"Modern Tracking" Short Course Taught at University of Hawaii

August 6-8, 2002

Topics OverviewTopics Overview

OverviewOverview

Mathematics Overview

– Linear Algebra and Linear Systems– Probability and Hypothesis Testing– State Estimation

Filtering Fundamentals

– Linear and Non-linear Filtering– Multiple Model Filtering

Tracking Basics

– Track Maintenance– Data Association Techniques– Activity Control

Mathematics ReviewMathematics Review

Mathematics ReviewMathematics Review

Linear Algebra and Linear Systems

– Definitions, Notations, Jacobians and Matrix Inversion Lemma– State-Space Representation (Continuous and Discrete) and Observability

Probability Basics

– Probability, Conditional Probability, Baye’s and Total Probability Theorem– Random Variables, Gaussian Mixture, and Covariance Matrices

Bayesian Hypothesis Testing

– Neyman-Pearson Lemma and Wald’s Theorem– Chi-Square Distribution

Estimation Basics

– Maximum Likelihood (ML) and Maximum A Posteriori (MAP) Estimators– Least Squares (LS) and Minimum Mean Square Error (MMSE) Estimators– Cramer-Rao Lower Bound, Fisher Information, Consistency and Efficiency

Vector and Matrix BasicsVector and Matrix Basics

Definitions and NotationsDefinitions and Notations

n

i

a

a

a

aa

2

1

nT aaaa 21

nmnn

m

m

ij

aaa

aaa

aaa

aA

21

22221

11211

nmmm

n

n

jiT

aaa

aaa

aaa

aA

21

22212

12111

Basic Matrix and Vector PropertiesBasic Matrix and Vector Properties

Symmetric and Skew Symmetric Matrix

Matrix Product (NxS = [NxM] [MxS]):

Transpose of Matrix Product

Matrix Inverse

TT AAAA

m

kkjikij baABcC

1

m

kkijk

TTTji

T abABABcC1

)(

IAA 1

Inner Product (Vectors must have equal length)

Outer Product (NxM = [N] [M])

Matrix Trace

Trace of Matrix Product

n

iii

T bababa1

,

jiijT bacCab

)()(1

Tn

iii ATraATr

Basic Matrix and Vector PropertiesBasic Matrix and Vector Properties

)()( BATrABTr TT BBAABATrA

TB

A

AB

Matrix Inversion LemmaMatrix Inversion Lemma

In Estimation Theory, the following complicated inverse appears:

The Matrix Inversion Lemma yields an alternative expression which does not depend on the inverses of the matrices in the above expression:

An alternative form of the Matrix Inversion Lemma is:

111 HRHP T

HPRHPHPHP TT 1

1111111 ABCBABBAABCBA TTT

The GradientThe Gradient

The Gradient operator with respect to an n-dimensional vector “x” is:

Thus the gradient of a scalar function “f” is:

The gradient of an m-dimensional vector-valued function is:

T

nx xx

1

T

nx x

f

x

ff

1

NxMxfxfxx

f m

T

n

Tx

)()(11

The Jacobian MatrixThe Jacobian Matrix

The Jacobian Matrix is a matrix of derivatives describing a linear mapping from one set of coordinates to another. This is the transpose of the gradient of a vector-valued function (p. 24):

This is typically used as part of a Vector Taylor Expansion for approximating a transformation.

n

mm

n

n

m

x

x

x

x

x

x

x

x

xxx

xxx

x

xxxJ

1

1

1

1

21

21

),...,,(

),...,,(,

o

xo xx

x

xxxx

o

)(

The Jacobian Matrix: An ExampleThe Jacobian Matrix: An Example

The conversion from Spherical to Cartesian coordinates yields:

)()()()()( eSinreCosbCosreCosbSinrx

ebrx

)(0)(

)()()()()()(

)()()()()()(

,

eCosreSin

eSinbCosreCosbSinreCosbCos

eSinbSinreCosbCosreCosbSin

xxJ

Ix

x

x

xxxJxxJ

),(,

Linear Systems BasicsLinear Systems Basics

Dirac Delta FunctionDirac Delta Function

The Dirac Delta Function is defined by:

This function is defined by its behavior under integration:

In general, the Dirac Delta Function has the following “sifting” behavior:

The discrete version of this is called the Kronnecker Delta:

tt 0)(

badttb

a,1)(

bafdtttfb

a,)()()(

ji

jiij 1

0

State-Space Representation (Continuous)State-Space Representation (Continuous) A Dynamic Equation is typically expressed in the standard form (p. 27):

While the Measurement Equation is expressed in the standard form:

)()()()()( tutBtxtAtx

)(tx

is the state vector of dimension “nx”

)(tu

is the control input vector of dimension “ny”

)(tA is the system matrix of dimension “nx x nx”

)(tB is the input gain matrix of dimension “nx x ny”

)()()( txtCtz

)(tz

is the measurement vector of dimension “nz”

)(tC is the observation matrix of dimension “nz x nx”

Example State-Space SystemExample State-Space System

A typical (simple) example is the constant velocity system:

This system is not yet in state-space form:

And suppose that we only have position measurements available:

0)( t

2

1

00

00

00

10

u

u

)()()()()( tutBtxtAtx

01meas

)()()( txtCtz

State-Space Representation (Discrete)State-Space Representation (Discrete) A continuous state-space system can also be written in discrete form (p. 29):

While the Measurement Equation is expressed in the discrete form:

1111 kkkkk uGxFx

kx

is the state vector of dimension “nx” at time “k”

ku

is the control input vector of dimension “ny” at time “k”

kF is the transition matrix of dimension “nx x nx” at time “k”

kG is the input gain matrix of dimension “nx x ny” at time “k”

kkk xHz

kz

is the measurement vector of dimension “nz” at time “k”

kH is the observation matrix of dimension “nz x nx” at time “k”

Example Revisited in Discrete TimeExample Revisited in Discrete Time

The constant velocity discrete time model is given by:

Since there is no time-dependence in the measurement equation, it is a trivial extension to the continuous example:

12

11

1

11

00

00

10

1

k

k

k

kkk

k

k

u

utt

k

kmeask

01

1111 kkkkk uGxFx

kkk xHz

State Transition MatrixState Transition Matrix

We wish to be able to convert a continuous linear system to a discrete time linear system. Most physical problems are easily expressible in the continuous form while most measurements are discrete. Consider the following time-invariant homogeneous linear system (pp. 180-182):

We have the solution:

If we add a term, making an inhomogeneous linear system, we obtain:

kkkttA

kkk tttfortxeAsILtxttFtx k ,,)( 1111

1111

kk tttforAtAwheretxtAtx ,)()()()( 1

kk tttforBtBwheretutBtxtAtx ,)()()()()()( 1

Matrix Superposition IntegralMatrix Superposition Integral

Then, the state transition matrix is applied to the additive term and integration is performed to obtain the generalized solution:

Consider the following example:

kk

t

t kkkk tttforduBtFtxttFtxk

,)()(,,)( 111111

)(2

0

0

102

2

1

2

1 tux

x

x

x

u(t)=(t) x2 x1

s

22s

1

Observability CriteriaObservability Criteria A system is categorized as observable if the state can be determined

from a finite number of observations, assuming that the state-space model is correct.

For a time-invariant linear system, the observability matrix is given by:

Thus, the system is observable if this matrix has a rank equal to “nx” (pp. 25,28,30).

1xnFH

FH

H

Observability Criteria: An ExampleObservability Criteria: An Example

For the nearly constant velocity model described above, we have:

The rank of this matrix is “2” only if the delta time interval is non-zero. Thus, we can only estimate position and velocity both (using only position measurements) if these position measurements are separated in time.

The actual calculation of rank is a subject for a linear algebra course and leads to ideas such as linear independence and singularity (p. 25)

t

t1

01

10

101

01

Probability BasicsProbability Basics

Axioms of ProbabilityAxioms of Probability

Suppose that “A” and “B” denote random events, then the following axioms hold true for probabilities:

– Probabilities are non-negative:

– The probability of a certain event is unity:

– Additive for mutually exclusive events:

AAP 0

1SP

BPAPBAPthenBAPIf 0

Mutually Exclusive

Conditional ProbabilityConditional Probability

The conditional probability of an event “A” given the event “B” is:

For example, we might ask the following tracking related questions:

– Probability of observing the current measurement given the previous estimate of the track state

– Probability of observing a target detection within a certain surveillance region given that a true target is present

Formulating these conditional probabilities is the foundation of track initiation, deletion, data association, SNR detection schemes…

BP

BAPBAP

|

Total Probability TheoremTotal Probability Theorem

Assume that we have a set of events “Bi” which are mutually exclusive:

And exhaustive:

Then the Total Probability Theorem states:

jiBBP ji 0

11

n

iiBP

n

iii

n

ii BPBAPBAPAP

11

|

Baye’s TheoremBaye’s Theorem

We can work the conditional probability definition in order to obtain the reverse conditional probability:

This conditional probability “Bi” is called the Posterior Probability while the unconditional probability of “Bi” is called the Prior Probability.

In the case of “Bi” being mutually exclusive and exhaustive, we have (p. 47):

AP

BPBAP

AP

ABPABP iii

i

||

n

jjj

iii

BPBAP

BPBAPABP

1

|

||Posterior Probability

Prior Probability

Likelihood Function

Gaussian (Normal) Random VariablesGaussian (Normal) Random Variables

The Gaussian Random Variable is the most well-known, well-investigated type because of its wide application in the real world and its tractable mathematics.

A Gaussian Random Variable is one which has the following probability density function (PDF) :

and is denoted:

),(~ 2Nx

2

2

2

2

2

2

1),;()(

x

exNxp

Gaussian (Normal) Random VariablesGaussian (Normal) Random Variables

The Expectation and Second Central Moment of this distribution are:

These are only with respect to scalar random variables…what about vector random variables?

222

2222

2

2

2

2

2

2][][]])[[(

2][

dxex

xExExExE

dxex

xE

x

x

Mean Square Variance

Mean

Vector Gaussian Random VariablesVector Gaussian Random Variables

The vector generalization is straight forward:

The Expectation and Second Central Moment of this distribution are:

Notice that the Variance is now replaced with a matrix called a Covariance Matrix.

If the vector “x” is a zero-mean error vector than the covariance matrix is called the Mean Square Error.

2

1

2

1),;()(

xPx T

eP

PxNxp

PxExxExE

xET

]])[])([[(

][

Baye’s Theorem: Gaussian CaseBaye’s Theorem: Gaussian Case

The “noise” of a device, denoted “x”, is observed. Normal functionality is denoted by event “B1” while a defective device is denoted by event “B2”:

The conditional probability of defect is (using Baye’s Theorem):

Using the two distributions, we have:

),0;(),0;( 222

211 xNBxNB

22

112211

222

||

1

1

||

||

BPBxPBPBxPBPBxPBPBxP

BPBxPxBP

22

2

21

2

22

21

12

2

1

1|

xx

eBPBP

xBP

Baye’s Theorem: Gaussian CaseBaye’s Theorem: Gaussian Case

If we assume the diffuse prior, that the probability of each event is equal, then we have a simplified formula:

If we further assume that 2 = 4 1 and that x = 2, then we have:

Note that the likelihood ratio largely dominates the result of this calculation. This quantity is crucial in inference and statistical decision theory and often called “evidence from the data”.

22

2

21

2

22

1

2

2

1

1|

xx

e

xBP

998.0|2 xBP

2

121 |

|,

BxP

BxPBB

Gaussian MixtureGaussian Mixture

Suppose we have “n” possible events “Aj” which are mutually exclusive and exhaustive. And further suppose that each event has a Gaussian PDF as follows (pp. 55-56):

Then, the total PDF is given by the Total Probability Theorem:

This mixture can be approximated as another Gaussian once the mixed moments are computed.

jjjjj pAPandPxNxA

,~

n

jij APAxpxp

1

)|()(


The first moment (mean) is easily derived as:

The covariance matrix is more complicated, but we simply apply the definition:

n

jjj

n

jjj

n

jij

xpx

pAxpEAPAxpExpEx

1

11

)|()|()(

n

jjj

Tjjjj

n

jjj

TT

pAxxxxxxxxE

pAxxxxExxxxEP

1

1

|

|


Continuing the insanity:

The spread of the means term inflates the covariance of the final mixed random variable to account for the differences between each individual mean and the mixed mean.

n

jj

Tjj

n

jjj

n

jj

Tjj

n

jjj

Tjj

pxxxxpP

pxxxxpAxxxxEP

11

11

|

Spread of the Means

Bayesian Hypothesis TestingBayesian Hypothesis Testing

Bayesian Hypothesis TestingBayesian Hypothesis Testing

We consider two competing hypotheses about a parameter “” defined as:

We also define standard definitions concerning the decision errors:

11

00

:

:

H

HNull Hypothesis

Alternate Hypothesis

trueHHacceptPP

trueHHacceptPP

II

I

e

e

10

01

|

|Type I Error (False Alarm)

Type II Error (Miss)

Neyman-Pearson LemmaNeyman-Pearson Lemma

The power of the hypothesis test is defined as:

The Neyman-Pearson Lemma states that the optimal decision (most powerful test) rule subject to a fixed Type I Error () is the Likelihood Ratio Test (pp.72-73):

1| 11 trueHHacceptPTest Power (Detection)

IePHHHP

H

H

HzP

HzPHH

0001

00

01

0

101

|,

;

;

|

|,

Likelihood Functions

Sequential Probability Ratio TestSequential Probability Ratio Test

Suppose, we have a sequence of independent identically distributed (i.i.d.) measurements “Z={zi}” and we wish to perform a hypothesis test. We can formulate this in a recursive form as follows:

000

101

0

101 |

|,

HPHZP

HPHZP

ZHP

ZHPHHPR

Likelihood Functions a priori Probabilities

n

ii

n

i i

in HHHHPR

HzP

HzP

HP

HPHHPR

101010

1 0

1

00

1001 ,,

|

|,

n

iin HHHHPRHHPR

10101001 ,ln,ln,ln

Sequential Probability Ratio TestSequential Probability Ratio Test

So, the recursive for of the SPRT is:

Using Wald’s Theorem, we continue to test this quantity against two thresholds until a decision is made:

Wald’s Theorem applies when the observations are an i.i.d. sequence.

0101101 ,ln,ln,ln HHHHPRHHPR kkk

1ln

1ln

;

;

;

,ln

12

10

21

21

01

TandT

TH

TandTcontinue

TH

HHPRk

Chi-Square DistributionChi-Square Distribution

The chi-square distribution with “n” degrees of freedom has the following functional form:

It is related to an “n” dimensional vector Gaussian distribution as follows:

More generally, the sum of squares of “n” independent zero-mean, unity variance random variables is distributed as a chi-square with “n” degrees of freedom (pp.58-60).

22

2

2

2

22

1)(

xn

nn exn

x

21 ~ nT xxPxx

Chi-Square DistributionChi-Square Distribution

The chi-square distribution with “n” degrees of freedom has the following statistical moments:

The sum of two independent random variables which are chi-square are also chi-square:

nxExEnxE 2]])[[(][ 2

221

22

21

21

21

~

~~

nn

nn

qq

qq

Estimation BasicsEstimation Basics

Parameter EstimatorParameter Estimator

A parameter estimator is a function of the observations (measurements) that yields an estimate of a time-invariant quantity (parameter). This estimator is typically denoted as:

We also denote the error in the estimate as:

kjj

kkk zZwhereZkxx

1,ˆˆ

Estimate Estimator Observations

kk xxx ˆ~

EstimateTrue

Estimation ParadigmsEstimation Paradigms

Non-Bayesian (Non-Random):

– There is no prior PDF incorporated– The Likelihood Function PDF is formed– This Likelihood Function PDF is used to estimate the parameter

Bayesian (Random):

– Start with a prior PDF of the parameter– Use Baye’s Theorem to find the posterior PDF– This posterior PDF is used to estimate the parameter

xpxZp

cZp

xpxZpZxp |

1||

Posterior Likelihood Prior

xZpxZ |)(

Estimation MethodsEstimation Methods

Maximum Likelihood Estimator (Non-Random):

Maximum A Posteriori Estimator (Random):

xpxZpZxx

MAP |maxarg)(ˆ

xZpZxx

ML |maxarg)(ˆ

0

|

ˆ

MLxdx

xZdp

Unbiased EstimatorsUnbiased Estimators


Bayesian (Random):

General Case:

0)|( 0)](ˆ[ xZxE

xxZp

kk k

)(][ˆ xpZxpk

k xEZxE k

0~ kk ZxE

Estimation Comparison ExampleEstimation Comparison Example

Consider a single measurement of an unknown parameter “x” which is susceptible to additive noise “w” that is zero-mean Gaussian:

The ML approach yields:

Thus, the MLE is the measurement itself because there is no prior knowledge.

2,0~ Nwwxz

zxx

exzNxzpx

x

ML

xz

)(maxargˆ

2

1),;()|()(

2

2

2

2

2

Estimation Comparison ExampleEstimation Comparison Example

The MAP, with a Gaussian prior, approach yields:

Thus, the MAPE is a linear combination of the prior information and the observation and it is weighted based upon the variance of each.

NOTE: The MLE and MAPE are equivalent for a diffuse prior !

)()|(maxargˆ

111)(

2)(2)(

)()|()|(

),;()(

20

221

220

21

21

2

))((

0

2

)(

2

)(

20

21

2

20

2

2

2

zzxpx

andzx

z

e

zp

e

zp

xpxzpzxp

xxNxp

x

MAP

zxxxxz

PriorInformation

MeasurementInformation

Batch Estimation ParadigmsBatch Estimation Paradigms

Consider that we now have a set of observations available for estimating a parameter and that in general these observations are corrupted by measurement noise:

Least Squares (Non-Random)

Minimum Mean Square Error (Random):

k

jjj

x

LSk xhzx

1

2)(minargˆ

kjjjj

k wxhzZ,,1

)(

dxZxpxZxEx

ZxxEx

kkMMSEk

k

x

MMSEk

||ˆ

|ˆminargˆ 2

ˆ

Unbiasedness of ML and MAP EstimatorsUnbiasedness of ML and MAP Estimators

Maximum Likelihood Estimate:

Maximum A Posterior Estimate:

000 ][][][]ˆ[ xwExwxEzExE MLk

][

])[(][

][ˆ

20

2

20

20

2

2

20

2

20

20

2

2

20

2

20

20

2

2

20

2

20

20

2

2

20

2

20

20

2

2

xExxx

wExxwxEx

zExzxExE MAPk

Estimation ErrorsEstimation Errors


Bayesian (Random):

General Case:

])(ˆ[])](ˆ[)(ˆ[)](ˆ[2

0

2xZxEZxEZxEZxVar k

kk

kk

kk

k

casesallZxMSE

randomnonxandunbiasedxZxZxE

kk

kkk

k ˆ

ˆˆvar~ 2

]|)(ˆ[])(ˆ[)](ˆ[22 kk

kk

kk

k ZxZxEExZxEZxMSE

Variances of ML and MAP EstimatorsVariances of ML and MAP Estimators

Maximum Likelihood Estimate:

Maximum A Posterior Estimate:

The MAPE error is less than the MLE error since the MAPE incorporates prior information.

220

2

0 ][]ˆ[]ˆvar[ xzExxEx MLk

MLk

MLk

MAPk

MAPk xxxEx ˆvarˆˆvar 2

20

2

20

22

Cramer-Rao Lower BoundCramer-Rao Lower Bound

The Cramer-Rao Lower Bound states that a limit on the ability to estimate a parameter.

Not surprisingly, this lower limit is related to the likelihood function which we recall as the “evidence from the data”. This limit is called the Fisher Information Matrix.

When equality holds, the estimator is called efficient. An example of this is the MLE estimate we have been working with.

1ˆˆˆ k

Tkk

kk

kk JxZxxZxEZxMSE

xkTxxk xZpEJ |ln

Filtering FundamentalsFiltering Fundamentals

Filtering FundamentalsFiltering Fundamentals

Linear Filtering

– Linear Gaussian Assumptions, Kalman Filter, Kalman Properties– Direct Discrete-Time, Discretized Continuous-Time, Steady State Gains

Non-Linear Filtering

– Non-Linear Dynamics & Measurements, Extended Kalman Filter– Iterated Extended Kalman Filter

Multiple-Model Filtering

– Need for Multiple Models, Adaptive Filtering– Switching Multiple Model & Interacting Multiple Model Filter– Variable Structure IMM

Linear FilteringLinear Filtering

Kalman-Bucy ProblemKalman-Bucy Problem A stochastic discrete-time linear dynamic system:

The measurement equation is expressed in the discrete form:

111111 kkkkkkk uGxFx

kx

is the state vector of dimension “nx” at time “k”

kkuG

is the control input of dimension “nx” at time “k”

kF is the transition matrix of dimension “nx x nx” at time “k”

kk is the plant noise of dimension “nx” at time “k”

kkkk wxHz

kz

is the measurement vector of dimension “nz” at time “k”

kH is the observation matrix of dimension “nz x nx” at time “k”

kw

is the measurement noise of dimension “nz” at time “k”

Kalman-Bucy ProblemKalman-Bucy Problem

The Linear Gaussian Assumptions are:

The measurement and plant noises are uncorrelated:

The conditional mean is:

The estimation error is denoted by:kkx |ˆ

jkk

Tjkk

jkkTjkk

RwwEwE

QEE

0

0

0][ kkwE

kizZZxEx ikk

jkj

,|ˆ |

1|ˆ kkxFiltered State Estimate Extrapolated State Estimate

kjjkj xxx || ˆ~

Kalman-Bucy ProblemKalman-Bucy Problem

The estimate covariance is defined as:

The predicted measurement is given by:

The measurement residual or innovation is denoted by:

kkP |

kTkjkjkj ZxxEP |~~|||

1| kkPFiltered Error Covariance Extrapolated Error Covariance

1|1111

1| ˆ||||ˆ

kkkk

kk

kkk

kkkk

kkk xHZwEZxEHZwxHEZzEz

1|1| ˆˆ

kkkkkkkk xHzzz

Kalman-Bucy ApproachKalman-Bucy Approach

Recall that the MMSE is equivalent to the MAPE in the Gaussian case.

Recall that the MAPE, with a Gaussian prior, is a linear combination of the measurement and the prior information.

Recall that the prior information was, more specifically, the expectation of the random variable prior to receiving the measurement.

If we consider the Kalman Filter to be a recursive process which applies a static Bayesian estimation (MMSE) algorithm at each step, we are compelled to consider the following linear combination.

kkkkkkk zKxKx 1|| ˆˆ

Prior StateInformation

ObservationInformation

Kalman Filter - UnbiasednessKalman Filter - Unbiasedness We start with the proposed linear combination:

We wish to ensure that the estimate is unbiased, that is:

Given the proposed linear combination, we determine the error to be:

Applying the unbiasedness constraint, we have:

kkkkkkk zKxKx 1|| ˆˆ

0~| kkxE

kkkkkkkkkkk wKxKxIHKKx 1||

~~

][]~[][0]~[ 1|| kkkkkkkkkkk wEKxEKxEIHKKxE

kkk HKIK

Kalman Filter – Kalman GainKalman Filter – Kalman Gain So, we have the following simplified linear combination:

We also desire the filtered error covariance, so that it can be minimized:

If we minimize the trace of this expression with respect to the gain:

kkkkkk Kxx 1|| ˆˆ

T

kkkT

kkkkkkkk

Tkkkkkk

KRKHKIPHKIP

xxEP

1||

|||~~

1

1|1|

kTkkkk

Tkkkk RHPHHPK

Kalman Filter - RecipeKalman Filter - Recipe

Extrapolation:

Update:

Tkkk

Tkkkkkk

kkkkkkk

QFPFP

uGxFx

11111|111|

111|111| ˆˆ

Tkkk

Tkkkkkkkk

kkkkkk

KRKHKIPHKIP

Kxx

1||

1|| ˆˆ

Kalman Filter – InnovationsKalman Filter – Innovations

The innovations are zero-mean, uncorrelated (p. 213) and have covariance:

The normalized innovation squared or statistical distance is chi-square distributed:

So, we expect that the innovations should have a mean and variance of:

The Kalman Gain can now be written as:

The state errors are correlated:

212 ~znkk

Tkk Sd

kTkkkk

Tkkk RHPHES 1|

zizi ndndE 2]var[][ 22

11|

1

1|1|

kTkkkk

Tkkkk

Tkkkk SHPRHPHHPK

1|111|1|~~

kkkkkT

kkkk PFHKIxxE

Kalman Filter – Likelihood FunctionKalman Filter – Likelihood Function

We wish to compute the likelihood function given the dynamics model used:

Which has the explicit form:

Alternatively, we can write:

k

kkTk

kkk

S

SZzp

2det

21

exp|

1

1

kkkkkkkkkkkkkk

k SNSzzNSzzNxzpZzp ,0;,0;ˆ,ˆ;ˆ|| 1|1|1|1

kkkrk

kk

kk SdS

dZzp

2detln

2

1

2

1ln

2det

21

exp| 2

2

1

Kalman Filter – Measurement ValidationKalman Filter – Measurement Validation

Suppose our Kalman filter has the following output at a given time step:

Suppose that we now receive 3 measurements of unknown origin:

Evaluate the consistency of these measurements for this Kalman filter model. This procedure is called gating and is the basis for data association.

15

10ˆ

10

01

160

05|11|1 kkkkk xHP

25

19,

5

16,

20

7

90

04 31

21

111 kkk

ik zzzR

2.9%996%95

138222

22

31

21

21

21

11

21

kkkkkk zdzdzd��

Kalman Filter – InitializationKalman Filter – Initialization

The true initial state is a random variable distributed as:

It is just as important that the initial covariance and estimate realistically reflect the actual accuracy. Thus, the initial estimate should satisfy:

If the initial covariance is too small, then the Kalman gain will initially be small and the filter will take a longer time to converge.

Ideally, the initial state estimate should be within one standard deviation (indicated by the initial covariance) of the true value. This will lead to optimal convergence time.

0|00|00 ,ˆ PxNx

%95~~ 20|0

10|00|0 xn

T xPx

Kalman Filter – InitializationKalman Filter – Initialization

In general, a batch weighted least-squares curve fit can be used (Chapter 3):

This initialization will always be statistically consistent so long as the measurement errors are properly characterized.

1

0

121010

0

0

,,,,

x

x

xxx

n

init

Tnnninit

Tninit

R

R

R

FHHHzzz

110|0

1110|0ˆ

initinitTinitinitinit

Tinitinitinit

Tinit HRHPzRHHRHx

Kalman Filter – SummaryKalman Filter – Summary

The Kalman Gain:

– Proportional to the Predicted Error– Inversely Proportional to the Innovation Error

The Covariance Matrix:

– Independent of measurements– Indicates the error in the state estimate assuming that all of the

assumptions/models are correct

The Kalman Estimator:

– Optimal MMSE state estimator (Gaussian)– Best Linear MMSE state estimator (Non-Gaussian)– The state and covariance completely summarize the past

Kalman Filter – SummaryKalman Filter – Summary

Kalman Filter:Direct Discrete Time Example

Kalman Filter:Direct Discrete Time Example

Consider the simplest example of the nearly constant velocity (CV) dynamics model:

The recursive estimation process is given by the Kalman equations derived above.

How do we select “q”?

rwEwxz

qExT

Tx

Tx

kkkk

kk

kkkkk

2

21

2

1

01

2

10

1

Discrete WhiteNoise Acceleration

2maxaq

Kalman Filter:Other Direct Discrete Time Models

Kalman Filter:Other Direct Discrete Time Models

For nearly constant acceleration (CA) models, the Discrete Weiner Process Acceleration (DWPA) model is commonly used:

Notice the simple relationship between the “q-value” and the physical parameter that is one derivative higher than that which is estimated.

2max

2

23

234

21

2

1

2

12

3

234

1

2

100

10

21

aq

TT

TTT

TTT

qqQ

qExT

T

xT

TT

x

Tkkk

k

k

k

k

kkkk

Kalman Filter:Discretized Continuous-Time Models


These models are derived from continuous time representations using the matrix superposition integral. Ignoring the control input:

Thus, the process noise covariance is found by:

dvDevandeF

where

vxFx

tqvtvEtvDtxAtx

T TAk

AT

kkkk

)(~

~)(~)(~)(~)(

0

)(

111

110 1

0 212210 1

~

)(~)(~

dTFDqDTF

ddTFDvvEDTFvvEQ

Tk

TT

k

T Tk

TT

kTkkk



Continuous White Noise Acceleration (CWNA) for CV Model:

Continuous Weiner Process Acceleration (CWPA) for CA Model:

Singer [IEEE-AES, 1970] developed the Exponentially Correlated Acceleration (ECA) for CA model (p. 187 & pp.321-324):

TaqTT

TTqQtvt k

2max2

23~

2

23~)(~)(

Taq

TTT

TTT

TTT

qQtvt k2max

23

234

345

~

26

238

6820~)(~)(

)(~)()( tvtt

Kalman Filter:Time Consistent Extrapolation

Kalman Filter:Time Consistent Extrapolation

So, what is the difference between the Direct Discrete-Time and Discretized Continuous-Time models for CV or CA models. Which one should be used?

Thus, for the Continuous-Time model, 2 extrapolations of 1 second yields the same result as 1 extrapolation of 2 seconds.

In general, the Continuous-Time models have this time consistent property.

This is because the process noise covariance is derived using the transition matrix, while the Direct Discrete-Time is arbitrary.

112

112

T

CWNAT

T

CWNAT

T

CWNAT

T

DWNAT

T

DWNAT

T

DWNAT

QFQFPFFQFPF

QFQFPFFQFPF

Kalman Filter:Steady State Gains

Kalman Filter:Steady State Gains

If we iterate the Kalman equations for the covariance indefinitely, the updated covariance (and thus the Kalman gain) will reach steady state.

This is only true for Kalman models that have constant coefficients.

In this case, the steady-state solution is found using the Algebraic Matrix Riccati Equation (pp. 211 & 350):

The steady state Kalman gain becomes:

QFHPRHHPHPPFP Tss

Tss

Tssssss

1

1 RHHPHPK T

ssT

ssss

Kalman Filter:Steady State Biases

Kalman Filter:Steady State Biases

If a Kalman filter has reached steady-state, then it is possible to predict the filter’s bias resulting from un-modeled dynamics.

Consider the CV model with an un-modeled constant acceleration (p. 13):

The steady-state error is found to be:

T

KT

TGxGxFx ss

k

kkkkk

22

11

Un-ModeledAcceleration

T

TxGHKIxFHKIx ssssssssss 2

1

~~~

2

Kalman Filter – Summary #2Kalman Filter – Summary #2 The Kalman Gain:

– Reaches steady-state for constant coefficient models– Can determine steady-state errors for un-modeled dynamics

The Covariance Matrix:

– Is only consistent for when model matches true– Has no knowledge of the residuals

The Kalman Estimator:

– We need modifications for more general models– What about non-linear dynamics?

Non-Linear FilteringNon-Linear Filtering

Nonlinear Estimation ProblemsNonlinear Estimation Problems Previously, all dynamics and measurement models were linear. Now, we

consider a broader scope of estimation problems:

Nonlinear Dynamics:

– Ballistic Dynamics (TBM exo-atmospheric, Satellites, etc…)

– Drag/Thust Dynamics (TBM re-entry, TBM Boost, etc…)

Nonlinear Measurements:

– Spherical Measurements

– Angular Measurements

– Doppler Measurements

)(),()(

)(~)(),(

twtxhtz

tvtuDtxfx

EKF - Nonlinear DynamicsEKF - Nonlinear Dynamics

The state propagation can be done using numerical integration or a Taylor Series Expansion (linearization):

However, the linearization is necessary in order to propagate the covariance:

The state and covariance propagation are precisely as before:

1|11|11|1

1

ˆ2

221

ˆ1ˆ

)()(1

111111

2

)()(

kkkkkk

kk

xx

kk

xxkkxx

ttxfk

kkkkkkk

x

ftt

x

fttIeF

vuGxFx

Jacobian Matrix

Tkkk

Tkkkkkk

kkkkkkk

QFPFP

uGxFx

11111|111|

111|111| ˆˆ

Hessian Matrix

EKF - Nonlinear MeasurementsEKF - Nonlinear Measurements

We compute the linearization of the observation function:

The residual is thus:

The covariance update and Kalman gain are precisely as before (381-386):

1|1| ˆ2

2

ˆ

21|1|1| )ˆ()ˆ()ˆ()(

kkkk xx

kxx

k

kkkkkkkkkkk

x

hH

x

hH

xxHxxHxhxh

Jacobian Matrix

kkkkkkkkkkk wxHxhzzz

1|1|1|~)ˆ(ˆ

1

1|

1||

1|| ˆˆ

kTkkkk

Tkkk

Tkkkkkkkk

kkkkkk

SHPK

KRKHKIPHKIP

Kxx

Hessian Matrix

Polar MeasurementsPolar Measurements

Previously, we dealt with unrealistic observation models that assumed that measurements were Cartesian. Polar measurements are more typical. In this case, the observation function is nonlinear:

The Kalman Gain and Covariance Update only require the Jacobian of this observation function:

This Jacobian is evaluated at the extrapolated estimate.

Tkkkkkkkkk yyxxxyx

yxb

rxhwxhz

)(tan)()(

1

22

00

00

2222

2222

yx

x

yx

yyx

y

yx

x

x

hH

Ballistic DynamicsBallistic Dynamics

As a common example of nonlinear dynamics, consider the ballistic propagation equations specified in ECEF coordinates:

The gravitational acceleration components (to second order) are:

)(~22 22 tvDGzGyxyGxyx

zzyyxxT

zyx

T

22

23

22

23

22

23

532

31

512

31

512

31

R

z

R

RJ

R

z

R

z

R

RJ

R

y

R

z

R

RJ

R

x

G

G

G

G

e

e

e

z

y

x

Thrust and Re-entry DynamicsThrust and Re-entry Dynamics

As a common example of nonlinear dynamics, consider the ballistic propagation equations specified in ECEF coordinates:

The new states are the relative axial acceleration “a” and the relative mass depletion rate “”:

The process noise matrix (if extended to second order) becomes a function of the speed. Thus, a more rapidly accelerating target while have more process noise injected into the filter.

)(~2 tvDa

s

zaGz

s

yaay

s

xaax

azzyyxx

zBallisticy

Ballisticx

T

)(

21

)()()(tm

vvACTtatata

CD

dragthrust

)(

)()(

tm

tmt

Pseudo- MeasurementsPseudo- Measurements

In the case of the TBM dynamics, the ECEF coordinates are the most tractable coordinates.

However, typically the measurements are in spherical coordinates.

Furthermore, the Jacobian for the conversion from ECEF to RBE is extremely complicated.

Instead, we can convert the measurements into ECEF as follows:

However, since this is a linearization, we must be careful to make sure that this approximation holds.

Tmeaskmeaskkkxk JRJRwxIz '

33

Pseudo- MeasurementsPseudo- Measurements

The linearization is valid so long as (pp. 397-402):

4.02

r

br

Iterated EKFIterated EKF The IEKF iteratively computes the state “n” times during a single update.

This recursion is based on re-linearization about the estimate:

The state is updated iteratively with a re-linearized residual and gain:

Finally, the covariance is computed based upon the values of the final iteration:

1

ˆ1|ˆˆ1|

|1|ˆ|

1|1

|

|||

|

ˆˆ)ˆ(

ˆˆ

k

T

x

ikkkx

ik

T

x

ikkk

ik

ikkkkx

ik

ikkk

ik

ik

ikkk

ikk

RHPHHPK

xxHxhz

Kxx

ikk

ikk

ikk

ikk

nix

hHwhere

x

hH

kkikk x

kx

ik ,...,1,0

1|| ˆ

0

ˆ

Tnkk

nk

T

x

nk

nkkkx

nk

nkkk KRKHKIPHKIP

nkk

nkk

|| ˆ1|ˆ|

Multiple-Model FilteringMultiple-Model Filtering

Why Multiple Models?Why Multiple Models? When the target dynamics differ from the modeled dynamics, the state

estimates are subject to:

– Biases (Lags) in the state estimate

– Inconsistent covariance output

– In a tracking environment, this increases the probability of mis-association and track loss

In most tracking applications, the true target dynamics have an unknown time dependence.

To accommodate changing target dynamics, one can develop multiple target dynamics models and perform hypothesis testing.

This approach is called hybrid state estimation.

Why Multiple Models?Why Multiple Models?

Assuming a Constant

Velocity target dynamics, the

estimation errors become

inconsistent during an

acceleration

Assuming a Constant

Velocity target dynamics, the

estimation errors become

inconsistent during an

acceleration

1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800-60

-40

-20

0

20

40

60

80

1006-State Velocity Error (Meters/Sec) (x=red, y=blue, z=green, True Error=black)

Confidence intervalConfidence intervalGiven by KalmanGiven by Kalman

CovarianceCovariance

True state estimate errorTrue state estimate error

Why Multiple Models?Why Multiple Models?

A Constant Acceleration

model remains consistent,

however the steady-state estimation

error is larger

A Constant Acceleration

model remains consistent,

however the steady-state estimation

error is larger

1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800-200

-150

-100

-50

0

50

100

150





Adaptive Process NoiseAdaptive Process Noise

Since the normalized innovations squared indicate the consistency of the dynamics model, it can be monitored to detect deviations (pp. 424-426).

At each update, the perform the following threshold test:

Then, the process noise value is adjusted such that the statistical distance is equal to this threshold value:

The disadvantage is that false alarms result in sudden increases in error.

We can use a sliding window average of these residuals, however this can delay the detection of a maneuver (pp. 424-426).

max

2max

12kkk

Tkk dPSd

max1

qkk

Tk S

State Estimation – Multiple ModelsState Estimation – Multiple Models

We can further assume that the true dynamics is one of “n” models:

nrwxHz

nrvxFxrk

rk

rk

rk

rk

rk

rk

rk

,,2,1

,,2,1

1

111

Using Kalman filter outputs, each model likelihood function can be computed:

nr

S

S

rk

rk

rk

Trk

rk ,,2,1;

2det

21

exp1

At each filter update, the posterior model probabilities “ki ” are computed

recursively using Baye’s Theorem. The proper output can be selected using these probabilities (pp. 453-457).

State Estimation – SMMState Estimation – SMM

Each Kalman filter is updated independently and has no knowledge about the performance of any other filter.

This approach assumes that the target dynamics are time-independent

Measurement

Hypothesis 1

Hypothesis 2

Hypothesis “n”

1ˆkx

2ˆkx

nkx̂

HypothesisSelection

kx̂Most ProbableState Estimate

n

j

jk

jk

ik

iki

k

11

1

State Estimation – IMMState Estimation – IMM

Each Kalman filter interacts with others just prior to an update

This interaction allows for the possibility of a transition

This approach assumes that the target dynamics will change according to a Markov process. pp. 453-457.

Con

diti

onal

Pro

babi

lity

Upd

ate/

Sta

te E

stim

ate

Inte

ract

ion

Hypothesis 1

Hypothesis 2

Hypothesis “n”

ProbabilityUpdates

1ˆkx

2ˆkx

nkx̂

EstimateMixing

kx̂

11ˆ kx

21ˆ kx

nkx 1ˆ

IMM Estimate

Measurement

State Estimation – IMMState Estimation – IMM

Interaction

1|11

1|11 ,ˆ kkkk Px 1|1

21|1

2 ,ˆ kkkk Px

1|11,0

1|11,0 ,ˆ kkkk Px 1|1

2,01|1

2,0 ,ˆ kkkk Px

KalmanFilter

KalmanFilterkz

1k

kkkk Px |1

|1 ,ˆ

2

1

11

k

k

kkkk Px |2

|2 ,ˆ

Prob.Updates

2k kz

21

11, kk

EstimateMixing

kkkk Px || ,ˆ

State Estimation – Applied IMMState Estimation – Applied IMM

IMM adapts to changes in

target dynamics and

provides a consistent covariance

during these transitions

IMM adapts to changes in

target dynamics and

provides a consistent covariance

during these transitions

1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800-100

-80

-60

-40

-20

0

20

40

60

80





State Estimation – IMM Markov MatrixState Estimation – IMM Markov Matrix

The particular choice of the Markov Matrix is somewhat of an art.

Just like any filter tuning process, one can choose a Markov Matrix simply based upon observed performance.

Alternatively, this transition matrix has a physical relationship to the Mean Sojourn Time of a given dynamics state.

iscan

iii

iii E

T

NEp

pNE

1

11

1

1

State Estimation – VSIMMState Estimation – VSIMM

Air Targets: Adaptive Grid Coordinated Turning Model

TBM Targets: Constant Axial Thrust, Ballistic, Singer ECA

The SPRT is performed as follows:

Measurement

Air IMM

TBM IMM

Airkx̂

TBMkx̂

HypothesisSelection kx̂SPRT

1

1

;ˆ

;

;ˆ

12

1

21

2

TandT

Tx

TandTmixed

Tx

TBMk

Airk

TBMk

Airk

VSIMM SPRTVSIMM SPRT

The IMM model set probabilities are found using the Total Probability Theorem and Baye’s Theorem:

Airkn

i

ikij

n

j

jk

n

i

ikij

n

j

jk

n

i

ikij

n

j

jk

TBMkn

jj

jk

n

jj

jk

n

jj

jk

TBMk

Airkn

i

ikij

n

j

jk

n

i

ikij

n

j

jk

n

i

ikij

n

j

jk

Airkn

jj

jk

n

jj

jk

n

jj

jk

Airk

TBMTBMAirAir

TBMTBM

TBMAir

TBM

TBMTBMAirAir

AirAir

TBMAir

Air

pp

p

cc

c

pp

p

cc

c

1

11

111

1

11

11

11

1

1

11

111

1

11

11

11

1

IMMCombinedLikelihoodFunction

State Estimation – Applied IMMState Estimation – Applied IMM

TBM-IMM results for

Terrier-Lynx Target

TBM-IMM results for

Terrier-Lynx Target

Tracking BasicsTracking Basics

Tracking BasicsTracking Basics

Track Maintenance

– Track Score, Track While Scan– Confirmation, Deletion

Data Association

– Coarse Gating and Fine Gating– GNN, 2D Assignment, JPDA– Track Resolution

Activity Control

– Characterization of Clutter– Track Score Thresholds

Performance Results

Architecture OverviewArchitecture Overview

The tracker maintains bearing zones which are used to quickly access tracks in the database.

The association process is event driven based upon the receipt of an Unassociated Measurement Report (UMR).

Each UMR is processed by several layers of association and is tagged with a contact usage which indicates whether a contact is unused or an Associated Measurement Report (AMR).

Discriminating between UMR’s and AMR’s is the fundamental difference between state estimation and tracking. This is because the origin of each measurement is not known.

Important concepts in data association are coarse gating, fine gating, and track resolution.

The incorporation of concepts such as Probability of Detection (PD), Probability of False Alarm (PFA), and Clutter Density( ) makes a tracker more robust. These quantities are used by a process called Activity Control.

Track MaintenanceTrack Maintenance


The track maintenance component is responsible for taking measurements that have not been associated with any tracks and determining whether they form tracks.

This is done using the SPRT discussed before. As a result, the track maintenance function has the ability to confirm or delete potential tracks.

In this context, the Type I Error () is the probability of false confirmation while the Type II Error () is the probability of false deletion.

The probability ratio that we wish to entertain is the ratio of true target (T) to false detection (F):

TPFZP

TPTZP

ZFP

ZTPFTPR

0

0

|

|,


Recall that this can be re-writing in a recursive form as follows:

These likelihood functions assuming that a contact is obtained:

For example, for 3D spherical measurements:

n

kk

n

k k

kn FT

FP

TP

FzP

TzP

FP

TPFTPR

10

0

10

0 ),(|

|,

CDFAk

n

rr

rkk

VPPFzP

cTzP

|

|1

ContactVolume

ee

ee

bb

bb

rr

rr

ee

ee

bb

bb

rr

rrC dedbdrerdVV )cos(2


If a track is updated “with a miss”, that is no contact associated with it (track while scan):

The Probability Ratio can be recursively updated (in logarithm space):

FA

D

PFmissP

PTmissP

1|

1|

update

VPP

c

missP

P

FT

CDFA

n

rr

rk

FA

D

k

;ln

;1

1ln

),(ln1


The Natural Logarithm of the Probability Ratio is called the Track Score of a given track. This is monitored at each update and compared to the thresholds given by Wald’s Theorem:

Where the Type I Error is computed based upon a desired False Track Rate per Hour (NFC):

1ln

1ln

;

;

;

,ln

12

1

21

2

TandT

TF

TandTcontinue

TT

FTPRk

FA

FC

N

N

000,36

Number of False Alarmsper Second

PrefirmTracks

Data AssociationData Association

UMR Processing – Course GatingUMR Processing – Course Gating A rectangular coarse gate centered on the UMR is created.

To avoid extrapolating all tracks in the database, the coarse gate is made large enough to accommodate at least one scan period. In addition, it accommodates the maximum target velocity and maximum target acceleration, and measurement error:

Then, the series of bearing zones which enclose this coarse gate is determined:

Sensor Location

UMR

2maxmax

2maxmax

5.0

5.0

scanscanmeas

scanscanmeas

TaTsyy

TaTsxx

UMR Processing – Fine GatingUMR Processing – Fine Gating

We gather all tracks in these bearing zones and extrapolate them to the time of the UMR.

The residuals are then computed and the minimum statistical distance is obtained for each contact-track pairing “i-j ”.

This is compared to a gate “G” for a specified miss probability (Pmiss):

The volume of this gate “VG” is given by:

nrwhereGSd rij

rij

Trij

rij ,,2,1}{min

12

2exp22exp2

1110

22 GGdxxGdPPPG

ijGmiss

2122

)12( ijn

z

n

G SGn

V z

z

UMR Processing – GNNUMR Processing – GNN

The Global Nearest Neighbor (GNN) scheme is the simplest and most widely used method.

In the GNN approach, the UMR is associated with the track having the smallest minimal statistical distance.

In a multi-target environment, this is troublesome because the statistical distance becomes smaller with larger covariance. As a result, a poorer quality track can tend to “steal” measurements from other higher quality tracks.

To penalize poorer quality tracks, the UMR can be associated with the track having the smallest log-likelihood:

n

rr

rijijij cd

1

2 ln)ln(~

Combined IMM Likelihood

UMR Processing – GNN with IMMUMR Processing – GNN with IMM

Most of the benefit of the IMM filter is due to the consistent error covariance even during maneuvers.

More consistent covariance output means a reduction in the miss Probability.

In the case of closely spaced maneuvering targets, it is critical to have consistent errors to avoid “measurement swapping”.

An example of the decreased probability of “measurement swapping” as a result of IMM filtering is presented next.

Standard Kalman Filter TrackerStandard Kalman Filter Tracker

Maneuvering Track

Non-Maneuvering Track

Measurement

Filter Extrapolation (Kalman)

Extrapolation Error (Kalman)

Extrapolation Error (Kalman)

Tn 21102110

Tm 290290

Belongs to Non-Maneuvering Track

Standard Kalman Filter TrackerStandard Kalman Filter Tracker

6307.19ln~ 12

nnnTnn SSd

2307.19ln~ 12

mmmT

mm SSd

The maneuvering target will “steal” the measurement that belongs to the non-maneuvering target.

The extrapolation is incorrect (straight line) and the covariance is inconsistent (optimistic).

IMM Filter TrackerIMM Filter Tracker

Maneuvering Track

Non-Maneuvering Track

Measurement

Filter Extrapolation (IMM)

Extrapolation Error (IMM)

Extrapolation Error (IMM)

Tn 21102110

Tm 21052105

Belongs to Non-Maneuvering Track

IMM Filter TrackerIMM Filter Tracker

6307.19ln~ 12

nnnTnn SSd

8317.19ln~ 12

mmmT

mm SSd

The measurement is correctly associated with the non-maneuvering target.

The extrapolation is more correct and the covariance is consistent.

Is this scenario realistic?

4539.2exp22exp2

2

0

2 GGdxxPG

G

UMR Processing – 2D AssignmentUMR Processing – 2D Assignment

In this approach, the measurement-to-track association log-likelihood scores are used in combination with a penalty function to form a cost function.

These costs are then placed in a 2D assignment matrix as:

4321 TTTT

4

3

2

1

M

M

M

M

44434241

14131211

CCCC

CCCC

UMR Processing – 2D AssignmentUMR Processing – 2D Assignment

The particular form of the cost function depends upon the application and the types of data that are available.

The most common cost function used is derived by penalizing associations which deviation from the maximum likelihood gate “GML”:

This cost function makes it harder to make a new track than it is to accept a marginal association.

The optimal assignment problem defined above can be solved by a variety of algorithms such as: Auction Munkres, JVC, etc…

ijFAn

D

CDMLijMLij

SPP

VPjiGdjiGC

z 2

2

)2)(1(ln2),(

~),(

UMR Processing – JPDAUMR Processing – JPDA

In the GNN method, we have not considered the possibility of extraneous measurements or the probability of detection.

Consider the following non-trivial example:

Track #2Track #2

Measurement #2

Measurement #3Measurement #1


The JPDA association probabilities are found by considering all possible assignment possibilities and scoring each.

For “No” measurements and “NT” tracks, we have “h” hypothetical combinations with contact density “”:

Each of these hypotheses has the following association probability:

GoN VNandNh T

00 1

h

jj

ll

N

tGDD

N

j

ttjol

P

PPPPPP

T

tt

ojl

1

1

1

1

)( )1()()(

j

t

l

“1” if measurement “j” is assigned to a track

“1” if track “t” has an assignment

Number of measurements not assigned to any tracks


Hypothesis #1: All measurements are clutter; No measurements are assigned to a track.

Hypothesis #2: Measurement “3” and Track “2”.

Hypothesis #3: Measurement “1” and Track “1”, Measurement “2” and Track “2”:

231 )1( GDo PPP

30,00,0,0 1 tj

)1(232

2 GDDo PPPP

21,01,0,0 2 tj

222113 Do PP

11,10,1,1 3 tj

UMR Processing – JPDA MethodsUMR Processing – JPDA Methods

There are two ways in which these association probabilities can be used.

In a straight JPDA approach, these association probabilities are used as weighting factors for each residual and the filter is updated using an effective residual which is a combination of all measurements.

In the early 1990’s, this technique was found to cause a phenomenon known as track merging where closely spaced tracks would actually merge together and become a single track.

To avoid this phenomenon, the Nearest Neighbor JPDA (NNJPDA) updates the tracking filter with only the measurement having the highest association probability.

UMR Processing – JPDA MethodsUMR Processing – JPDA Methods

Kalman Filter & GNN

Kalman Filter & JPDA

IMM Filter & JPDA

UMR Processing – Track ResolutionUMR Processing – Track Resolution As the radar continues to scan, the tracker waits a given time period called

the resolution period after it would expect the radar to have visited a bearing zone before resolution is performed:

All tracks within the bearing zone are visited and they are updated with the “best” association (now called an AMR) given by the chosen algorithm.

If there are no associations that meet the minimum statistical distance threshold, then the track is updated with a miss and its track score is penalized accordingly.

Sensor Location

UMR

Sensor Pointing Position

resolve this bearing zone

Activity ControlActivity Control


The following inputs are used by Activity Control:

– Maintains radar performance parameters

Maximum range, Minimum range, Angular Limits Dimensionality of Measurements - nz Nominal Probability of Detection - PD

– Allows track maintenance requirements to be imposed

Probability of deleting a true track – Type II Error - Rate of confirming a false track – Type I Error per Hour - NFC


The following are computed by Activity Control:

– Given Contact Usage History of “n” scans:

Expected number of false alarms per second – NFA Track Confirmation Threshold – T2 Track Deletion Threshold – T1 False alarm probability - PFA

– Given Contact-to-Track Association Information:

Association Gate Size - VG Measurement Volume Size - VC Track disclosure quantities – Variance Reduction Ratio, etc…

Contact History

Activity Control

Nav History

Track Database

Scan Sync

scan periodinterpolated position

revisit periodnorth crossing

ownship ratesownship position

extrapolated position

correlation pairscorrelation triplets

prefirm trackscontact usage history

coarse gates

local clutter densityconfirm thresholddeletion threshold

prefirm tracksfirm tracks

track scoretrack disclosure

track promotiontrack deletion

state estimationcontact association

North CrossingInput

NavigationInput

Radar DetectionsInput

Tracker Subcomponents: Standalone Mode

Tracker Subcomponents: Standalone Mode

Track UpdatesAMR’s

Performance ResultsPerformance Results

Current Performance DataCurrent Performance Data

Tracker output (colored) compared to scenario

ground truth data (white)

62 Real Targets

62 Track Numbers

Range Acc ~ 80 mBearing Acc ~ 2 mrad

Elevation Acc ~ 15 mradScan Period ~ 5 sec

Tracker output (colored) compared to scenario

ground truth data (white)

62 Real Targets

62 Track Numbers

Range Acc ~ 80 mBearing Acc ~ 2 mrad

Elevation Acc ~ 15 mradScan Period ~ 5 sec


Crossing Targets are maintained without track

swapping

This is due to the combined IMM

likelihood function used for

association

Crossing Targets are maintained without track

swapping

This is due to the combined IMM

likelihood function used for

association


Tracker handles a wide range of

maneuver sizes

1-15 coordinated turns

No track loss

Negligible lag

“X” symbols indicate

extrapolated track updates

Tracker handles a wide range of

maneuver sizes

1-15 coordinated turns

No track loss

Negligible lag

“X” symbols indicate

extrapolated track updates

"Modern Tracking" Short Course Taught at University of Hawaii

Education

Transcript of "Modern Tracking" Short Course Taught at University of Hawaii