- Email: [email protected]
Choosing a track association method
Information Fusion 3 (2002) 119–133 www.elsevier.com/locate/inffus
Choosing a track association method Barbara F. La Scala b
a,*
, Alfonso Farina
b,1
a Preston Aviation Solutions, 488 Victoria Street, Richmond, Vic. 3121, Australia Systems Analysis Group, ALENIA Marconi Systems, Via Tiburtina Km. 12.400, Rome 00131, Italy
Received 10 April 2001; received in revised form 9 October 2001; accepted 13 November 2001
Abstract This paper examines the problem of selecting a track association method. This is the first stage of the track fusion process and the performance of such a method is essential to the overall success of this process. It is shown that more issues must be considered than just examining the probability of correct association when judging the performance of an association technique. A method that provides a high probability of correct association may well have poor performance in other areas. This paper examines several additional features that should be considered as they also have a significant effect on the quality of the combined tracks that are the final outcome of the fusion process. The problem is illustrated by examining several track-to-track association techniques for the problem of correlating radar tracks with electronic support measures (ESM) tracks for airborne sensors. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multisensor tracking; Track-to-track correlation; Sensor resolution; Non-homogenous sensor
1. Introduction The problem of fusing target tracks from multiple sensors is an important one in the sensor fusion field. Combining the information from multiple sensors has the potential to improve tracking accuracy and target acquisition rates. This problem has been studied in [2,5,6,11] among other works. The first, and a crucial step, in the fusion process is the association of tracks from different sensors. If the associations are made incorrectly then the fused track estimates will potentially be worse than those from a single sensor. Therefore, when selecting an association method, it is essential that the method performs well. This issue then becomes a matter of the criteria by which the performance of an association technique should be measured. This paper examines several such criteria for the evaluation of the performance of a track association method. Clearly, when designing a new association method or selecting between existing methods, an important factor to consider is the probability that two tracks from the same target are correctly associated. A method with a *
Corresponding author. Tel.: +61-3-9426-8223; fax: +61-3-9484-8975. E-mail addresses: bfl[email protected] (B.F. La Scala), [email protected] (A. Farina). 1 Tel: +39-06-4150-2279; fax: +39-06-4150-2665
low probability of correct association, PC , is plainly of little use. However, while this is the first performance measure that should be considered, it should not be the only one. For a method to be a robust tool for trackto-track association several other criteria must also be examined. In this paper we consider four other factors governing the performance of track association methods. These are: 1. the probability tracks from two different targets are falsely associated, PF ; 2. the effect of the correlation between two tracks from the same target and the consequences of neglecting this correlation (which is common practice); 3. the effect of limited sensor resolution and the loss of resolution in one or more sensors; and 4. the method for selecting a threshold for the association test. A fusion process that uses a track association method that performs poorly or is not robust when all these factors are taken into consideration may produce fused tracks with lower accuracy than the unfused input tracks. Each of the features we propose as being important measures of track association performance is examined in detail in following sections. The importance of each
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feature to the overall accuracy of track association, and hence the accuracy of the fused tracks that can be produced, is considered in detail. In addition, each is illustrated by considering an example track association problem. A number of different track association methods for this problem are considered and the performance of each of these against all the criteria mentioned is shown by simulation. The next section outlines the track association problem that will be used to illustrate our results while the following section describes the association methods that will be investigated. Section 4 outlines the standard test scenario that will be used for illustrating the features under discussion. Sections 5 and 6 consider the probability of correct and incorrect associations, respectively. Section 7 describes how the cross-covariance can be calculated for the two sensors, even though they have different state spaces. These sections extend results originally presented in [4] and [8]. Section 8 looks at the choice of the test statistic threshold while Section 9 considers the effects of a loss of sensor resolution.
2. Problem outline The track association problem used as our example is that of associating targets that are moving in two dimensions and which are tracked by two non-homogenous airborne sensors. That is, the two sensors are different types of sensors that operate differently and measure different aspects of the targets. The two sensors considered here are a passive, electronic support measures (ESM) sensor and an active radar sensor. It is assumed that the radar and ESM sensor are registered and synchronised, so the radar and ESM measurements can be treated as originating from the same location (the ‘‘own-ship’’) at the same time. Fig. 1 shows the geometric configuration of the sensor platform and a target. The target co-ordinates are ðnt ; mt Þ and the own-ship co-ordinates are ðno ; mo Þ, where the subscript t indicates these are the target co-ordinates, while o indicates the own-ship. The variables ðn; mÞ represent the 2D co-ordinate system. It is also assumed that the trajectory of the own-ship is known without error. The range of the target from the own-ship is denoted by r and its bearing by b. 2.1. ESM subsystem The ESM provides measurements of b only. Two state space models will be used to filter these measurements. The first (passive filter 1) simply smoothes the noise bearing measurements. The second (passive filter 2) uses modified polar co-ordinates (MPC) [1]. This coordinate basis is well suited to bearings-only tracking as
Fig. 1. Geometric configuration of the problem in two dimensions.
it automatically decouples the observable and unobservable elements of the estimated state vector [3]. The equations for the first passive filter are simply xp1 ðk þ 1Þ ¼ A1 xp1 ðkÞ þ B1 w1 ðkÞ;
ð1Þ
zp1 ðkÞ ¼ C1 xp1 ðkÞ þ vb ðkÞ;
ð2Þ
where 1 A1 ¼ 0
T ; 1
1 B1 ¼
2
T2 ; T
C1 ¼ ½ 1
0 ;
ð3Þ
and xp1 ¼ ½b b_ 0 , where ð0 Þ stands for the transpose operator. The subscript p indicates this is a filter for the passive sensor and the 1 that it is the first such filter. The sampling period is denoted by T and fw1 ðkÞg and fvb ðkÞg are independent, white, Gaussian processes with zero mean and variances given by r2p1 and r2bp respectively. The estimated bearing from this filter will be denoted by b^p1 . The state vector in MPC is " ! #0 _ 1 r xp2 ¼ b_ ; : ð4Þ ; b; r r The last term, the inverse range, is unobservable unless the own-ship manoeuvres appropriately. However, the second element, the inverse time-to-go, is always observable. This element is a measure of any perturbation from circular motion of the target. A discrete-time non-linear model xp2 ðk þ 1Þ ¼ A2 ðxp2 ðkÞÞ;
ð5Þ
zp2 ðkÞ ¼ C2 xp2 ðkÞ þ vbp ðkÞ;
ð6Þ
can be derived [4] where C2 ¼ ½0 0 1 0 and fvbp ðkÞg is an independent, white, Gaussian process with zero mean and variance given by r2bp . Again, p indicates this is a filter for the passive sensor while 2 shows that it is the second such filter. Note that both passive filters use the same bearing measurement noise as this is a property of
B.F. La Scala, A. Farina / Information Fusion 3 (2002) 119–133
the ESM sensor itself and hence is common to both filters. Note that when linearising the model (5) and (6) to construct an extended Kalman filter, it is necessary to add a noise term to the linearised state equation to allow for linearisation errors. If linearisation errors are not accounted for, the EKF can suffer from filter divergence – that is, it becomes increasingly confident of increasingly inaccurate estimates [9]. The non-linear filter which is built around (5) and (6) above yields estimates of bearing and inverse time-to-go ^ which will be labelled b^p2 and ðrr_Þp2 . If the target is not accelerating then the inverse time-to-go is directly proportional to the angular acceleration. Thus estimating the inverse time-to-go is equivalent to estimating the angular acceleration. Consequently, the estimate of the inverse time-to-go will be noisier than the estimates of b as we are estimating an acceleration term from noisy position measurements. 2.2. Radar subsystem The radar provides noisy measurements of the range, r, as well as the bearing, b. The state space model used for the radar measurement filter (the active filter) is nonlinear and is given by the equations xa ðk þ 1Þ ¼ Fxa ðkÞ þ Gwa ðkÞ;
ð7Þ
za ðkÞ ¼ Hðxa ðkÞÞ þ va ðkÞ;
ð8Þ
0
where xa ¼ ½nt n_t mt m_ t and 21 2 2 3 1 T 0 0 T 2 6 T 60 1 0 0 7 6 7 FðkÞ ¼ 6 4 0 0 1 T 5; GðkÞ ¼ 4 0 0 0 0 1 0
3 0 0 7 7 1 2 5: T 2 T
^r_ r
! ¼ g1 ð^ xa Þ ¼
121
ð^ x1 no Þð^ x2 n_o Þ þ ð^ x3 mo Þð^ x4 m_ o Þ x3 mo Þ2 ð^ x1 no Þ2 þ ð^
a
b^a ¼ g2 ð^ xa Þ ¼ tan 1
x^1 no x^3 mo
ð12Þ
!
ð13Þ
:
The error covariance matrix for these estimates of inverse time-to-go and bearing is given by Gð^ xa ÞPa G0 ð^ xa Þ to first order, where Pa is the solution to the Riccati equation for the active filter and ogi : ð14Þ GðxÞ ¼ ox j
^a x
3. Track association logic Given a sequence of state estimates from the active filter, f^ xia ðkÞg, and a sequence from one of the two passive filters, f^ xjp ðkÞg, we need to design a test to select between the hypotheses: H0 : the sequences are from the same target, i.e. i ¼ j. H1 : the sequences are from different targets, i.e. i 6¼ j. ^ia ðkÞ x ^jp ðkÞ, which is an estimate of Define ^ij ðkÞ ¼ x ij i j ðkÞ ¼ xa ðkÞ xp ðkÞ. Then the hypotheses become H0 : E½ij ¼ 0 and H1 : E½ij 6¼ 0. Let Rij ¼ E½ðij ^ij Þðij ^ij Þ0 ; i
j
ij
ð15Þ
ij 0
¼ P þ P P ðP Þ ; ð9Þ
The subscript a indicates that these variables relate to the filter for the active sensor. The non-linear measurement equation is 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðx1 no Þ2 þ ðx3 mo Þ2 5: Hðxa Þ ¼ 4 ð10Þ o tan 1 xx13 n
mo The noise processes fwa ðkÞg and fva ðkÞg are assumed to be independent, white, Gaussian processes with zero mean and variances given by 2 2 ra 1 0 rr 0 Qa ¼ ; Ra ¼ ; ð11Þ 0 r2ba 0 r2a2 respectively where rr is the standard deviation of the range measurement noise and rba is the standard deviation of the bearing measurement noise. ^a , estiFrom the extended Kalman filter estimates, x mates of bearing and inverse time-to-go are obtained by the zero memory non-linear transformations:
i
ð16Þ i 0
ij
^i Þðxi x ^ Þ and P ¼ E½ðxi x ^i Þ where P ¼ E½ðxi x ^j Þ0 [4]. In general, this cross-covariance term will ðxj x not be zero as the state estimate errors will be correlated if the trajectories are from the same target due to the common process noise in each filter. However, as a general rule it is assumed to be zero. Section 7 examines the effect of this assumption. Assuming Gaussian, independent noise processes in each filter, the quantities Pi and Pj can be approximated by the solutions of the Riccati equations in the (extended) Kalman filters used to compute the active and passive state estimates. Given the expressions for the covariance and crosscovariance terms, a test statistic for this hypothesis testing problem is Td ðkÞ ¼
d 1 X
0
1 ð^ ij Þ ðk rÞðRij Þ ^ij ðk rÞ
ð17Þ
r¼0
for some value of the time window length d, where k is the scan number. The distribution of this statistic will be approximately v2 with dnx degrees of freedom where nx is the dimension of the state, x, under the null hypothesis
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H0 . The accuracy of this v2 approximation is examined in Section 8. Three different methods for computing the test statistic will be compared. The first method is a commonly used approach. The remaining methods were proposed in [4]. These make use of MPC in the ESM filter. The three methods are: Method 1: Compare b^a with b^p1 from the first, linear passive filter. Method 2: Compare b^a with b^p2 from the second, MPC filter. ^ ^ Method 3: Compare ½b^a ; ðrr_Þa with ½b^p2 ; ðrr_Þp2 from the MPC filter. In Sections 5–9 the performance and robustness of each of the three methods are shown by simulation of the standard test scenario given in Section 4. These results illustrate why a method that has a high probability of correct association may still be a poor choice if it is not robust against other important criteria.
4. Standard test scenario A number of Monte Carlo studies were carried out to thoroughly evaluate the three track association methods described in Section 3. The same basic scenario was used in most cases. This basic scenario has the own-ship moving in a circle of radius 20 394 m, centred on a point at ()30 000, 29 606) m, with a constant turn rate of 0.2/s. In the case of studies with a single target, target A, the target trajectory is a straight line with the initial conditions nt ð0Þ ¼ 50 000 m;
ð18Þ
n_t ð0Þ ¼ 150 ms 1 ;
ð19Þ
mt ð0Þ ¼ 23 000 m;
ð20Þ
1
ð21Þ
m_ t ð0Þ ¼ 100 ms :
In studies with two targets there is an additional target, target B, also moving in a straight line, with the same velocity, but offset from the track of first target. For this second target the initial position is nt ð0Þ ¼ 50 000 m;
ð22Þ
mt ð0Þ ¼ 19 000 m:
ð23Þ
The sampling rate of both sensors is T ¼ 4 s. The default measurement noise standard deviations are rr ¼ 100 m and rbp ¼ rba ¼ 12 a degree. Note that we are using the same bearing noise variance for the radar and the ESM sensor for simplicity. In general, it is more likely that rba would be less than rbp . In all the examples it is assumed that the sensors are not misaligned. The effect of sensor misalignment, and
the spurious differences in track estimates that result are beyond the scope of this paper. However, we note that in [5] exact expressions for the probability of correct association, PC and incorrect association, PF have been derived when using maximum likelihood estimation (MLE) for this problem. These expressions describe PC and PF as functions of the sensor accuracy and sensor misalignment as well as the size of the batch used to calculate the ML track estimates.
5. Probability of correct association PC Clearly, the first consideration when selecting a track association method is how often will the test correctly associate two tracks from the same target. A method that cannot reliably associate tracks is plainly of little use. There are a number of features which affect the ability of a method to correctly associate tracks. These include: • • • •
the accuracy of the sensors; the accuracy of the individual track estimates; sensor misalignment; and the window length used when calculating the test statistic.
Inaccuracies in the sensors lead to inaccuracies in the tracks which can produce spurious differences in the track estimates. Even though sensor accuracy is not under the control of the designer of a track association method, it is still necessary to consider its effect on the performance of such a method. A particular technique may be very accurate but only when sensor accuracy is high. In a situation where accuracy is low another, more robust method may be a better choice even though it does not perform as well as the first method under ideal conditions. The accuracy of the track estimates is obviously strongly related to sensor accuracy but there are other factors also involved. One such factor is the choice of a state space model for the tracker. In [3], it is shown that for the same input a bearings-only tracker that uses MPC produces more accurate estimates than a simple smoother such as the first passive filter described in Section 2.1. This is due to the way the MPC filter decouples the observable and unobservable components of the system state. This allows the filter to extract the maximum possible information from the input. Another factor affecting track accuracy is the length of the target samples. The longer the target has been tracked, the more accurate the estimates, assuming it does not manoeuvre. Since this assumption is unrealistic, a track association method that quickly produces a definitive answer is to be desired.
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123
Fig. 2. Proportion of tracks associated over time for Scenario 1 (low bearing measurement noise and short test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution.
There are similar considerations when choosing the length of the window for the test statistic. If the motion of the target does not change, relative to the own-ship, then a longer window length will produce a test statistic with a lower variance. Again, however, assuming no manoeuvres are unrealistic, it is desirable to choose a method that is accurate with as short a window length as possible. A number of 1000 Monte Carlo runs were performed to examine the performance of each of the three test statistics when both sensors were tracking the same target. These simulations were based on the standard scenario outlined in Section 4. Four scenarios were considered. These are: S1: The basic scenario with the default measurement noise variances of rr ¼ 100 m and rba ¼ rbp ¼ 12 a degree and a window length of d ¼ 1. S2: The same noise variances, rr ¼ 100 m and rba ¼ rbp ¼ 12 a degree but a window length of d ¼ 3. S3: Increased noise variance in the bearing sensor, rba ¼ rbp ¼ 1 degree, the same range noise variance, rr ¼ 100 m, and a window length of d ¼ 1. S4: The same scenario as in S3 but with a window length of d ¼ 3. For each scenario the three test statistics were computed every five scans and compared to the appropriate 95% point from v2 tables as the test threshold. The cross-covariance term was assumed to be zero. The effect of including an estimate of the cross-covariance is examined in Section 7. The estimate of PC as a function of scan number for each test statistic in each of the four scenarios is shown in Figs. 2–5 below. From these figures it can be seen that
all three methods give a value of PC that is close to the desired value of 95%. Also, once the accuracy of the track estimates is sufficiently high, Method 3, which uses the MPC estimates of both bearing and inverse timeto-go, gives the highest probability of correct association. However, the methods that rely on MPC require higher track accuracy than the simple smoother. This is probably due to the fact that the MPC filter allows the angular acceleration estimate to vary, unlike the simple smoother which assumes this acceleration is zero. Thus the MPC methods yield a more variable estimate of bearing. This is confirmed by Figs. 4 and 5 which are the results for the scenarios with the increased noise in the bearing measurements. In these examples, the MPC methods take longer to converge. These figures suggest that the MPC association methods may not be the best choice in a situation where the accuracy of the sensors is relatively low and the targets are manoeuvring relatively quickly, due to the longer time these methods take to converge. Note, however, that the MPC methods actually work better with a shorter window length. This can be seen by comparing Fig. 2 with Fig. 3 and 4 with Fig. 5. In each pair, the second scenario differs from the first solely by having a longer test window length. Thus for a given degree of sensor accuracy, the vulnerability of the MPCbased methods to unanticipated manoeuvres can be reduced by selecting a minimal test statistic window length.
6. Probability of false association PF If the results of Section 5 were considered alone, one might conclude that the best choice of a track
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Fig. 3. Proportion of tracks associated over time for Scenario 2 (low bearing measurement noise and long test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution.
Fig. 4. Proportion of tracks associated over time for Scenario 3 (high bearing measurement noise and short test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution. The longer convergence time for the MPC-based methods reflects the higher degree of variation in the MPC bearing estimates.
association method for the sample problem was Method 1 and that the MPC-based methods were inferior. However, equally important to the selection of a track association method is the probability that a method will not falsely associate tracks from different targets. This is the ability of the method to discriminate. A good track association method must be just as capable of detecting differences between target tracks as it is able to detect when the tracks are from the same target. However, there is a trade-off between PC and PF . Increasing the value of PC for a particular method inevitably increases the value of PF also. This can be illustrated by considering the case of using a batch MLE
for this problem. In [5] an expression for PF as a function of the test threshold (which in turn is a function of PC ), distance between targets, sensor misalignment and sensor accuracy has been derived. Fig. 6 shows PF against scan number for four different values for PC . This shows that if a very high probability of correct association is required when using an MLE track association method, the track lengths must be long in order to overcome the initially high probability of falsely associating tracks from different targets. However, different track association methods will have different probabilities of incorrect association for a given value of PC . Thus when selecting a track associa-
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125
Fig. 5. Proportion of tracks associated over time for Scenario 4 (high bearing measurement noise and long test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution. The longer convergence time for the MPC-based methods reflects the higher degree of variation in the MPC bearing estimates.
0.6
0.5
Pfas
0.4
0.3
0.2
0.1
0
60
70
80
90
100
Scan Number
Pc = 0.6 Pc = 0.7 Pc = 0.8 Pc = 0.9
Fig. 6. PF versus scan number as a function of PC .
tion technique, it is desirable to use a method that still retains a relatively high power to discriminate between different targets even when the test threshold is chosen to give a high value of PC . To compare the three methods considered in this paper the same simulations that were used in Section 5 were repeated. However, now the ESM sensor was tracking a second target that was travelling parallel to the first, with a distance of 4000 m between the two. The radar continued to track the first target. Such a result could occur if, for example, only one target was emitting electromagnetic radiation. A more general example
would be when the radar and the ESM sensor both give rise to two tracks which have to be correctly associated. This case will not be considered here. Once again 1000 Monte Carlo runs were performed for each of the four scenarios outlined in Section 5 with the addition of the second target. The test statistics were again compared to the appropriate 95% point from v2 tables. This should yield a theoretical PF ¼ 0:05. Figs. 7– 10 show the results of the simulations. From these figures it can be seen that the first track association method, which uses the simple smoother for the ESM sensor, is not able to discriminate between the two tracks as readily as the MPC-based methods. This is particularly true when the noise bearing standard deviation is relatively high (1 degree). In this case, using the passive smoother on the ESM signal produces an estimated track for the second target which lies within the
3rba limits of the radar track of the first target as both targets get further away from the own-ship. Once this has occurred Method 1 effectively loses any power to discriminate between the tracks. It can also be seen from the figures that all three methods perform increasingly poorly as the number of scans increases. Again, this is a result of the increasing distance between the target and the own-ship as the scenario progresses. As this happens, the size of the target location uncertainty ellipses of both filters increases. Thus they overlap more often yielding an increase in PF . Note that in each case, increasing the window length improves performance. However, recall that from the previous section, this is at the expense of a corresponding reduction in the probability of correct association.
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Fig. 7. Proportion of tracks associated over time for Scenario 1 (low bearing measurement noise and short test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution.
Fig. 8. Proportion of tracks associated over time for Scenario 2 (low bearing measurement noise and long test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution.
The results of this section show that any conclusion reached by considering only the results in the previous section, Section 5, would, in fact, be erroneous. These results implied that the first method was preferable to those that relied on the MPC filter. The test scenario for each of the four figures in this section matches those in the previous section (e.g. Fig. 2 matches 7, 3 matches 8, and so on). For each pair, the test statistic threshold is the same, as are the window length and sensor accuracy. By comparing each pair of figures, it can be seen that the MPC-based methods retain a high discriminatory power even at high values of PC , while Method 1 does not. In fact, only the two MPC-based methods are capable of operating at close to the theoretical performance figures for both PC and PF making them a better choice than the first method.
Figs. 11 and 12 show the percentage of tracks falsely associated as the separation between the two target varies when rb ¼ 1 degree. These results confirm the greater discriminatory power of the MPC-based association methods as they can accurately distinguish between more closely spaced targets than Method 1. As the targets do not manoeuvre during the scenario, the power of all three association tests is increased as the window length increases.
7. Calculation of the cross-covariance It has been shown that the performance of trackto-track association methods may be improved by including the effect of the cross-covariance between the
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127
Fig. 9. Proportion of tracks associated over time for Scenario 3 (high bearing measurement noise and short test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution. The divergence over time is a result of the increased uncertainty in the bearing estimate as the distance to the targets increases.
0.6 Method 1 Method 2 Method 3
Proportion of Track Associated
0.5
0.4
0.3
0.2
0.1
0 40
60
80
100
120 Scan
140
160
180
200
Fig. 10. Proportion of tracks associated over time for Scenario 4 (high bearing measurement noise and long test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution. The divergence over time is a result of the increased uncertainty in the bearing estimate as the distance to the targets increases.
two track estimates when they are of the same target. Results for sensors of the same type are given in [2]. This was extended in [10] to the case when the sensor measurements, are dissimilar. More recently, [8] dealt with the case when the state spaces of the two filters, x1 and x2 are dissimilar. The scenario under consideration in this paper has dissimilar state spaces. Briefly, suppose there exists a one-to-one transformation between x1 and x2 or a transformation that can be constrained to be one-to-one using additional (perhaps physical) knowledge of the problem. That is, there exist g : Rn ! Rm and g 1 : Rm ! Rn such that gðx1 Þ ¼ x2 and g 1 ðx2 Þ ¼ x1 . Given such a transforma-
tion g it was shown in [8] that it was possible to write the state space model for the second filter in terms of the state vector of the first model, x1 . This can be done for both linear and non-linear models. For the problem being examined here x2 represents the state vector of the radar filter, xa , while x1 represents the state vector of either passive filter, xp1 or xp2 . Using this result, the filter for each sensor can be constructed using whichever state space model is most appropriate for that sensor, even if this means using different state spaces for different sensors. When tracks from two sensors are to be combined, the estimates from the second filter can be converted to the state space of
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Fig. 11. Proportion of tracks falsely associated as a function of the distance between the two targets. Test statistic window length ¼ 1.
Fig. 12. Proportion of tracks falsely associated as a function of the distance between the two targets. Test statistic window length ¼ 3.
the first filter using the transformation g 1 . As a result of this transformation, a recursion for the cross-covariance between the two filters can be calculated. To test the effect of including the cross-covariance term modified versions of the standard scenario given in Section 4 was used in two ways. In the first case, only one target was present, target A. The probability of correctly associating the tracks from each sensor was calculated, both with and without including the crosscovariance term. The results for the first association method are shown in Fig. 13. From this figure it can be seen that including the estimated cross-covariance term actually reduced the performance of the first association method. In general, the first method associated the tracks only 85% of the time when the cross-covariance
term was included rather than the expected 95% success rate. This result indicates that the test threshold was too high. The results for the two MPC-based methods are not shown as including the cross-covariance term did not make a significant difference in their performance. In the second test both targets A and B were present. However, in this case the second target’s initial position was ð40 000; 8000Þ m, i.e., it was offset from the first target in both dimensions. The radar track estimate for target A was compared to the ESM track for target B. As before, the three test statistics were calculated, both with and without the cross-covariance term. Once again, the results for the two MPC-based methods (Methods 2 and 3) did not change significantly. The results for the first method are shown in Fig. 14. The tracks should
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129
Fig. 13. Proportion of tracks associated over time. This should be approximately 0.95 if the test statistic has a v2 distribution. Results shown for first passive filter only.
have been incorrectly associated approximately 5% of the time. From the figure it can be seen that this method does not achieve this degree of accuracy. The spikes in the proportion of tracks associated correspond to significant changes in the bearing-rate. The state space model for the ESM filter using Method 1 assumes a nearly constant bearing-rate and there is clearly a drop in performance when this assumption is violated. The performance does not improve until the ESM filter has had time to adjust to the change in bearing-rate. However, from the figure it can also be seen that including the cross-covariance term does reduce the false association rate of this method. However, this improvement in performance could simply be the result of the fact that
including the cross-covariance term meant that the actual distribution of the test statistic was less like a v2 distribution than before. This possibility is considered in more detail in the following section. From these results it can be seen that the MPC-based methods are robust to deviations from the (false) assumption that the cross-covariance, P ij , is zero. This makes these methods a more desirable choice for practical use as it means their implementation can be kept relatively simple. In contrast, the association method based on the simple smoother is not robust to the common, if false, assumption that P ij is zero. While its performance can be improved by the inclusion of P ij , this requires careful selection of the test statistic
Fig. 14. Proportion of tracks falsely associated over time. This should be approximately 0.05 as the test statistic should have a v2 distribution. The spikes correspond to significant changes in the true bearing-rate. Results shown for first passive filter only which assumes a constant bearing-rate.
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threshold. This once again confirms that selecting a method by considering PC alone may result in a poor choice for a track association method.
8. Test statistic threshold All the results discussed above have made the standard assumption that the test statistic, Td , has a v2 distribution under the null hypothesis, that is, when the two tracks are from the same target. This assumption is routine, however, as the results of including an estimate of cross-covariance shown in the previous section, it is not always an accurate one. If the threshold chosen is too high, then the probability of correct association, PC , will be less than desired. On the other hand, if the threshold is set too low then the number of falsely associated tracks will be higher than anticipated. Thus another feature to consider when selecting an association method is the ease with which an appropriate threshold can be selected that will result in the association technique having a performance, in practice, that is close to its theoretical performance. When using MLE batch estimation for the airborne sensor problem, an analytical expression for PC as a function of k, the test statistic threshold, can be derived as was shown in [5]. The value of PC , when the sensors are not misaligned and are tracking the same target, is given by the equation k k PC ¼ 1 1 þ exp
: ð24Þ 2 2 There is no closed form expression for k as a function of PC but the equation can be inverted numerically. Thus the selection of an appropriate test threshold for an MLE batch association method is straightforward. However, this is not the case for recursive estimators of the type considered in this paper. Instead, it is common to assume that the v2 distribution is a sufficiently good approximation to the actual distribution of such test statistics. Simulations were used to investigate the validity of the v2 approximation for the recursive estimation case. These used the basic scenario described in Section 4. The value of the test statistic for each of the three association methods was calculated at the end of the 500 scans for 1000 Monte Carlo runs. This produced a sample distribution for the test statistic for each method. This sample distribution was produced for each method with window lengths of 1, 2 and 3. The results are shown in Tables 1–3 along with the corresponding point from v2 tables for comparison. From these tables we can see that the assumption that the test statistics have a v2 distributions is, fortunately, most accurate around the 90–95% range. Also, the approximation improves as the window length increases.
Table 1 Test statistic threshold for a window length of 1 75%
90%
95%
99%
Method 1 Method 2 v2
1.05 1.11 1.32
2.22 2.12 2.71
3.01 3.03 3.84
5.59 4.67 6.64
Method 3 v2
1.42 2.77
2.56 4.61
3.60 5.99
5.27 9.21
Table 2 Test statistic threshold for a window length of 2 75%
90%
95%
Method 1 Method 2 v2
2.12 2.19 2.77
3.81 4.36 4.61
5.48 6.05 5.99
99% 9.47 9.98 9.21
Method 3 v2
2.88 5.39
5.29 7.78
6.97 9.49
10.90 13.28
Table 3 Test statistic threshold for a window length of 3 75%
90%
95%
99%
Method 1 Method 2 v2
3.16 3.42 4.11
5.60 6.34 6.25
7.62 8.90 7.82
11.80 14.20 11.34
Method 3 v2
4.48 7.84
7.52 10.64
10.10 12.59
16.10 16.81
However, in general, the sample threshold is below that predicted by v2 tables. This is to be expected given the results in previous sections. This indicates that using v2 tables to select a test threshold may result in a fusion process that falsely associates tracks more often than was desired. Consider the results of Section 7. Here the number of tracks associated when the cross-covariance term was included for Method 1, reduced both the correct and false association rates when the value of the test statistic was compared to a v2 threshold. This implies including the cross-covariance term reduces the accuracy of the v2 approximation as the v2 -based threshold is clearly too high. By contrast, the MPC-based methods, particularly Method 2, are more closely approximated by the v2 distribution as can be seen by Tables 1–3. These results, combined with those in Sections 5 and 6, show that it is easier to select a test statistic threshold for the MPCbased methods that will result in a track association technique that has an actual performance that closely matches its theoretical performance.
9. Effect of sensor resolution A complete model of the resolution capacity of a given sensor is very difficult. However, the probability,
B.F. La Scala, A. Farina / Information Fusion 3 (2002) 119–133
PR , that a sensor resolves two closely spaced targets will clearly depend on the distance between the targets in addition to other features. In [7], a simple but intuitively appealing model was proposed for PR . From [7] PR can be modelled as a Gaussian-like function 1 0 PR ¼ 1 exp ðy1 y2 Þ D 1 ðy1 y2 Þ ; ð25Þ 2 where yi is the noiseless measurements of the ith target and D is a positive definite, symmetric matrix that describes the extension and spatial orientation of the sensor resolution cells. In general, D will depend on the target–sensor geometry. Following [7], we will assume that the range resolution ara of the radar sensor is essentially determined by the length of the emitted pulse. The angular resolution of both the radar and ESM sensors, aba and abp , is limited by the beamwidth of each sensor. Also, we will assume that the probability of unresolved targets is decreasing for increasing distances ðdr; dbÞ between the targets. For the radar sensor, assuming that the range resolution is independent of the angular resolution, "1 # 0 ara
1 D ¼ ; ð26Þ 0 a1b a
while for the ESM sensor, D 1 ¼ 1=abp . Suppose that there are two targets but a sensor fails to resolve them. Then, instead of measuring y1 or y2 , the sensor measures the mean target position 1 ym ¼ ðy1 þ y2 Þ: 2
ð27Þ
In effect, the sensor measures the position of a third, synthetic target that is midway between the two real targets.
131
Recall from Section 3, assuming that both sensors resolved the targets, under the null hypothesis the expected value of the test statistic Td was zero. Under H1 (the two tracks are from different targets) the expected value of the test statistic will be some non-zero value, say s. Suppose now, one sensor fails to resolve the targets, then one track will be a track of a true target while the other is of the third, synthetic target. From (27) and (17) it can be shown that under these circumstances the expected value of the test statistic under H1 will only be 1 s. This reduction in E½Td increases the probability of 4 falsely associating the two tracks. This effect was examined in more detail in [8]. To test the effects of sensor resolution the standard test scenario, given above, was modified to include a range resolution for the radar of ar ¼ 100 m and an angular resolution of both sensors of ab ¼ 2 degrees. In addition, the initial position of the second target, target B, was changed to nt ð0Þ ¼ 40 000 m;
ð28Þ
mt ð0Þ ¼ 8000 m:
ð29Þ
In this scenario, the radar was able to resolve both targets at all times. However, the ESM sensor could only resolve the targets initially. As the two targets moved away from the own-ship, the difference in the bearing measurements eventually became too small. Fig. 15 shows the probability of resolution over time for the ESM sensor. The probability does not decay monotonically as the target moves away from the sensor platform, but instead displays some periodicity due to the circular motion of the own-ship. Two cases were examined. In the first case, both sensors were initially tracking target A. After scan 230, the ESM sensor could no longer resolve targets A and B and began to return the track of the third, synthetic
Fig. 15. Probability of resolution for the ESM sensor. This decreases as the targets move further away from the sensor.
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Fig. 16. Proportion of tracks associated over time. The tracks should associate up to scan 230 as the ESM sensor has a better than 50% chance of resolving the targets up to this time.
target midway between A and B. The calculation of the test statistics did not take into account the possibility that a sensor may have failed to resolve the targets. The results are shown in Fig. 16. Until scan 230 the tracks are from the same target, and this is detected by all three methods. Immediately after resolution is lost, all methods stop associating the tracks, but only the methods using MPC in the passive filter (Methods 2 and 3) continue to recognise that the sensors are tracking different targets after scan 230. The simple passive smoother of the bearing (Method 1) is unable to deal with the step change in the bearing of the ESM measurements when the sensor loses resolution. It can only put this change down to impulsive noise in the system and con-
tinues to erroneously associate the tracks, that are now from two different targets, a high proportion of the time. The second case examined resolution effects on the probability of incorrect association PF . This case was tested in a similar manner as above, except that in this situation the ESM sensor was initially tracking target B. The results are shown in Fig. 17. Once again it can be seen that while all three methods give acceptable performance when the sensors are able to resolve the targets, that is, all methods recognise that the tracks are from different targets at a rate that is close to their desired theoretical performance rate. However, only Methods 2 and 3 continue to operate at close to this performance rate once resolution is lost.
Fig. 17. Proportion of tracks falsely associated over time. This increases as the probability of resolution decreases.
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Once again, the results of this section show that considering more features than just the probability of correct association can lead to the selection of a different track association method than the one that may have been selected when considering PC alone. In Section 5, if the results of that section are considered in isolation, then the first track association method is superior to the MPC-based methods. However, as it can be seen from Figs. 16 and 17, this conclusion is not the case when the effect of the loss of sensor resolution is considered. Only the MPC-based methods were able to cope with the loss of resolution and continue to associate tracks from the same target and discriminate between those from different targets.
10. Conclusions This paper has examined a number of features that need to be considered when selecting a track-to-track association method. It has been demonstrated that it is not sufficient to simply consider the probability of correct association given by a method. Other features such as the power of the method to distinguish between different tracks; the ability to easily select the appropriate test threshold; and the effects of the loss of sensor resolution can have a significant effect on the performance of a track association method. It was argued that all these features must be considered together when selecting a track association method. If this is not done, then the combined tracks produced by a fusion process that uses a poorly chosen track association technique may be less accurate than the original, unfused tracks. The effect of all the features discussed in this paper was illustrated by considering the problem of trackto-track association for dissimilar airborne sensors. It was seen that, for this problem, filters using MPC were more robust than another, more common technique. While the MPC-based techniques did not provide as
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high a value of PC as the more common technique, they did perform well when all the five criteria discussed in the paper were examined. In fact, this paper has shown that the method that provides the best probability of correct association is the least robust and actually performs more poorly than the MPC-based methods when measured against other, equally important criteria, of track association performance.
References [1] V.J. Aidala, S.E. Hammel, Utilization of modified polar coordinates for bearings-only tracking, IEEE Transactions on Automatic Control 20 (3) (1983) 283–294. [2] Y. Bar-Shalom, X.-R. Li, Multitarget–Multisensor Tracking: Principles and Techniques, YBS, Connecticut, USA, 1995. [3] A. Farina, Target tracking with bearings-only measurements, Signal Processing 78 (1999) 61–78. [4] A. Farina, B. La Scala, Methods for the association of active and passive tracks for airborne sensors, in: International Radar Symposium IRS-98, Munich, Germany, 15–17 September 1998, pp. 725–734. [5] A. Farina, R. Miglioli, Association between active and passive tracks for airborne sensors, Signal Processing 69 (1998) 209–217. [6] A. Farina, F.A. Studer, Radar Data Processing, vol. 2, Research Studies Press, 1985. [7] W. Koch, G. vanKeuk, Multiple hypothesis track maintenance with possibly unresolved measurements, IEEE Transactions on Aerospace and Electronic Systems 33 (3) (1997) 883–892. [8] B. La Scala, A. Farina, Effects of cross-covariance and resolution on track association, in: International Conference on Information Fusion, Paris, France, 10–13 July 2000. [9] B.F. La Scala, R.R. Bitmead, M.R. James, Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem, Mathematics of Control, Signals and Systems 8 (1) (1995) 1–26. [10] R.K. Saha, Track-to-track fusion with dissimilar sensors, IEEE Transactions on Aerospace and Electronic Systems 32 (3) (1996) 1021–1028. [11] G.V. Trunk, J.D. Wilson, Association of df bearing measurements with radar tracks, IEEE Transactions on Aerospace and Electronic Systems 23 (4) (1987) 438–447.
Choosing a track association method Barbara F. La Scala b
a,*
, Alfonso Farina
b,1
a Preston Aviation Solutions, 488 Victoria Street, Richmond, Vic. 3121, Australia Systems Analysis Group, ALENIA Marconi Systems, Via Tiburtina Km. 12.400, Rome 00131, Italy
Received 10 April 2001; received in revised form 9 October 2001; accepted 13 November 2001
Abstract This paper examines the problem of selecting a track association method. This is the first stage of the track fusion process and the performance of such a method is essential to the overall success of this process. It is shown that more issues must be considered than just examining the probability of correct association when judging the performance of an association technique. A method that provides a high probability of correct association may well have poor performance in other areas. This paper examines several additional features that should be considered as they also have a significant effect on the quality of the combined tracks that are the final outcome of the fusion process. The problem is illustrated by examining several track-to-track association techniques for the problem of correlating radar tracks with electronic support measures (ESM) tracks for airborne sensors. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multisensor tracking; Track-to-track correlation; Sensor resolution; Non-homogenous sensor
1. Introduction The problem of fusing target tracks from multiple sensors is an important one in the sensor fusion field. Combining the information from multiple sensors has the potential to improve tracking accuracy and target acquisition rates. This problem has been studied in [2,5,6,11] among other works. The first, and a crucial step, in the fusion process is the association of tracks from different sensors. If the associations are made incorrectly then the fused track estimates will potentially be worse than those from a single sensor. Therefore, when selecting an association method, it is essential that the method performs well. This issue then becomes a matter of the criteria by which the performance of an association technique should be measured. This paper examines several such criteria for the evaluation of the performance of a track association method. Clearly, when designing a new association method or selecting between existing methods, an important factor to consider is the probability that two tracks from the same target are correctly associated. A method with a *
Corresponding author. Tel.: +61-3-9426-8223; fax: +61-3-9484-8975. E-mail addresses: bfl[email protected] (B.F. La Scala), [email protected] (A. Farina). 1 Tel: +39-06-4150-2279; fax: +39-06-4150-2665
low probability of correct association, PC , is plainly of little use. However, while this is the first performance measure that should be considered, it should not be the only one. For a method to be a robust tool for trackto-track association several other criteria must also be examined. In this paper we consider four other factors governing the performance of track association methods. These are: 1. the probability tracks from two different targets are falsely associated, PF ; 2. the effect of the correlation between two tracks from the same target and the consequences of neglecting this correlation (which is common practice); 3. the effect of limited sensor resolution and the loss of resolution in one or more sensors; and 4. the method for selecting a threshold for the association test. A fusion process that uses a track association method that performs poorly or is not robust when all these factors are taken into consideration may produce fused tracks with lower accuracy than the unfused input tracks. Each of the features we propose as being important measures of track association performance is examined in detail in following sections. The importance of each
1566-2535/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 6 - 2 5 3 5 ( 0 2 ) 0 0 0 5 0 - 7
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feature to the overall accuracy of track association, and hence the accuracy of the fused tracks that can be produced, is considered in detail. In addition, each is illustrated by considering an example track association problem. A number of different track association methods for this problem are considered and the performance of each of these against all the criteria mentioned is shown by simulation. The next section outlines the track association problem that will be used to illustrate our results while the following section describes the association methods that will be investigated. Section 4 outlines the standard test scenario that will be used for illustrating the features under discussion. Sections 5 and 6 consider the probability of correct and incorrect associations, respectively. Section 7 describes how the cross-covariance can be calculated for the two sensors, even though they have different state spaces. These sections extend results originally presented in [4] and [8]. Section 8 looks at the choice of the test statistic threshold while Section 9 considers the effects of a loss of sensor resolution.
2. Problem outline The track association problem used as our example is that of associating targets that are moving in two dimensions and which are tracked by two non-homogenous airborne sensors. That is, the two sensors are different types of sensors that operate differently and measure different aspects of the targets. The two sensors considered here are a passive, electronic support measures (ESM) sensor and an active radar sensor. It is assumed that the radar and ESM sensor are registered and synchronised, so the radar and ESM measurements can be treated as originating from the same location (the ‘‘own-ship’’) at the same time. Fig. 1 shows the geometric configuration of the sensor platform and a target. The target co-ordinates are ðnt ; mt Þ and the own-ship co-ordinates are ðno ; mo Þ, where the subscript t indicates these are the target co-ordinates, while o indicates the own-ship. The variables ðn; mÞ represent the 2D co-ordinate system. It is also assumed that the trajectory of the own-ship is known without error. The range of the target from the own-ship is denoted by r and its bearing by b. 2.1. ESM subsystem The ESM provides measurements of b only. Two state space models will be used to filter these measurements. The first (passive filter 1) simply smoothes the noise bearing measurements. The second (passive filter 2) uses modified polar co-ordinates (MPC) [1]. This coordinate basis is well suited to bearings-only tracking as
Fig. 1. Geometric configuration of the problem in two dimensions.
it automatically decouples the observable and unobservable elements of the estimated state vector [3]. The equations for the first passive filter are simply xp1 ðk þ 1Þ ¼ A1 xp1 ðkÞ þ B1 w1 ðkÞ;
ð1Þ
zp1 ðkÞ ¼ C1 xp1 ðkÞ þ vb ðkÞ;
ð2Þ
where 1 A1 ¼ 0
T ; 1
1 B1 ¼
2
T2 ; T
C1 ¼ ½ 1
0 ;
ð3Þ
and xp1 ¼ ½b b_ 0 , where ð0 Þ stands for the transpose operator. The subscript p indicates this is a filter for the passive sensor and the 1 that it is the first such filter. The sampling period is denoted by T and fw1 ðkÞg and fvb ðkÞg are independent, white, Gaussian processes with zero mean and variances given by r2p1 and r2bp respectively. The estimated bearing from this filter will be denoted by b^p1 . The state vector in MPC is " ! #0 _ 1 r xp2 ¼ b_ ; : ð4Þ ; b; r r The last term, the inverse range, is unobservable unless the own-ship manoeuvres appropriately. However, the second element, the inverse time-to-go, is always observable. This element is a measure of any perturbation from circular motion of the target. A discrete-time non-linear model xp2 ðk þ 1Þ ¼ A2 ðxp2 ðkÞÞ;
ð5Þ
zp2 ðkÞ ¼ C2 xp2 ðkÞ þ vbp ðkÞ;
ð6Þ
can be derived [4] where C2 ¼ ½0 0 1 0 and fvbp ðkÞg is an independent, white, Gaussian process with zero mean and variance given by r2bp . Again, p indicates this is a filter for the passive sensor while 2 shows that it is the second such filter. Note that both passive filters use the same bearing measurement noise as this is a property of
B.F. La Scala, A. Farina / Information Fusion 3 (2002) 119–133
the ESM sensor itself and hence is common to both filters. Note that when linearising the model (5) and (6) to construct an extended Kalman filter, it is necessary to add a noise term to the linearised state equation to allow for linearisation errors. If linearisation errors are not accounted for, the EKF can suffer from filter divergence – that is, it becomes increasingly confident of increasingly inaccurate estimates [9]. The non-linear filter which is built around (5) and (6) above yields estimates of bearing and inverse time-to-go ^ which will be labelled b^p2 and ðrr_Þp2 . If the target is not accelerating then the inverse time-to-go is directly proportional to the angular acceleration. Thus estimating the inverse time-to-go is equivalent to estimating the angular acceleration. Consequently, the estimate of the inverse time-to-go will be noisier than the estimates of b as we are estimating an acceleration term from noisy position measurements. 2.2. Radar subsystem The radar provides noisy measurements of the range, r, as well as the bearing, b. The state space model used for the radar measurement filter (the active filter) is nonlinear and is given by the equations xa ðk þ 1Þ ¼ Fxa ðkÞ þ Gwa ðkÞ;
ð7Þ
za ðkÞ ¼ Hðxa ðkÞÞ þ va ðkÞ;
ð8Þ
0
where xa ¼ ½nt n_t mt m_ t and 21 2 2 3 1 T 0 0 T 2 6 T 60 1 0 0 7 6 7 FðkÞ ¼ 6 4 0 0 1 T 5; GðkÞ ¼ 4 0 0 0 0 1 0
3 0 0 7 7 1 2 5: T 2 T
^r_ r
! ¼ g1 ð^ xa Þ ¼
121
ð^ x1 no Þð^ x2 n_o Þ þ ð^ x3 mo Þð^ x4 m_ o Þ x3 mo Þ2 ð^ x1 no Þ2 þ ð^
a
b^a ¼ g2 ð^ xa Þ ¼ tan 1
x^1 no x^3 mo
ð12Þ
!
ð13Þ
:
The error covariance matrix for these estimates of inverse time-to-go and bearing is given by Gð^ xa ÞPa G0 ð^ xa Þ to first order, where Pa is the solution to the Riccati equation for the active filter and ogi : ð14Þ GðxÞ ¼ ox j
^a x
3. Track association logic Given a sequence of state estimates from the active filter, f^ xia ðkÞg, and a sequence from one of the two passive filters, f^ xjp ðkÞg, we need to design a test to select between the hypotheses: H0 : the sequences are from the same target, i.e. i ¼ j. H1 : the sequences are from different targets, i.e. i 6¼ j. ^ia ðkÞ x ^jp ðkÞ, which is an estimate of Define ^ij ðkÞ ¼ x ij i j ðkÞ ¼ xa ðkÞ xp ðkÞ. Then the hypotheses become H0 : E½ij ¼ 0 and H1 : E½ij 6¼ 0. Let Rij ¼ E½ðij ^ij Þðij ^ij Þ0 ; i
j
ij
ð15Þ
ij 0
¼ P þ P P ðP Þ ; ð9Þ
The subscript a indicates that these variables relate to the filter for the active sensor. The non-linear measurement equation is 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðx1 no Þ2 þ ðx3 mo Þ2 5: Hðxa Þ ¼ 4 ð10Þ o tan 1 xx13 n
mo The noise processes fwa ðkÞg and fva ðkÞg are assumed to be independent, white, Gaussian processes with zero mean and variances given by 2 2 ra 1 0 rr 0 Qa ¼ ; Ra ¼ ; ð11Þ 0 r2ba 0 r2a2 respectively where rr is the standard deviation of the range measurement noise and rba is the standard deviation of the bearing measurement noise. ^a , estiFrom the extended Kalman filter estimates, x mates of bearing and inverse time-to-go are obtained by the zero memory non-linear transformations:
i
ð16Þ i 0
ij
^i Þðxi x ^ Þ and P ¼ E½ðxi x ^i Þ where P ¼ E½ðxi x ^j Þ0 [4]. In general, this cross-covariance term will ðxj x not be zero as the state estimate errors will be correlated if the trajectories are from the same target due to the common process noise in each filter. However, as a general rule it is assumed to be zero. Section 7 examines the effect of this assumption. Assuming Gaussian, independent noise processes in each filter, the quantities Pi and Pj can be approximated by the solutions of the Riccati equations in the (extended) Kalman filters used to compute the active and passive state estimates. Given the expressions for the covariance and crosscovariance terms, a test statistic for this hypothesis testing problem is Td ðkÞ ¼
d 1 X
0
1 ð^ ij Þ ðk rÞðRij Þ ^ij ðk rÞ
ð17Þ
r¼0
for some value of the time window length d, where k is the scan number. The distribution of this statistic will be approximately v2 with dnx degrees of freedom where nx is the dimension of the state, x, under the null hypothesis
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H0 . The accuracy of this v2 approximation is examined in Section 8. Three different methods for computing the test statistic will be compared. The first method is a commonly used approach. The remaining methods were proposed in [4]. These make use of MPC in the ESM filter. The three methods are: Method 1: Compare b^a with b^p1 from the first, linear passive filter. Method 2: Compare b^a with b^p2 from the second, MPC filter. ^ ^ Method 3: Compare ½b^a ; ðrr_Þa with ½b^p2 ; ðrr_Þp2 from the MPC filter. In Sections 5–9 the performance and robustness of each of the three methods are shown by simulation of the standard test scenario given in Section 4. These results illustrate why a method that has a high probability of correct association may still be a poor choice if it is not robust against other important criteria.
4. Standard test scenario A number of Monte Carlo studies were carried out to thoroughly evaluate the three track association methods described in Section 3. The same basic scenario was used in most cases. This basic scenario has the own-ship moving in a circle of radius 20 394 m, centred on a point at ()30 000, 29 606) m, with a constant turn rate of 0.2/s. In the case of studies with a single target, target A, the target trajectory is a straight line with the initial conditions nt ð0Þ ¼ 50 000 m;
ð18Þ
n_t ð0Þ ¼ 150 ms 1 ;
ð19Þ
mt ð0Þ ¼ 23 000 m;
ð20Þ
1
ð21Þ
m_ t ð0Þ ¼ 100 ms :
In studies with two targets there is an additional target, target B, also moving in a straight line, with the same velocity, but offset from the track of first target. For this second target the initial position is nt ð0Þ ¼ 50 000 m;
ð22Þ
mt ð0Þ ¼ 19 000 m:
ð23Þ
The sampling rate of both sensors is T ¼ 4 s. The default measurement noise standard deviations are rr ¼ 100 m and rbp ¼ rba ¼ 12 a degree. Note that we are using the same bearing noise variance for the radar and the ESM sensor for simplicity. In general, it is more likely that rba would be less than rbp . In all the examples it is assumed that the sensors are not misaligned. The effect of sensor misalignment, and
the spurious differences in track estimates that result are beyond the scope of this paper. However, we note that in [5] exact expressions for the probability of correct association, PC and incorrect association, PF have been derived when using maximum likelihood estimation (MLE) for this problem. These expressions describe PC and PF as functions of the sensor accuracy and sensor misalignment as well as the size of the batch used to calculate the ML track estimates.
5. Probability of correct association PC Clearly, the first consideration when selecting a track association method is how often will the test correctly associate two tracks from the same target. A method that cannot reliably associate tracks is plainly of little use. There are a number of features which affect the ability of a method to correctly associate tracks. These include: • • • •
the accuracy of the sensors; the accuracy of the individual track estimates; sensor misalignment; and the window length used when calculating the test statistic.
Inaccuracies in the sensors lead to inaccuracies in the tracks which can produce spurious differences in the track estimates. Even though sensor accuracy is not under the control of the designer of a track association method, it is still necessary to consider its effect on the performance of such a method. A particular technique may be very accurate but only when sensor accuracy is high. In a situation where accuracy is low another, more robust method may be a better choice even though it does not perform as well as the first method under ideal conditions. The accuracy of the track estimates is obviously strongly related to sensor accuracy but there are other factors also involved. One such factor is the choice of a state space model for the tracker. In [3], it is shown that for the same input a bearings-only tracker that uses MPC produces more accurate estimates than a simple smoother such as the first passive filter described in Section 2.1. This is due to the way the MPC filter decouples the observable and unobservable components of the system state. This allows the filter to extract the maximum possible information from the input. Another factor affecting track accuracy is the length of the target samples. The longer the target has been tracked, the more accurate the estimates, assuming it does not manoeuvre. Since this assumption is unrealistic, a track association method that quickly produces a definitive answer is to be desired.
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123
Fig. 2. Proportion of tracks associated over time for Scenario 1 (low bearing measurement noise and short test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution.
There are similar considerations when choosing the length of the window for the test statistic. If the motion of the target does not change, relative to the own-ship, then a longer window length will produce a test statistic with a lower variance. Again, however, assuming no manoeuvres are unrealistic, it is desirable to choose a method that is accurate with as short a window length as possible. A number of 1000 Monte Carlo runs were performed to examine the performance of each of the three test statistics when both sensors were tracking the same target. These simulations were based on the standard scenario outlined in Section 4. Four scenarios were considered. These are: S1: The basic scenario with the default measurement noise variances of rr ¼ 100 m and rba ¼ rbp ¼ 12 a degree and a window length of d ¼ 1. S2: The same noise variances, rr ¼ 100 m and rba ¼ rbp ¼ 12 a degree but a window length of d ¼ 3. S3: Increased noise variance in the bearing sensor, rba ¼ rbp ¼ 1 degree, the same range noise variance, rr ¼ 100 m, and a window length of d ¼ 1. S4: The same scenario as in S3 but with a window length of d ¼ 3. For each scenario the three test statistics were computed every five scans and compared to the appropriate 95% point from v2 tables as the test threshold. The cross-covariance term was assumed to be zero. The effect of including an estimate of the cross-covariance is examined in Section 7. The estimate of PC as a function of scan number for each test statistic in each of the four scenarios is shown in Figs. 2–5 below. From these figures it can be seen that
all three methods give a value of PC that is close to the desired value of 95%. Also, once the accuracy of the track estimates is sufficiently high, Method 3, which uses the MPC estimates of both bearing and inverse timeto-go, gives the highest probability of correct association. However, the methods that rely on MPC require higher track accuracy than the simple smoother. This is probably due to the fact that the MPC filter allows the angular acceleration estimate to vary, unlike the simple smoother which assumes this acceleration is zero. Thus the MPC methods yield a more variable estimate of bearing. This is confirmed by Figs. 4 and 5 which are the results for the scenarios with the increased noise in the bearing measurements. In these examples, the MPC methods take longer to converge. These figures suggest that the MPC association methods may not be the best choice in a situation where the accuracy of the sensors is relatively low and the targets are manoeuvring relatively quickly, due to the longer time these methods take to converge. Note, however, that the MPC methods actually work better with a shorter window length. This can be seen by comparing Fig. 2 with Fig. 3 and 4 with Fig. 5. In each pair, the second scenario differs from the first solely by having a longer test window length. Thus for a given degree of sensor accuracy, the vulnerability of the MPCbased methods to unanticipated manoeuvres can be reduced by selecting a minimal test statistic window length.
6. Probability of false association PF If the results of Section 5 were considered alone, one might conclude that the best choice of a track
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Fig. 3. Proportion of tracks associated over time for Scenario 2 (low bearing measurement noise and long test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution.
Fig. 4. Proportion of tracks associated over time for Scenario 3 (high bearing measurement noise and short test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution. The longer convergence time for the MPC-based methods reflects the higher degree of variation in the MPC bearing estimates.
association method for the sample problem was Method 1 and that the MPC-based methods were inferior. However, equally important to the selection of a track association method is the probability that a method will not falsely associate tracks from different targets. This is the ability of the method to discriminate. A good track association method must be just as capable of detecting differences between target tracks as it is able to detect when the tracks are from the same target. However, there is a trade-off between PC and PF . Increasing the value of PC for a particular method inevitably increases the value of PF also. This can be illustrated by considering the case of using a batch MLE
for this problem. In [5] an expression for PF as a function of the test threshold (which in turn is a function of PC ), distance between targets, sensor misalignment and sensor accuracy has been derived. Fig. 6 shows PF against scan number for four different values for PC . This shows that if a very high probability of correct association is required when using an MLE track association method, the track lengths must be long in order to overcome the initially high probability of falsely associating tracks from different targets. However, different track association methods will have different probabilities of incorrect association for a given value of PC . Thus when selecting a track associa-
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Fig. 5. Proportion of tracks associated over time for Scenario 4 (high bearing measurement noise and long test window length). This should be approximately 0.95 as the test statistic should have an approximate v2 distribution. The longer convergence time for the MPC-based methods reflects the higher degree of variation in the MPC bearing estimates.
0.6
0.5
Pfas
0.4
0.3
0.2
0.1
0
60
70
80
90
100
Scan Number
Pc = 0.6 Pc = 0.7 Pc = 0.8 Pc = 0.9
Fig. 6. PF versus scan number as a function of PC .
tion technique, it is desirable to use a method that still retains a relatively high power to discriminate between different targets even when the test threshold is chosen to give a high value of PC . To compare the three methods considered in this paper the same simulations that were used in Section 5 were repeated. However, now the ESM sensor was tracking a second target that was travelling parallel to the first, with a distance of 4000 m between the two. The radar continued to track the first target. Such a result could occur if, for example, only one target was emitting electromagnetic radiation. A more general example
would be when the radar and the ESM sensor both give rise to two tracks which have to be correctly associated. This case will not be considered here. Once again 1000 Monte Carlo runs were performed for each of the four scenarios outlined in Section 5 with the addition of the second target. The test statistics were again compared to the appropriate 95% point from v2 tables. This should yield a theoretical PF ¼ 0:05. Figs. 7– 10 show the results of the simulations. From these figures it can be seen that the first track association method, which uses the simple smoother for the ESM sensor, is not able to discriminate between the two tracks as readily as the MPC-based methods. This is particularly true when the noise bearing standard deviation is relatively high (1 degree). In this case, using the passive smoother on the ESM signal produces an estimated track for the second target which lies within the
3rba limits of the radar track of the first target as both targets get further away from the own-ship. Once this has occurred Method 1 effectively loses any power to discriminate between the tracks. It can also be seen from the figures that all three methods perform increasingly poorly as the number of scans increases. Again, this is a result of the increasing distance between the target and the own-ship as the scenario progresses. As this happens, the size of the target location uncertainty ellipses of both filters increases. Thus they overlap more often yielding an increase in PF . Note that in each case, increasing the window length improves performance. However, recall that from the previous section, this is at the expense of a corresponding reduction in the probability of correct association.
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Fig. 7. Proportion of tracks associated over time for Scenario 1 (low bearing measurement noise and short test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution.
Fig. 8. Proportion of tracks associated over time for Scenario 2 (low bearing measurement noise and long test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution.
The results of this section show that any conclusion reached by considering only the results in the previous section, Section 5, would, in fact, be erroneous. These results implied that the first method was preferable to those that relied on the MPC filter. The test scenario for each of the four figures in this section matches those in the previous section (e.g. Fig. 2 matches 7, 3 matches 8, and so on). For each pair, the test statistic threshold is the same, as are the window length and sensor accuracy. By comparing each pair of figures, it can be seen that the MPC-based methods retain a high discriminatory power even at high values of PC , while Method 1 does not. In fact, only the two MPC-based methods are capable of operating at close to the theoretical performance figures for both PC and PF making them a better choice than the first method.
Figs. 11 and 12 show the percentage of tracks falsely associated as the separation between the two target varies when rb ¼ 1 degree. These results confirm the greater discriminatory power of the MPC-based association methods as they can accurately distinguish between more closely spaced targets than Method 1. As the targets do not manoeuvre during the scenario, the power of all three association tests is increased as the window length increases.
7. Calculation of the cross-covariance It has been shown that the performance of trackto-track association methods may be improved by including the effect of the cross-covariance between the
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Fig. 9. Proportion of tracks associated over time for Scenario 3 (high bearing measurement noise and short test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution. The divergence over time is a result of the increased uncertainty in the bearing estimate as the distance to the targets increases.
0.6 Method 1 Method 2 Method 3
Proportion of Track Associated
0.5
0.4
0.3
0.2
0.1
0 40
60
80
100
120 Scan
140
160
180
200
Fig. 10. Proportion of tracks associated over time for Scenario 4 (high bearing measurement noise and long test window length) when two targets are present. This should be approximately 0.05 as the test statistic should have an approximate v2 distribution. The divergence over time is a result of the increased uncertainty in the bearing estimate as the distance to the targets increases.
two track estimates when they are of the same target. Results for sensors of the same type are given in [2]. This was extended in [10] to the case when the sensor measurements, are dissimilar. More recently, [8] dealt with the case when the state spaces of the two filters, x1 and x2 are dissimilar. The scenario under consideration in this paper has dissimilar state spaces. Briefly, suppose there exists a one-to-one transformation between x1 and x2 or a transformation that can be constrained to be one-to-one using additional (perhaps physical) knowledge of the problem. That is, there exist g : Rn ! Rm and g 1 : Rm ! Rn such that gðx1 Þ ¼ x2 and g 1 ðx2 Þ ¼ x1 . Given such a transforma-
tion g it was shown in [8] that it was possible to write the state space model for the second filter in terms of the state vector of the first model, x1 . This can be done for both linear and non-linear models. For the problem being examined here x2 represents the state vector of the radar filter, xa , while x1 represents the state vector of either passive filter, xp1 or xp2 . Using this result, the filter for each sensor can be constructed using whichever state space model is most appropriate for that sensor, even if this means using different state spaces for different sensors. When tracks from two sensors are to be combined, the estimates from the second filter can be converted to the state space of
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Fig. 11. Proportion of tracks falsely associated as a function of the distance between the two targets. Test statistic window length ¼ 1.
Fig. 12. Proportion of tracks falsely associated as a function of the distance between the two targets. Test statistic window length ¼ 3.
the first filter using the transformation g 1 . As a result of this transformation, a recursion for the cross-covariance between the two filters can be calculated. To test the effect of including the cross-covariance term modified versions of the standard scenario given in Section 4 was used in two ways. In the first case, only one target was present, target A. The probability of correctly associating the tracks from each sensor was calculated, both with and without including the crosscovariance term. The results for the first association method are shown in Fig. 13. From this figure it can be seen that including the estimated cross-covariance term actually reduced the performance of the first association method. In general, the first method associated the tracks only 85% of the time when the cross-covariance
term was included rather than the expected 95% success rate. This result indicates that the test threshold was too high. The results for the two MPC-based methods are not shown as including the cross-covariance term did not make a significant difference in their performance. In the second test both targets A and B were present. However, in this case the second target’s initial position was ð40 000; 8000Þ m, i.e., it was offset from the first target in both dimensions. The radar track estimate for target A was compared to the ESM track for target B. As before, the three test statistics were calculated, both with and without the cross-covariance term. Once again, the results for the two MPC-based methods (Methods 2 and 3) did not change significantly. The results for the first method are shown in Fig. 14. The tracks should
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Fig. 13. Proportion of tracks associated over time. This should be approximately 0.95 if the test statistic has a v2 distribution. Results shown for first passive filter only.
have been incorrectly associated approximately 5% of the time. From the figure it can be seen that this method does not achieve this degree of accuracy. The spikes in the proportion of tracks associated correspond to significant changes in the bearing-rate. The state space model for the ESM filter using Method 1 assumes a nearly constant bearing-rate and there is clearly a drop in performance when this assumption is violated. The performance does not improve until the ESM filter has had time to adjust to the change in bearing-rate. However, from the figure it can also be seen that including the cross-covariance term does reduce the false association rate of this method. However, this improvement in performance could simply be the result of the fact that
including the cross-covariance term meant that the actual distribution of the test statistic was less like a v2 distribution than before. This possibility is considered in more detail in the following section. From these results it can be seen that the MPC-based methods are robust to deviations from the (false) assumption that the cross-covariance, P ij , is zero. This makes these methods a more desirable choice for practical use as it means their implementation can be kept relatively simple. In contrast, the association method based on the simple smoother is not robust to the common, if false, assumption that P ij is zero. While its performance can be improved by the inclusion of P ij , this requires careful selection of the test statistic
Fig. 14. Proportion of tracks falsely associated over time. This should be approximately 0.05 as the test statistic should have a v2 distribution. The spikes correspond to significant changes in the true bearing-rate. Results shown for first passive filter only which assumes a constant bearing-rate.
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threshold. This once again confirms that selecting a method by considering PC alone may result in a poor choice for a track association method.
8. Test statistic threshold All the results discussed above have made the standard assumption that the test statistic, Td , has a v2 distribution under the null hypothesis, that is, when the two tracks are from the same target. This assumption is routine, however, as the results of including an estimate of cross-covariance shown in the previous section, it is not always an accurate one. If the threshold chosen is too high, then the probability of correct association, PC , will be less than desired. On the other hand, if the threshold is set too low then the number of falsely associated tracks will be higher than anticipated. Thus another feature to consider when selecting an association method is the ease with which an appropriate threshold can be selected that will result in the association technique having a performance, in practice, that is close to its theoretical performance. When using MLE batch estimation for the airborne sensor problem, an analytical expression for PC as a function of k, the test statistic threshold, can be derived as was shown in [5]. The value of PC , when the sensors are not misaligned and are tracking the same target, is given by the equation k k PC ¼ 1 1 þ exp
: ð24Þ 2 2 There is no closed form expression for k as a function of PC but the equation can be inverted numerically. Thus the selection of an appropriate test threshold for an MLE batch association method is straightforward. However, this is not the case for recursive estimators of the type considered in this paper. Instead, it is common to assume that the v2 distribution is a sufficiently good approximation to the actual distribution of such test statistics. Simulations were used to investigate the validity of the v2 approximation for the recursive estimation case. These used the basic scenario described in Section 4. The value of the test statistic for each of the three association methods was calculated at the end of the 500 scans for 1000 Monte Carlo runs. This produced a sample distribution for the test statistic for each method. This sample distribution was produced for each method with window lengths of 1, 2 and 3. The results are shown in Tables 1–3 along with the corresponding point from v2 tables for comparison. From these tables we can see that the assumption that the test statistics have a v2 distributions is, fortunately, most accurate around the 90–95% range. Also, the approximation improves as the window length increases.
Table 1 Test statistic threshold for a window length of 1 75%
90%
95%
99%
Method 1 Method 2 v2
1.05 1.11 1.32
2.22 2.12 2.71
3.01 3.03 3.84
5.59 4.67 6.64
Method 3 v2
1.42 2.77
2.56 4.61
3.60 5.99
5.27 9.21
Table 2 Test statistic threshold for a window length of 2 75%
90%
95%
Method 1 Method 2 v2
2.12 2.19 2.77
3.81 4.36 4.61
5.48 6.05 5.99
99% 9.47 9.98 9.21
Method 3 v2
2.88 5.39
5.29 7.78
6.97 9.49
10.90 13.28
Table 3 Test statistic threshold for a window length of 3 75%
90%
95%
99%
Method 1 Method 2 v2
3.16 3.42 4.11
5.60 6.34 6.25
7.62 8.90 7.82
11.80 14.20 11.34
Method 3 v2
4.48 7.84
7.52 10.64
10.10 12.59
16.10 16.81
However, in general, the sample threshold is below that predicted by v2 tables. This is to be expected given the results in previous sections. This indicates that using v2 tables to select a test threshold may result in a fusion process that falsely associates tracks more often than was desired. Consider the results of Section 7. Here the number of tracks associated when the cross-covariance term was included for Method 1, reduced both the correct and false association rates when the value of the test statistic was compared to a v2 threshold. This implies including the cross-covariance term reduces the accuracy of the v2 approximation as the v2 -based threshold is clearly too high. By contrast, the MPC-based methods, particularly Method 2, are more closely approximated by the v2 distribution as can be seen by Tables 1–3. These results, combined with those in Sections 5 and 6, show that it is easier to select a test statistic threshold for the MPCbased methods that will result in a track association technique that has an actual performance that closely matches its theoretical performance.
9. Effect of sensor resolution A complete model of the resolution capacity of a given sensor is very difficult. However, the probability,
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PR , that a sensor resolves two closely spaced targets will clearly depend on the distance between the targets in addition to other features. In [7], a simple but intuitively appealing model was proposed for PR . From [7] PR can be modelled as a Gaussian-like function 1 0 PR ¼ 1 exp ðy1 y2 Þ D 1 ðy1 y2 Þ ; ð25Þ 2 where yi is the noiseless measurements of the ith target and D is a positive definite, symmetric matrix that describes the extension and spatial orientation of the sensor resolution cells. In general, D will depend on the target–sensor geometry. Following [7], we will assume that the range resolution ara of the radar sensor is essentially determined by the length of the emitted pulse. The angular resolution of both the radar and ESM sensors, aba and abp , is limited by the beamwidth of each sensor. Also, we will assume that the probability of unresolved targets is decreasing for increasing distances ðdr; dbÞ between the targets. For the radar sensor, assuming that the range resolution is independent of the angular resolution, "1 # 0 ara
1 D ¼ ; ð26Þ 0 a1b a
while for the ESM sensor, D 1 ¼ 1=abp . Suppose that there are two targets but a sensor fails to resolve them. Then, instead of measuring y1 or y2 , the sensor measures the mean target position 1 ym ¼ ðy1 þ y2 Þ: 2
ð27Þ
In effect, the sensor measures the position of a third, synthetic target that is midway between the two real targets.
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Recall from Section 3, assuming that both sensors resolved the targets, under the null hypothesis the expected value of the test statistic Td was zero. Under H1 (the two tracks are from different targets) the expected value of the test statistic will be some non-zero value, say s. Suppose now, one sensor fails to resolve the targets, then one track will be a track of a true target while the other is of the third, synthetic target. From (27) and (17) it can be shown that under these circumstances the expected value of the test statistic under H1 will only be 1 s. This reduction in E½Td increases the probability of 4 falsely associating the two tracks. This effect was examined in more detail in [8]. To test the effects of sensor resolution the standard test scenario, given above, was modified to include a range resolution for the radar of ar ¼ 100 m and an angular resolution of both sensors of ab ¼ 2 degrees. In addition, the initial position of the second target, target B, was changed to nt ð0Þ ¼ 40 000 m;
ð28Þ
mt ð0Þ ¼ 8000 m:
ð29Þ
In this scenario, the radar was able to resolve both targets at all times. However, the ESM sensor could only resolve the targets initially. As the two targets moved away from the own-ship, the difference in the bearing measurements eventually became too small. Fig. 15 shows the probability of resolution over time for the ESM sensor. The probability does not decay monotonically as the target moves away from the sensor platform, but instead displays some periodicity due to the circular motion of the own-ship. Two cases were examined. In the first case, both sensors were initially tracking target A. After scan 230, the ESM sensor could no longer resolve targets A and B and began to return the track of the third, synthetic
Fig. 15. Probability of resolution for the ESM sensor. This decreases as the targets move further away from the sensor.
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Fig. 16. Proportion of tracks associated over time. The tracks should associate up to scan 230 as the ESM sensor has a better than 50% chance of resolving the targets up to this time.
target midway between A and B. The calculation of the test statistics did not take into account the possibility that a sensor may have failed to resolve the targets. The results are shown in Fig. 16. Until scan 230 the tracks are from the same target, and this is detected by all three methods. Immediately after resolution is lost, all methods stop associating the tracks, but only the methods using MPC in the passive filter (Methods 2 and 3) continue to recognise that the sensors are tracking different targets after scan 230. The simple passive smoother of the bearing (Method 1) is unable to deal with the step change in the bearing of the ESM measurements when the sensor loses resolution. It can only put this change down to impulsive noise in the system and con-
tinues to erroneously associate the tracks, that are now from two different targets, a high proportion of the time. The second case examined resolution effects on the probability of incorrect association PF . This case was tested in a similar manner as above, except that in this situation the ESM sensor was initially tracking target B. The results are shown in Fig. 17. Once again it can be seen that while all three methods give acceptable performance when the sensors are able to resolve the targets, that is, all methods recognise that the tracks are from different targets at a rate that is close to their desired theoretical performance rate. However, only Methods 2 and 3 continue to operate at close to this performance rate once resolution is lost.
Fig. 17. Proportion of tracks falsely associated over time. This increases as the probability of resolution decreases.
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Once again, the results of this section show that considering more features than just the probability of correct association can lead to the selection of a different track association method than the one that may have been selected when considering PC alone. In Section 5, if the results of that section are considered in isolation, then the first track association method is superior to the MPC-based methods. However, as it can be seen from Figs. 16 and 17, this conclusion is not the case when the effect of the loss of sensor resolution is considered. Only the MPC-based methods were able to cope with the loss of resolution and continue to associate tracks from the same target and discriminate between those from different targets.
10. Conclusions This paper has examined a number of features that need to be considered when selecting a track-to-track association method. It has been demonstrated that it is not sufficient to simply consider the probability of correct association given by a method. Other features such as the power of the method to distinguish between different tracks; the ability to easily select the appropriate test threshold; and the effects of the loss of sensor resolution can have a significant effect on the performance of a track association method. It was argued that all these features must be considered together when selecting a track association method. If this is not done, then the combined tracks produced by a fusion process that uses a poorly chosen track association technique may be less accurate than the original, unfused tracks. The effect of all the features discussed in this paper was illustrated by considering the problem of trackto-track association for dissimilar airborne sensors. It was seen that, for this problem, filters using MPC were more robust than another, more common technique. While the MPC-based techniques did not provide as
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high a value of PC as the more common technique, they did perform well when all the five criteria discussed in the paper were examined. In fact, this paper has shown that the method that provides the best probability of correct association is the least robust and actually performs more poorly than the MPC-based methods when measured against other, equally important criteria, of track association performance.
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