A variable structure multiple model particle filter for GMTI tracking

A variable structure multiple model particle filter for GMTI tracking
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  A Variable Structure Multiple Model ParticleFilter for GMTI Tracking M. Sanjeev ArulampalamDSTO, Adelaide, OrtonCambridge University, GordonQinetiq, Malvern, U.K.njgordon@taz.qinetiq.comBranko RisticDSTO, Adelaide, Abstract –  The problem of tracking ground targets with GMTI sensors has received some attention in the recent past. In addition to standard GMTI sensor mea-surements, one is interested in using non-standard in- formation such as road maps, and terrain-related visi-bility conditions to enhance tracker performance. The conventional approach to this problem has been to use the Variable structure IMM (VS-IMM), which uses the concept of directional process noise to model motion along particular roads. In this paper, we present a par-ticle filter based approach to this problem which we call Variable structure Multiple model Particle filter (VS-MMPF). Simulation results show that the performance of the VS-MMPF is much superior to that of VS-IMM. Keywords:  GMTI Tracking, Variable StructureIMM, Particle Filter. 1 Introduction In standard tracking problems, the only inputs avail-able to the tracker are sensor measurements obtainedthrough one or more sensors. However, in some ap-plications 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 con-straints. 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 ques-tion is, can this information be used by the tracker toproduce better estimates of the target state?It turns out that incorporating such non-standardinformation in conventional Kalman filter based track-ers is not an easy task. The reason is that, in gen-eral, incorporating non-standard information leads tohighly non-Gaussian posterior densities, and conven-tional trackers cannot easily handle propagation of non-Gaussian densities in a dynamic state estimationframework. However, there has been some attempt atincorporating such non-standard information within aKalman filter based tracker. The most common of these is the Variable Structure Interacting MultipleModel (VS-IMM) algorithm [4].The VS-IMM uses a modified version of the standardIMM, where the number, and structure, of the multi-ple models active at any particular time are allowed tovary. The various models may represent motion un-der different conditions of visibility, road constraintsand target speeds. Although the VS-IMM has beenshown to produce better results than methods thatdon’t use such non-standard information, it still hasmajor drawbacks. In particular, the non-standard in-formation available to the tracker will lead to highlynon-Gaussian posterior pdfs which are approximatedby a finite mixture of Gaussians. In addition, the VS-IMM does not have a mechanism to incorporate hardconstraints on position and speed. Because of theseweaknesses, the use of VS-IMM has only resulted inmodest improvement in accuracy over methods thatdo not use such non-standard information.In this project we propose a new algorithm basedon Sequential Monte Carlo methods, which we termVariable Structure Multiple Model Particle Filter (VS-MMPF). The basic principle is to use particles (ran-dom samples) to represent the posterior density of thestate of a target in a dynamic state estimation frame-work where non-standard 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 partic-ular, the non-standard 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 ve-hicles. The proposed algorithm is tested on simulateddata and compared with the performance of the VS-IMM.The organisation of the paper is as follows. Sec-tion 2 describes the GMTI tracking problem and itsmathematical formulation. Section 3 reviews the VS-IMM algorithm followed by Section 4 which presentsthe VS-MMPF algorithm. Finally, simulation resultsare presented in Section 5. 2 Problem Description and For-mulation 2.1 Problem Description This section describes the problem of GMTI track-ing with non-standard 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 par-ticular road has a high probability of continuing itsmotion constrained along that road. Or, an off-roadtarget 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 on-road target at a junction cancontinue only in one of the roads meeting at the junc-tion. 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 lo-cation, terrain features such as hills and tunnels mayhide the target from the sensor’s view. Thus, the vary-ing 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 gener-ally determined by the surrounding terrain conditions,for example, rivers, open fields or ditches. The roadsegment TU represents a tunnel with zero target de-tection probability.The complete road map with terrain conditions canbe specified as follows. Each road segment is rep-resented by two way-points as shown in Table 1.These way-points determine the direction, location andlength of each road. The visibility is defined as a bi-nary valued probability, and the entry/exit conditionis given by a Boolean variable. The entire specifica-tion 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 pro-ceeds 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 fol-lows. In addition to standard GMTI measurementsof range and azimuth, the above topography informa-tion is available to the tracker. The objective is to ex-ploit this non-standard 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 trajec-tory 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 on-road targets, the road constraints mean moreuncertainty along the road than orthogonal to it. Sup-pose  σ 2 a  and  σ 2 o  are the variances along and orthogonalto the road, respectively, and the road direction is spec-ified by the angle  ψ  (measured clockwise from  y -axis).Then, the process noise covariance matrix correspond-ing to on-road 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 con-cept, 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 veloc-ity components of the target in the Cartesian coordi-nates. 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 depen-dent white Gaussian process noise sequence with co-variance  Q ( m k ), and  x k  is constrained to be in themode-dependent set Ξ( m k ). Physically,  m k  representsthe type of motion (off-road, on-road in a specific di-rection, etc.,) of target. This is modelled as a Markovprocess whose states belong to a variable set. Sup-pose 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 off-road 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 prob-lem 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 co-variance 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 range-rate mea-surements, and only consider a binary valued  P  sD . Theinclusion of range-rate and a realistic modelling of   P  sD will be addressed in future work.Equation (4) is nonlinear and thus each module of the VS-IMM would require an EKF. Alternatively, aswill be done in this paper, we use a Converted Mea-surement 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 measure-ments, 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 , esti-mate the target state ˆ x k | k  =  E [ x k | Z  k , R ]. 3 Variable Structure IMM The Variable Structure IMM (VS-IMM) 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 X-Y plane. 929  As described above,  S  k  denotes the set of modes ac-tive in the interval ( k  − 1 ,k ]. Then, the mode proba-bility 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 asso-ciated covariance of filter module  s  ∈ S  k  are denotedby ˆ x sk | k  and  P sk | k , respectively. In addition, in the VS-IMM 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 de-noted by  p rs [ S  k − 1 , S  k ]. 2 With the above definitions, the five steps of eachcycle of the VS-IMM are as follows.1. Step 1:  Mode set update  . Based on the state esti-mate 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 co-variances from the previous scan are combined toobtain the initial condition for the mode-matchedfilters. 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 ma-trix 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 VS-IMM, these transition probabilities are deter-mined on the basis of sojourn times of the modes. 3. Step 3:  Mode-conditioned filtering  . A convertedmeasurement Kalman filter is used for each mod-ule to compute the mode-conditioned state esti-mate 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 co-variance, respectively, of the measurement  z k  inmodule  s .4. Step 4:  Mode Probability Update.  The mode prob-abilities 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 condi-tioned estimates and their covariances are com-bined to give the overall state estimate and co-variance 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 VS-IMM 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 X-Yplane 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, ev-ery 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 corre-sponding to that road is added to S  k . Finally, base-linemodels that correspond to off-road 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 vali-dated, and (c) none of the road segments is validated. 4 Variable Structure MultipleModel Particle filter The particle filter (PF) is a technique for implement-ing a recursive Bayesian filter by Monte-Carlo simula-tions [2]-[3]. The key idea of the PF is to representthe required posterior density function by a set of ran-dom 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 al-gorithm is termed Variable Structure Multiple ModelParticle Filter (VS-MMPF). The key features of theVS-MMPF algorithm are a) The number of models ac-tive at any particular time, and b) the state transitions,vary depending on current state and topography.To see the operation of the VS-MMPF, consider a setof particles  { ( x ik − 1 ,m ik − 1 ) } N i =1  that represents the pos-terior 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 no-tation 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 seg-ment  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 trans-formation 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 off-road particle.Consider case (1) where  m ik − 1  =  m  is a road parti-cle. Before predicting, it must be determined whetherthis particle executes a road motion or off-road mo-tion 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 en-sures  x ∗ ki ∈ G ∩R  m . However, if it is determined thatthe particle executes an off-road motion, then an in-termediate state  x  i k − 1  is formed whose velocity is per-pendicular to the current velocity of   x ik − 1 . This inter-mediate 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 corre-sponding to off-road 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 junc-tion  J  , then the following is carried out. Suppose junc-tion  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 J  . This is done as 931
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