Principles of Radar Tracking

Using the Kalman Filter to locate targets

Abstract

Problem-Tracking moving targets, minimize radar noise

Solution-Use the Kalman Filter to largely eliminate noise when determining the velocities and distances

Noise

• Error (noise) is described by an ellipse– Defined by variance and covariance in x

and y

• Two kinds of error– State– Measurement

TeamsReciproverse

Brian DaiJoshua NewmanMichael Sobin

LextenStephen ChanAdam LloydJonathan MacMillanAlex Morrison

History of the Kalman Filter

• Problem: 1960’s, Apollo command capsule

• Dr. Kalman and Dr. Bucy– Make highly adaptable iterative

algorithm– No previous data storage– Estimates non-measured quantities

(velocity)

• Later found to be useful for other applications, such as air traffic control

Dr. Kalman

Model

kkk

kkk

rHxy

qΦxx

1

xk: position and velocity (state) of the target at time k (k+1 is next time step)Φ: state transition matrix qk: uncertainty in the state due to “noise” (e.g. wind variation and pilot error)

yk: measurement at time kH: term that gets rid of velocity in Xr: measurement noise, dictated by our devices

Other Important Matrices

• P: error covariance matrix– Describes estimate accuracy

• K: Kalman gain matrix– Intermediate weighting factor between

measured and predicted

• I: identity matrix

Some Matrices

y

y

x

x

x

2222

2222

2222

2222

yyyyxyx

yyyyxxy

yxyxxxx

yxxyxxx

P

m

m

y

xy

22

22

mmm

mmm

yyx

yxx

R

Kalman Filter: Predict

kkk xΦx ˆˆ |1

QΦΦPP T

kkk |1

Kalman Filter: Correct

1|1|| ˆˆˆ kkkkkkkk xHyKxx

1|| kkkkk PHKIP

1

1|1|

RHHPHPK Tkk

Tkkk

Tools: Visual Basic• Matlib- an external matrix operations

library• Input format – text files, simulated

radar data• Console- data output

Tools: Excel Track Charts

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Truth

Our Results

Raw Data

Tools: Excel Residual Analysis

Residual Plot

0

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1.5

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2.5

3

Time (hr)

Re

sid

ua

l (m

i)

Our ResidualRaw Data residual

Filter Development: Cartesian Coordinates

• Filter Implemented• Test: Residual Analysis• Does it work?

Cartesian Residuals

Residual, Case 2

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Time (hours)

Res

idu

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es)

Our Residual

Raw Data residual

Filter Development: Polar Coordinates

• Prefiltering• Polar to

Cartesian conversion

• More appropriate data feed

• Error matrices– Redefine R

][2sin2

1

cossin

sincos

22222

222222

222222

mmmm

mmm

mmm

R

R

R

Ryx

Ry

Rx

Filter Development: Multiple Radars

• Mapping coordinates to absolute coordinate plane

• Two radars means a smaller error ellipse

• Note drop in residual– Switch to second

radar

Residual, Case 4

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33

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00

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Time (hours)

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idu

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mil

es)

Our Residual

Raw Data Residual

Multiple Radar Residuals

Radar 2 starts

Radar 1

Radar 2 to end

Maneuvering Targets

• Filter Reinitialization– 3σ error ellipse

(~98%)– If three consecutive

data points outside ellipse, reinitialize filter

– Should happen upon maneuvering

• Prevents biased prediction matrix

3σ

GOOD

Predicted point

BAD

Maneuvering UFO

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Truth Track

Our Data

Raw Data

Maneuvering Target Tracks

Maneuvering Target Residuals

Residual, Case 5

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0.00

00

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48

Time (hrs)

Re

sid

ua

l (m

i)

Residual of Us

Residual of Data

Interception

• Give interceptor path using filter– Interceptor: constant velocity– Intercept UFO

• Cross target path before designated time

• Solve using Law of Cosines

Interception Triangles

vt (from filter)

Dist plane-UFO

630t

Intercept pt

Current plane pt

Current UFO pt

β

θ

Δy

Δx

Interceptor Equations

vt

Dist

Current UFO pt

β

x

y

x

y

v

v

Dist

Distarctanarctan

Disty

Distx

vy

vx

Current plane pt

Intercept pt

Interceptor Equations

vt

Dist630t

Current UFO pt

β

22

2222

22222

630

)(630cos)(cos)(

cos)(2630:Cosines of Law

v

distvdistvdistvt

distvtdisttvt

Intercept pt

Current plane pt

Interceptor Equations

630t (course

of plane)

Intercept pt

Current plane pt

θ

Δy

Δx

x

yarctan

Interceptor Track

Maneuvering Plane with Interceptor

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Data

Our Results

Interceptor

Multiple Targets

• Tracking multiple targets lends itself to an object oriented approach

• Why is it useful? Collision avoidance

Target Class

Methods:

•Initialize

•Predict

•Correct

Matrices

•X

•Y

•P

•R

Target Object

Target Object

Collision Avoidance

Collision Avoidance Math

Express position at a future time t:

tvyy

tvxx

y

x

111

111

ˆ

ˆ

Plane 1:

tvyy

tvxx

y

x

222

222

ˆ

ˆ

Plane 2:

Collision Avoidance Math

1ˆˆˆˆ 212

212 yyxx

12

12

12

12

yyy

xxx

vvv

vvv

yyy

xxx

Determine if planes will be within one mile at any such time:

Make some substitutions to simplify the expression:

Collision Avoidance Math

Arrive at inequality describing dangerous time interval:

The solution to this inequality is the time intervalwhen the planes will be in danger

01 2 22222 yxtvyvxtvv yxyx

1

222

22

yxc

vyvxb

vva

yx

yx

02 cbtat

Collision Tracks

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Plane 1Plane 2Estimated Collision Interval

Conclusion

• Using the Kalman filter, we were able to minimize radar noise and analyze target tracking scenarios.

• We solved: plane collision avoidance, interception, tracking multiple aircraft

• Still relevant today: several space telescopes use the Kalman Filter as a low powered tracking device

Acknowledgements

• Mr. Randy Heuer• Zack Vogel• Dr. Paul Quinn• Dr. Miyamoto• Ms. Myrna Papier• NJGSS ’07 Sponsors

Works Cited

• http://www.physics.utah.edu/~detar/phycs6720/handouts/curve_fit/curve_fit/img147.gif

• http://www.afrlhorizons.com/Briefs/Mar02/OSR0106.html

• http://www.cs.unc.edu/~welch/kalman/media/images/kalman-new.jpg

• http://www.combinatorics.org/Surveys/ds5/gifs/5-VD-ellipses-labelled.gif