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/
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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
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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 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
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Other axioms
True
A BA B
A∩B
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Random variables
( )P x
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x
Continuous random variables
( )( )p x
xa b
Pr( )x
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a b
Univariate Gaussian distribution
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Normal distribution
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Normal distribution
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Multi‐variate Gaussian distribution
Mean vectorCovariance matrix Mean vectorCovariance matrix
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Joint and conditional probabilities
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Marginal and total Probability
Di tDiscrete
Continuous
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Posterior probability and Bayes rule
Prior probability distribution
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
This is an important rule in probabilistic robotics. It applies whenever a variable y carries no information
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about a variable x, if the value z of another variable is known
Conditional independence ≠ absolute independence
conditional independenceconditional independence
andand
absolute independenceabsolute independence
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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
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Covariance
Covariancemeasures the squared expected deviation from the mean
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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
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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
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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)
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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 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.
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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.
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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
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Measurement probability
Dynamical stochastic system
Temporal generative model Hidden Markov model (HMM)
Dynamic Bayesian network (DBN)Dynamic Bayesian network (DBN)
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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
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State belief (posterior)
Correction/update
Bayes filter
Basic algorithm
Prediction
U d tUpdate
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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 th t b (O) l d (C)that can be open (O) or closed (C)
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 , , 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
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Thank you.
Any question?
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PhD course 2010 Outline pptxPhD_course_2010‐Outline.pptx
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