Probabilistic Robotics Bayes Filter Implementations Particle filters Samplebased
Probabilistic Robotics Bayes Filter Implementations Particle filters
Sample-based Localization (sonar) 2
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 3
Importance Sampling Weight samples: w = f / g 4
Importance Sampling with Resampling: Landmark Detection Example 5
Distributions 6
Distributions Wanted: samples distributed according to p(x| z 1, z 2, z 3) 7
This is Easy! We can draw samples from p(x|zl) by adding noise to the detection parameters. 8
Importance Sampling with Resampling 9
Importance Sampling with Resampling Weighted samples After resampling 10
Particle Filters 11
Sensor Information: Importance Sampling 12
Robot Motion 13
Sensor Information: Importance Sampling 14
Robot Motion 15
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 16
Particle Filter Algorithm draw xit-1 from Bel(xt-1) draw xit from p(xt | xit-1, ut-1) Importance factor for xit: 17
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’. 18
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 19
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’ 20 Also called stochastic universal sampling
Motion Model Reminder Start 21
Proximity Sensor Model Reminder Laser sensor Sonar sensor 22
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Sample-based Localization (sonar) 42
Initial Distribution 43
After Incorporating Ten Ultrasound Scans 44
After Incorporating 65 Ultrasound Scans 45
Estimated Path 46
Using Ceiling Maps for Localization 47
Vision-based Localization P(z|x) z h(x) 48
Under a Light Measurement z: P(z|x): 49
Next to a Light Measurement z: P(z|x): 50
Elsewhere Measurement z: P(z|x): 51
Global Localization Using Vision 52
Robots in Action: Albert 53
Application: Rhino and Albert Synchronized in Munich and Bonn 54 [Robotics And Automation Magazine, to appear]
Localization for AIBO robots 55
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)? 56
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). 57
Random Samples Vision-Based Localization 936 Images, 4 MB, . 6 secs/image Trajectory of the robot: 58
Odometry Information 59
Image Sequence 60
Resulting Trajectories Position tracking: 61
Resulting Trajectories Global localization: 62
Global Localization 63
Kidnapping the Robot 64
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. 66
- Slides: 64