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

Muhammad Moeen YaqoobPage 1

Moment-Matching Trackers for Difficult Targets

Muhammad Moeen Yaqoob

Supervisor:

Professor Richard Vinter


Talk Outline

• Introduction

• Background

• The Shifted Rayleigh Filter

• Performance of the Shifted Rayleigh Filter

• Geometry of Bearings-Only Tracking

• Summary


Talk Outline

• Introduction– What is Target Tracking?– Research Goals

• Background




• Summary


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, ...


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


Talk Outline

• Introduction

• Background– The Dynamic System Model– Bayesian Approach to Target Tracking– The Kalman Filter– Sub-Optimal Filters– The Bearings-Only Problem




• Summary


The Dynamic System Model

111 ,, kskkkk vuxfx

kmkkkk wuxhy ,,

•System Equation: (a first order Markov process)

•Measurement Equation:

kx : target state vector, sku : system input vector

kv : system noise sequence with covariance matrix skQ

kf : vector-valued state-transition function (possibly non-linear)

ky : measurement vector, mku : measurement input vector

kw : measurement noise sequence with covariance matrix

kh : vector-valued measurement function (possibly non-linear)

mkQ


Prediction Step

1:11 kk yxp 1:1 kk yxp kk yxp :1 kk yxp :11

ky

Time step k

Observed Measurement

Correction Step Prediction Step

•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

kk yxp :1

kkkk yxEx :1|

•Problem: Only a theoretical solution; integrals are not tractable •Solution does exist in highly restrictive cases e.g. the Kalman filter


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

111 kskkkk vuxFx k

mkkkk wuxHy

kkx

kkP

Calculation of Conditional Mean

and Covariance of the State

Measurement ky

time 1k time k

Normal conditional density of the state


The Kalman Filter (cont’d)

•Correction Step:

1

1

11

kkkkkk

kkmkkkkkkkk

kTkkkk

PHKIP

yKuKxHKIx

VHPK

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: s

kTkkkkkk

skkkkkk

QFPFP

uxFx

1111

1111

mk

Tkkkkk QHPHV 1

Predicted state mean

Predicted state covariance

Predicted measurement covariance


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

N-Particles

True posterior pdf

The Particle Filter


Sub-Optimal Filters (cont’d)

•Moment-Matching Filters:

–Aim: approximate state distribution by a fixed number of moments

–1st and 2nd 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!


•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

North

kk wky

kxarctan

)(kx )(ky

kw : sensor noise (zero mean, Gaussian)


Talk Outline

• Introduction

• Background

• The Shifted Rayleigh Filter– Overview– Formulation– Comparison of the Kalman Filter and the SRF



• Summary


•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 statisticsNormal Approximation of target stateNon-Normal conditional density of target state

Normal conditional density of target state

Calculation of Exact Conditional Mean

and Covariance of the Target State

Measurement kb

time 1k time k

Overview


kmkkk wuHxy

kk VmN ,

ky

Unitcircle

kr

kb

km

original distribution

k

•System equation:

•Bearing vector:

•Measurement equation:

•Virtual measurement:

•State using virtual measurement

where

2

2

12 2

01

2

0

s z

k ks z

s e ds

E s z

se ds

Formulation

1 1 1s

k k k k kx F x u v

sin , cosT

k k kb

kmkkk wuHxb

kx

kkmkkkkkkk yKuKxHKIx 1)(

1,0~ kkkk PHKIN

2

2

12 2

01

0

2

s z

s zk k

s e ds

E s

s

z

dse

•But kkk bry

Shifted Rayleigh density!

kkkVs 21k

ky

transformed distribution

222/1 , ImVN kk


Unitcircle

2-D sensor noise density

0

actual target position

0b

sensor noise density amplified by range estimate

actual target position

•Sensor Noise:

•Traditional Filters: 1-D scalar noise

•Shifted Rayleigh Filter: 2-D vector noise

Formulation (cont’d)

kk wky

kxarctan

20,N

22 20,N I

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

1.2

The angle in radians

bearing densitynormal density

•Measurement Noise:

22

2

12

IuHxEQQ m

kkkktrk

mk


Comparison of the Kalman Filter and the SRFThe Kalman Filter The Shifted Rayleigh Filter

•Prediction Step:

sTkkkk

skkkkk

QFFPP

uxFx

111

1111

mk

Tkkk QHHPV 1

•Prediction Step:

sTkkkk

skkkkk

QFFPP

uxFx

111

1111

mk

Tkkk QHHPV 1

kkkkkTkk

Tk

Tkkkkkkkkk

zzzbVb

KbbKPHKIP

211

1

2

1 kkkkk PHKIP

•Correction Step:

•Correction Step: 1

1

kT

kkk VHPK11

k

Tkkk VHPK

1

1 21

1 21 11

ˆ ˆ

ˆ

mk k k k k kk k k k

Tk k k k k

T T mk k k k k k kk k

x I K H x K u K b

b V b z

z b V b b V Hx u

kkmkkkkkkk yKuKxHKIx 1ˆˆ


Talk Outline

• Introduction

• Background


• Performance of the Shifted Rayleigh Filter– Single Sensor Scenarios– Multiple Sensor Scenarios


• Summary


Single Sensor Scenarios (Scenario- I)

0 20 40 1008060 1200

5

25

20

15

10

30

Platform Motion

Target Motion

Horizontal Displacement (metres)

Ver

tical

Dis

plac

emen

t (m

etre

s)

0 20

State Vector: 2-D System Noise: N(0, 0.01) Platform perturbation: N(0,1) Tracking Period: 20 sec

Standard Deviation of Sensor Noise: 3o


0 5 10 15 20 250

5

10

15

20

25

time (seconds)

RM

S ta

rget

pos

ition

err

or (

met

res)

Pseudomeasurement FilterShifted Rayleigh Filter

Single Sensor Scenarios (Scenario- I)


-30 -20 -10 0 10 20 30 40

-20

-10

0

10

20

30

40

Horizontal Displacement (metres)

Ver

tical

Dis

plac

emen

t (m

etre

s)

mp mt

Target PositionPlatform PositionConfidence Region of TargetConfidence Region of Platform

Single Sensor Scenarios (Scenario- II) State Vector: 2-D

Tracking Period: 200 sec Sensor Noise: 0o


0 50 100 150 200 2500

2

4

6

8

10

12

time (seconds)

RM

S ta

rget

pos

ition

err

or (

met

res)

x-error (Pseudomeasurement Filter)x-error (Shifted Rayleigh Filter)y-error (Pseudomeasurement Filter)y-error (Shifted Rayleigh Filter)

Single Sensor Scenarios (Scenario- II)


0 50 100 150 200 2500

20

40

60

80

100

120

time (seconds)

RM

S ta

rget

pos

ition

err

or (

met

res)

x-error (Pseudomeasurement Filter)x-error (Shifted Rayleigh Filter)y-error (Pseudomeasurement Filter)y-error (Shifted Rayleigh Filter)

Single Sensor Scenarios (Scenario- II)


-300 -200 -100 0 100 200 300

-300

-200

-100

0

100

200

300

x position (metres)

y po

sitio

n (m

etre

s)

Target Trajectory

Drifting Sensor

CentralMonitor

Sensor-Target Bearing

Monitor-Sensor Bearing

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: 8o


0 25 50 750

10

20

30

40

50

60

70

80

90

time (seconds)

RM

S ta

rget

x-p

ositi

on e

rror

(m

etre

s)PF estimate errorSRF estimate error

Multiple Sensor Scenarios (Scenario- I)


0 25 50 750

50

100

150

time (seconds)

RM

S ta

rget

x-p

ositi

on e

rror

(m

etre

s)PF estimate errorSRF estimate error

Multiple Sensor Scenarios (Scenario- I)


-400 -300 -200 -100 0 100 200 300 400

-400

-300

-200

-100

0

100

200

300

400

x position (metres)

y po

sitio

n (m

etre

s)

Target Trajectory

Central Monitor

Drifting Sensor

Sensor-TargetBearing

Monitor-Sensor Bearing

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: 16o Bulk Drift Perturbation Noise: N(0, 0.02) Std.Dev. of Monitor Sensor Noise: 0.8o


Talk Outline

• Introduction

• Background



• Geometry of Bearings-Only Tracking– Problem Formulation– Classification of Target-Observer Configurations– Summary

• Summary


•Target Characteristics:

–A single target following a constant turn-rate model

•Observer Characteristics:

–Fixed Observer Platform

•Measurements:

–Regular, noise-less measurements

Problem Formulation

3x

2x1x

13 x

kx

ky

Observer

Plane of Manoeuvre

Focal Plane

are obtained by central projection of target position onto the sensor ‘focal’ plane

0321 kkkk xaxaxx


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

Target Trajectory

Observer

configurations:

Target Trajectory

Observer

Target Trajectory

Observer

Target Trajectory

Observer


•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

a


Talk Outline

• Introduction

• Background




• Summary– Conclusion


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 !


Questions ?

Thank You!

Documents

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