Report

Using the Kalman Filter to Estimate the state of a Maneuvering Aircraft ECEn -670 Stochastic Process Prepared By: Kevin Meier Alok Desai Instructor: Dr. Brian Mazzeo 11/29/2011 ECEn -670 Stochastic Process 1 Outlines • Kalman filter • Correlation Between the Process and Measurement Noise • Application of KF for estimating Bearing and Range • Simulation results 11/29/2011 ECEn -670 Stochastic Process 2 Kalman Filter • Purpose: It is to use measurements observed over time, containing noise (random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values. • When system model and measurement model equations are linear, then to estimate the state vector recursively. 11/29/2011 ECEn -670 Stochastic Process 3 Estimating States • System dynamic model: • Measurement model: 11/29/2011 ECEn -670 Stochastic Process 4 Kalman Filter Estimation 11/29/2011 ECEn -670 Stochastic Process 5 Kalman Filter (Cont.) • State estimation: • Error covariance (a priori): • Kalman Gain: • Error covariance update (a posteriori): • State estimate update: 11/29/2011 ECEn -670 Stochastic Process 6 Correlation Between the Process and Measurement Noise • Correlation be given by • Prediction equation remain unchanged. • Measurement equation 11/29/2011 ECEn -670 Stochastic Process 7 Range and Bearing Estimation • Radars are used to track aircraft. 11/29/2011 ECEn -670 Stochastic Process 8 • Range = ct/2 11/29/2011 ECEn -670 Stochastic Process 9 How the Kalman filter applies to Radar • Radar is used to track the state of an aircraft • The state is the range, range rate, bearing and bearing rate 11/29/2011 ECEn -670 Stochastic Process 10 How to model the aircraft with no acceleration data • Model the acceleration as a uniform random variable using the singer model. Where the acceleration is correlated from sample to sample 11/29/2011 ECEn -670 Stochastic Process 11 How the Kalman filter applies to Radar • The radar uses sensors to measure the Range and Bearing angle. In this process there is sensor measurement noise 11/29/2011 ECEn -670 Stochastic Process 12 How the Kalman filter applies to Radar • The process and measurement noise are zeromean white Gaussian random variables 11/29/2011 ECEn -670 Stochastic Process 13 11/29/2011 ECEn -670 Stochastic Process 14 Error Covariance for Range Error covariance (One prediction) 11/29/2011 Error covariance (Multiple prediction) ECEn -670 Stochastic Process 15 Error Covariance of Bearing Error covariance (One prediction) 11/29/2011 Error covariance (Multiple prediction) ECEn -670 Stochastic Process 16 Bearing Angle Bearing Angle (One prediction) 11/29/2011 Bearing Angle (Multiple prediction) ECEn -670 Stochastic Process 17 Vehicle Range Vehicle Range (One Prediction) 11/29/2011 Vehicle Range (Multiple Prediction) ECEn -670 Stochastic Process 18 Range Error Range Error (One Prediction) c 11/29/2011 Vehicle Range (Multiple Prediction) ECEn -670 Stochastic Process 19 Bearing Rate Bearing ( one prediction ) 11/29/2011 ECEn -670 Stochastic Process Bearing (multiple prediction ) 20 Range Range (One prediction ) 11/29/2011 Range (Multiple prediction ) ECEn -670 Stochastic Process 21 Range Error and Range Rate with correlated noise Range Rate Range Error 11/29/2011 ECEn -670 Stochastic Process 22 Questions?? Thank you ! 11/29/2011 ECEn -670 Stochastic Process 23