Slides for the book Probabilistic Robotics Authors Sebastian
Slides for the book: Probabilistic Robotics • Authors: • Sebastian Thrun • Wolfram Burgard • Dieter Fox • Publisher: • MIT Press, 2005. • Web site for the book & more slides: http: //www. probabilistic-robotics. org/ 1
Probabilistic Robotics Bayes Filter Implementations Particle filters
Sample-based Localization (sonar) 3
Particle Filters § § Represent belief by random samples § Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter § § § Filtering: [Rubin, 88], [Gordon et al. , 93], [Kitagawa 96] Estimation of non-Gaussian, nonlinear processes Computer vision: [Isard and Blake 96, 98] Dynamic Bayesian Networks: [Kanazawa et al. , 95]d 4
Importance Sampling Weight samples: w = f / g 5
Importance Sampling with Resampling: Landmark Detection Example 6
Distributions 7
Distributions Wanted: samples distributed according to p(x| z 1, z 2, z 3) 8
This is Easy! We can draw samples from p(x|zl) by adding noise to the detection parameters. 9
Importance Sampling with Resampling 10
Importance Sampling with Resampling Weighted samples After resampling 11
Particle Filters 12
Sensor Information: Importance Sampling 13
Robot Motion 14
Sensor Information: Importance Sampling 15
Robot Motion 16
Particle Filter Algorithm 1. Algorithm particle_filter( St-1, ut-1 zt): 2. 3. For Generate new samples 4. Sample index j(i) from the discrete distribution given by wt-1 5. Sample from using and 6. Compute importance weight 7. Update normalization factor 8. Insert 9. For 10. Normalize weights 17
Particle Filter Algorithm draw xit-1 from Bel(xt-1) draw xit from p(xt | xit-1, ut-1) Importance factor for xit: 18
Resampling • Given: Set S of weighted samples. • Wanted : Random sample, where the probability of drawing xi is given by wi. • Typically done n times with replacement to generate new sample set S’. 19
Resampling wn Wn-1 wn w 1 w 2 Wn-1 w 3 • Roulette wheel • Binary search, n log n w 1 w 2 w 3 • Stochastic universal sampling • Systematic resampling • Linear time complexity • Easy to implement, low variance 20
Resampling Algorithm 1. Algorithm systematic_resampling(S, n): 2. 3. For Generate cdf 4. 5. Initialize threshold 6. For 7. 8. 9. 10. While ( ) Draw samples … Skip until next threshold reached Insert Increment threshold 11. Return S’ 21 Also called stochastic universal sampling
Motion Model Reminder Start 22
Proximity Sensor Model Reminder Laser sensor Sonar sensor 23
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Sample-based Localization (sonar) 43
Initial Distribution 44
After Incorporating Ten Ultrasound Scans 45
After Incorporating 65 Ultrasound Scans 46
Estimated Path 47
Using Ceiling Maps for Localization 48
Vision-based Localization P(z|x) z h(x) 49
Under a Light Measurement z: P(z|x): 50
Next to a Light Measurement z: P(z|x): 51
Elsewhere Measurement z: P(z|x): 52
Global Localization Using Vision 53
Robots in Action: Albert 54
Application: Rhino and Albert Synchronized in Munich and Bonn 55 [Robotics And Automation Magazine, to appear]
Localization for AIBO robots 56
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)? 57
Approaches • Randomly insert samples (the robot can be teleported at any point in time). • Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops). 58
Random Samples Vision-Based Localization 936 Images, 4 MB, . 6 secs/image Trajectory of the robot: 59
Odometry Information 60
Image Sequence 61
Resulting Trajectories Position tracking: 62
Resulting Trajectories Global localization: 63
Global Localization 64
Kidnapping the Robot 65
Summary • Particle filters are an implementation of • • recursive Bayesian filtering They represent the posterior by a set of weighted samples. 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. 67
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