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Gaussian Filters
§ The Kalman filter and its variants can
- nly model Gaussian distributions
Robot Mapping Short Introduction to Particle Filters and Monte - - PowerPoint PPT Presentation
Robot Mapping Short Introduction to Particle Filters and Monte - - PowerPoint PPT Presentation
Robot Mapping Short Introduction to Particle Filters and Monte Carlo Localization Cyrill Stachniss 1 Gaussian Filters The Kalman filter and its variants can only model Gaussian distributions 2 Motivation Goal: approach for
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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 an interval, the higher its probability density
How to obtain such samples?
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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 à 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
- f 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
- 3. Resampling: “Replace unlikely
samples by more likely ones”
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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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Application: Particle Filter for Localization (Known Map)
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Resampling
§ Survival of the fittest: “Replace unlikely samples by more likely ones” § “Trick” to avoid that many samples cover unlikely states § Needed as we have a limited number
- f samples
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w2 w3 w1 wn Wn-1
Resampling
w2 w3 w1 wn Wn-1
§ Roulette wheel § Binary search § O(n log n) § Stochastic universal sampling § Low variance § O(n)
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Low Variance Resampling
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initialization
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- bservation
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resampling
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motion update
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measurement
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weight update
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resampling
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motion update
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measurement
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weight update
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resampling
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motion update
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measurement
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weight update
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resampling
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motion update
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measurement
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Summary – Particle Filters
§ Particle filters are non-parametric, recursive Bayes filters § Posterior is represented by a set of weighted samples § Not limited to Gaussians § Proposal to draw new samples § Weight to account for the differences between the proposal and the target § Work well in low-dimensional spaces
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