Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization

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1 Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello 1

2 Motivation Recall: Discrete filter Discretize the continuous state space High memory complexity Fixed resolution (does not adapt to the belief) Particle filters are a way to efficiently represent non-gaussian distribution Basic principle Set of state hypotheses ( particles ) Survival-of-the-fittest 2

3 Sample-based Localization (sonar)

4 Mathematical Description Set of weighted samples State hypothesis Importance weight The samples represent the posterior 4

5 Function Approximation Particle sets can be used to approximate functions The more particles fall into an interval, the higher the probability of that interval How to draw samples from a function/distribution? 5

6 Rejection Sampling Let us assume that f(x)<1 for all x Sample x from a uniform distribution Sample c from [0,1] if f(x) > c keep the sample otherwise reject the sample c x f(x) x f(x ) c OK 6

7 Importance Sampling Principle We can even use a different distribution g to generate samples from f By introducing an importance weight w, we can account for the differences between g and f w = f / g f is called target g is called proposal Pre-condition: f(x)>0 à g(x)>0 Derivation: See webpage 7

8 Importance Sampling with Resampling: Landmark Detection Example

9 Distributions

10 Distributions Wanted: samples distributed according to p(x z 1, z 2, z 3 ) 10

11 This is Easy! We can draw samples from p(x z l ) by adding noise to the detection parameters.

12 Importance Sampling Target distribution f : p(x z 1, z 2,..., z n ) = k p(z k x) p(z 1, z 2,..., z n ) Sampling distribution g: p(x z l ) = p(z l x)p(x) p(z l ) p(x) Importance weights w: f g = p(x z 1, z 2,..., z n ) p(x z l ) = p(z l ) p(z k x) k l p(z 1, z 2,..., z n )

13 Importance Sampling with Resampling Weighted samples After resampling

14 Particle Filters

15 Sensor Information: Importance Sampling Bel( x) w α p( z x) Bel α p( z x) Bel Bel ( x) ( x) ( x) = α p( z x)

16 Robot Motion Bel (x) p(x u, x') Bel(x') d x'

17 Sensor Information: Importance Sampling Bel(x) α p(z x) Bel (x) w α p(z x) Bel (x) Bel (x) = α p(z x)

18 Robot Motion Bel (x) p(x u, x') Bel(x') d x'

19 Particle Filter Algorithm Sample the next generation for particles using the proposal distribution Compute the importance weights : weight = target distribution / proposal distribution Resampling: Replace unlikely samples by more likely ones 19

20 Particle Filter Algorithm 1. Algorithm particle_filter( S t-1, u t, z t ): 2. S t 3. For i =1,, n Generate new samples 4. Sample index j(i) from the discrete distribution given by w t-1 i j(i) 5. Sample x t from p(x t x t 1,u t ) using x t 1 and 6. w i t = p(z t x i t ) Compute importance weight i 7. η = η + w t Update normalization factor 8. S t = S t {< x i t, w i t >} Add to new particle set 9. For =, η = 0 i =1,, n 10. w i t = w i t /η Normalize weights u t 20

21 Particle Filter Algorithm Bel(x t ) = η p(z t x t ) p(x t x t 1,u t ) Bel(x t 1 ) dx t 1 Importance factor for x i t : w t i = draw x i t-1 from Bel(x t-1 ) draw x i t from p(x t xi t-1,u t ) target distribution proposal distribution = η p(z t x t ) p(x t x t 1,u t ) Bel (x t 1 ) p(x t x t 1,u t ) Bel (x t 1 ) p(z t x t )

22 Resampling Given: Set S of weighted samples. Wanted : Random sample, where the probability of drawing x i is given by w i. Typically done n times with replacement to generate new sample set S.

23 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, n log n Stochastic universal sampling Systematic resampling Linear time complexity Easy to implement, low variance

24 Resampling Algorithm 1. Algorithm systematic_resampling(s,n): 1 2. S ' =, c1 = w 3. For i = 2 n Generate cdf i 4. c i = ci 1 + w 1 5. u ~ U]0, n ], i 1 Initialize threshold 1 = 6. For j =1 n Draw samples 7. While ( u j > c i ) Skip until next threshold reached 8. i = i { i 1 S' = S' < x, n > } Insert 10. u = u 1 n Increment threshold j+ 1 j Return S Also called stochastic universal sampling

25 Mobile Robot Localization Each particle is a potential pose of the robot Proposal distribution is the motion model of the robot (prediction step) The observation model is used to compute the importance weight (correction step) [For details, see PDF file on the lecture web page] 25

26 Motion Model Reminder end pose start pose According to the estimated motion

27 Motion Model Reminder rotation translation rotation Decompose the motion into Traveled distance Start rotation End rotation

28 Motion Model Reminder Uncertainty in the translation of the robot: Gaussian over the traveled distance Uncertainty in the rotation of the robot: Gaussians over start and end rotation For each particle, draw a new pose by sampling from these three individual normal distributions

29 Motion Model Reminder Start

30 Proximity Sensor Model Reminder Laser sensor Sonar sensor

31 Mobile Robot Localization Using Particle Filters (1) Each particle is a potential pose of the robot The set of weighted particles approximates the posterior belief about the robot s pose (target distribution) 31

32 Mobile Robot Localization Using Particle Filters (2) Particles are drawn from the motion model (proposal distribution) Particles are weighted according to the observation model (sensor model) Particles are resampled according to the particle weights 32

33 Mobile Robot Localization Using Particle Filters (3) Why is resampling needed? We only have a finite number of particles Without resampling: The filter is likely to loose track of the good hypotheses Resampling ensures that particles stay in the meaningful area of the state space 33

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51 Sample-based Localization (sonar) 51

52 Initial Distribution 52

53 After Incorporating Ten Ultrasound Scans 53

54 After Incorporating 65 Ultrasound Scans 54

55 Estimated Path 55

56 Using Ceiling Maps for Localization [Dellaert et al. 99]

57 Vision-based Localization z P(z x) h(x)

58 Under a Light Measurement z: P(z x):

59 Next to a Light Measurement z: P(z x):

60 Elsewhere Measurement z: P(z x):

61 Global Localization Using Vision

62 Limitations The approach described so far is able to track the pose of a mobile robot and to globally localize the robot How can we deal with localization errors (i.e., the kidnapped robot problem)? 63

63 Approaches Randomly insert a fixed number of samples This assumes that the robot can be teleported at any point in time Alternatively, insert random samples proportional to the average likelihood of the particles 64

64 Summary Particle Filters Particle filters are an implementation of recursive Bayesian filtering They represent the posterior by a set of weighted samples They can model non-gaussian distributions Proposal to draw new samples Weight to account for the differences between the proposal and the target Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter 65

65 Summary PF Localization In the context of localization, the particles are propagated according to the motion model. They are then weighted according to the likelihood of the observations. In a re-sampling step, new particles are drawn with a probability proportional to the likelihood of the observation. 66

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