Probabilistic Robotics Sebastian Thrun-- Stanford

Transcription

1 robabilisic Roboics Sebasian Thrn-- Sanford Inrodcion robabiliies Baes rle Baes filers

2 robabilisic Roboics Ke idea: Eplici represenaion of ncerain sing he calcls of probabili heor ercepion sae esimaion Acion ili opimiaion 2

3 Aioms of robabili Theor ra denoes probabili ha proposiion A is re. 0 r A r Tre r False 0 r A B r A + r B r A B 3

4 A Closer Look a Aiom 3 r A B r A + r B r A B Tre A A B B B 4

5 5 Using he Aioms r r 0 r r r r r r r r r r A A A A False A A Tre A A A A A A + + +

6 Discree Random Variables X denoes a random variable. X can ake on a conable nmber of vales in { 2 n }. X i or i is he probabili ha he random variable X akes on. vale i. is called probabili mass fncion. 6

7 Coninos Random Variables X akes on vales in he coninm. px or p is a probabili densi fncion. E.g. r a b p d p b a 7

8 Join and Condiional robabili X and Y If X and Y are independen hen is he probabili of given / If X and Y are independen hen 8

9 Law of Toal robabili Marginals Discree case Coninos case p d p p d p p p d 9

10 0 Baes Formla evidence prior likelihood

11 Normaliaion η η a : a a : η η Algorihm:

12 2 Condiioning Law of oal probabili: d d d

13 Baes Rle wih Backgrond Knowledge 3

14 4 Condiioning Toal probabili: d d d

15 Condiional Independence eqivalen o and 5

16 Simple Eample of Sae Esimaion Sppose a robo obains measremen Wha is open? 6

17 Casal vs. Diagnosic Reasoning open is diagnosic. open is casal. Ofen casal knowledge is easier o obain. Baes rle allows s o se casal knowledge: open con freqencies! open open 7

18 Eample open 0.6 open 0.3 open open 0.5 open open open open p open + open p open open raises he probabili ha he door is open. 8

19 Combining Evidence Sppose or robo obains anoher observaion 2. How can we inegrae his new informaion? More generall how can we esimae... n? 9

20 20 Recrsive Baesian Updaing n n n n n n Markov assmpion: n is independen of... n- if we know n i i n n n n n n n n η η

21 2 Eample: Second Measremen 2 open open 0.6 open 2/ open open open open open open open 2 lowers he probabili ha he door is open.

22 Acions Ofen he world is dnamic since acions carried o b he robo acions carried o b oher agens or js he ime passing b change he world. How can we incorporae sch acions? 23

23 Tpical Acions The robo rns is wheels o move The robo ses is maniplaor o grasp an objec lans grow over ime Acions are never carried o wih absole cerain. In conras o measremens acions generall increase he ncerain. 24

24 Modeling Acions To incorporae he ocome of an acion ino he crren belief we se he condiional pdf This erm specifies he pdf ha eecing changes he sae from o. 25

25 Eample: Closing he door 26

26 Sae Transiions for close door : open closed 0 If he door is open he acion close door scceeds in 90% of all cases. 27

27 Inegraing he Ocome of Acions Coninos case: ' ' d' Discree case: ' ' 28

28 29 Eample: The Resling Belief ' ' ' ' closed closed closed open open open open open open closed closed closed open open closed closed closed

29 Baes Filers: Framework Given: Sream of observaions and acion daa : Sensor model. Acion model. d { rior probabili of he ssem sae. Waned: Esimae of he sae X of a dnamical ssem. The poserior of he sae is also called Belief: Bel } 30

30 Markov Assmpion p 0 : : : p p : : : p Underling Assmpions Saic world Independen noise erfec model no approimaion errors 3

31 32 d Bel η Baes Filers η Baes observaion acion sae Bel Markov η Markov d η d η Toal prob. Markov d η

32 Bel η Bel d Baes Filer Algorihm. Algorihm Baes_filer Beld : 2. η0 3. If d is a percepal daa iem hen 4. For all do For all do Else if d is an acion daa iem hen 0. For all do. 2. Rern Bel Bel ' Bel η η + Bel' Bel' η Bel' Bel' ' Bel ' d' 33

33 Baes Filers are Familiar! Bel η Bel d Kalman filers aricle filers Hidden Markov models Dnamic Baesian neworks ariall Observable Markov Decision rocesses OMDs 34

34 Smmar Baes rle allows s o compe probabiliies ha are hard o assess oherwise. Under he Markov assmpion recrsive Baesian pdaing can be sed o efficienl combine evidence. Baes filers are a probabilisic ool for esimaing he sae of dnamic ssems. 35