Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
3, N
o
. 1
,
Febr
u
a
r
y
201
3,
pp
. 93
~101
I
S
SN
: 208
8-8
7
0
8
93
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
/
IJECE
Perform
a
nce E
v
aluati
on of Unscented Kal
m
an Filt
er for
Gaussian and Non-Gaussian
Tracking Application
Leela Kum
a
ri
. B
*
,
P
a
dm
a R
a
ju
. K
**
*
Department of Electronics
and
communi
cation
Engineering, G.V.P. College of
Engineering, Vis
a
khapatnam, Ind
i
a
**
Department of
Electronics an
d Communication Engineering,
J.
N.
T.
U.
KAKINADA,
Ka
kinada, India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 30, 2012
Rev
i
sed
D
ec 29
, 20
12
Accepte
d
Ja
n 10, 2013
S
t
ate es
tim
at
ion
theor
y
is
one
of the bes
t
m
a
them
atic
al appro
aches
to
anal
yz
e var
i
an
ts
in the s
t
a
t
es
of
the s
y
s
t
em
or proces
s
.
The s
t
a
t
e of the
s
y
stem is d
e
fined b
y
a set of va
r
i
ab
les
that provide a complete
represent
a
tion
o
f
the intern
al c
ondition at an
y
given instant of
tim
e.
Filtering of Ran
dom
processes i
s
referre
d to as Estim
ation
,
and is a well
defined s
t
at
is
tic
al techn
i
que
. T
h
ere are two t
y
pes
of s
t
ate e
s
tim
ation
processes, Linear and Nonlin
ear
. Linear
es
tim
ati
on of a s
y
s
t
em
c
a
n e
a
s
i
l
y
be an
al
yz
ed b
y
using Kalm
an F
ilter
(KF) but
i
s
optim
al on
l
y
when the
m
odel is
linear .
B
ut M
o
s
t
of the s
t
ate es
tim
at
ion
problem
s
are nonline
a
r,
thereb
y lim
iting
the
pra
c
t
i
ca
l
app
lic
ati
ons of
the
KF and EKF. Unscented
Kalm
an filter and Particle fi
lter
ar
e best known for nonlinear estim
ates.
The appro
ach in
this
paper is
to
anal
yz
e th
e alg
o
rithm
for m
a
neuvering
target track
ing using bearing onl
y
m
eas
ure
m
ents
for both Gaus
s
i
an
/Nongaussian distributions wher
e UKF pr
ovides
better probability
of state
es
tim
ation
.
M
ontec
arlo com
put
er s
i
m
u
lations
a
r
e us
ed to anal
ys
e th
e
performance
.The simulations r
e
sults
showed that UKF provid
e
s better
perform
ance f
o
r Gaus
s
i
an
dis
t
ributed m
odels
com
p
ared
to the
nongaussian models.
Keyword:
Kalm
an
filte
Ex
tend
ed
Kalman
filter
Un
scen
ted
Kalman
filter
Gaus
sian
/No
n
Gaus
si
an
di
st
ri
but
i
o
n
Copyright ©
201
3 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
:
LeelaKum
ari. B
Depa
rt
m
e
nt
of
El
ect
roni
cs
an
d c
o
m
m
uni
cat
ion
en
gi
nee
r
i
n
g
,
G
.
V
.
P. C
o
llege of
En
g
i
n
eer
i
n
g,
Vi
sak
h
a
p
at
na
m
,
Indi
a.
Em
ai
l
:
l
eel
a882
1@y
a
ho
o.c
o
m
1.
INTRODUCTION
C
ont
r
o
l
o
f
a
n
y
pr
ocess m
odel
i
n
g
,
o
b
t
a
i
n
e
d
f
r
o
m
a pr
io
ri
k
n
o
w
led
g
e
of certain
ob
serv
ab
le p
a
ram
e
ters
is standard pra
c
tice for Engineers.
For m
a
n
y
o
f
th
e app
licatio
n
s
sim
p
le
m
o
d
e
ls with
li
n
ear ap
prox
imatio
n
s
aro
u
nd a
desi
gn
p
o
i
n
t
s
u
f
f
i
ce t
h
e re
qui
re
m
e
nt
. Si
nce al
l th
e n
a
tural ph
eno
m
en
a are n
on-lin
ear, it is v
e
ry
im
port
a
nt
t
o
st
udy
t
h
e
no
nl
i
n
ear m
odel
s
an
d
t
h
ei
r c
o
nt
rol
f
o
r
t
h
e
fol
l
o
wi
n
g
reaso
n
s:
1)
Som
e
sy
st
em
s ha
ve a l
i
n
ea
r
ap
pr
oxi
m
a
t
i
on t
h
at
i
s
n
on
cont
rol
l
a
bl
e
n
ear i
n
t
e
re
st
i
n
g
w
o
r
k
i
n
g
poi
n
t
s.
Linearization i
s
ineffective
e
v
en locally for s
u
ch cases.
2)
Eve
n
i
f
t
h
e l
i
n
eari
zed m
odel
i
s
cont
rol
l
a
bl
e
one m
a
y
wi
sh
t
o
ext
e
nd t
h
e
ope
rat
i
o
nal
d
o
m
ai
n bey
o
nd t
h
e
v
a
lid
ity d
o
m
ain
in
t
o
n
o
n
lin
ear reg
i
on
fo
r
b
e
t
t
er pred
iction
.
3)
Som
e
control
problem
s
are
external
t
o
t
h
e
pr
ocess a
nd
cann
o
t
be a
n
s
w
ere
d
by a linearly approac
h
e
d
m
odel
.
T
h
e s
u
ccess
of the
linear model i
n
ide
n
tifi
cation
or
in c
o
ntrol has
its ca
use i
n
the
good unde
rstanding of it.
A
b
e
tter m
a
ste
r
y of inv
a
rian
ts of
n
o
n
lin
ear mo
d
e
ls fo
r so
m
e
tran
sform
a
t
i
o
n
s is a
p
r
erequ
i
site to
a tru
e
th
eo
ry
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 3
,
N
o
. 1
,
Feb
r
u
a
ry
2
013
:
93
–
10
1
94
o
f
non
lin
ear i
d
en
tificatio
n
and
co
n
t
ro
l. And
all n
o
n
lin
ear
s
y
ste
m
s are suppos
ed to
ha
ve
a state space
of finite
d
i
m
e
n
s
io
n
.
State Esti
m
a
tio
n
tech
n
i
q
u
e
s are
h
a
nd
led b
y
filtering
techn
i
que m
o
d
e
ls for
perfo
r
m
a
n
ce.
A co
mm
o
n
app
r
o
a
ch
to
ov
erco
m
e
th
is p
r
ob
lem
is to
lin
earize th
e system b
e
fore u
s
i
n
g
th
e Kalm
an
filter, resu
ltin
g in
an ex
ten
d
ed Kalm
an
filter. Th
is lin
earizatio
n
do
es h
o
wev
e
r po
se
so
m
e
p
r
ob
lem
s
,
e.g
.
it
can
resu
lt in
no
nrealistic esti
mate
s [1,
2
]
o
v
e
r a p
e
ri
o
d
of ti
m
e
. Th
e
d
e
v
e
lopmen
t o
f
b
e
tter
esti
m
a
to
r alg
o
r
ith
m
s
for nonline
a
r Syste
m
s has therefore attr
acted a great deal of interest in
the scientific comm
unity, because the
i
m
p
r
ov
em
en
ts will u
ndo
ub
tedly h
a
v
e
g
r
eat i
m
p
act in
a wide rang
e
o
f
engin
eering
field
s
. Th
e EKF
h
a
s
b
een
considere
d
t
h
e
standard in
the theory of
nonlinear stat
e esti
m
a
tio
n
.
Th
is
p
a
p
e
r
d
eals
with
how to estimate a
n
o
n
lin
ear m
o
d
e
l with
un
scen
ted
k
a
lm
a
n
filter (UKF
). Th
e appro
ach
i
n
th
is p
a
p
e
r is to an
alyze
Unscented
Kal
m
an
filter wh
ere UKF pro
v
i
des b
e
tter
prob
ab
ility
o
f
state esti
m
a
t
i
o
n
for a
b
eari
n
g on
ly passiv
e
targ
et track
i
ng
.
2.
UNSCE
N
TED KAL
M
AN
FILTER
Inst
ea
d o
f
l
i
n
e
a
ri
si
ng t
h
e f
u
n
c
t
i
ons,
UK
F t
r
ansf
o
r
m
uses a set
of
poi
nt
s and
pr
o
p
ag
at
es t
h
em
t
h
rou
g
h
th
e
actu
a
l n
o
n
l
in
ear fun
c
tion
,
eli
m
in
atin
g
linearizatio
n
alto
gethe
r
. T
h
e
points are
chos
e
n
suc
h
that thei
r
mean,
cova
ri
ance
an
d
hi
g
h
er
o
r
d
e
r
m
o
m
e
nt
s
m
a
tch t
h
e
G
a
ussi
a
n
ran
d
o
m
vari
abl
e
. M
e
a
n
a
n
d c
ova
ri
ance
c
a
n
be
recalculated
from
the propa
gated points
, to
yield
m
o
re acc
urate
res
u
lts com
p
ared t
o
Ta
ylor’s
series
ordina
ry
fun
c
tion
lin
earizatio
n
.
Sel
ect
i
on
of
s
a
m
p
l
e
p
o
i
n
t
s
i
s
n
o
t
a
r
bi
t
r
a
r
y
.
G
a
ussi
a
n
ra
n
dom
vari
a
b
l
e
i
n
N
di
m
e
nsi
ons
uses
2
N
+
1
sam
p
l
e
poi
nt
s.
M
a
t
r
i
x
squa
re
ro
ot
and C
o
v
a
ri
ance de
fi
ni
t
i
ons a
r
e use
d
t
o
sel
ect
si
gm
a poi
nt
s i
n
s
u
c
h
a way
that their c
o
variance is sam
e
as
the Ga
ussian random
variabl
e
.
The
unsc
e
nt
e
d
Tran
sf
orm
appr
oac
h
has
t
h
e
adva
nt
a
g
e t
h
a
t
noi
se i
s
t
r
eat
ed as a
n
onl
i
n
ear f
u
n
c
t
i
on t
o
account for non Gaussia
n
or
non a
d
d
itive noises. The st
ra
tegy for
doi
ng
so i
n
volves propa
g
ation
of
noise
t
h
r
o
u
g
h
f
unct
i
ons
by
fi
rst
augm
ent
i
ng t
h
e
st
at
e vect
or t
o
i
n
cl
u
d
e
n
o
i
s
e so
urces
. Si
g
m
a poi
nt
s a
r
e
t
h
en
selected from
the augm
ented state,
wh
ich in
clud
es
no
ise
v
a
lu
es also
. Th
e
n
e
t resu
lt i
s
th
at an
y nonlin
ear
effects of proc
ess and m
easure
m
ent noise are capture
d
with
the sam
e
acc
uracy as the re
st of the state, whic
h
in
turn in
creases esti
m
a
tio
n
accu
racy
i
n
p
r
esen
ce
o
f
add
itive no
ise so
urces.
3.
AMODELLING E
X
AMPLE FOR MANEUVERIN
G TARGET T
R
ACKING
USING BE
ARING
ONLY
MEAS
URE
MENTS
There
are
m
a
ny
m
e
t
hods
a
v
a
i
l
a
bl
e t
o
obt
ai
n
t
a
rget
m
o
t
i
on
param
e
t
e
rs i
n
son
a
r
si
g
n
al
pr
ocessi
n
g
[
3
-
8]
.Tar
get
i
s
assum
e
d
m
ovi
ng
at
const
a
nt
co
urse a
nd c
onst
a
nt
spee
d. It
s m
o
ti
on i
s
up
da
t
e
d every
seco
nd
. Th
e
o
w
n
sh
ip is
also
assu
m
e
d
to
b
e
statio
nary. It is
assu
m
e
d
th
at no
ise in
o
n
e
b
e
aring
m
easu
r
emen
t is
u
n
c
orrelated
with
th
at o
f
th
e o
t
h
e
r. An
o
t
h
e
r assu
m
p
tio
n
is th
at th
e
mean
v
a
lu
e o
f
th
e
n
o
i
se is zero. In
th
e
sim
u
l
a
t
o
r, ra
n
dom
num
bers are gene
rat
e
d
usi
n
g cent
r
al
l
i
m
i
t
t
h
eorem
.
The o
u
t
p
ut
of
Gaus
si
an r
a
nd
om
gene
rat
o
r i
s
used as Gaus
si
an
noi
se fo
r t
h
e B
eari
ng
m
easurem
ents. The raw beari
n
gs are corrupted wi
th the
Gaus
si
an
n
o
i
s
e
.
T
h
e
out
put
o
f
an
ot
he
r
Gaus
s
i
an ra
n
dom
ge
nerat
o
r
wi
t
h
gi
ven
pe
rce
n
t
a
ge
i
n
p
u
t
e
r
r
o
r
i
s
use
d
to corrupt the
freque
ncy m
eas
urem
ents.
T
h
e
obtained
beari
n
g
is
m
odified
a
ccordin
g
to the q
u
adran
t
i
n
wh
ich
it ex
ists su
ch
th
at its ran
g
e
is
fro
m
0
-
360
d
e
g
.
(clo
ck
wise p
o
s
itiv
e). Th
e b
ear
i
n
g is con
s
id
ered
with
resp
ect to
No
rt
h
.
Tar
g
et
pa
ram
e
ters
[R,
B,
C
a
n
d
S]
an
d
Ow
n shi
p
param
e
t
e
rs [oc
r
and
osp
d
]
ar
e read an
d t
a
k
e
n as
in
pu
t b
y
th
e sim
u
la
to
r. Assumed
erro
r in
B
earing
m
easurement (sigm
a
_b) and ra
nge m
easurem
ent (sigma_r)
are also fe
d as
input.
Assumpti
ons:
Fo
llowing
are t
h
e assu
m
p
tio
n
s
m
a
d
e
in
th
e si
m
u
la
to
r.
1)
At start,
own sh
ip
is at th
e
orig
in
.
2)
Target
i
s
m
ovi
ng
at
co
nst
a
nt
vel
o
ci
t
y
an
d
3)
All angles are
considere
d
with respect
t
o
Y-axis.
3.
1. Ow
n
shi
p
moti
on
The o
w
n shi
p
m
o
ti
on i
s
i
n
t
r
o
duce
d
as f
o
l
l
o
ws. C
o
nsi
d
er t
h
e fi
g
2 s
h
o
w
n bel
o
w
.
The
ow
n shi
p
i
s
m
ovi
ng wi
t
h
a
vel
o
ci
t
y
v
0,
x
0
i
s
t
h
e di
st
ance
of t
h
e
ow
n s
h
i
p
fr
om
t
h
e x-c
o
o
r
di
nat
e
, y
0
is the distance of the
ow
n s
h
i
p
f
r
o
m
t
h
e y
an
d
Oc
r
i
s
t
h
e a
n
gl
e m
a
ki
n
g
wi
t
h
no
rt
h.
Fr
om
Fi
g 1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Perfo
r
man
ce Eva
l
ua
tio
n o
f
Un
scen
ted
K
a
lma
n
Filter fo
r
Ga
u
s
sian
a
n
d
N
o
n-Gau
ssian
… (Leela
Kuma
ri. B)
95
Sin (oc
r
)
= x
0
/ v
0
(1
)
Co
s (o
cr
) = y
0
/ v
0
(2
)
For e
v
ery
seco
nd c
h
an
ge i
n
X an
d Y com
p
o
n
e
n
t
of
ow
n
shi
p
p
o
si
t
i
on
i
s
fou
n
d
an
d adde
d t
o
t
h
e
pre
v
ious X,
Y com
pone
nts of
own
shi
p
position.
For
t
s
=1s
ec
dX
0
=v
0
*si
n
(O
cr)
*
ts
(3
)
dY
0
=v
0
*
c
o
s
(Ocr
)*
ts
(
4
)
Whe
r
e dX
0
is
change i
n
X-c
o
m
pone
nt of
own
shi
p
position i
n
1 sec.
dY
0 is
chan
ge i
n
Y-c
o
m
pone
nt
o
f
own
sh
ip po
sitio
n in
1 sec.
v
0
is o
w
n
sh
ip
v
e
l
o
city.Ocr
is own
sh
i
p
co
urse.
(X
0
Y
0
) is
o
w
n sh
i
p
po
sitio
n.
The
n
X
0
=
(X
0
+dX
0)
&
Y
0
=
(
Y
0
+dY
0
)
(
5
)
3.
2.
Initia
l ta
rg
et po
sitio
n
Fro
m
in
pu
t b
e
aring
,
in
itial p
o
s
ition
o
f
targ
et is know
n as
fo
llow
s
.Co
n
s
i
d
eri
n
g Fi
g
2
.
Show
n b
e
l
o
w.
R-Range
T-Tar
g
et
O-
Obser
v
er
For
t
s
=1sec
X
t
=
r
a
n
g
e
*
s
i
n
(
b
e
a
r
i
n
g
)
(
6
)
Y
t
=
r
a
n
g
e
*
c
o
s
(
b
e
a
r
i
n
g
)
(
7
)
Whe
r
e (X
t
, Y
t
)
is targ
et
p
o
sitio
n with resp
ect
to
o
w
n
sh
i
p
as th
e
o
r
i
g
in
3.
3.
Ta
rg
et Mo
tion
The
t
a
rget
m
o
ti
on i
s
i
n
t
r
od
uc
ed as
f
o
l
l
o
w
s
.
C
onsi
d
er
t
h
e
F
i
g 3
.
s
h
o
w
n
bel
o
w
.
Y
(T
r
u
e
North)
y
0
o1
oc
r
X
Fig 1
Tc
r
T2
y
t
X
T1
Fi
g 3.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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08
I
J
ECE
Vo
l. 3
,
N
o
. 1
,
Feb
r
u
a
ry
2
013
:
93
–
10
1
96
Fro
m
in
pu
t rang
e an
d Bearing in
itial p
o
s
ition of targ
et is
k
n
o
w
n
,
X
t
=Range*sin(
B
earing)
(8)
Y
t
=R
an
ge*c
os
(B
eari
n
g)
(9
)
(X
t,
Y
t
) is target p
o
sitio
n
with
resp
ect t
o
o
w
n
sh
ip as th
e orig
in
.
F
o
r ev
e
r
y
1
s
e
c,
c
h
ang
e
in X
t,
and Y
t
are calcu
lated
an
d ad
ded
to prev
iou
s
targ
et
p
o
s
ition
.
dX
t
=
v
t
*
s
in
(Tcr
)*
ts
(
1
0
)
dY
t
=v
t
*c
os
(Tc
r
)
*
t
s
(1
1)
X
t
=
(X
t
+dX
t
)an
d
Y
t
= (Y
t
+dY
t
)
(
1
2
)
Whe
r
e dX
t
is c
h
ange i
n
X-com
ponent
of target position i
n
1 sec
dY
t
i
s
chan
ge i
n
Y
-
c
o
m
pone
nt
o
f
targ
et p
o
s
ition
in
1
sec.
V
t
is targ
et v
e
l
o
city.Tcr is target co
urse
with
resp
ect t
o
tru
e
nort
h
.
X
t
=
(X
t
+dX
t)
and Y
t
= (Y
t
+dY
t
).
The t
a
r
g
et
i
s
a
s
sum
e
d t
o
m
a
i
n
t
a
i
n
fi
xe
d c
o
ur
se an
d
vel
o
ci
t
y
t
h
r
o
ug
h t
h
e
ob
servat
i
o
n
d
u
rat
i
on.
3.
4.
Tar
g
et
t
r
a
c
ki
ng
a
n
d
m
a
t
h
emat
i
c
al
m
o
d
e
l
i
n
g
State and m
eas
urem
ent equations
:
Th
e
targ
et is
assu
m
e
d
to
be
m
o
v
i
n
g
with
co
nstan
t
v
e
lo
ci
ty
as
sho
w
n i
n
t
h
e
fi
g1
.
An
d i
s
de
fi
ned
t
o
ha
ve t
h
e
st
at
e vect
o
r
.
Xs (k)
= [
x
(k
)
y
(k)
R
x
(k
)
R
y
(k
)
W
x
(k
)
W
y
(k
)
]
T
(
1
3
)
Whe
r
e
R
x
(k)
R
y
(k
)
denot
e t
h
e
r
e
l
a
t
i
v
e ran
g
e c
o
m
pone
nt
s
bet
w
een
o
b
se
rve
r
and
t
a
r
g
et
. T
h
e
o
b
ser
v
e
r
state is sim
ilarly defi
ned as
X
0
= [
x
0
y
0
x
0
y
0
]
T
(
1
4
)
Fi
g 4.Ta
rget
an
d obs
er
ver
enc
o
u
n
t
e
r
The t
a
r
g
et
st
at
e dy
nam
i
c equ
a
t
i
on i
s
gi
ve
n
b
y
Xs (k+
1
)
=
(k
+1
/k
) X
s
(k
)
+b
(
k
+1
) +W
(k
)
(
1
5
)
Whe
r
e
(k
+1
/k
),
b
(k
+1
) and
W
(k
)
ar
e tr
ansien
t m
a
tr
ix
,
de
termin
istic v
ecto
r
and
p
l
an
t
n
o
i
se resp
ectively.
Th
e tran
sien
t
matrix
is g
i
v
e
n b
y
(
K
+
1
/
k
)
=
1 0 0 0
t
s
0
0
0
1
0
0
0 t
s
0
t
s
0
1 0 t
s
2
/2 0
0
0 t
s
0 1 0
t
s
2
/2
0
(16
)
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Perfo
r
man
ce Eva
l
ua
tio
n o
f
Un
scen
ted
K
a
lma
n
Filter fo
r
Ga
u
s
sian
a
n
d
N
o
n-Gau
ssian
… (Leela
Kuma
ri. B)
97
Whe
r
e t
s
is sam
p
le ti
m
e
an
d b
(k
+1
) is
g
i
v
e
n
b
y
b
(
k
+1
)
=
[0
0
-(X
(k
+1)
-
X(
k)
)
-(
y(
k
+
1)
-
y
(k
))
0
0
0
]
T
(1
7)
w (k) is
a zero m
ean gaussia
n
no
ise
v
ecto
r
with
E [W
(k
)
W (k
)
T
] =Q
kj
.
It is assum
e
d tha
t
the m
easurement
noi
se
an
d
pl
ant
n
o
i
s
e are
u
n
c
o
rrel
a
t
e
d.
T
h
e
bearing m
easure
m
ent, m
odeled as
B
m
(k+1)
=
tan
(
1
8
)
Whe
r
e
kj
t
h
e Kr
on
ecke
r
del
t
a funct
i
o
n, a
nd
(k) is error in the m
easur
em
en
t an
d
th
is erro
r is
assum
e
d to be
zero m
ean Ga
ussian wit
h
va
ri
ance
2
.The
measurem
ent and
plant nois
es are
to be uncorrelated
to each ot
her.
Kr
on
ecke
r
Del
t
a
Fu
nct
i
on:
kj =
1
if
k=j
0
i
f
k
≠
j
(19
)
4.
FILTER MODEL
FORMULATION
4.
1.
Au
gmen
ta
tion
o
f
State
Vector
Th
e filter start
s
b
y
au
g
m
en
tin
g
th
e state v
e
cto
r
to
N
Dim
e
n
s
ion
s
,
wh
ere
N is th
e su
m
o
f
d
i
m
e
n
s
io
n
s
in
th
e orig
in
al state-
v
ector
, m
o
d
e
l
no
ise an
d
measurem
ent noise.
Th
e co
v
a
rian
ce m
a
trix
is similarly au
g
m
e
n
ted
t
o
a
N
2
matrix
. Tog
e
ther th
is
form
s t
h
e au
g
m
en
ted
state
estim
a
te
vector a
n
d covariance m
a
trix
:
(2
0)
(2
1)
4.
2.
Crea
ti
n
g
2N+
1
si
gma
-
p
o
i
n
t
s
The
m
a
t
r
i
x
i
s
c
hos
en
t
o
c
o
nt
ai
n t
h
e
s
e
poi
nt
s,
and its col
u
m
n
s are calc
u
lated as
follows:
(2
2)
Subscri
p
t ‘i’meansi
th
col
u
m
n
of t
h
e s
qua
re ro
ot
of
t
h
ecovariance m
a
trix. The pa
ram
e
te
r
α
, in
th
e
in
terv
al 0
<
α
<1
, d
e
term
in
es sig
m
a-p
o
i
n
t
sp
read
. Th
is p
a
ram
e
ter is t
y
p
i
cally q
u
ite
lo
w,
n
o
rm
all
y
arou
nd
0.
00
1, t
o
a
voi
d
no
n
-
l
o
cal
ef
fe
ct
s. The
res
u
l
t
i
ng m
a
t
r
i
x
can
n
o
w
b
e
d
eco
m
p
o
s
ed
v
e
rtically in
to
th
e
rows, which contains t
h
e state
;
The rows
, w
h
i
c
h c
ont
ai
n
sam
p
l
e
d
p
r
oces
s
n
o
i
s
e a
n
d
The rows
, which c
ontain sampled m
easurem
ent noise.
4.
3.
Weig
htag
e
in Estimatio
n
Each
sig
m
a-p
o
in
t is also
assi
g
n
e
d
a weigh
t
. Th
e resu
ltin
g weig
h
t
s for mean
an
d
co
varian
ce (C
)
esti
m
a
tes th
en
b
eco
m
e
:
(2
3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 3
,
N
o
. 1
,
Feb
r
u
a
ry
2
013
:
93
–
10
1
98
4.
4.
Estima
tion
Th
e filter
th
en
p
r
ed
icts n
e
x
t
state
b
y
prop
ag
atin
g the sig
m
a-p
o
i
n
t
s th
rou
g
h
t
h
e state and
measurem
ent m
odels, and then calcu
lating
weigh
t
ed
av
erag
es and
co
v
a
rian
ce m
a
trices o
f
th
e
resu
lts:
(
2
4
)
4.
5.
Mea
n
and
C
o
vari
a
nce
The
pre
d
ictions are t
h
en updated
with
ne
w m
eas
urem
e
n
ts
by first ca
lculating t
h
e
measurem
ent
cova
riance
and state
m
easure
m
ent cross
correlation m
a
tric
es, which are t
h
en use
d
t
o
det
e
rm
ine Kalm
a
n
gain
-
New state
of t
h
e system
; - Its asso
ciated c
ova
riance
- E
xpecte
d
obse
rv
atio
n
;
-
Cro
s
s-correlation
matrix
-
Kalm
an Gain
(2
5)
(2
6)
(2
7)
(2
8)
(2
9)
-
New state
of
the system;
- Its ass
o
ciate
d
c
ova
riance
-
E
xpect
e
d
obs
er
vat
i
on;
- Cro
ss-correlatio
n
m
a
trix
-Kalm
a
n Gain
Th
e p
r
o
p
erties o
f
th
is a
l
go
rithm:
1)
Since the m
ean and c
o
varia
n
c
e
of
x are ca
pt
ure
d
preci
sely up t
o
the sec
ond order, t
h
e calculated val
u
es
of the m
ean and covaria
n
ce of Nonlinea
r function
(Yi
= f
[Xi]) are corre
c
t to the second orde
r as well.
This m
eans that the
m
ean is cal
culated to a higher
orde
r of accu
racy
than the
EKF, whe
r
eas the
cova
riance is
calculated to the sam
e
order of
acc
uracy.
Howe
ver, there are furthe
r
perform
a
nce
bene
fi
t
s
.
Si
nce
t
h
e
di
st
ri
b
u
t
i
o
n
of
x
i
s
bei
n
g
app
r
oxi
m
a
t
e
d r
a
t
h
er t
h
an
t
h
e
f
unct
i
o
n,
i
t
s
se
r
i
es ex
pan
s
i
o
n
i
s
not
t
r
un
cat
ed i
n
a part
i
c
ul
ar or
der
.
It
can
be sh
own
th
at th
e u
n
s
cen
t
ed alg
o
r
ith
m
is
a
b
le to
p
a
rtially
i
n
co
rp
orat
e i
n
f
o
rm
at
i
on f
r
om
t
h
e hi
ghe
r
or
de
rs, lea
d
ing to e
v
en greater
acc
uracy.
2)
The si
gm
a poi
nt
s capt
u
re t
h
e
sam
e
m
ean and c
o
vari
a
n
ce
irres
p
ective of the choice of
matrix squa
re
ro
ot
whi
c
h
i
s
u
s
ed.
3)
The m
ean and
cova
ri
ance a
r
e
cal
cul
a
t
e
d usi
n
g st
an
dar
d
vec
t
or a
nd m
a
t
r
i
x
ope
rat
i
o
ns. T
h
i
s
m
eans t
h
at
th
e algo
rith
m
is su
itab
l
e for an
y
c
hoi
ce
o
f
p
r
ocess m
o
d
e
l
,
an
d i
m
pl
em
ent
a
t
i
on i
s
e
x
t
r
em
el
y
rapi
d
because it is
not necessa
ry to e
v
aluate t
h
e Jac
obea
n
s
which
are
neede
d
i
n
a
n
E
K
F.
5.
RESULTS
The res
u
l
t
s
are anl
y
sed fo
r U
K
F
i
n
t
h
e
p
r
ese
n
ce of
Ga
ussi
a
n
noi
se
an
d N
o
nga
ussi
a
n
noi
s
e
f
o
r
t
h
e
in
itial
co
nd
itio
n
s
g
i
v
e
n
b
e
low.
Pa
ra
m
e
ter
Scen
ario
1
Initial range,
m
e
t
e
rs
5000
Initial bearing, deg
0
Target speed,
m
e
t
e
rs/sec
2
T
a
r
g
et cour
se,
d
eg
135
Observer speed,
meters/sec
10
Obser
v
er
cour
se, deg
90
E
r
r
o
r in the bear
in
g,
deg(
one
sig
m
a
)
0.
33
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Perfo
r
man
ce Eva
l
ua
tio
n o
f
Un
scen
ted
K
a
lma
n
Filter fo
r
Ga
u
s
sian
a
n
d
N
o
n-Gau
ssian
… (Leela
Kuma
ri. B)
99
Fig
5. e
s
tim
a
tion
er
ro
rs
o
f
Ta
rget M
o
tion
Pa
ram
e
ters
(Range,Bearing,C
o
urse and
Spee
d) in the
prece
nce
of
Gaus
sian n
o
ise
Analysis:
Du
rat
i
o
n of r
u
n:
1
8
0
0
sec.
Tim
e
taken
for conve
rgence
of Range
fo
r
m
a
n
e
uv
er
targ
et i
s
10
39
sec.
Ti
m
e
tak
e
n
f
o
r con
v
e
rg
en
ce of
Cour
se
f
o
r
man
e
uv
er
targ
et
is 13
19
sec.
Tim
e
t
a
ken
fo
r
co
nve
r
g
ence
o
f
S
p
ee
df
or
m
a
neu
v
e
r
t
a
r
g
et
i
s
9
1
7
sec.
Fig
6. e
s
tim
a
tion
er
ro
rs
o
f
Ta
rget M
o
tion
Pa
ram
e
ters
(Range,Bearing,C
o
urse and
Spee
d) in the
prece
nce
of
N
ong
au
ssian
no
ise
Analysis:
Du
rat
i
o
n of r
u
n:
1
8
0
0
sec.
Tim
e
taken
for conve
rgence
of Range
fo
r
m
a
n
e
uv
er
targ
et i
s
16
47
sec.
Ti
m
e
tak
e
n
f
o
r con
v
e
rg
en
ce of
Cour
se
f
o
r
man
e
uv
er
targ
et
is 16
30
sec.
Tim
e
t
a
ken
fo
r
co
nve
r
g
ence
o
f
S
p
ee
df
or
m
a
neu
v
e
r
t
a
r
g
et
i
s
1
6
4
6
sec
.
0
10
00
20
0
0
0
20
0
0
40
0
0
60
0
0
80
0
0
X
:
1
039
Y
:
125.
5
Er
r
o
r
i
n
R
a
n
g
e
Es
t
i
m
a
t
e
t
i
m
e
i
n
se
con
d
s
R
a
n
g
e
E
r
r
o
r
i
n
M
e
t
e
rs
0
10
00
20
0
0
0
0.
5
1
1.
5
Er
r
o
r
i
n
Be
a
r
i
n
g
Es
t
i
m
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t
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m
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i
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nd
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B
e
ar
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r
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gr
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s
0
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Y
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929
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r
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r
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e
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t
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t
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u
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r
r
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r
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De
g
r
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s
0
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00
20
0
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40
60
80
10
0
X:
9
1
7
Y
:
2.
521
Er
r
o
r
i
n
Sp
e
e
d
E
s
t
i
m
a
t
e
t
i
m
e
i
n
se
co
nd
s
S
p
e
e
d
E
rro
r i
n
k
n
o
t
s
0
10
0
0
20
00
0
20
00
40
00
60
00
80
00
X
:
1
647
Y:
1
0
.6
3
Er
r
o
r
i
n
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t
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m
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t
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r
r
o
r i
n
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e
te
rs
0
10
0
0
20
00
0
1
2
3
E
r
ro
r i
n
B
e
a
r
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s
t
i
m
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t
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ti
m
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i
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c
o
n
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s
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e
ar
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r
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D
e
gr
e
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s
0
10
0
0
20
00
0
50
10
0
15
0
20
0
X
:
1
630
Y:
0
.
8
5
9
8
E
r
ro
r i
n
C
o
u
r
s
e
E
s
t
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co
nd
s
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ur
se
E
r
r
o
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D
e
gr
ee
s
0
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0
0
20
00
0
2
4
6
8
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X
:
1
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Y:
0
.
0
4
1
3
2
Er
r
o
r
i
n
Sp
e
e
d
E
s
t
i
m
a
t
e
ti
m
e
i
n
s
e
c
o
n
d
s
Sp
e
e
d
Er
r
o
r
i
n
k
n
o
t
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 3
,
N
o
. 1
,
Feb
r
u
a
ry
2
013
:
93
–
10
1
10
0
Fi
g
7. C
o
m
p
ari
s
on
U
K
F
f
o
r
G
a
ussi
an
an
d
N
o
n-
Ga
ussi
an
di
s
r
i
b
ut
i
o
n
6.
CO
NCL
USI
O
NS
Application
of
KF to
nonlinea
r syste
m
s results in
highly inaccurate estim
ates
.
This pape
r looks
int
o
th
e n
e
ed
to con
s
isten
tly p
r
edict th
e n
e
w state and
ob
se
rv
ati
o
n of th
e
system
with
th
e p
r
esen
tatio
n
of UKF
for
no
nl
i
n
ea
r sy
st
em
s. W
e
ha
ve use
d
th
e non
lin
ear algorith
m, UKF th
at h
a
s
two great advantages
ove
r the KF.
First, it is abl
e
to
pre
d
ict the st
ate of t
h
e syste
m
m
o
re accurately. Second, it is
m
u
ch less difficult to
im
pl
em
ent
.
Th
e be
nefi
t
s
of
t
h
e al
go
ri
t
h
m
were
dem
ons
trated in a
realistic exam
p
l
e, b
ear
i
n
g on
ly
passiv
e
target trac
king. This
pa
per
ha
s conside
r
e
d
one s
p
ecific
fo
r
m
of t
h
e u
n
sce
n
t
e
d t
r
ans
f
orm
fo
r
o
n
e
part
i
c
ul
ar se
t
o
f
assu
m
p
tio
n
s
. It is sh
own
that th
e
n
u
m
b
e
r
o
f
sig
m
a p
o
i
n
t
s can be ex
tend
ed to yield
a
Filter wh
ich match
e
s
m
o
m
e
nt
s u
p
t
o
t
h
e
f
o
u
r
t
h
or
der
.
T
h
i
s
hi
ghe
r
or
de
r ext
e
nsi
o
n e
ffect
i
v
el
y
de
-bi
a
ses
alm
o
st
al
l
co
m
m
on
no
nl
i
n
ea
r c
o
o
r
di
nat
e
t
r
a
n
s
f
o
r
m
a
t
i
ons.
Th
e
p
a
p
e
r b
e
gan
with
the si
m
u
la
tio
n
o
f
t
h
e m
o
tio
n
of the targ
et and
determin
in
g
t
h
e in
itial targ
et
p
a
ram
e
ter n
a
mely
b
earing
.
Th
is p
a
ram
e
t
e
r was th
en
co
rrup
ted
with
n
o
i
se (Th
e
noise is assu
m
e
d
to
be
Gaus
sian a
n
d
nongaussian a
n
d the
results
are c
o
m
p
ared a
nd a
n
alyze
d
for two
cases) to
g
e
t th
e no
isy
measu
r
em
en
ts. Ex
ten
d
e
d
Kal
m
an
Filter can
filter th
e
noi
sy
m
easu
r
emen
ts an
d ex
tend
th
e targ
et
m
o
t
i
o
n
p
a
ram
e
ters bu
t is h
a
v
i
ng
com
p
u
t
atio
n
a
l d
i
fficu
lties.
Th
e un
scen
ted
Kal
m
an
Filter alg
o
rith
m
red
u
c
es th
is
d
i
fficu
lty.
Su
bsequ
e
n
tly,
m
a
n
e
uv
ering
o
f
th
e o
w
n
sh
i
p
was d
e
tected
u
s
ing
relativ
e Bearin
g
algo
ri
th
m
and
1030
1647
0
500
1000
1500
2000
convergence
of
range
Ukf
for
Gaussian
distribution
Ukf
for
Non
Gaussian
distribution
1319
1630
0
500
1000
1500
2000
Convergence
for
Course
UKF
for
gaussian
distribution
UKF
for
nongaussian
distribution
917
1646
0
500
1000
1500
2000
Convergence
of
Speed
UKF
for
gaussian
distribution
UKF
for
nongaussian
distribution
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Perfo
r
man
ce Eva
l
ua
tio
n o
f
Un
scen
ted
K
a
lma
n
Filter fo
r
Ga
u
s
sian
a
n
d
N
o
n-Gau
ssian
… (Leela
Kuma
ri. B)
10
1
CPA algorithm
.
The
n
the state of th
e target
was correcte
d
accordingly afte
r the detection of correct
own s
h
ip
evasi
o
n.
M
o
nt
e-C
a
rl
o
si
m
u
l
a
t
i
on
was ca
rri
e
d
out
i
n
t
h
e
e
n
d i
n
a
num
ber
of
scena
r
i
o
s
.
Th
e resu
lts con
f
irm
th
at th
e
failu
re rate of UKF is in
sig
n
i
fican
t. Fo
r th
e
UKF th
e in
itial errors in
x
p
o
s
ition
were m
o
re th
an 16
0
m
u
n
d
e
r the assu
m
p
tio
n
o
f
Gau
ssian
no
ise, and
are
m
o
re in
th
e case
of
no
n
g
au
ssi
an di
st
ri
but
i
o
ns
,
i
.
e t
h
et
arget
m
o
t
i
on param
e
t
e
rs conve
r
ges
at an
earlier tim
e
Under t
h
e ass
u
m
p
tion
o
f
G
a
u
ssian
no
ise th
an
Non
g
a
u
ssian
n
o
i
se. Th
er
ef
or
e
we m
a
y co
n
c
lude th
at UK
F is
r
obu
st algo
r
ithm
f
o
r
Gaus
si
an
di
st
ri
but
i
o
ns
t
h
a
n
f
o
r N
o
ng
aussi
a
n
di
st
ri
b
u
t
i
o
n
s
.
The pe
rf
orm
a
nce can f
u
rt
he
r be i
m
prove
d
i
n
t
h
e prese
n
ce of n
o
nga
us
si
an n
o
i
s
e by
t
a
ki
ng m
o
re
n
u
m
b
e
r
o
f
samp
le po
in
ts and
i
s
re
ferred to
as Particle filter.
R
EFERENCES
[1]
Simon Julier and Jeffrey
Uhlmann. A
new extension of
th
e kalman
f
ilter
to nonlin
ear s
y
stems.
Int.Sym
p
.
Aerospace/Defense Sensing, Si
m
u
l. AndControls,
Orlando, FL
, 19
97.
[2]
N.J. Gordon, D.J. Salmond,
an
d A.F.M. Smit
h. A novel approach to
nonlin
ear/non-Gaussian Bay
e
sian state
estimation
.
In
IEE Proceedings o
n
Radar
and
Sig
n
al Processing, v
o
lu
me 140, pages 107{113, 1993
.
[3]
A G Lindgren ,and K F Gong,
position and vel
o
cit
y
est
i
m
a
tion
via bear
ing observat
i
ons, IEE
E
Transactions on
Aerospace
and Electronic
s
y
stems,
AES-14, pp-56
4-77,jul.1978.
[4]
V J Aidala,Kal
m
a
n Filter beha
viour in bear
ing
onl
y
tr
ack
ing a
pplic
ations, I
E
E
E
Transa
ctions
on
Aerospace
and
Electronic s
y
s
t
ems,AE
S-15, pp-29-39,Jan.1979
.
[5]
S C Nardone,A. G Lindgren
,an
d
K F Gong fundemental prope
r
ties and
perform
ance of conv
entional b
earing
on
ly
target motion
an
aly
s
is, I
E
EE Tr
ansactions
on
Automatic Con
t
rols,AC-29, pp-775-
87,Sep.1984
.
[6]
V J Aidala
and
S
E Ham
m
e
l ,Ut
iliz
ation
of m
odi
fied pol
ar
coordi
nates for
bear
ing
onl
y tr
acking
,
I
EEE
Transa
ctio
ns
on Automatic C
ontrols,AC-28, p
p
-283-94,Mar.1
983.
[7]
T L Song ,and
J L Sp
y
e
r
,
A Stochastic analy
s
is of a modifi
ed
gain ex
tended
kalman filt
er with applications to
estim
ation with
bearing onl
y
m
easur
em
ents, IEEE Tr
ansac
tio
ns on Autom
a
tic Controls,AC-
30,No.10, pp-94
0-
9,Oct.1985
.
[8]
W Gro
s
sman, bearing only
tracking:A h
y
brid coo
r
dinate
s
y
stem approach,J
.Guidan
ce, Vol.17,No-
3
, pp-451-9,May-
jun.1994.
Evaluation Warning : The document was created with Spire.PDF for Python.