A Variable Structure Multiple Model Particle Filter for GMTI Tracking M. Sanjeev Arulampalam DSTO, Adelaide, Australia.
[email protected]
Neil Gordon Qinetiq, Malvern, U.K.
[email protected]
Matthew Orton Cambridge University, U.K
[email protected]
Branko Ristic DSTO, Adelaide, Australia
[email protected]
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 non-standard information such as road maps, and terrain-related visibility 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 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 VS-MMPF is much superior to that of VS-IMM. Keywords: GMTI Tracking, IMM, Particle Filter.
1
Variable
Structure
Introduction
In standard tracking problems, the only inputs available to the tracker are sensor measurements obtained through one or more sensors. However, in some applications there may be some additional information available which could be exploited in the estimation process. For instance, one may have some knowledge of the environment in which the target is being tracked or there may be some knowledge of some constraints on the dynamic motion of the target, such as speed constraints. An example application in a military context is 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 to produce better estimates of the target state? It turns out that incorporating such non-standard information in conventional Kalman filter based track-
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ers is not an easy task. The reason is that, in general, incorporating non-standard information leads to highly non-Gaussian posterior densities, and conventional trackers cannot easily handle propagation of non-Gaussian densities in a dynamic state estimation framework. However, there has been some attempt at incorporating such non-standard information within a Kalman filter based tracker. The most common of these is the Variable Structure Interacting Multiple Model (VS-IMM) algorithm [4]. The VS-IMM uses a modified version of the standard IMM, where the number, and structure, of the multiple models active at any particular time are allowed to vary. The various models may represent motion under different conditions of visibility, road constraints and target speeds. Although the VS-IMM has been shown to produce better results than methods that don’t use such non-standard information, it still has major drawbacks. In particular, the non-standard information available to the tracker will lead to highly non-Gaussian posterior pdfs which are approximated by a finite mixture of Gaussians. In addition, the VSIMM does not have a mechanism to incorporate hard constraints on position and speed. Because of these weaknesses, the use of VS-IMM has only resulted in modest improvement in accuracy over methods that do not use such non-standard information. In this project we propose a new algorithm based on Sequential Monte Carlo methods, which we term Variable Structure Multiple Model Particle Filter (VSMMPF). The basic principle is to use particles (random samples) to represent the posterior density of the state of a target in a dynamic state estimation framework where non-standard information is utilised. Since particle filtering methods have no restrictions on the type of models, including the noise distributions used,
one can choose rather complex models to represent ground vehicle motion in a GMTI context. In particular, the non-standard information available through road maps, speed constraints, etc., is modelled by a generalised Jump Markov system with constraints on the 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 simulated data and compared with the performance of the VSIMM. The organisation of the paper is as follows. Section 2 describes the GMTI tracking problem and its mathematical formulation. Section 3 reviews the VSIMM algorithm followed by Section 4 which presents the VS-MMPF algorithm. Finally, simulation results are presented in Section 5.
2 2.1
Problem Description and Formulation Problem Description
This section describes the problem of GMTI tracking with non-standard information. We consider the problem of tracking ground targets from measurements obtained using a single sensor. The surveillance region includes road networks and varying terrain conditions, such as hills, tunnels, open fields, etc. Depending on the target’s present location, its motion is constrained by these external factors, i.e., the road network and terrain conditions. For example, a target on a particular road has a high probability of continuing its motion constrained along that road. Or, an off-road target travelling in the open field is free to move in any direction, however, it may enter a road only at certain locations due to constraints such as a river or a hill. Likewise, an on-road target at a junction can continue only in one of the roads meeting at the junction. Thus, road networks and terrain conditions result in constrained target motion capabilities. The target motion is also constrained by speed restrictions which may be known. In addition to target motion constraints, the terrain conditions can also influence the measurement process in the following way. Depending on the target’s location, terrain features such as hills and tunnels may hide the target from the sensor’s view. Thus, the varying obscuration conditions of targets needs to be taken into consideration. A typical road map is shown in Figure 1 with four roads, AJ, BJ, CJ, and DJ, meeting at junction J. Road segments with solid lines allow entry into or exit from the roads while those with broken lines (eg., TU and BJ) indicate that the targets on the road cannot get
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off or those off the road cannot get on. In a typical GMTI tracking problem, such restrictions are generally determined by the surrounding terrain conditions, for example, rivers, open fields or ditches. The road segment TU represents a tunnel with zero target detection probability. The complete road map with terrain conditions can be specified as follows. Each road segment is represented by two way-points as shown in Table 1. These way-points determine the direction, location and length of each road. The visibility is defined as a binary valued probability, and the entry/exit condition is given by a Boolean variable. The entire specification is summarised in Table 1, and in the sequel this topography information will be denoted by R. Figure 1 also shows a sample target trajectory through the road network described above. A target starts at point I, merges with road AJ at M, then proceeds to junction J through tunnel TU, branches off to road JD, exits road JD at E, and terminates at F in the open field. The GMTI tracking problem of interest is as follows. In addition to standard GMTI measurements of range and azimuth, the above topography information is available to the tracker. The objective is to exploit this non-standard information to yield enhanced tracker performance. I A 10000 M target trajectory 9000 T U
8000 J
B
C 7000
F
E
6000
5000
D 5000
6000
7000
8000
9000
10000
11000
12000
Figure 1: An example road network and target trajectory
2.2
Problem Formulation
The problem of target tracking with road constraints is handled using the concept of directional process noise. To see this, note that the standard motion models assume that the target can move in any direc-
S a is the set of all possible motion models. Then, the Markov process mk is characterised by the following transition probabilities that depend on Sk−1 , Sk , and xk−1 , i.e.,
Table 1: Specification of road map Road Segments 1 AT 2 BJ 3 CJ 4 DJ 5 JU 6 TU
Waypoints A and T B and J C and J D and J J and U T and U
Visibility 1.0 1.0 1.0 1.0 1.0 0.0
Allow Entry/Exit True False True True True False
prs [Sk−1 , Sk , xk−1 ] = P(mk = s ∈ Sk |mk−1 = r ∈ Sk−1 , xk−1 ). (3) The measurement equation applicable to this problem is given by1
zk = h(xk ) + nk (4) p where h(xk ) = [ x2k + yk2 , tan−1 (xk /yk )]0 , nk ∼ tion with equal probability, therefore use equal process N (0, Rk ) and Rk is a 2 × 2 diagonal measurement conoise variance in both x and y directions, i.e., σx2 = σy2 . variance matrix with elements equal to the variances For on-road targets, the road constraints mean more in range and azimuth, σ 2 and σ 2 , respectively. For r θ uncertainty along the road than orthogonal to it. Sup- model s, this measurement is received with detection 2 2 pose σa and σo are the variances along and orthogonal probability P s , which is chosen to be either 0 or 1 D to the road, respectively, and the road direction is spec- depending on its visibility specified in Table 1. Note ified by the angle ψ (measured clockwise from y-axis). that in this paper, we do not include range-rate meaThen, the process noise covariance matrix correspond- surements, and only consider a binary valued P s . The D s ing to on-road motion is given by [4] inclusion of range-rate and a realistic modelling of PD 2 will be addressed in future work. − cos ψ sin ψ σo 0 − cos ψ sin ψ Equation (4) is nonlinear and thus each module of Q= sin ψ cos ψ 0 σa2 sin ψ cos ψ the VS-IMM would require an EKF. Alternatively, as (1) will be done in this paper, we use a Converted MeaWith the above directional process noise matrix consurement Kalman Filter [1], where (4) is linearised as cept, the GMTI tracking problem is formulated in a Jump Markov System (JMS) framework. Let the state 1 0 0 0 0 zk = xk + n0k , (5) xk = (xk , yk , x˙ k , y˙ k )0 consist of the position and veloc0 1 0 0 ity components of the target in the Cartesian coordinates. Then, the discrete time kinematic model for the and n0k ∼ N (0, R0k ), problem is given by 2 0 σx (Rk ) σxy (R0k ) 0 Rk = , (6) xk = F xk−1 + Gvk−1 (mk ), xk ∈ Ξ(mk ), (2) σyx (R0k ) σy2 (R0k ) where
1 0 0 1 F = 0 0 0 0
T 0 1 0
0 T , 0 1
T 2 /2 0 G= T 0
0 T 2 /2 , 0 T
T is the sampling time, mk is the mode in effect in the interval (k − 1, k], vk−1 (mk ) is a mode dependent white Gaussian process noise sequence with covariance Q(mk ), and xk is constrained to be in the mode-dependent set Ξ(mk ). Physically, mk represents the type of motion (off-road, on-road in a specific direction, etc.,) of target. This is modelled as a Markov process whose states belong to a variable set. Suppose there are R road segments in the network and let mk ∈ {0, 1, . . . , R}. Here mk = m ∈ {1, . . . , R} implies a motion model corresponding to road m and mk = 0 refers to off-road motion. Now let Sk ⊆ S a denote the set of modes active in the interval (k − 1, k], where
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σx2 (R0k ) σy2 (R0k ) σxy (R0k )
= rk2 σθ2 cos2 θk + σr2 sin2 θk , = rk2 σθ2 sin2 θk + σr2 cos2 θk ,
(7) (8)
= σyx (R0k ) = (σr2 − rk2 σθ2 ) sin θk cos θk . (9)
Here rk and θk 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 = {z1 , . . . , zk } and the topography information R, estiˆ k|k = E[xk |Z k , R]. mate the target state x
3
Variable Structure IMM
The Variable Structure IMM (VS-IMM) algorithm for 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 origin of the X-Y plane.
As described above, Sk denotes the set of modes active in the interval (k − 1, k]. Then, the mode probability of mode s ∈ Sk is defined as 4 µsk = P mk = s ∈ Sk |Z k .
(10)
Also, the mode conditioned state estimate and associated covariance of filter module s ∈ Sk are denoted ˆ sk|k and Psk|k , respectively. In addition, in the VSby x IMM described in [4], prs [Sk−1 , Sk , xk−1 ] defined in (3) has no explicit dependence on xk−1 , and so will be denoted by prs [Sk−1 , Sk ].2 With the above definitions, the five steps of each cycle of the VS-IMM are as follows. 1. Step 1: Mode set update. Based on the state estimate at k − 1 and the topography, the mode set of the IMM is updated as Sk
= =
s ∈ S a |Sk−1 , R, Z k−1 n s ∈ S a |Sk−1 , R, {ˆ xrk−1|k−1 , Prk−1|k−1 ,
r ∈ Sk−1 }} .
(11)
This step is described in more detail in Section 3.1. 2. Step 2: Mode interaction or mixing The mode conditioned state estimates and the associated covariances from the previous scan are combined to obtain the initial condition for the mode-matched filters. The initial condition in scan k for the filter module s ∈ Sk is computed using ˆ 0s x k−1|k−1
=
X
r|s ˆ rk−1|k−1 , µk−1|k−1 x
3. Step 3: Mode-conditioned filtering. A converted measurement Kalman filter is used for each module to compute the mode-conditioned state estimate and covariance, given the initial conditions (12) and (14). In addition, depending on whether or not a measurement was received and based s on the detection probability PD corresponding to module s, we compute the likelihood function Λsk for each module s as s =1 N [νks ; 0, Ssk ], zk received and PD s 0, zk received and PD =0 s Λk = s 1, no z and P = 0 k D s 0, no zk and PD =1 (15) where νks and Ssk are the innovation and its covariance, respectively, of the measurement zk in module s. 4. Step 4: Mode Probability Update. The mode probabilities are updated based on the likelihood of each mode using P Λsk `∈Sk−1 p`s [Sk−1 , Sk ]µ`k−1 s P µk = P (16) r ` r∈Sk `∈Sk−1 Λk p`r [Sk−1 , Sk ]µk−1
5. Step 5: State Combination. The mode conditioned estimates and their covariances are combined to give the overall state estimate and covariance as follows: X ˆ k|k = ˆ sk|k x µsk x (17) s∈Sk
Pk|k =
(12)
s∈Sk
r∈Sk−1
where r|s µk−1|k−1
3.1 = P{mk−1 = r|mk = s, Z =
k−1
}
prs [Sk−1 , Sk ]µrk−1 P
`∈Sk−1
p`s [Sk−1 , Sk ]µ`k−1
(, 13)
are the mixing probabilities. The covariance matrix associated with (12) is given by P0s k−1|k−1
=
X
r∈Sk−1
r|s µk−1|k−1 Prk−1|k−1
˜ rs · (d ˜ rs )0 . + d k−1 k−1
X
(14)
˜ rs = [ˆ ˆ 0s where d xrk−1|k−1 − x k−1 k−1|k−1 ].
2 In the VS-IMM, these transition probabilities are determined on the basis of sojourn times of the modes.
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ˆ k|k ][ˆ ˆ k|k ]0 . µsk Psk|k + [ˆ xsk|k − x xsk|k − x (18)
Mode Set Update
The VS-IMM adaptively updates the set of active modes Sk based on the current estimate, its covariance and the topography. We present a brief review of it below although details can be found in [4]. Let L` ∈ R denote the `-th road in the network. Then, a model corresponding to this road is included in Sk by testing whether any segment of this road lies within a certain neighborhood ellipse centered at the predicted location (ˆ xk|k−1 , yˆk|k−1 ). The ellipse Ek is the region in the X-Y plane which satisfies the following condition: ( 0 4 x x−x ˆk|k−1 Ek = : y y − yˆk|k−1 " #−1 11 12 Pk|k−1 Pk|k−1 x−x ˆk|k−1 × ≤ α (19) 21 22 Pk|k−1 Pk|k−1 y − yˆk|k−1
where α is the “gate threshold” and " # P11 P12 pos k|k−1 k|k−1 Pk|k−1 = P21 P22 k|k−1 k|k−1
vectors such that the position components lie on road m, i.e., (20)
is the position submatrix of the prediction covariance Pk|k−1 . A road is deemed validated if any segment of it belongs to Ek . The mode set update proceeds as follows. First, every junction Jj ∈ R is subject to the above test. If Jj ∈ Ek , then a model corresponding to each road meeting Jj is added to Sk . Next, all road segments that 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 Sk . Finally, base-line models that correspond to off-road motion are added if any one of the following criteria are satisfied: (a) A road (with entry/exit) is validated, (b) A road (with no entry/exit) is validated and one if its endpoints from which the target can get into open field is also validated, and (c) none of the road segments is validated.
4
Variable Structure Multiple Model Particle filter
The particle filter (PF) is a technique for implementing a recursive Bayesian filter by Monte-Carlo simulations [2]-[3]. The key idea of the PF is to represent the required posterior density function by a set of random samples or “particles”, and to compute the state estimate based on these samples. The original particle filter proposed by Gordon et al [2] was developed for a single dynamic model using the Sampling Importance Resampling (SIR) algorithm to propagate and update the particles. Extensions to multiple models have been reported and the underlying theory can be found in [5, 6]. In this paper we present a modified version of the multiple model particle filter that is suited to GMTI tracking. The resulting algorithm is termed Variable Structure Multiple Model Particle Filter (VS-MMPF). The key features of the VS-MMPF 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 VS-MMPF, consider a set of particles {(xik−1 , mik−1 )}N i=1 that represents the posterior pdf p(xk−1 , mk−1 |Z k−1 ) of the state and mode at k−1. Now, suppose at k we have some measurement zk .3 It is required to construct a sample {(xik , mik )}N i=1 which characterises the posterior pdf p(xk , mk |Z k ) at k. This is carried out in two steps: prediction and update. Before describing these in detail, the following notation is introduced. Let Rm denote the set of state 3 Note
that zk can also be null indicating no measurements.
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4
Rm = {x : (x, y) is on road segment m}.
(21)
Similarly, 4
R0m = {x : (x, y) is on ‘extended’ road segment m}. (22) where ‘extended’ road segment m is the road obtained by 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 speed constraints, i.e., 4
G = {x : |v|min ≤
p x˙ 2 + y˙ 2 ≤ |v|max }.
(23)
Finally, let R90 and R−90 denote the rotational transformation matrices corresponding to rotations through 90◦ and −90◦ , respectively.
4.1
Prediction Step
Here we describe the prediction phase of a specific particle (xik−1 , mik−1 ). Two cases will be considered: (1) mik−1 = m ∈ {1, . . . , R}, i.e., a road particle, and (2) mik−1 = 0, i.e., an off-road particle. Consider case (1) where mik−1 = m is a road particle. Before predicting, it must be determined whether this particle executes a road motion or off-road motion at k. This is determined by the model parameter pm , which is the probability of executing road motion at k, given the target was on road m at k − 1. If it is determined that this particle executes road motion at k, then the prediction is carried out by generating i vk−1 ∼ N ∗ (0, Q(m)) such that i x∗k i = F xik−1 + Gvk−1 (m), x∗k i ∈ G ∩ R0m
(24)
where N ∗ (·, ·) is a truncated Gaussian density that ensures x∗k i ∈ G ∩ R0m . However, if it is determined that the particle executes an off-road motion, then an in0i termediate state xk−1 is formed whose velocity is perpendicular to the current velocity of xik−1 . This intermediate state is then predicted according to 0i
i x∗k i = F xk−1 + Gvk−1 (0), x∗k i ∈ G
(25)
i where vk−1 ∼ N ∗ (0, Q(0)) is a process noise corresponding to off-road motion. Predictions near road junctions is a bit tricky. Specifically, if it turns out that in the transition xik−1 → x∗k i given in (24), the particle crosses a junction J, then the following is carried out. Suppose junction J has road segments j1 , . . . , jnJ connecting to it. Then, the predicted particle is placed in one of these road segments with probability 1/nJ . This is done as
follows. Let Rm jn denote the rotational transformation matrix corresponding to a rotation about the junction that will place a road m particle4 on road jn . Then, the particle is predicted to road jn by pre-multiplying it by Rm jn . The above procedure can be summarised in the following algorithm.
where p∗ is the probability of transitioning to road j when dij = 0 and τ=
5 2 T σx 4
is a distance measure chosen such that P(d > τ ) ≈ 0 where d is the distance travelled due to process noise with variance σx2 and T is the sampling time. i Based on the computed transition probabilities it is Prediction Algorithm (part A): mk−1 = m 6= 0 determined if the particle transitions to any of the road • Generate u ∼ U[0,1] . segments. Suppose it transitions to road m. Then, i • If u ≤ pm , set m0k = m, the prediction step simply involves placing the partii i i i ◦ Set x0k = F xik−1 + Gvk−1 (m0k ), x0k ∈ G ∩ R0m , cle in the coordinates of the shortest distance point i s ◦ If x0k has crossed a junction J (xsim , yim ) to road m. The velocity components are i 1. sample u ∼ U[0,1] , initialised to be of the same magnitude as (x˙ ik−1 , y˙ k−1 ) 2. find n : u ∈ ((n − 1)/nJ , n/nJ ], but with direction along the road. If however it is den ∈ {1, . . . nJ }, termined that the particle remains an off-road one, a 0i ∗i 3. set x∗k i = Rm similar prediction to (25) is executed. jn xk and mk = jn , i The above procedure can be summarised by the Else if x0k ∈ Rj for some j ∈ {1, . . . , R}, then following algorithm. 1. Set m∗k i = j, i 2. Set x∗k i = x0k , Prediction Algorithm (part B): mik−1 = 0 Else i
1. Set x∗k i = x0k , 2. Set m∗k i = 0. End If Else ◦ Set m∗k i = 0, ◦ Sample u ∼ U[0,1] , ◦ If u ≤ 1/2, set RT = R90 , else RT = R−90 , 0i 0i i ◦ Set (x˙ k−1 , y˙ k−1 ) = RT · [xik−1 , yk−1 ]0 , 0i 0i 0i i , (x˙ k−1 , y˙ k−1 )]0 , ◦ Set xk−1 = [xik−1 , yk−1 0i i ∗i (m∗k i ), x∗k i ∈ G. ◦ Set xk = F xk−1 + Gvk−1 End If
i
Now consider the case (2) where mik−1 = 0 which corresponds to an off-road particle. Before prediction it must be determined whether at k this particle transitions to a road particle or not. To do so, we first i perform a preliminary prediction of xik−1 to x0k with zero process noise. Then, based on the proximity of i x0k to each road j, a probability of the ith particle transitioning to road j is computed as follows. Let cij be defined as i 4 1 if particle i crossed road j in xik−1 → x0k cij = 0 otherwise (26) i Also let dij be the shortest distance from x0k to road j. Then, we construct a linearly varying probability Pij of the ith particle transitioning to road j as ∗ cij = 1 p 0 cij = 0, dij > τ Pij = (27) ∗ p (τ − dij )/τ otherwise.
4 Here we refer to a road m particle that has crossed junction J in the transition (24).
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• Set x0k = F xik−1 . • For j = 1...R ◦ Compute cij and dij , s ◦ Compute (xsij , yij ), ◦ Compute Pij , ◦ Sample u ∼ U[0,1] , ◦ If u ≤ Pij , set m∗k i = j, else set m∗k i = 0, ◦ If m∗k i ∈ {1, . . . , R} exit loop. End For • If m∗k i = m ∈ {1, . . . , R} s ◦ Set (xik , yki ) = (xsim , yim ), ◦ Sample u ∼ U[0,1] , ◦ If u ≤ 1/2 set α = 1 else α = −1, i i 0 ◦ Set (x˙ ∗k , y˙ k∗ ) = q[α|vi | sin ψm , α|vi | cos ψm ] , where |vi | =
i (x˙ ik−1 )2 + (y˙ k−1 )2 , i
i
◦ Set x∗k i = (xik , yki , x˙ k∗ , y˙ k∗ ), Else i ◦ Set x∗k i = F xik−1 + Gvk−1 (m∗k i ), x∗k i ∈ G. End If
4.2
Update Step
This step involves updating the predicted particles {(x∗k i , m∗k i )}N i=1 with the new measurement zk . Note that zk can also be null which indicates no measurements due to obscurations. To update, each particle is assigned a weight proportional to its likelihood: i N (zk ; x∗k i , R) zk 6= ∅, PD =1 i 0 z = 6 ∅, P i k D =0 q˜ki = p(zk |x∗k ) = i 1 z = ∅, P k D =0 i 0 zk = ∅, PD =1 (28)
i where PD is the detection probability corresponding to mode m∗k i . Each particle i, i = 1, . . . , N receives the normalised weight,
q˜i qki = PN k
˜kj j=1 q
.
Next, we resample N times with replacement from i N the set {(x∗k i , m∗k i )}N i=1 with weights {qk }i=1 to obi i N tain a new set of samples {(xk , mk )}i=1 such that P((xik , mik ) = (x∗k j , m∗k j )) = qkj .
4.3
Initialisation
To initialise the VS-MMPF we use a two-point initialisation to obtain x0 and P0 . Then, we sample N times from N (x0 , P0 ) to obtain {xi0 }N i=1 . Also, the initial mode is set to mi0 = 0 for i = 1, . . . N . Thus 0 we have {(xi0 , mi0 }N i=1 which characterises p(x0 , m0 |Z ) 0 where Z refers to “prior” information.
4.4
VS-MMPF Algorithm Summary
The entire algorithm is summarised below. 1. Set k = 0. Generate N samples {xi0 }N i=1 ∼ N (x0 , P0 ) where x0 and P0 are determined from 2-point initialisation. Set mi0 = 0, i = 1, ..., N . Increment k. 2. Predict to next (xik−1 , mik−1 )
step:
For
each
particle
if mik−1 = m ∈ {1, ..., R} predict to (x∗k i , m∗k i ) using Prediction Algorithm (Part A), else predict to (x∗k i , m∗k i ) using Prediction Algorithm (Part B). end. 3. Form weights q˜ki = p(zk |x∗k i ) using (28). Normalise weights : q˜i qki = PN k
˜kj j=1 q
.
4. Compute Estimate,
ˆ k|k = x
N X
qki x∗k i .
i=1
5. Resample a new set {(xik , mik )}N i=1 by resampling with replacement N times from the discrete set i i ∗j ∗j (x∗k i , m∗k i )N i=1 where P((xk , mk ) = xk , mk )) = j qk . 6. Increment k and move to step 2.
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5
Simulation Results
In this section we compare the performance of the two trackers described in Sections 3 and 4. The scenario used for the simulations is identical to the one shown in Figure 1. A ground target, initially at (10, 10.5)km travels at speed 12 m/s along the path indicated by the dotted line. The width of the road network is 8 m and it includes a tunnel in segment TU where no measurements are received. The receiver, for the purpose of simulation, was considered static and located at the origin. Measurements of range and azimuth are received at a sampling rate of T = 5 sec for a total of 110 scans. The sensor accuracies of range and azimuth measurements are σr = 20 m and σθ = 0.5◦ , respectively. The parameters of the VS-IMM algorithm are as follows. This filter is initialised with two second order white noise acceleration models with equal process noise levels in both the x and y directions. For these base-line models, σx = σy were 0.6 and 2.4 m/s2 , which correspond to constant velocity and maneuver modes, respectively. In addition, six road motion models (one corresponding to each segment in Table 1) were used with parameters σa = 0.6 m/s2 and σ0 = 0.0001 m/s2 . The sojourn times of the base line constant velocity and maneuver models are 100 sec and 7.1 sec, respectively while that of road models is 100 sec. These sojourn times are used in the computation of the Markov chain transition matrix [1]. The VS-MMPF used 7 models — one for each road segment and one constant velocity base line model with σx = σy = 0.6 m/s2 for off-road motion. The road segment models are identical to that of VS-IMM. The parameter pm described in Section 4.1 was set to 0.95 m = 1, 3, 5, 6 pm = (29) 1 m = 2, 4 where mode m corresponds to motion along road segment m in Table 1. The number of particles for the VS-MMPF was chosen to be N = 1000. Figure 2 shows a comparison of the RMS error performance for the two trackers averaged over 100 MC runs. The dotted vertical lines in the graph correspond to mode transition times. The VS-MMPF shows a much reduced position error compared to VS-IMM, particularly in the region where the target is on road. Specifically, the average RMS error for the on-road target is about 70 m and 25 m for the VS-IMM and VS-MMPF respectively. This implies that VS-MMPF provides a 65% improvement over VS-IMM. Observe that while the target is within the tunnel (time periods k = 40 − 50) the RMS error increases due to lack of measurements. Also, notice that while the target is off-road (k ≤ 10 and k > 90) the performance of
300
300
speed constraint: 20 km/h < speed < 60 km/h
VS−IMM VS−MMPF
VS−IMM: Error (on road) ∼ 70 m VS−MMPF: Error (on road) ∼ 25 m
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Figure 2: Comparison of RMS Error performance
the two filters is comparable. For this region, since no explicit hard constraints are introduced, the posterior pdf will tend to be nearly Gaussian, which is handled equally well by both filters. Next, we compare the RMS error for the case where the tracker has knowledge of speed constraints. Suppose we know that the speed of the target is in the range 6 ≤ |v| ≤ 17 m/s. Such information can be incorporated in the VS-MMPF which ensures that particles not satisfying the above constraints are rejected. On the contrary, it is difficult to incorporate such nonstandard information in the VS-IMM framework. Figure 3 shows a comparison of RMS error performance for this case in which this speed constraint information is used only by the VS-MMPF algorithm. We notice that the VS-MMPF performance is further improved here. Specifically, the average error on road is reduced from 25 m to 16 m leading to a 77% improvement over the VS-IMM.
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Conclusions
In this paper, we presented a new algorithm termed Variable Structure Multiple Model Particle Filter (VSMMPF) for GMTI tracking with non-standard information. The RMS error performance of this new filter is compared with that of the VS-IMM algorithm. For the scenario considered, the VS-MMPF outperformed the VS-IMM giving 65% improvement in error performance for a road target. The VS-MMPF performance is further improved with other non-standard information such as speed constraints. In particular, for a case with speed constraint knowledge, the VS-MMPF showed a 77% improvement in RMS error compared to the VS-IMM.
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Figure 3: Comparison of RMS Error performance – case with speed constraints 6 ≤ |v| ≤ 17 m/s
Acknowledgements The authors wish to thank Ms. Melanie Webb for her help in the preparation of the manuscript. The authors also acknowledge Dr. Mahendra Mallick for his useful comments on the paper.
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