Recursive Bayes Filtering Advanced AI

Transcription

1 Recursive Bayes Filering Advanced AI Wolfram Burgard

2 Tuorial Goal To familiarie you wih probabilisic paradigm in roboics! Basic echniques Advanages ifalls and limiaions! Successful Applicaions! Open research issues

3 Roboics Yeserday

4 Roboics Today

5 RoboCup

6 hysical Agens are Inherenly Uncerain! Uncerainy arises from four major facors:! Environmen sochasic unpredicable! Robo sochasic! Sensor limied noisy! Models inaccurae

7 Naure of Sensor Daa Odomery Daa Range Daa

8 robabilisic Techniques for hysical Agens Key idea: Eplici represenaion of uncerainy using he calculus of probabiliy heory ercepion sae esimaion Acion uiliy opimiaion

9 Advanages of robabilisic aradigm! Can accommodae inaccurae models! Can accommodae imperfec sensors! Robus in real-world applicaions! Bes known approach o many hard roboics problems

10 ifalls! Compuaionally demanding! False assumpions! Approimae

11 Ouline! Inroducion! robabilisic Sae Esimaion! Robo Localiaion! robabilisic Decision Making! lanning! Beween MDs and OMDs! Eploraion! Conclusions

12 Aioms of robabiliy Theory ra denoes probabiliy ha proposiion A is rue.! 0 r A! r True r False 0! r A B r A + r B r A B

13 A Closer Look a Aiom 3 r A B r A + r B r A B True A A B B B

14 Using he Aioms r r 0 r r r r r r r r r r A A A A False A A True A A A A A A + + +

15 Discree Random Variables! X denoes a random variable.! X can ake on a finie number of values in { 2 n }.! X i or i is he probabiliy ha he random variable X akes on value i.!. is called probabiliy mass funcion.! E.g. Room

16 Coninuous Random Variables! X akes on values in he coninuum.! px or p is a probabiliy densiy funcion.! E.g. r [ a b] p d p b a

17 Join and Condiional robabiliy! X and Yy y! If X and Y are independen hen y y! y is he probabiliy of given y y y / y y y y! If X and Y are independen hen y

18 Law of Toal robabiliy Marginals Discree case Coninuous case y y y y y p d p p y dy p p y p y dy

19 Bayes Formula evidence prior likelihood y y y y y y y

20 Normaliaion y y y y y y η η y y y y y au : au au : η η Algorihm:

21 Condiioning! Toal probabiliy:! Bayes rule and background knowledge: y y y d y y y

22 Simple Eample of Sae Esimaion! Suppose a robo obains measuremen! Wha is open?

23 Causal vs. Diagnosic Reasoning! open is diagnosic.! open is causal.! Ofen causal knowledge is easier o obain.! Bayes rule allows us o use causal knowledge: open coun frequencies! open open

24 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.

25 Combining Evidence! Suppose our robo obains anoher observaion 2.! How can we inegrae his new informaion?! More generally how can we esimae... n?

26 Recursive Bayesian Updaing n n n n n n " " " " Markov assumpion: n is independen of... n- if we know n i i n n n n n n n n η η " " " "

27 Eample: Second Measuremen! 2 open open 0.6! open 2/ open open open open open open open 2 lowers he probabiliy ha he door is open.

28 A Typical ifall! Two possible locaions and 2! ! p2 d p d p d Number of inegraions

29 Acions! Ofen he world is dynamic since! acions carried ou by he robo! acions carried ou by oher agens! or jus he ime passing by change he world.! How can we incorporae such acions?

30 Typical Acions! The robo urns is wheels o move! The robo uses is manipulaor o grasp an objec! lans grow over ime! Acions are never carried ou wih absolue cerainy.! In conras o measuremens acions generally increase he uncerainy.

31 Modeling Acions! To incorporae he oucome of an acion u ino he curren belief we use he condiional pdf u! This erm specifies he pdf ha eecuing u changes he sae from o.

32 Eample: Closing he door

33 Sae Transiions u for u close door : open closed 0 If he door is open he acion close door succeeds in 90% of all cases.

34 Inegraing he Oucome of Acions Coninuous case: u u ' ' d' Discree case: u u ' '

35 Eample: The Resuling Belief ' ' ' ' u closed closed closed u open open open u open u open u open closed closed u closed open open u closed u closed u closed

36 Bayes Filers: Framework! Given:! Sream of observaions and acion daa u: d { u 2 " u! Sensor model.! Acion model u.! rior probabiliy of he sysem sae.! Waned:! Esimae of he sae X of a dynamical sysem.! The poserior of he sae is also called Belief: Bel u 2 " u }

37 Markov Assumpion Z - Z Z + u - X - X X + p d d... d0 d + d p d d... d0 p d d +... d d2... d p d d +... p u d... d p u Underlying Assumpions! Saic world! Independen noise! erfec model no approimaion errors u 2 0

38 Bayes Filers 2 2 u u u u " " η Bayes observaion u acion sae 2 u u Bel " Markov 2 u u " η d Bel u η Markov 2 d u u u " η 2 2 d u u u u " " η Toal prob.

39 Bel η u Bel d Bayes Filer Algorihm. Algorihm Bayes_filer Beld : 2. η0 3. if d is a percepual daa iem hen 4. For all do For all do else if d is an acion daa iem u hen 0. For all do. 2. reurn Bel Bel ' Bel η η + Bel' Bel' η Bel' Bel' u ' Bel ' d'

40 Bayes Filers are Familiar! Bel η u Bel d! Kalman filers! aricle filers! Hidden Markov models! Dynamic Bayes neworks! arially Observable Markov Decision rocesses OMDs

41 Applicaion o Door Sae Esimaion! Esimae he opening angle of a door! and he sae of oher dynamic objecs! using a laser-range finder! from a moving mobile robo and! based on Bayes filers.

42 Resul

43 Lessons Learned! Bayes rule allows us o compue probabiliies ha are hard o assess oherwise.! Under he Markov assumpion recursive Bayesian updaing can be used o efficienly combine evidence.! Bayes filers are a probabilisic ool for esimaing he sae of dynamic sysems.

44 Tuorial Ouline! Inroducion! robabilisic Sae Esimaion! Localiaion! robabilisic Decision Making! lanning! Beween MDs and OMDs! Eploraion! Conclusions

45 The Localiaion roblem Using sensory informaion o locae he robo in is environmen is he mos fundamenal problem o providing a mobile robo wih auonomous capabiliies. [Co 9]! Given! Map of he environmen.! Sequence of sensor measuremens.! Waned! Esimae of he robo s posiion.! roblem classes! osiion racking! Global localiaion! Kidnapped robo problem recovery

46 Represenaions for Bayesian Robo Localiaion Discree approaches 95 Topological represenaion 95 uncerainy handling OMDs occas. global localiaion recovery Grid-based meric represenaion 96 global localiaion recovery aricle filers 99 sample-based represenaion global localiaion recovery AI Kalman filers lae-80s? Gaussians approimaely linear models posiion racking Roboics Muli-hypohesis 00 muliple Kalman filers global localiaion recovery

47 Wha is he Righ Represenaion?! Kalman filers! Muli-hypohesis racking! Grid-based represenaions! Topological approaches! aricle filers

48 Gaussians : ~ σ µ πσ σ µ e p N p -σ σ µ Univariae 2 2 / 2 / 2 : ~ µ Σ µ Σ µ Σ e p Ν p d π µ Mulivariae

49 Kalman Filers Esimae he sae of processes ha are governed by he following linear sochasic difference equaion. + A + Bu + v C + w The random variables v and w represen he process measuremen noise and are assumed o be independen whie and wih normal probabiliy disribuions. 49

50 Kalman Filers [Schiele e al. 94] [Weiß e al. 94] [Borensein 96] [Gumann e al ] [Arras 98]

51 Kalman Filer Algorihm. Algorihm Kalman_filer <µσ> d : 2. If d is a percepual daa iem hen T 3. T K ΣC CΣC + Σ obs 4. µ µ + K Cµ 5. Σ I KC Σ 6. Else if d is an acion daa iem u hen µ A µ + Bu Σ AΣA 9. Reurn <µσ> T + Σ ac

52 Non-linear Sysems! Very srong assumpions:! Linear sae dynamics! Observaions linear in sae! Wha can we do if sysem is no linear?! Linearie i: EKF! Compue he Jacobians of he dynamics and observaions a he curren sae.! Eended Kalman filer works surprisingly well even for highly non-linear sysems.

53 Kalman Filer-based Sysems! [Gumann e al ]:! Mach LRF scans agains map! Highly successful in RoboCup mid-sie league Couresy of S. Gumann

54 Kalman Filer-based Sysems 2 Couresy of S. Gumann

55 Kalman Filer-based Sysems 3! [Arras e al. 98]:! Laser range-finder and vision! High precision <cm accuracy Couresy of K. Arras

56 Mulihypohesis Tracking [Co 92] [Jensfel Krisensen 99]

57 Localiaion Wih MHT! Belief is represened by muliple hypoheses! Each hypohesis is racked by a Kalman filer! Addiional problems:! Daa associaion: Which observaion corresponds o which hypohesis?! Hypohesis managemen: When o add / delee hypoheses?! Huge body of lieraure on arge racking moion correspondence ec. See e.g. [Co 93]

58 MHT: Implemened Sysem! [Jensfel and Krisensen 990]! Hypoheses are eraced from LRF scans! Each hypohesis has probabiliy of being he correc one: H i ˆ { i i i Σ H}! Hypohesis probabiliy is compued using Bayes rule H s i! Hypoheses wih low probabiliy are deleed! New candidaes are eraced from LRF scans C j s Hi Hi s { j j R}

59 MHT: Implemened Sysem 2 Couresy of. Jensfel and S. Krisensen

60 MHT: Implemened Sysem 3 Eample run # hypoheses H bes Map and rajecory Hypoheses vs. ime Couresy of. Jensfel and S. Krisensen

61 iecewise Consan [Burgard e al. 9698] [Fo e al. 99] [Konolige e al. 99]

62 iecewise Consan Represenaion bel < y θ >

63 Grid-based Localiaion

64 Tree-based Represenaions Idea: Represen densiy using a varian of Ocrees

65 Xavier: Localiaion in a Topological Map [Simmons and Koenig 96]

66 aricle Filers # Represen densiy by random samples # Esimaion of non-gaussian nonlinear processes # Mone Carlo filer Survival of he fies Condensaion Boosrap filer aricle filer # Filering: [Rubin 88] [Gordon e al. 93] [Kiagawa 96] # Compuer vision: [Isard and Blake 96 98] # Dynamic Bayesian Neworks: [Kanaawa e al. 95]

67 MCL: Global Localiaion

68 MCL: Sensor Updae Bel w α p Bel α p Bel Bel α p

69 MCL: Robo Moion Bel p u ' Bel ' d '

70 MCL: Sensor Updae Bel w α p Bel α p Bel Bel α p

71 MCL: Robo Moion Bel p u ' Bel ' d '

72 aricle Filer Algorihm. Algorihm paricle_filer S - u - : 2. S 3. For i " n Generae new samples 4. Sample inde ji from he discree disribuion given by w - i j i 5. Sample from p u using and i i 6. w p Compue imporance weigh i 7. η η + w Updae normaliaion facor i i 8. S S { < w > } Inser 9. For η 0 i " n u i i 0. w w /η Normalie weighs

73 Resampling! Given: Se S of weighed samples.! Waned : Random sample where he probabiliy of drawing i is given by w i.! Typically done n imes wih replacemen o generae new sample se S.

74 Resampling W n- w n w w 2 W n- wn w w 2 w 3 w 3 Roulee wheel Binary search log n Sochasic universal sampling Sysemaic resampling Linear ime compleiy Easy o implemen low variance

75 Moion Model p a - - Model odomery error as Gaussian noise on α β and δ

76 Moion Model p a - - Sar

77 Model for roimiy Sensors The sensor is refleced eiher by a known or by an unknown obsacle: Laser sensor Sonar sensor

78 MCL: Global Localiaion Sonar [Fo e al. 99]

79 Using Ceiling Maps for Localiaion [Dellaer e al. 99]

80 Vision-based Localiaion s s h

81 MCL: Global Localiaion Using Vision

82 Localiaion for AIBO robos

83 Adapive Sampling

84 KLD-sampling Idea: Assume we know he rue belief. Represen his belief as a mulinomial disribuion. Deermine number of samples such ha we can guaranee ha wih probabiliy - δ he KL-disance beween he rue poserior and he sample-based approimaion is less han ε. Observaion: For fied δ and ε number of samples only depends on number k of bins wih suppor: n 2ε k δ k 2ε 2 9 k 2 9 k 2 Χ + δ 3

85 MCL: Adapive Sampling Sonar

86 aricle Filers for Robo Localiaion Summary! Approimae Bayes Esimaion/Filering! Full poserior esimaion! Converges in O/ #samples [Tanner 93]! Robus: muliple hypoheses wih degree of belief! Efficien in low-dimensional spaces: focuses compuaion where needed! Any-ime: by varying number of samples! Easy o implemen

87 Localiaion Algorihms - Comparison Non- Gaussian Non-Gaussian Feaures Gaussian Gaussian Sensors +/ Robusness ++ +/ Accuracy +/++ -/ Efficiency memory Yes +/o o/+ iecewise consan Grid-based fied/variable +/ Efficiency ime ++ + o + Implemenaion Samples iecewise consan Muli-modal Gaussian oserior Yes Yes Yes No Global localiaion Mulihypohesis racking aricle filer Topological maps Kalman filer

88 Localiaion: Lessons Learned! robabilisic Localiaion Bayes filers! aricle filers: Approimae poserior by random samples! Eensions:! Filer for dynamic environmens! Safe avoidance of invisible haards! eople racking! Recovery from oal failures! Acive Localiaion! Muli-robo localiaion