A Variable Structure Multiple Model ParticleFilter for GMTI Tracking
M. Sanjeev ArulampalamDSTO, Adelaide, Australia.sanjeev.arulampalam@dsto.defence.gov.auMatthew OrtonCambridge University, U.Kmro20@eng.cam.ac.ukNeil GordonQinetiq, Malvern, U.K.njgordon@taz.qinetiq.comBranko RisticDSTO, Adelaide, Australiabranko.ristic@dsto.defence.gov.au
Abstract –
The problem of tracking ground targets with GMTI sensors has received some attention in the recent past. In addition to standard GMTI sensor measurements, one is interested in using nonstandard in formation such as road maps, and terrainrelated visibility conditions to enhance tracker performance. The conventional approach to this problem has been to use the Variable structure IMM (VSIMM), which uses the concept of directional process noise to model motion along particular roads. In this paper, we present a particle ﬁlter based approach to this problem which we call Variable structure Multiple model Particle ﬁlter (VSMMPF). Simulation results show that the performance of the VSMMPF is much superior to that of VSIMM.
Keywords:
GMTI Tracking, Variable StructureIMM, Particle Filter.
1 Introduction
In standard tracking problems, the only inputs available to the tracker are sensor measurements obtainedthrough one or more sensors. However, in some applications there may be some additional informationavailable which could be exploited in the estimationprocess. For instance, one may have some knowledgeof the environment in which the target is being trackedor there may be some knowledge of some constraints onthe dynamic motion of the target, such as speed constraints. An example application in a military contextis Ground Moving Target Indicator (GMTI) tracking,where one may have some information of the terrain,such as road maps and visibility conditions. The question is, can this information be used by the tracker toproduce better estimates of the target state?It turns out that incorporating such nonstandardinformation in conventional Kalman ﬁlter based trackers is not an easy task. The reason is that, in general, incorporating nonstandard information leads tohighly nonGaussian posterior densities, and conventional trackers cannot easily handle propagation of nonGaussian densities in a dynamic state estimationframework. However, there has been some attempt atincorporating such nonstandard information within aKalman ﬁlter based tracker. The most common of these is the Variable Structure Interacting MultipleModel (VSIMM) algorithm [4].The VSIMM uses a modiﬁed version of the standardIMM, where the number, and structure, of the multiple models active at any particular time are allowed tovary. The various models may represent motion under diﬀerent conditions of visibility, road constraintsand target speeds. Although the VSIMM has beenshown to produce better results than methods thatdon’t use such nonstandard information, it still hasmajor drawbacks. In particular, the nonstandard information available to the tracker will lead to highlynonGaussian posterior pdfs which are approximatedby a ﬁnite mixture of Gaussians. In addition, the VSIMM does not have a mechanism to incorporate hardconstraints on position and speed. Because of theseweaknesses, the use of VSIMM has only resulted inmodest improvement in accuracy over methods thatdo not use such nonstandard information.In this project we propose a new algorithm basedon Sequential Monte Carlo methods, which we termVariable Structure Multiple Model Particle Filter (VSMMPF). The basic principle is to use particles (random samples) to represent the posterior density of thestate of a target in a dynamic state estimation framework where nonstandard information is utilised. Sinceparticle ﬁltering methods have no restrictions on thetype of models, including the noise distributions used,
927
ISIF © 2002
one can choose rather complex models to representground vehicle motion in a GMTI context. In particular, the nonstandard information available throughroad maps, speed constraints, etc., is modelled by ageneralised Jump Markov system with constraints onthe state. In addition, the transition probabilities of the Markov process are designed to be state dependent,thus allowing for realistic characteristics of ground vehicles. The proposed algorithm is tested on simulateddata and compared with the performance of the VSIMM.The organisation of the paper is as follows. Section 2 describes the GMTI tracking problem and itsmathematical formulation. Section 3 reviews the VSIMM algorithm followed by Section 4 which presentsthe VSMMPF algorithm. Finally, simulation resultsare presented in Section 5.
2 Problem Description and Formulation
2.1 Problem Description
This section describes the problem of GMTI tracking with nonstandard information. We consider theproblem of tracking ground targets from measurementsobtained using a single sensor. The surveillance regionincludes road networks and varying terrain conditions,such as hills, tunnels, open ﬁelds, etc. Depending onthe target’s present location, its motion is constrainedby these external factors, i.e., the road network andterrain conditions. For example, a target on a particular road has a high probability of continuing itsmotion constrained along that road. Or, an oﬀroadtarget travelling in the open ﬁeld is free to move inany direction, however, it may enter a road only atcertain locations due to constraints such as a river ora hill. Likewise, an onroad target at a junction cancontinue only in one of the roads meeting at the junction. Thus, road networks and terrain conditions resultin constrained target motion capabilities. The targetmotion is also constrained by speed restrictions whichmay be known.In addition to target motion constraints, the terrainconditions can also inﬂuence the measurement processin the following way. Depending on the target’s location, terrain features such as hills and tunnels mayhide the target from the sensor’s view. Thus, the varying obscuration conditions of targets needs to be takeninto consideration.A typical road map is shown in Figure 1 with fourroads, AJ, BJ, CJ, and DJ, meeting at junction J. Roadsegments with solid lines allow entry into or exit fromthe roads while those with broken lines (eg., TU andBJ) indicate that the targets on the road cannot getoﬀ or those oﬀ the road cannot get on. In a typicalGMTI tracking problem, such restrictions are generally determined by the surrounding terrain conditions,for example, rivers, open ﬁelds or ditches. The roadsegment TU represents a tunnel with zero target detection probability.The complete road map with terrain conditions canbe speciﬁed as follows. Each road segment is represented by two waypoints as shown in Table 1.These waypoints determine the direction, location andlength of each road. The visibility is deﬁned as a binary valued probability, and the entry/exit conditionis given by a Boolean variable. The entire speciﬁcation is summarised in Table 1, and in the sequel thistopography information will be denoted by
R
.Figure 1 also shows a sample target trajectorythrough the road network described above. A targetstarts at point I, merges with road AJ at M, then proceeds to junction J through tunnel TU, branches oﬀ toroad JD, exits road JD at E, and terminates at F inthe open ﬁeld.The GMTI tracking problem of interest is as follows. In addition to standard GMTI measurementsof range and azimuth, the above topography information is available to the tracker. The objective is to exploit this nonstandard information to yield enhancedtracker performance.
5000 6000 7000 8000 9000 10000 11000 120005000600070008000900010000
A M I T U J C B D E F target trajectory
Figure 1: An example road network and target trajectory
2.2 Problem Formulation
The problem of target tracking with road constraintsis handled using the concept of
directional
processnoise. To see this, note that the standard motionmodels assume that the target can move in any direc
928
Table 1: Speciﬁcation of road mapRoad AllowSegments Waypoints Visibility Entry/Exit1 AT A and T 1.0 True2 BJ B and J 1.0 False3 CJ C and J 1.0 True4 DJ D and J 1.0 True5 JU J and U 1.0 True6 TU T and U 0.0 Falsetion with equal probability, therefore use equal processnoise variance in both
x
and
y
directions, i.e.,
σ
2
x
=
σ
2
y
.For onroad targets, the road constraints mean moreuncertainty along the road than orthogonal to it. Suppose
σ
2
a
and
σ
2
o
are the variances along and orthogonalto the road, respectively, and the road direction is speciﬁed by the angle
ψ
(measured clockwise from
y
axis).Then, the process noise covariance matrix corresponding to onroad motion is given by [4]
Q
=
−
cos
ψ
sin
ψ
sin
ψ
cos
ψ
σ
2
o
00
σ
2
a
−
cos
ψ
sin
ψ
sin
ψ
cos
ψ
(1)With the above directional process noise matrix concept, the GMTI tracking problem is formulated in aJump Markov System (JMS) framework. Let the state
x
k
= (
x
k
,y
k
,
˙
x
k
,
˙
y
k
)
consist of the position and velocity components of the target in the Cartesian coordinates. Then, the discrete time kinematic model for theproblem is given by
x
k
=
F
x
k
−
1
+
G
v
k
−
1
(
m
k
)
,
x
k
∈
Ξ(
m
k
)
,
(2)where
F
=
1 0
T
00 1 0
T
0 0 1 00 0 0 1
, G
=
T
2
/
2 00
T
2
/
2
T
00
T
,T
is the sampling time,
m
k
is the mode in eﬀect inthe interval (
k
−
1
,k
],
v
k
−
1
(
m
k
) is a mode dependent white Gaussian process noise sequence with covariance
Q
(
m
k
), and
x
k
is constrained to be in themodedependent set Ξ(
m
k
). Physically,
m
k
representsthe type of motion (oﬀroad, onroad in a speciﬁc direction, etc.,) of target. This is modelled as a Markovprocess whose states belong to a variable set. Suppose there are
R
road segments in the network and let
m
k
∈ {
0
,
1
,...,R
}
. Here
m
k
=
m
∈ {
1
,...,R
}
impliesa motion model corresponding to road
m
and
m
k
= 0refers to oﬀroad motion. Now let
S
k
⊆ S
a
denotethe set of modes active in the interval (
k
−
1
,k
], where
S
a
is the set of all possible motion models. Then, theMarkov process
m
k
is characterised by the followingtransition probabilities that depend on
S
k
−
1
,
S
k
, and
x
k
−
1
, i.e.,
p
rs
[
S
k
−
1
,
S
k
,
x
k
−
1
] =
P
(
m
k
=
s
∈ S
k

m
k
−
1
=
r
∈ S
k
−
1
,
x
k
−
1
)
.
(3)The measurement equation applicable to this problem is given by
1
z
k
=
h
(
x
k
) +
n
k
(4)where
h
(
x
k
) = [
x
2
k
+
y
2
k
,
tan
−
1
(
x
k
/y
k
)]
,
n
k
∼ N
(
0
,
R
k
) and
R
k
is a 2
×
2 diagonal measurement covariance matrix with elements equal to the variancesin range and azimuth,
σ
2
r
and
σ
2
θ
, respectively. Formodel
s
, this measurement is received with detectionprobability
P
sD
, which is chosen to be either 0 or 1depending on its visibility speciﬁed in Table 1. Notethat in this paper, we do not include rangerate measurements, and only consider a binary valued
P
sD
. Theinclusion of rangerate and a realistic modelling of
P
sD
will be addressed in future work.Equation (4) is nonlinear and thus each module of the VSIMM would require an EKF. Alternatively, aswill be done in this paper, we use a Converted Measurement Kalman Filter [1], where (4) is linearised as
z
k
=
1 0 0 00 1 0 0
x
k
+
n
k
,
(5)and
n
k
∼ N
(
0
,
R
k
),
R
k
=
σ
2
x
(
R
k
)
σ
xy
(
R
k
)
σ
yx
(
R
k
)
σ
2
y
(
R
k
)
,
(6)
σ
2
x
(
R
k
) =
r
2
k
σ
2
θ
cos
2
θ
k
+
σ
2
r
sin
2
θ
k
,
(7)
σ
2
y
(
R
k
) =
r
2
k
σ
2
θ
sin
2
θ
k
+
σ
2
r
cos
2
θ
k
,
(8)
σ
xy
(
R
k
) =
σ
yx
(
R
k
) = (
σ
2
r
−
r
2
k
σ
2
θ
)sin
θ
k
cos
θ
k
.
(9)Here
r
k
and
θ
k
are the range and azimuth measurements, respectively, at
k
.The problem is, for the JMS system deﬁned by(2) and (5), given a sequence of measurements
Z
k
=
{
z
1
,...,z
k
}
and the topography information
R
, estimate the target state ˆ
x
k

k
=
E
[
x
k

Z
k
,
R
].
3 Variable Structure IMM
The Variable Structure IMM (VSIMM) algorithmfor GMTI tracking has been presented in detail in [4].Here we brieﬂy review its basic concepts.
1
In this paper, we assume a static sensor, located in the srcinof the XY plane.
929
As described above,
S
k
denotes the set of modes active in the interval (
k
−
1
,k
]. Then, the mode probability of mode
s
∈ S
k
is deﬁned as
µ
sk
=
P
m
k
=
s
∈ S
k

Z
k
.
(10)Also, the mode conditioned state estimate and associated covariance of ﬁlter module
s
∈ S
k
are denotedby ˆ
x
sk

k
and
P
sk

k
, respectively. In addition, in the VSIMM described in [4],
p
rs
[
S
k
−
1
,
S
k
,
x
k
−
1
] deﬁned in (3)has no explicit dependence on
x
k
−
1
, and so will be denoted by
p
rs
[
S
k
−
1
,
S
k
].
2
With the above deﬁnitions, the ﬁve steps of eachcycle of the VSIMM are as follows.1. Step 1:
Mode set update
. Based on the state estimate at
k
−
1 and the topography, the mode setof the IMM is updated as
S
k
=
s
∈ S
a
S
k
−
1
,
R
,Z
k
−
1
=
s
∈ S
a
S
k
−
1
,
R
,
{
ˆ
x
rk
−
1

k
−
1
,
P
rk
−
1

k
−
1
,r
∈ S
k
−
1
}}
.
(11)This step is described in more detail in Section3.1.2. Step 2:
Mode interaction or mixing
The modeconditioned state estimates and the associated covariances from the previous scan are combined toobtain the initial condition for the modematchedﬁlters. The initial condition in scan
k
for the ﬁltermodule
s
∈ S
k
is computed usingˆ
x
0
sk
−
1

k
−
1
=
r
∈S
k
−
1
µ
r

sk
−
1

k
−
1
ˆ
x
rk
−
1

k
−
1
,
(12)where
µ
r

sk
−
1

k
−
1
=
P
{
m
k
−
1
=
r

m
k
=
s,Z
k
−
1
}
=
p
rs
[
S
k
−
1
,
S
k
]
µ
rk
−
1
∈S
k
−
1
p
s
[
S
k
−
1
,
S
k
]
µ
k
−
1
,
(13)are the mixing probabilities. The covariance matrix associated with (12) is given by
P
0
sk
−
1

k
−
1
=
r
∈S
k
−
1
µ
r

sk
−
1

k
−
1
P
rk
−
1

k
−
1
+ ˜
d
rsk
−
1
·
(˜
d
rsk
−
1
)
.
(14)where ˜
d
rsk
−
1
= [ˆ
x
rk
−
1

k
−
1
−
ˆ
x
0
sk
−
1

k
−
1
].
2
In the VSIMM, these transition probabilities are determined on the basis of sojourn times of the modes.
3. Step 3:
Modeconditioned ﬁltering
. A convertedmeasurement Kalman ﬁlter is used for each module to compute the modeconditioned state estimate and covariance, given the initial conditions(12) and (14). In addition, depending on whetheror not a measurement was received and basedon the detection probability
P
sD
corresponding tomodule
s
, we compute the likelihood function Λ
sk
for each module
s
asΛ
sk
=
N
[
ν
sk
;0
,
S
sk
]
, z
k
received and
P
sD
= 10
, z
k
received and
P
sD
= 01
,
no
z
k
and
P
sD
= 00
,
no
z
k
and
P
sD
= 1(15)where
ν
sk
and
S
sk
are the innovation and its covariance, respectively, of the measurement
z
k
inmodule
s
.4. Step 4:
Mode Probability Update.
The mode probabilities are updated based on the likelihood of each mode using
µ
sk
=Λ
sk
∈S
k
−
1
p
s
[
S
k
−
1
,
S
k
]
µ
k
−
1
r
∈S
k
∈S
k
−
1
Λ
rk
p
r
[
S
k
−
1
,
S
k
]
µ
k
−
1
(16)5. Step 5:
State Combination.
The mode conditioned estimates and their covariances are combined to give the overall state estimate and covariance as follows:ˆ
x
k

k
=
s
∈S
k
µ
sk
ˆ
x
sk

k
(17)
P
k

k
=
s
∈S
k
µ
sk
P
sk

k
+ [ˆ
x
sk

k
−
ˆ
x
k

k
][ˆ
x
sk

k
−
ˆ
x
k

k
]
.
(18)
3.1 Mode Set Update
The VSIMM adaptively updates the set of activemodes
S
k
based on the current estimate, its covarianceand the topography. We present a brief review of itbelow although details can be found in [4]. Let
L
∈ R
denote the
th road in the network. Then, a modelcorresponding to this road is included in
S
k
by testingwhether any segment of this road lies within a certainneighborhood ellipse centered at the predicted location(ˆ
x
k

k
−
1
,
ˆ
y
k

k
−
1
). The ellipse
E
k
is the region in the XYplane which satisﬁes the following condition:
E
k
=
xy
:
x
−
ˆ
x
k

k
−
1
y
−
ˆ
y
k

k
−
1
×
P
11
k

k
−
1
P
12
k

k
−
1
P
21
k

k
−
1
P
22
k

k
−
1
−
1
x
−
ˆ
x
k

k
−
1
y
−
ˆ
y
k

k
−
1
≤
α
(19)
930
where
α
is the “gate threshold” and
P
posk

k
−
1
=
P
11
k

k
−
1
P
12
k

k
−
1
P
21
k

k
−
1
P
22
k

k
−
1
(20)is the position submatrix of the prediction covariance
P
k

k
−
1
. A road is deemed
validated
if any segment of it belongs to
E
k
.The mode set update proceeds as follows. First, every junction
J
j
∈ R
is subject to the above test. If
J
j
∈ E
k
, then a model corresponding to each roadmeeting
J
j
is added to
S
k
. Next, all road segmentsthat did not pass during the junction test is tested. If any point on the road satisﬁes the test, a model corresponding to that road is added to
S
k
. Finally, baselinemodels that correspond to oﬀroad motion are addedif any one of the following criteria are satisﬁed: (a) Aroad (with entry/exit) is validated, (b) A road (with noentry/exit) is validated and one if its endpoints fromwhich the target can get into open ﬁeld is also validated, and (c) none of the road segments is validated.
4 Variable Structure MultipleModel Particle ﬁlter
The particle ﬁlter (PF) is a technique for implementing a recursive Bayesian ﬁlter by MonteCarlo simulations [2][3]. The key idea of the PF is to representthe required posterior density function by a set of random samples or “particles”, and to compute the stateestimate based on these samples.The srcinal particle ﬁlter proposed by Gordon etal [2] was developed for a single dynamic model usingthe Sampling Importance Resampling (SIR) algorithmto propagate and update the particles. Extensions tomultiple models have been reported and the underlyingtheory can be found in [5, 6]. In this paper we presenta modiﬁed version of the multiple model particle ﬁlterthat is suited to GMTI tracking. The resulting algorithm is termed Variable Structure Multiple ModelParticle Filter (VSMMPF). The key features of theVSMMPF algorithm are a) The number of models active at any particular time, and b) the state transitions,vary depending on current state and topography.To see the operation of the VSMMPF, consider a setof particles
{
(
x
ik
−
1
,m
ik
−
1
)
}
N i
=1
that represents the posterior pdf
p
(
x
k
−
1
,m
k
−
1

Z
k
−
1
) of the state and modeat
k
−
1. Now, suppose at
k
we have some measurement
z
k
.
3
It is required to construct a sample
{
(
x
ik
,m
ik
)
}
N i
=1
which characterises the posterior pdf
p
(
x
k
,m
k

Z
k
) at
k
. This is carried out in two steps: prediction andupdate.Before describing these in detail, the following notation is introduced. Let
R
m
denote the set of state
3
Note that
z
k
can also be null indicating no measurements.
vectors such that the position components lie on road
m
, i.e.,
R
m
=
{
x
: (
x,y
) is on road segment
m
}
.
(21)Similarly,
R
m
=
{
x
: (
x,y
) is on ‘extended’ road segment
m
}
.
(22)where ‘extended’ road segment
m
is the road obtainedby arbitrarily extending both end points of road segment
m
to give an inﬁnitely long road. Next, let
G
denotes the set of state vectors satisfying some speedconstraints, i.e.,
G
=
{
x
:

v

min
≤
˙
x
2
+ ˙
y
2
≤ 
v

max
}
.
(23)Finally, let
R
90
and
R
−
90
denote the rotational transformation matrices corresponding to rotations through90
◦
and
−
90
◦
, respectively.
4.1 Prediction Step
Here we describe the prediction phase of a speciﬁcparticle (
x
ik
−
1
,m
ik
−
1
). Two cases will be considered:(1)
m
ik
−
1
=
m
∈ {
1
,...,R
}
, i.e., a road particle, and(2)
m
ik
−
1
= 0, i.e., an oﬀroad particle.Consider case (1) where
m
ik
−
1
=
m
is a road particle. Before predicting, it must be determined whetherthis particle executes a road motion or oﬀroad motion at
k
. This is determined by the model parameter
p
m
, which is the probability of executing road motionat
k
, given the target was on road
m
at
k
−
1. If itis determined that this particle executes road motionat
k
, then the prediction is carried out by generating
v
ik
−
1
∼ N
∗
(
0
,
Q
(
m
)) such that
x
∗
ki
=
F
x
ik
−
1
+
G
v
ik
−
1
(
m
)
,
x
∗
ki
∈ G ∩R
m
(24)where
N
∗
(
·
,
·
) is a truncated Gaussian density that ensures
x
∗
ki
∈ G ∩R
m
. However, if it is determined thatthe particle executes an oﬀroad motion, then an intermediate state
x
i
k
−
1
is formed whose velocity is perpendicular to the current velocity of
x
ik
−
1
. This intermediate state is then predicted according to
x
∗
ki
=
F
x
i
k
−
1
+
G
v
ik
−
1
(0)
,
x
∗
ki
∈ G
(25)where
v
ik
−
1
∼ N
∗
(
0
,
Q
(0)) is a process noise corresponding to oﬀroad motion.Predictions near road junctions is a bit tricky.Speciﬁcally, if it turns out that in the transition
x
ik
−
1
→
x
∗
ki
given in (24), the particle crosses a junction
J
, then the following is carried out. Suppose junction
J
has road segments
j
1
,...,j
n
J
connecting to it.Then, the predicted particle is placed in one of theseroad segments with probability 1
/n
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. This is done as
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