Particle Filtering

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Particle Filtering

• Frank Dellaert
• Describes joint work with Dieter Fox, Wolfram
  Burgard, and Sebastian Thrun
• Presented by Ananth Ranganathan

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(No Transcript)
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Filtering

• Maintaining Knowledge over Time
• Tracking a quantity given ALL previous
  measurements
• Representing uncertainty using Gaussians or
  samples

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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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Minerva
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Motivation

• Crowded public spaces
• Unmodified environments

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Museum Application
Desired Location
Exhibit
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Global Localization

• Where in the world is Minerva the Robot ?
• Vague initial estimate
• Noisy and ambiguous sensors

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Local Tracking

• Sharp initial estimate
• Noisy and ambiguous sensors

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The Bayesian Paradigm

• Knowledge as a probability distribution

60 Rain
40 dry
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Probability of Robot Location
P(Robot Location)
Y
State space 2D, infinite states
X
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Bayesian Filtering

• Two phases
• 1. Prediction Phase
• 2. Measurement Phase

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1. Prediction Phase
u
xt-1
xt
P(xt) ? P(xtxt-1,u) P(xt-1)
Motion Model
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2. Measurement Phase
z
xt
P(xtz) k P(zxt) P(xt)
Sensor Model
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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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What sensor ?

• Sonar ?
• Laser ?
• Vision ?

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Problem Large Open Spaces

• Walls and obstacles out of range
• Sonar and laser have problems
• One solution Coastal Navigation

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Problem Large Crowds

• Horizontally mounted sensors have problems
• One solution Robust filtering

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Solution Ceiling Camera

• Upward looking camera
• Model of the world Ceiling Mosaic

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Global Alignment (other talk)
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Ceiling Mosaic
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Large FOV Problems

• 3D ceiling -gt 3D Model ?
• Matching whole images slow

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Small FOV Solution

• Model orthographic mosaic
• No 3D Effects
• Very fast

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Vision based Sensor
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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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Hidden Markov Models
A
B
C
D
E
A B C D E
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Kalman Filter

• Very powerful
• Gaussian, unimodal

sensor
motion
motion
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Kalman Filter Very Easy

• Think adding quadratics
• Then Minimize
• Dynamics Enlarge Quadratic

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Under Light
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Next to Light
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Elsewhere
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Markov Localization

• Fine discretization over x,y,theta
• Very successful Rhino, Minerva, Xavier

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Dynamic Markov Localization

• Burgard et al., IROS 98
• Idea use Oct-trees

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Sampling as Representation
P(Robot Location)
Y
X
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Samples ltgt Densities

• Density gt samplesObvious
• Samples gt densityHistogram, Kernel Density
  Estimation

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Sampling Advantages

• Arbitrary densities
• Memory O(samples)
• Only in Typical Set
• Great visualization tool !
• minus Approximate

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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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Disclaimer

• Handschin 1970 (!)lacked computing power
• Bootstrap filter 1993 Gordon et al.
• Monte Carlo filter 1996 Kitagawa
• Condensation 1996 Isard Blake

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Added Twists

• Camera moves, not object
• Global localization

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Particle Filtering
weighted Sk
Sk
Sk-1
Sk
Predict
Weight
Resample
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1. Prediction Phase
u
P(xt ,u)
Motion Model
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2. Measurement Phase
P(zxt)
Sensor Model
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3. Resampling Step
O(N)
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A more in depth look
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Bayes Law, new look

• Densitiesupdate prior p(x) to p(xz) via l(xz)
• Samplesupdate a sample from p(x) to a sample
  from the posterior p(xz) through l(xz)

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Bayes Law Problem

• We really want p(xz) samples
• But we only have p(x) samples !
• How can we upgrade p(x) to p(xz) ?

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More General Problem

• We really want h(x) samples
• But we only have g(x) samples !
• How can we upgrade g(x) to h(x) ?

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Solution Importance Sampling

• 1. generate xi from g(x)
• 2. calculate wi h(xi)/g(xi)
• 3. assign weight qi wi/ ?wi
• Still works if h(x) only known up to
  normalization factor

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Mean and Weighted Mean

• Fair sampleobtain samples xi from
  p(xz)Em(x)z ? m(xi)/N
• Weighted sample obtain weighted samples (xi,qi)
  from p(xz) Em(x)z ? qi m(xi)

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Bayes Law using Samples

• 1. generate xi from p(x)
• 2. calculate wi l(xiz)
• 3. assign weight qi wi/ ? wi
• Indeed wip(xz)/p(x) l(xz) p(x) /p(x)
  l(xz)
• 4. if you want, resample from (xi,qi)

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Particle Filtering
weighted Sk
Sk
Sk-1
Sk
Predict
Weight
Resample
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Outline

• Robot Localization
• Sensor ?
• Density Representation ?
• Monte Carlo Localization
• Results

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Video A

• Office Environment
• Sonar Sensors
• Global Localization
• Symmetry confusion

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Global Localization
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Global Localization (2)
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Global Localization (3)
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Reference Path
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Accuracy
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Video B

• Smithsonian Museum of American History
• Ceiling Camera, Global Localization

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Odometry Only
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Using Vision
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More Coffee
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Fast Internet Morning
Odometry only
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Video C

• UW Sieg Hall
• Laser

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Take Home Messages

• Particle FilteringPowerful yet
  efficientSignificantly less memory and CPUVery
  simple to implement
• Representing uncertainty using samples is
  powerful, fast, and simple !