Data Fusion and Multiple Models Filtering for Launch Vehicle Tracking and Impact Point Prediction: The Alcântara Case – Julio Cesar Bolzani de Campos Ferreira

Data Fusion and Multiple Models Filtering for Launch Vehicle Tracking and Impact Point Prediction: The Alcântara Case – Julio Cesar Bolzani de Campos Ferreira 15/12/2004

Data Fusion and Multiple Models Filtering Data Fusion and Multiple Models Filtering for Launch Vehicle Tracking and Impact for Launch Vehicle Tracking and Impact Point Prediction: The Alcântara CasePoint Prediction: The Alcântara Case

Julio Cesar Bolzani de Campos FerreiraJulio Cesar Bolzani de Campos FerreiraProfessional Master Dissertation – 15/12/2004Professional Master Dissertation – 15/12/2004


ContentsContents

Introduction

Kalman Filter

ProposedApproach

Reference CoordinateTransformations

Multiple HypothesisTesting

Data FusionProcess

Comparison ofData Fusion Methods

Impact PointPrediction

Target Models


IntroductionIntroduction



Addressed Technology Potential ApplicationsAddressed Technology Potential Applications

Air Traffic Control (ATC)Air Traffic Control (ATC)

Robot GuidanceRobot GuidanceAir and Ground SurveilanceAir and Ground Surveilance

Remote SensingRemote Sensing

Launch Vehicle TrackingLaunch Vehicle Tracking


Poor estimates may cause payload loss

Demands accurate estimation of position and velocity for prediction of the orbit parameters

Poor estimates may cause payload loss

Demands accurate estimation of position and velocity for prediction of the orbit parameters


Payload Orbital InjectionPayload Orbital Injection



Impact Point PredictionImpact Point Prediction

IPP has a fundamental role in safety-of-flight

Relies on vehicle position and velocity estimates

IPP has a fundamental role in safety-of-flight

Relies on vehicle position and velocity estimates


ProposedApproach


Proposed ApproachProposed Approach

Problem OverviewProblem Overview

PropelledPropelledFlightFlight

Free FlightFree Flight

ParachuteParachuteDeployedDeployed

Long DistanceLong Distance

Short DistanceShort Distance



CI FusionCI Fusion

ADOUR

ATLAS

CIFUSION

OUTPUT

Exploits the complementary characteristics of SHORT and LONG range radars.



Kalman FilteringKalman Filtering

ADOUR

ATLAS

CIFUSION

OUTPUT

KalmanFilter

KalmanFilter

Provides position, velocity, and acceleration estimates and

their corresponding covariance.


CIFUSION

OUTPUT


Multiple Hypothesis TestingMultiple Hypothesis Testing

ADOUR

ATLAS

KFBALLISTIC

KFPROPELLED

MHT

KFBALLISTIC

KFPROPELLED

MHT

Multiple models cover both propelled and ballistic flight

behaviors.



Reference Frame TransformationsReference Frame Transformations

CIFUSION

OUTPUT

KFBALLISTIC

KFPROPELLED

MHT

KFBALLISTIC

KFPROPELLED

MHT

ADOUR

ATLAS

Fusion is performed in a common reference frame, demanding local-level estimates to be rotated and

translated.



De-biased Spherical-to-Cartesian TransformationDe-biased Spherical-to-Cartesian Transformation

CIFUSION

OUTPUT

KFBALLISTIC

KFPROPELLED

MHT

KFBALLISTIC

KFPROPELLED

MHT

ADOUR

ATLAS

Cartesian coordinates are appropriate to accomplish the

necessary rotation and translation transformations.


ReferenceReferenceCoordinateCoordinate

FramesFrames


Reference Coordinate FramesReference Coordinate Frames


Subtracting the bias…

)1(sin

)1(sincos

)1(coscos

2/

2/2/

2/2/

22

2222

2222

eer

eer

eer

z

y

xDe-biased

transformation

BiasedTransformation

sin

sincos

coscos

r

r

r

z

y

xPURE GEOMETRICAL

TREATEMENT




Target distance and signal-to-noise ratio (SNR) affect the slant range variance.

2 2 2, ,radar rdiag R

2

22 2 2 60 1, , 50.10

2radar cte

cdiag D

SNR

R

SNR data

D

Transforming from uncorrelated spherical measurement errors into de-biased cartesian ones gives rise to correlated measurement errors.



Radar Frame to Launch-Pad Frame TransformationRadar Frame to Launch-Pad Frame Transformation

Rotação +

Translação

Z

Y

X

(1,1)

Y

X

Z

(2,2)




11 radar radar

radar radar

1 0 0

0 cos( ) sin( )

0 -sin( ) cos( )

R

lp radar lp radar

21

lp radar lp radar

cos( - ) 0 -sin( - )

0 1 0

sin( - ) 0 cos( - )

R

12 lp lp

lp lp

1 0 0

0 cos( ) -sin( )

0 sin( ) cos( )

R

12 21 11radarlp D R R R

xy

z

xy

z

xy

z




Tx y z x y z x y zX

, ,lp radar radar radarradar lp lp lpdiag D D DD

0 0 0 0 0 0Tradar

lp x y z T

radar radarlp lp radar lp X D X T

radar radar radarTlp lp radar lpP D P D


TargetTargetModelsModels


Target ModelsTarget Models

Singer’s Classical ModelSinger’s Classical Model

PMAX PMAX

-AMAX AMAX0

P0

p(a)

a

01 2

2MAX

MAX

P P

A

00

1 2

2MAX

MAX MAX MAX MAXMAX

P Pf x P x A P P x A

A

xf x dx

22 x f x dx

2

201 4

3MAX

MAX

AP P

= 0




22 2

11 11 ( 1 )1 ( 1 ) 1 ( 1 )

1 1 10 1 (1 ) , 0 1 (1 ) , 0 1 (1 )

0 00 0 0 0

yx z

yx z

zxy

TT Tyx z

yx z

TT T

x y z

TT T

T T eT T e T T e

diag e e e

ee e

Φ

Tx x x y y y z z zX

T

T

n

Tn

n

n

n

n

n

e

e

eTT

T

00

)1(1

10

)1(1

1

),(

2

State Transition MatrixState Transition Matrix



Singer’s Adapted ModelsSinger’s Adapted Models


Single Side p.d.f.Single Side p.d.f.

Propulsion

Ballistic

Shifted Gate p.d.f.Shifted Gate p.d.f.

Propulsion

Ballistic



Single Side P.D.F. / Shifted Gate P.D.F.Single Side P.D.F. / Shifted Gate P.D.F.

PMAX

AMAX0

P0

p(a)

a

01 MAX

MAX

P P

A

012MAX

MAX

AP P

2

2 20 01 2 1

3MAX

MAX MAX

AP P A P

PMAX

A0

p(a)

a

1 MAX

MAX MIN

P

A A

AMAXAMIN

2 21( )

2 2MAX MAX MIN

MAXMAX MIN

P A Ax AP

A A

3 3

2 2 21

3 3MAX MAX MIN

MAXMAX MIN

P A AA P

A A

Since acceleration mean for both models is non-zero it must be considered in the target equation of motion. Thus, an inhomogeneous driving input must be calculated.



Inhomogeneous Driving Input for Biased ModelsInhomogeneous Driving Input for Biased Models

1 Taccu k e

2

( 1)

1 ( 1) 1 1 1 exp 10

0 1 1 1 exp 1 0

10 0 exp 1

k T

kT

k T k T k T

k k T w d

k T

u


KalmanKalmanFilterFilter


The Kalman FilterThe Kalman Filter

State VectorState Vector

Tx y z x y z x y zX

Used for coordinate frame transformations, implementing rotations through a 9x9 block

diagonal matrix.

Used for coordinate frame transformations, implementing rotations through a 9x9 block

diagonal matrix.

Tx x x y y y z z zX

Used for filtering, also through a 9x9 block diagonal matrix.

Used for filtering, also through a 9x9 block diagonal matrix.



Singer’s Classical Model – ParametersSinger’s Classical Model – Parameters

0.05 PMAX=0.05

-10m/s2 AMAX=10m/s20

P0=0.1

p(a)

a

0.04

HorizontalHorizontalAxisAxis

0.05 PMAX=0.05


P0=0.1

p(a)

a

0.008

VerticalVerticalAxisAxis



Single Side P.D.F. – ParametersSingle Side P.D.F. – Parameters

PropulsionPropulsionModelModel

0.05 PMAX=0.05


P0=0.1

p(a)

a

0.04

0.7

80m/s20

p(a)

a

0.0038

VerticalVerticalChannelChannel

Horizontal Horizontal ChannelChannel

BallisticBallisticModelModel



0.05 PMAX=0.05

-5m/s2 AMAX=5m/s20

P0=0.1

p(a)

a

0.04

-10m/s2 0

1

p(a)

a



Single Side P.D.F. – Parameter AdjustmentSingle Side P.D.F. – Parameter AdjustmentA

tla

s R

ad

ar

Atl

as

Ra

da

rA

do

ur

Rad

ar

Ad

ou

r R

ada

r

Propelled PhasePropelled Phase Ballistic PhaseBallistic Phase



Shifted Gate P.D.F. – ParametersShifted Gate P.D.F. – Parameters

PropulsionPropulsionModelModel



0.9

90m/s20

0

a

0.005

70m/s2 75m/s2

0.05 PMAX=0.05


P0=0.1

p(a)

a

0.04

BallisticBallisticModelModel



-10m/s2 0

0.3

p(a)

a

0.07

-5m/s2-15m/s2

0.05 PMAX=0.05

-5m/s2 AMAX=5m/s20

P0=0.1

p(a)

a

0.04



Shifted Gate P.D.F. – Parameter AdjustmentShifted Gate P.D.F. – Parameter AdjustmentA

tla

s R

ad

ar

Atl

as

Ra

da

rA

do

ur

Rad

ar

Ad

ou

r R

ada

r

Propelled PhasePropelled Phase Ballistic PhaseBallistic Phase


MultipleMultipleModelsModels


Multiple ModelsMultiple Models

ConceptConcept

Applicability of model 3



State space of interest



Multiple Hypothesis Testing (MHT)Multiple Hypothesis Testing (MHT)

Sensors

Filter 1

Filter 2

Filter n

ProbabilityCalculation

CombineEstimates Output

estimate



Multiple Hypothesis Testing (MHT)Multiple Hypothesis Testing (MHT)

11

22

1 1exp | 1

22 det | 1

Ti i i im

i

k k k k kk k

r S rS

1

ˆ ˆ|n

i iik k k

y y

,1ˆ ˆ ˆ ˆ| | | | | |

Tn

yy i i yy i iik k k k k k k k k k k k k

P P y y y y



MHT Probability Along TrajectoryMHT Probability Along Trajectory

Radar AdourRadar Adour Radar AtlasRadar Atlas



MHT Covariance Output AnalysisMHT Covariance Output AnalysisMultiple ModelsMultiple Models

MHT Covariance Output AnalysisMHT Covariance Output Analysis



Switching ModelsSwitching Models

11 1 when 1 211 0 when 1 2

1 1 1

Propelled

Propelled

Ballistic Propelled

Output Propelled

Output Propelled

Output Output

k k

k k

k k




MHT Covariance Output AnalysisMHT Covariance Output Analysis



MHT Vertical Acceleration ResultsMHT Vertical Acceleration Results



DataDataFusionFusionProcessProcess


The Data Fusion ProcessThe Data Fusion Process

Issues on System’s StatisticsIssues on System’s Statistics

a bc W a+W bT T T T

acc a aa a a ab b b ba b bb bP = W P W +W P W +W P W +W P W

Linear Update and CovarianceLinear Update and Covariance

T T T Tcc a aa a a ab b b ba a b bb bP = W P W +W P W +W P W +W P W

True CovarianceTrue Covariance

0 ab abP P

0abP

Consistency AssuredConsistency Assured

Consistency NOT AssuredConsistency NOT Assured



Covariance Intersection – Geometric InterpretationCovariance Intersection – Geometric Interpretation

Kalman Filter(independence between

Paa and Pbb)

CovarianceIntersection

Pcc for many choices of Pab



Covariance Intersection EquationsCovariance Intersection Equations

CI EquationsCI Equations

1 1 11 ... n

1 1 n ncc a a a aP P P

1 1 11 1 ...

n nn n 1 1cc a a a aP c P a P a

11 ini

The The nn parameters are used to minimize the determinant parameters are used to minimize the determinant

of Pof Pcccc and is recalculated for every update. and is recalculated for every update.



CI Results – Singer’s Classical ModelCI Results – Singer’s Classical Model















CI Results – Shifted Gate P.D.F. ModelCI Results – Shifted Gate P.D.F. Model














Comparison ofComparison ofData FusionData Fusion

MethodsMethods


Comparison of Data Fusion MethodsComparison of Data Fusion Methods

Measurement FusionMeasurement Fusion

Fusion

xk-1|k-1 xk|k-1 xk|k

Kalman Filtering

Prediction Correction

z-1

zk1 zk

2



Measurement FusionMeasurement Fusion

ADOUR

ATLAS

Spherica-to-cartesian transformation

rotation andtranslation

Spherical-to-cartesian transformation

Kalman Filtering

(Singer’s Classical)

rotation andtranslation

MeasurementFusion

FusedEstimate

11 1 1 2 2 1k k k k k k kz z R R R z z

11 11 2

k k kR R R



Measurement Fusion ResultsMeasurement Fusion Results



Measurement Fusion ResultsMeasurement Fusion Results



Track-to-track FusionTrack-to-track Fusion

Fu

sio

n

xk-1|k-1 xk|k-1 xk|k

Kalman Filter #2

Prediction Correction

z-1

xk-1|k-1 xk|k-1 xk|kPrediction Correction

z-1

Kalman Filter #1



Track-to-track FusionTrack-to-track Fusion

11 1 12 1 2 12 21 2 1| | | | | | | | | |ˆ ˆ ˆ ˆk k k k k k k k k k k k k k k k k k k kx x P P P P P P x x

12 1 1 12 2 2 1 1 2 2| 1 1| 1 1 1 1 1

T TT Tk k k k k k k k k k k k k k k k kP I K H F P F I K H I K H Q I K H

ADOUR

ATLAS


Kalman filtering(Singer’s Classical)rotation and translation


Kalman filtering(Singer’s Classical)

rotation and translation

Track-to-track Fusion

Fused Estimate



Track-to-track Fusion ResultsTrack-to-track Fusion Results



Track-to-track Fusion ResultsTrack-to-track Fusion Results



Computational EffortComputational Effort

Measurement Fusion

Fusion and filteringTotal:19.2s

Cost:1.019.2s

Track-to-track Fusion

KFAdour KFAtlas Fusion Total:30.5s

Cost:1.62.0s 3.8s 24.7s

Multiple Models and

CI

KFAdour

Ballistic

KFAdour

PropulsionMHTAdour Fusion

Total:71.3s

Cost:3.7

3.5s 3.6s 3.9s

60.0sKFAtlas

Ballistic

KFAtlas

PropulsionMHTAtlas

3.5s 3.6s 4.2s

NOTE: These CPU times consider a 500s tracking period for radar Atlas instead of 963s since raw data for both radars shall correspond to a same time interval in order to be fused.


Impact PointImpact PointPredictionPrediction



Reference System TransformationReference System Transformation

' ' '

' ' '

' '

ˆsin cos sin cos cos

ˆsin sin cos cos sin

ˆcos sin

x y z

x y z

x z

X X X

X X X

X X

AX x

y

z

ˆ ˆ ˆcos cos cos sin sinTR BR x y z

ˆ' sin cos sin sin cos '

ˆsin cos '

ˆcos cos cos sin sin '

x y z

x y

x y z

X X X

X X

X X X

X x

y

z

NED > Earth FrameNED > Earth Frame

When Transforming Position RB shall be addedWhen Transforming Position RB shall be added

Earth Frame > NEDEarth Frame > NED



Impact Point CalculationImpact Point Calculation

Mi

Ni

Mti

Ri

i

ti

Rti

O (centro da Terra)

Vi

dRi

dt

Ri didt



Covariance EllipsoidsCovariance Ellipsoids

1P

( )-1Position Covariance

( )-1Velocity CovarianceCovariance

Eigeinvalues providesellipsoid axis

Eigeinvectors providesellipsoid orientation




2

23

2( )

2

kkP k erf ke



Covariance EllipsoidsCovariance EllipsoidsVelocityVelocityEllipsoidEllipsoid

AccelerationAccelerationVectorVector

Velocity Vector

Velocity Vector

121 vertices per ellipsoid

14,641 (1212) impact pointson Earth’s surface

PositionPositionEllipsoidEllipsoid




PositionPositionEllipsoidEllipsoid

VelocityVelocityEllipsoidEllipsoid

AccelerationAccelerationVectorVector

Velocity Vector

Velocity Vector

Total of 121 impact pointson Earth’s surface



Effect of Neglecting Position CovarianceEffect of Neglecting Position Covariance



Impact Point Maximum UncertaintyImpact Point Maximum Uncertainty



Impact Area – Propelled FlightImpact Area – Propelled Flight

Ellipsoids Magnified 1000XEllipsoids Magnified 1000XImpact Area Magnified 20XImpact Area Magnified 20X



Impact Area – TrajectoryImpact Area – Trajectory

Ellipsoids Magnified 1000XEllipsoids Magnified 1000XImpact Area Magnified 20XImpact Area Magnified 20X


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Data Fusion and Multiple Models Filtering for Launch Vehicle Tracking and Impact Point Prediction: The Alcântara Case – Julio Cesar Bolzani de Campos Ferreira