Probability: Review. Pieter Abbeel UC Berkeley EECS. Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics

Transcription

1 robabilit: Review ieter Abbeel UC Berkele EECS Man slides adapted from Thrun Burgard and Fo robabilistic Robotics

2 Wh probabilit in robotics? Often state of robot and state of its environment are unknown and onl nois sensors available robabilit provides a framework to fuse sensor information à Result: probabilit distribution over possible states of robot and environment Dnamics is often stochastic hence can t optimie for a particular outcome but onl optimie to obtain a good distribution over outcomes robabilit provides a framework to reason in this setting à Result: abilit to find good control policies for stochastic dnamics and environments

3 Eample : Helicopter State: position orientation velocit angular rate Sensors: GS : nois estimate of position sometimes also velocit Inertial sensing unit: nois measurements from i 3-ais gro [angular rate sensor] ii 3-ais accelerometer [measures acceleration + gravit; e.g. measures 000 in free-fall] iii 3-ais magnetometer Dnamics: Noise from: wind unmodeled dnamics in engine servos blades

4 Eample 2: Mobile robot inside building State: position and heading Sensors: Odometr sensing motion of actuators: e.g. wheel encoders Laser range finder: Measures time of flight of a laser beam between departure and return Return is tpicall happening when hitting a surface that reflects the beam back to where it came from Dnamics: Noise from: wheel slippage unmodeled variation in floor

5 Aioms of robabilit Theor 0 r A r! r! 0 ra! B ra+ rb" ra# B ra denotes probabilit that the outcome ω is an element of the set of possible outcomes A. A is often called an event. Same for B. Ω is the set of all possible outcomes. ϕ is the empt set. 5

6 A Closer Look at Aiom 3 ra! B ra+ rb" ra# B! A A! B B 6

7 Using the Aioms ra!" \ A ra+ r" \ A# ra$" \ A r" ra+ r" \ A# r! ra+ r" \ A# 0 r" \ A # ra 7

8 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 probabilit that the random variable X takes on value i.. is called probabilit mass function. E.g. X models the outcome of a coin flip head 2 tail

9 Continuous Random Variables X takes on values in the continuum. px or p is a probabilit densit function. r a b p d b a E.g. p 9

10 Joint and Conditional robabilit X and Y If X and Y are independent then is the probabilit of given / If X and Y are independent then Same for probabilit densities just à p 0

11 Law of Total robabilit Marginals Discrete case Continuous case p d p p d p p p d

12 2 Baes Formula evidence prior likelihood

13 3 Normaliation η η au : au : au η η Algorithm:

14 4 Conditioning Law of total probabilit: d d d

15 Baes Rule with Background Knowledge 5

16 Conditional Independence equivalent to and 6

17 Simple Eample of State Estimation Suppose a robot obtains measurement What is open? 7

18 Causal vs. Diagnostic Reasoning open is diagnostic. open is causal. Often causal knowledge is easier to obtain. Baes rule allows us to use causal count knowledge: frequencies! open open open 8

19 Eample open 0.6 open 0.3 open open 0.5 open openopen open open open open p open + open p open open raises the probabilit that the door is open. 9

20 Combining Evidence Suppose our robot obtains another observation 2. How can we integrate this new information? More generall how can we estimate... n? 20

21 2 Recursive Baesian Updating n n n n n n Markov assumption: n is independent of... n- if we know. n n n! n n!! n n!!...n i i...n " # $ % & '

22 22 Eample: Second Measurement 2 open open 0.6 open 2/ open open open open open open open 2 lowers the probabilit that the door is open.

23 A Tpical itfall Two possible locations and p2 d p d p d Number of integrations 23