I ntroduction to Mobile Robotics Probabilistic Robotics Wolfram - PowerPoint PPT Presentation

I ntroduction to Mobile Robotics Probabilistic Robotics Wolfram Burgard 1

Probabilistic Robotics Key idea: Explicit representation of uncertainty (using the calculus of probability theory)  Perception = state estimation  Action = utility optimization 2

Axiom s of Probability Theory P (A) denotes probability that proposition A is true.    3

A Closer Look at Axiom 3 True A A ∧ B B B 4

Using the Axiom s 5

Discrete Random Variables  X denotes a random variable  X can take on a countable number of values in {x 1 , x 2 , …, x n }  P ( X=x i ) or P ( x i ) is the probability that the random variable X takes on value x i  P ( ) is called probability mass function .  E.g. 6

Continuous Random Variables  X takes on values in the continuum.  p ( X=x ) or p ( x ) is a probability density function  E.g. p ( x ) x 7

“ Probability Sum s up to One ” Discrete case Continuous case 8

Joint and Conditional Probability  P ( X=x and Y=y ) = P ( x,y )  If X and Y are independent then P ( x,y ) = P ( x ) P ( y )  P ( x | y ) is the probability of x given y P ( x | y ) = P ( x,y ) / P ( y ) P ( x,y ) = P ( x | y ) P ( y )  If X and Y are independent then P ( x | y ) = P ( x ) 9

Law of Total Probability Discrete case Continuous case 10

Marginalization Discrete case Continuous case 11

Bayes Form ula 12

Norm alization Algorithm : 13

Bayes Rule w ith Background Know ledge 14

Conditional I ndependence = P ( x , y z ) P ( x | z ) P ( y | z ) = P ( x z ) P ( x | z , y )  Equivalent to = P ( y z ) P ( y | z , x ) and  But this does not necessarily mean = P ( x , y ) P ( x ) P ( y ) (independence/ marginal independence) 15

Sim ple Exam ple of State Estim ation  Suppose a robot obtains measurement z  What is P ( open | z ) ? 16

Causal vs. Diagnostic Reasoning  P ( open|z ) is diagnostic  P ( z|open ) is causal  In some situations, causal knowledge is easier to obtain count frequencies!  Bayes rule allows us to use causal knowledge: P ( z | open ) P ( open ) = P ( open | z ) P ( z ) 17

Exam ple P ( z| ¬ open ) = 0.3  P ( z|open ) = 0.6  P ( open ) = P ( ¬ open ) = 0.5 P ( z | open ) P ( open ) = P ( open | z ) + ¬ ¬ P ( z | open ) p ( open ) P ( z | open ) p ( open ) ⋅ 0 . 6 0 . 5 0 . 3 = = = P ( open | z ) 0 . 67 ⋅ + ⋅ + 0 . 6 0 . 5 0 . 3 0 . 5 0 . 3 0 . 15  z raises the probability that the door is open 18

Com bining Evidence  Suppose our robot obtains another observation z 2  How can we integrate this new information?  More generally, how can we estimate P ( x | z 1 , ..., z n ) ? 19

Recursive Bayesian Updating Markov assum ption: z n is independent of z 1 ,...,z n-1 if we know x 20

Exam ple: Second Measurem ent P ( z 2 | ¬ open ) = 0.3  P ( z 2 |open ) = 0.25  P ( open|z 1 ) =2/3 • z 2 lowers the probability that the door is open 21

Actions  Often the world is dynam ic since  actions carried out by the robot ,  actions carried out by other agents ,  or just the tim e passing by change the world  How can we incorporate such actions ? 23

Typical Actions  The robot turns its w heels to move  The robot uses its m anipulator to grasp an object  Plants grow over tim e …  Actions are never carried out w ith absolute certainty  In contrast to measurements, actions generally increase the uncertainty 24

Modeling Actions  To incorporate the outcome of an action u into the current “ belief ” , we use the conditional pdf P ( x | u, x’ )  This term specifies the pdf that executing u changes the state from x’ to x . 25

Exam ple: Closing the door 26

State Transitions P ( x | u, x’ ) for u = “ close door ” : 0.9 0.1 open closed 1 0 If the door is open, the action “close door ” succeeds in 90% of all cases 27

I ntegrating the Outcom e of Actions Continuous case: Discrete case: We will make an independence assumption to get rid of the u in the second factor in the sum. 28

Exam ple: The Resulting Belief 29

Bayes Filters: Fram ew ork  Given:  Stream of observations z and action data u:  Sensor model P ( z | x )  Action model P ( x | u, x’ )  Prior probability of the system state P ( x )  W anted:  Estimate of the state X of a dynamical system  The posterior of the state is also called Belief : 30

Markov Assum ption Underlying Assum ptions  Static world  Independent noise  Perfect model, no approximation errors 31

z = observation u = action Bayes Filters x = state Bayes Markov Total prob. Markov Markov 32

∫ = η Bayes Filter Algorithm Bel ( x ) P ( z | x ) P ( x | u , x ) Bel ( x ) dx − − − t t t t t t 1 t 1 t 1 Algorithm Bayes_ filter ( Bel(x) , d ): 1. η = 0 2. 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then For all x do 10. 11. Return Bel ’ (x) 12. 33

Bayes Filters are Fam iliar! ∫ = η Bel ( x ) P ( z | x ) P ( x | u , x ) Bel ( x ) dx − − − t t t t t t 1 t 1 t 1  Kalman filters  Particle filters  Hidden Markov models  Dynamic Bayesian networks  Partially Observable Markov Decision Processes (POMDPs) 34

Probabilistic Localization

Probabilistic Localization

Sum m ary  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. 37