Practical Course WS12/13 Introduction to Monte Carlo Localization

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1 Practical Course WS12/13 Introduction to Monte Carlo Localization Cyrill Stachniss and Luciano Spinello 1

2 State Estimation Estimate the state of a system given observations and controls Goal: 2

3 Bayes Filter General framework for state estimation Prediction step based on controls Correction step based on measurements Recursive structure: 3

4 Bayes Filter Derivation 4

5 Prediction and Correction Step Bayes filter is often formulated as a two step process Prediction step Correction step 5

6 Motion and Observation Model Prediction step motion model Correction step sensor or observation model 6

7 Different Realizations The Bayes filter is a framework for recursive state estimation There are different realizations Different properties Linear vs. non-linear models for motion and observation models Gaussian distributions only? Parametric vs. non-parametric filters 7

8 Motion Model 8

9 Robot Motion Models Robot motion is inherently uncertain How can we model this uncertainty? 9

10 Probabilistic Motion Models Specifies a posterior probability that action u carries the robot from x to x. 10

11 Typical Motion Models In practice, one often finds two types of motion models: Odometry-based Velocity-based Odometry-based models for systems that are equipped with wheel encoders Velocity-based when no wheel encoders are available 11

12 Odometry Model Robot moves from to. Odometry information 12

13 Probability Distribution Noise in odometry Example: Gaussian noise 13

14 Examples (Odometry-Based) 14

15 Sensor Model 15

16 Model for Laser Scanners Scan z consists of K measurements. Individual measurements are independent given the robot position 16

17 Beam-Endpoint Model 17

18 Beam-Endpoint Model map likelihood field 18

19 Ray-cast Model Ray-cast model considers the first obstacle long the line of sight Mixture of four models 19

20 Particle Filter Realization of the Bayes filter Non-parametric approach Models arbitrary distributions 20

21 Motivation Goal: approach for dealing with arbitrary distributions 21

22 Key Idea: Samples Use multiple samples to represent arbitrary distributions samples 22

23 Particle Set Set of weighted samples state hypothesis importance weight The samples represent the posterior 23

24 Particles for Approximation Particles for function approximation The more particles fall into an interval, the higher its probability density 24

25 Particle Filter Recursive Bayes filter Non-parametric approach Models the distribution by samples Prediction: propagate the samples given the motion model (proposal) Correction: weighting by considering the observation The more samples we use, the better is the estimate! 25

26 Monte Carlo Localization Each particle is a pose hypothesis Proposal is the motion model Correction via the observation model Resampling: Replace unlikely samples by more likely ones 26

27 Particle Filter for Localization 27

28 Application: Particle Filter for Localization (Known Map) 28

29 Resampling Survival of the fittest: Replace unlikely samples by more likely ones Trick to avoid that many samples cover unlikely states Needed as we have a limited number of samples 29

30 Resampling W n-1 w n w 1 w 2 W n-1 w n w 1 w 2 w 3 w 3 Roulette wheel Binary search O(n log n) Stochastic universal sampling Low variance O(n) 30

31 Low Variance Resampling 31

32 initialization 32

33 observation 33

34 resampling 34

35 motion update 35

36 measurement 36

37 weight update 37

38 resampling 38

39 motion update 39

40 measurement 40

41 weight update 41

42 resampling 42

43 motion update 43

44 measurement 44

45 weight update 45

46 resampling 46

47 motion update 47

48 measurement 48

49 Summary Particle Filters Particle filters are non-parametric, recursive Bayes filters Posterior is represented by a set of weighted samples Not limited to Gaussians Proposal to draw new samples Weight to account for the differences between the proposal and the target Work well in low-dimensional spaces 49

50 Summary PF Localization Particles are propagated according to the motion model They are weighted according to the likelihood of the observation Called: Monte-Carlo localization (MCL) MCL is the gold standard for mobile robot localization today 50

51 Literature On Monte Carlo Localization Thrun et al. Probabilistic Robotics, Chapter 8.3 On the particle filter Thrun et al. Probabilistic Robotics, Chapter 3 On motion and observation models Thrun et al. Probabilistic Robotics, Chapters 5 & 6 51

52 Key Questions What does our map look like? What kind of sensor do we use? How to obtain a sensor model? How to describe the motion? 52

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