
d
in
g
th
e
wh
ee
ls
attac
h
ed
to
th
e
s
h
af
t
o
f
th
e
d
r
iv
in
g
DC
g
ea
r
m
o
to
r
s
.
T
h
e
m
ea
s
u
r
ed
a
n
g
u
lar
s
p
ee
d
s
an
d
,
o
b
tain
e
d
f
r
o
m
th
e
ac
tu
al
r
o
b
o
t
o
r
its
eq
u
iv
alen
t
m
o
d
el
ar
e
th
en
f
ed
to
th
e
f
o
r
war
d
k
in
em
atics
b
lo
c,
wh
ich
in
tu
r
n
d
eliv
er
s
th
e
m
ea
s
u
r
ed
lin
ea
r
an
d
an
g
u
lar
s
p
ee
d
s
1
an
d
Ω
1
o
f
th
e
r
o
b
o
t
ch
as
s
is
.
T
h
ese
two
s
ig
n
als
ar
e
th
en
u
s
ed
to
cl
o
s
e
th
e
o
u
te
r
lo
o
p
m
ak
in
g
th
e
wh
o
le
s
y
s
tem
wo
r
k
as d
escr
ib
ed
.
3
.
2
.
E
x
perim
ent
a
l r
esu
lt
s
Af
ter
a
f
ew
tr
ial
-
an
d
-
er
r
o
r
s
ettin
g
s
in
s
im
u
latio
n
,
t
h
e
p
r
o
p
o
s
ed
co
n
tr
o
ller
p
ar
am
eter
s
1
an
d
2
h
av
e
b
ee
n
s
et
to
t
h
e
v
al
u
es
lis
ted
in
T
ab
le
1
.
T
h
e
p
e
r
f
o
r
m
an
ce
o
f
t
h
is
co
n
tr
o
ller
h
as
th
e
n
b
ee
n
test
ed
th
r
o
u
g
h
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
33
-
42
38
s
ev
er
al
s
ce
n
ar
io
s
,
am
o
n
g
wh
i
ch
th
e
r
esu
lts
o
f
a
n
ex
am
p
le
u
s
in
g
a
L
is
s
ajo
u
s
cu
r
v
e
ty
p
e
tr
ajec
to
r
y
,
d
e
f
in
ed
b
y
th
e
p
ar
am
etr
ic
eq
u
atio
n
s
:
2
(
)
=
2
(
cos
50
−
1
)
+
0
.
25
an
d
2
(
)
=
2
s
in
25
,
ar
e
s
h
o
wn
in
Fig
u
r
es
3
an
d
4.
A
v
is
u
al
in
s
p
ec
tio
n
o
f
th
eses
p
lo
ts
s
h
o
ws
th
at
th
is
tr
ac
k
in
g
ex
p
er
im
en
t
h
as
b
ee
n
s
u
cc
ess
f
u
lly
ac
h
iev
e
d
.
Af
ter
less
th
an
2
0
s
ec
o
n
d
s
,
th
e
ac
tu
al
r
o
b
o
t
tr
ajec
to
r
y
was
n
ea
r
ly
s
u
p
er
im
p
o
s
ed
o
n
th
e
r
ef
e
r
en
c
e
p
ath
d
r
aw
n
b
y
th
e
v
ir
tu
al
r
o
b
o
t.
T
h
is
ca
n
b
e
v
er
i
f
ied
in
th
e
s
ec
o
n
d
p
lo
t o
f
Fig
u
r
e
3
,
in
wh
ich
th
e
later
al
er
r
o
r
is
s
h
o
wn
to
b
e
v
er
y
clo
s
e
to
ze
r
o
.
As
ca
n
b
e
o
b
s
er
v
ed
in
Fig
u
r
e
4
,
th
e
ac
tu
al
r
o
b
o
t
tr
ajec
to
r
y
h
as
b
ee
n
s
lig
h
tl
y
d
ev
iated
f
r
o
m
th
e
r
ef
er
en
ce
p
at
h
af
ter
th
e
p
o
s
itio
n
in
d
icate
d
b
y
th
e
(
-
2
.
4
,
1
.
2
3
)
co
o
r
d
in
ates.
T
h
is
is
th
e
r
esu
lt
o
f
an
ex
ter
n
al
d
is
tu
r
b
an
ce
ca
u
s
ed
b
y
a
m
an
u
al
h
ar
d
p
u
s
h
o
n
th
e
r
i
g
h
t w
h
ee
l,
p
r
ev
en
tin
g
it f
r
o
m
r
o
tatin
g
f
r
ee
ly
d
u
r
i
n
g
a
s
h
o
r
t
p
er
io
d
,
s
tar
tin
g
at
th
e
7
0
s
ec
o
n
d
tim
e
s
am
p
le.
Fro
m
th
e
t
wo
last
p
lo
ts
o
f
Fig
u
r
e
3
,
o
n
e
ca
n
s
ee
th
at
t
h
e
co
n
s
eq
u
en
ce
s
o
f
s
lo
win
g
d
o
wn
th
e
r
ig
h
t
wh
ee
l
h
as
a
d
i
r
ec
t
ef
f
ec
t
o
n
th
e
lef
t
wh
ee
l
,
m
ea
n
in
g
t
h
at
th
e
p
r
o
p
o
s
ed
o
u
ter
lo
o
p
co
n
tr
o
ller
h
as b
ee
n
ab
le
to
p
o
s
itiv
ely
r
e
ac
t a
g
ain
s
t th
is
u
n
ex
p
ec
ted
p
e
r
tu
r
b
atio
n
,
allo
win
g
th
e
r
o
b
o
t to
b
e
b
ac
k
o
n
th
e
r
ef
er
en
ce
p
ath
a
f
ew
s
ec
o
n
d
s
later
.
T
h
is
is
also
d
u
e
to
th
e
im
p
o
r
tan
t r
o
le
p
lay
ed
b
y
th
e
two
PID
co
n
tr
o
ller
s
,
wh
ic
h
ar
e
r
esp
o
n
s
ib
le
f
o
r
m
atch
in
g
th
e
m
ea
s
u
r
ed
r
o
tatio
n
al
s
p
e
ed
s
o
f
b
o
th
w
h
ee
ls
with
th
o
s
e
co
m
p
u
ted
b
y
th
e
m
ain
co
n
tr
o
ller
.
T
h
is
ca
n
also
b
e
v
er
if
ied
i
n
th
e
last
two
p
lo
ts
o
f
Fig
u
r
e
3
,
wh
ich
s
h
o
ws th
at
th
ese
r
eq
u
ir
em
en
ts
ar
e
ef
f
ec
tiv
ely
f
u
lf
illed
.
Fig
u
r
e
2
.
B
lo
c
k
d
iag
r
am
o
f
th
e
co
m
p
lete
co
n
tr
o
l sy
s
tem
T
ab
le
1
.
PID
an
d
tr
ajec
to
r
y
co
n
tr
o
ller
s
’
p
ar
am
ete
r
s
1
2
0
.
0
1
2
7
0
.
1
5
7
3
0
0
.
1
2
.
5
tim
e
p
lo
ts
o
f
th
e
ac
tu
al
a
n
d
v
ir
tu
al
r
o
b
o
t
p
o
s
tu
r
es
later
al
er
r
o
r
r
esp
o
n
s
e
r
ig
h
t w
h
ee
ls
s
p
ee
d
s
lef
t w
h
ee
ls
s
p
ee
d
s
Fig
u
r
e
3
.
R
ea
l
-
tim
e
d
ata
ac
q
u
i
r
ed
d
u
r
in
g
t
h
e
tr
ac
k
in
g
o
f
a
L
i
s
s
ajo
u
s
ty
p
e
tr
ajec
to
ry
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
E
xp
erimen
ta
l v
a
lid
a
tio
n
o
f a
t
r
a
jecto
r
y
tr
a
ck
in
g
co
n
tr
o
ller
fo
r
a
tw
o
-
w
h
ee
led
mo
b
ile
r
o
b
o
t
(
B
o
u
a
lem
K
a
z
ed
)
39
Fig
u
r
e
4
.
E
x
p
er
im
e
n
tal
r
esu
lts
f
o
r
a
L
is
s
ajo
u
s
ty
p
e
tr
ajec
to
r
y
tr
ac
k
in
g
with
e
x
ter
n
al
d
is
tu
r
b
an
ce
s
I
n
a
s
ec
o
n
d
test
,
wh
er
e
r
o
b
o
t
h
as
b
ee
n
m
o
v
in
g
o
n
0
.
2
5
-
m
eter
g
r
id
m
ar
k
ed
s
u
r
f
ac
e,
th
e
s
n
ap
s
h
o
ts
s
h
o
wn
in
Fig
u
r
e
5
h
av
e
b
ee
n
s
elec
ted
f
r
o
m
a
1
5
0
s
ec
o
n
d
s
v
id
eo
r
ec
o
r
d
ed
d
u
r
in
g
th
is
ex
p
er
im
en
t.
Fig
u
r
e
6
g
iv
es
a
m
o
r
e
d
etailed
v
iew
o
f
s
o
m
e
o
f
th
e
s
ig
n
als
ac
q
u
ir
ed
d
u
r
in
g
th
is
p
r
o
ce
s
s
.
Fo
r
th
is
ex
am
p
le,
th
e
r
o
b
o
t
r
ef
er
en
ce
tr
ajec
to
r
y
is
a
h
y
p
o
tr
o
ch
o
id
d
ef
in
e
d
b
y
th
e
p
ar
am
etr
ic
eq
u
atio
n
s
:
2
(
)
=
0
.
36
c
os
0
.
2
+
0
.
164
c
os
0
.
3
an
d
2
(
)
=
0
.
36
s
in
0
.
2
−
0
.
164
s
in
0
.
3
.
As
ca
n
b
e
n
o
ticed
in
Fig
u
r
e
5
,
th
is
tr
ac
k
in
g
ex
p
er
im
en
t
s
h
o
ws
th
at
af
ter
an
in
itial
tr
an
s
itio
n
o
f
ab
o
u
t
6
0
s
ec
o
n
d
s
,
d
u
r
in
g
wh
ich
th
e
r
o
b
o
t
h
as
alm
o
s
t
co
m
p
leted
th
e
f
ir
s
t
r
o
u
n
d
,
p
as
s
in
g
th
r
o
u
g
h
th
e
p
o
s
itio
n
s
lab
eled
0
to
7
,
th
e
later
al
er
r
o
r
2
h
as
b
ee
n
g
r
ad
u
ally
d
ec
r
ea
s
in
g
m
ak
i
n
g
th
e
r
o
b
o
t m
o
v
e
v
e
r
y
clo
s
e
t
o
th
e
tar
g
et
t
r
ajec
to
r
y
.
Fr
o
m
th
e
u
p
p
e
r
p
a
r
t
o
f
Fig
u
r
e
6
,
we
ca
n
s
ee
h
o
w
th
e
r
o
b
o
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p
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.
4.
CO
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N
T
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p
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p
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.
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f
r
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m
th
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th
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l
p
o
in
t
o
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v
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is
k
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d
o
f
m
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an
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s
is
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er
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d
if
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icu
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to
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v
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th
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est
m
eth
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p
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d
ex
p
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s
h
o
wn
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at
t
h
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t
ak
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r
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u
ite
l
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n
g
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co
m
p
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to
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p
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test
.
T
h
is
ca
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b
e
ex
p
lain
ed
b
y
t
h
e
f
ac
t
t
h
at
d
u
r
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n
g
th
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in
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s
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o
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tr
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h
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y
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f
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W
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p
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tech
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th
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p
r
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p
o
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c
o
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tr
o
ller
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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41
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g
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6
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d
z
.
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