robabilistic Robotics Slides from Autonomous Robots Siegwart and Nourbaksh Chapter 5 robabilistic Robotics S. Thurn et al.
Today Overview of probability Representing uncertainty ropagation of uncertainty Bayes Rule Application to Localiation and Mapping CS 685 2
robabilistic Robotics Key idea: Eplicit representation of uncertainty using the calculus of probability theory! erception state estimation! Action utility optimiation
Aioms of robability Theory ra denotes probability that proposition A is true 0 r A r True r False 0 r A B r A + r B r A B
A Closer Look at Aiom 3 r A B r A + r B r A B True A A B B B
Synta! Basic element: random variable! Similar to propositional logic: possible worlds defined by assignment of values to random variables.! Boolean random variables e.g. Cavity do I have a cavity?! Discrete random variables e.g. Weather is one of <sunnyrainycloudysnow>! Domain values must be ehaustive and mutually eclusive! Elementary proposition constructed by assignment of a value to a random variable: e.g. Weather sunny Cavity false abbreviated as cavity! Comple propositions formed from elementary propositions and standard logical connectives e.g. Weather sunny Cavity false
Synta! Atomic event: A complete specification of the state of the world about which the agent is uncertain E.g. if the world consists of only two Boolean variables Cavity and Toothache then there are 4 distinct atomic events: Cavity false Toothache false Cavity false Toothache true Cavity true Toothache false Cavity true Toothache true! Atomic events are mutually eclusive and ehaustive
rior probability! rior or unconditional probabilities of propositions Cavity true 0. and Weather sunny 0.72 correspond to belief prior to arrival of any new evidence! robability distribution values for all possible assignments: Weather <0.720.0.080.> normalied i.e. sums to! Joint probability distribution for a set of random variables gives the probability of every atomic event on those random variables WeatherCavity a 4 2 matri of values: Weather sunny rainy cloudy snow Cavity true 0.44 0.02 0.06 0.02 Cavity false 0.576 0.08 0.064 0.08
Joint Distribution! Every question about the domain can be answered from joint probability distribution! Two eamples of joint distributions Weather sunny rainy cloudy snow Cavity true 0.44 0.02 0.06 0.02 Cavity false 0.576 0.08 0.064 0.08
Conditional probability! Definition of conditional probability: a b a b b! roduct rule gives an alternative formulation: a b a bb b aa! A general version holds for whole distributions e.g. WeatherCavity Weather Cavity Cavity! View as a set of 4 2 equations not matri multiplication! Chain rule is derived by successive product rule X X n X...X n- X n X...X n- X...X n-2 X n- X...X n-2 X n X...X n- n X i X X i i
Inference by enumeration! Start with the joint probability distribution:! Given a proposition calculate what is its probability! For any proposition φ sum the atomic events where it is true: φ Σ ω:ω φ ω
Inference by enumeration! Start with the joint probability distribution:! For any proposition φ sum the atomic events where it is true: φ Σ ω:ω φ ω! toothache 0.08 + 0.02 + 0.06 + 0.064 0.2
Inference by enumeration! Start with the joint probability distribution:! Can also compute conditional probabilities: cavity toothache cavity toothache toothache 0.06 + 0.064 0.08 + 0.02 + 0.06 + 0.064 0.4
Discrete Random Variables! X denotes a random variable.! X can take on a countable number of values in { 2 n }.! X i or i is the probability that the random variable X takes on value i.! is called probability mass function.! E.g. Recognie which room you robot is in! This is just shorthand for Room office Room kitchen Room bedroom Room corridor Room 0.70.20.080.02
Continuous Random Variables! X takes on values in the continuum.! px or p is a probability density function.! E.g. r a b p d p b a
Joint and Conditional robability! X and Yy y! If X and Y are independent then y y! y is the probability of given y y y / y y y y! If X and Y are independent then y verify using definitions of conditional probability and independence
Law of Total robability Marginals Discrete case Law of total probability Marginaliation y y y y y Continuous case p d p p y dy p p y p y dy
Bayes Formula evidence prior likelihood y y y y y y y
Normaliation y y y y y y η η y y y y y au : au : au η η Algorithm:
Bayes' Rule! roduct rule Bayes' rule:! or in distribution form a b a bb b aa Y X a b X YY X b aa b! Useful for assessing diagnostic probability from causal probability: CauseEffect EffectCause Cause / Effect Note: posterior probability of meningitis still very small!
Bayes' Rule! Baye s rule Y X X YY X! More general version conditionalied on some evidence X Y ey e Y X e X e! E.g. let M be meningitis S be stiff neck: m s s mm s! Normaliation same for m and ~m Y X αx Y Y 0.8 0.00 0. 0.008
Bayes Rule with Background Knowledge y y y
Conditional Independence y y! Equivalent to y and y y! But this does not necessarily mean y y independence/marginal independence
Simple Eample of State Estimation! Suppose a robot obtains measurement! What is open?
Causal vs. Diagnostic Reasoning! open is diagnostic! open is causal! Often causal knowledge is easier to obtain! Bayes rule allows us to use causal knowledge: count frequencies! open open open
Eample! open 0.6 open 0.3! open open 0.5 open open open open p open + open p open open 0.6 0.5 0.6 0.5 + 0.3 0.5 0.3 0.3 + 0.5 0.67 " raises the probability that the door is open
Combining Evidence! Suppose our robot obtains another observation 2! How can we integrate this new information?! More generally how can we estimate... n?
Eample: Second Measurement! 2 open 0.5 2 open 0.6! open 2/3 0.625 8 5 5 8 3 5 3 3 3 5 3 3 2 2 3 2 2 2 2 2 2 + + + open open open open open open open 2 lowers the probability that the door is open
Recursive Bayesian Updating n n n n n n Markov assumption: n is independent of... n- if we know...... n i i n n n n n n n n η η
Actions! Often the world is dynamic since! actions carried out by the robot! actions carried out by other agents! or just the time passing by change the world! How can we incorporate such actions?
Typical Actions! The robot turns its wheels to move! The robot uses its manipulator to grasp an object! lants grow over time! Actions are never carried out with absolute certainty! In contrast to measurements actions generally increase the uncertainty
Modeling Actions! To incorporate the outcome of an action u into the current belief we use the conditional pdf u! This term specifies the pdf that eecuting u changes the state from to.
Eample: Closing the door
State Transitions u for u close door : 0.9 0. open closed 0 If the door is open the action close door succeeds in 90% of all cases
Integrating the Outcome of Actions Continuous case: u u ' ' d' Discrete case: u u ' '
Eample: The Resulting Belief 6 8 3 0 8 5 0 ' ' 6 5 8 3 8 5 0 9 ' ' u closed closed closed u open open open u open u open u open closed closed u closed open open u closed u closed u closed + + + +
Bayes Filters: Framework! Given:! Stream of observations and action data u:! Sensor model! Action model u! rior probability of the system state! Wanted: d t { u u! Estimate of the state X of a dynamical system! The posterior of the state is also called Belief: t t } Bel t t u u t t
Markov Assumption p t 0:t :t u :t p t t p t :t :t u :t p t t u t Underlying Assumptions! Static world! Independent noise! erfect model no approimation errors
Bayes Filters observation u action state Bel t t u u t t Bayes η t t u u t t u u t Markov η t t t u u t Total prob. η t t t u u t t t u u t d t Markov η t t t u t t t u u t d t Markov η t t t u t t t u t d t η t t t u t t Bel t d t
Bayes Filter Algorithm. Algorithm Bayes_filter Beld : 2. η0 3. If d is a perceptual data item then 4. For all do 5. 6. 7. For all do 8. 9. Else if d is an action data item u then 0. For all do. Bel t η t t t u t t 2. Return Bel Bel ' Bel η η + Bel' Bel' η Bel' Bel' u ' Bel ' Bel t d t d'
Bayes Filters are Familiar! Bel t η t t t u t t Bel t d t! Kalman filters! article filters! Hidden Markov models! Dynamic Bayesian networks! artially Observable Markov Decision rocesses OMDs
Summary! Bayes rule allows us to compute probabilities that are hard to assess otherwise.! Under the Markov assumption recursive Bayesian updating can be used to efficiently combine evidence.! Bayes filters are a probabilistic tool for estimating the state of dynamic systems.
Belief Representation! a Continuous map with single hypothesis! b Continuous map with multiple hypothesis! d Discretied map with probability distribution! d Discretied topological map with probability distribution R. Siegwart I. Nourbakhsh
Belief Representation: Characteristics! Continuous! recision bound by sensor data! Typically single hypothesis pose estimate! Lost when diverging for single hypothesis! Compact representation and typically reasonable in processing power.! Discrete! recision bound by resolution of discretisation! Typically multiple hypothesis pose estimate! Never lost when diverges converges to another cell! Important memory and processing power needed. not the case for topological maps R. Siegwart I. Nourbakhsh
Bayesian Approach: A taonomy of models More general Bayesian rograms Bayesian Networks Courtesy of Julien Diard S: State O: Observation A: Action S t S t- DBNs S t S t- A t Markov Chains Bayesian Filters Markov Loc MDs article Filters discrete HMMs semi-cont. HMMs continuous HMMs MCML OMDs More specific S t S t- O t Kalman Filters S t S t- O t A t R. Siegwart I. Nourbakhsh