Probabilistic Fundamentals Probabilistic Fundamentals in Robotics in - PDF document

5/21/2012 Probabilistic Fundamentals Probabilistic Fundamentals in Robotics in Robotics Basic Conc e pts in Pr Basic Conc e pts in Pr obability obability 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 1

5/21/2012 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 Basilio Bona 3 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 Basilio Bona 4 2

5/21/2012 Other axioms True A A B B A Ç B Basilio Bona 5 Random variables P x ( ) x Basilio Bona 6 3

5/21/2012 Continuous random variables p x ( ) Pr( ) x x a b Basilio Bona 7 Univariate Gaussian distribution Basilio Bona 8 4

5/21/2012 Normal distribution Basilio Bona 9 Normal distribution Basilio Bona 10 5

5/21/2012 Multi ‐ variate Gaussian distribution Covariance matrix Mean vector Basilio Bona 11 Joint and conditional probabilities Basilio Bona 12 6

5/21/2012 Marginal and total Probability Discrete Continuous Basilio Bona 13 Posterior probability and Bayes rule Prior probability distribution Posterior probability distribution Basilio Bona 14 7

5/21/2012 Bayes rule conditioned by another variable Basilio Bona 15 Normalization Basilio Bona 16 8

5/21/2012 Marginal probability Marginal probability Basilio Bona 17 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 Basilio Bona 18 9

5/21/2012 Conditional independence ¹ absolute independence conditional independence and absolute independence Basilio Bona 19 Expectation of a random variable  Features of probabilistic distributions are called statistics statistics  Expectation of a random variable (RV) X is defined as Basilio Bona 20 10

5/21/2012 Covariance  Covariance measures the squared expected deviation  Covariance measures the squared expected deviation from the mean Basilio Bona 21 Entropy  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 observing x Basilio Bona 22 11

5/21/2012 Robot environment interaction LOCALIZATION LOCALIZATION PLANNING PLANNING PERCEPTION PERCEPTION ACTION ACTION Environment Environment Basilio Bona 23 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) Basilio Bona 24 12

5/21/2012 State Basilio Bona 25 Complete state Basilio Bona 26 13

5/21/2012 Stochastic process Basilio Bona 27 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 Basilio Bona 28 14

5/21/2012 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. Basilio Bona 29 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 y p State transition probability  Measurements are generated, according to probabilistic laws from the present state only laws, from the present state only Measurement probability Basilio Bona 30 15

5/21/2012 Dynamical stochastic system Temporal generative model Hidden Markov model (HMM) Dynamic Bayesian network (DBN) Basilio Bona 31 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) Prediction State belief (prior) Correction/update State belief (posterior) Basilio Bona 32 16

5/21/2012 Bayes filter  Basic algorithm Prediction Update Basilio Bona 33 Mathematical formulation of the Bayesian filter (1) the state is complete Basilio Bona 34 17

5/21/2012 Mathematical formulation of the Bayesian filter (2) the state is complete ... ... Basilio Bona 35 Mathematical formulation of the Bayesian filter (3) The filter requires three probability distributions Basilio Bona 36 18

5/21/2012 Bayes filter recursion Basilio Bona 37 Causal vs. diagnostic reasoning A rover obtains a measurement z from a door that can be open ( O ) or closed ( C ) Easier to obtain Easier to obtain Basilio Bona 38 19

5/21/2012 Example Basilio Bona 39 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 Basilio Bona 40 20

5/21/2012 Thank you. Any question? Basilio Bona 41 PhD_course_2010 ‐ Outline.pptx Basilio Bona 42 21