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Probabilistic Robotics

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Algorithm particle_filter( St-1, ut-1 zt): For Generate new samples ... draw xit from p(xt | xit-1,ut-1) Importance factor for xit: Particle Filter Algorithm ... – PowerPoint PPT presentation

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Title: Probabilistic Robotics


1
Probabilistic Robotics
Bayes Filter Implementations Particle filters
2
Sample-based Localization (sonar)
3
Particle Filters
  • Represent belief by random samples
  • Estimation of non-Gaussian, nonlinear processes
  • Monte Carlo filter, Survival of the fittest,
    Condensation, Bootstrap filter, Particle filter
  • Filtering Rubin, 88, Gordon et al., 93,
    Kitagawa 96
  • Computer vision Isard and Blake 96, 98
  • Dynamic Bayesian Networks Kanazawa et al., 95d

4
Importance Sampling
Weight samples w f / g
5
Importance Sampling with ResamplingLandmark
Detection Example
6
Distributions
7
Distributions
Wanted samples distributed according to p(x z1,
z2, z3)
8
This is Easy!
We can draw samples from p(xzl) 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
  • Algorithm particle_filter( St-1, ut-1 zt)
  • For
    Generate new samples
  • Sample index j(i) from the discrete
    distribution given by wt-1
  • Sample from using
    and
  • Compute importance weight
  • Update normalization factor
  • Insert
  • For
  • Normalize weights

17
Particle Filter Algorithm
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
  • Stochastic universal sampling
  • Systematic resampling
  • Linear time complexity
  • Easy to implement, low variance
  • Roulette wheel
  • Binary search, n log n

20
Resampling Algorithm
  1. Algorithm systematic_resampling(S,n)
  2. For Generate cdf
  3. Initialize threshold
  4. For Draw samples
  5. While ( ) Skip until next threshold
    reached
  6. Insert

  7. Increment threshold
  8. Return S

Also called stochastic universal sampling
21
Motion Model Reminder
Start
22
Proximity Sensor Model Reminder
Sonar sensor
Laser sensor
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Sample-based Localization (sonar)
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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
49
Under a Light
Measurement z
P(zx)
50
Next to a Light
Measurement z
P(zx)
51
Elsewhere
Measurement z
P(zx)
52
Global Localization Using Vision
53
Robots in Action Albert
54
Application Rhino and Albert Synchronized in
Munich and Bonn
Robotics And Automation Magazine, to appear
55
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 SamplesVision-Based Localization
  • 936 Images, 4MB, .6secs/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
Recovery from Failure
66
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.
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