probabilistic robotics - cornell university · particle-filters.ppt author: ashutosh created date:...
TRANSCRIPT
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Probabilistic Robotics
Bayes Filter Implementations
Particle filters
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Sample-based Localization (sonar)
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§ 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., 95]d
Particle Filters
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Weight samples: w = f / g
Importance Sampling
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Importance Sampling with Resampling
Weighted samples After resampling
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Particle Filters
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)|()(
)()|()()|()(
xzpxBel
xBelxzpw
xBelxzpxBel
ααα
=←
←
−
−
−
Sensor Information: Importance Sampling
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∫←− 'd)'()'|()( , xxBelxuxpxBel
Robot Motion
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)|()(
)()|()()|()(
xzpxBel
xBelxzpw
xBelxzpxBel
ααα
=←
←
−
−
−
Sensor Information: Importance Sampling
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Robot Motion
∫←− 'd)'()'|()( , xxBelxuxpxBel
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draw xit-1 from Bel(xt-1)
draw xit from p(xt | xi
t-1,ut-1)
Importance factor for xit:
)|()(),|(
)(),|()|(ondistributi proposal
ondistributitarget
111
111
tt
tttt
tttttt
it
xzpxBeluxxp
xBeluxxpxzp
w
∝
=
=
−−−
−−−η
1111 )(),|()|()( −−−−∫= tttttttt dxxBeluxxpxzpxBel η
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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w2
w3
w1 wn
Wn-1
Resampling
w2
w3
w1 wn
Wn-1
• Roulette wheel • Binary search, n log n
• Stochastic universal sampling • Systematic resampling
• Linear time complexity • Easy to implement, low variance
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Start
Motion Model Reminder
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Proximity Sensor Model Reminder
Laser sensor Sonar 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
P(z|x)
h(x)
z
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Under a Light
Measurement z: P(z|x):
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Next to a Light
Measurement z: P(z|x):
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Elsewhere
Measurement z: P(z|x):
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Global Localization Using Vision
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Robots in Action: Albert
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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 Samples Vision-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.