A Multiple Model Structure for Tracking by Variable Rate ... - CiteSeerX

A Multiple Model Structure for Tracking by Variable Rate Particle Filters Yener Ulker, Bilge Gunsel, Serap Kirbiz Multimedia Signal Processing and Pattern Recognition Lab., Istanbul Technical University, Dep. of Electronics and Communications Engineering 34469 Maslak Istanbul, Turkey {yeneru, gunselb, kirbiz}@itu.edu.tr

Abstract In contrast to the fixed rate modeling of the conventional methods, recently introduced variable rate particle filters (VRPF) achieves to track maneuvering objects with a small number of states by imposing a probability distribution on state arrival times. Although this enables VRPF an appealing method, representing the target motion dynamics with a single model hinders the capability of estimating maneuver parameters precisely. To overcome this weakness we have incorporated multiple model approach with the variable rate model structure. The introduced model referred as Multiple Model Variable Rate Particle Filter (MM-VRPF) utilizes a parsimonious representation for smooth regions of trajectory while it adaptively locates frequent state points at high maneuver regions, resulting in a much more accurate tracking. Simulation results obtained in a bearings-only target tracking problem show that the proposed model outperforms the conventional VRPF, the fixed rate multiple model particle filters (MMPF) and interacting multiple model using extended Kalman filters (IMM-EKF).

1. Introduction Maneuvering target tracking has taken much attention in the past decade with the development of numerous numerical techniques. Multiple model approaches which characterize target motion dynamics with a set of models, have been the most widely used techniques in the field. In maneuvering target tracking, Kalman filter based interacting multiple models (IMM) [1] and multiple model particle filters (MMPF) [2] are well known multiple model structures used for estimation of the fixed rate state distribution. Unlike the fixed rate standard tracking models, recently variable rate particle filters (VRPF), which models the state arrival times as a semi-Markovian random process is introduced in [3]. Although the variable-

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rate structure using curvilinear motion dynamics enables tracking of a wide range of motion characteristics, similar to the conventional models, the VRPF with its single mode structure results in poor estimate of the target trajectory especially at highly maneuvering regions. In order to overcome this drawback, in this paper, we introduce a Multiple Model Variable Rate Particle Filtering (MM-VRPF) scheme that integrates the multiple model structure with the variable rate filtering. The MM-VRPF is capable of switching between discrete state of candidate motion modes which can successfully model the maneuver parameters as well as the state arrival times. In our work we preserve the continuous deterministic process proposed in [3] while adapting a multiple model structure to variable rate framework. In [4], two sojourn distributions having different driving noise are defined to classify the type of the maneuvering objects. In our model we are proposing usage of different sojourn distributions, but unlike [4], our objective is efficiently modeling the parameters and arrival times of the maneuver. Results obtained for the bearings-only tracking application demonstrate that the proposed structure finely locates the states onto critical change points thus improves the VRPF’s trajectory estimation performance.

2. Variable rate models In conventional fixed rate state-space models a state variable xt evolves with time index t. A variable rate state is defined as xk = (τk , θk ) where k is a discrete index, τk denotes the state arrival time for state k and θk denotes the vector of variables that parameterizes the target state. Variable state sequence follows a Markovian process such that states are independently generated according to a density function, xk ∼ p(xk |xk−1 ) where τk is finite and τk > τk−1 [3]. Unlike the fixed rate models, allocation of states in a variable rate model is asynchronous with the timing of observations. Indeed, under the assumption that an observation yt is independent of

all other data points except the neighborhood states xNt = {xk ; k ∈ Nt (x0:∞ )} , the likelihood is defined as a density distribution for consecutive values of t as, yt ∼ p(yt |x0:∞ ) = p(yt |xNt ).

(1)

Assume that the largest and smallest elements of neighborhood set Nt is defined as Nt+ and Nt− , respectively. Let y0:t = (y0 , . . . , yt ) and x0:N + = (x0 , . . . , xN + ) t t denote the observations received until time t and the corresponding target states, respectively. At each time t, the VRPF model approximates to the optimal filtering distribution that can be expressed as a combination of Np multi-dimensional dirac delta functions, each representing a particle, as in Eq.(2). p(x0:N + , Nt+ |y0:t ) t

≈

Np X

wti δ(xi0:N + − x0:N + ) (2) t

t

i=1

i This is achieved by updating the particle weight wt−1 of particle i according to Eq.(3) to obtain new weight wti , whenever a new observation arrives.

wti

∝

i wt−1

p(yt |xiNt )p(xiN +

+ t−1 +1:Nt

q(xiN +

+ t−1 +1:Nt

|xiN + ) t−1

|xiN + , y0:t )

For t = 1 . . . T where T is the index to the last observation, the prior state distribution in Eq.(5) which is equal to the proposal distribution is sampled.   p xN + +1:N + |xiN + , i = 1 . . . Np (5) t−1

(3)

If the prior state distribution is chosen as the proposal distribution q(.), the updating rule can be simplified as in Eq.(4) which is referred as the bootstrap version of variable rate filtering model [3]. (4)

3. Multiple model VRPF Conventially VRPF defines the target motion and state arrival times with a single model. However when a maneuver is undertaken or a straight motion is in progress motion parameters and arrival times always show diversity due to nature of the tracking problem. Thus it is often not suitable to estimate the state arrival times and maneuver parameters with a single model. To overcome drawbacks of VRPF, we proposed a multiple structured variable rate model which defines maneuver parameters and arrival times by using a mode dependent dynamic model with three sojourn time distributions. In the following we will describe the incorporation of multiple model structure into variable rate models. A. Initialization At time t = 0, Np samples are drawn from predefined initial state distribution p(x0 ). Particles are generated for the next state until each particle contains a valid neighborhood xN0 . Initialized samples xiN0 , i = 1/Np . i = 1 . . . Np , are equally weighted, wt=0

t

t−1

Note that generation of particles from the proposal distribution is necessary to fill out the neighborhood Nt of an observation yt thus no particles will be generated when the neighborhood is already complete. Unlike the VRPF, in MM-VRPF model, a new discrete state variable mk which denotes mode of the motion dynamics is included into the variable rate state vector represented as, xk = [θk , τk , mk ] ,

mk ∈ {1, . . . r} .

(6)

Thus the conditional state distribution of proposed multiple model structure is expressed as in Eq.(7). p(xk |xk−1 )

t−1

i wti ∝ wt−1 p(yt |xiNt )

B. Prediction

= p(θk |θk−1 , τk , τk−1 , mk−1 , mk ) p(mk |mk−1 )p(τk |τk−1 , mk−1 )(7)

In our work the curvilinear motion model used in [3] which represents the maneuvering trajectory as a deterministic continuous-time process is used. Therefore motion kinematics vector θk is represented by the vector [TT,k TP,k v(τk ) ψ(τk ) z(τk )] where TT,k and TP,k are tangential and perpendicular forces applied to target, v(τk ) and ψ(τk ) are the target speed and course at state k, and z(τk ) = [l1 l2 ] denotes the position vector at l1 and l2 coordinates. Sojourn time distribution p(τk |τk−1 , mk−1 ), shown in Eq.(7) is conditioned on discrete mode variable and previous state arrival times are represented by a shifted Gamma distribution according to, τk − τk−1 − τn ∼ G(αn , βn ),

(8)

where n = mk−1 , mk−1 ∈ {1 . . . r}, is the index to mode of the motion model, and τn is the sojourn time shifting parameter. Therefore new arrival times τki , i = 1 . . . Np , conditioned on mode mk and previous i state arrival time τk−1 , are generated by using Eq.(8). Note that, unlike the VRPF, the MM-VRPF allows modeling the sojourn times by r different distributions. This requires probabilistic modeling of the motion transitions p(mk |mk−1 ) shown in Eq.(7). Following the conventional notation, in our multiple model, p(mk |mk−1 ) is defined by the time invariant mode transition probability matrix P where each element p(mk = h|mk−1 = l) = phl denotes the transition probability from mode h to mode l. Set of predicted

model indices mik , i = 1 . . . Np , are generated from the previous model set particles mik−1 according to the matrix P. Prediction step is accomplished by generating particles θki constituing tangential and perpendicular forces TT,k and TP,k which are assumed as Gaussian distributions conditioned on mode variable mk as described by Eq.(9), p(TT,k |mk ) ∼ p(TP,k |mk ) ∼

2 N (µT,n , σT,n ) 2 N (0, σP,n )

(9)

where n = mk , mk ∈ {1 . . . r} is the mode index. Parameters of the Gaussian distributions are chosen to match the characteristics of the object being tracked. The rest of the state variables, position z(τk ), speed v(τk ), and course ψ(τk ) are deterministically determined from the previous state variable xk−1 by using motion dynamics. C. Updating the Particle Weights Update process is accomplished according to simple bootstrap version of the weight update equation given by Eq.(4) . Thus the likelihood p(yt |xNt ) employed in updating the particle weights wt is defined as in Eq.(10), p(yt |xNt ) = p(yt |θˆt )

(10)

where θˆt is the interpolated continuous time state process at time t which is a deterministic function of xNt thus θˆt = ft (xNt ). Consequently the state vector θˆt includes the interpolated particles of xiNt . In fact, true posterior p(xNt , Nt+ |y0:t ) is approximated by the par Np  Np ticles xiNt i=1 and associated weights wti i=1 . Following the weight update procedure, resampling is being carried out, if necessary. Same evaluation, except the initialization, has been carried out for each new observation. Note that under the variable rate framework [3], a regeneration step is introduced which augments each particle representing the posterior distribution by fixing the past state values. Omitting regeneration causes poor estimation of the arrival times thus often reduces the filter performance drastically. However ability of the MM-VRPF in modeling state arrival times removes the necessity of the regeneration and avoids us computational burden of the process.

4. Experimental results and conclusions In this section we present experimental results obtained by the proposed MM-VRPF and compared its performance with the conventional VRPF, MMPF and IMM-EKF in a bearings-only maneuvering target tracking application. In the bearings-only target tracking

problem, an observation (a measurement) at time instant t is represented as in Eq.(11),   l1 − l1o yt = arctan + vt (11) l2 − l2o where z = [l1 l2 ]T is the target position vector, vt ∼ N (0, σθ2 ) is a Gaussian distributed sensor noise and [l1o l2o ]T refers to the sensor position which is known by the observer [1]. Two performance metrics, defining the instant root mean square position error (RMSEt ) and time averaged root mean square position error (RMSE) are used in the evaluation of tracking performance. Reported results are obtained by running 100 Monte Carlo simulations for each filter. For ith run, let (ˆl1 it , ˆl2 it ) and (l1 it , l2 it ) denote the true and estimated positions obtained at time t, respectively. RMSEt and RMSE values are computed by Eq.(12) where T is the index to the last observation. v u L u1 X RMSEt = t (ˆl1 i − l1 it )2 + (ˆl2 it − l2 it )2 (12) L i=1 t v u T X L u 1 X RMSE = t (ˆl1 i − l1 it )2 + (ˆl2 it − l2 it )2 T L t=1 i=1 t

Performance of the MM-VRPF is tested on a number of maneuvering bearings-only test scenarios. Results are reported for the test scenario illustrated in Fig.1(a) which is akin to the one synthesized in [3]. In the scenario, ownship and target move at a constant speed of 5 and 4 knots, respectively. Target executes a maneuver between minutes 20 − 25 with a constant turn rate of 24◦ /min. Observation period is equal to 1 minute and total number of 40 observations are assumed to be perturbed by zero mean Gaussian noise having a standard deviation of σθ = 1.5◦ . In order to observe behavior of all methods clearly, initial conditions are set to the true values. Therefore, all filters are initialized with their bearings and range centered on their true mean values of a Gaussian with variance σr = 100m and σθ = 1.5◦ , respectively. Initial values of heading and velocity are also set to the true values. Model Parameters used for evaluation of the MM-VRPF and VRPF are shown in Table-1. Np = 3000 particles are used in the evaluation. We chose the matrix P as in Eq.(13),   0.5 0.25 0.25 P =  0.4 0.5 0.1  (13) 0.4 0.1 0.5 considering the sojourn time distribution parameters of each mode given in Table-1. Thus the elements of matrix P is different than the one used for fixed rate models

0

MM-VRPF 149.68

−1

VRPF 224.91

MMPF 200.15

IMM-EKF 365.34

2

l coordimate (km)

Table 2: RMS position error (RMSE) for VRPF, MM-VRPF, MMPF and IMM-EKF

Target Observer

1

1

2 3 4 l coordinate (km)

MM−VRPF st. MM−VRPF trj.

1 0

−1

6

VRPF st. VRPF trj.

1 0

−1

2

−2

5

1

l coordimate (km)

l2 coordimate (km)

−2 0 (a)

3 4 5 (b) l coordinate (km) 1

−2

3 4 5 (c) l coordinate (km) 1

Figure 1: (a) Target and observer trajectories for the scenario. Trajectories and states of a particle generated by (b) the MM-VRPF, and (c) the VRPF. 700 MM−VRPF VRPF PFMM IMM−EKF

600

400

t

RMSE (meter)

500

300

200

100

0

0

5

10

15 20 25 Observation index (t)

30

35

40

Figure 2: RMS position error versus time index t. described in [1]. Probabilities of the transitions from maneuvering models do not change, however in contrast to fixed rate models straight motion model indexed n = 1 switch to either of the adaptive maneuvering models indexed n = {2, 3} with higher probabilities. Note that the MMPF and IMM-EKF use nearly constant turn (NCT) model with 3 possible turn rates where parameters are chosen for each mode as w1 = 0, w2 = 0.5, w3 = −0.5 rad/sn (w ∈ Ω) and state noise covariance matrix is set to 2 × 10−6 I2 . Fig.1(b) and (c) illustrate the trajectory and state arrival points of a single particle generated by the VRPF and MM-VRPF, respectively. As it is shown in Fig.1(b), the proposed MM-VRPF is capable of locating frequent states at high maneuvering regions while using a parsimonious state representation for the smooth regions of the trajectory owing to the sojourn time distribution parameters of the adaptive models. When a maneuver Table 1: MM-VRPF and VRPF parameters

µT,n , σT,n µP,n , σP,n αn , βn τn

mode-1 0, 100 (0, 500) (1.5, 4) 0

MM-VRPF mode-2,3 0, 100 (±11000, 3000) (0.5, 0.35) 0.5

VRPF 0, 100 (0, 5000) (0.5, 6.5) 0

is undertaken continuous representation of the motion dynamics enables MM-VRPF to locate frequent state points even higher than the observation period . This yields better characterization of the maneuver parameters and arrival times independently from the observation time. It can be concluded that the proposed MMVRPF, using a more flexible rate estimation procedure, is capable of estimating the target trajectory precisely. Fig.2 plots the RMSEt versus observation time index t for four different models: the VRPF, MM-VRPF, MMPF and IMM-EKF. These results are attained by using 40 observations acquired for the scenario described above. As it is shown in the figure, the proposed MMVRPF is able to track the target before and after the maneuver with lower RMSEt values compared to the other filters. The VRPF which uses a single curvilinear motion model and the fixed rate model MMPF perform worse after the maneuver while the RMSEt of IMM-EKF utilizing NCT motion model is high through the trajectory. This shows the ability of MM-VRPF to characterize the straight as well as maneuvering trajectories with the same set of parameters. RMSE values reported in Table-2 also show that the MM-VRPF is superior to the other filters in terms of mean estimation errors. In this work we adapted multiple model approach with variable rate model structure and compared the performance of introduced MM-VRPF with the conventional VRPF, MMPF and IMM-EKF algorithms. It is shown that the proposed multiple model structure utilizing three distinct dynamic motion models and sojourn time distributions enables efficient characterization of maneuver parameters as well as arrival times.

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