U sing the Kalman Filter to Estimate the state of a Maneuvering Aircraft

1. Using the Kalman Filter to Estimate the state of a Maneuvering Aircraft Prepared By: Kevin Meier Alok Desai ECEn -670 Stochastic Process Instructor: Dr. Brian Mazzeo ECEn -670 Stochastic Process

2. Outlines • Kalman filter • Correlation Between the Process and Measurement Noise • Application of KF for estimating Bearing and Range • Simulation results ECEn -670 Stochastic Process

3. 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. ECEn -670 Stochastic Process

4. Estimating States • System dynamic model: • Measurement model: ECEn -670 Stochastic Process

5. Kalman Filter Estimation ECEn -670 Stochastic Process

6. Kalman Filter (Cont.) • State estimation: • Error covariance (a priori): • Kalman Gain: • Error covariance update (a posteriori): • State estimate update: ECEn -670 Stochastic Process

7. Correlation Between the Process and Measurement Noise • Correlation be given by • Prediction equation remain unchanged. • Measurement equation ECEn -670 Stochastic Process

8. Range and Bearing Estimation • Radars are used to track aircraft. ECEn -670 Stochastic Process

9. Range = ct/2 ECEn -670 Stochastic Process

10. 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 ECEn -670 Stochastic Process

11. 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 ECEn -670 Stochastic Process

12. 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 ECEn -670 Stochastic Process

13. How the Kalman filter applies to Radar • The process and measurement noise are zero-mean white Gaussian random variables ECEn -670 Stochastic Process

14. ECEn -670 Stochastic Process

15. Error Covariance for Range Error covariance (One prediction) Error covariance (Multiple prediction) ECEn -670 Stochastic Process

16. Error Covariance of Bearing Error covariance (One prediction) Error covariance (Multiple prediction) ECEn -670 Stochastic Process

17. Bearing Angle Bearing Angle (One prediction) Bearing Angle (Multiple prediction) ECEn -670 Stochastic Process

18. Vehicle Range Vehicle Range (One Prediction) Vehicle Range (Multiple Prediction) ECEn -670 Stochastic Process

19. Range Error Range Error (One Prediction) c Vehicle Range (Multiple Prediction) ECEn -670 Stochastic Process

20. Bearing Rate Bearing ( one prediction ) Bearing (multiple prediction ) ECEn -670 Stochastic Process

21. Range Range (One prediction ) Range (Multiple prediction ) ECEn -670 Stochastic Process

22. Range Error and Range Ratewith correlated noise Range Error Range Rate ECEn -670 Stochastic Process

23. Questions?? Thank you ! ECEn -670 Stochastic Process