ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino - - PowerPoint PPT Presentation

ROBOTICS 01PEEQW

Basilio Bona DAUIN – Politecnico di Torino

Probabilistic Fundamentals in Robotics

Non-parametric Filters

Course Outline

Basic mathematical framework Probabilistic models of mobile robots Mobile robot localization problem Robotic mapping Probabilistic planning and control Reference textbook

Thrun, Burgard, Fox, “Probabilistic Robotics”, MIT Press, 2006 http://www.probabilistic-robotics.org/

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Probabilistic models of mobile robots

Recursive state estimation

Basic concepts in probability Robot environment Bayes filters

Gaussian filters

Kalman filter Extended Kalman Filter Unscented Kalman filter Information filter

Nonparametric filters

Histogram filter Particle filter

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Introduction

Nonparametric filters do not rely on fixed functional form of the posterior probability They approximate posteriors over continuous state spaces (CSS) with finitely many values

Decomposition of CSS in finitely many regions and representation of the posterior by a histogram: histogram filters Representation of CSS by finitely many samples: particle filters

Nonparametric filters represents well multimodal distributions, i.e., distinct hypotheses (as in mobile robotics) Computational complexity

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Histogram filter

The state space can be discrete or continuous The random variable Xt can take finitely many values Examples of discrete spaces are:

Grid element in a grid map: occupied/free Door: open/closed Terrain slope: none/mid/high Terrain characteristics: sand/rock/grass/…

Discrete Bayes filters can be used for this type of problems

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Discrete Bayes filter

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Discrete probability distributio

Continuous State

Discrete Bayes filter can be used to approximate continuous state spaces They are called histogram filters Space is divided into mutually non-overlapping intervals (bins or grid elements)

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Approximation

If state is discrete, the following are well defined

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If state is continuous, approximation is necessary; e.g.,

Mean state value (normalized)

Histogram transformation

The histogram of the transformed random variable is computed by passing multiple points from each histogram bin through the nonlinear function

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Decomposition of the state space

Bins definition: for the histogram approach a decomposition of the state space is necessary Static decomposition partition the state in a fixed pre-define number

• f mutually non-overlapping subsets: easier to implement, high

computational resources Dynamic decomposition adapt the decomposition to the shape of the posterior distribution reduced computational resources, added algorithmic complexity Density trees are an example of dynamic decomposition

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Density trees

• Space decomposition is recursive
• Adapts the resolution to the posterior probability: the

less likely is a region, the coarser the decomposition

• Compact representation: higher approximation quality

with the same number of bins

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Topological and grid maps

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Static state and binary Bayes filter

Binary Bayes filters are used when the state is both static and binary

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Binary Bayes filter

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Particle filters: key concepts

Another nonparametric implementation of the Bayes filter Approximate the posterior by a finite number of parameters (as in histogram filters) Key idea: the posterior belief is represented by a set of state samples drawn from the distribution The state samples are called particles A particle is a hypothesis as to what the true world state may be at time t The likelihood for a state hypothesis xt to be included in the particle set shall be proportional to the its Bayes filter posterior bel(xt)

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Particle filters: examples

The denser a region is populated by samples, the more likely is that the true state belongs to this region

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Normal distribution Multimodal distribution

Particle filters: mathematical description

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Particle filter algorithm

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Importance factor, used to incorporate the measurement into the particle set

Resampling aka importance sampling M particles are drawn with replacement from the temporary set The probability is given by the importance weight. At the end of the process …

Hypothetical state is generated

Resampling

Resampling is an important step to correctly approximate the posterior belief It can be seen as a probabilistic implementation of the “survival-of- the-fittest” model It focuses the particles on regions of space with high posterior probability We will discuss resampling in more details

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Importance sampling

Starting with samples coming from a distribution g, we want to compute an expectation over a probability function f

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Importance sampling

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This is what we want Target distribution

Importance sampling

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This is what we have Proposal distribution

Importance sampling

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This is what we obtain

Practical considerations and properties

Density estimation or density extraction: we may want continuous description of belief, not discrete approximations given by particles Sampling variance: statistics from particles is different from statistics from original densities Resampling Sampling bias, particle deprivation: not treated here

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Density estimation

Methods

Gaussian approximation: simple but captures only approximated distribution; unimodality only Histogram approximation: can represent multimodal distributions, computationally highly efficient, the complexity of computing density in any state point is independent of the number of particles Kernel density approximation: can represent multimodal distributions, smoothness and algorithmic simplicity, the complexity of computing density in any state point is linear in the number of particles

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Gaussian approximation

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Histogram approximation

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Kernel approximation

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Sampling variance

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250 particles 25 particles

Resampling

Sampling variance is amplified through repetitive resampling Look at step 3. It may happen that no command signal ut is applied No new states are introduced at successive steps The particles are erased and new ones are not created M identical copies of a single particle will survive The variance of the particle set decreases, but … the variance of the particle set as an estimator of the true belief increases

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Summary and conclusions

The histogram filter decomposes the state space in finitely many convex regions. It represents the cumulative posterior probability of each region by a single numerical value Many state space decomposition techniques exist. The granularity of decomposition may or may not depend on the structure of the

• environment. When it does, the decomposition is called topological

An alternative nonparametric technique is the particle filter

• algorithm. They are easy to implement and, with due care, are the

most versatile of all Bayes filter algorithms. Specific strategies exist to reduce the error in particle filters

• Reduction of the variance of the estimate that arises from the randomness of the algorithm
• Adaptation of the number of particles in accordance to the complezity of the posterior

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