Localization and Map Making

Transcription

1 Localiaion and Map Making My old office DILab a UTK ar of he following noes are from he book robabilisic Roboics by S. Thrn W. Brgard and D. Fo

2 Two Remaining Qesions Where am I? Localiaion Where have I been? Map-making

3 Moivaion Can make opological map localie relaive o landmarks More desirable: meric maps localie a any poin more readable by a hman GS isn he answer Localiaion error is meer Indoor Wan o know he feares of he environmen no js he robo There are ncerainies ha ms be facored in: probabilisic roboics

4 Localiaion Eample

5 Coasal Navigaion Eample

6 robabilisic Roboics Key idea: represen ncerainy eplicily sing he calcls of probabiliy heory Insead of relying on a single bes gess we have a whole space of gesses Topics: Basic conceps in probabiliy Bayes filers Localiaion Mapping

7 Discree robabiliies X denoes a random variable. X can ake on a conable nmber of vales in { 2 n }. X= i or i is he probabiliy ha he random variable X akes on vale i. E.g. flip a coin Discree probabiliies sm o one

8 Coninos Random Variables X akes on vales in he coninm. px= or p is a probabiliy densiy fncion pdf. r a b p d b a p p d

9 Gassian Fncion Gassian fncion is a common - dimensional disribion N; μ; σ 2 p = 2πσ 2 -/2 ep{--μ 2 /2σ 2 } Mean; variance Defines widh of bell Defines heigh of bell Defines posiion of bell s peak From wikipedia

10 Join and Condiional robabiliy X= and Y=y = y If X and Y are independen hen y = y y y is he probabiliy of given y y = y / y or y = y y If X and Y are independen hen y =

11 Law of Toal robabiliy Discree case y y y y y Coninos case p d p p y p p y p y dy dy B 3 B 2 B 5 B 4 p A B A B A i i B B 7 B 6

12 Bayes Formla smmaries he knowledge we have regarding X prior o incorporaing he daa y y is he poserior probabiliy disribion over X y describes how sae variables X case sensor measremen Y evidence prior likelihood y y y y y y y y

13 Normaliaion Since y does no depend on Symbol η indicaes ha he final resl has o be normalied o y y y y y y

14 Condiioning I s perfecly fine o condiion any rle discssed so far on arbirary random variables sch as he variable Z Condiioning of probabiliies will pdae hem o ake accon of possible new informaion We eend he Bayes rle: y y y

15 Condiional Independence Condiion he rle for combining probabiliies of independen random variables y y eqivalen o y and y y I applies whenever a variable y carries no informaion abo a variable if anoher variable s vale is known

16 Law of Toal robabiliy d y y y d d

17 Robo Environmen Ineracion

18 Environmens and Saes Environmens are characeried by sae which incldes: robo pose y yaw locaion and feares of srronding objecs in he environmen landmarks disincive places locaion and velociies of moving objecs and people ec. The sae a ime will be denoed which will be called complee if i is he bes predicor of he fre Compleeness enails ha knowledge of pas saes measremens or conrols carry no addiional informaion ha wold help s predic he fre more accraely

19 Environmen Ineracion Environmen sensor measremen he daa a ime is denoed Z :2 = Z Z + Z 2 Conrol acions change he sae of he world conrol daa a ime is denoed :2 = + 2 robabilisic Generaive Laws The sae is generaed sochasically from - Sae ransiion probabiliy: p 0:- :- : = p - Measremen probabiliy: p 0:- :- : = p

20 Dynamic Sochasic Sysem Hidden Markov Model HMM Or Dynamic Bayes Nework DBN Sae ransiion probabiliy + measremen probabiliy

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

22 Casal vs. Diagnosic Reasoning open is diagnosic. open is casal. Ofen casal knowledge is easier o obain. Bayes rle allows s o se casal knowledge: open open open

23 Eample open = 0.6 open = 0.3 open = open = 0.5 open open open open p open open p open open raises he probabiliy ha he door is open.

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

25 Recrsive Bayesian 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 recrsively

26 Eample: Second Measremen 2 open = open = 0.6 open =2/ open open open open open open open 2 lowers he probabiliy ha he door is open.

27 Incorporaing 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.

28 Eample: Closing he door

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

30 Inegraing he Ocome of Acions Coninos case: ' ' d' Discree case: ' '

31 Eample: The Resling Belief ' ' ' ' closed closed closed open open open open open open closed closed closed open open closed closed closed

32 Belief Disribion A belief reflecs robo s inernal knowledge abo sae of environmen A belief disribion assigns a probabiliy o each possible hypohesis wih regards o he re sae We denoe belief over by bel an abbr. for he poserior bel = p.... Can be calclaed from he following bel by incorporaing he sensor measremen hs he correcion or measremen pdae Calclae a poserior before incorporaing js afer eecing conrol bel = p..-.. I predics he sae a ime based on previos sae hs he predicion sep

33 Bayes Filers: Framework Given: Sream of observaions and acion daa : Sensor model. Acion model. rior probabiliy of he sysem sae. Waned: d { Esimae of he sae X of a dynamical sysem. The poserior of he sae is also called Belief: } Bel

34 Markov Assmpion Underlying Assmpions Saic world Independen noise erfec model no approimaion errors : : : p p : : : 0 p p

35 Bayes Filers = observaion = acion = sae d Bel Bayes Bel Markov Markov d d Toal prob. Markov d

36 Bayes Bel Filer Algorihm Bel d. Algorihm Bayes_filer Beld : 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 Bel' Bel' Bel' Bel' ' Bel ' d'

37 Eample work on board

38 Markov Assmpion Violaions: Unmodeled dynamics no inclded in Inaccracies in he prob. Models error in he map for a localiing robo Approimaion errors Gassian disribion ec. Incomplee sae represenaions are preferable o more complee ones o redce compleiy of he Bayes filer Bayes filer is pracically robs o sch violaions Need o careflly define

39 Bayes Filer Applicaions Robo localiaion: Observaions are range readings Saes are posiions on a map Speech recogniion: Observaions are acosic signals Saes are specific posiions in specific words Machine ranslaion: Observaions are words Saes are ranslaion opions

40 Eample: Robo Localiaion

41 Eample: Robo Localiaion

42 Eample: Robo Localiaion

43 Eample: Robo Localiaion

44 Eample: Robo Localiaion

45 Eample: Robo Localiaion

46 Smmary Bayes rle allows s o compe probabiliies ha are hard o assess oherwise. Under he Markov assmpion recrsive Bayesian pdaing can be sed o efficienly combine evidence. Bayes filers are a probabilisic ool for esimaing he sae of dynamic sysems.