Previously on TT, Target Tracking: Lecture 2 Single Target Tracking Issues. Lecture-2 Outline. Basic ideas on track life

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1 REGLERTEKNIK Previously on TT, AUTOMATIC CONTROL Target Tracing: Lecture 2 Single Target Tracing Issues Emre Özan emre@isy.liu.se Division of Automatic Control Department of Electrical Engineering Linöping University Linöping, Sweden Fundamentals of Target Tracing: Sensor models Measurement models Target models Structure of a basic tracer October 29, 2014 E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 Lecture-2 Outline Basic ideas on trac life Trac life Gating M/N logic based trac life Score based trac life Confirmed trac deletion Single Target Data Association Nearest neighbor (NN) Probabilistic data association (PDA) False alarms and clutter complicate target tracing even if we now that there is at most one target. Trac initiation: Form tentative tracs and confirm only the persistent ones (ones getting data). Trac maintenance: Feed the confirmed tracs with only relevant data. Closely related to data association. Measure quality if possible. Trac deletion: Delete tracs with persistent absence of data or with low quality. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

2 Curse of combinations! We cannot consider all target-measurement combinations when the number of involved quantities is 10. The combinatorics and related probabilities becomes trouble in this case (even if most of the probabilities turn out to be almost zero on machine precision). Combinations If there are 10 targets, 100 measurements, P D = 1, and if one can compute 1 million associations per second; It taes about 2 million years to compute all possible association hypothesis. Gates and Gating Gating is the hard decisions made about which measurements are considered as valid (feasible) measurements for a target. The region in the measurement space that the feasible measurements for a target is allowed to be is called as target s gate. E. Özan Target Tracing October 29, / 31 Gating Illustration y y y 1 2 ŷ 1 3 y 3 y y 4 5 ŷ 1 1 ŷ 1 2 y 7 y 6 x ŷ 1 i : Predicted measurement position for the ith target y j : jth measurement E. Özan Target Tracing October 29, / 31 Gating Choices Ellipsoidal Gating Demystified When the target state and measurement models are correct Many different choices are possible which are basically a form of distance measuring between the predicted measurement and the measurements depending on the target uncertainty. Rectangular (low computation) y x ŷx 1 κσx 1 y y ŷy 1 κσy 1 where κ is generally 3. Ellipsoidal (most common) Suppose S 1 = U U T where ỹ T S 1 ỹ y ŷ 1 N (0 ny, S 1 ) where U is invertible. Then, 1ỹ = ỹ T U T U 1 ỹ = U 1 ỹ 2 2 u 2 2 u U 1 ỹ N (0 nx, I nx ) (y ŷ 1 ) T S 1 1 (y ŷ 1 ) γ G where γ G is the gate threshold. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

3 Ellipsoidal Gating Demystified Ellipsoidal Gating Demystified When the target state and measurement models are correct ỹ y ŷ 1 N (0 ny, S 1 ) 1 Gate statistic ỹ T S 1 1ỹ is χ 2 n y -distributed. 1 Suppose S 1 = U U T ỹ T S 1 where U is invertible. Then, 1ỹ = ỹ T U T U 1 ỹ = U 1 ỹ 2 2 u γg = 4.7 n=1 n=2 n=3 n=4 n= PG = 0.9 where u U 1 ỹ N (0 nx, I nx ) n=1 n=2 n=3 n=4 n=5 Chi-square distribution Sum of squares of n i.i.d. scalars distributed with N (0, 1) is χ 2 -distributed with degrees of freedom n chi2pdf(.,n) chi2cdf(.,n) E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 Ellipsoidal Gating Demystified Gating at = 1 Gate statistic ỹ T S 1 1ỹ is χ 2 n y -distributed γg = 4.7 n=1 n=2 n=3 n=4 n= chi2pdf(.,n) PG = 0.9 n=1 n=2 n=3 n=4 n= chi2cdf(.,n) Gating should be possible after getting the first measurement y 0 at = 0. In other words, at = 1, the gating should be applied on y 1. For this, ŷ 1 0 and S 1 0 should be calculated. With a single measurement, it might generally not be possible to form a proper state covariance P 0 0. Use measurement covariance and prior info about the target to form an initial state covariance P 0 0. Magic Command Use γ G =chi2inv(p G,n y ) to get your unitless threshold. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

4 Gating at = 1 Gating at = 1 Example-1: Cartesian state, position only measurements: x [ x y v x v y ] T ] T y [ ] T [ x y + e x e y where e x N (0, σ2 x) and e y N (0, σ2 y). Suppose we got y 0, what would be ˆx 0 0 and P 0 0? ˆx 0 0 = [ y0 x y y ] T P 0 0 = diag ( σx, 2 σy, 2 (vmax/κ) x 2, (vmax/κ) y 2) This is called single point initiation [Mallic (2008)] in the literature. Example-1: An alternative is Do the first gating with y 1 y 0 T vmax κ. When the gate is satisfied, form the state estimate and covariance as [ ˆx 1 1 = P 1 1 = y x 1 y y 1 y x 1 yx 0 T y y 1 yy 0 T ] T σ 2 x 0 σ 2 x/t 0 0 σ 2 y 0 σ 2 y/t σ 2 x/t 0 2σ 2 x/t σ 2 y/t 0 2σ 2 y/t 2 This method is called as two-point difference initiation in the literature [Mallic (2008)]. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 Trac Initiation & Deletion: M/N logic Trac Initiation & Deletion: M/N logic Example: 2/2&2/3 logic Tentative trac (initiator) Deleted initiator Confirmed initiator Measurement in gate No measurement in gate Simple counting based low-computation trac initiation. Trac (for not tentative but confirmed ones) deletion happens in the case of a predefined consecutive number of misses (no measurement in the gate). Independent of the trac quality as long as gate criteria is satisfied. Position These two tracs are equivalent according to M/N logic. tim E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

5 Score based approach (SPRT) Score based approach (SPRT) Suppose we have a measurement sequence y 0: (y i can be φ empty set.). Consider two hypotheses about it. H 0 : H 1 : All measurements originate from F A All measurements originate from a single NT Define the score (denoted as LPR meaning log probability ratio) of a (possibly tentative) trac (or target) as LPR = log P (H 1 y 0: ) P (H 0 y 0: ) Since P (H 0 y 0: ) = 1 P (H 1 y 0: ), we have P (H 1 y 0: ) = exp(lpr ) 1 + exp(lpr ) How to calculate the score: With the initial data y 0 φ (first data of an initiator) LPR 0 = log p(y 0 H 1 )P (H 1 ) p(y 0 H 0 )P (H 0 ) = log β NT β F A + C where β NT and β F A are new target and false alarms rates respectively. P D is the detection probability. Setting LPR 0 = 0 is also a common method. With new measurement y φ LPR =LPR 1 + log p(y y 0: 1, H 1 ) p(y y 0: 1, H 0 ) = LPR 1 + log P Dp 1 (y ) β F A where p 1 (y ) N (y ; ŷ 1, S 1 ) is the innovation (measurement prediction) lielihood. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 Score based approach (SPRT) Score based approach (SPRT) How to calculate the score: With missing (no) measurement i.e., y = φ LPR =LPR 1 + log(1 P D P G ) Maing decisions based on LPR Design parameters: P F C The probability with which you can tolerate your tracer accept (confirm) false tracs. i.e., False trac confirmation probability. P T M The probability with which you can tolerate your tracer delete (reject) true tracs. i.e., True trac miss probability. Maing decisions based on LPR Finding thresholds for LP R γ high = log 1 P T M Probability of accepting a true trac = log P F C Probability of accepting a false trac P T M Probability of rejecting a true trac γ low = log = log 1 P F C Probability of rejecting a false trac Decision mechanism: LP R γ high : Confirm the tentative trac. i.e., Accept H 1. LP R γ low : Delete the tentative trac. i.e., Accept H 0. γ low < LP R < γ high : Not enough evidence, continue testing, i.e., leave the tentative trac as it is. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

6 How to Delete Confirmed Tracs: M/N-logic case How to Delete Confirmed Tracs: Score case Confirmed tracs are, most of the, deleted if they do not get any gated measurements for a while. In a scenario where one uses M/N-logic for trac initiation, one can delete tracs if the trac is not updated with a gated measurement for N D scans. Another M/N logic can also be constructed also for confirmed trac deletion. Example: 2/2&2/3 trac deletion logic: Delete the confirmed trac if the trac is not updated for two consecutive scans and not updated in at least 2 of the three following scans. When KF s are not updated for several scans, the innovation covariances (determining the gate size) also become larger. A chec on the gate size can also be useful in deletion. Score calculation also taes into account the suitability of the target behavior to the model used for target state dynamics. If the target gets still some measurements inside the gate but the incoming measurements are quite wrongly predicted, a score based criterion can still be used for deleting confirmed tracs. Problem is that absolute score value becomes highly biased towards previous good measurements. It taes forever to delete a target that has long been traced. Either use a sliding window or a fading memory LP R = i= N+1 LP R i LP R = αlp R 1 + LP R with a careful threshold selection (note that α < 1). E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 How to Delete Confirmed Tracs: Score case Higher level trac initiation logic What Blacman proposes in the boo is to use the decrease from the maximum value reached. With this method: Thin of an hypothetical event that you would lie to delete the target e.g. 5 consecutive no measurements in the gate events and calculate the decrease in the score that would happen D LP R = 5 log(1 P G P D ) (a negative quantity) Keep the maximum values reached by the scores of every trac during the tracs life. Call this value Delete the trac j if M j LP R, = max 0 i 1 LP Rj i LP R j < M j LP R, + D LP R Now that we now how to process a sequence of measurements y 0: to decide whether it is a trac or not. Generating such sequence of measurements is the job of a higher level processing. This level requires some data abstraction for the illustration of an example algorithm. We consider objects called Initiator each one of them corresponding to tentative tracs and eeping some level of its own history. An example can be Initiator: Age Last update State estimate State covariance M/N logic state Score E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

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9 I 5 I 5 Higher level logic: Verbal Description Higher level logic: Implementation Advice Time = 0: Initialize tentative tracs from all measurements. Time > 0: Gate the measurements with the current initiators. Mae a simple association of gated measurements to the current initiators. Notice that with the tentative tracs, the association problem is not as critical as it is in confirmed tracs. Process the current initiators with their associated measurements. Update their inematic estimates, covariances, M/N-logic states, or scores etc.. Start new initiators from un-used measurements. Chec updated M/N-logic states or scores and confirm, delete those initiators (tentative tracs) satisfying the related criteria. In the implementation of exercises, you can use whatever type of programming you lie. The following are general advice if you want some. Try to be as object oriented as possible. Define an initiator (or tentative) object (struct or class). Write the following functions Initator=Initializer(InitiatingMeasurement) (constructor) GateDecisions=Gating(InitiatorArray,MeasurementSet) AssociationDecisions=Associate(InitiatorArray......,MeasurementSet,GateDecisions) UpdatedInitiatorArray=Update(InitiatorArray......,MeasurementSet,AssociationDecisions) flags=checconfirm(initiatorarray) flags=checdelete(initiatorarray) E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

10 Single Target Data Association: NN Single Target Data Association: NN Nearest neighbor association rule: Choose the closest measurement to the measurement prediction in the gate in the sense that i = arg where m G gate. min 1 i m G (y i ŷ 1) T S 1 1 (yi ŷ 1) is the number of measurement in the y 5 y 1 y 3 y 2 ŷ 1 The target is updated with the selected measurement and the measurement is not further processed by either initiators or other targets (if any). The measurements outside the gate are certainly sent to higher level initiator logic to be processed. y 4 Other measurements in the gate: If there are other measurements in the gate than the one selected by the NN rule, their transfer to the initiation logic is application dependent. If it is nown that there might be closely spaced targets, and the sensor reports are accurate, then they can be sent to the higher initiation logic to be further processed. i.e., to be started with tentative tracs. If the sensor reports are noisy, or the targets are extended with respect to sensor resolution, somes people choose to forget about them forever. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31 Single Target Data Association: PDA Single Target Data Association: PDA Probabilistic data association: NN is a hard decision mechanism. Soft version of it is to not mae a hard decision but use all of the measurements in the gate to the extent that they suit the prediction. in the gate are shown as Y = {y i }m i=1. We have the following hypotheses about these measurements θ 0 ={All of Y is FA i.e., no target originated measurement in the gate.} θ i ={Measurement y i belongs to target, all the rest are FA.} for i = 1,..., m. Then, the estimated density p(x Y 0: ) can be calculated using total probability theorem as m p(x Y 0: ) = p(x θ i, Y 0: ) p(θ i Y 0: ) }{{} i=0 µ i y 5 y 1 y 3 y 2 ŷ 1 y 4 p(x θ i, Y 0: ) = { p(x Y 0: 1 ), i = 0 p(x Y 0: 1, y i ), otherwise In the special case of a Kalman filter { ˆx i = ˆx 1 i = 0 ˆx 1 + K (y i ŷ 1), otherwise { Σ i = Σ 1 i = 0 Σ 1 K S 1 K T, otherwise Note that the quantities Σ i and K, are the same for i = 1,..., m. E. Özan Target Tracing October 29, / 31 E. Özan Target Tracing October 29, / 31

11 Single Target Data Association: PDA References In the KF case, the overall state estimate ˆx can be calculated as where m ˆx = µ i ˆxi = ˆx 1 + K (y eq i=0 = µ0ŷ 1 + y eq m µ i yi i=1 is the equivalent measurement. The covariance Σ corresponding to ˆx is given by m µ i i=0 Σ = ŷ 1) [Σ i + (ˆxi ˆx )(ˆx i ˆx ) T ] Calculation of the probabilities µ i is quite involved. E. Özan Target Tracing October 29, / 31 Y. Bar-Shalom, Dimensionless score function for multiple hypothesis tracing IEEE Transactions on Aerospace and Electronic Systems, vol.43, no.1, pp , Jan S. Blacman and R. Popoli, Design and Analysis of Modern Tracing Systems. Norwood, MA: Artech House, Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracing: Principles, Techniques. Storrs, CT: YBS Publishing, M. Mallic, B. La Scala, Comparison of single-point and two-point difference trac initiation algorithms using position measurements, Acta Automatica Sinica, vol.34, no.3, pp , Mar E. Özan Target Tracing October 29, / 31