CSE-571 Probabilistic Robotics

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Gervais Dennis
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1 CSE-57 robabilistic Robotics robabilistic Robotics robabilities Baes rule Baes filters robabilistic Robotics Ke idea: Eplicit represetatio of ucertait usig the calculus of probabilit theor erceptio state estimatio Actio utilit optimiatio Discrete Radom Variables X deotes a radom variable. X ca take o a coutable umber of values i { }. X i or i is the probabilit that the radom variable X takes o value i.. is called probabilit mass fuctio. E.g. Room Cotiuous Radom Variables X takes o values i the cotiuum. px or p is a probabilit desit fuctio. b r a b p d E.g. a p

2 Joit ad Coditioal robabilit Law of Total robabilit Margials X ad Y If X ad Y are idepedet the is the probabilit of give / If X ad Y are idepedet the Discrete case Cotiuous case p d p p d p p p d Baes Formula Normaliatio likelihood prior evidece Ofte causal kowledge is easier to obtai tha diagostic kowledge. Baes rule allows us to use causal kowledge. η η ' ' '

3 Simple Eample of State Estimatio Suppose a robot obtais measuremet What is ope? Eample ope 0.6 ope 0.3 ope ope 0.5 ope ope ope ope p ope + ope p ope ope raises the probabilit that the door is ope. Normaliatio η η Algorithm: : au η ' au : η au ' ' Coditioig Baes rule ad backgroud kowledge:??? d d d 3

4 4 Coditioig Baes rule ad backgroud kowledge: d Coditioal Idepedece Equivalet to ad Simple Eample of State Estimatio Suppose our robot obtais aother observatio. What is ope? Recursive Baesia Updatig Markov assumptio: is coditioall idepedet of... - give. η η... i i...

5 Eample: Secod Measuremet ope 0.5 ope 0.6 ope / 3 ope / 3 ope ope ope ope ope + ope ope lowers the probabilit that the door is ope. Baes Filters: Framework Give: Stream of observatios ad actio data u: dt { u ut t} Sesor model. Actio model u. rior probabilit of the sstem state. Wated: Estimate of the state X of a damical sstem. The posterior of the state is also called Belief: Bel t t u ut t Baes Filters observatio u actio state Baes Bel ηfilter Algorithm u Bel t t t t t t t dt Bel u u Baes t t t t η u u u u t t t t t Markov η t t t u ut Total prob. Markov η t t t u u t t t u u t d t η t t t ut t t u ut dt η t t t ut t Bel t dt. Algorithm Baes_filter Beld : If d is a perceptual data item the 4. For all do 5. Bel ' Bel 6. η η + Bel' 7. For all do 8. Bel' η Bel' 9. Else if d is a actio data item u the 0. For all do. Bel ' u ' Bel' d'. Retur Bel 5

Markov Assumptio Damic Eviromets Two possible locatios ad 0.99 0.09 0.

: t : t u : t p t t p t : t : t u: t p t t ut Uderlig Assumptios Static world Idepedet oise erfect model o approimatio errors p d 0.

6 Markov Assumptio Damic Eviromets Two possible locatios ad p t 0 : t : t u : t p t t p t : t : t u: t p t t ut Uderlig Assumptios Static world Idepedet oise erfect model o approimatio errors p d p d p d Number of itegratios Represetatios for Baesia Robot Localiatio Baes Filters for Robot Localiatio Discrete approaches 95 Topological represetatio 95 ucertait hadlig OMDs occas. global localiatio recover Grid-based metric represetatio 96 global localiatio recover article filters 99 sample-based represetatio global localiatio recover AI Kalma filters late-80s Gaussias uimodal approimatel liear models positio trackig Robotics Multi-hpothesis 00 multiple Kalma filters global localiatio recover 6

7 Baes Filters are Familiar! Bel t η t t t ut t Bel t dt Kalma filters article filters Hidde Markov models Damic Baesia etworks artiall Observable Markov Decisio rocesses OMDs Summar Baes rule allows us to compute probabilities that are hard to assess otherwise. Uder the Markov assumptio recursive Baesia updatig ca be used to efficietl combie evidece. Baes filters are a probabilistic tool for estimatig the state of damic sstems. 7

Many Techniques Developed. Uncertainty. Aspects of Uncertainty. Decision Theory = Probability + Utility Theory

Many Techniques Developed. Uncertainty. Aspects of Uncertainty. Decision Theory = Probability + Utility Theory Ucertait CSE 47 Ma Techiques Developed Fu Logic Certait Factors No-mootoic logic robabilit Ol oe has stood the test of time! UW CSE AI Facult Aspects of Ucertait Suppose ou have a flight at oo Whe should