Manoeuvring Target Tracking with the IMM-VDA Algorithm Barbara La Scala
Graham W. Pulford
Dept. of Electrical and Electronic Engineering University of Melbourne Victoria 3010, Australia
[email protected]
GIS/FST, Qinetiq Cody Technology Park Ively Rd, Farnborough GU14 0LX, United Kingdom
[email protected]
Abstract— This paper describes an algorithm for tracking a manoeuvring target in heavy clutter and/or with a low probability of detection. It is known that when tracking under such adverse conditions multi-scan tracking algorithms, such as the multi-hypothesis tracker (MHT), provide improved performance over single-scan trackers. This paper uses a computationally efficient algorithm for multi-scan target tracking based on the Viterbi algorithm, known as the Viterbi Data Association (VDA) algorithm. In this paper it is shown how the VDA algorithm can be combined with the well-known Interacting Multiple Model (IMM) method to create an effective multi-scan manoeuvring target tracker. The performance of the IMM-VDA algorithm is shown by simulation. It is compared to another manoeuvring target algorithm based on the VDA approach which uses a hard decision manoeuvre detection scheme. In addition, it is compared to a single-scan tracking algorithm based on the Probabilistic Data Association method.
I. I NTRODUCTION Under benign conditions, such as when the target signal to noise ratio is high or the density of clutter is low and the target is not manoeuvring, it is not difficult to obtain good tracking performance. However, a more challenging problem occurs when the probability of detection is low or there are large amounts of clutter or both. In such situations, tracking algorithms that consider the association between detections and targets over multiple scans typically outperform so-called single-scan algorithms, such as the Probabilistic Data Association (PDA) method [1], [2]. One such multi-scan tracking algorithm is the VDA algorithm [3], which applies the Viterbi algorithm to the data association problem. The Viterbi algorithm (VA) [4] is a dynamic programming technique for finding the shortest path through a trellis in a computationally efficient manner. The use of the Viterbi algorithm for solving the data association problem when tracking targets in clutter was first suggested by Quach and Farooq [5]. Their approach finds the approximate maximum likelihood assignment of measurements to the target. The problem they considered was that of tracking a single, manoeuvring target assumed to be present surveillance region. This method was modified in [3] to suit the problem of tracking multiple, nonmanoeuvring targets in heavy clutter and was applied to tracking with an Over-The-Horizon radar (OTHR). The method of [3] also incorporated a target existence model for automatic track maintenance in scenarios where targets may enter or exit
0-7803-9286-8/05/$20.00 © 2005 IEEE
the surveillance region at arbitrary times. Extensive testing showed that the Viterbi Data Association (VDA) algorithm is more effective at tracking than conventional, single-scan algorithms, such as PDA approaches when the probability of target detection is moderate to low or there is heavy clutter. Standard implementations of tracking algorithms assume a single model for the motion of the target. Typically it is assumed that the target moves with constant velocity. While this is not always true, it is often the case that the motion of the target can be modelled this way with manoeuvres representing occasional abrupt deviations from this behaviour. An alternative approach to dealing with manoeuvring targets is to extend the system description to include multiple models for the motion of the target. Each model, representing a different type of target motion, is run in parallel and the probability of the validity of each model is evaluated. The output of the algorithm is then a weighted sum of the individual model filters. A highly effective variation of this filter bank approach is the Interacting Multiple Model (IMM) algorithm, first described in [6]. This paper presents an extension of the VDA algorithm of [3] to incorporate the IMM algorithm for tracking a single, manoeuvring target. This combination of algorithms has the potential to provide accurate tracking of manoeuvring targets in heavily cluttered environments. Simulations are used to compare the IMM-VDA tracker to a number of alternative approaches. One approach is a variant of the VA-based algorithm of Quach and Farooq [5] which is also designed to track manoeuvring targets but does so using a single target motion model [7]. We will refer to the method of [7] as the VA-QF algorithm in this paper. In addition, the performance of two single-scan trackers are considered. These are the IPDA algorithm [8] which is a PDA-based algorithm that also incorporates a target existence model but assumes a single model for target motion; and the extension of that algorithm to include the IMM method, IPDA-IMM [9]. In the next section the standard VDA algorithm is described. The derivation of the IMM-VDA algorithm is given in Section III, while the VA-QF method of [7] is briefly described in Section IV and the two IPDA methods are covered in Section V. Simulation results showing the comparative performance of all five algorithms in heavy clutter are given in Section VI.
II. T HE V ITERBI DATA A SSOCIATION A LGORITHM Consider the problem of tracking a single target in clutter. Suppose we have a target whose motion can be described by a linear state space model of the form x(k + 1) = F(k)x(k) + w(k)
(1)
The state process noise w(k) is a zero-mean, white noise process with known covariance denoted by Q(k). Given a non-zero probability of a false alarm, in general more than one measurement may be received at any given time. Let the probability the target is detected be given by PD . Let Z(k) = {z1 (k), . . . , zn(k) (k)} be the set of measurements at time k where H(k)x(k) + v(k) if target z(k) = (2) clutter otherwise where v(k) is a zero-mean, white Gaussian noise process with covariance matrix R(k). The clutter detections, for simplicity, are assumed to be uniformly distributed throughout the measurement space. The number of clutter detections received is assumed to be given by a discrete-time Poisson process. The VDA algorithm of [3] is designed to find the approximate maximum likelihood association of measurements to the target, when the target may enter or leave the surveillance region at arbitrary times. This maximum likelihood data association problem can be expressed as follows. Given a set of n(k) measurements at time k the set of all possible events {θ j (k)} can be written as • θ−1 (k) = the target is not in the surveillance region • θ0 (k) = the target is in the surveillance region but is not detected • θ j (k) = the target is in the region, is detected and z j (k) is the target measurement, j = 1, . . . , n(k). Given a sequence of scans from k = 1, . . . , N, and denoting the set of possible associations by Γ = {θ j1 , . . . , θ jN } the data association problem is then to find the Γ∗ that satisfies max p(Z(1), . . . , Z(N), θ j1 , . . . , θ jN |x(1), . . . , x(N)) Γ
j1 ,..., jk−1
(4)
where the associations over scans i = 1, . . . , k−1 are given by θ ji (i) it can be shown that [10] d¯j (k) =
max
i=−1,...,n(k−1)
a¯i j (k)d¯i (k−1)
△
ai j (k) = − ln a¯i j (k).
(7)
The Viterbi algorithm [4] can then be used to find the minimum cost path through the trellis in a computationally efficient manner. Given the definition of the transitions costs in (7), this path yields a solution to the maximum likelihood data association problem (3). To calculate the transition costs (7) exactly it would be necessary to know the true state sequence {X k }. Since this is not available, the VDA algorithm makes use of the estimated state sequence {Xˆ k } obtained by running a Kalman filter along each path through the trellis. Thus the VDA algorithms of both [5] and [3] are approximate maximum likelihood techniques. An example of a trellis from the VDA algorithm is shown in Figure 1. Time 1
Time 2
Time k-2
Time k-1
^x 2
x^ k-2
x^ k-1
Time k
z
-1
z 0
z 1
z 2 ^x k
^x 1
Fig. 1. Example of a track through a VDA trellis. The circles represent detections and the squares the dummy target does not exist (z−1 ) and target not detected (z0 ) nodes.
(3)
Write Z k for the sequence of measurements {Z(1), . . . , Z(k)} and X k for the equivalent sequence of states {x(1), . . . , x(k)} and consider an arbitrary sequence of associations ending with the association θ j (k), written as Θ j (k). Defining △ d¯j (k) = max p(Z k , Θ j (k)|X k ), j = −1, 0, . . . , n(k)
maximum likelihood data association problem to a sequential scalar minimisation problem. Consider a trellis consisting of n(k) + 2 nodes at time k, where each node represents a possible association event θ j (k), j = −1, . . . , n(k). Let the transition cost from node θi (k−1) to θ j (k) be given by
(5)
where
Expressions for the transition costs can be derived as follows. First, let the event, E(k), that the target is in the surveillance region at time k, evolve according to a two state Markov chain [8] with transition probabilities P(E(k)|E(k−1)) = δ0 ¯ P(E(k)|E(k−1)) = δ1
(8) (9)
where P(·) is the probability of a discrete random variable, then the a priori probability of track existence at time k is given by PE (k|k−1) = δ0 PE (k−1|k−1) + δ1 {1 − PE (k−1|k−1)} (10)
△
a¯i j (k) = p(Z(k), θ j (k)|Z k−1 , Θi (k−1), X k )
(6)
and p(·) is used to represent a probability density function (PDF). Taking the negative logarithm of (5) reduces the
Neglecting gating, and assuming a Poisson distribution for the number of clutter points in the surveillance region, the transitions costs for the VDA algorithm are given by the
equations a¯i,−1 (k) = ∆−1 λ(1 − PE (k|k−1)) a¯i,0 (k) = ∆−1 λ(1 − PD)PE (k|k−1) a¯i j (k) ≈ ∆
−1
PD PE (k|k−1) × N(z j (k); zˆi (k|k−1), Si (k|k−1))
(11) (12) (13)
where ∆ is a normalisation constant and • λ is the spatial clutter density; • zˆi (k|k−1) is the predicted value of y(k) given the track ending with the association θi (k−1); • N(·; µ, Σ) is a multivariate Gaussian PDF with mean µ and covariance Σ; and • Si (k|k−1) is the predicted measurement error covariance from the Kalman filter. Note, the VDA algorithm described in [3] includes gating to reduce the computational complexity but this is not included here for simplicity. The normalisation constant for the VDA algorithm, ∆, can be calculated by noting that, conditioned on the past data and associations, the current associations are mutually exclusive and exhaustive:
△
P(mi (k − 1)|m j (k), Z k−1 ) 1 pi j µi (k−1) cj
=
µi| j
=
(19)
for i, j = 1, . . . , R where c j is the normalisation constant R
c j = ∑ pi j µi (k−1)
(20)
i=1
The mixed initial conditions for each filter are then R
xˆ0 j (k−1|k−1) = ∑ µi| j (k−1|k−1)xˆi(k−1|k−1) (21) i=1 R P0 j (k−1|k−1) = ∑ µi| j (k−1|k−1) Pi (k−1|k−1) i=1
n(k)
∑
algorithm consists of the following steps. This outline assumes that exactly one measurement is made at each scan and that it is the target detection. 1) Mixed Initial Conditions Prior to collecting the k-th measurement, mixed initial conditions are generated for each filter. The mixing probabilities are given by
Pr(θ j (k)|Z
k−1
, Θi (k − 1)) = 1
(14)
j=−1
III. M ANOEUVRING TARGET T RACKING A. The IMM Algorithm The Interacting Multiple Model approach to tracking assumes that the target motion can be represented by one of a finite set of models, called “modes”, at any given time. That is, the target dynamics are given by x(k + 1) = F(m(k))x(k) + w(m(k), k)
(15)
z(k) = H(m(k))x(k) + v(m(k), k) (16) 1 2 where m(k) ∈ M , M , . . . , M R . It also assumes the switching between these modes occurs according to a Markov chain with known transition probabilities. The key feature of the IMM approach is the manner in which the track estimates from these multiple models are combined. The IMM method mixes the previous cycle’s mode-conditioned estimates to initialise the current cycle of each mode-conditioned filter. The use of these mixed initial conditions for each of the R filters in the filter bank allows the IMM to achieve performance similar to that of a GPB(2) algorithm which uses R2 filters [11]. Let m j (k) be the event that the target motion was given by mode M j at time k for j = 1, . . . , R. Define µ(k) = (µ j (k)) j=1,...,R as the vector of a posteriori mode probabilities, i.e. µ j (k) = P(m j (k)|Z k ) (17) Let the model transition probabilities be given by pi j = P(m j (k)|mi (k−1))
(18)
and for each mode i assume that the a posteriori track state estimates xˆi (k−1|k−1) and associated state error covariance matrix Pi (k−1|k−1) at scan k−1 are known, then the IMM
xˆi (k−1|k−1) − xˆ0 j (k−1|k−1) ′ × xˆi (k−1|k−1) − xˆ0 j (k−1|k−1)
+
(22)
2) Mode-Matched Filtering Given a measurement z(k), the updated track estimates, xˆ j (k|k) and P j (k|k), are generated via a standard Kalman filter (or extended Kalman filter in the case of a nonlinear model) using the mixed initial conditions. The likelihood of the measurement given each mode j is computed as Λ j (k)
△
=
p(z(k)|m j (k), Z k−1 ) j
(23) j
= N z(k); zˆ (k|k−1), S (k|k−1)
where zˆ j and S j are the predicted measurement and associated error covariance from the j-th filter using the mixed initial conditions computed in Step 1. 3) Mode Probability Update The a posteriori probability of each mode is then given by µ j (k)
△
= P(m j (k)|Z k )
(24)
R
=
1 Λ j (k) ∑ pi j µi (k−1) c i=1
where the normalising constant is R
c=
∑ Λ j (k)c j
(25)
j=1
4) Combined Track Estimates For output, the first and second moments of the IMM mixture PDF are respectively R
x(k|k) ˆ =
∑ µ j (k)xˆ j (k|k) j=1
(26)
and R
∑ µ j (k)
P(k|k) =
j=1
P j (k|k)+
(27)
j ′ xˆ j (k|k) − x(k|k) ˆ xˆ (k|k) − x(k|k) ˆ
B. IMM-VDA Algorithm
To derive the IMM-VDA algorithm it is necessary to calculate the transition costs when there are multiple models for the target motion. Recall that the definition of the transition cost from node θi (k−1) to θ j (k) is given by △
ai j (k) = − ln a¯i j (k)
(28)
does not include a model for the existence of the target so its set of data association events is simply • θ0 (k) = the target is not detected • θ j (k) = the target is detected and z j (k) is the target measurement, j = 1, . . . , n(k). In addition, the VA-QF algorithm of [7] and [5] assume a single model for the target motion. Manoeuvring targets are tracked by heuristic modifications to the transition costs, similar in effect to increasing the process noise covariance in the filter. The transition costs for the VA-QF algorithm of [7], when the target is not manoeuvring, are given by ai j (k) = 0
where △
a¯i j (k) = p(Z(k), θ j (k)|Z k−1 , Θi (k−1), X k )
(29)
ai j (k)
R
∑ p(Z(k)θ j (k)|ms (k)Z k−1 , Θi (k−1)) s=1
×P(ms (k)|Z k−1 , Θi (k−1))
(30)
∆
PD PE (k|k−1)N(z j (k); zˆsi (k|k−1), Sis (k|k−1))
1 δi j (k|k−1) = z˜i j (k|k−1)′ Si−1 (k|k−1)˜zi j (k|k−1) 2 (31)
for j = 1, . . . , n(k) where is the predicted measurement at time k given the path to node i at time k−1 using mode s and Sis (k|k−1) is the associated measurement error covariance matrix. The first term in (30) for j = −1 and j = 0 and for all s is given by the equivalent equations for the standard VDA, (11) and (12), respectively. Using the assumption that the system modes form a Markov chain, the second term on the right-hand side of (30) is given by (32)
R
=
∑ P(ms (k)|mr (k−1))P(mr (k−1)|Z k−1 , Θi (k−1)) r=1
=
prs µir (k−1)
where µir (k −1) for r = 1, . . . , R are the a posteriori mode probabilities for the sequence of associations ending with node i at time k−1. Thus the IMM-VDA algorithm is implemented by running the IMM filter along each possible path through the trellis (which corresponds to a unique sequence of data associations). The basic structure of the VDA algorithm is unchanged, except that the transition probabilities for the standard algorithm, (11)–(13), are replaced by (30). IV. T HE VA-QF A LGORITHM The algorithm described in this section is a modified version [7] of the technique first described in [5]. Both variants use the Viterbi algorithm to perform data association in a similar fashion to the VDA algorithm. However, the VA-QF algorithm
(34)
(35)
and
zˆsi (k|k−1)
P(ms (k)|Z k−1 , Θi (k−1))
PDV = δi j (k|k−1) − ln 1 − PD 1 + ln ((2π)nz |Si (k|k−1)|) 2
where nz is the dimension of the measurement space and
The first term on the right-hand side of (30) is given by −1
(33)
and for j 6= 0 and i = 0, . . . , n(k−1)
Conditioning on the mode hypothesis at scan k we have a¯i j (k) =
j = 0, i = 0, . . . , n(k−1)
z˜i j (k|k−1) = z j (k) − zˆi (k|k−1)
(36)
It can be seen that these costs are the normalised negative log likelihood of the measurement z j (k) given the path to node i at time k − 1, assuming PE (k) = 1 for all k. When tracking a target moving in two dimensions and with prior knowledge of typical target behaviour, manoeuvres are detected and tracked using a validation test and penalty terms. The process can be summarised as follows. 1) At the end of scan k, check if the track corresponding to the node with the minimum cost at time k − 1 is associated with at least one measurement. That is, the best path from the node with the minimum cost at k−1 leads to a detection at time k. 2) If the track corresponding to the best path is not updated with a detection for 3 consecutive scans then add a penalty term to the transition costs ai j (k) for j = 1, . . . , n(k). That is, replace ai j (k) with a˜i j (k) where a˜i j (k) = ai j (k) − κ(k)
(37)
2
(38)
and 1 κ(k) = 2 △
v(k) ˆ − v¯ σv
where v¯ is the known average target velocity and v(k) ˆ is the estimated target velocity at time k and σv is the corresponding estimated error standard deviation. 3) If the track is updated with a detection after using the modified transition costs (37) then the algorithm returns to using the original costs (35).
For reasons of space, only a very brief description of these methods will be given here. Probabilistic data association based trackers are widely used. At each scan, a gate is drawn around the predicted target location and each measurements within that gate produces an updated track state estimate. These estimates are then combined to produce a single, weighted track estimate. The weights are the data association probabilities, i.e. the probability that the detection was generated by the target. The original formulation of this PDA approach is given in [1], where it was assumed that the target is always present in the surveillance region. This was then extended in [8] to incorporate the simultaneous estimation of the a posteriori probability of target existence, P(E(k)|Z k ), along with the track state. This algorithm is called the Integrated Probabilistic Data Association (IPDA) tracker. This track existence probability estimate can be used as a measure of track quality for the confirmation and deletion of tracks. The standard IPDA algorithm assumes a single model for target motion. In [9] it was extended to incorporate the IMM algorithm. VI. S IMULATIONS The performance of the five algorithms, the standard VDA, the IMM-VDA, the VA-QF algorithm, standard IPDA and the IPDA-IMM method were tested using simulations of a generic scenario. The target was moving in two dimensions with the state vector giving the position and speed in each dimension. The sampling rate was T = 2s and, for simplicity, the target measurements were taken to be in Cartesian co-ordinates with measurement error variances in the x and y co-ordinates of σx = σy = 25m. The nominal target speed was 25 m/s and this value was used for v¯ in (38) for the VA-QF algorithm. The target motion was generally constant velocity with two constant speed turns. The first turn occurred between scans 20 to scan 25 at a rate of 3◦ /s, while the second occurred between scans 40 to 45 at a rate of 6◦ /s. The clutter in the surveillance region was uniform with a false alarm rate of PFA = 10−6 while the probability of target detection was only PD = 0.6. Both the VDA-based algorithms and both the IPDA-based algorithms used the same two state Markov chain model for predicting target existence (8)-(9) with δ0 = 0.9 and δ1 = 0.0. All tracks were initiated using two-point initialisation with the true target detections. All algorithms were tuned in the same way for all common parameters, so that the only differences were in the manner in which they handled manoeuvres and/or data association. The IMM-VDA algorithm used R = 5 models, each of which was a constant turn rate model [11] where ω ∈ {−6, −3, 0, 3, 6} degrees per second. Note that ω = 0 corresponds to a constant velocity target. A turn rate of 3◦ /s is typical for a commercial aircraft in a holding pattern. The initial mode probabilities were taken as ′ µ = 0.05 0.05 0.80 0.05 0.05 (39)
while the mode transition probability matrix was 0.83 0.0 0.17 0.0 0.0 0.0 0.83 0.17 0.0 0.0 0.075 0.075 0.7 0.075 0.075 [pi j ] = 0.0 0.0 0.17 0.83 0.0 0.0 0.0 0.17 0.0 0.83
(40)
The average sojourn time in state i of a Markov chain can be shown to be given by 1/(1 − pii ) [11] therefore this choice for the mode transition probabilities assumes that the target spends approximately 6 scans turning before returning to straight line motion. The performance of all five algorithms was evaluated using 200 Monte Carlo simulations. Typical estimated tracks from each of the algorithms are shown in Figure 2. The average RMS position and velocity errors for each are shown in Figures 3–6. These values were only calculated while the algorithm had a track on the target. The Viterbi-based algorithms were considered to have lost track once the track no longer used any of the available target detections to update the track for the remainder of the track’s lifetime. The PDAbased algorithms were considered to have lost track once their estimated probability of target existence was less than 0.2. 1800 IPDA VDA IPDA−IMM IMM−VDA VA−QF Truth
1600
1400
1200 Y Coordinate
V. T HE IPDA M ETHODS
1000
800
600
400
200
0
0
500
1000
1500 2000 X Coordinate
2500
3000
3500
Fig. 2. Estimated tracks from each of the five algorithms in the X-Y plane. The thicker line is the true target trajectory.
The proportion of true tracks over time for each of the five algorithms is shown in Figure 7. A track was classified as true if the algorithm used at least two-thirds of the target detections to update the track over the lifetime of the track. From this it can be seen that the standard VDA and IPDA algorithms are unable to track through the first manoeuvre. This is to be expected since they were designed on the assumption that the target motion is always constant velocity. These two algorithms are included so they can be compared to the equivalent algorithms which have been augmented with the IMM method. The results for the VA-QF algorithm show that it is only able to track through the first manoeuvre around a third of the time. The structure of the manoeuvre detector means it takes at least 3 scans to detect the start of any manoeuvre. Due to the
4
35
10
IPDA VDA IPDA−IMM IMM−VDA VA−QF
IPDA VDA IPDA−IMM IMM−VDA VA−QF
30
25 3
X Position RMS Errors
X Velocity RMS Errors
10
20
15
2
10
10
5
1
10
0 0
Fig. 3.
10
20
30 Scan
40
50
60
RMS errors in the x position estimates for each algorithm.
0
10
20
30 Scan
40
50
60
Fig. 5. RMS errors in velocity estimates in the x coordinate for each algorithm.
4
10
30
IPDA VDA IPDA−IMM IMM−VDA VA−QF
IPDA VDA IPDA−IMM IMM−VDA VA−QF
25
3
10
Y Velocity RMS Errors
Y Position RMS Errors
20
2
10
15
10
5
1
10
Fig. 4.
0
10
20
30 Scan
40
50
60
RMS errors in the y position estimates for each algorithm.
low probability of detection, the manoeuvre detector often fires later than this. As a result, the algorithm is unable to regain lock on the target, even once the manoeuvre is detected. The sharp drop in the number of true tracks shortly after the tracks are initialised is due to the manoeuvre detector being falsely triggered by clutter detections. The results for the VA-QF algorithm illustrate two major problems of explicit manoeuvre detection – false triggering of the detector due to clutter; and missed manoeuvres. In contrast, both the IMM-VDA and IPDA-IMM algorithm track through both manoeuvres in a larger proportion of cases. The IMM approach avoids the problem of explicit detection of the onset and termination of manoeuvres by its probabilistic weighting approach. The most significant cause of track loss for the IPDA-IMM algorithm is the low probability of target detection. This can be seen from the steady decline in the number true tracks for this method, in comparison to the marked drops at the two turns for the other algorithms. The estimated mode probabilities for the IMM-VDA and IPDA-IMM algorithms are shown in Figures 8 and 9. These show that while there is uncertainty about the exact turn rate during the manoeuvres, when taken with the RMS plots, it
0
0
10
20
30 Scan
40
50
60
Fig. 6. RMS errors in velocity estimates in the y coordinate for each algorithm.
can be seen that the IMM-based algorithms are still able to maintain track on the target due to the power of the IMM approach. Note, that the number of tracks classified as true at scan 60 is significantly reduced even for the IMM-VDA algorithm. In this case this is due to the algorithm overestimating the time spent turning. If the scenario is run for longer a large proportion of those tracks regain the true track, as is illustrated in Figure 10. In this figure, crosses (’x’) indicate the track was updated with the target detection. Circles (’o’) indicate where the target detection was not used, either because it was not available or because the track was updated using a clutter detection. VII. C ONCLUSIONS This paper presents a highly effective algorithm for tracking manoeuvring targets in adverse clutter environments. It combines two powerful tools for tracking under such conditions. The first is the use of the Viterbi algorithm to provide multiscan tracking performance, while the second is the versatile IMM algorithm for tracking manoeuvres without the need
0.9
1 IPDA VDA IPDA−IMM IMM−VDA VA−QF
0.9
0.8
0.7 IPDA−IMM Mode Probabilities
0.7 Proportion of True Tracks
−6 deg/s −3 deg/s 0 deg/s 3 deg/s 6 deg/s
0.8
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3 0.2
0.2
0.1
0.1
0
0
10
20
30 Scan
40
50
0
60
Fig. 7. Proportion of true tracks as a function of time for all five algorithms. The first turn occurs from scan 20 to 25. The second turn occurs from scan 40 to 45.
0
10
20
30 Scan
40
50
60
Fig. 9. Estimated mode probabilities from the IPDA-IMM algorithm. The first turn occurs during scans 20 to 25 at a rate of 3◦ /s. The second turn occurs during scans 40 to 45 at a rate of 6◦ /s.
1
2500
0.9
2000
0.7 −6 deg/s −3 deg/s 0 deg/s 3 deg/s 6 deg/s
0.6
0.5
1500 Y Position
IMM−VDA Mode Probabilities
0.8
0.4
1000
0.3
0.2
500
0.1
0
0
10
20
30 Scan
40
50
60
0
0
500
1000
1500
2000
2500
X Position
Fig. 8. Estimated mode probabilities from the IMM-VDA algorithm. The first turn occurs during scans 20 to 25 at a rate of 3◦ /s. The second turn occurs during scans 40 to 45 at a rate of 6◦ /s.
Fig. 10. Estimated track from the IMM-VDA algorithm when the scenario length is extended. The true target trajectory is given by the solid line and the dashed line gives the estimated track.
for the explicit detection of manoeuvre onset and termination times. Simulations have shown the ability of the combined IMM-VDA algorithm to track manoeuvring targets accurately, even when manoeuvres occur in relatively heavy clutter. The method for incorporating the IMM into the VDA framework that is described here is not the only one possible. A number of other approaches can be contemplated. The fundamental consideration is whether to use a Gaussian mixture PDF or a single Gaussian at each node in the trellis, and then to define suitable transition probabilities. We hope to report on these ideas in a subsequent article.
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ACKNOWLEDGMENTS The authors wish to thank Prof. Yaakov Bar-Shalom for his helpful suggestions in the early stages of this work. R EFERENCES [1] Y. Bar-Shalom and E. Tse. Tracking in a cluttered environment with probabilistic data association. Automatica, 11:451–460, 1975. [2] Y. Bar-Shalom and X.-R. Li. Estimation and Tracking: Principles, Techniques and Software. Artech House, 1993.
