Probabilistic Fundamentals Probabilistic Fundamentals in Robotics in - - PDF document

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Probabilistic Fundamentals Probabilistic Fundamentals in Robotics in Robotics

Basic Conc e pts in Pr

• bability

Basic Conc e pts in Pr

• bability

Basilio Bona DAUIN – Politecnico di Torino

Course Outline

• Motivations
• Basic mathematical framework
• Probabilistic models of mobile robots

Probabilistic models of mobile robots

• Mobile robot localization problem
• Robotic mapping
• Probabilistic planning and control

Reference textbook [TBF2006] Thrun, Burgard, Fox, “Probabilistic Robotics”, MIT Press, 2006 http://www.probabilistic‐robotics.org/

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Basic mathematical framework

• Basic concepts in probability
• Recursive state estimation

– Robot environment – Bayes filters

• Gaussian filters

– Kalman filter – Extended Kalman Filter – Unscented Kalman filter Information filter – Information filter

• Nonparametric filters

– Histogram filter – Particle filter

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Basic concepts in probability

• In binary logic, a proposition about the state of the world

is only True or False; no third hypothesis is considered

• Bayesian probability is a measure of the degree of

y p y g f belief of a proposition, or an objective degree of rational belief, given the evidence

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Other axioms

A B

True

A B AÇB

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Random variables

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( ) P x x

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Continuous random variables

( ) p x

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x a b Pr( ) x

Univariate Gaussian distribution

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Normal distribution

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Normal distribution

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Multi‐variate Gaussian distribution

Mean vector Covariance matrix

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Joint and conditional probabilities

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Marginal and total Probability

Discrete Continuous

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Posterior probability and Bayes rule

Prior probability distribution

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Posterior probability distribution

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Bayes rule conditioned by another variable

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Normalization

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Marginal probability

Marginal probability

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Conditional independence

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This is an important rule in probabilistic robotics. It applies whenever a variable y carries no information about a variable x, if the value z of another variable is known

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Conditional independence ¹ absolute independence

conditional independence

and

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absolute independence

Expectation of a random variable

• Features of probabilistic distributions are called statistics

statistics

• Expectation of a random variable (RV) X is defined as

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Covariance

• Covariance measures the squared expected deviation
• Covariance measures the squared expected deviation

from the mean

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Entropy

• Entropy measures the expected information that the

value of x carries

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In discrete case is the number of bits required to encode x using an optimal encoding, assuming that p(x) is the probability of observing x

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Robot environment interaction

LOCALIZATION LOCALIZATION PLANNING PLANNING PERCEPTION PERCEPTION ACTION ACTION

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Environment Environment

Robot environment interaction

• World or environment is a dynamical system that has an

internal state

• Robot sensors can acquire information about the world

q internal state

• Sensors are noisy and often complete information cannot

be acquired

• A belief measure about the state of the world is stored by

the robot

• Robot influences the world through its actuators (e.g.,

they make it move in the environment)

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State

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Complete state

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Stochastic process

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Markov chains

• a Markov chain is a discrete random process with the

p Markov property

• A stochastic process has the Markov property if the

conditional probability distribution of future states of the process depend only upon the present state; that is, given the present, the future does not depend on the past. p , p p

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Environment interaction

• Measurements: are perceptual interaction between the

robot and the environment obtained through specific equipment (called also perceptions).

• Control actions: are change in the state of the world
• btained through active asserting forces.
• Odometer data: are of perceptual data that convey the

information about the robot change of status; as such they are not considered measurements, but control data, since they measure the effect of control actions.

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Probabilistic generative laws

• Evolution of state is governed by probabilistic laws.
• If state is complete and Markov, then evolution depends
• nly on present state and control actions

y p

• Measurements are generated, according to probabilistic

laws from the present state only

State transition probability

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laws, from the present state only

Measurement probability

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Dynamical stochastic system

Temporal generative model Hidden Markov model (HMM) Dynamic Bayesian network (DBN)

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Belief distribution

• What is a belief: it is a measure of the robot’s internal knowledge

about the true state of the environment

• Belief is traditionally expressed as conditional probability

distributions distributions.

• Belief distribution: assigns a probability (or a density) to each

possible hypothesis about the true state, based upon available data (measurements and controls)

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State belief (posterior) State belief (prior) Prediction Correction/update

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Bayes filter

• Basic algorithm

Prediction Update

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Mathematical formulation of the Bayesian filter (1)

the state is complete

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Mathematical formulation of the Bayesian filter (2)

the state is complete

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... ...

Mathematical formulation of the Bayesian filter (3)

The filter requires three probability distributions

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Bayes filter recursion

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Causal vs. diagnostic reasoning

A rover obtains a measurement z from a door that can be open (O) or closed (C)

Easier to obtain

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Easier to obtain

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Example

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References

• Many textbooks on Probability Theory and Statistics

– Bertsekas, D. P., and J. N. Tsitsiklis. Introduction to Probability. Athena Scientific Press, 2002. Grimmett G R and D R Stirzaker Probability and Random Processes 3rd – Grimmett, G. R., and D. R. Stirzaker. Probability and Random Processes. 3rd ed., Oxford University Press, 2001. – Ross S., A First Course in Probability. 8th ed., Prentice Hall, 2009.

• Other materials

– http://cs.ubc.ca/~arnaud/stat302.html: slides from the course by A. Doucet, University of British Columbia – video course: http://academicearth.org/lectures/introduction‐probability‐ video course: http://academicearth.org/lectures/introduction probability and‐counting: UCLA/MATHEMATICS – Introduction: Probability and Counting, by Mark Sawyer | Math and Probability for Life Sciences

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Thank you. Any question?

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PhD_course_2010‐Outline.pptx

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