DAUIN Politecnico di Torino July 2010

Transcript

1 Probabilistic Fundamentals in Robotics Basic Concepts in Probability Basilio Bona DAUIN Politecnico di Torino July 2010

2 Motivations Course Outline Basic mathematicalframework 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, robotics.org/ probabilistic robotics org/ Basilio Bona July

3 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 Basilio Bona July

4 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 Basilio Bona July

5 Other axioms True A B A B Basilio Bona July

6 Random variables Px ( ) Basilio Bona July x

7 Continuous random variables px ( ) a Pr( x) b x Basilio Bona July

8 Univariate Gaussian distribution Basilio Bona July

9 Normal distribution Basilio Bona July

10 Normal distribution Basilio Bona July

11 Multi variate Gaussian distribution Covariance matrix Mean vector Basilio Bona July

12 Joint and conditional probabilities Basilio Bona July

13 Discrete Marginal and total Probability Continuous Basilio Bona July

14 Posterior probability and Bayes rule Prior probability distribution Posterior probability distribution Basilio Bona July

15 Bayes rule conditioned by another variable Basilio Bona July

16 Normalization Basilio Bona July

17 Marginal probability Marginal probability Basilio Bona July

18 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 July

19 Conditional independence absolute independence conditional independence and absolute independence Basilio Bona July

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

21 Covariance Covariance measures the squared expected deviation from the mean Basilio Bona July

22 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 July

23 Robot environment interaction LOCALIZATION PLANNING PERCEPTION ACTION Environment Basilio Bona July

24 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) Basilio Bona July

25 State Basilio Bona July

26 Complete state Basilio Bona July

27 Stochastic process Basilio Bona July

28 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. Basilio Bona July

29 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 tthe robot change of status; tt as such they are not considered measurements, but control data, since they measure the effect of control actions. Basilio Bona July

30 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 State t transition probability bilit Measurements are generated, according to probabilistic laws, from the present state only Measurement probability Basilio Bona July

31 Dynamical stochastic system Temporal generative model Hidden Markov model (HMM) Dynamic Bayesian network (DBN) Basilio Bona July

32 Belief distribution What is a blifi belief: it is a measure of the robot s internal lknowledge 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) Prediction State belief (prior) Correction/update State belief (posterior) Basilio Bona July

33 Bayes filter Basic algorithm Prediction Update Basilio Bona July

34 Mathematical formulation of the Bayesian filter (1) the state is complete Basilio Bona July

35 Mathematical formulation of the Bayesian filter (2) the state is complete Basilio Bona July

36 Mathematical formulation of the Bayesian filter (3) The filter requires three probability distributions Basilio Bona July

37 Bayes filter recursion Basilio Bona July

38 Causal vs. diagnostic reasoning A rover obtains a measurement z from a door that t can be open (O) or closed d(c) Easier to obtain Basilio Bona July

39 Example Basilio Bona July

40 References Many textbooks on Probability Theory and Statistics Bertsekas, D. P., and J. N. Tsitsiklis. Introduction to Probability. Athena Scientific Press, Grimmett, G. R., and D. R. Stirzaker. Probability and Random Processes. 3rd ed., Oxford University Press, Ross S., A First Course in Probability. 8th ed., Prentice Hall, Other materials slides from the course by A. Doucet, University of British Columbia video course: probabilityand counting: UCLA/MATHEMATICS Introduction: Probability and Counting, by Mark Sawyer Math and Probability for Life Sciences Basilio Bona July

41 Thank you. Any question? Basilio Bona July

42 PhD_course_2010 Outline.pptx Basilio Bona July