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- 1. Probabilistic State Estimation With Application To Vehicle Navigation Matthew Kirchner Naval Air Warfare Center – Weapons Division Department of ECEE – University of Colorado at Boulder November 1, 2010
- 2. Topics • Why? • Review • Background • Kalman Filter • Particle Filter • SLAM
- 3. Why Probability? • Real sensors have uncertainty • May have multiple sensors • Some states are not directly observable • Ambiguous sensor observations
- 4. Importance of Bayes Rule Bayes Rule Recursive Bayesian Estimation Linear Kalman Filter Unscented Kalman Filter Extended Kalman Filter EKF SLAM FastSLAMParticle Filter FastSLAM
- 5. Bayes Rule )( )()|( )|( BP APABP BAP Prior Likelihood Posterior Normalizing Constant
- 6. Equivalent Bayes Rule )()|()|( APABPBAP )()|()|( APABPBAP
- 7. Bayes Rule Example • School: 60% boys and 40% girls • All boys wear pants • Half of girls wear skirts, half wear pants • You see a random student and can only tell they are wearing pants. • Based on your observation, what is the probability the student you saw is a girl?
- 8. Bayes Rule Example • School: 60% boys and 40% girls • All boys wear pants • Half of girls wear skirts, half wear pants • You see a random student and can only tell they are wearing pants. • Based on your observation, what is the probability the student you saw is a girl? • We want to find: – P(Student=Girl | Clothes=Pants) • Prior? – P(Student=Girl) = 0.4 • Likelihood? – P(Clothes=Pants | Student=Girl) = 0.5 • Normalizing Constant? – P(Clothes=Pants) = 0.8 • Bayes Rule! – P(Student=Girl | Clothes=Pants) = (0.5)*(0.4)/(0.8) = 0.25
- 9. Gaussian 1D 2D
- 10. Gaussian 1D 2D
- 11. Gaussian • 2D Probability density function described by mean vector and covariance matrix • 1D Probability density function described by mean and variance ),(~ 2 Nx 2 221 12 2 1 2 1 2 1 ),(~ NX x x X
- 12. Functions of Random Variables
- 13. Functions of Random Variables • Linear function • Mean: • Covariance: • Linear functions only! xFy T xy FF xFy
- 14. Kalman Filter • Introduced in 1960 by Rudolf Kalman • Many applications: – Vehicle guidance systems – Control systems – Radar tracking – Object tracking in video – Atmospheric models
- 15. Kalman Filter ),|( ttt uzxPWe want to find:
- 16. Kalman Filter Process Model Process Model Observation Model Observation Model Observation Model
- 17. Kalman Filter • Process Model: – Deterministic: – Probabilistic: • Used to calculate prior distribution • Observation Model: – Deterministic: – Probabilistic: • Used to calculate likelihood distribution ),( 1 ttt uxfx )( tt xfz ),,(),|( 11 tttttt wuxfuxxP ),()|( tttt vxfxzP
- 18. Kalman Filter: Assumptions • Underlying system is modeled as Markov – • All beliefs are Gaussian distributions – Additive zero mean Gaussian noise • Linear process and observation models – Process Model: • – Observation Model: • ),,,|()|( 3211 tttttt xxxxPxxP ttt vHxz tttt wBuFxx 1 ),0(~ QNwt ),0(~ RNvt
- 19. Kalman Filter Steps 1. Using process model, previous state, and controls, find prior – Sometimes called ‘predict’ step, or ‘a priori’ 2. Using prior and observation model, find sensor likelihood – If I knew state, what should sensors read? 3. Find observation residual – Difference between actual sensor values and what was calculated in step 2. Also sometimes called ‘innovation’.
- 20. Kalman Filter Steps 4. Compute Kalman gain matrix 5. Using Kalman gain and prior, calculate posterior – Sometimes called ‘correct’ step or ‘a posteriori’
- 21. Kalman Filter Steps • Algorithm Kalman_Filter( ) – 1a: – 1b: – 2: – 3: – 4: – 5a: – 5b: • Return( ) ttttt uBxFx 1 ˆ t T tttt QFF 1 ˆ ttt xHz ˆˆ ttt zzz ˆ 1 )ˆ(ˆ t T ttt T ttt RHHHK tttt zKxx ˆ tttt HKI ˆ)( tttt zux ,,, 11 ttx ,
- 22. Kalman Filter
- 23. Kalman Filter Example 1txtx
- 24. Kalman Filter Example Root Mean Squared Error (RMSE) KF 1.0030m GPS Only 3.6840m
- 25. Kalman Filter Example GPS Lost GPS Reacquired
- 26. Kalman Filter Example
- 27. Kalman Filter • O(k^2.4+n^2) • Many real systems are non linear – Extended Kalman filter – Unscented Kalman filter – Particle filter • Some systems are non-Gaussian – Particle filter
- 28. Extended Kalman Filter • Linearize process and observation models – By finding Jacobian matrices – Analytically or numerically • Then use regular Kalman filter algorithm • Sub-optimal
- 29. 29 EKF Linearization
- 30. EKF Linearization
- 31. Unscented Kalman Filter
- 32. Unscented Kalman Filter EKF UKF
- 33. Particle Filter
- 34. Particle Filter • Represent distribution as set of randomly generated samples, called ‘particles’. • Functions can be nonlinear and non-gaussian • Multi-hypothesis belief propagation
- 35. Particle Filter • Sample the prior • Compute likelihood of particles given measurement – Also called particle ‘weights’ • Sample posterior: Sample from particles proportional to particle weights – Also called ‘resampling’ or ‘importance sampling’
- 36. Simultaneous Localization and Mapping • SLAM Problem: – Need map to localize – Need location to make map • Brainstorming: How can we solve this problem? – Map could be locations of landmarks or occupancy grid • Kalman filter based: landmark positions part of state variables • Particle filter based: landmark positions or occupancy map included in each particle ),|,( ttt uzmxP
- 37. Feature-based SLAM
- 38. FastSLAM - Example
- 39. Other Probabilistic Applications
- 40. Other Probabilistic Applications

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