lecture3-2010

# lecture3-2010 - CS 547 Sensing and Planning in Robotics...

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Unformatted text preview: CS 547: Sensing and Planning in Robotics Gaurav S. Sukhatme Computer Science Robotic Embedded Systems Laboratory University of Southern California [email protected] http://robotics.usc.edu/~gaurav Bayes Filters: Framework • Given: • Stream of observations z and action data u: • Sensor model P(z|x). • Action model P(x|u,x’) . • Prior probability of the system state P(x). • Wanted: • Estimate of the state X of a dynamical system. • The posterior of the state is also called Belief : ) , , , | ( ) ( 1 1 t t t t z u z u x P x Bel K = } , , , { 1 1 t t t z u z u d K = Slide courtesy of S. Thrun, D. Fox and W. Burgard Recursive Bayesian Updating Markov assumption : z n is independent of d 1 ,...,d n-1 if we know x. Bel new ( x ) = P ( x | d 1 ,..., d n- 1 , z n ) = P ( z n | x , d 1 ,..., d n- 1 ) P ( x | d 1 ,... d n- 1 ) P ( z n | d 1 ,..., d n- 1 ) = P ( z n | x ) P ( x | d 1 ,... d n- 1 ) P ( z n | d 1 ,..., d n- 1 ) = hP ( z n | x ) P ( x | d 1 ,..., d n- 1 ) = hP ( z n | x ) Bel old ( x ) Markov Assumption Underlying Assumptions • Static world • Independent noise • Perfect model, no approximation errors ) , | ( ) , , | ( 1 : 1 : 1 1 : 1 t t t t t t t u x x p u z x x p-- = ) | ( ) , , | ( : 1 : 1 : t t t t t t x z p u z x z p = Slide courtesy of S. Thrun, D. Fox and W. Burgard Recursive Bayesian Updating • Action Update Bel new ( x ) = P ( x | d 1 ,..., d n- 1 , u n ) = P ( x | ò d 1 ,..., d n- 1 , u n , x n- 1 ) P ( x n- 1 | d 1 ,..., d n- 1 , u t ) dx t- 1 = P ( x | u n ò , x n- 1 ) P ( x n- 1 | d 1 ,..., d n- 1 ) dx t- 1 = P ( x | u n ò , x n- 1 ) Bel old ( x n- 1 ) dx t- 1 Bayes Filter Algorithm 1. Algorithm Bayes_filter ( Bel(x),d ): 2. η= 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then 10. For all x do 11. 12. Return Bel’(x) ) ( ) | ( ) ( ' x Bel x z P x Bel = ) ( ' x Bel + = η η ) ( ' ) ( ' 1 x Bel x Bel- = η ' ) ' ( ) ' , | ( ) ( ' dx x Bel x u x P x Bel ∫ = 1 1 1 ) ( ) , | ( ) | ( ) (--- ∫ = τ τ τ τ τ τ τ τ δξ ξ Βελ ξ υ ξ Π ξ ζ Π ξ Βελ η Slide courtesy of S. Thrun, D. Fox and W. Burgard 7...
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## This note was uploaded on 10/17/2010 for the course CSCI 547 at USC.

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lecture3-2010 - CS 547 Sensing and Planning in Robotics...

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