Probabilistic Fundamentals in Robotics - polito.it · Probabilistic Fundamentals in Robotics Basic Concepts in Probability Basilio Bona DAUIN – Politecnico di Torino July 2010.

Probabilistic FundamentalsProbabilistic Fundamentalsin Roboticsin Roboticsin Roboticsin Robotics

Basic Concepts in ProbabilityBasic Concepts in Probability

Basilio Bonaasilio onaDAUIN – Politecnico di Torino

July 2010

Course Outline

Motivations

Basic mathematical frameworkBasic 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 PressThrun, Burgard, Fox, Probabilistic Robotics , MIT Press, 2006

http://www probabilistic‐robotics org/http://www.probabilistic‐robotics.org/

Basilio Bona July 2010 2

Basic mathematical framework

Basic concepts in probabilityRecursive state estimationRecursive state estimation– Robot environment– Bayes filtersBayes filters

Gaussian filters– Kalman filterKalman filter– Extended Kalman Filter– Unscented Kalman filter– Information filter

Nonparametric filtersp– Histogram filter– Particle filter


Basic concepts in probability

In binary logic, a proposition about the state of the world is only True or False; no third hypothesis is consideredis 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 rationalbelief of a proposition, or an objective degree of rational belief, given the evidence


Other axioms

True

A BA B

A∩B


Random variables

( )P x


x

Continuous random variables

( )( )p x

xa b

Pr( )x


a b

Univariate Gaussian distribution


Normal distribution


Normal distribution


Multi‐variate Gaussian distribution

Mean vectorCovariance matrix Mean vectorCovariance matrix


Joint and conditional probabilities


Marginal and total Probability

Di tDiscrete

Continuous


Posterior probability and Bayes rule

Prior probability distribution

Posterior probability distribution


Bayes rule conditioned by another variable


Normalization


Marginal probability

Marginal probability


Conditional independence

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

Conditional independence ≠ absolute independence

conditional independenceconditional independence

andand

absolute independenceabsolute independence


Expectation of a random variable

Features of probabilistic distributions are called statisticsstatistics

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


Covariance

Covariancemeasures the squared expected deviation from the mean


Entropy

Entropymeasures the expected information that the value of x carriesvalue of x carries

In discrete case is the number of bits required to encode x


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

Robot environment interaction

LOCALIZATIONLOCALIZATION PLANNINGPLANNING

PERCEPTIONPERCEPTION ACTIONACTION

E i tE i t


EnvironmentEnvironment

Robot environment interaction

World or environment is a dynamical system that has an internal stateinternal state

Robot sensors can acquire information about the world internal stateinternal state

Sensors are noisy and often complete information cannot b i dbe acquired

A beliefmeasure 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)


State


Complete state


Stochastic process


Markov chains

a Markov chain is a discrete random process with the Markov propertyMarkov 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.


Environment interaction

Measurements: are perceptual interaction between the robot and the environment obtained through specificrobot 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 i f ti b t th b t h f t t hinformation about the robot change of status; as such they are not considered measurements, but control data, i th th ff t f t l tisince they measure the effect of control actions.


Probabilistic generative laws

Evolution of state is governed by probabilistic laws.

If state is complete and Markov then evolution dependsIf state is complete and Markov, then evolution depends only on present state and control actions

St t t iti b bilit

Measurements are generated, according to probabilistic

State transition probability

g , g plaws, from the present state only

Measurement probability


Measurement probability

Dynamical stochastic system

Temporal generative model Hidden Markov model (HMM)

Dynamic Bayesian network (DBN)Dynamic Bayesian network (DBN)


Belief distribution

h i b li f i i f h b ’ i l k l dWhat 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)

State belief (prior)Prediction

Correction/update


State belief (posterior)

Correction/update

Bayes filter

Basic algorithm

Prediction

U d tUpdate


Mathematical formulation of the Bayesian filter (1)

the state is complete



the state is complete


......


The filter requires three probability distributions


Bayes filter recursion


Causal vs. diagnostic reasoning

A rover obtains a measurement z from a door th t b (O) l d (C)that can be open (O) or closed (C)

Easier to obtain


Example


References

Many textbooks on Probability Theory and Statistics– Bertsekas, D. P., and J. N. Tsitsiklis. Introduction to Probability. Athena , , y

Scientific Press, 2002.

– Grimmett, G. R., and D. R. Stirzaker. Probability and Random Processes. 3rd ed Oxford University Press 2001ed., Oxford University Press, 2001.

– Ross S., A First Course in Probability. 8th ed., Prentice Hall, 2009.

Other materialsOther 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 Sciencesby Mark Sawyer | Math and Probability for Life Sciences


Thank you.

Any question?


PhD course 2010 Outline pptxPhD_course_2010‐Outline.pptx


Probabilistic Fundamentals in Robotics - polito.it · Probabilistic Fundamentals in Robotics Basic Concepts in Probability Basilio Bona DAUIN – Politecnico di Torino July 2010.

Documents