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

Teams Reciproverse Brian Dai Joshua Newman Michael Sobin Lexten Stephen Chan Adam Lloyd Jonathan MacMillan Alex 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 x k : position and velocity (state) of the target at time k (k+1 is next time step) Φ: state transition matrix q k : uncertainty in the state due to “noise” (e.g. wind variation and pilot error) y k : measurement at time k H: term that gets rid of velocity in X r: 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

Kalman Filter: Predict

Kalman Filter: Correct

Tools: Visual Basic Matlib- an external matrix operations library Input format – text files, simulated radar data Console- data output

Tools: Excel Track Charts

Tools: Excel Residual Analysis

Filter Development: Cartesian Coordinates Filter Implemented Test: Residual Analysis Does it work?

Cartesian Residuals

Filter Development: Polar Coordinates Prefiltering Polar to Cartesian conversion More appropriate data feed Error matrices –Redefine R

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

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 Target Tracks

Maneuvering Target Residuals

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 ΔyΔy ΔxΔx

Interceptor Equations vt Dist Current UFO pt β Dist y Dist x vyvy vxvx Current plane pt Intercept pt

Interceptor Equations vt Dist 630t Current UFO pt β Intercept pt Current plane pt

Interceptor Equations 630t (course of plane) Intercept pt Current plane pt θ ΔyΔy ΔxΔx

Interceptor Track

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

Collision Avoidance

Collision Avoidance Math Express position at a future time t: Plane 1:Plane 2:

Collision Avoidance Math 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 interval when the planes will be in danger

Collision Tracks

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 cs6720/handouts/curve_fit/curve_fit/img 147.gif 02/OSR0106.html media/images/kalman-new.jpg ds5/gifs/5-VD-ellipses-labelled.gif