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
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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
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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, ...
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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
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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
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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
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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
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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
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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
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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
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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!
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•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)
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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
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•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
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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
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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
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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
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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
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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
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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)
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-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
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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)
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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)
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-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
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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)
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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)
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-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
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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
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•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
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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
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•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
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Talk Outline
• Introduction
• Background
• The Shifted Rayleigh Filter
• Performance of the Shifted Rayleigh Filter
• Geometry of Bearings-Only Tracking
• Summary– Conclusion
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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 !
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Questions ?
Thank You!