5/21/2012 1 Probabilistic Fundamentals Probabilistic Fundamentals in Robotics in Robotics Basic Concepts in Probability Basic Concepts in Probability 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/ Basilio Bona 2
Basic Concepts in ProbabilityBasic Concepts in Probability
Basilio BonaDAUIN – Politecnico di Torino
Course Outline
Motivations
Basic mathematical framework
Probabilistic models of mobile robotsProbabilistic 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 fbelief 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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xa 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 vectorCovariance 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 statisticsstatistics
Expectation of a random variable (RV) X is defined as
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Covariance
Covariancemeasures the squared expected deviation Covariancemeasures the squared expected deviation from the mean
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Entropy
Entropymeasures 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
LOCALIZATIONLOCALIZATION PLANNINGPLANNING
PERCEPTIONPERCEPTION ACTIONACTION
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EnvironmentEnvironment
Robot environment interaction
World or environment is a dynamical system that has an internal state
Robot sensors can acquire information about the world qinternal state
Sensors are noisy and often complete information cannot be 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 pMarkov 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 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 actionsy 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 distributionsdistributions.
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 probabilityand‐counting: UCLA/MATHEMATICS – Introduction: Probability and Counting, by Mark Sawyer | Math and Probability for Life Sciences
