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Robot Mapping
Short Introduction to Particle Filters and Monte Carlo Localization
Gian Diego Tipaldi, Wolfram Burgard
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Gaussian Filters
The Kalman filter and its variants can only model Gaussian distributions
Courtesy: K. Arras
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Motivation
Goal: approach for dealing with arbitrary distributions
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Key Idea: Samples
Use multiple samples to represent arbitrary distributions
samples
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Particle Set
Set of weighted samples
The samples represent the posterior
state hypothesis
importance weight
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Particles for Approximation
Particles for function approximation
The more particles fall into a region, the higher the probability of the region
How to obtain such samples?
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Closed Form Sampling is Only Possible for a Few Distributions
Example: Gaussian
How to sample from other distributions?
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Importance Sampling Principle
We can use a different distribution g
to generate samples from f
Account for the “differences between
g and f ” using a weight w = f / g
target f
proposal g
Pre-condition:
f(x)>0 g(x)>0
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Importance Sampling Principle
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Particle Filter
Recursive Bayes filter
Non-parametric approach
Models the distribution by samples
Prediction: draw from the proposal
Correction: weighting by the ratio of target and proposal
The more samples we use, the better is the estimate!
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Particle Filter Algorithm
1. Sample the particles using the
proposal distribution
2. Compute the importance weights
Resampling: Draw sample with probability and repeat times
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Particle Filter Algorithm
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Monte Carlo Localization
Each particle is a pose hypothesis
Proposal is the motion model
Correction via the observation model
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Particle Filter for Localization
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Resampling
Draw sample with probability . Repeat times.
Informally: “Replace unlikely samples by more likely ones”
Survival of the fittest
“Trick” to avoid that many samples cover unlikely states
Needed as we have a limited number of samples
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w2
w3
w1 wn
Wn-1
Resampling
w2
w3
w1 wn
Wn-1
Roulette wheel
Binary search
O(J log J)
Stochastic universal sampling
Low variance
O(J)
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Low Variance Resampling
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initialization Courtesy: Thrun, Burgard, Fox
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observation Courtesy: Thrun, Burgard, Fox
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resampling Courtesy: Thrun, Burgard, Fox
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motion update Courtesy: Thrun, Burgard, Fox
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measurement Courtesy: Thrun, Burgard, Fox
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weight update Courtesy: Thrun, Burgard, Fox
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resampling Courtesy: Thrun, Burgard, Fox
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motion update Courtesy: Thrun, Burgard, Fox
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measurement Courtesy: Thrun, Burgard, Fox
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weight update Courtesy: Thrun, Burgard, Fox
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resampling Courtesy: Thrun, Burgard, Fox
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motion update Courtesy: Thrun, Burgard, Fox
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measurement Courtesy: Thrun, Burgard, Fox
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weight update Courtesy: Thrun, Burgard, Fox
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resampling Courtesy: Thrun, Burgard, Fox
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motion update Courtesy: Thrun, Burgard, Fox
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measurement Courtesy: Thrun, Burgard, Fox
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Summary – Particle Filters
Particle filters are non-parametric, recursive Bayes filters
Posterior is represented by a set of weighted samples
Proposal to draw the samples for t+1
Weight to account for the differences between the proposal and the target
Work well in low-dimensional spaces
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Summary – PF Localization
Particles are propagated according to the motion model
They are weighted according to the likelihood of the observation
Called: Monte-Carlo localization (MCL)
MCL is the gold standard for mobile robot localization today
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Literature
On Monte Carlo Localization
Thrun et al. “Probabilistic Robotics”, Chapter 8.3
On the particle filter
Thrun et al. “Probabilistic Robotics”, Chapter 3
On motion and observation models
Thrun et al. “Probabilistic Robotics”, Chapters 5 & 6
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Slide Information
These slides have been created by Cyrill Stachniss as part of the robot mapping course taught in 2012/13 and 2013/14. I created this set of slides partially extending existing material of Edwin Olson, Pratik Agarwal, and myself.
I tried to acknowledge all people that contributed image or video material. In case I missed something, please let me know. If you adapt this course material, please make sure you keep the acknowledgements.
Feel free to use and change the slides. If you use them, I would appreciate an acknowledgement as well. To satisfy my own curiosity, I appreciate a short email notice in case you use the material in your course.
My video recordings are available through YouTube: http://www.youtube.com/playlist?list=PLgnQpQtFTOGQrZ4O5QzbIHgl3b1JHimN_&feature=g-list
Cyrill Stachniss, 2014 [email protected]
bonn.de