Principles of Radar Target TrackingThe Kalman Filter: Mathematical Radar Analysis

Problems with Radar Radar can’t measure velocity Radar has measurement error:

“noise” Measurement Noise

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

10 15 20 25 30 35 40

X - Position (miles)

Y -

Po

siti

on

(m

iles

)

Raw Data

Purpose of Kalman Filter Transform data input from

radar trackers into usable form Reduce measurement error

(“noise”) of target’s position and velocity

Predict future state of target using previous state estimate and new data

Lightweight, robust, and expandable program

Rudolph Kalman Rudolph E.

Kalman was the “inventor” of the Kalman Filter

Began research on control theory in 1958

Blended earlier works

Worked with partner R.S. Bucy

http://www.rpi.edu/~kracua/seminar/det.html

Overview of Kalman Filter

Initialize Matrices

Read Data

Predict

Update

Output Results

Finish

Correct Measurement Covariance

Introduction to Project Part 1

2 Team Scenario, competing for government contract

Similar Projects Individual

Programs, Analyses, Graphs required

Part 2 Teams Merge Written

Component

Problems Getting Started

Problems

New programming language

Unfamiliar algorithm

Matrix Algebra

Solutions

Looked at help files and API’s

Teamwork in research

Matrix library

Kalman Model State Model

Measurement Model

kkk qXX 1

kkk rHXY

2

2

2

2

yyyyxyx

yyyyxyx

yxyxxxx

yxyxxxx

P

Steps of Kalman Filter Predict

QPP Tkkkkkk 1

kkkkk XX 1

y

x

v

y

v

x

X

1000

100

0010

001

t

t

Steps of Kalman Filter Correct

1

11

kT

kkT

kkk RHHPHPK

1)( kkkkk PHKIP

2

2

yyx

yxxR

m

m

y

xY

11 kkkkkkkk HXYKXX

y

x

v

y

v

x

X

Programming Made using

Visual Basic .NET

Read data file Convert

coordinates Predict location Output to Excel Graph flight

path

Case Studies:Basic Kalman Filter Filter noise from a basic,

linear data Limited functionality, based

solely on Cartesian coordinates

Built to be expandable, adaptable

Challenges First experience with Kalman

Filter tracking

Case Studies:How to Read Graphs Data Analysis

Comparison of raw data, estimated state, and truth

Filter takes noisy data and projects a path close to the truth

Position Residual Comparison of

mean squared error

Estimate v. Truth should decrease as filter gains accuracy relative to the Raw Data v. Truth

Data Analysis - Basic Filter

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

10 15 20 25 30 35 40

X - Position (miles)

Y -

Po

sit

ion

(m

iles

)

Raw Data

Estimate

Truth

Position Residual - Basic Filter

0

0.5

1

1.5

2

2.5

3

0.025 0.075 0.125 0.175 0.225

Time (hours)

Dis

tan

ce

(m

iles

) Raw Data v.Truth

Estimate v.Truth

Case Studies:Basic Filter

Data Analysis - Basic Filter

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

10 15 20 25 30 35 40

X - Position (miles)

Y -

Po

sit

ion

(m

iles

)

Raw Data

Estimate

Truth

Case Studies:Basic Filter

Position Residual - Basic Filter

0

0.5

1

1.5

2

2.5

3

0.025 0.075 0.125 0.175 0.225

Time (hours)

Dis

tan

ce

(m

iles

)

Raw Data v.Truth

Estimate v.Truth

Case Studies:Filter with Polar Coordinates Data inputted in range and

bearing Challenges

Transformation of measurement data from Polar to Cartesian coordinates

Error ellipse based on accuracy of range and bearing

σr

σθ

Case Studies:Filter with Polar Coordinates Filter

incorporates past and current data

Increased accuracy with more data

Position Residual (Estimate v. Truth) should decrease relative to noise

Data Analysis - Filter with Polar Coordinates

-40

-30

-20

-10

0

10

20

30

40

-20 -10 0 10 20 30

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Position Residual - Filter with Polar Coordinates

0

0.5

1

1.5

2

2.5

0.025 0.075 0.125 0.175 0.225

Time (hours)

Dis

tan

ce

(m

iles

)

Raw Data v. Truth

Estimate v. Truth

Case Studies:Filter with Polar Coordinates

Data Analysis - Filter with Polar Coordinates

-40

-30

-20

-10

0

10

20

30

40

-20 -10 0 10 20 30

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Case Studies:Multiple Targets Code rewrite

necessary Object-oriented

rather than structured programming

Handles each target individually and allows the same steps to be used for each target

Data Analysis - Multiple Targets

-30

-20

-10

0

10

20

30

40

-20 -10 0 10 20 30 40

X-Position (miles)

Y-P

osi

tio

n (

mile

s) Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Truth - Plane 1

Truth - Plane 2

Position Residual - Multiple Targets

0

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2 0.25

Time (hours)

Dis

tan

ce (

mile

s)

Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Case Studies:Multiple Targets

Data Analysis - Multiple Targets

-30

-20

-10

0

10

20

30

40

-20 -10 0 10 20 30 40

X-Position (miles)

Y-P

osi

tio

n (

mile

s) Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Truth - Plane 1

Truth - Plane 2

Case Studies:Collision Avoidance Use data on

multiple targets

Predict collisions based on probable courses

Alert target aircraft if within a certain range

Data Analysis - Collision Avoidance

-50

-40

-30

-20

-10

0

10

20

30

40

-100 -50 0 50 100

X - Position (miles)

Y -

Po

sit

ion

(m

iles

)

Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Truth - Plane 1

Truth - Plane 2

Position Residual - Collision Avoidance

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25

Time (hours)

Dis

tan

ce

(m

iles

)

Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Case Studies:Collision Avoidance

Data Analysis - Collision Avoidance

-50

-40

-30

-20

-10

0

10

20

30

40

-100 -50 0 50 100

X - Position (miles)

Y -

Po

sit

ion

(m

iles

)

Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Truth - Plane 1

Truth - Plane 2

Case Studies:Collision Avoidance

Position Residual - Collision Avoidance

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25

Time (hours)

Dis

tan

ce

(m

iles

)

Raw Data - Plane 1

Raw Data - Plane 2

Estimate - Plane 1

Estimate - Plane 2

Case Studies:Maneuver Detection

Comparison of projected path and measured data

When target deviates from projected course, reinitialize tracking

Additional coding necessary

Data Analysis - Manuever Detection

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-30 -20 -10 0 10 20 30 40 50

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Position Residual - Maneuver Detection

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Time (hours)

Dis

tan

ce (

mile

s)

Raw Data v. Truth

Estimate v. Truth

Case Studies:Maneuver Detection

Data Analysis - Manuever Detection

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-30 -20 -10 0 10 20 30 40 50

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Case Studies:Maneuver Detection

Position Residual - Maneuver Detection

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Time (hours)

Dis

tan

ce (

mile

s)

Raw Data v. Truth

Estimate v. Truth

Case Studies:Interceptor Includes

maneuver detection algorithms

Direct interceptor towards earliest projected interception

Reinitialize tracker and plane’s course after maneuvers

Data Analysis - Interceptor

-140

-120

-100

-80

-60

-40

-20

0

20

40

-60 -40 -20 0 20 40 60

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Interceptor Estimate

Position Residual - Interceptor

0

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time (hours)

Dis

tan

ce (

mile

s)

Raw Data v. Truth

Estimate v. Truth

Case Studies:Interceptor

Data Analysis - Interceptor

-140

-120

-100

-80

-60

-40

-20

0

20

40

-60 -40 -20 0 20 40 60

X - Position (miles)

Y -

Po

siti

on

(m

iles)

Raw Data

Estimate

Truth

Interceptor Estimate

Conclusion Visual Basic .NET successfully

handled the Kalman equations Kalman Filter successfully

reduced noise in all scenarios Position Residual graphs

confirms that the filter gains accuracy over time

Basic Filter proved expandable and advanced features were successfully incorporated in later scenarios

Thank You

References [IEEE] Institute of Electrical and Electronics Engineers. 2003

Jan 23. Rudolf E. Kalman, 1930-. IEEE History Center. <http://www.ieee.org/web/aboutus/history_center/biography/kalman.html> Accessed 2006 Aug 3.

Department of Computer Science at University of North Carolina. 2001 Jan 31. Rudolph Emil Kalman. <http://www.cs.unc.edu/~welch/kalman/kalmanBiblio.html> Accessed 2006 Aug 3.

Blackman, Samuel S. 1986. Multiple-Target Tracking with Radar Applications. Artech House, Inc. Norwood, MA.

Bishop G, Welch G. 2006. An Introduction to the Kalman Filter. <http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf>. Accessed 2006 Aug 3.

Anas SA. 2003 Jan 18. Matrix operations library .NET.