Monte Carlo Localization Tutorial

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Monte Carlo Localization (Tutorial)

• Kriara Lito

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Introduction

• Monte Carlo localization (MCL) is a Monte Carlo
  method to determine the position of a robot given
  a map of its environment based on Markov
  localization.
• implementation of the Particle filter

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Initialization

• Random initialization of configuration space
• Like in particle filters method

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Probability update

• With each sensor update, the probability that
  each hypothetical configuration is correct is
  updated based on a statistical model of the
  sensors and Bayes' theorem.

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Re-initialization

• When the probability of a hypothetical
  configuration becomes very low, it is replaced
  with a new random configuration.

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Comparison with particle filters

• In each iteration there is a conditional
  probability (Bayes Theorem) given the prior
  position of the user.

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Algorithm 6

• Density
• by a set of N random samples / particles
• Prediction Phase The predictive density over xk
  is then obtained by integration
• Update Phase
• use a measurement model to incorporate
  information from the sensors to obtain the
  posterior PDF
• We assume that the measurement is
  conditionally independent of earlier measurements
  given , and that the measurement model
  is given in terms of a likelihood
• The posterior density over xk is obtained using
  Bayes theorem
• The process is repeated recursively

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Monte Carlo method(Tutorial)

• Lito Kriara

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Introduction

• Monte-Carlo algorithms always return a result,
  not an approach of the exact solution.
• The result of a Monte Carlo method may not be
  correct, but the probability of the algorithms
  success increases proportionally to its
  execution time.

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Introduction

• Class of computational algorithms that rely on
  repeated random sampling to compute their results
  
• They rely on repeated computation and random or
  pseudo-random numbers
• used when it is infeasible or impossible to
  compute an exact result with a deterministic
  algorithm

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Goal

• Derive algorithms that, starting from some
  initial value and applying iterative methods,
  produce a sequence that
• It is unpredictable for the uninitiated (relation
  with Chaotic dynamical systems)
• It passes a battery of standard statistical tests
  of randomness (like Kolmogorov-Smirnov test,
  ARMA(p,q), etc).

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Example

• player makes random shots
• player applies algorithms
• player can determine likely locations of other
  player's ships
• based on outcome of
• random sampling
• algorithm

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Monte Carlo methods pattern

• Define domain of possible inputs
• Generate inputs randomly from domain
• perform deterministic computation on them
• Aggregate results of individual computations into
  final result

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• Examples of Monte Carlo use

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Difficult to even approximate
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Classical Monte Carlo Integration

• Assume we know how to generate draws from
• What does it mean to draw from ?

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Basic Component

• Multiplicative Congruential Generator
• xi takes values on 0, 1, ...,M
• Transformation into a generator on 0, 1 with
• x0 is called the seed

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Choices of Parameters

• Period and performance depends crucially on
• a, b, and M
• Pick a 13, c 0, M 31, and x0 1
• Let us run badrandom.m
• Historical bad examples
• IBM RND from the 1960s

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A Good Choice

• A traditional choice
• a 75 16807, c 0, m 231 - 1
• Period bounded by
• M. 32 bits versus 64 bits hardware

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Real Life

• Code your own random number generator
• Matlab implements such functions
• http//stat.fsu.edu/pub/diehard/

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Example - Acceptance Sampling

• ? with support C,
• is called the target density
• z g (z) with support C ? C,
• g is called the source density
• We require
• We know how to draw from g
• Condition

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Acceptance Sampling Algorithm

• Steps
• u U(0, 1)
• If return to step 1.
• Set

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Conclusion

• Monte Carlo method is a basic idea (slide 5)
  transformed according to the context used each
  time.
• Such a transformation is for localization issues
• Monte Carlo Localization

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References

• http//en.wikipedia.org/wiki/Monte_Carlo_localizat
  ion
• http//en.wikipedia.org/wiki/Monte_Carlo_method
• http//www.chem.unl.edu/zeng/joy/mclab/mcintro.htm
  l
• http//www.econ.upenn.edu/jesusfv/LectureNotes_5_
  montecarlo
• A. Doucet, On sequential simulation-based
  methods for Bayesian filtering, Tech. Rep.
  CUED/F-INFENG/TR.310, Department of Engineering,
  University of Cambridge, 1998.
• F. Dellaert, D. Fox, W. Burgard and S. Thrun,
  Monte Carlo Localization for Mobile Robots,
  IEEE International Conference on Robotics
  Automation, 1998