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1. 1. An ISO 9001:2008 Registered Company UNCLASSIFIED Kalman FilterDistribution Restrictions: < Enter any appropriate distribution restrictions in title master, (eg. Distribution D)>Data Rights: <Enter any applicable date rights restrictions, (eg. SBIR data rights or other similar information>;DP-FM-016, Rev 2Effective Date: 22 February 2012
2. 2. Kalman Filter Facts  Dr. Rudolf Kalman is alive and well today (82 years old)  Important and used everywhere: GPS (predict update location), surface to air missiles (hit target), machine vision (track targets), brain computer interface  Not really a filter, it is an optimal estimator (infers parameters of interest from indirect, noise measurements)  It is recursive – so when a new measurement arrives it is processed and you get a new estimate  Performs Data Fusion usually between measured and UNCLASSIFIED estimated states22FEB122 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
3. 3. Conceptual Overview – Example Definition y  Lost on the 1-dimensional line, boat is not moving  Imagine that you are guessing your position by looking at the stars using sextant UNCLASSIFIED  Position function of time: y(t)  Assume Gaussian distributed measurements (errors)3
4. 4. Conceptual Overview - Prediction 0.16  Sextant Measurement 0.14 at t1: Mean = z1 and 0.12 Variance = z1 0.1 probability ŷ(t1) = z1  Optimal estimate of 0.08 Predicted Position position is: ŷ(t1) = z1 0.06  Variance of error 0.04 [y(t1) - ŷ(t1)] estimate: 0.02 2 (t ) = 2 0 0 10 20 30 40 50 60 70 80 90 100 e 1 z1 z  Boat in same position UNCLASSIFIED at time t2 - Predicted  What if we also had a position is z1 GPS unit?4
5. 5. Conceptual Overview - Measurement 0.16 0.14 prediction ŷ-(t2) State (by looking 0.12 at the stars at t2) 0.1 0.08 Measurement using GPS z(t2) 0.06 0.04 0.02 0 0 10 20 30 40 50 60 70 80 90 100 • So we have the prediction ŷ-(t2) • GPS Measurement at t2: Mean = z2 and Variance = z2 UNCLASSIFIED • Need to correct the prediction by Sextant due to measurement to get ŷ(t2)5
6. 6. Conceptual Overview – Data Fusion 0.16  Kalman filter: fuse corrected optimal 0.14 estimate ŷ(t2) measurement and 0.12 prediction ŷ-(t2) prediction based on 0.1 confidence 0.08 measurement z(t2)  Corrected mean is 0.06 the new optimal 0.04 0.02 estimate of position 0 0 10 20 30 40 50 60 70 80 90 100  New variance is smaller than either UNCLASSIFIED What if the boat is of the previous two moving? variances6
7. 7. Conceptual Overview – Prediction Model 0.16 ŷ(t2)  At time t3, boat 0.14 moves with velocity 0.12 Naïve Prediction dy/dt=u 0.1 (sextant) ŷ-(t3)  Naïve approach: 0.08 Shift probability to 0.06 0.04 the right to predict 0.02  This would work if 0 0 10 20 30 40 50 60 70 80 90 100 we knew the velocity exactly (perfect UNCLASSIFIED Try and predict where model) it winds up.7
8. 8. Conceptual Overview – Prediction Model 0.16 ŷ(t2)  But you may not be 0.14 so sure about the 0.12 Naïve Prediction exact velocity 0.1 (sextant) ŷ-(t3)  Better to assume 0.08 Prediction ŷ-(t3) imperfect model by 0.06 0.04 adding Gaussian 0.02 noise 0 0 10 20 30 40 50 60 70 80 90 100  dy/dt = u + w  Distribution for UNCLASSIFIED Assumptions: prediction is prediction moves linear, noise is Gaussian and spreads out8
9. 9. Conceptual Overview – Update 0.16 Corrected optimal estimate ŷ(t3) • Now we take a 0.14 Updated Sextant position using GPS measurement (GPS) 0.12 at t3 0.1 Measurement z(t3) GPS • Need to once again 0.08 correct the 0.06 Prediction ŷ-(t3) Sextant 0.04 prediction (fusion) 0.02 • Recursive – rinse 0 0 10 20 30 40 50 60 70 80 90 100 and repeat as time goes on UNCLASSIFIED Update, recursively9
10. 10. Conceptual Overview  Optimal estimator only if:  Prediction model is linear (function of measurements)  All error (noise) is Gaussian: model error, measurement error  Why is Kalman Filter so popular  Good results in practice due to optimality and structure.  Convenient form for online real time processing.  Easy to formulate and implement given a basic understanding.  Measurement equations need not be inverted. UNCLASSIFIED10
11. 11. State Space Equations Estimated Estimated Control State State Input (now) (before) UNCLASSIFIED Observed Measurement How do you find A,B,H?22FEB12 AWGN?11 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
12. 12. Input Output Update Equations Place holder Description Equation State Prediction Where do we end up Covariance Prediction When we get there, how much error Innovation Compare Reality to Prediction Innovation Covariance Compare real error to predicted error Kalman Gain What do you trust more? State Update UNCLASSIFIED New estimate of where we are22FEB12 Covariance Update New estimate of error12 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
13. 13. Algorithm Correction (Measurement Update) Prediction (Time Update) (1) Compute the Kalman Gain (1) Project the state ahead (2) Update estimate with measurement zk (2) Project the error covariance ahead (3) Update Error Covariance UNCLASSIFIED22FEB1213 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
14. 14. Measuring Constant Voltage (Classic Example 1) UNCLASSIFIED22FEB1214 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
15. 15. Predicting Trajectory of Projectile (Angry Bird) UNCLASSIFIED22FEB1215 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
16. 16. Equations UNCLASSIFIED22FEB1216 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
17. 17. Simulation Results UNCLASSIFIED22FEB1217 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
18. 18. Modified TWS example State : y   { x , y , x , y} Cov : Q E [ ww *] UNCLASSIFIED22FEB1218 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
19. 19. Derivation  UNCLASSIFIED22FEB1219 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.
20. 20. What if Assumptions don’t hold  UNCLASSIFIED22FEB1220 Notice: Use or disclosure of data contained on this sheet is subject to the restriction on the title page of this document.