Lecture Outline. Target Tracking: Lecture 3 Maneuvering Target Tracking Issues. Maneuver Illustration. Maneuver Illustration. Maneuver Detection

Transcription

1 REGLERTEKNIK Lecture Outline AUTOMATIC CONTROL Target Tracking: Lecture 3 Maneuvering Target Tracking Issues Maneuver Detection Emre Özkan emre@isy.liu.se Division of Automatic Control Department of Electrical Engineering Linköping University Linköping, Sweden Multiple Model Approaches Non-switching multiple models Switching multiple models Generalized pseudo Bayesian (GPB) methods Interacting multiple model (IMM) algorithm November 2, 24 E. Özkan Target Tracking November 2, 24 / 23 E. Özkan Target Tracking November 2, 24 2 / 23 Maneuver Illustration Maneuver Illustration A simple illustration of the maneuver problem with simplistic parameters. P D =. P G = P F A = KF with CV model We try different process noise standard deviations σ a =.,, m/s Target A simple illustration of the maneuver problem with simplistic parameters. P D =. P G = P F A = KF with CV model We try different process noise standard deviations σ a =.,, m/s Target E. Özkan Target Tracking November 2, 24 3 / 23 E. Özkan Target Tracking November 2, 24 3 / 23

2 Maneuver Illustration Maneuver Illustration 22 σa =.m/s 2 22 σa = m/s 2 22 σa = m/s 2 22 Gates σa =.m/s 2 22 Gates σa = m/s 2 22 Gates σa = m/s E. Özkan Target Tracking November 2, 24 4 / 23 E. Özkan Target Tracking November 2, 24 4 / 23 Maneuvers Maneuver Detection: Low-Pass High-Pass Normalized innovation square again comes into the picture. Maneuvers are the model mismatch problems in tracking. Using a high order kinematic model that allows versatile tracking is not a solution in the case where data origin uncertainty is present. Instead it makes the gates unnecessarily large and makes susceptible to clutter. Hence maneuvers should be detected and compensated. A maneuver should be detected both when the switches to a higher order model than we use in our KF, and when it switches to a lower order model than we use in the KF. We know that ɛỹk χ 2 n y. ɛỹk = ỹ T k S k k ỹk This is also the gating statistics. So we should check this quantity in a window to avoid false alarms. Use a sliding window or a recursive forgetting. ɛ s k = k i=k N+ ɛỹi or ɛ r k = αɛr k + ɛ ỹ k where α <. E. Özkan Target Tracking November 2, 24 5 / 23 E. Özkan Target Tracking November 2, 24 6 / 23

3 Maneuver Detection: Low-Pass High-Pass Maneuver Detection: Low-Pass High-Pass Use one of the statistics ɛ s k = k i=k N+ ɛỹi or ɛ r k = αɛr k + ɛ ỹ k In the case of perfect model match, we have ɛ s k χ2 Nn y and ɛ r k χ2 α ny where the second distribution is an approximation at the steady state (effective window length α ). A maneuver is declared when the maneuver statistics ɛ k exceeds a threshold ɛ max. The threshold ɛ max is adjusted such that in the case of no maneuver P (ɛ k ɛ max ) = PF maneuver A }{{} During a low-pass to high-pass transition detected from an ɛ k that is obtained by summing ɛỹi over a window of length N (or effective window length α ), there accumulates considerable amount of error in the estimates. These should be compensated when such a detection happens. Generally last estimates in the (effective) window are recalculated. For this purpose, some previous history of estimates and measurements are kept in memory. E. Özkan Target Tracking November 2, 24 7 / 23 E. Özkan Target Tracking November 2, 24 8 / 23 Maneuver Detection: High-Pass Low-Pass The decision in the reverse direction (from high-pass to low-pass) can be given with the same statistics if the statistics ɛ k gets lower than a threshold ɛ min. The threshold ɛ min is adjusted such that in the case of correct model P (ɛ k ɛ min ) = P maneuver miss Maneuver Detection: High-Pass Low-Pass The decision in the reverse direction (from high-pass to low-pass) can be given with the same statistics if the statistics ɛ k gets lower than a threshold ɛ min. The threshold ɛ min is adjusted such that in the case of correct model P (ɛ k ɛ min ) = P maneuver miss PFA = Pmiss =.5 pdf of χ 2 4 cdf of χ ɛmin = 4 ɛmax = 8 ɛmin = 4 ɛmax = 8 E. Özkan Target Tracking November 2, 24 9 / ɛk ɛk chi2pdf(.,n) chi2cdf(.,n) E. Özkan Target Tracking November 2, 24 9 / 23

4 Multiple Model Approaches Multiple Model Approaches: JMLS Target motions can generally be classified into a number of predefined number of modes e.g. Constant Coordinated turn (circular motion with constant speed and angular rate) Constant acceleration Using maneuver detection is a type of making a hard decision between these models i.e., serial use of models (use one model first then switch to another one etc.). The soft version uses all the models at the same (parallel use of models) and combines their results to the extent that they suit to the measurements collected probabilistically. Jump Markov linear systems (JMLS): give a useful framework for using multiple models x k =A(r k )x k + B(r k )w k y k =C(r k )x k + D(r k )v k x k is the state that we would like estimate from y k. This state is called as base state. r k {, 2,..., N r } represents model number and is called as mode (or modal) state. Note that r k is also unknown and must be estimated from measurements y k. A( ), B( ), C( ) and D( ) are mode dependent parameters. E. Özkan Target Tracking November 2, 24 / 23 E. Özkan Target Tracking November 2, 24 / 23 Multiple Model Approaches: JMLS Multiple Model Approaches: Optimal Solution Multiple model approaches can be classified into two broad categories as Non-switching models Switching models Non-Switching case: The underlying model r k is unknown but fixed for all s, i.e., r k = r, k =, 2,.... This type of approaches is useful in system identification with finite number of model alternatives but not very suitable for TT. Switching case: The underlying model r k can jump between different values in {, 2,..., N r } The behavior of r k is generally modeled as first order homogeneous Markov chain with a fixed transition probability matrix. Suppose we started estimation at and now we are at k. There are a total of Nr k different model histories r :k that might have occurred in this period. We show these by {r i :k }N k r i=. When a specific model history r:k i is given we can calculate the estimated density of the state x k as p(x k y :k, r i :k ) = N (x k; ˆx i k k, Σi k k ) which is given by a KF that is matched to the model history. The overall MMSE estimate ˆx k k is then given as where µ i k P (ri :k y :k). N k r ˆx k k = µ i k ˆxi k k i= E. Özkan Target Tracking November 2, 24 2 / 23 E. Özkan Target Tracking November 2, 24 3 / 23

5 Multiple Model Approaches: Optimal Solution Multiple Model Approaches: Optimal Solution r 3 = ˆx 3 3 Storage and computation requirements of the optimal increase exponentially. ˆx r = r = 2 ˆx ˆx 2 r 2 = r 2 = 2 r 2 = ˆx ˆx 2 ˆx 3 ˆx 4 r 3 = 2 r 3 = r 3 = 2 r 3 = r 3 = 2 r 3 = r3 = 2 ˆx ˆx ˆx ˆx ˆx ˆx The posterior density of the state at k is given as N k r p(x k y :k ) = µ i k N (x k; ˆx i k k, Σi k k ) i= The number of components in the Gaussian mixture should be decreased. Some approaches use pruning (discarding low probability terms) periodically. ˆx We here will consider the most popular approach merging. 2 3 E. Özkan Target Tracking November 2, 24 4 / 23 E. Özkan Target Tracking November 2, 24 5 / 23 Multiple Model Approaches: Mixture Reduction Multiple Model Approaches: GPB Generalized pseudo Bayesian algorithms (GPB) The Gaussian mixture given by can be approximated as ˆx k N π iˆx i k, i= p(x k ) = N π i N (x k ; ˆx i k, Σi k ) i= p(x k ) N (x k ; ˆx k, Σ k ), Σ k where, N [ π i Σ i k + (ˆx i k ˆx k)(ˆx i k ˆx k) T ] i= This is a moment matching approximation and called as merging. The second term in the covariance approximation (brackets) is called as the spread of the means. The above choice for the mean and the covariance minimizes the Kullback-Leibler divergence between the original mixture and its approximation. E. Özkan Target Tracking November 2, 24 6 / 23 GPB Approximation: p(x k y :k ) N (x k ; ˆx k k, Σ k k ) Storage: mean and covariance Computation: N r Kalman s Merge with probabilities µ i k P (r k = i y :k ). ˆx r = r = 2 ˆx ˆx r2 = ˆx ˆx ˆx 2 ˆx 2 2 E. Özkan Target Tracking November 2, 24 7 / 23

6 Multiple Model Approaches: GPB2 Multiple Model Approaches: GPB2 vs. IMM Generalized pseudo Bayesian algorithms (GPB) GPB2 Approximation: p(x k y :k ) µ i k N (x k; ˆx i k k, Σi k k ) Storage: N r means and covariances Computation: N 2 r Kalman s i= ˆx ˆx 2 r = r = 2 r = r = 2 ˆx ˆx 2 ˆx 2 ˆx GPB2 ˆx 2 r2 = r2 = ˆx ˆx 2 ˆx 2 ˆx ˆx 2 2 ˆx ˆx 2 r = r = 2 r = r = 2 ˆx ˆx 2 ˆx 2 ˆx ˆx 2 r2 = r2 = ˆx ˆx 2 ˆx 2 ˆx ˆx 2 2 E. Özkan Target Tracking November 2, 24 8 / 23 E. Özkan Target Tracking November 2, 24 9 / 23 Multiple Model Approaches: GPB2 vs. IMM Multiple Model Approaches: IMM ˆx ˆx 2 r = r = 2 r = r = 2 ˆx ˆx 2 ˆx 2 ˆx GPB2 ˆx 2 r2 = r2 = ˆx ˆx 2 ˆx 2 ˆx ˆx 2 2 IMM Interacting Multiple Models IMM Approximation: p(x k y :k ) µ i k N (x k; ˆx i k k, Σi k k ) Same approximation as GPB2 Storage: N r means and covariances Computation: N r Kalman s i= ˆx ˆx 2 r = r = 2 ˆx ˆx 2 r 2 = r 2 = 2 ˆx ˆx 2 ˆx ˆx 2 r = r = 2 ˆx ˆx 2 r 2 = r 2 = 2 ˆx ˆx E. Özkan Target Tracking November 2, 24 9 / 23 E. Özkan Target Tracking November 2, 24 2 / 23

7 Multiple Model Approaches: IMM IMM Illustration Gating and Data Association with IMM At each step, one can just calculate the following overall predicted measurement ŷ k k and innovation covariance S k k ŷ k k = µ i k ŷi k k S k k = [S ] k k i + (ŷi k k ŷ k k )( ) T i= i= µ i k We can do the gating and data association with these quantities. An alternative is to do individual gating for each model and then to take the union of the gated measurements from all models. In this case, the overall likelihood for association is formed from individual likelihoods as p(y k y :k ) = i= µ i k p(y k y :k, r k = i) }{{} individual likelihood from ith KF The same illustration with now IMM. P D =. P G = P F A = IMM uses two CV models same except for σ a =.m/s 2. σ 2 a = m/s IMM E. Özkan Target Tracking November 2, 24 2 / 23 E. Özkan Target Tracking November 2, / 23 IMM Illustration References 22 Gates of IMM The same illustration with now IMM. P D =. P G = P F A = IMM uses two CV models same except for σ a =.m/s 2. σ 2 a = m/s Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation. New York: Wiley, 2. X. Rong Li and V.P. Jilkov, Survey of maneuvering tracking. Part I. Dynamic models, IEEE Transactions on Aerospace and Electronic Systems, vol.39, no.4, pp , Oct. 23. X. Rong Li and V.P. Jilkov, Survey of maneuvering tracking. Part V. Multiple-model methods, IEEE Transactions on Aerospace and Electronic Systems, vol.4, no.4, pp , Oct E. Özkan Target Tracking November 2, / 23 E. Özkan Target Tracking November 2, / 23