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.
4, N
o
. 1
,
Mar
c
h
20
15
,
pp
. 53
~62
I
S
SN
: 208
9-4
8
5
6
53
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
The Con
t
rol Desi
gn of Ship Form
ation with
the P
r
es
ence of a
Lead
er
Misw
an
to
1
, I.
Pran
ot
o
2
, H
.
M
u
ha
mma
d
3
, D
.
Ma
ha
ya
na
4
1
Departement of
Mathematics, Airlangg
a
Univ
ersity
(UNAIR), Su
rabay
a
, Indon
esia
2
Departem
ent
of
Mathem
at
ics, In
stitut
Tekno
logi Bandung
(ITB)
, Bandung,
Indon
esia
3
Aeronauti
c
s an
d Astronauti
c
s, I
n
stitut
Tekno
log
i
Bandung
(ITB)
, Bandung
, Indo
nesia
4
Control S
y
s
t
em
and Com
puter, I
n
stitut
Tekno
log
i
Bandung
(ITB)
, Bandung
, Indo
nesia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mei 28, 2014
Rev
i
sed
Au
g
20
, 20
14
Accepted
Sep 10, 2014
Fo
rm
atio
n
contro
l is an
i
m
p
o
r
tan
t
b
e
h
a
v
i
or for m
u
lti-ag
en
ts syste
m
(swa
rm
). Thi
s
pape
r ad
dre
sse
s t
h
e opt
i
m
al tracki
ng c
ont
r
o
l
pro
b
l
e
m
f
o
r
s
w
a
r
m w
h
o
s
e
ag
e
n
ts
ar
e s
h
ip
s
mo
v
i
ng to
g
e
th
er
in
a s
p
e
c
i
f
i
c
g
e
o
m
etr
y
f
o
r
m
atio
n
.
W
e
study f
o
r
m
atio
n
con
t
ro
l of
t
h
e swar
m
m
o
d
e
l
whi
c
h co
nsi
s
t
s
of t
h
ree a
g
ent
s
and
one a
g
e
n
t has a role a
s
a leader.
The a
g
ents
of
swarm
are
m
oving to
fo
llow th
e
lead
er p
a
th.
First, we
desi
g
n
t
h
e c
ont
rol
of
t
h
e
l
ead
er
wi
t
h
P
o
nt
ry
agi
n
M
a
xi
m
u
m
Pri
n
ci
pl
e.
The c
ont
rol
of
t
h
e l
eade
r
i
s
desi
g
n
e
d
f
o
r t
r
acki
n
g t
h
e
des
i
red
pat
h
.
We s
h
ow t
h
at
the trac
king e
r
ror
of
th
e p
a
th
o
f
th
e
le
ad
er
tr
a
c
i
ng
a
d
e
sired
p
a
th
is sufficien
tly small.
After that, ge
om
etry a
p
proach is
u
s
ed
to
d
e
sign
th
e co
n
t
ro
l of th
e o
t
h
e
r.
We sh
ow th
at th
e po
sitio
n
i
ng
and t
h
e o
r
i
e
nt
a
t
i
on o
f
eac
h ag
ent
can
be co
n
t
rol
l
e
d
depe
n
d
e
nt
o
n
t
h
e
lead
er.
Th
e
sim
u
la
tio
n
resu
lts sho
w
to
illu
strate o
f
t
h
is meth
od
at the
l
a
st
sect
i
on
of
t
h
i
s
pape
r.
Keyword:
Dy
nam
i
c Sy
st
em
of Shi
p
M
odel
of
Sw
arm
Nu
m
e
rical Si
mu
latio
n
Pontryagi
n
Ma
xim
u
m
Principle
Tracki
n
g Error
Copyright ©
201
5 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
:
Misw
an
to,
Depa
rtem
ent of Mathem
atics,
Facul
t
y
o
f
Sci
e
nces a
n
d Tec
h
nol
ogy
,
Airlangga Uni
v
ersity,
K
a
m
p
u
s
C Jl.
Mu
lyo
r
ej
o Su
rab
a
ya 601
15
, In
don
esia.
Em
a
il: miswanto
@fst.un
a
ir.ac.id
1.
INTRODUCTION
In
rece
nt years
,
there ha
ve
be
en a
n
inc
r
easing
num
ber
of
re
searche
s
on t
h
e su
bj
ect
o
f
un
deract
uat
e
d
vehicle.
The e
x
am
ple of
underactuated ve
hi
cle we will
foc
u
s
on is a surface vessel
(ship) m
oving t
o
t
r
ack a
desire
d path. The
trac
king problem
is
a
challengi
ng
problem
in surface vessel.
Some researc
h
ers ha
ve
discuss
e
d t
h
e c
ont
rol
design
of a s
u
rface
ves
s
el to trac
k a
de
sired
path in
[1], [3], [5],
[7]
.
In [1], the aut
h
ors
study the trac
king
problem
of
underactuated surface ve
ssel
using ada
p
tiv
e
cont
rol. T
h
ey
design a c
ontinuous
ti
m
e
-v
aryin
g
track
i
n
g con
t
roller th
at forces th
e po
siti
on/
ori
e
nt
at
i
o
n
t
r
a
c
ki
n
g
e
r
r
o
r to an arb
itrarily sm
al
l
n
e
igh
borhoo
d
ab
ou
t zero
in th
e presen
ce
o
f
u
n
c
ertain
ty in
t
h
e hy
dr
o
d
y
n
a
m
i
c
dam
p
i
ng c
o
ef
fi
ci
ent
s
.
In
[3]
,
t
h
e
au
tho
r
s stud
y t
h
e
p
r
ob
lem
o
f
p
o
s
ition
track
i
n
g
of
un
d
e
r
act
u
a
ted
v
e
h
i
cles
in
bo
th two
and
three-d
i
m
e
n
s
io
n
a
l
spaces. T
h
e main contri
bution is a
de
sign methodol
ogy
to
construct
a
nonlinear trac
ki
ng cont
roller
that yields
global stability and expone
ntial conve
rgence
of the position tracki
ng erro
r to a neighborhood
of the origi
n
t
h
at
can
be
m
a
de a
r
bi
t
r
a
r
i
l
y
sm
all
.
Furt
he
rm
ore, t
h
e
des
i
red t
r
a
j
ect
ory
d
o
es
n
o
t
nee
d
t
o
be
a t
r
i
m
m
i
ng
t
r
aject
o
r
y
an
d can be a
n
y
suf
f
i
ci
ent
l
y
sm
ooth t
i
m
e
-vary
i
n
g
bo
un
de
d cu
rv
e, i
n
cl
u
d
i
n
g t
h
e dege
nerat
e
c
a
se of a
con
s
t
a
nt
t
r
a
j
ec
t
o
ry
(set
-p
oi
nt
). I
n
[5]
,
t
h
e a
u
t
h
ors
st
u
d
y
t
h
e c
ont
r
o
l
des
i
gn a
n
d t
h
e t
r
acki
n
g
pr
obl
e
m
for a
no
nl
i
n
ea
r
un
de
ract
uat
e
d
sy
st
em
. They
descri
be
ho
w t
o
use
back
st
ep
pi
n
g
t
o
devel
o
p
co
nt
rol
l
a
ws t
o
pe
r
f
o
r
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
4,
No
. 1,
M
a
rc
h 20
1
5
:
5
3
– 62
54
trajectory trac
king for a
n
onlinear, underac
tuated surfac
e
vessel.
Hi
s res
earch e
x
tends
earlier bac
k
ste
ppi
ng
desi
g
n
s f
o
r
un
deract
uat
e
d ve
ssel
s
by
expl
ai
ni
n
g
h
o
w t
o
s
e
l
ect
out
p
u
t
s
whe
n
ge
ne
ral
i
zed f
o
rces act
on t
h
e
vessel.
In [7],
the aut
h
ors c
o
nside
r
trac
king control of a s
u
rface
vessel
with
only two
cont
rol i
n
puts. The
y
d
i
scu
s
s track
i
ng
co
n
t
ro
l
o
f
both
th
e
po
sitio
n v
a
riab
les an
d
th
e course an
gle o
f
t
h
e su
rface v
e
ssel. Th
ey th
u
s
seek t
o
c
ont
r
o
l
angl
e
de
grees
of
f
r
eed
om
wi
t
h
onl
y
t
w
o c
o
n
t
rol
i
n
p
u
t
s
va
ri
abl
e
.
In
[1
0
]
, t
h
e au
tho
r
stud
ies th
e
p
r
o
b
l
em
o
f
con
t
ro
lling
the p
l
an
ar
p
o
sitio
n and
o
r
ien
t
atio
n
o
f
an
autonom
ous
s
u
rface
vessel
using two i
n
depende
n
t thrus
t
er
s.
He s
h
ows that although the
system
is not
asy
m
p
t
o
tically
stab
ilizab
le to a g
i
v
e
n equ
ilib
ri
u
m
so
lu
tio
n u
s
ing
a tim
e-i
n
v
a
rian
t con
tinu
o
u
s
feed
b
a
ck
, it is
strongly accessible and sm
all-ti
m
e
lo
cally controllable
at any equilibri
um
and, he
nce, t
h
e syste
m
is
asy
m
p
t
o
tically
stab
ilizab
le to
a d
e
sired
equ
ilib
riu
m
u
s
in
g
t
i
m
e
-in
v
a
rian
t d
i
scon
tinu
o
u
s
feedb
a
ck
laws. In
[2
],
th
e au
t
h
ors con
s
id
er a lin
ear syste
m
with
d
e
lay in
state and control
with bo
t
h
m
a
tched and
unm
atched
pert
ur
bat
i
o
n
s
.
They
ap
pl
y
t
h
e bl
oc
k co
nt
r
o
l
t
echni
q
u
e t
o
desi
g
n
a sl
i
d
i
ng m
ode re
g
u
l
a
t
o
r t
h
at
g
u
ara
n
t
ees
asym
pt
ot
i
c
ref
e
rence
t
r
acki
n
g f
o
r a cl
ass
of
l
i
n
ear
del
a
y
e
d
sy
st
em
s wi
t
h
di
st
ur
ba
nces.
T
h
i
s
cl
ass
of
sy
st
em
s i
s
t
hose
p
r
esent
e
d i
n
so
-cal
l
e
d
bl
oc
k c
ont
rol
l
a
bl
e f
o
rm
wi
t
h
del
a
y
.
T
h
e bl
o
c
k c
ont
r
o
l
t
ech
ni
q
u
e i
s
use
d
t
o
deri
v
e
a slid
in
g
m
a
n
i
f
o
ld
on
wh
ich
th
e m
o
tio
n
o
f
th
e clo
s
ed-
l
oop syste
m
is
stab
le, an
d
th
e tr
ack
i
ng
er
ro
r
is zer
o
e
d
.
In [
4
]
,
an ada
p
t
i
v
e t
r
acki
n
g cont
rol
p
r
o
b
l
e
m
i
s
st
udi
ed fo
r a fou
r
w
h
eel
m
obi
l
e
robot
.
The aut
h
o
r
s pr
op
ose a
form
u
l
atio
n
for th
e ad
ap
tiv
e
track
ing
p
r
ob
l
e
m
th
at m
e
e
t
s th
e n
a
t
u
ral
prerequ
i
site su
ch
th
at it redu
ces
to
th
e
st
at
e feedbac
k
t
r
acki
n
g
pr
obl
e
m
i
f
t
h
e param
e
t
e
rs are k
n
o
w
n
. T
h
ey
de
ri
ve
a gene
ral
m
e
t
h
od
ol
o
g
y
f
o
r s
o
l
v
i
n
g
t
h
ei
r pr
obl
em
.
In
[9], t
h
e aut
h
ors
study the form
at
i
on co
nt
r
o
l
o
f
swa
r
m
whos
e age
n
ts are Dubin’s ca
rs. The a
g
ent
s
of s
w
arm
are m
ovi
ng t
o
t
r
ac
k a desi
re
d pat
h
. T
h
ey
co
nsi
d
er th
e swarm
m
o
d
e
l with
p
r
e
s
ence of a leader. First,
they design t
h
e control
of the leader
with t
r
acki
ng e
r
ror dynamics. The
cont
rol
of t
h
e l
eader is
desi
gned for
t
r
acki
n
g t
h
e d
e
si
red pat
h
. A
f
t
e
r t
h
at
, ge
om
et
ry
appr
oac
h
i
s
used t
o
de
si
gn t
h
e c
ont
rol
of t
h
e ot
her
.
In [
8
]
,
Miswanto et al. study the t
r
ac
king
pr
obl
em
of a
s
w
arm
m
odel
wi
t
h
t
h
e
pr
es
ence
of a lea
d
er
by
using t
h
e least
squ
a
re m
e
t
hod
. That
m
odel
is a cont
r
o
l
sy
st
em
whi
c
h co
n
s
i
s
t
s
of m
a
ny
agent
s
a
nd
o
n
e
agent
ha
s a ro
l
e
as a
l
eader. T
h
e c
o
nt
r
o
l
of
o
p
t
i
m
a
l
m
o
t
i
on o
f
t
h
e
l
eader i
s
o
b
t
a
i
n
ed
by
u
s
i
n
g t
h
e l
east
sq
uare
m
e
t
hod.
I
n
pa
r
t
i
c
ul
ar,
this control ste
e
rs the leader t
o
trace a
desired pat
h
.
In
[6], Tang et al. study op
tim
a
l output trac
king c
o
ntrol
(OOTC)
prob
l
e
m
fo
r a class o
f
b
ilin
ear
syste
m
s with
a q
u
a
dratic p
e
rfo
rm
an
ce in
d
e
x u
s
ing
a su
ccessiv
e
app
r
oxi
m
a
t
i
on app
r
oac
h
(S
A
A
). T
h
ey
de
ve
l
op a desi
gn
p
r
oces
s of t
h
e
OOTC
l
a
w
ba
sed o
n
t
h
e S
A
A fo
r
b
ilin
ear system.
In t
h
i
s
pa
per
we co
nsi
d
er
f
o
rm
at
i
on co
nt
r
o
l
o
f
the s
w
arm
m
odel whose
age
n
ts are
ships m
oving
t
oget
h
e
r
i
n
a s
p
eci
fi
c ge
om
etry
fo
rm
at
i
on. I
n
t
h
i
s
m
ode
l,
one s
h
i
p
has a
role as a leader. T
h
e control
of t
h
e
leader is
desi
gned for tracki
n
g the
de
si
r
e
d
p
a
th
.
W
e
show
th
at t
h
e tr
ack
i
ng
error
of t
h
e
path
of the
leader
tracin
g
a d
e
si
red
p
a
th
is suffi
cien
tly s
m
a
ll a
n
d
th
e
d
i
stan
ce
between the le
ader s
h
ip pat
h
and the
desire
d path
i
s
pr
eser
ve
d.
I
n
t
h
e
next
sect
i
on,
t
h
e
f
o
rm
al
pr
o
b
l
e
m
for
m
ul
at
i
on i
s
de
scri
be
d.
I
n
sec
t
i
on
3,
we
des
i
gn
t
h
e
cont
rol of the leader s
h
ip usi
n
g Pontryagi
n
Maxim
u
m Pr
inciple. In section
4, we
de
sign the control of each
agent followe
r using ge
om
etr
y
approach
.
In
sectio
n
5
,
we sh
ow
nu
m
e
rical
sim
u
lat
i
o
n
s
t
o
illu
strate ou
r
resu
lts.
2.
PROBLEM FORMUL
ATION
In th
is section
,
we in
t
r
odu
ce t
h
e
d
y
n
a
m
i
c sy
ste
m
o
f
th
e m
o
d
e
l sh
ip
as sh
own in
Figu
re 1.
Fi
gu
re 1.
The
m
odel
shi
p
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0
8
The C
o
nt
rol
D
e
si
gn
of
F
o
r
m
a
t
i
on
Shi
p w
i
t
h
t
h
e Pre
s
ence
of
a Le
a
d
er (
M
i
s
w
ant
o)
55
In t
h
i
s
pa
pe
r, t
h
e
dy
nam
i
cs sy
st
em
of shi
p
i
s
t
a
ked
f
r
om
Tzeng
an
d C
h
en
m
odel
.
They
onl
y
di
scuss
o
n
e s
h
i
p
and
i
n
t
h
i
s
pap
e
r
di
scuss
f
o
r
t
h
ree
shi
p
s
w
h
i
c
h
descri
bed
as
:
tcos
tsin
tsin
tcos
0
0
0.0562
1
,2,3
1
Whe
r
e
,
∈
re
pre
s
ent
s
t
h
e
p
o
si
t
i
on
of t
h
e
i
-
t
h shi
p
, a
n
d
∈
0
,2
represen
ts the
ori
e
nt
at
i
on of
t
h
e
i
-th
s
h
ip.
are v
e
lo
cities in
su
rg
e
o
f
th
e
i
-t
h s
h
ip,
are v
e
l
o
cities in
sway
of
t
h
e
i
-th
ship,
are
yaw
rate
of
t
h
e
i
-th
shi
p
a
n
d
are
ru
der
a
ngl
e
o
f
t
h
e
i
-t
h s
h
i
p
.
I
n
t
h
i
s
pape
r,
t
h
e
desi
r
e
d
pat
h
th
at would
b
e
t
r
ack
ed
by th
e lead
er ship
is ob
tain
ed
u
s
ing
calcu
l
u
s
v
a
riation
a
l m
e
t
h
od
. Th
e p
a
t
h
is
d
e
no
ted b
y
,
.
In
t
h
is
p
a
p
e
r, th
ere are t
w
o pro
b
l
em
s wh
ich
will b
e
d
i
sc
u
ssed
.
First,
we
desig
n
th
e con
t
ro
l of the lead
er
shi
p
fo
r t
r
ac
ki
ng t
h
e
desi
re
d
pat
h
by
P
ont
ry
agi
n
M
a
xi
m
u
m
Pri
n
ci
pl
e.
F
u
rt
herm
ore,
w
e
desi
g
n
t
h
e
c
ont
rol
of
t
h
e ot
her
age
n
t
s
by
ge
om
et
ry
app
r
oach
t
o
fol
l
ow t
h
e leader'
s
pat
h
with a c
e
rtain
distance.
3.
THE CONT
ROL DESIGN
OF THE LE
ADER
SHIP
We c
o
nside
r
a
m
odel of the
leader s
h
i
p
, s
u
ch as
(1
).
We
d
e
sign
t
h
e co
ntro
l
o
f
th
e lead
er sh
i
p
b
y
Pontryagi
n
Ma
xim
u
m
Principle for m
i
ni
mizing the tracki
n
g error
i
n
o
r
de
r t
o
kee
p
t
h
e po
si
t
i
on of
t
h
e
l
e
ader
sh
ip
close to
the d
e
sired
p
a
th
.
We de
fine a tr
ackin
g er
ro
r
e
(
t
) as the difference bet
w
een t
h
e actual leade
r
ship
pat
h
a
n
d t
h
e
de
si
red
pat
h
,
1
.
2
Thus,
e
(
t
) liv
es in
for
ev
er
y
t
. Th
e
o
r
ig
i
n
al
prob
lem
is tran
slated
to
t
h
e
fo
llo
wi
n
g
o
p
tim
al
co
n
t
ro
l
prob
lem
.
W
e
sear
ch fo
r
th
at m
a
k
e
s th
e
fo
llowing
fun
c
tio
n
a
l m
i
n
i
miz
e
d
,
1
2
‖
‖
.
3
The term
represen
ts th
e to
tal co
st of t
h
e co
n
t
ro
l
u
s
ed
b
y
th
e lead
er sh
i
p
an
d
‖
‖
represen
ts th
e t
o
tal
er
ro
r.
In
th
is pap
e
r,
th
e
v
a
lu
e o
f
th
e con
s
tant
k
is restricted
to
1
.
No
w,
th
e Ham
i
lto
n
i
an
fu
n
c
tion
of
th
e
syste
m
is
1
2
‖
‖
.4
Using
t
h
is fun
c
tio
n
,
we
bu
ild
t
h
e
Ham
i
lto
n
i
an
system
:
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I
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56
IJR
A
V
o
l
.
4,
No
. 1,
M
a
rc
h 20
1
5
:
5
3
– 62
56
1
1
1
t
co
s
1
1
t
sin
1
2
1
1
t
sin
1
1
t
co
s
1
3
1
0
4
1
0
5
1
1
6
1
0.0562
1
1
1
1
0
1
1
2
0
1
1
3
1
co
s
1
2
sin
1
5
1
4
1
sin
1
2
co
s
1
1
5
1
1
1
sin
1
1
1
co
s
1
2
1
1
co
s
1
1
1
sin
1
1
6
5
0
.0562
6
B
y
t
h
e Pont
ry
agi
n
M
a
xi
m
u
m
Princi
ple,
the value of
H
m
u
st b
e
op
ti
m
i
ze
d
with
resp
ect to
th
e con
t
ro
l
.
Thus
0
, sinc
e
0
p
m
u
st
be const
a
nt
a
nd
n
e
gat
i
v
e,
wi
t
h
o
u
t
l
o
ss
o
f
g
e
n
e
rality, we
let
0
1
p
. Th
us,
we
obt
ai
n
t
h
e
c
o
nt
rol
. Th
en, th
is co
n
t
ro
l
is substitu
ted
in
(5).
Thu
s
, we ob
tain
a
system
o
f
d
i
fferen
tial
eq
u
a
tion
s
:
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7
0
8
The C
o
nt
rol
D
e
si
gn
of
F
o
r
m
a
t
i
on
Shi
p w
i
t
h
t
h
e Pre
s
ence
of
a Le
a
d
er (
M
i
s
w
ant
o)
57
1
1
t
co
s
1
1
t
sin
1
1
1
t
sin
1
1
t
co
s
1
1
0
1
0
1
1
1
0.0562
1
6
1
1
2
1
3
1
co
s
1
2
sin
1
4
1
sin
1
2
cos
1
5
1
1
1
sin
1
1
1
co
s
1
2
1
1
co
s
1
1
1
sin
1
6
0
.0562
6
5
6
In
itial and final con
d
ition
s
o
f
th
e state
variab
les
,
,
,
,
,
ar
e kn
own
.
How
e
ver
,
t
h
e
costate equations
;
1,2,
…
6
do
es no
t
h
a
v
e
in
itial co
nd
itio
n. So
, th
e so
lu
tion
o
f
th
e system o
f
di
ffe
re
nt
i
a
l
equat
i
on i
s
di
ffi
cul
t
t
o
obt
ai
n
.
Thi
s
pa
per
pr
op
oses t
h
e st
eepest
ga
di
ent
desce
n
t
m
e
t
hod. T
h
i
s
m
e
thod is use
d
to approxim
ate the
initial
condition
of t
h
e costate va
riab
les in the s
y
stem
of diffe
rential
eq
u
a
tion
.
Fi
rst, th
e in
itial v
a
l
u
e
of state v
a
riab
les are
g
i
v
e
n
b
y
0
;
1
,2
,3
,
4
,5
,6
an
d t
h
e i
n
itial
value
of
costat
e varia
b
les are
arbitra
r
ily guessed
by
0
;
1
,2
,3
,4
,5
,6
. The
values a
r
e
us
ed to
so
lv
e th
e syste
m
o
f
d
i
fferentia
l
equat
i
on. Ne
xt
, we
cal
cul
a
t
e
6
2
1
0
20
30
40
50
6
0
1
,,
,,
,
(
)
ii
T
i
F
q
qq
qq
q
x
T
x
, whe
r
e
()
i
x
T
i
s
obt
ai
ned fr
om
t
h
e sol
u
t
i
o
n of t
h
e
sy
st
em
of di
ff
erent
i
a
l
equat
i
on a
nd
iT
x
is th
e fin
a
l co
nd
itio
n
s
o
f
th
e st
ate v
a
riab
les.
Afterward
s
, we
d
e
term
in
e th
e v
a
lu
e
o
f
th
e n
e
w
10
20
30
40
50
60
,,
,,
,
qq
q
q
q
q
by
usi
n
g t
h
e st
ee
pest
gra
d
i
e
nt
desc
ent
m
e
t
hod,
s
u
ch
as i
n
T
j
a
h
jana
[7]
.
The
val
u
e
i
s
use
d
t
o
m
a
ke t
h
e
new
10
20
30
40
5
0
6
0
,,
,,
,
F
qq
q
q
q
q
less
th
an
th
e
o
l
d
10
20
30
40
5
0
6
0
,,
,,
,
F
qq
q
q
q
q
. Th
e
p
r
o
cess is d
o
n
e
repeated
ly u
n
til th
e v
a
lu
e
o
f
10
20
30
40
5
0
6
0
,,
,,
,
F
qq
q
q
q
q
is sm
a
ll en
o
ugh
.
4
.
THE CONTROL DESIGN
OF THE
FOLLOWING AGENTS
In t
h
i
s
sect
i
o
n
,
we desi
gn t
h
e cont
r
o
l
o
f
t
h
e fol
l
o
wer
usi
ng
geom
et
ry
app
r
oach
. Fi
g
u
r
e
2 sh
ows t
h
e
p
o
s
ition
o
f
th
e th
ree
sh
ip
s. Wh
ere
1 an
d
2 are th
e d
i
stan
ce o
f
th
e
fo
llo
wer s
h
ip t
o
the l
eader shi
p
.
and
are th
e orien
t
at
io
n
o
f
th
e
fo
llower sh
ip
t
o
positio
n
of th
e lead
er sh
ip. In
this p
a
p
e
r,
and
are assum
e
d to
be c
o
n
s
t
a
nt
.
4.
1.
T
h
e C
o
nt
rol
De
si
gn
of The
First Foll
ower
Ship
From
t
h
e Fi
gu
r
e
2,
we
ha
ve
sin
co
s
(
7
)
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56
IJR
A
V
o
l
.
4,
No
. 1,
M
a
rc
h 20
1
5
:
5
3
– 62
58
Fig
u
re 2
.
Th
e Po
sition
o
f
Three
Sh
i
p
s
Differen
tiatin
g th
e eq
u
a
ti
o
n
s
ab
ov
e
with
resp
ect to
tim
e, we ob
tain
cos
sin
,
8
because
assum
e
d to be constant, then the system
of equations (8) above can be written
as
cos
sin
9
Thu
s
, we ob
tain
cos
sin
10
Differen
tiatin
g th
e eq
u
a
ti
o
n
s
(1
0) abo
v
e
with resp
ect to
tim
e
,
we ob
tain
,
co
s
sin
sin
cos
0.0562
11.
Sub
s
titu
tin
g syste
m
(1
) fo
r i = 2
i
n
th
e equ
a
ti
o
n
(11
)
, on
e
obtain
0
.0562
2
sin
co
s
sin
cos
sin
cos
12
Th
en
, th
is con
t
ro
l
is sub
s
titu
ted
to th
e system
(1
) with
2
. Thu
s
,
we
ob
tain
a syste
m
o
f
th
e
d
i
fferen
tia
l
eq
u
a
tion
s
of the first
fo
llower
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8
The C
o
nt
rol
D
e
si
gn
of
F
o
r
m
a
t
i
on
Shi
p w
i
t
h
t
h
e Pre
s
ence
of
a Le
a
d
er (
M
i
s
w
ant
o)
59
tcos
tsin
tsin
tcos
0
0
13
2
sin
2
co
s
2
sin
cos
sin
cos
4.
2. T
h
e
Con
t
rol
De
si
gn
of
The Sec
o
nd F
o
llower
From
t
h
e fi
gu
r
e
2,
we
ha
ve
sin
co
s
14
Using sim
ilar steps s
u
ch as i
n
4.1., one m
a
y
desi
g
n
t
h
e c
ont
rol
of
t
h
e
seco
nd
f
o
l
l
o
wer
0
.0562
2
sin
2
cos
2
sin
cos
sin
cos
15
Th
en
, t
h
is con
t
ro
l
is su
b
s
titu
t
e
d
to th
e
system
(1
) with
3
. Thu
s
,
we
ob
tain
a syste
m
o
f
t
h
e
d
i
fferen
tial
eq
u
a
tion
s
of the second
fo
llower:
tcos
tsin
tsin
tcos
0
0
16
2
sin
2
cos
2
sin
cos
sin
co
s
5.
N
U
M
E
RICAL SOMU
LATION
In
t
h
is section
,
so
m
e
n
u
m
erical si
m
u
latio
n
s
to
illu
stra
te th
e syste
m
(6
), (13
)
and
(16
)
are reported
.
Th
e
co
n
t
ro
l d
e
si
gn
o
f
th
e lead
er sh
ip
used
th
e Po
n
t
ryag
in
Maxi
m
u
m
Prin
cip
l
e fo
r
o
p
tim
al c
o
n
t
ro
l prob
lems. To
ap
pro
ach
t
h
e in
itial v
a
lu
e of
th
e co-state v
a
riab
les used
st
eep
est grad
ien
t
d
e
scen
t m
e
th
o
d
. Fu
rt
h
e
rm
o
r
e, th
e
cont
rol
desi
gn
of t
h
e f
o
l
l
o
w
e
r shi
p
used
g
e
om
et
ry
appro
ach (M
i
s
wa
nt
o
,
et
al
., 201
2)
.
Fi
rst
of al
l
gi
ven a
desire
d pat
h
traced by the le
ader
ship. The
Start position and e
n
d
posit
i
on
of t
h
e desired
path is
0
0,
10
da
n
20
2
.
78,
143
.14
. Th
e
d
e
sired p
a
t
h
is
writ
ten
as
fo
llows:
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56
IJR
A
V
o
l
.
4,
No
. 1,
M
a
rc
h 20
1
5
:
5
3
– 62
60
cos
0.15
sin
0.15
sin
0.15
cos
0.15
17
Th
e pro
b
l
em
i
n
th
is section
is to
d
e
sign
the co
n
t
ro
l fo
r t
h
ree sh
ip
s, so
th
e th
ree sh
ip
m
o
v
e
to
fo
llow the
d
e
sired
p
a
th fro
m
th
e
in
itial p
o
s
ition
(
0
sec
o
nds
)
t
o
t
h
e
en
d
(
2
0
second
s).
First
o
f
all is
d
e
sign
ed
m
o
ti
on co
nt
r
o
l
of t
h
e l
eade
r
shi
p
f
o
r t
r
ac
k a desi
red
p
a
t
h
. Desi
gn c
ont
rol
o
f
t
h
e
l
eader s
h
i
p
us
ed t
h
e
Pon
t
riag
y
n
Max
i
m
u
m
Prin
cip
l
e. Tab
l
e
1
b
e
lo
w
shows st
art
p
o
sitio
n
and
en
d
po
sitio
n of
th
e lead
er sh
i
p
. Th
e
lead
er sh
ip is ex
p
ected
to m
a
n
e
u
v
e
r tr
aci
ng t
h
is
path as
clos
e as
possi
ble.
Tab
l
e 1. Start po
sitio
n (
0
seco
nd) an
d end
p
o
s
ition
(
2
0
second)
of the lea
d
er s
h
ip.
x
(0
)
m
y
(0
)
m
u
(0
)
v
(0
)
(0)
deg
r
(0
)
x
(2
0)
m
y
(2
0)
m
u
(20
)
v
(2
0)
(2
0)
deg
r
(2
0)
0
10
10
0.
5
90
0
2.
78
14
3.
1
4
10
0.
5
-9
0
-9
The desi
re
d pa
t
h
i
n
e
q
uat
i
o
n
(17
)
is sub
s
titu
ted
i
n
(6) wit
h
th
e p
a
ram
e
te
r
1
. As e
x
plained a
b
ove, the
initial value of the co-state variable
approxim
ated by the m
e
thod
of stee
pe
st gradie
nt descent. Fi
g. 3
bel
o
w
sh
ows th
e t
r
aj
ecto
ry
o
f
t
h
e lead
er sh
i
p
traci
ng
th
e d
e
sired path
b
y
u
s
i
n
g the m
e
th
o
d
.
Fr
o
m
Figu
r
e
3 abov
e sh
ow
s
th
at th
e t
r
aject
ory
of the
lea
d
er shi
p
(so
lidlin
e) can
track th
e
d
e
sired
p
a
th (circle)
fro
m
th
e strart
po
sitio
n
0
second
s t
o
t
h
e end
po
sitio
n
2
0
second
s
w
ith a sm
al
l eno
ugh
d
i
stan
ce. Th
is
mean
s th
at th
e lead
er sh
ip
can
m
o
v
e
fr
o
m
o
n
e area (th
e
startin
g
po
sition
)
t
o
ano
t
h
e
r area
(fin
a
l
p
o
s
ition
)
,
h
o
p
e
o
f
t
h
e resu
lts
o
f
nu
m
e
rical si
m
u
la
tio
n
s
ar
e t
h
e fi
rst fo
llo
wer sh
i
p
a
n
d
t
h
e seco
nd
fo
llower sh
ip
also
m
o
v
e
to
fo
llow t
h
e
traj
ectory of
t
h
e leader shi
p
from
the
start position
0second
s
to t
h
e
end
p
o
s
ition
20seconds
. Tra
j
ectory error be
tween the
t
r
aje
c
tories of the l
eader
sh
ip
with
d
e
sired
p
a
th can
be see
n
i
n
Fi
gu
re
4.
F
i
g
u
r
e
3
.
T
r
aj
ec
to
r
y
o
f
th
e
le
ad
e
r
s
h
ip
an
d
t
h
e de
si
red
pat
h
.
Fi
gu
re 4.
Tra
j
e
c
t
o
ry
Err
o
r
bet
w
een the t
r
ajec
tory
of t
h
e lea
d
er
shi
p
an
d desi
re
d pat
h
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
The C
o
nt
rol
D
e
si
gn
of
F
o
r
m
a
t
i
on
Shi
p w
i
t
h
t
h
e Pre
s
ence
of
a Le
a
d
er (
M
i
s
w
ant
o)
61
Nex
t
, we sh
ow th
e
nu
m
e
rical
si
m
u
latio
n
s
to
illu
strate m
o
d
e
l (1)
in the
two dim
e
nsional s
p
ace.
The
m
ove
m
e
nt
of t
h
e t
w
o ag
e
n
t
s
are
desc
ri
b
e
d by
t
h
e sy
st
em
s (13
)
a
nd
(
1
6
)
a
nd t
h
e l
e
ader i
s
desc
ri
b
e
d by
t
h
e sy
st
em
(6).
Distance
b
e
tween
t
h
e in
itial
p
o
s
ition of th
e first
fo
llower
sh
ip to th
e i
n
itial po
sitio
n
o
f
lead
er sh
ip is
50
m
and the
distance
b
e
t
w
een th
e in
itial po
sitio
n
of t
h
e seco
nd
fo
llo
wer
sh
ip to
i
n
itial p
o
s
ition
o
f
the
lead
er sh
ip is
5
0
m
.
Orien
t
atio
n
ang
l
e
o
f
t
h
e fi
rst fo
llower sh
ip
and
seco
nd
fo
ll
o
w
er
sh
ip
to
t
h
e
l
eader s
h
i
p
i
s
36
de
grees
an
d
10
de
gree
s. I
n
t
h
i
s
si
m
u
l
a
t
i
o
n
,
v
e
lo
city o
f
th
ree sh
ip is assu
m
e
d
to
equ
a
l. Th
e
in
itial co
nd
itio
n
s
o
f
th
e t
h
ree
sh
ip
s are g
i
v
e
n as fo
llows.
Tab
l
el 2. In
itial p
o
sitio
n
o
f
three sh
ip
Kapal
x
(0
)
m
y
(0
)
m
u
(0
)
v
(0
)
(0)
Deg
r
(0
)
M
a
st
er
0
10
10
0.
5
90
0
Sl
ave
I
40
-2
0
10
0.
5
90
0
Slav
e
I
I
-
40
-
20
1
0
0
.
5 9
0
0
Nex
t
,
we show th
e traj
ect
o
r
y o
f
th
e m
o
d
e
l (1
) fo
r thre
e
ship. Traject
ory of the th
ree
ship can be seen in
Fi
gu
re 4.
Fi
gu
re
5.
Tra
j
e
c
t
o
ry
of t
h
ree
s
h
i
p
f
o
rm
at
i
on
Th
e resu
lts of n
u
m
erical si
m
u
la
tio
n
s
in
Fig
u
re 4
abo
v
e sh
ows th
at
two
fo
llower
sh
ip
s can
fo
llo
w the
traj
ectory
o
f
the lead
er sh
ip mo
v
e
s
fro
m
th
e i
n
itial p
o
sitio
n
0
secon
d
s
) t
o
end
po
sitio
n (
2
0
sec
o
nds
)
.
Fro
m
Fig
u
re 4, it can
b
e
seen
th
at th
e triang
le form
at
i
on of
three s
h
ip are
prese
r
ve
d
with th
e lead
er
p
o
sitio
n
s
o
n
th
e po
in
ting
p
a
rt, bu
t t
h
e
measu
r
em
en
t o
f
th
e t
r
iang
le
is no
t preserv
e
d
.
Th
e
m
easu
r
e
m
en
t o
f
th
e trian
g
l
e
form
at
io
n
is smaller.
W
e
su
sp
ect th
at t
h
is is cau
se
d
by
t
h
e
at
t
r
act
i
on
fu
nct
i
on t
h
at
i
s
t
o
o s
t
ro
ng
.
6.
CO
NCL
USI
O
N
A Fro
m
th
e numerical si
m
u
la
tio
n
resu
lts ab
ov
e, it can
b
e
seen
th
at th
e track
i
ng
erro
r
o
f
t
h
e p
a
th
of th
e
lead
er sh
ip
tracin
g
a
d
e
sired
path
is
sufficiently s
m
a
ll and the dista
n
ce bet
w
een t
h
e pat
h
of lea
d
er s
h
ip
and t
h
e
desi
re
d pat
h
i
s
prese
r
ved
.
A g
e
om
et
ry
appr
o
ach fo
r f
o
rm
at
ion
c
ont
r
o
l
of
a gr
o
up o
f
shi
p
i
s
i
nvest
i
g
at
ed i
n
t
h
i
s
pape
r. T
h
e si
m
u
l
a
t
i
on o
n
t
h
ree s
h
i
p
fo
r
m
at
i
on dem
ons
trates that the propose
d
meth
od
is effect
iv
e and
feasib
le.
In the fu
t
u
re
wo
rk
s, we
will d
i
scu
ss t
h
e m
o
v
e
men
t
co
n
t
ro
l
of m
o
d
e
l swarm
co
n
s
istin
g
of sev
e
ral
ships
with a s
p
ecific ge
om
etr
y
form
ation.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
4,
No
. 1,
M
a
rc
h 20
1
5
:
5
3
– 62
62
ACKNOWLE
DGE
M
ENTS
Au
t
h
ors
wo
u
l
d lik
e to
t
h
ank
th
e m
y
in
sti
t
u
tio
n
(Ai
r
lang
g
a
Un
iv
ersity) for fi
n
a
n
c
ial su
pp
orting
th
is
researc
h
.
The
aut
h
o
r
s
wo
ul
d al
s
o
l
i
k
e t
h
ank
s
Di
rect
o
r
a
t
e Gene
ral
of
Hi
g
h
e
r
E
duc
at
i
on
Depa
rt
m
e
nt
o
f
Nat
i
onal
E
d
uc
at
i
on,
In
d
onesi
a. Thi
s
re
sear
ch was
su
p
p
o
r
t
e
d by
U
n
gg
ul
an Per
g
ur
ua
n
Ti
ng
gi
(
U
PT
)
gra
n
t
un
de
r
c
ont
ract
num
ber 7
6
7
3
/
U
N
3
/
K
R
/
20
1
3
,
2
M
a
y
2
0
1
3
.
REFERE
NC
ES
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Behal,
D.
M. Dawson,
B.
Xi
an,
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c
k
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a
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d S
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s
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