Muhammad Moeen YaqoobPage 1 Moment-Matching Trackers for Difficult Targets Muhammad Moeen Yaqoob Supervisor: Professor Richard Vinter

Muhammad Moeen YaqoobPage 2 Talk Outline Introduction Background The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking Summary

Muhammad Moeen YaqoobPage 3 Talk Outline Introduction –What is Target Tracking? –Research Goals Background The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking Summary

Muhammad Moeen YaqoobPage 4 What is Target Tracking? To process noisy sensor measurements received from one or more sensors (radar, sonar, etc.) and estimate the state of an object. Applications: –Military applications: Air defence systems, military surveillance, … –Civilian applications: Air traffic control, policing,...

Muhammad Moeen YaqoobPage 5 Research Goals Development of new algorithms for ‘difficult’ tracking problems where conventional trackers fail or give poor performance Difficult tracking problems: –Tracking manoeuvring targets –Bearings-only tracking, Range-only tracking Traditional approaches: –computationally very expensive (the particle filter) –poor results because of approximations involved in the tracker design (the extended Kalman filter, etc.) A new algorithm solves the bearings-only tracking problem with highly reduced computational complexity

Muhammad Moeen YaqoobPage 6 Talk Outline Introduction Background –The Dynamic System Model –Bayesian Approach to Target Tracking –The Kalman Filter –Sub-Optimal Filters –The Bearings-Only Problem The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking Summary

Muhammad Moeen YaqoobPage 7 The Dynamic System Model System Equation: (a first order Markov process) Measurement Equation: : target state vector,: system input vector : system noise sequence with covariance matrix : vector-valued state-transition function (possibly non-linear) : measurement vector,: measurement input vector : measurement noise sequence with covariance matrix : vector-valued measurement function (possibly non-linear)

Muhammad Moeen YaqoobPage 8 Target motion, sensor observations models; stochastic processes Aim: Construct the posterior probability density function (pdf) Complete solution to the estimation problem through the pdf. e.g. Minimum Mean Square Error (MMSE) estimate: A recursive filter (updates estimates with each new measurement) has two steps: Prediction step and Correction Step Bayesian Approach to Target Tracking Problem: Only a theoretical solution; integrals are not tractable Solution does exist in highly restrictive cases e.g. the Kalman filter

Muhammad Moeen YaqoobPage 9 Normal density of the state The Kalman Filter Exact/optimal solution to the state estimation problem Assumptions: –System noise and measurement noise are ‘Gaussian’ –System model and measurement model are ‘linear’ Generates estimates of the conditional mean and conditional covariance Calculation of Conditional Mean and Covariance of the State Measurement time Normal conditional density of the state

Muhammad Moeen YaqoobPage 10 The Kalman Filter (cont’d) Correction Step: Kalman gain Corrected state mean Corrected state covariance What is wrong with the Kalman filter ? –Too restrictive; optimal only for linear Gaussian models Is there any other acceptable solution for the rest of the models? –Use sub-optimal approximations to approximate the exact solution to the state estimation problem Prediction Step: Predicted state mean Predicted state covariance Predicted measurement covariance

Muhammad Moeen YaqoobPage 11 Sub-Optimal Filters Two Categories: 1)Density approximation filters Density Approximation Filters: –Aim: Direct approximation of the conditional densities of the state Approximate the posterior pdf by N weighted ‘particles’ or ‘random’ samples The posterior pdf approaches the ‘true’ pdf as N → ∞ Advantage: Versatility! Disadvantage: Computationally expensive! 2)Moment-matching filters  The Particle Filter

Muhammad Moeen YaqoobPage 12 Sub-Optimal Filters (cont’d) Moment-Matching Filters: –Aim: approximate state distribution by a fixed number of moments –1 st and 2 nd moments; to exploit Kalman filtering framework Linearises all non-linear models and uses a simple Kalman Filter Represents prior density by ‘deterministically’ chosen sample points Propagate points through non-linear functions to get predicted moments Transforms the non-linear measurement into a linear form –Advantage: –Disadvantage:  The Extended Kalman Filter (EKF)  The Unscented Kalman Filter (UKF)  The Pseudomeasurement Filter Computationally inexpensive! Inflexible!, less accurate!

Muhammad Moeen YaqoobPage 13 Determine position and velocity of a target from noise corrupted bearing or angle measurements only Applications: –Submarine tracking (using passive sonar) –Aircraft surveillance (using radar in passive mode) Highly non-linear measurement model Severely ill-conditioned for some target-sensor configurations, : coordinates of relative position of target w.r.t sensor The Bearings-Only Problem : sensor noise (zero mean, Gaussian)

Muhammad Moeen YaqoobPage 14 Talk Outline Introduction Background The Shifted Rayleigh Filter –Overview –Formulation –Comparison of the Kalman Filter and the SRF Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking Summary

Muhammad Moeen YaqoobPage 15 A moment-matching filter for single-target tracking, where –Target and sensor dynamics can be represented by linear models –Noisy bearing measurements are made from one or more sensors Incorporates a novel measurement model Aim: To estimate the conditional mean and covariance of the state Based on exact calculations of conditional statistics Normal Approximation of target state Non-Normal conditional density of target state Normal conditional density of target state Calculation of Exact Conditional Mean and Covariance of the Target State Measurement time Overview

Muhammad Moeen YaqoobPage 16 System equation: Bearing vector: Measurement equation: Virtual measurement: State using virtual measurement where Formulation But Shifted Rayleigh density!

Muhammad Moeen YaqoobPage 17 Sensor Noise: Traditional Filters: 1-D scalar noise Shifted Rayleigh Filter: 2-D vector noise Formulation (cont’d) Measurement Noise:

Muhammad Moeen YaqoobPage 18 Comparison of the Kalman Filter and the SRF The Kalman FilterThe Shifted Rayleigh Filter Prediction Step: Correction Step:

Muhammad Moeen YaqoobPage 19 Talk Outline Introduction Background The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter –Single Sensor Scenarios –Multiple Sensor Scenarios Geometry of Bearings-Only Tracking Summary

Muhammad Moeen YaqoobPage 20 Single Sensor Scenarios (Scenario- I) State Vector: 2-D System Noise: N(0, 0.01) Platform perturbation: N(0,1) Tracking Period: 20 sec Standard Deviation of Sensor Noise: 3 o

Muhammad Moeen YaqoobPage 21 Single Sensor Scenarios (Scenario- I)

Muhammad Moeen YaqoobPage 22 Single Sensor Scenarios (Scenario- II) State Vector: 2-D Tracking Period: 200 sec Sensor Noise: 0 o

Muhammad Moeen YaqoobPage 23 Single Sensor Scenarios (Scenario- II)

Muhammad Moeen YaqoobPage 24 Single Sensor Scenarios (Scenario- II)

Muhammad Moeen YaqoobPage 25 Multiple Sensor Scenarios (Scenario- I) State Vector: 16-D Noise on Target Dynamics: N(0, 0.4) Tracking Period: 72 sec Sensor Perturbation Noise: N(0, 16) Standard Deviation of Sensor Noise: 8 o

Muhammad Moeen YaqoobPage 26 Multiple Sensor Scenarios (Scenario- I)

Muhammad Moeen YaqoobPage 27 Multiple Sensor Scenarios (Scenario- I)

Muhammad Moeen YaqoobPage 28 Multiple Sensor Scenarios (Scenario- II) State Vector: 12-D Noise on Target Dynamics: N(0, 0.16) Tracking Period: 100 sec Sensor Perturbation Noise: N(0, 1) Std.Dev. of Sensor Noise: 16 o Bulk Drift Perturbation Noise: N(0, 0.02) Std.Dev. of Monitor Sensor Noise: 0.8 o

Muhammad Moeen YaqoobPage 29 Talk Outline Introduction Background The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking –Problem Formulation –Classification of Target-Observer Configurations –Summary Summary

Muhammad Moeen YaqoobPage 30 Target Characteristics: –A single target following a constant turn-rate model Observer Characteristics: –Fixed Observer Platform Measurements: –Regular, noise-less measurements Problem Formulation are obtained by central projection of target position onto the sensor ‘focal’ plane

Muhammad Moeen YaqoobPage 31 Classification of Target-Observer Configurations Three different target-observer –The Generic Configuration: Plane of target manoeuvre does not contain the observer –The Singular Configuration: The target track is on a circle that is co-planar with, though does not pass through, the observer –The Sub-Generic Configuration: The track lies either on a circle passing through the observer or on a straight line configurations:

Muhammad Moeen YaqoobPage 32 A sliding-window algorithm for three-dimensional bearings- only tracking Identification of target-observer configuration requires five observations Evaluation of turn-rate parameter needs five, seven or four observations Algorithm quite sensitive to noise Summary

Muhammad Moeen YaqoobPage 33 Talk Outline Introduction Background The Shifted Rayleigh Filter Performance of the Shifted Rayleigh Filter Geometry of Bearings-Only Tracking Summary –Conclusion

Muhammad Moeen YaqoobPage 34 Conclusion Moment-matching methodology has an important role to play in complex tracking problems. But its successful application depends on the manner in which it is carried out !

Muhammad Moeen YaqoobPage 35 Questions ? Thank You!