of EEE Dept of EEE EWRD University of Melbourne University of Technology Sydney DSTO Vic NSW SA Australia Australia Australia dmusickieemuozau schallaengutseduau sofiasuvorovadstodefencegovau Abstract Automatic track initiation of manoeuvring target ID: 25562 Download Pdf

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of EEE Dept of EEE EWRD University of Melbourne University of Technology Sydney DSTO Vic NSW SA Australia Australia Australia dmusickieemuozau schallaengutseduau sofiasuvorovadstodefencegovau Abstract Automatic track initiation of manoeuvring target

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Automatic Track Initiation of Manoeuvring Target in Clutter Darko Mu sicki Subhash Challa Soﬁa Suvorova Dept. of EEE, Dept. of EEE, EWRD University of Melbourne, University of Technology Sydney, DSTO Vic NSW SA Australia Australia Australia d.musicki@ee.mu.oz.au schalla@eng.uts.edu.au sofia.suvorova@dsto.defence.gov.au Abstract Automatic track initiation of manoeuvring targets in clutter is considered and an approximate solution is derived. The proposed approach integrates target ex- istence uncertainty and target dynamical model un- certainty using independent Markov

Chains within the general Bayesian Framework. Using Gaussian approx- imations an IPDA-IMM algorithm is derived and its performance is compared with other PDA based solu- tions - IMM–PDA and IMM–IPDA. 1 Introduction Tracking manoeuvring targets in clutter is a problem of immense practical interest. The most popular frame- work for maneuvering target trajectory estimation is Interacting Multiple Model (IMM) [1] algorithm for estimating processes with switching coeﬃcients. Most of recently published works focus on the variable struc- ture approaches aimed at making tracking models and

approximations model adaptable [2] and extensions of such approaches to scenarios involving clutter [3]. Initiating tracks in clutter results in true tracks which follow targets, as well as false tracks which do not. To perform the false track discrimination, a measure of track quality is necessary. There are only a few tech- niques that address the target existence problem along with target tracking in solid Bayesian formalism [4, 5]. Integrated Probabilistic Data Association (IPDA) [6] ﬁlter addresses the target tracking and target existence problems simultaneously. It extends the PDA

[7] ap- proximation with the notion of target existence. How- ever, most of the earlier IPDA eﬀorts focused only on non-manoeuvring targets. Integrating target existence and target model uncer- tainty using the IPDA was ﬁrst considered in IMM- This research has been supported by the Centre of Expertise in Networked Decision and Sensor Systems and funded by the Defence Science and Technology Organisation Australia IPDA [8]. The presented solution incorporates a bank of IPDA ﬁlters, one for each trajectory model, within an IMM framework. We revisit the problem and pro- pose

an alternative method of integrating target ex- istence and manoeuvre uncertainty along with appro- priate Gaussian assumptions. Our solution consists of IMM maneuvering ﬁlter within the IPDA framework. We refer to this as the IPDA-IMM ﬁlter. IMM-PDA is another technique that addresses the problem of target existence, albeit, in an indirect way [9]. It takes the proof by contradiction approach; ﬁrst it assumes that the target exists and deﬁnes the idea of target detectability, then it argues, that if the prob- ability of target detectability is low then the target

does not exist. In contrast, IPDA explicitly models the target existence as a random variable. In non- manoeuvring target tracking approaches, IPDA ap- pears to be the preferred method [10]. IMM-PDA con- sists of a bank of PDA ﬁlters within the IMM frame- work; one PDA ﬁlter for each trajectory model as well as one PDA ﬁlter for no-target model ( = 0). The probability that a true track is being followed is the complement of the probability of no-target model. Thus, IMM-PDA seamlessly integrates maneuvering target tracking with tracking in clutter (data associ- ation). The

performance of IPDA-IMM, IMM-PDA and IMM- IPDA ﬁlters are compared in a simulated maneuvering target scenario in an environment of heavy and non- uniform clutter, both in terms of the estimation accu- racy and false track discrimination property. This paper is organized as follows. Following the in- troduction, the problem deﬁnition and modelling issues are considered in Section 2. IMM approximation in the context of data association along with the IPDA-IMM speciﬁc equations are considered in Section 3. The gat- ing issues along with a method of gate volume calcula- tion

are dealt with in Section 4. IPDA approximation is considered in Section 5. A complete description of the IPDA-IMM algorithm is presented in Section 6. Simulations are described in Section 7 followed by the

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concluding remarks in Section 8. 2 Problem Formulation and Modelling Without loss of generality, we assume that the dynamic models and the sensor measurement processes are lin- ear and described by (1) where =1 ,M denotes the dynamic model at time is the state of the target trajectory, is the measurement, (1) ,F ) are the state propagation matrices, (1) ,H ) are the mea-

surement matrices. Process noise (1) , and measurement noise (1) , )arezero mean white and uncorrelated Gaussian noise se- quences with covariance matrices (1) ,Q and (1) ,R ) respectively. In target track- ing environment and do not usually depend on , here the dependency is retained for the purpose of generalization. It is assumed that changes in target trajectory can be modelled as a Markov Chain with given transitional probabilities: η, η, [1 ,M (2) We assume that the trajectory of the target can be described at any time by one of dynamic models. For example, a set of dynamic

models can include a uniform motion trajectory, both positive and negative acceleration and a coordinate turn. Switching between the dynamic models allows tracking of maneuvering targets. In the cluttered environment, at each time zero or more measurements are received, target measurements with the probability of detection and (or) spurious - clut- ter measurements. We refer to the target measurement as the true measurement, and to the clutter measure- mentsasthefalsemeasurements. Thetruemeasure- ment is a priori unknown. At each time validation (gating) process selects a pos- sibly empty

subset of received measurements for track estimation. If the target is detected, its’ measurement will be validated with probability . The validation of measurements is described in Section 4 of this paper. Let denote the set of validated measurements at time ,andlet k,i ;( =1 ,m 0) denote the -th measurement of ,where is the number of validated measurements at time . Denote by the set of all validated measurements up to and including time Given the dynamics (1) and measurements the problem is to obtain the a posteriori probability of tar- get existence (denoted by ), the state estimate de-

noted by ) and the state estimate error covariance (denoted by )atanytime 3 The IMM Filter IMM ﬁlter [1] consists of one (usually Kalman) ﬁlter for each dynamic model; the ﬁlter estimates interact with each other to allow soft model switching. PDA [7] based algorithms approximate state probabil- ity density function (pdf) with a single Gaussian pdf after each measurement update. In IPDA-IMM, this approximation is carried out for each model as in equa- tions (4). Total IMM state estimate and prediction are approximated as a weighted sum of Gaussian pdfs: )= =1 η,Z +1

θ,Z )= =1 +1 η, η,Z (3) +1 )= =1 +1 +1 +1 η,Z where each probability density function on the right hand side is approximated with a single Gaussian pdf and +1 η, +1 θ,Z +1 +1 When used within the IPDA-IMM, the IMM at time has the following structure: IMM initialization starts with the state predic- tion and the state prediction error covariance for each model ,P ). From IPDA we obtain the set of validated measure- ments, the set of data association probabilities and the a posteriori probability that the model is correct for each model, i.e , k,i )and IMM Step 1 is

the data association update of state estimate and state estimate error covariance, i.e )and ) for each IMM model using data as- sociation estimate. Due to the PDA [7] approxima- tion, the estimate of state is approximated with a single Gaussian pdf with mean ) and covariance

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): )= =0 k,i ) k,i ) (4) )= =0 k,i k,i )+ k,i ) k,i ) where k,i )and k,i )for =1 ,m are state estimates and state estimate covariances calcu- lated assuming that the -th measurement is true. For = 0, i.e assuming that there is no true measurement, k, )= k, )= (5) where ) is the state prediction and is the

corrected state prediction error covariance ma- trix. When gating probability is 99, is approximately equal to the state prediction error covariance matrix [11]. The operation described by equations (4) and (5) is denoted by D( θ,k ). IMM Step 2 estimates the state of the tracker (this is the tracker output at time ) by combining the state estimates for each model [1]. =1 ) =1 )+ ) (6) IMM Step 3 calculates state estimates at time given the model at time +1.Thisstepisreferredto as “IMM mixing” [1]. +1 )= =1 η, +1 η, )= η, +1 )= =1 +1 η, ) ) (7) )= =1 +1 η,

)+ ) ) IMM Step 4 calculates the predicted state and error covariance for each model +1 )= +1 ) ) (8) +1 )= +1 +1 +1 The IMM Output for IPDA for each model con- sists of the state prediction, error covariance and the prediction of the model probability, i.e +1 ,P +1 , +1 4 Gating Algorithm The algorithm that validates measurements is referred to as “the gating algorithm”. Gating is done for each IMM model separately in the usual manner [11], i.e basedontheapriorimeasurementpdf θ,Z ) and the gating probability . First, we cal- culate the predicted measurement and innovation co- variance

matrix for each model as follows: )= ) ); )= (9) A validation gate is constructed around the predicted measurement ), such that the probability of the true measurement (if detected) falling in the gate is . For each model we select the validated measure- ments (i.e those inside the validation gate) )and calculate the volume ) of the validation gate. All IMM models must operate on the same set of measure- ments. The combined validation gate with a single set of measurements is obtained as the union of all validation gates, i.e =1 ); =1 The exact calculation for the volume of combined val-

idation gate (consisting of the union of overlapping gates for each model), is generally complex. One pos- sible approach is described in [12]. In this paper we use approach, ﬁrst described in [13], for calculating the volume of combined validation gate. We use the following approximation: k, app =1 =1 =max k, app max )) for (10) where ) is the number of selected measurements ) for model and is the number of measure- ments in . This approximation appears to work well in simulations. The volume of the validation gate for = 0 (i.e when there is no selected measurement) is irrelevant,

since does not appear in the formulae for =0. Approximation (10) is more precise for denser clutter environment (where association needs to be better).

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If better precision is required, a few additional ran- dom ”measurements” can be placed in the gates before using the equation (10). These measurements should then be removed. In our simulations these additional measurements were not used. The conditional pdf of each selected measurement is )= k,i θ,Z ); k,i 0 otherwise =1 (11) for each model and total respectively, where θ,Z )= ; ,S )). 5 IPDA Filter We model the

target existence as a Markov process, and distinguish between two cases [6]. Markov Chain One Markov process takes two discrete states - target exists and target does not exist. If the target exists, it is always detectable; i.e. at each time its measurement is present with probability .The Markov Chain One was described and used in [6, 9, 14]. The Markov Chain Two process is described in [6, 14, 15] and it takes three discrete states - the target exists and is detectable, the target exists and is temporarily not detectable, and the target does not exist. In this paper we use Markov Chain One

formulae only. Let denote the event that the target exists at time and let be the estimated probability of target existence at time 1. Let be the a priori probability of target existence at time . The relationship between the estimated probability of target existence at time 1 and predicted probability of target existence at time is 11 21 (1 (12) The transition probabilities ij are assumed known. Each algorithm can be parametric or non-parametric. Parametric algorithms assume knowledge of the statis- tical mean of the clutter measurement density at each point of observation space [16]. Clutter

density at the location of measurement k,i will be denoted with Non-parametric algorithms assume no prior knowledge of clutter density. In this case we assume uniform clut- ter density in the gate and estimate it as (13) where is the expected number of clutter measure- ments, i.e for (14) The IPDA part of the ﬁlter has the following steps: IPDA input from previous scan consists of the predicted probability of target existence, IPDA input from previous scan IMM ,for each IMM model, consists of the predicted model probabilities, ), and predicted state pdf, usually parametrized with )and

), from which measurement prediction pdf θ,Z ), usually parametrized with )and are calculated using (9). Measurement Selection is carried out as described in Section 4. Probability of target existence and model prob- abilities update )= (1 =1 ); =0 (1 =1 ); =0 (1 )= )(1 )) =1 )(1 )) (15) where 1 ) is the likelihood ratio for model ,1 is the likelihood ratio [6]. Data Association Probabilities . We denote by the k,i =1 ,m a posteriori probability that -th measurement is the true measurement and by k, posteriori probability that the true measurement is not present in , conditioned on target

existence. We have k, )= k,i )= (1 )) =1 ,m (16) Target existence prediction probability, +1 is calculated using equation (12). IPDA output for IMM consists of the set of selected measurements , the set of data association probabil- ities k,i =1 ,m , and the set of model a poste- riori probabilities, ).

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IPDA output for next scan consists of the target existence prediction probability +1 . A posteriori probability of target existence, is used for false track discrimination. 6 IPDA-IMM Algorithm — Complete Description At any time IPDA-IMM algorithm has the following structure:

Inputs Delivered by the IPDA : the predicted probability of target existence, i.e Delivered by the IMM : The predicted state mean and the error covariance, and the predicted model state probability for each IMM model , Delivered by the sensor : measurements. Step 1 is gating, in which a suitable measurement set is chosen from the measurements collected at time . The conditional pdfs of selected measurements is also calculated in this step. It is detailed in Section 4. Step 2 is the IPDA calculation of probability of target existence which is an output of the ﬁlter. IMM data association

probabilities k,i ) and model prob- abilities ) are also calculated in this step. These calculations are detailed in Section 5. Step 3 is the IMM algorithm detailed in Section 3. It calculates the ﬁlter output state estimate which con- sists of the combined state estimate and the error co- variance, i.e and The predicted mean and error covariance, and mode probabilities +1 ,P +1 , +1 for each model =1 ,M ,usedas inputs of IPDA- IMM at time + 1 are also calculated in this step. Step 4 is the prediction of the probability of target ex- istence at time +1,given +1 It is calculated using

equation (12). 7 Simulations A simulation study is used to compare three algo- rithms: IPDA-IMM described in this paper, IMM- IPDA of [8] and IMM-PDA of [9]. The target trajec- tory is simulated as shown in Figure 1. It is constructed as follows: Figure 1: Test Track and Clutter Distribution 10 seconds of uniform motion with constant ve- locity 18 m s, 10 seconds of exponential acceleration, i.e as exp( αt where is the initial velocity and =0 05 s 10 seconds of exponential deceleration i.e as exp( αt where is the initial velocity and 05 s 10secondsofarightturnwithangularvelocity rad

s; 10 seconds of exponential acceleration with 05 s 10 seconds of exponential deceleration with 05 s 10 seconds of a left turn with angular velocity rad sand 10 seconds of uniform motion. Clutter measurements are generated in clusters as shown in Figure 1. We simulate two rectangular clus- ters of heavy clutter. The rest of the clutter is dis- tributed uniformly over the surveillance area. Every second the number of clutter measurements is selected from Poisson distribution to maintain average clut- ter density 2e in the light clutter regions and 1e in the heavy clutter. Target motion model

set (1) consists of: 1. Uniform motion: target moves on a straight line with constant velocity. 2. Acceleration: target moves with constant accel- eration. 3. Target is executing a left coordinated turn with constant angular velocity rad

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4. Target is executing a right coordinated turn with constant angular velocity rad The state vector consists of (17) where ( ξ, ) denote Cartesian coordinates. All models have white plant noise with covariance matrix: 00 00 00 00 0010 00 01 (18) where =1s. IMM transition probability matrix for IPDA-IMM and IMM-IPDA is: IPDA-IMM 91 0 03 0

03 0 03 03 0 91 0 03 0 03 05 0 05 0 90 05 0 05 0 0 (19) Target existence is modelled by Markov Chain One with the following transition probabilities: 11 21 98 0 (20) For IMM-PDA ﬁlter the set of motion models is aug- mented by the ”target does not exist” model. Transi- tion probability matrix is IMM-PDA 89 0 03 0 03 0 03 0 02 03 0 89 0 03 0 03 0 02 05 0 05 0 8800 02 05 0 0500 88 0 02 00001 (21) A two dimensional radar system is modelled and mea- surements are provided every 1 s. The measurement noise is simulated with standard deviation of 5 m in range and 1 mrad in bearing at the range

5 km. Prob- ability of target detection is =0 9. For all three algorithms tracks are initiated at each time step, using two-point diﬀerencing [11] with ini- tial probability of target existence/detectability as de- scribedin[16]. Each algorithm simulation consists of 500 runs. The position and velocity root mean square errors are shown in Figure 2, where all three ﬁlters show similar performance. Averages over time of RMSE in position for IPDA-IMM, IMM-PDA, and IMM-IPDA were 7.4, 7.6, and 8.1 m respectively. The averages over time of RMSE in velocity for IPDA-IMM, IMM-PDA, and

IMM-IPDA were 4.4, 4.6, and 5.3 m/s respectively. Figure 2: Position and Velocity RMSE False track discrimination performances are shown in Figure 3, where the number of conﬁrmed true and falsetracksovertimearedisplayed. Averagesover time of the number of conﬁrmed true tracks for IPDA- IMM, IMM-PDA, and IMM-IPDA were 450, 425, and 438 respectively. Averages over time of the number of conﬁrmed false tracks for IPDA-IMM, IMM-PDA, and IMM-IPDA were 1.53, 17.8, and 6.59 respectively. 8 Conclusion A new integrated approach to manoeuvring target tracking in clutter with

automatic track initiation is considered in this paper. Using IMM and PDA ap- proximations, we derive a new algorithm, referred to as IPDA-IMM and compare its performance with IMM- IPDA and IMM-PDA algorithms in simulated scenar- ios. The proposed IPDA-IMM appears to outper- forms IMM-PDA and IMM-IPDA in the simulated en- vironment both in terms of estimation accuracy and true/false track statistics.

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Figure 3: Number of Conﬁrmed True and False Tracks References [1] Henk Blom and Yaakov Bar-Shalom. The inter- acting multiple model algorithm for systems with markovian

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