principles of radar target tracking the kalman filter: mathematical radar analysis
TRANSCRIPT
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
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X - Position (miles)
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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
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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
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X - Position (miles)
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Estimate
Truth
Position Residual - Basic Filter
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0.025 0.075 0.125 0.175 0.225
Time (hours)
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Estimate v.Truth
Case Studies:Basic Filter
Data Analysis - Basic Filter
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X - Position (miles)
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Truth
Case Studies:Basic Filter
Position Residual - Basic Filter
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Time (hours)
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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
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X - Position (miles)
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Raw Data
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Truth
Position Residual - Filter with Polar Coordinates
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Time (hours)
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Raw Data v. Truth
Estimate v. Truth
Case Studies:Filter with Polar Coordinates
Data Analysis - Filter with Polar Coordinates
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X - Position (miles)
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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
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X-Position (miles)
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mile
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Raw Data - Plane 2
Estimate - Plane 1
Estimate - Plane 2
Truth - Plane 1
Truth - Plane 2
Position Residual - Multiple Targets
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Time (hours)
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Raw Data - Plane 1
Raw Data - Plane 2
Estimate - Plane 1
Estimate - Plane 2
Case Studies:Multiple Targets
Data Analysis - Multiple Targets
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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
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X - Position (miles)
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Raw Data - Plane 1
Raw Data - Plane 2
Estimate - Plane 1
Estimate - Plane 2
Truth - Plane 1
Truth - Plane 2
Position Residual - Collision Avoidance
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Time (hours)
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Raw Data - Plane 1
Raw Data - Plane 2
Estimate - Plane 1
Estimate - Plane 2
Case Studies:Collision Avoidance
Data Analysis - Collision Avoidance
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X - Position (miles)
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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
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Time (hours)
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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
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X - Position (miles)
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Truth
Position Residual - Maneuver Detection
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Time (hours)
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Raw Data v. Truth
Estimate v. Truth
Case Studies:Maneuver Detection
Data Analysis - Manuever Detection
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Case Studies:Maneuver Detection
Position Residual - Maneuver Detection
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Time (hours)
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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
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Estimate
Truth
Interceptor Estimate
Position Residual - Interceptor
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Time (hours)
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Raw Data v. Truth
Estimate v. Truth
Case Studies:Interceptor
Data Analysis - Interceptor
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X - Position (miles)
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Estimate
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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.