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

ROBOTICS 01PEEQW

Basilio Bona DAUIN – Politecnico di Torino

Probabilistic Fundamentals in Robotics

Basic Concepts in Probability

Course Outline

Motivations Basic mathematical framework 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

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 belief of a proposition, or an objective degree of rational belief, given the evidence

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Other axioms

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A B A∩B

True

Random variables

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( ) P x x

Continuous random variables

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( ) p x 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

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Mean vector Covariance matrix

Joint and conditional probabilities

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Marginal and total Probability

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Discrete Continuous

Posterior probability and Bayes rule

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Posterior probability distribution Prior probability distribution

Bayes rule conditioned by another variable

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Normalization

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Marginal probability

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Marginal probability

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

• f another variable is known

Conditional independence ≠ absolute independence

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and

conditional independence absolute independence

Expectation of a random variable

Features of probabilistic distributions are called statistics Expectation of a random variable (RV) X is defined as

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Covariance

Covariance measures the squared expected deviation from the mean

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Entropy

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Entropy measures the expected information that the value of x carries

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

• bserving x

Robot environment interaction

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LOCALIZATION PLANNING PERCEPTION ACTION

Environment

Robot environment interaction

World or environment is a dynamical system that has an internal state Robot sensors can acquire information about the world 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 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.

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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 obtained 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 only on present state and control actions

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Measurements are generated, according to probabilistic laws, from the present state only

State transition probability Measurement probability

Dynamical stochastic system

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

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. 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

Bayes filter

Basic algorithm

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Prediction Update

Mathematical formulation of the Bayesian filter (1)

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the state is complete

Mathematical formulation of the Bayesian filter (2)

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the state is complete ... ...

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

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

Easier to

• btain

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 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-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?