TELKOM
NIKA
, Vol.13, No
.1, March 2
0
1
5
, pp. 211~2
2
0
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i1.791
211
Re
cei
v
ed O
c
t
ober 2
3
, 201
4; Revi
se
d Ja
nuary 16, 20
1
5
; Acce
pted Janua
ry 3
0
, 20
15
Nonlinear Filtering with IMM Algorithm for Coastal
Radar Target Tracking System
Rika Sus
t
ika
1
*, Joko Sury
ana
2
1
Research C
e
nter for Informatics, Indon
esi
an Institute of Scienc
es (LIPI)
Jl Cisitu 2
1
/15
4
D, Band
un
g 4
013
5, Indon
esi
a
2
Bandu
ng Insti
t
ute of
T
e
chnol
og
y (IT
B
),
Jl. Ganesha N
o
10 Ban
d
u
ng
401
35, Indo
ne
sia
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: rika00
2@l
i
pi.
go.id
A
b
st
r
a
ct
T
h
is paper pr
esents a p
e
rformanc
e eval
u
a
tion of
n
onl
in
ear filteri
ng w
i
th Interacting
Multipl
e
Mode
l (IMM) algorit
hm
f
o
r i
m
ple
m
entatio
n o
n
Indo
nesi
an
coastal r
adar t
a
rget tracki
ng
system
. On thi
s
radar,
targ
et motion is mo
del
e
d
usi
ng Cartesi
an
co
or
d
i
nate
but targ
et pos
ition
meas
ure
m
ents are
prov
id
ed
in po
lar co
ordi
nate (ran
ge
an
d a
z
i
m
uth). F
o
r this impl
e
m
e
n
tation, w
e
inv
e
stigate
d
tw
o types of no
nli
n
ear
filterin
g, Co
nv
erted Me
asur
e
m
e
n
t
Kal
m
an
F
ilter (CMKF
)
and
Unsc
en
ted Kal
m
an F
ilter (UKF
). IMM
alg
o
rith
m is us
ed to antici
pat
e target motio
n
uncerta
inty. Many si
mul
a
tio
n
s on rad
a
r target tracking a
r
e
deve
l
op
ed
un
d
e
r assu
mptio
n
that no
ise c
h
aracteristic
is
know
n. In this
pap
er, the
perf
o
rmanc
e of IM
M-
CMKF and IMM-UKF is evaluated for co
n
d
i
tion that rad
a
r
doesn
’
t kn
ow
noi
s
e
characte
ristic and there
is
m
i
s
m
atch on noise
m
o
deling. Result
s from
simulation show that
I
MM-CMKF has better performanc
e than
IMM-UKF whe
n
tracking
maneuver
ing trajec
tory. Result
s also show that IMM-CMK
F is m
o
r
e
robust than
IMM-UKF
w
hen there is
m
i
s
m
atch on n
o
ise
m
o
de
lin
g.
Ke
y
w
ords
:
C
M
KF
, filtering, IMM, radar, UKF
,
target tracking
1. Introduc
tion
Primary obj
e
c
tive of rada
r target tra
cki
ng is
to e
s
timate state trajecto
rie
s
of a moving
target accu
ra
tely by using noi
sy measurem
ent. There are man
y
algorithms
for rada
r target
tracking. Kal
m
an filter i
s
t
he mo
st po
pu
lar meth
od in
mode
rn targe
t
tracking
syst
ems
be
cau
s
e
of
its simpli
city and computat
ional efficie
n
cy [1].
One of the issue
s
in the desi
gn of target tr
acking
system i
s
the choi
ce of the targe
t
motion mo
del
[2]. On the
simple
st a
pproximation, th
e mod
e
l is a
s
sumed
a
s
t
he tru
e
dyn
a
m
ic
target an
d a
singl
e filter ru
ns ba
se
d on i
t. This
app
ro
ach h
a
s
seve
ral obvio
us fl
aws be
cau
s
e
the
estimation
do
es not ta
ke in
to account a
possibl
e
mismatch b
e
twe
en the re
al target dynami
c
and
the filter mod
e
l [3]. To solve this p
r
obl
em, H.
A.P Bloom introdu
ced a
safe
ad
aptation o
r
soft
swit
chin
g met
hod that
kno
w
n a
s
Interacting Multip
le
Model
(IMM)
[4]. IMM use
a ban
k of filter to
estimate
state variabl
es o
f
dynamic sy
stem. Each
filter used different mo
del t
o
cha
r
a
c
teri
ze a
spe
c
ific motio
n
of a target, whi
c
h ma
ke
s it po
ssible to
describ
e a whole motion.
By using more
than one mo
d
e
l, the IMM algorithm i
s
mo
re ca
pabl
e
to track targ
ets
with motion u
n
ce
rtainty [3].
In tracking
appli
c
ation, target motio
n
is
usu
a
lly best mode
led usi
ng Cartesi
a
n
coo
r
din
a
tes [
5
]. When o
b
s
ervatio
n
is
also in
Ca
rte
s
ian
coo
r
di
n
a
te, system
can b
e
mod
e
led
usin
g linea
r
model. In thi
s
conditio
n
, IMM with
Kal
m
an filter
ca
n be u
s
ed to
track the ta
rget.
Unfortu
nately
,
on coa
s
tal
rada
r sy
ste
m
, the targe
t
position m
easure
m
ent
s are commo
nly
provide
d
in p
o
lar
coo
r
di
na
te, in terms
o
f
range
and
azimuth
with
respe
c
t to th
e rad
a
r l
o
cation.
The differe
nces bet
wee
n
tracking
coo
r
di
nate and
m
e
asu
r
em
ent co
ordin
a
te ma
ke nonlin
earity
of
the sy
stem [
6
]. This matt
er m
a
kes Ka
lman filter
ca
n’t
be used
without modif
i
cation
be
ca
u
s
e
Kalman filter
only wo
rk o
n
linear
system
[8]. To so
lve this pro
b
lem,
there a
r
e so
me altern
atives.
The sim
p
lest
one is by co
nver
ting the
measurement
s to a Cart
e
s
ian frame, an
d then Kalma
n
filter is used as filtering
algorith
m
. Th
is m
e
thod
kn
own
a
s
Con
v
erted M
e
a
s
urem
ent Kal
m
an
Filter (CMKF
)
[5]. Another
solutio
n
is by
usin
g n
onlin
ear filteri
ng
b
a
se
d on
Kal
m
an Filte
r
. T
h
e
ben
chma
rk of
nonli
nea
r filt
ering
b
a
sed
on Kalm
an
filter i
s
Extend
ed Kalm
an
Fi
lter (EKF
) [9].
In
EKF, the nonlinea
r sy
ste
m
and me
asurem
ent mo
d
e
ls a
r
e si
mpl
y
lineari
z
ed
aournd the
state
curre
n
tly esti
mated. It is
well
kno
w
n
that du
e to
th
e erro
rs intro
duced
by the
linea
rization,
the
EKF is a
su
b-optimal
and
b
i
ase
d
e
s
timat
o
r, a
nd th
e calcul
ating of
t
he Ja
cobi
an matrix
is
al
wa
ys
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 1, March 2
015 : 211 – 2
2
0
212
a very difficul
t
and error-p
r
one p
r
o
c
e
ss
[10]. On
199
5, Simon J. Julier a
nd Jeffrey K. Uhlma
nn
prop
osed a
n
o
ther al
gorith
m
that kno
w
n
as Unsce
n
te
d Kalman Filt
er (UKF). In their p
ape
r th
ey
said th
at UK
F is mo
re
accurate an
d l
e
ss difficult t
o
implem
ent
than EKF a
s
ben
chma
rk
on
nonlin
ear filte
r
ing [7].
In this pa
pe
r, we
pre
s
e
n
t the pe
rfor
mance evalu
a
tion of IM
M method
u
s
ing t
w
o
nonlin
ear filtering
algo
rith
ms b
a
sed
on
Kalman
filte
r
; those
are IMM-CMKF a
nd IMM-UKF,
for
impleme
n
tation on
coa
s
t
a
l rad
a
r targ
et tracking
system. Evaluation is do
n
e
to find out the
perfo
rman
ce
of these algo
rithms in con
d
ition wh
e
n
radar d
o
e
s
n’t kno
w
the re
al
target dyna
mic
and re
al noise cha
r
a
c
teri
stic. Execution time is
also u
s
ed a
s
evalu
a
tion parame
t
er to know t
h
e
comp
ari
s
o
n
of computati
onal efficien
cy of the al
gorithms. Resul
t
from this evaluation ca
n
be
use
d
for impl
ementation o
n
Indone
sia
n
co
a
s
tal ra
da
r target tra
cki
n
g
system.
2. Model and
IMM Algorithm
2.1. Basic M
odel
The trackin
g
of the
singl
e targ
et is b
a
se
d on
the
choi
ce
of a
model to
de
scrib
e
the
dynamic of
a
target. T
he
si
mplest
targ
et motion
mo
d
e
l is d
e
scri
be
d in
the
Ca
rt
esia
n
coo
r
din
a
te
system by lin
ear di
screte
-time differen
c
e
equation
with additive noi
se a
s
[5]
(1)
whe
r
e A is the state transit
ion matrix
(b
ase
d
on mod
e
l of target dynamic),
is state vector on
time index k.
The state ve
ctor (
) con
s
i
s
ts of position a
nd velocity
or accele
ration
of the moving
target
on Ca
rtesian coo
r
di
nate,
i.e,
=[
]
T
.
is p
r
o
c
e
s
s noise th
at is assume
d to
be white a
nd
zero mea
n
G
aussia
n
with
covari
an
ce Q
.
The targ
et is tracked by
grou
nd ba
se
d r
ada
r an
d provide
s
me
asu
r
em
ent of range
(r)
and a
z
imuth (
θ
). The me
asurem
ent mod
e
l is given a
s
[5]
(2)
whe
r
e
is time index,
is measurement
, and
is me
asu
r
em
ent n
o
ise. Fo
r this case, wh
en
tracking
is do
ne o
n
Carte
s
ian
coo
r
din
a
te but
me
as
ure
m
e
n
t
on
po
la
r
c
o
or
d
i
n
a
t
e, me
as
u
r
e
m
en
t
model can be
given as [5]
⁄
,
,
(3)
whe
r
e k is time index,
is ran
ge of the target,
is
azimuth of the target,
and
are target
positio
n
on Cartesi
an coo
r
dinate,
an
d
and
are m
e
asu
r
em
ent n
o
ise
on p
o
lar coo
r
din
a
te,
that is assum
ed to be whit
e and zero m
ean Ga
ussia
n
with cova
ria
n
ce
R.
,
(4)
whe
r
e
is ra
n
ge mea
s
u
r
e
m
ent stan
da
rd deviation a
nd
is azim
uth mea
s
u
r
em
ent stan
dard
deviation.
2.2. Interacting Multiple Model
Interacting M
u
ltiple Model
(IMM)
algo
rithm is a
solution for target
dynamic un
certainty
o
r
to handle
a p
o
ssible mi
sm
atch b
e
twe
e
n
real ta
rget d
y
namic a
nd fi
lter mod
e
l. The ba
si
c idea
of
IMM is assu
me a set of model
s a
s
po
ssi
ble candi
d
a
tes of the re
al dynami
c
target, ru
n a ba
nk of
elemental filters, e
a
ch ba
sed o
n
a uni
que mod
e
l in
the set, and
gene
rate the
overall e
s
tim
a
tes
by a process based o
n
the results of these eleme
n
tal filters. On this pap
er, we
used two filters
with u
n
iqu
e
model
on
ea
ch filte
r
; on
e
model
u
s
ed
co
nsta
nt vel
o
city (CV) an
d the
othe
r u
s
ed
con
s
tant a
c
celeratio
n
(CA) model.
IMM method
con
s
i
s
ts
of fo
ur m
a
jor ste
p
s
: in
teractio
n
or mixing, filt
ering,
upd
ate
model
prob
ability, and com
b
inati
on. The equ
a
t
ions for ea
ch
step are as f
o
llows [8]:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Nonli
nea
r Filtering
with IMM Algorithm
for Co
asta
l
Ra
dar Ta
rget T
r
acking Syste
m
(Rika Susti
k
a)
213
Interaction
The mixing probabilitie
s
|
fo
r each mod
e
l are calculate
d
as
̅
∑
(5)
|
̅
(6)
Whe
r
e
is the probability of model M
i
and
̅
is nomaliza
t
ion factor. Mixed inputs (m
ean
s and
covari
an
ce
s)
of each filter
are calculate
d
as
∑
|
(7)
∑
|
(8)
Filtering
As described
before on t
he introd
ucti
on, for imple
m
entation on
coa
s
tal rad
a
r wh
en
tracking on Cartesi
an coo
r
dinate
a
nd
m
easure
m
ent on
pol
ar
coo
r
dinate, filter t
hat used o
n
this
IMM method
is no
nlinea
r f
iltering. T
w
o
filter
ing meth
od ha
s b
een
evaluated, t
hose are CM
KF
and UKF. Th
ese meth
od
s will be de
scri
bed latter.
Update Model Probability
In addition to
mean and
covarian
ce, we comp
ute th
e likelih
ood o
f
the measu
r
ement for
each filter as
Λ
v
;0
,S
(9)
and proba
bilites of ea
ch m
odel at
time step k are cal
c
ulated a
s
∑
Λ
̅
(10
)
Λ
̅
(11)
whe
r
e c i
s
no
rmali
z
ing fa
ctor.
Combination
Combi
nation
step is to co
mpute state
m
ean an
d co
varian
ce final
, and comp
uted with
these e
quatio
ns
∑
(12
)
∑
(13
)
3. Filtering Algorithms
The expl
anati
on bel
ow i
s
a
bout two
filtering al
g
o
rithm
s
that h
a
ve b
een eval
uate
d
on thi
s
r
e
sear
ch.
3.1. Conv
erted Meas
ure
m
ent Kalma
n
Filter
Conve
r
ted M
easure
m
ent
Kalman Filte
r
is
a
n
alte
rnative app
ro
ach fo
r tra
c
king i
n
Carte
s
ia
n co
ordin
a
te u
s
in
g pola
r
me
asurem
ents. Po
lar
coo
r
din
a
te
mea
s
ureme
n
t is tra
n
sfo
r
med
to Ca
rtesi
a
n
coo
r
di
nate
system
s,
the
n
co
nvention
a
l Kalman fil
t
er is
appli
e
d. Kalman f
ilter
con
s
i
s
ts of two main step
s,
these
a
r
e time update a
nd
measurement
update.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 1, March 2
015 : 211 – 2
2
0
214
Time update
[8]
:
State matrix predi
ction
|
|
(14
)
Covari
an
ce
matrix
predi
ct
ion
|
|
(15
)
Measur
e
men
t
upda
te
[8]:
Innovation (re
s
idu
a
l) covari
ance
|
(16
)
Kalman gai
n update
|
(17
)
State estimation upd
ate usi
ng last mea
s
urem
ent
|
|
|
(18
)
Erro
r cova
ria
n
ce u
pdate
|
|
(19)
With the conv
erted me
asurement Ka
lma
n
filter, polar
coo
r
din
a
te m
easure
m
ent first converte
d
to
Carte
s
ia
n co
ordin
a
te mea
s
ureme
n
t usi
ng these equ
ation.
cos
(20)
sin
(21
)
Next, the me
asu
r
em
ent error m
a
trix, R,
neede
d
to b
e
adju
s
ted
si
nce
data wa
s mea
s
u
r
e
d
as
rang
e and a
z
i
m
uth [9].
(22)
(23
)
W
h
en
(24)
(25)
(26
)
3.2. Unsce
nted Kalman F
ilter
Un
scented K
a
lman Filte
r
use
s
u
n
sce
n
ted tra
n
sfo
r
m
to give Ga
ussian
app
roxi
mation to
the filtering
solution
s of n
o
n
linea
r filterin
g problem
.
In un
scented
tran
sform,
we
determi
nisti
c
ally
cho
s
e
a fixed numb
e
r of
sigma
point
s, whi
c
h
ca
pture the
de
sire
d mom
e
n
t
s (me
an a
n
d
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Nonli
nea
r Filtering
with IMM Algorithm
for Co
asta
l
Ra
dar Ta
rget T
r
acking Syste
m
(Rika Susti
k
a)
215
covari
an
ce
) of distributio
n of a variable exac
tly. After that we prop
agate
the sigma points
throug
h the n
online
a
r fun
c
tion and e
s
tim
a
te the mo
me
nts of tran
sformed varia
b
le
from them.
For
appli
c
ati
on
o
n
co
asta
l radar, whe
n
tracking do
ne on Ca
rtesian coo
r
din
a
te whil
e
measurement
on polar co
ordin
a
te, time update
u
s
ed equatio
n on Kalman filter time upd
ate.
Un
scented transfo
rm is u
s
ed on mea
s
u
r
eme
n
t updat
e becau
se no
nlinea
rity is on measurem
ent
equatio
n.
These equ
ations b
e
lo
w are measureme
n
t updat
e on
Un
scented K
a
lman Filter a
l
gorithm [8].
Gene
ration of
sigma poi
nts
χ
|
X
|
(27
)
χ
|
X
n
λ
P
|
,
i
1
,…,n
(28)
χ
|
X
n
λ
P
|
,
i
n
1,
…
,
2n
(29)
Map sig
m
a p
o
ints to mea
s
urem
ent sp
ace
y
|
h
χ
|
, i=
0,...,2n
(30)
Predi
ct
Z
|
,
cov
a
rian
ce
S
, and cross
covari
an
ce of state da
n measureme
n
t
P
.
Z
|
∑
w
y
|
(31
)
S
∑
w
y
|
Z
|
y
|
Z
|
R
|
(32)
P
∑
w
χ
|
X
|
y
|
Z
|
(33
)
Filter
gain
K
P
S
(34
)
Upd
a
te state
dan cova
rian
ce e
s
timation
X
|
X
|
K
Z
Z
|
(35)
|
|
(36
)
4. Simulation Scenario
Simulation was do
ne to e
v
aluate the p
e
rform
a
n
c
e o
f
two nonlin
e
a
r filtering m
e
thod
s
usin
g IMM algorithm
s for
con
d
ition that
rada
r doe
sn
’
t
know th
e dynamic
of targ
et. Two types of
trajecto
ry were si
mulate
d; those are nonma
n
e
u
vering a
n
d
maneuveri
ng traje
c
tori
es.
Nonm
ane
uve
r
ing traj
ecto
ry is gene
rat
ed usi
ng
CV
(co
n
sta
n
t velocity) mo
d
e
l. Maneuve
r
ing
trajecto
ry i
s
gene
rated
u
s
ing
CV mo
de
l, combi
ned
with
CA (co
n
s
tant a
c
cele
ration)
mod
e
l
as
maneuve
r
dy
namic. Sam
p
ling interval
on simul
a
tion
is 1 se
con
d
and a total o
f
200 se
con
d
s
.
The target i
s
tracke
d by a ground b
a
se
d rad
a
r
with positio
n
on center
coo
r
din
a
te (0,0).
Measurement
data a
r
e
ge
nerate
d
by a
dding
Gau
s
si
an noi
se
to real traj
ecto
ry. The valu
es of
para
m
eters that are
con
s
i
dere
d
in t
he simulation is p
r
esented o
n
Table 1.
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015 : 211 – 2
2
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216
Table 1. Trajectory ge
ne
ra
tion para
m
ete
r
s
Paramete
r
Value
Initial position [x y]
[-1000 10
00] (m
)
Initial veloc
i
ty
[v
x
v
y
]
[4 3] (m/s)
Psd of process noise (q_cv = q_ca)
0.1
Measurement
no
ise:
Range standa
rd
deviation (
σ
r
)
Azimuth standard deviation (
σ
θ
)
10 (m)
1
o
Time index of m
a
neuvering
1-50 and
71-12
0
(s)
For initial e
s
timation we
u
s
ed o
ne poi
n
t
appro
a
ch when the first measurement
data
use
d
as fi
rst
estimation
of
and
. Firs
t es
timation of v
e
loc
i
ty
and
on one
point
approa
ch
is assig
ned
as zero [9]. The perfo
rm
ance is
mea
s
ured by the
percentag
e fit error (PFE
) of
positio
n estim
a
tion [11]:
PFE
100
∗
(37
)
PFE
100
∗
(38
)
PFE
P
F
E
(39
)
whe
r
e
and
are targ
et position estimat
i
on and
and
are real target position.
Every param
eter i
s
sim
u
la
ted usi
ng 5
0
runs
monte
ca
rlo
simulatio
n
.
For evalu
a
tion, in this
pa
per
we say that perform
an
ce is good en
oug
h
for implemen
tation if PFE is und
er 3%.
Simulation
s h
a
ve done
with three
sce
na
rios,
1
st
sce
na
rio
First sce
nari
o
is simul
a
tio
n
whe
n
there
is no misma
t
ch on noi
se
modeling. Its mean tha
t
noise p
a
ra
m
e
ter that
is u
s
ed
by filter
is th
e
same
as
real
noi
se
that was u
s
ed o
n
d
a
ta
trajecto
ry and
data measurement gen
eration as
can
be se
en on T
able 1.
2
nd
s
c
e
na
r
i
o
Secon
d
scen
ario i
s
simul
a
tion on
co
ndi
tion when th
ere i
s
mi
sm
a
t
ch o
n
noi
se
modelin
g. Its
mean that no
ise pa
ram
e
te
r that is
used
by filter is different with real noi
se tha
t
is used o
n
measurement
trajecto
ry ge
neratio
n. Tab
l
e 2 sh
ows some pa
ram
e
ters fo
r filterin
g step o
n
2
nd
scena
rio.
Table 2. Filtering Param
e
te
rs, 2
nd
Scen
ario
Parameters
Value
Psd of process noise (q_cv = q_ca)
0.01
Measurement
no
ise:
Range standa
rd
deviation (
σ
r
)
Azimuth standard deviation (
σ
θ
)
10 m
4
0
Probabilities of s
w
itching model o
n
IMM
0.95
0.05
0.05
0.95
Pr
ior
pr
obability
of IMM
[0.9 0.1]
3
rd
sce
nari
o
Third
scena
ri
o is sim
u
latio
n
whe
n
proce
ss n
o
ise and
measurement
noise i
s
vari
ed. The third
scena
rio i
s
d
one to evalu
a
te the ro
bu
stness of
the
algorith
m
s
when the
r
e i
s
mismat
ch o
n
noise mo
deli
ng. we h
a
ve
simulate
d the
filtering
algo
rithms with
variation
of p
r
oce
s
s n
o
ise
power
spe
c
tral den
sity (q
) and m
e
a
s
urement noi
se
stand
ard
devi
a
tion. On thi
s
pape
r, only
stand
ard
d
e
viation of
a
z
im
uth (
θ
) was varied.
Tabl
e
3 sho
w
s the
filtering
para
m
eters o
n
3
rd
scena
rio.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
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930
Nonli
nea
r Filtering
with IMM Algorithm
for Co
asta
l
Ra
dar Ta
rget T
r
acking Syste
m
(Rika Susti
k
a)
217
Table 3. Filtering Param
e
te
rs, 3
rd
scen
ari
o
Param
e
ters
Value
Psd of process noise (q_cv = q_ca)
0.01 until 1
w
i
th
interval 0.01
Measurement
no
ise:
Range standa
rd
deviation (
σ
r
)
Azimuth standard deviation (
σ
θ
)
10 (m)
0.1
o
until 4
o
w
i
t
h
interval 0.1
o
5. Simulation Resul
t
s
Some figures and table
s
belo
w
are
si
mulation result for two trajecto
rie
s
usi
ng thre
e
simulatio
n
scenari
o
s. Ea
ch
of simulation
has do
ne u
s
i
ng monte carl
o simulatio
n
with 50 ru
ns.
5.1. Non ma
neuv
ering trajectory
Figure 1 sh
ows pe
rform
ance of CM
KF
method
with IMM al
gorithm
(IMM-CMK
F
)
comp
are wit
h
UKF m
e
th
od with
IMM
algo
rithm (I
MM-UK
F) o
n
nonm
ane
uvering
traje
c
to
ry.
Figure 1.a is trajecto
ry estimation for first
scena
ri
o whe
n
there is no mi
smatch o
n
no
ise
modelin
g, an
d Fig
u
re
1.b i
s
traje
c
tory e
s
timation
for
se
con
d
scen
ario,
whe
n
th
ere
is mismat
ch
on noi
se mod
e
ling.
a. 1
st
s
c
e
nario
b. 2
nd
sce
nario
Figure 1. Estimation of non
maneuve
r
ing
trajecto
ry
From
Figu
re
1, the
re
sult
of simul
a
tion
usi
ng
1
st
an
d 2
nd
sce
na
ri
o ne
arly the
same.
A
target
can
be
tra
c
ked
by IMM-CMKF a
nd IMM
-
UK
F
with g
ood
pe
rforma
nce. E
s
timation
of n
o
n
maneuve
r
ing
trajecto
ry almost coi
n
ci
d
e
s with r
eal
trajecto
ry. To see the
differen
c
e
s
more
clea
rly, we co
unt the PFE as ca
n be see
n
on Table 4.
Table 4. PFE of nonman
eu
vering traj
ect
o
ry estimatio
n
A
l
gori
t
h
m
PFE (
%
)
1
s
t
scenari
o
2
n
d
scenario
IMM-CMKF
1.4541
1.7877
IMM-UKF
1.3574
1.7413
From
Tabl
e 4
we
ca
n
see
that pe
rfor
m
a
nce
of IMM-CMKF and
IM
M-UKF
on
1
st
scen
ario
is b
e
tter the
n
pe
rform
a
n
c
e on
2
nd
sce
nario.
On
all
of the
scen
ario
s, IMM-CMKF ha
s b
e
tter
perfo
rman
ce
than IMM-UK
F. Results al
so
sho
w
t
hat
on this type
of trajecto
ry, perfo
rman
ce
s of
all scena
rio
s
are go
od with
PFE fewer than 3%.
To see
rob
u
stness of the
s
e filtering
me
thods,
simul
a
tion u
s
ing
3
rd
scen
ario
ha
s d
one.
We have sim
u
lated
va
riati
on
of noi
se mismat
ch
on
noi
se m
odel
ing. Paramet
e
rs that u
s
e
d
on
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930
TELKOM
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Vol. 13, No. 1, March 2
015 : 211 – 2
2
0
218
this simul
a
tio
n
can
be
see
n
on
Ta
ble
3.
Simulation result for non
maneuve
r
ing
trajecto
ry can
be
see
n
on Fi
gu
re 2. Figu
re
2.a is si
mulat
i
on re
sult
if p
r
ocess n
o
ise
is varried, an
d Figu
re 2.b
is
simulatio
n
if measurement
noise i
s
varri
ed.
a.
Proce
ss n
o
ise varia
t
ion
b. Measurem
ent noi
se variation (
σ
θ
)
Figure 2. PFE of position estimation o
n
nonman
euve
r
ing traj
ecto
ry
As we ca
n see
from
Fi
gu
re 2,
no
si
gn
ific
ant diffe
re
nce
s
betwee
n
all t
w
o
ki
n
d
s
of the
filtering alg
o
r
ithms o
n
nonma
neuve
r
ing traj
e
c
tory. Figure 2
sho
w
s th
a
t
on tracki
ng
nonma
neuve
r
ing targ
et, mismatch o
n
n
o
ise
model
i
n
g is n
o
t a bi
g problem
b
e
ca
use the t
w
o
algorith
m
s h
a
v
e good pe
rfo
r
man
c
e, with
PFE is under 3%.
5.2. Maneuv
ering traje
c
tor
y
Position
esti
mation u
s
in
g
CMKF
an
d
UKF
with IM
M algo
rithm
to tra
c
k man
euverin
g
target
can
b
e
se
en o
n
F
i
gure
3. Figu
re 3.a i
s
e
s
ti
mation of m
aneuve
r
ing t
r
ajecto
ry on
1
st
scena
rio an
d Figure 3.b is
estimati
on
of maneuve
r
ing trajecto
ry
on 2
nd
scen
a
ri
o.
a. 1
st
s
c
enario
b. 2
nd
s
c
e
na
r
i
o
Figure 3. Position estimatio
n
on mane
uvering traje
c
tory
Targ
et
m
a
neuver
i
ng
Targ
et
m
a
neuver
i
ng
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Nonli
nea
r Filtering
with IMM Algorithm
for Co
asta
l
Ra
dar Ta
rget T
r
acking Syste
m
(Rika Susti
k
a)
219
We can see
from Figu
re
3 that erro
r e
s
tima
tion is l
a
rge
whe
n
ta
rget is ma
ne
uvering.
Figure 3.b
show th
at mismat
ch on
n
o
ise m
ode
lli
ng ma
ke e
r
rors
on traje
c
tory e
s
timat
i
on
increa
se, e
s
p
e
cially
whe
n
target i
s
ma
n
euverin
g. IMM-CMK
F
filtering m
a
kes l
a
rge
r
e
r
ror th
an
IMM-UKF o
n
this co
ndition
. The result can
see
clea
rer by co
untin
g the PFE as can b
e
se
en
on
Table 5.
Table 5. PFE of maneuve
r
i
ng traje
c
tory
estimation
A
l
gori
t
h
m
PFE (
%
)
1
s
t
scenari
o
2
n
d
scenario
IMM-CMKF
1.6475
2.9946
IMM-UKF
1.6312
3.0287
Table 5 sho
w
s that PFE of IMM-CMKF
and IMM-UK
F almost the
same fo
r 1
st
scen
ario.
On 2
nd
scena
rio, IMM-CM
KF is better than IMM-UK
F when ta
rget
is maneuve
r
i
ng.
From si
mulat
i
on usin
g 3
rd
sce
nari
o
, we have got result a
s
can
be se
en on
Figure 4.
Figure 4 sho
w
s,
whe
n
there i
s
a ma
n
euver o
n
target dynami
c
, mismat
ch o
n
noi
se mo
d
e
ling
(process noi
se o
r
mea
s
urem
ent noi
se) ma
ke si
g
n
ificant influ
ence to traj
ectory e
s
tim
a
tion
accuracy, e
s
peci
a
lly wh
e
n
filtering
u
s
i
ng IMM-UK
F.
From
50
ru
ns m
onte
ca
rlo sim
u
lation
for
each pa
ram
e
ter, there
are
some
sim
u
la
tions that
the
trajecto
ry e
s
timation is di
vergen. O
n
this
situation,
rad
a
r
can’t tra
c
k
the obje
c
t. Th
is di
vergen
ce
on some
sim
u
lation
s ma
kes m
ean
of PFE
from 50 run
s
Monte Ca
rl
o simul
a
tion
usin
g IMM-
UKF is more than 3%. O
n
this conditi
on
,
estimation
u
s
ing IMM-CM
KF ha
s b
e
tter pe
rforma
n
c
e. Estimation
error (den
oted a
s
perce
n
t
age
fit erro
r
or P
F
E) fro
m
filte
r
ing
usi
ng IM
M-CMK
F
i
s
l
o
we
r tha
n
e
s
timation e
rro
r usi
ng IMM
-
UKF.
From thi
s
si
mulation
we
can
say that
IMM-CMKF
is mo
re rob
u
st than
UK
F algo
rithm
for
con
d
ition whe
n
there is mi
smatch on n
o
i
s
e mod
e
ling.
a.
Process noi
se variation
b. Measure
m
ent noise (
σ
θ
) variation
Figure 4. PFE of position estimation o
n
maneuve
r
ing
trajecto
ry
5.3 Executio
n
Time
Information a
bout execution time is u
s
eful for m
a
king trade
-offs bet
ween
e
s
timatio
n
accuracy a
n
d
comp
utation
a
l time wh
en
sele
cting
a
suitable alg
o
rit
h
m for a
spe
c
ific a
pplication
[12]. We
cou
n
t the exe
c
uti
on time
to ev
aluate
co
m
p
u
t
ational
comp
lexity of these
algo
rithm
s
. T
h
e
averag
es of e
x
ecution time
s for the two
approa
ch
e
s
from simul
a
tio
n
s are given i
n
Table 6.
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Vol. 13, No. 1, March 2
015 : 211 – 2
2
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220
Table 6. Execution time
Algorithm
Execution time (s
econd)
IMM-CMKF
0.077
IMM-UKF
0.285
Table
6
sho
w
s that exe
c
ution
time
of
IMM-CMK
F
is sm
aller than IMM
-
UKF. The
executio
n time for IMM-UKF is about
3.7 times
mo
re than IMM
-
CMKF. This
result sho
w
s t
hat
IMM-CMKF
method is
si
mpler than I
MM-UK
F. IMM-
CMK
F
is si
mple on this
appli
c
ation b
e
ca
use
the main p
r
o
c
e
ss i
s
only converting m
e
asu
r
em
ent
from pola
r
to Carte
s
ia
n co
ordin
a
te, and
then
the system
ru
ns Kalm
an filter ba
se
d on it
. There i
s
n
o
lineari
z
atio
n step on thi
s
filtering
proce
ss.
On IMM-UKF
,
there i
s
un
scented t
r
an
sform
a
tion
p
r
oce
s
s t
hat
t
a
ke
s mo
re t
i
m
e
co
mpa
r
e
w
i
t
h
conve
r
ting th
e measureme
n
t from polar
to Carte
s
ian
coo
r
din
a
te.
6. Conclusio
n
Interactin
g M
u
ltiple Mo
del
usin
g
Conv
erted
Mea
s
u
r
ement Kalm
a
n
Filter
(MM
-
CMKF)
and Interacti
ng Multiple
Model u
s
i
ng Unsce
n
te
d Kalman
F
ilter (IMM
-UKF) have
b
een
con
s
id
ere
d
fo
r implem
enta
t
ion on coa
s
tal rad
a
r, e
s
p
e
cially for Ind
one
sian
coa
s
tal rada
r targ
et
tracking
syste
m
. All two
types
algo
rithm,
whe
n
n
o
mi
smatch
on
noi
se
modeli
ng,
are
able
to track
the target wit
h
good d
e
g
r
e
e
of accuracy. IMM-
UKF algorith
m
is
a little better than IMM-CMKF
algorith
m
wh
en no mi
smat
ch on
noi
se
modelin
g, but
with long
er
executio
n time. When th
ere is
mismat
ch
on
noi
se m
ode
ling, IMM-CMKF algo
rith
m ha
s b
e
tter perfo
rma
n
ce
than IMM
-
UKF
algorithm. IMM-UK
F still
has
good performance
when no maneuver on ta
rget dynamic but t
h
e
perfo
rman
ce i
s
ba
d when t
here i
s
m
ane
uver on
targ
e
t
dynamic. IM
M-CMK
F
is
more
rob
u
st t
han
IMM-UKF on this conditio
n
.
Com
putation
a
l
co
mplexi
ty
of IMM-CMKF is
also le
ss
than IMM-UK
F.
From
this resull
s, it
ca
n be
con
c
lu
ded th
at IM
M-CMK
F
i
s
better
than
IMM-UKF
for
impleme
n
tation on
coa
s
tal ra
dar targ
et trackin
g
system, and
IMM-CMKF i
s
suita
b
le to
be
impleme
n
ted
on Indon
esi
a
n coa
s
tal
ra
d
a
r targ
et tracking
system.
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ces
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nad
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r
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