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On Constraints Exploitation for ParticleFiltering Based Target Tracking
F. Papi, M. Podt, Y. BoersTHALES Nederland B.V.
Sensors-TBU Radar EngineeringHengelo, The Netherlands
Email: [email protected]
G. Battistello, M. UlmkeFraunhofer FKIE
Sensor Data and Information Fusion Dept.Wachtberg, Germany
Email: [email protected]
Abstract—Nonlinear target tracking is a well known problem,and its Bayes optimal solution, based on particle filtering tech-niques, is nowadays applied in high performance surveillancesystems. Nonetheless, the practical application of Particle Filters(PFs) may still be difficult, so that possibly available externalknowledge can be exploited to increase the tracking performance.
In this paper we assume such knowledge be formalized interms of constraints on target dynamics. Hence, a Constrainedversion of the Filtering problem has to be solved. We first treatthe case of perfectly known hard constraints, and show thatexploitation of knowledge in the prediction or in the update stepof the Bayesian filtering recursion are equivalent.
We then focus on the case of soft constraints. Here, the lack ofinformation on when and how the target violates the constraintsmakes the filtering problem much more difficult. Simulationresults show that a straightforward extension of the Pseudo-Measurements approach is not sufficient. However, detecting theviolation of constraints is possible if the knowledge is processedusing an Interactive Multiple Models (IMM) scheme.
I. INTRODUCTION
Many real life problems require the estimation of the stateof a system that changes over time using a sequence of noisymeasurements made on the system. Target tracking based onmeasurements collected by a radar or a similar sensor is animportant application example.
For many years, the Kalman Filter (KF) [1] has beenconsidered the work horse algorithm when dealing with linearGaussian systems. In the last twenty years, accurate modelinghas required the inclusion of nonlinearity and non-Gaussianityin the equations used for estimation purposes, thus making theKF inapplicable in its basic form. Classical methods for non-linear filtering are, e.g., the Extended Kalman Filter (EKF) [2]and the Unscented Kalman Filter (UKF) [3]. They are basedon deterministic approximations of the a posteriori probabilitydensity function (PDF), and are oftentimes sufficient when theeffects of nonlinearity and non-Gaussianity are mild.
Bayesian methods provide a rigorous framework for dy-namic state estimation problems. The Bayesian approach isto construct the PDF of the state based on all the availableinformation, and then find an approximation of such a pos-teriori PDF, which does not usually admit a closed form.Particle Filter (PF) algorithms, which are Monte Carlo basedapproximations of the Bayesian recursion, nowadays representthe state of art in nonlinear filtering [4]. In fact, from the
seminal paper of Gordon [5] in 1993, PF techniques haverevolutionized the way nonlinear tracking problems are tack-led. They operate by propagating particles that are distributedaccording to the approximately true PDF of the state, andpractically represent an efficient numerical approximation ofthe full recursive Bayesian estimation problem. In addition,convergence to the true a posteriori distribution is guaranteedfor a sufficiently large number of particles [6].
Many sources of external information about the observedscenario may be available to the tracking system. For instance,the output of a classification algorithm might provide valu-able information concerning the type of target observed (e.g.friendly, hostile, etc.), its radar cross section, etc. The exploita-tion of a known road network has been proven effective intracking of ground vehicles [7]. Similarly for a maritime sce-nario, the knowledge on shipping lanes and sea/land distinctioncan improve the tracking and detection performance [8].
This work is part of a project in which we are interested onthe improvements achievable through Bayes optimal exploita-tion of external knowledge [9]. In this paper we consider theConstrained Filtering problem. We first focus on the case ofperfectly known hard constraints, and show that exploitationof knowledge in the prediction or in the update step of theBayesian filtering recursion are equivalent. We then focuson the case of soft constraints, and show through simula-tion results that a straightforward extension of the Pseudo-Measurements technique is not sufficient. However, if thedetection of such anomalies is of interest, the problem canbe solved by using Interactive Multiple Models (IMM).
The paper is organized as follows: in section II we discussrelated works; in section III we briefly review the Bayesian Fil-tering problem; in section IV we consider the Constrained Fil-tering problem for the case of hard constraints; in section V wereview the two main techniques for constraints exploitation inparticle filtering. We then consider the case of soft constraintsin section VI, and propose a PF-based IMM approach for thespecific application of detecting the occurrence of unmodeledbehaviors. Finally, section VII collects our conclusions.
II. RELATED WORKS
In this section we consider related works on the exploitationof external knowledge that may be available in the form of
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equality and/or inequality constraints. The idea of using stateconstraints to improve the tracking performance dates backto the 90s, when attempts to exploit hard linear equality con-straints in Kalman Filtering led to the definition of the Pseudo-Measurements approach [10]. The key idea is to interpret theconstraints as additional measurements, and is proven to beoptimal for linear systems subject to hard constraints [11].
In [12] the use of particle filtering for road-constrainedtarget tracking is considered. The proposed algorithm prop-agates the joint PDF of the target kinematic state and targetID in a road-constrained environment. Inequality constraintson target speed and off-road distance are treated using a min-max saturation approach, which requires a low computationalload while leading to suboptimal constraints satisfaction.
Road map assisted ground targets tracking is also consideredin [13]. Here the author proposes the use of a Gaussiansum algorithm within a Variable Structure Multiple Models(VSMM) scheme. As long as the predicted estimate is insidethe same road, a Kalman filter is used to perform the updatestep. When the target approaches a junction, an on-roadprojection is necessary, and a multiple hypotheses approach isfollowed. Good results are presented, but the approach cannotbe extended to nonlinear inequality constraints.
Ground targets tracking with airborne GMTI sensor mea-surements is considered in [14]. A refined GMTI sensormodel with state dependent detection probability and infor-mation about the clutter notch is proposed. Both equalityand inequality constraints are used to model the known roadnetwork. The latter are used to model non-zero width roads.The authors investigate the performance of both Gaussian sumand particles based approximations, in which the predictionstep is performed in road coordinates, while the update stepis carried out in the 2D Cartesian space.
Hard inequality state constraints are considered in [15]. Thepaper is focused on tracking of airplanes with a known flightenvelope (i.e. minimum and maximum velocities). The authorspropose to obtain samples from a truncated distribution using aRejection-Sampling approach, where particles are reproposedif they do not meet the constraints. The proposed PF algorithmconverges to the true a posteriori PDF for a sufficientlylarge number of particles, but might be unfeasible due to thecomputational load required.
In [16] the use of PF for littoral tracking is proposed.The authors formulate the problem as Joint Tracking andClassification (JTC), where a target class is assigned for eachisolated land or water region. A similar approach is followedin [17], where the authors propose a modified version of theJTC-PF algorithm that uses class-dependent speed likelihoods.
In [18] a VSMM-PF is used for tracking of ground targetswith GMTI sensors. The information available through aroad map is modeled using a Jump Markov system withstate dependent transition probabilities. Each road segment isrepresented by two way-points determining direction, location,and length of each road. The visibility is defined as a binaryvalued probability, and an entry/exit condition is given by aBoolean variable for a subset of the roads. The inequality
constraints on target speed are applied in prediction throughthe generation of random variables from a truncated Gaussian.
III. BAYESIAN FILTERING PROBLEM
In this section we briefly describe the Bayesian FilteringProblem and recall a convergence result [19] for its particlefiltering based solution. In the next section we will focus onthe exploitation of external knowledge formalized in terms ofconstraints on the target dynamics.
Suppose the system is described by the following state andmeasurement equations:
xk+1 = fk(xk) + wk (1)zk = hk(xk) + vk (2)
where xk ∈ Rnx is the system state, zk ∈ Rnz themeasurement vector, wk ∼ pwk(w) the process noise, andvk ∼ pvk(v) the measurement noise. The Markov propertyholds for the system (1)-(2), i.e.,
p(xk|xk−1,xk−2, . . . ,x0) = p(xk|xk−1) (3)
where p(xk|xk−1) is known as the transition Kernel.Let Zk
4={
z0 z1 . . . zk}
be the sequence of mea-surements up to and including time k. Hence, the measurementzk at time k is independent from past states, i.e.,
p(zk|zk−1, . . . , z1,xk,xk−1, . . . ,x0) = p(zk|xk) (4)
where p(zk|xk) is known as the likelihood function.Given a realization of Zk, the filtering problem aims at
computing the conditional probability density p(xk|Zk). Letus assume that at time step k − 1 the PDF p(xk−1|Zk−1) isavailable, then the problem is solved using the recursion:• Prediction Step
p(xk|Zk−1) =∫p(xk|xk−1) p(xk−1|Zk−1) dxk−1 (5)
where p(xk|Zk−1) is the predictive density at time k.• Update Step
p(xk|Zk) =p(zk|xk) p(xk|Zk−1)
p(zk|Zk−1)(6)
where p(zk|Zk−1) is the Bayes normalization constant.Various state estimators are obtained from the a posteriori
PDF p(xk|zk), e.g. the minimum variance (MV) estimator,i.e.,
xMVk
4=∫
Rnxxk p(xk|Zk) dxk (7)
or the maximum a posteriori (MAP) estimator, i.e.,
xMAPk
4=
arg maxxk ∈ Rnx p(xk|Zk) (8)
More in general, if φ(xk) : Rnx → Rnφ is a function of thestate we want to estimate, most estimation algorithms computean approximation of the conditional expectation:
E(φ(xk)|Zk
)=∫
φ(xk) p(xk|Zk) dxk (9)
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The particle filter computes an approximation of (9) using theempirical filtering density [5]:
pN (xk|zk) =N∑i=1
wik δxik(xk) (10)
where each particle xik has an importance weight wik associ-ated to it, and δxik(·) denotes the delta-Dirac mass locatedat xik. Convergence results for the mean square error inapproximating eq. (9) have been given in [19], i.e.,
Let us assume that the likelihood function p(zk|·) is boundedin the argument xk ∈ Rnx , and that the system Kernel isweakly dependent on past state values, then for all k ≥ 0 thereexist a constant c such that for any function φ ∈ B(Rnx):
E[((pN ,φ)− (p,φ))2
]≤ c ‖φ‖
N(11)
where N is the number of particles, B(·) the set of Borelbounded functions in Rnx , and we used the notation (p,φ)
4=∫
pφ. This roughly means that if the true optimal filter isquickly mixing, then uniform convergence in time of theparticle filtering method is ensured. However, a sufficientlylarge number of particles is required in practice. Analyses onthe minimal number of particles are reported in [20].
IV. CONSTRAINED BAYESIAN FILTERING
As already mentioned, oftentimes additional informationabout the state is available. In fact, the state variables usuallycorrespond to physical quantities of interest, and validityregions may be helpful for the filter design. For instance, inground target tracking constraints on target position are easilyobtained from sea/land distinction, while speed constraints canbe defined based on the solution of the classification problemand/or based on the type of terrain/sea traveled at the moment.
Here we specifically focus on Constrained Bayesian Fil-tering for the case of hard constraints. Despite being purelytheoretic at first glance, the case of perfectly known hardconstraints is oftentimes encountered in practice. Examples arethe tracking of ground vehicles moving on a road network,or the tracking of ships traveling on canals. In general, allthe constraints arising from physical laws, e.g., the massconservation for chemical reactions, are of this family.
Let us assume that external information is available in termsof nonlinear inequality constraints:
ak ≤ Ck(xk) ≤ bk (12)
where Ck : Rnx → Rnc , and the inequality sign holds forall elements. For convenience, let Ck be the set of all statessatisfying the inequality constraint (12), i.e.
Ck4= {xk : xk ∈ Rnx ,ak ≤ Ck(xk) ≤ bk} (13)
and Ck 4= {C0, C1, . . . , Ck} be the sequence of Ck up to
time k. From a Bayesian viewpoint, exploitation of externalknowledge boils down to finding an approximation of:
p(xk|Zk, Ck) ∝
{p(xk|Zk), if xk ∈ Ck
0, otherwise (14)
where conditioning is performed also with respect to thesequence Ck of constrained state variables. Let us assume thatwe are able to define a two step recursion such that:
p(xk−1|Zk−1, Ck−1) Prediction−−−−−−−−−→ p(xk|Zk−1, Ck)
p(xk|Zk−1, Ck) Update−−−−−−→
p(xk|Zk, Ck)
Then, for a sufficiently large number of particles, a particle fil-tering approximation pN (xk|Zk, Ck) of the above a posterioriPDF will converge to the exact a posteriori distribution.
In the following we define two Bayesian recursions for Con-strained Filtering, in which knowledge about the constraints isused in the prediction step in one case, and in the update stepin the other case. We show that from a Bayesian viewpointthe two recursions are equivalent.
A. Using Knowledge in the Prediction Step
In order to exploit information in the prediction step, wedefine the following predictive PDF:
p(xk|Zk−1, Ck) =
=∫p(xk,xk−1|Zk−1, Ck) dxk−1
=∫p(xk|xk−1,Zk−1, Ck) p(xk−1|Zk−1, Ck) dxk−1
=∫p(xk|xk−1, Ck) p(xk−1|Zk−1, Ck−1) dxk−1 (15)
Proceeding, in the update step of the Bayesian recursion wedefine the following a posteriori distribution:
p(xk|Zk, Ck) =p(xk, zk,Zk−1, Ck)p(zk,Zk−1, Ck)
=p(zk|xk,Zk−1, Ck) p(xk|Zk−1, Ck) p(Zk−1, Ck)
p(zk|Zk−1, Ck) p(Zk−1, Ck)
=p(zk|xk) p(xk|Zk−1, Ck)
p(zk|Zk−1, Ck)(16)
B. Using Knowledge in the Update Step
Here we define the predictive distribution without condi-tioning on the most recent set Ck, i.e.
p(xk|Zk−1, Ck−1)
=∫p(xk,xk−1|Zk−1, Ck−1) dxk−1
=∫p(xk|xk−1,Zk−1, Ck−1) p(xk−1|Zk−1, Ck−1) dxk−1
=∫p(xk|xk−1) p(xk−1|Zk−1, Ck−1) dxk−1 (17)
And then update using both zk and Ck, i.e.
p(xk|Zk, Ck) =p(xk, zk,Zk−1, Ck, Ck−1)p(zk,Zk−1, Ck, Ck−1)
=p(zk|xk) p(Ck|xk) p(xk|Zk−1, Ck−1)
p(zk|Zk−1, Ck) p(Ck|Ck−1)(18)
457
Let us now compare the a posteriori PDFs defined by eqs.(16) and (18). We conclude that the two recursions coincidefrom a Bayesian viewpoint if the following equivalence holds:
p(xk|Zk−1, Ck) =p(Ck|xk) p(xk|Zk−1, Ck−1)
p(Ck|Ck−1)(19)
which can be proven using Bayes theorem. Hence, for a suffi-ciently large number of particles, and for the case of perfectlyknown hard constraints, a particle filtering approximationto the constrained filtering recursion will provide the sameresults independently from processing the available externalknowledge in the prediction or update step.
V. PARTICLE FILTERING METHODS
In this section we review the two main methods for con-straints exploitation in particle filtering: (a) the Rejection-Sampling PF which carries out the processing of section IV-A,and (b) the Pseudo-Measurements PF which carries out theprocessing described in section IV-B.
A. Rejection-Sampling (Prediction Step)
A procedure to perform constrained sampling was intro-duced in [15]. Consider the conditional probability theorem:
p(x|A) =p(x,A)p(A)
(20)
where A represents constraints on x such that A ={x : x ≤ a}. Then the following are true:
p(x|A) = p(x|x ≤ a) =p(x, x ≤ a)p(x ≤ a)
(21)
p(x|A) =
p(x)
p(x ≤ a), if x ≤ a
0, otherwise(22)
In other words, the constrained PDF is the original p(x)restricted to A and normalized. Thus the particle filter ofAlgorithm 1 conceptually solves the problem. The solution istheoretically correct and extremely simple, but the computa-tional load required generally makes the approach unfeasible.
B. Pseudo-Measurements (Update Step)
The Pseudo-Measurements approach interprets the con-straints as additional measurements. The main step is thedefinition of a constraint based likelihood function:
p(Ck|xik) ={
1, if ak ≤ Ck(xik) ≤ bk0, otherwise (23)
Thus, the only necessary modification is the use of an addi-tional likelihood function in the evaluation of the weights:
wik = wik−1
p(zk|xik) p(Ck|xik) p(xik|xik−1)qk(xik|xik−1, zk)
(24)
where the ∼ sign is used to refer to unnormalized weights.Furthermore, if we use the transition kernel as proposaldistribution, we have the common simplification:
wik = wik−1 p(zk|xik) p(Ck|xik) (25)
Algorithm 1: Rejection-Sampling Particle Filter
Input:{xi
k−1, wik−1
}N
i=1and the new measurement zk
Output:{xi
k, wik
}N
i=1
while i = 1, 2, . . . , N (Prediction Step) dowhile xi
k 6∈ Ck (Rejection-Sampling) doGenerate a New Particle: xi
k ∼ pk(xik|x
ik−1)
endendwhile i = 1, 2, . . . , N (Update Step) do
Compute Weights: wik = wi
k−1 p(zk|xik) ;
end
Normalization Step: wik = wi
k/∑N
i=1 wik ∀ i ;
Effective Sample Size: Neff = 1/∑N
i=1(wik)2 ;
if Neff ≤ βN (Resampling Step) thenNew Particles
{xi
k, 1/N}N
i=1s.t. P (xi
k = xik) = wi
kend
which leads us to the definition of Algorithm 2.The approach generally requires a higher number of parti-
cles since many of them are discarded at each step. However,experience tells that the computational load is usually stronglyreduced compared to Rejection-Sampling PF of Algorithm 1.
Algorithm 2: Pseudo-Measurements Particle Filter
Input:{xi
k−1, wik−1
}N
i=1and the new measurement zk
Output:{xi
k, wik
}N
i=1
while i = 1, 2, . . . , N (Prediction Step) doGenerate a New Particle: xi
k ∼ pk(xik|x
ik−1)
endwhile i = 1, 2, . . . , N (Update Step) do
Compute Weights: wik = wi
k−1 p(zk|xik) p(Ck|xk) ;
end
Normalization Step: wik = wi
k/∑N
i=1 wik ∀ i ;
Effective Sample Size: Neff = 1/∑N
i=1(wik)2 ;
if Neff ≤ βN (Resampling Step) thenNew Particles
{xi
k, 1/N}N
i=1s.t. P (xi
k = xik) = wi
kend
C. Discussion about the two filtersLet us now discuss on the algorithms, and show that they
provide the same results also from a practical viewpoint. TheRejection-Sampling PF is in principle an optimal solution.In fact, all the particles verify the constraints at every timestep thanks to an efficient approximation of the constrainedpredictive PDF p(xk|Zk−1, Ck). The Pseudo-MeasurementsPF requires resampling at every time step if all the particlesneed to be constrained. The computational load per particle isdrastically reduced, but a larger number of particles is required.
Let Xk 4={
x0 . . . xk}
be the sequence of systemstates. Hence, the general expression for the weights is:
wk =p(Zk|Xk) p(Xk)
p(Zk)(26)
458
which for sequential constrained filtering becomes:
wk = wk−1p(zk|xk) p(xk|xk−1, Ck)
q(xk|Xk−1,Zk)(27)
where we assume that we are able to evaluate the constrainedtransition Kernel p(xk|xk−1, Ck).In Rejection-Sampling the following holds true:
q(xk|Xk−1,Zk) = p(xk|xk−1, Ck) (28)
which yields for the evaluation of the weights:
wRSk = wRSk−1 p(zk|xk) (29)
In the Pseudo-Measurements approach we choose:
q(xk|Xk−1,Zk) = p(xk|xk−1) (30)
and use the hard constrained likelihood:
p(Ck|xk) =p(xk|xk−1, Ck)p(xk|xk−1)
={
1 xk ∈ Ck0 otherwise (31)
which yields for the evaluation of the weights:
wPSk = wPSk−1 p(zk|xk) p(Ck|xk) (32)
which for the subset of particles verifying the constraintscoincides with eq. (29). Hence, in the case of hard constraintsand for a sufficiently large number of particles, the twomethods provide the same results.
A different reasoning can be followed in order to verifythe correctness of the Pseudo-Measurements PF. The con-straints in eq. (12) affect the target dynamics from a physicalviewpoint. Hence, exploitation of such knowledge in theprediction step as described in section IV-A is Bayes optimalif we can evaluate the constrained Kernel p(xk|xk−1, Ck). Inaddition, the convergence results of PFs are not affected bythe chosen importance function q(xk|Xk−1,Zk) as long as theweights are re-scaled using eq. (27). Hence, we can use theunconstrained transition Kernel as importance function andevaluate the weights as:
wk = wk−1p(zk|xk) p(xk|xk−1, Ck)
p(xk|xk−1)(33)
which exactly coincides with the equation used by Algo-rithm 2, thus proving again the correctness of the Pseudo-Measurements PF in the case of hard constraints.
D. Comparisons using the KL DivergenceLet us now consider a simple 2D tracking example and
perform simulative comparisons of the algorithms. We havealready proven that the two methods are equivalent from atheoretical viewpoint, and we are now interested in showingthat they yield equivalent empirical distributions.
In information theory, the concept of differential entropyis related to the information content of a continuous randomvariable. The relative entropy, or Kullback-Leibler Divergence(KLD), is a non-symmetric measure of the diversity existingbetween two continuous probability distributions. Let a and b
be two continuous densities on Rd, then the KLD DKL(a, b)between a and b is a scalar number in [0,∞] such that:
DKL(a, b) ={
0 if a ≡ b+∞ if support(b) 6⊆ support(a) (34)
The KLD is always positive and equal to zero if and only ifthe two densities coincide. Hence, it is an appropriate meansto evaluate the closeness of a density to another.
Let {x1,x2, . . .xn} and {y1,y2, . . . ,ym} be i.i.d. samplesdrawn from a and b, respectively. An asymptotically unbiasedand mean square consistent estimator for the KLD DKL(a, b)was introduced in [21]. Furthermore, in [22] the estimatorwas shown to be effective for comparisons of particle filters.Here we follow a similar approach. In particular, we areinterested in showing that the particle-based KLD betweenthe empirical PDFs obtained from the Rejection-Sampling PFand the Pseudo-Measurements PF converges towards zero foran increasing number of particles. Notice that since the twomethods are equivalent, the true KLD is 0.
Fig. 1. Scenario: Rejection-Sampling PF vs. Pseudo-Measurements PF
We consider a simple 2D tracking problem as the one depictedin fig. 1, where a ship is traveling inside a known shippinglane. The chosen state vector is xk =
[xk yk xk yk
],
where (xk, yk) and (xk, yk) are position and velocity vectors,respectively. A Nearly Constant Velocity (NCV) model is usedfor the dynamics, and we assume that a Radar positioned atthe Cartesian origin collects measurements of range, azimuth,and range rate. Hence, the nonlinear measurement functionhk(·) in (2) takes the form:
hk(xk)4=
√
(xk)2 + (yk)2
atan2(yk, xk)
xk xk + yk yk√(xk)2 + (yk)2
(35)
and vk is zero-mean Gaussian noise. Parameters used in sim-ulations are reported in Table I. Knowledge on the shippinglane and target dynamics is modeled by the constraints:
Ck4= {xk ∈ Rnx : 45 ≤ yk ≤ 55, vk ≤ 10} (36)
vk4=√
(xk)2 + (yk)
2
and implies the following hard constrained likelihood:
p(Ck|xik) ={
1, if xik ∈ Ck0, otherwise (37)
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Parameter Symbol ValueRange Std. Deviation σr 25m
Azimuth Std. Deviation σθ 1 degRange Rate Std. Deviation σr 1m/s
Sampling Time Ts 1s
TABLE IPARAMETERS USED IN SIMULATION
Fig. 2. Mean Value and Variance of the estimated Kullback-LeiblerDivergence over 100 Monte Carlo runs
In fig. 2 we report the results obtained over 100 MonteCarlo trials for an increasing number of particles. We reportthe Mean Value and the Variance for the estimated Kullback-Leibler Divergence for the two constrained PFs. From fig. 2it is possible to verify the convergence towards 0 for both theMean and the Variance of the estimated KLD. The analysisis of interest since it verifies the equivalence of the twomethods from a practical viewpoint. However, a sufficientlylarge number of particles is required to achieve satisfactorylow values of the KLD.
VI. SOFT CONSTRAINED FILTERING
In this section we consider the case of Soft Constraints. Asalready mentioned, we define a constraint as being soft whenthe tracker knows exactly the set Ck in eq. (13), but the targetcan violate the constraints. A good example is a maritimescenario with a known shipping lane. On average ships willtravel within the shipping lane, possibly following the middleline of it. However, ships can travel outside the shipping lane.How should we model such information?
We are interested in studying the performance of thePseudo-Measurements PF in the presence of soft constraints.In particular, our simulation results show that when unmod-eled behaviors occur, the performance degradation in usinga straightforward implementation of constrained filtering is
important. Such problem cannot be solved without focusingon a specific application. In fact, in the second part of thissection we show that if the detection of unmodeled behaviorsis of interest, then a suitably designed Interactive MultipleModels (IMM) PF can effectively solve the problem.
Let us consider the scenario of fig. 3, where a target ismoving from the initial position (x0, y0) = (50, 52.5)m at aconstant speed of 5m/s along the x component. Knowledge
Fig. 3. Considered scenario. The ship travels from left to right at a constantvelocity of 5m/s along the x component. For time instants k ∈ [20, 40] theship violates the soft constraint defined by the shipping lane.
on the shipping lane and target dynamics is given by (36). Theconstraints are soft since the target travels outside the shippinglane for k ∈ [20, 40]. The chosen state vector contains againthe Cartesian components, and a Nearly Constant Velocity(NCV) model is used for the target dynamics. A radar collectsmeasurements of range, azimuth, and range rate. Hence, thenonlinear measurement function in (35) is used once more, aswell as the parameters reported in Table I.
A. Soft Pseudo-Measurements
We now consider a straightforward extension of the Pseudo-Measurements PF for the case of soft constraints. If theconstraints are given by eq. (36), an extension is obtainedusing the following knowledge-based likelihood:
p(Ck|xik) ={
1, if xik ∈ Ckα, otherwise (38)
where α < 1 can be chosen based on the frequency at whichthe constraint has been violated so far. Compared to eq. (23),instead of using a 0−1 likelihood, we define a fuzzy member-ship alike function. In the tracking literature, this approach isconsidered a reasonable solution to soft constrained filtering,and has already been used with promising results [23].
We tested the Unconstrained SIR-PF and the Pseudo-Measurements PF over 200 Monte Carlo runs. For the Pseudo-Measurements PF we considered different values of the param-eter α = {0.2, 0.3, . . . , 0.9}. Both filters evaluate the MAPestimate of eq. (8), which is implemented using the approachdescribed in [24]. The tracking performance is measured interms of the absolute value of the Mean Error (ME), and interms of the Root Mean Square Error (RMSE).
The results are reported in figs. 4 and 5. During the con-straint violation period, i.e., for k ∈ [21, 40], the unconstrainedPF has better results. For k ∈ (41, 80], the ME and RMSE
460
Fig. 4. Absolute Value of the Mean Error, and Root Mean Square Error forthe MAP Estimate along the x component.
Fig. 5. Absolute Value of the Mean Error, and Root Mean Square Error forthe MAP Estimate along the y component.
decrease but time is needed before good performance isreached again.
To better understand the problem, we run the two filtersusing N = 105 particles over a single trial. In such a case, theempirical distribution given by the Unconstrained PF is a goodapproximation of the true a posteriori distribution. In fig. 6 wereport the empirical distributions at time instant k = 40, whichis the end of the constraint violation period. The chosen αparameter for this case was α = 0.8. Notice how the empiricaldistribution obtained from the Pseudo-Measurements PF isshifted towards lower values of the x component. Hence, thebiases in figs. 4 and 5 are not due to symmetry reasons, but tothe fact that the Pseudo-Measurements PF is targeting a PDF
Fig. 6. Empirical Distributions and true target position at k = 40 forN = 105 particles. N is sufficiently high to say that the SIR-PF gives thetrue a posteriori distribution. From the ’SIR PseudoMeas’ filter we obtain adistribution which is shifted towards lower values of x, this meaning that boththe MV and MAP estimators will give biased estimates.
that does not represent the true target behavior.
B. Detecting Constraints ViolationAt the beginning of this section we asked ourselves how
we should model the information relative to soft constraints.The problem is that we cannot model the constraints violationssince they are unknown. This is a common problem in model-ing, and it is usually solved through increasing of the processnoise intensity. The above presented extension of the Pseudo-Measurements PF is based on the same idea. However, ouranalyses showed that such an approach might not be sufficient.
We are interested in exploiting the known constraints on thetarget dynamics while being able of detecting the occurrenceof unmodeled behaviors (constraint violation). This is a spe-cific application that has great importance in both military andcivil environments. Hence, we propose an Interactive MultipleModels (IMM) PF using the following models:• mk = 1, Constrained Motion Model: a straight line NCV
model implemented through the Pseudo-MeasurementsPF using the binary likelihood of eq. (23);
• mk = 2, Anomalies Detection Model: a straight lineNCV model which is used to predict particles outsidethe shipping lane. Also this model is implemented usingthe Pseudo-Measurements PF with the likelihood:
p(Ck|xik) ={
0, if 45 ≤ xik ≤ 551, otherwise (39)
where mk is the mode variable. The IMM-PF used in simu-lation is based on the algorithm proposed in [25].
We tested the Pseudo-Measurements IMM-PF and the stan-dard Unconstrained PF over 200 Monte Carlo runs usingN = 1e4 particles. In particular, for the IMM-PF we usehalf of the particles for each model. The results in terms ofposition RMSE are depicted in fig. 7. The behavior is similar
461
Fig. 7. Position RMSE over time for the standard Unconstrained PF and thePseudo-Measurements IMM-PF. The filter does not overcome the increase inerror due to the constraint violation for k ∈ [21, 40].
Fig. 8. Average Posterior Mode Probabilities for the IMM-PF. Notice howthe filter is able to detect the constraint violation occurring for k ∈ [21, 40].
to what is shown in figs. 4 and 5. However, from fig. 8 wenotice that the filter is able to detect the constraint violation.
VII. CONCLUSION
This paper addresses fundamental issues arising when track-ing a target that is subject to known constraints. The ParticleFilter is the workhorse for such cases. If constraints are knownand correctly modeled, then the PF converges to the correct aposteriori PDF. In particular, we formally showed that in suchcase, processing of external knowledge in the prediction or inthe update step of the filtering recursion are equivalent from aBayesian viewpoint. This means that we can use the Pseudo-Measurements technique for optimal processing of externalknowledge. The equivalence is also shown from a practicalviewpoint through a simple 2D tracking example. Here aparticle approximation of the Kullback-Leibler Divergence isused as a measure of closeness between empirical densities.
We then focused on the case of soft constraints, anddemonstrated through simulations that if unmodeled behaviorsoccur, the performance degradation is important. Such problemcannot be solved in general without focusing on a specificapplication. In particular, we showed that when the detectionof unmodeled behaviors is of interest, then an IMM-PF withsuitably chosen models effectively solves the problem.
VIII. ACKNOWLEDGMENT
The research leading to these results has received fundingfrom the EUs Seventh Framework Programme under grantagreement no 238710. The research has been carried out inthe MC IMPULSE project: https://mcimpulse.isy.liu.se
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