Probabilistic Robotics

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Probabilistic Robotics
Bayes Filter Implementations Particle filters
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Sample-based Localization (sonar)
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Importance Sampling
Weight samples w f / g
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Distributions
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Distributions
Wanted samples distributed according to p(x z1,
z2, z3)
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This is Easy!
We can draw samples from p(xzl) by adding noise
to the detection parameters.
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Importance Sampling with Resampling
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Importance Sampling with Resampling
Weighted samples
After resampling
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Particle Filters
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Sensor Information Importance Sampling
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Robot Motion

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Sensor Information Importance Sampling
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Robot Motion
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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

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Particle Filter Algorithm
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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.

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Resampling Implementation

• Stochastic universal sampling
• Systematic resampling
• Linear time complexity
• Easy to implement, low variance

• Roulette wheel
• Binary search, n log n

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Resampling Algorithm

1. Algorithm systematic_resampling(S,n)
3. For Generate cdf
5. Initialize threshold
6. For Draw samples
7. While ( ) Skip until next threshold
   reached
9. Insert
10. 
    Increment threshold
11. Return S

Also called stochastic universal sampling
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Motion Model Reminder
Start
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Proximity Sensor Model Reminder
Sonar sensor
Laser sensor
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Sample-based Localization (sonar)
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Initial Distribution
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After Incorporating Ten Ultrasound Scans
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After Incorporating 65 Ultrasound Scans
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Estimated Path
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Using Ceiling Maps for Localization
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Vision-based Localization
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Under a Light
Measurement z
P(zx)
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Next to a Light
Measurement z
P(zx)
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Elsewhere
Measurement z
P(zx)
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Global Localization Using Vision
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Application Rhino and Albert Synchronized in
Munich and Bonn
Robotics And Automation Magazine, to appear
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Localization for AIBO robots
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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)?

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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).

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Random SamplesVision-Based Localization

• 936 Images, 4MB, .6secs/image
• Trajectory of the robot

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Odometry Information
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Image Sequence
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Resulting Trajectories
Position tracking
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Resulting Trajectories
Global localization
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Global Localization
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Kidnapping the Robot
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Recovery from Failure
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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.