7630 Autonomous Robotics Probabilistic Localisation

Transcription

1 7630 Auonomous Roboics Probabilisic Localisaion Principles of Probabilisic Localisaion Paricle Filers for Localisaion Kalman Filer for Localisaion Based on maerial from R. Triebel, R. Käsner, R. Siegwar, D. Scaramuzza

2 The Bayes Filer (Bayes) (Markov) (To. prob.) (Markov) (Markov)

3 The Bayes Filer Algorihm Bayes_filer : 1. if is a sensor measuremen hen for all do for all do 7. else if is an acion hen 8. for all do 9. reurn

4 Bayes Filer Varians The Bayes filer principle is used in: Kalman filers Paricle filers (Mone-Carlo-Localizaion) Hidden Markov models Dynamic Bayesian neworks Parially Observable Markov Decision Processes (POMDPs)

5 LOCALISATION

6 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 6 The localisaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen. As i sars o move, say from a precisely known locaion, i migh keep rack of is locaion using odomery. However, afer a cerain movemen he robo will ge very uncerain abou is posiion => updae using an observaion of is environmen Observaion leads also o an esimae of he robo posiion which can hen be fused wih he odomeric esimaion o ge he bes possible updae of he robos acual posiion.?

7 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 8 Descripion of he probabilisic localizaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen.

8 5 9 Descripion of he probabilisic localizaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen. As i sars o move, say from a precisely known locaion, i can keep rack of is moion using odomery. R. Siegwar, D. Scaramuzza ETH Zurich - ASL

9 5 10 Descripion of he probabilisic localizaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen. As i sars o move, say from a precisely known locaion, i can keep rack of is moion using odomery. Due o odomery uncerainy, afer some movemen he robo will become very uncerain abou is posiion. R. Siegwar, D. Scaramuzza ETH Zurich - ASL

10 5 11 Descripion of he probabilisic localizaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen. As i sars o move, say from a precisely known locaion, i can keep rack of is moion using odomery. Due o odomery uncerainy, afer some movemen he robo will become very uncerain abou is posiion. To keep posiion uncerainy from growing unbounded, he robo mus localize iself in relaion o is environmen map. To localize, he robo migh use is on-board exerocepive sensors (e.g. ulrasonic, laser, vision sensors) o make observaions of is environmen Sensor reference frame R. Siegwar, D. Scaramuzza ETH Zurich - ASL

11 5 12 Descripion of he probabilisic localizaion problem 5 - Localizaion and Mapping Consider a mobile robo moving in a known environmen. As i sars o move, say from a precisely known locaion, i can keep rack of is moion using odomery. Due o odomery uncerainy, afer some movemen he robo will become very uncerain abou is posiion. To keep posiion uncerainy from growing unbounded, he robo mus localize iself in relaion o is environmen map. To localize, he robo migh use is on-board exerocepive sensors (e.g. ulrasonic, laser, vision sensors) o make observaions of is environmen The robo updaes is posiion based on he observaion. Is uncerainy shrinks. Sensor reference frame R. Siegwar, D. Scaramuzza ETH Zurich - ASL

12 5 13 Acion and percepion updaes 5 - Localizaion and Mapping In robo localizaion, we disinguish wo updae seps: 1. ACTION updae: he robo moves and esimaes is posiion hrough is propriocepive sensors. During his sep, he robo uncerainy grows. 2. PERCEPTION updae: he robo makes an observaion using is exerocepive sensors and correc is posiion by opporunely combining is belief before he observaion wih he probabiliy of making exacly ha observaion. During his sep, he robo uncerainy shrinks. Probabiliy of making his observaion Robo belief before he observaion R. Siegwar, D. Scaramuzza ETH Zurich - ASL

13 5 14 The soluion o he probabilisic localizaion problem 5 - Localizaion and Mapping Solving he probabilisic localizaion problem consiss in solving separaely he Acion and Percepion updaes R. Siegwar, D. Scaramuzza ETH Zurich - ASL

14 5 15 The soluion o he probabilisic localizaion problem 5 - Localizaion and Mapping A probabilisic approach o he mobile robo localizaion problem is a mehod able o compue he probabiliy disribuion of he robo configuraion during each Acion and Percepion sep. The ingrediens are: 1. The iniial probabiliy disribuion p ( x ) 0 2. The saisical error model of he propriocepive sensors (e.g. wheel encoders) 3. The saisical error model of he exerocepive sensors (e.g. laser, sonar, camera) 4. Map of he environmen (If he map is no known a priori hen he robos needs o build a map of he environmen and hen localizing in i. This is called SLAM, Simulaneous Localizaion And Mapping) R. Siegwar, D. Scaramuzza ETH Zurich - ASL

15 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 16 The soluion o he probabilisic localizaion problem 5 - Localizaion and Mapping How do we solve he Acion and Percepion updaes? Acion updae uses he Theorem of Toal probabiliy (or convoluion) Percepion updae uses he Bayes rule

16 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 17 Acion updae 5 - Localizaion and Mapping In his phase he robo esimaes is curren posiion based on he knowledge of he previous posiion and he odomeric inpu Using he Theorem of Toal probabiliy, we compue he robo s belief afer he moion as which can be wrien as a convoluion

17 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 18 Percepion updae 5 - Localizaion and Mapping In his phase he robo correcs is previous posiion (i.e. is former belief) by opporunely combining i wih he informaion from is exerocepive sensors p ( x z ) p ( z x p ( z ) p ( x ) ) bel ( x ) p ( z x ) bel ( x ) where η is a normalizaion facor which makes he probabiliy inegrae o 1.

18 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 19 Percepion updae 5 - Localizaion and Mapping Explanaion of bel ( x ) p ( z x ) bel ( x ) This is he probabiliy of observing z given ha he robo is a posiion x, ha is: p z x ) ( p ( x ) p ( x z ) p ( z x p ( z ) p ( x ) ) p ( x )

19 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 20 Illusraion of probabilisic bap based localizaion 5 - Localizaion and Mapping Iniial probabiliy disribuion p ( x ) 0 x p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x ) Acion updae p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x )

20 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 21 Illusraion of probabilisic bap based localizaion 5 - Localizaion and Mapping Iniial probabiliy disribuion p ( x ) 0 x p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x ) Acion updae p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x )

21 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 22 Illusraion of probabilisic bap based localizaion 5 - Localizaion and Mapping Iniial probabiliy disribuion p ( x ) 0 x p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x ) Acion updae p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x )

22 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 23 Illusraion of probabilisic bap based localizaion 5 - Localizaion and Mapping Iniial probabiliy disribuion p ( x ) 0 x p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x ) Acion updae p ( z x ) Percepion updae bel ( x ) p ( z x ) bel ( x )

23 MARKOV LOCALISATION

24 5 25 Markov localizaion 5 - Localizaion and Mapping Markov localizaion uses a grid space represenaion of he robo configuraion. For sake of simpliciy le us consider a robo moving in a one dimensional environmen (e.g. moving on a rail). Therefore he robo configuraion space can be represened hrough he sole variable x. Le us discreize he configuraion space ino 10 cells R. Siegwar, D. Scaramuzza ETH Zurich - ASL

25 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 26 Markov localizaion 5 - Localizaion and Mapping Le us discreize he configuraion space ino 10 cells Suppose ha he robo s iniial belief is a uniform disribuion from 0 o 3. Observe ha all he elemens were normalized so ha heir sum is 1.

26 5 27 Markov localizaion 5 - Localizaion and Mapping Iniial belief disribuion Acion phase: Le us assume ha he robo moves forward wih he following saisical model This means ha we have 50% probabiliy ha he robo moved 2 or 3 cells forward. Considering wha he probabiliy was before moving, wha will he probabiliy be afer he moion? R. Siegwar, D. Scaramuzza ETH Zurich - ASL

27 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 28 Markov localizaion: Acion updae 5 - Localizaion and Mapping The soluion is given by he convoluion of he wo disribuions *

28 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 30 Markov localizaion: Percepion updae 5 - Localizaion and Mapping Le us now assume ha he robo uses is onboard range finder and measures he disance from he origin. Assume ha he saisical error model of he sensors is This plo ells us ha he disance of he robo from he origin can be equally 5 or 6 unis. Wha will he final robo belief be afer his measuremen? The answer is again given by he Bayes rule: bel x ) p ( z x ) bel ( x ) (

29 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 31 Grid-based Represenaion - Muli Hypohesis Grid size around 20 cm Localizaion and Mapping Couresy of W. Burgard

30 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 32 Markov Localizaion: Case Sudy 2 Grid Map (5) 5 - Localizaion and Mapping Example 2: Museum Laser scan 1 Couresy of W. Burgard

31 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 33 Markov Localizaion: Case Sudy 2 Grid Map (6) 5 - Localizaion and Mapping Example 2: Museum Laser scan 2 Couresy of W. Burgard

32 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 34 Markov Localizaion: Case Sudy 2 Grid Map (7) 5 - Localizaion and Mapping Example 2: Museum Laser scan 3 Couresy of W. Burgard

33 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 35 Markov Localizaion: Case Sudy 2 Grid Map (8) 5 - Localizaion and Mapping Example 2: Museum Laser scan 13 Couresy of W. Burgard

34 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 36 Markov Localizaion: Case Sudy 2 Grid Map (9) 5 - Localizaion and Mapping Example 2: Museum Laser scan 21 Couresy of W. Burgard

35 5 37 Drawbacks of Markov localizaion 5 - Localizaion and Mapping In he more general planar moion case he grid-map is a hree-dimensional array where each cell conains he probabiliy of he robo o be in ha cell. In his case, he cell size mus be chosen carefully. During each predicion and measuremen seps, all he cells are updaed. If he number of cells in he map is oo large, he compuaion can become oo heavy for real-ime operaions. The convoluion in a 3D space is clearly he compuaionally mos expensive sep. As an example, consider a 30x30 m environmen and a cell size of 0.1 m x 0.1 m x 1 deg. In his case, he number of cells which need o be updaed a each sep would be 30x30x100x360=32.4 million cells! Fine fixed decomposiion grids resul in a huge sae space Very imporan processing power needed Large memory requiremen R. Siegwar, D. Scaramuzza ETH Zurich - ASL

36 5 38 Drawbacks of Markov localizaion 5 - Localizaion and Mapping Reducing complexiy Various approached have been proposed for reducing complexiy One possible soluion would be o increase he cell size a he expense of localizaion accuracy. Anoher soluion is o use an adapive cell decomposiion insead of a fixed cell decomposiion. Unimodal disribuion / Kalman Filer Use a single Normal disribuion o represen he disribuion Assumes measuremen and acion process are also normally disribued Closed form soluion. Randomized Sampling / Paricle Filer The main goal is o reduce he number of saes ha are updaed in each sep Approximaed belief sae by represening only a represenaive subse of all saes (possible locaions) E.g updae only 10% of all possible locaions The sampling process is ypically weighed, e.g. pu more samples around he local peaks in he probabiliy densiy funcion However, you have o ensure some less likely locaions are sill racked, oherwise he robo migh ge los R. Siegwar, D. Scaramuzza ETH Zurich - ASL

37 KALMAN FILTER FOR LOCALISATION

38 5 40 Markov Kalman Filer Localizaion 5 - Localizaion and Mapping Markov localizaion localizaion saring from any unknown posiion recovers from ambiguous siuaion. However, o updae he probabiliy of all posiions wihin he whole sae space a any ime requires a discree represenaion of he space (grid). The required memory and calculaion power can hus become very imporan if a fine grid is used. Kalman filer localizaion racks he robo and is inherenly very precise and efficien. However, if he uncerainy of he robo becomes o large (e.g. collision wih an objec) he Kalman filer will fail and he posiion is definiively los. Recommended reading: Inroducion o he Kalman Filer (Welch & Bishop): hp:// R. Siegwar, D. Scaramuzza ETH Zurich - ASL

39 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 41 Inroducion o Kalman Filer (1) 5 - Localizaion and Mapping Two measuremens Weighed leas-square Finding minimum error Afer some calculaion and rearrangemens

40 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 42 Inroducion o Kalman Filer (2) 5 - Localizaion and Mapping In Kalman Filer noaion

41 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 43 Inroducion o Kalman Filer (3) 5 - Localizaion and Mapping Dynamic Predicion (robo moving) Moion u = velociy w = noise Combining fusion and dynamic predicion

42 Prediced feaures observaions percepion R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 44 The Five Seps for Map-Based Localizaion 5 - Localizaion and Mapping Encoder Predicion Measuremen and Posiion (odomery) posiion esimae Esimaion (fusion) Map daa base YES mached predicions and observaions Maching 1. Predicion based on previous esimae and odomery 2. Observaion wih on-board sensors 3. Measuremen predicion based on predicion and map 4. Maching of observaion and map 5. Esimaion -> posiion updae (poseriori posiion) Observaion on-board sensors raw sensor daa or exraced feaures

43 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 - Localizaion and Mapping 5 46 Kalman Filer Localizaion: Predicion updae b s s b s s s s b s s s s y x u x f x l r l r l r l r l r ) 2 sin( 2 ) 2 cos( 2 ), ( ˆ T u u T x x F Q F F P F P l l r r s k s k Q 0 0 Odomery

44 5 47 Kalman Filer for Mobile Robo Localizaion: Observaion 5 - Localizaion and Mapping The second sep i o obain he observaion Z (measuremens) from he robo s sensors a he new locaion The observaion usually consiss of a se n 0 of single observaions z j exraced from he differen sensors signals. I represens feaures like lines, doors or any kind of landmarks. The parameers of he arges are usually observed in he sensor frame {S}. Therefore he observaions have o be ransformed o he world frame {W} or he measuremen predicion have o be ransformed o he sensor frame {S}. This ransformaion is specified in he funcion h j (see laer). R. Siegwar, D. Scaramuzza ETH Zurich - ASL

45 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 48 Observaion 5 - Localizaion and Mapping Raw Dae of Laser Scanner Exraced Lines Exraced Lines in Model Space l in e j j r j z j R r j j Sensor (robo) frame R, j r r rr j

46 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 49 Measuremen Predicion 5 - Localizaion and Mapping In he nex sep we use he prediced robo posiion generae muliple prediced observaions ẑ They have o be ransformed ino he sensor frame zˆ j h j xˆ j xˆ and he map o We can now define he measuremen predicion as he se conaining all n i prediced observaions Zˆ zˆ 1 j n i The funcion h j is mainly he coordinae ransformaion beween he world frame and he sensor frame j

47 5 50 Measuremen Predicion 5 - Localizaion and Mapping For predicion, only he walls ha are in he field of view of he robo are seleced. This is done by linking he individual lines o he nodes of he pah R. Siegwar, D. Scaramuzza ETH Zurich - ASL

48 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 - Localizaion and Mapping 5 51 Kalman Filer for Mobile Robo Localizaion: Measuremen Predicion: Example The generaed measuremen predicions have o be ransformed o he robo frame {R} According o he figure in previous slide he ransformaion is given by and is Jacobian by

49 5 52 Kalman Filer for Mobile Robo Localizaion: Maching 5 - Localizaion and Mapping Assignmen from observaions z j (k+1) (gained by he sensors) o he arges z (sored in he map) For each measuremen predicion for which an corresponding observaion is found we calculae he innovaion: and is innovaion covariance found by applying he error propagaion law: The validiy of he correspondence beween measuremen and predicion can e.g. be evaluaed hrough he Mahalanobis disance: R. Siegwar, D. Scaramuzza ETH Zurich - ASL

50 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 53 Maching 5 - Localizaion and Mapping

51 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 54 Kalman Filer for Mobile Robo Localizaion: Maching: Example 5 - Localizaion and Mapping To find correspondence (pairs) of prediced and observed feaures we use he Mahalanobis disance wih

52 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 - Localizaion and Mapping 5 55 Kalman Filer for Mobile Robo Localizaion: Esimaion: Applying he Kalman Filer Kalman filer gain: Updae of robo s posiion esimae: The associae variance

53 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 56 Kalman Filer for Mobile Robo Localizaion: Esimaion: 1D Case 5 - Localizaion and Mapping For he one-dimensional case wih we can show ha he esimaion corresponds o he Kalman filer for one-dimension presened earlier.

54 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 57 Esimaion 5 - Localizaion and Mapping Kalman filer esimaion of he new robo posiion : x By fusing he predicion of robo posiion (magena) wih he innovaion gained by he measuremens (green) we ge he updaed esimae of he robo posiion (red)

55 R. Siegwar, D. Scaramuzza ETH Zurich - ASL 5 58 Markov versus Kalman: summary 5 - Localizaion and Mapping Remember: boh mehods solve he convoluion and Bayer rules for he acion and he percepion updae respecively, bu:

56 PARTICLE FILTER FOR LOCALISATION

57 Sampling Mehods Sampling Mehods are widely used in Compuer Science as an approximaion of a deerminisic algorihm o represen uncerainy wihou a parameric model o obain higher compuaional efficiency wih a small approximaion error Sampling Mehods are also ofen called Mone Carlo Mehods Example: Mone-Carlo Inegraion Sample in he bounding box Compue fracion of inliers Muliply fracion wih box size

58 Non-Parameric Represenaion Probabiliy disribuions (e.g. a robo s belief) can be represened: Paramerically: e.g. using mean and covariance of a Gaussian Non-paramerically: using a se of hypoheses (samples) drawn from he disribuion Advanage of non-parameric represenaion: No resricion on he ype of disribuion (e.g. can be muli-modal, non- Gaussian, ec.)

59 Non-Parameric Represenaion The more samples are in an inerval, he higher he probabiliy of ha inerval Bu: How o draw samples from a funcion/disribuion?

60 Inversion sampling Rejecion sampling Imporance sampling SAMPLING FROM A DISTRIBUTION

61 Sampling from a Disribuion There are several approaches: Probabiliy ransformaion Uses inverse of he c.d.f h Rejecion Sampling Imporance Sampling... Probabiliy ransformaion: Sample uniformly in [0,1] Transform using h-1 Cumulaive disribuion Funcion Bu: Needs calculaion of h and is inverse

62 Rejecion Sampling Simplificaion: Assume for all z Sample z uniformly Sample c from If : keep he sample oherwise: rejec he sample p(z) z

63 Rejecion Sampling 2. General case: Find proposal disribuion q Easy o sample from q Find k wih Sample uniformly from q Sample unif. from [0,kq(z 0 )] Rejec if (unnormalized) Rejecion area Bu: Rejecion sampling is inefficien.

64 Imporance Sampling Idea: assign an imporance weigh w o each sample Wih he imporance weighs, we can accoun for he differences beween p and q p is called arge q is called proposal (as before)

65 Imporance Sampling Explanaion: The prob. of falling in an inerval A is he area under p This is equal o he expecaion of he indicaor funcion A Requiremen: Approximaion wih samples drawn from q:

66 Exercise d X P X 0 = N 5,1 P X k+1 X k = N X k + 1,1 P Z k X k = N( X k, 1 5 X k ) Z 0 = 5, Z 1 = 4, Z 2 = 3.1, Z 3 = 1.9, Z 4 = 1 Compue P(X k Z k,, Z 0 )

67 An alernaive o EKF for non-linear funcions, Non-gaussian noise APPLICATION: THE PARTICLE FILTER

68 The Paricle Filer Non-parameric implemenaion of Bayes filer Represens he belief (poserior) by a se of random sae samples. This represenaion is approximae. Can represen disribuions ha are no Gaussian. Can model non-linear ransformaions. Basic principle: Se of sae hypoheses ( paricles ) Survival-of-he-fies

69 Example

70 Mahemaical Descripion Se of weighed samples: Sae hypoheses Imporance weighs The samples represen he probabiliy disribuion: Poin mass disribuion ( Dirac )

71 The Paricle Filer Algorihm Algorihm Paricle_filer : for o do Sample from proposal Compue sample weighs 6. for o do Resampling 7. reurn

72 APPLICATION TO LOCALISATION

73 Localizaion wih Paricle Filers Each paricle is a poenial pose of he robo Proposal disribuion is he moion model of he robo (predicion sep) The observaion model is used o compue he imporance weigh (correcion sep) Randomized algorihms are usually called Mone Carlo algorihms, herefore we call his: Mone-Carlo Localizaion

74 Back o Our Example The iniial belief is a uniform disribuion (global localizaion). This is represened by an (approximaely) uniform sampling of iniial paricles.

75 Sensor Informaion The sensor model imporance weighs: is used o compue he new

76 Robo Moion Afer resampling and applying he moion model he paricles are disribued more densely a hree locaions.

77 Sensor Informaion Again, we se he new imporance weighs equal o he sensor model.

78 Robo Moion Resampling and applicaion of he moion model: One locaion of dense paricles is lef. The robo is localized.

79 A closer look a he algorihm Algorihm Paricle_filer : for o do Sample from proposal Compue sample weighs 6. for o do Resampling 7. reurn

80 Sampling from Proposal This can be done in he following ways: Adding he moion vecor o each paricle direcly (his assumes perfec moion) Sampling from he moion model, e.g. for a 2D moion wih ranslaion velociy v and roaion velociy w we have: Posiion Orienaion

81 Sampling from Moion Model Algorihm sample_moion_model_vel : robo specific parameers : reurn zero-mean, Gaussian sampling wih variance

82 Moion Model Sampling (Example) Sar

83 Compuaion of Imporance Weighs Compuaion of he sample weighs: Proposal disribuion: We sample from ha using he moion model. Targe disribuion (new belief): We can no direcly sample from ha: we need imporance sampling. Compuaion of imporance weighs:

84 Proximiy Sensor Models How can we obain he sensor model? Sensor Calibraion: Laser sensor Sonar sensor

85 Resampling for o do Given: Se of weighed samples. Waned : Random sample, where he probabiliy of drawing x i is equal o w i. Typically done M imes wih replacemen o generae new sample se.

86 Resampling W n-1 w n w 1 w 2 W n-1 w n w 1 w 2 w 3 w 3 Roulee wheel Binary search: O(n log n) Sochasic universal sampling Sysemaic resampling Linear ime complexiy Easy o implemen, low variance

87 Example

88 Limiaions The approach described so far is able o rack he pose of a mobile robo and o globally localize he robo. How can we deal wih localizaion errors (i.e., he kidnapped robo problem)?

89 The Kidnapped Robo Problem Possible Approaches: Randomly inser samples (he robo can be elepored a any poin in ime). Inser random samples proporional o he average likelihood of he paricles (he robo has been elepored wih higher probabiliy when he likelihood of is observaions drops).

90 Iniial Disribuion

91 Afer Ten Ulrasound Scans

92 Afer 65 Ulrasound Scans

93 Esimaed Pah

94 CONCLUSION

95 Conclusion Probabilisic localisaion is considered he sandard way of localising in roboics Uncerainy on he sensor daa and moion process Variaion around he Bayes filer based on workspace and disribuions: KF for linear gaussian processes (rare) EKF for non-linear (no oo much) gaussian processes PF for generic processes and kidnapped robos Discreized Markov Loc for exhausive represenaions UKF, combinaions, variaions, Remember he imporance of Daa Associaion