Track-to-Track Fusion in a Heterogeneous Sensory Environment - ISIF

Track-to-Track Fusion in a
Heterogeneous Sensory Environment
D. Gendron K. Benameur M. Farooq
Department of Electrical & Surface Radar Section Department of Electrical
Computer Engineering Defence Research Establishment Ottawa Computer Engineering
Royal Military College of Canada 3701 Carling Avenue Royal Military College of Canada
PO Box 17000 Stn Forces Ottawa, Ontario, Canada K1A 0Z4 PO Box 17000 Stn Forces
Kingston, Ontario, K7K 7B4 kaouthar.benameur@dreo.dnd.ca Kingston, Ontario, K7K 7B4
gendron-d@rmc.ca
farooq-m@rmc.ca
ABSTRACT
In this paper, we present different approaches
for the association of tracks for airborne
sensors. The proposed approaches explore the
effects of the choice of coordinate systems on the
tracking filters and the association process. The
performance of the association techniques is
analysed in terms of the probability of correct
classification (P c ) and the probability of false
association (P fa ). This practical aspect of the
multi-target multi-sensor tracking problem is
presented for the association of radar tracks to
ESM tracks in different scenarios.
1. INTRODUCTION
In this paper, we consider two sensors
on board surveillance aircraft, henceforth
referred as the observer. The sensors on board
carry out surveillance over a space where
multiple targets are presumed to exist. The first
sensor is a radar that provides range and bearing
measurements and the second sensor is an ESM
(or IR) system that yields bearing only
measurements. It is assumed that the targets are
flying at a constant velocity and at a constant
heading.
Using dissimilar information from both
the radar and ESM (or IR) systems , different
tracking and association techniques have been
presented in the literature [1-5]. In [4], Saha
presented an algorithm based on constructing an
azimuthal gate centred on the filtered ESM track.
The radar track candidates within this gate are
then enumerated and a closeness score is
computed. The track pairing is derived as a
function of the closeness score. Gendron et al.
[5] have demonstrated that the classical data
association technique yields acceptable results.
Farina and La Scala [3] presented different
association methods for tracks from dissimilar
sensors, which are derived in different coordinate
systems. In this paper, we examine the
performance of these association methods in
relation to the choice of coordinate frames for
various scenarios. The performance of the
association techniques is analysed in terms of the
probability of correct classification (P c ) and the
probability of false association (P fa ). This
practical aspect of the multi-target multi-sensor
tracking problem is presented for the association
of radar tracks to ESM tracks for different
scenarios.
In Section 2 of this paper, we present a
mathematical formulation of the problem. The
association approaches are defined in Section 3,
which involve the selection of the coordinate
systems for the trackers and the decision rules.
In Section 4, we define the three different
scenarios used to evaluate the effect of the
selection of the coordinate system on the
association process. The simulation results are
presented in Section 5. Finally, in Section 6 the
paper is summarised.
2. PROBLEM FORMULATION
In most target tracking applications, the
target motion can be best modelled in Cartesian
coordinates. With the active radar system, the
measurement of a target’s position is typically
reported in polar or spherical coordinates. If the
motion equations are linear in the Cartesian
coordinates and the observations are in Polar
Coordinates (PC), one can obtain a polar-to-
Cartesian conversion. This conversion makes it
possible to use the Kalman filter without the
linearisation problems inherent in the Extended
1

Kalman Filter (EKF), however, the nonlinearities
are imbedded in the noise processes.
The resulting filter is called the Converted
Measurement Kalman Filter (CMKF).
2.1 Converted Measurement Kalman
Filter
The active sensor provides noisy
measurements of the target(s) bearing (β), as
well as the target(s) range (r). The state space
model used for the active sensor measurement
filter is represented by the following equations:
where
x ( k 1) = Φx(
k ) + w(
k ) + U(
k)
+ , (1)
[ x x&
y y&
] T
x =
(2)
is the relative state vector and where the
superscript "T" represents the transpose
operation. F defines the state transition matrix,
w(k) is a zero-mean, white, Gaussian noise
sequence with assumed covariance Q, and U(k)
is the assumed deterministic input such as the
relative position change (acceleration) of the
observer. The measurement vector z is modelled
as
where
z ( k)
Cx(
k ) + v(
k )
= , (3)
⎡x
( k)
= ⎢
⎣y
m
m
( k)
⎤
( k)
⎥
⎦
z , (4)
⎡1
= ⎢
⎣0
0
0
0
1
0⎤
0
⎥
⎦
C , and (5)
⎛cos(
β
( k)
=
⎜
⎝ sin( β
)
−r
⎞⎛η
m
sin( βm)
⎟⎜
r β
m
cos(
m)
⎠⎝
m
r
v (6)
η
m)
β
where the subscript "m" denotes the measured
quantities, and η r and η β are zero -mean, white,
Gaussian noise processes for the range and the
bearing, respectively. In the CMKF, the polar
measurements, the measured range and bearing
m
2
m
r x + y
m
2
⎞
⎟
⎠
= , and (7)
−1⎛
y ⎞
m
β =
⎜
⎟
m
tan
(8)
⎝ xm
⎠
are converted into the Cartesian form by the
standard coordinate transformation
x
= cos( β ) m
, and (9)
m
r m
ym r m
sin( β ) m
= . (10)
2.2 Extended Kalman Filter with
Modified Polar Coordinates
The EKF is basically an extension of
the Linear Kalman Filter (LKF). The difference
is that the EKF is employed to handle the case of
non-linear measurement process and/or nonlinear
target dynamics. The basic idea is to
linearise the non-linearities about each estimate
once it has been computed. As soon as a new
state estimate is produced, a new and hopefully
more accurate reference state trajectory is
incorporated into the estimation process. So, the
update equation becomes
xˆ(
k | k)
= xˆ(
k | k −1)
+
K(
k )[ z(
k)
− h( xˆ(
k | k −1))]
(11)
where h( x ˆ( k | k −1))
is a non-linear
measurement function. The gain is defined as
K(
k)
= P(
k | k −1)
H
[ H
x
( k)
P(
k | k −1)
H
T
x
T
x
( k)
( k)
+ R(
k)]
−1
(12)
where H x (k) is the linearised measurement
matrix expressed as
H . (13)
x( k)
= ∇
xh(
x,
k)
x=
xˆ(
k|
k−1)
Aidala and Hammel [6] have proposed
a different set of equations for the state and
measurement formulated in terms of Modified
Polar Coordinates (MPC), while the algorithm
itself is configured as an EKF to solve the
bearing only Target Motion Analysis (TMA)
problem. The state vector in MPC is
[ β & r&
β 1 ] T
y =
, (14)
r r
2

where β & represents the relative bearing rate,
r& is the ratio of the relative range rate and the
r
relative range, and is the measure of the
closeness of the observer to the target, which is
called Inverse-Time-To-Go (ITTG), β is the
relative bearing and (1 / r) represents the inverse
of the relative range. This coordinate frame is
well suited for TMA as it automatically
decouples the observable and unobservable
components of the state vector. It is to be noted
that the first three terms of the state vector are
always observable while, the last term, (1 / r), is
not observable until the observer performs a
manoeuvre. The analysis in [7] reveals that the
observer must manoeuvre to render the system
observable and that certain manoeuvres are more
appropriate than others for a given observer
speed and profile. The target course is defined by
the following non-linear differential equations:
⎡−
2y1
y2
+ y4(
a
⎢
2 2
dy
⎢y1
− y2
+ y4
( a
=
dt ⎢
⎢
y1
⎢⎣
− y1y4
x
x
cos( y3)
− a
y
sin( y3))
⎤
⎥
sin( y3)
− a
y
cos( y3))
⎥
⎥
⎥
⎥⎦
(15)
where a x and a y are the Cartesian components of
the observer’s acceleration.
2.3 Polar Coordinates
The algorithm that follows is
represented in Polar Coordinate (PC). In this
coordinate frame, the filter simply smoothes the
noisy bearing measurements. The state vector
for the PC is as follows:
p
[ β & r&
β r] T
x =
. (16)
The non-linear differential equations describing
the evolution of the target course in this
coordinate system are described in equation (17):
dx
p
=
dt
⎡ 2xp2
⎢
−
⎢
2
⎢xp1xp
4
⎢
⎢xp1
⎢
⎣xp
2
(17)
x
p1
x
p4
( a
+
+ ( a sin( x
x
x
p3
cos( x
) − a
y
p3
) − a
cos( x
y
p3
2.4 Classical Data Association
Technique
sin( x
))
p3
))
x
Assuming the two target tracks
are xˆ
radar
and xˆ
esm
for the active and passive
cases, respectively, the association logic test,
referred to as the statistical distance test statistic,
is
ij −1
T
(ˆ x − xˆ
)( P ) (ˆ x − xˆ
≤ λ ,
radar esm
radar esm
)
where
P
(18)
ij i j c c T
= Pa
+ Pp
−P
−(P
) , (19)
and P i j
a and P p are the solutions of the Ricatti
equations for the Kalman filters [3] used for the
active and passive cases, respectively, assuming
the process noise is Gaussian and independent
for each filter. In general, the cross-covariance
term P c will not be zero as the state estimate
errors will be correlated if the trajectories are
from the same target due to the common process
noise in each filter. In the current study, the
cross-covariance is assumed to be zero. The
expression in equation (18) represents a χ 2
distribution with n degrees of freedom and λ is
selected from the χ 2 tables.
3. ASSOCIATION RULES
Five track-to-track association
approaches proposed in [3] are considered here.
The first two techniques are commonly used
methods while the three re maining were
introduced in [3]. Two of the newer association
techniques make use of the ITTG (G = r?/ r)
element available from the MPC filter. The five
association methodologies, as presented in
Figure 1, are:
p4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3

a) Method 1 compares the bearing
estimates from the active filter
βˆ with the bearing estimates
radar
from the PC filter
ˆβ ;
1esm
b) Method 2 compares βˆ radar
with
the bearing estimates from the
MPC filter
ˆβ 2 ;
esm
c) Method 3 compares the bearing and
the ITTG estimates from the active
ˆβ with the
filter [
radar,
Ĝ radar
]
bearing and ITTG estimates from
the MPC filter [ 2esm,
G2 ˆ
esm
]
ˆβ ;
d) Method 4 compares βˆ
radar
, with
the measured bearing
ESM system; and
β
m
from the
e) Method 5 compares
[ ˆβ
radar,
Ĝ radar
] with β
m
and
ˆ .
G2 esm
4. SIMULATION SCENARIOS
4.1 Profile A
In what follows, we describe three
different scenarios analysed in this study. Under
the assumption that the ESM and radar
measurements are synchronised, Profile A
represents a scenario where the targets are
moving toward the observer at a constant speed
and heading as shown in Figure 2. We also
assume that the observer’s trajectory is known.
The initial position and velocity in
Cartesian coordinates of the observer and the two
targets for Profile A, are respectively: (0 km, 50
km, 200 m/s, 0 m/s), (150 km, 3 km, -150 m/s, 0
m/s), and (150 km, 7 km, -150 m/s, 0 m/s). The
distance between the two targets is 4 km for each
profile.
4.2 Profile B
Profile B shows a scenario where the
targets are moving away from the observer at a
constant velocity and bearing as shown in Figure
3.
The initial position and velocity in
Cartesian coordinates of the observer and the two
targets for Profile A, are respectively: (0 km, 0
km, 200 m/s, 0 m/s), (50 km, 50 km, 150 m/s, 0
m/s), and (50 km, 54 km, 150 m/s, 0 m/s).
4.3 Profile C
The final scenario is Profile C. In this
profile, the targets are still moving away from
the observer at a constant speed and heading, but
the targets are flying at a 45 degree angle as
shown in Figure 4.
This geometry offers a good
compromise between increased bearing rate and
reduced relative range allowing the velocity
errors to be minimised. This, in turn, will benefit
the ESM filters as the bearing rate is increased
when the observer crosses the target’s Line Of
Sight (LOS), which results in a substantially
improved estimation of the bearing [7]. The
initial position and velocity in Cartesian
coordinates of the observer and the two targets
for Profile A, are respectively: (0 km, 0 km, 200
m/s, 0 m/s), (60 km, 40 km, 150cos(45°) m/s,
150sin(45°) m/s), and (60 km, 44 km,
150cos(45°) m/s, 150sin(45°) m/s).
5. SIMULATION RESULTS
A number of Monte Carlo simulations
have been carried out for the three defined
profiles to determine the performance of the five
different association techniques. The sampling
period (T) is chosen to be 4 seconds. One
hundred runs were performed for every case
under consideration. The measurement noises
were set as follows:
a) Range noise σ
r
= 10 m;
b) Bearing noise σ
βa
for the active
sensor = 1 degree; and
c) Bearing noise σ
βp
for the passive
sensor = 3 degrees.
The performance of these techniques is
compared in terms of Probability of False
Association (P fa ) and Probability of Correct (P c )
association.
Simulation results reveal that P c
presents a similar performance for Profiles B and
C. However, we observe a strong deterioration
4

in the P c performance for Profile A in the case of
Method 1. This poor performance is attributed to
the PC model, which is a not a good choice for
extrapolating target position between sensor
updates. This is particularly true at shorter
ranges, where the pseudo-accelerations created
by constant speed/constant course targets,
introduce very large errors in polar tracking [8],
which is the case of Profile A (Figure 5), where
the targets are moving toward the observer
causing the range to decrease rapidly.
Considering P fa , simulation results show
that the performance depends strongly on the
profile and the association rules. Figures 6, 7,
and 8 clearly demonstrate that the P fa
performance is a function of the tracker
coordinate system as well as the profile
geometry. Note that the P c is 95 % in these
simulations.
To further evaluate the performance of
the five track-to-track association methods in
terms of P fa , the curves illustrating the
percentage of false association versus the
distance separating the two targets is shown in
Figures 9, 10, and 11 for each profile. These
plots allow us to determine if the approaches can
distinguish between two closely spaced targets.
The results presented in Figures 9 to 11
demonstrate that Methods 1, 2, and 3, which use
bearing estimates, are better than Methods 4 and
5, which use noisy bearing measurements in
distinguishing between two targets. Moreover,
the performance of each method improves as the
distance between the targets increases as
expected. Unlike the results presented in [3], the
results of this study do not demonstrate that the
methods (Methods 2 and 3) using the MPC filter
are better than Method 1. The dissimilar results
are attributed to the type of profiles utilised in
this study.
while Methods 4 and 5, which rely on measured
ESM bearing for the association between tracks,
are not reliable. The study also demonstrates
that association Method 1, using the PC based
estimates, and Method 2, using the MPC based
estimates, have comparable performances.
REFERENCES
[1] A. Farina and R. Miglioli, "Association of
active and passive tracks for airborne sensors",
Signal Processing 69, pp. 209-217, 1998.
[2] F. R. Castella, "Theoretical performance of a
multisensor track-to-track correlation technique",
IEE Proceeding, Radar, Sonar and Navigation,
Vol. 142, No. 6, December 1995.
[3] A. Farina and B. La Scala, "Methods for the
Association of Active and Passive Tracks for
Airborne Sensors".
[4] R. K. Saha, “Analytical evaluation of an
ESM/radar track association algorithm”, SPIE,
Vol. 1698, Signal and Data Processing of Small
Targets, 1992.
[5] D. Gendron, M. Farooq, K. Benameur,
“Track-to-Track Fusion in a Multisensory
Environment”, SPIE Proceedings, Vol. 4380,
Signal Processing, Sensor Fusion, and Target
Recognition , Orlando April 2001.
[6] V. J. Aidala and S. E. Hammel, “Utilization
of Modified Polar Coordinates for Bearing-Only
Tracking”, IEEE Transactions on Automatic
Control, Vol. AC-28, No. 3, March 1983.
[7] D. Van Huyssteen and M. Farooq,
“Performance analysis of bearings-only tracking
algorithm”, 1998 Aerosense Conference of SPIE,
Orlando, FL, April 1998.
[8] Y. Bar-Shalom, W. D. Blair, “Multitarget-
Multisensor Tracking: Applications and
Advances”, Volume III, Artech House, 2000.
6. CONCLUSION
In this paper, we explored the problem
of associating ESM tracks with one or more
possible radar tracks derived in different
coordinate systems. Simulations results show
that the choice of the coordinate system is a
complex issue, which depends not only on the
sensors but also on the scenario. One important
outcome of this study is that Methods 1,2 and 3,
which use bearing estimates to compute the
track-to-track association, are capable of
correctly associating two closely spaced targets,
5

Figure 1, Association Rules [3]
6 x 104 Profile A, True Tracks
5
Observer
Target A
Target B
4
3
2
Y position in Meters
1
0
0 2 4 6 8 10 12 14 16
X position in Meters
x 10 4
Figure 2, True Tracks for Profile A
Profile B, True Tracks
6 x 104 X position in Meters
5
4
3
2
Y position in Meters
1
Observer
Target A
Target B
0
0 2 4 6 8 10 12 14 16
x 10 4
Figure 3, True Tracks for Profile B
6

x 10 4
Profile C, True Tracks
12
10
Observer
Target A
Target B
8
6
Y position 4 in Meters
2
0
0 5 10 15
X position in Meters
x 10 4
Figure 4, True Tracks for Profile C
Figure 7, P fa for Profile B
7 x 104 Profile A, True Tracks & ESM (Polar) Track Estimates
6
5
Observer
Target A
TA Estimates
Target B
TB Estimates
4
3
Y position
2
in Meters
1
0
0 5 10 15
X position in Meters
x 10 4
Figure 5, PC Filter with Profile A
Figure 8, P fa for Profile C
Figure 6, P fa for Profile A
Figure 9, P fa vs Distance Between Targets for Profile A
7

Figure 10, P fa vs Distance Between Targets for Profile B
Figure 11, P fa vs Distance Between Targets for Profile C
8