robabilit: Review ieter Abbeel UC Berkele EECS Man slides adapted from Thrun Burgard and Fo robabilistic Robotics
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
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
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
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
A Closer Look at Aiom 3 ra! B ra+ rb" ra# B! A A! B B 6
Using the Aioms ra!" \ A ra+ r" \ A# ra$" \ A r" ra+ r" \ A# r! ra+ r" \ A# 0 r" \ A # ra 7
Discrete Random Variables X denotes a random variable.! 2 4 3 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 0.5 2 0.5 8
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
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
Law of Total robabilit Marginals Discrete case Continuous case p d p p d p p p d
2 Baes Formula evidence prior likelihood
3 Normaliation η η au : au : au η η Algorithm:
4 Conditioning Law of total probabilit: d d d
Baes Rule with Background Knowledge 5
Conditional Independence equivalent to and 6
Simple Eample of State Estimation Suppose a robot obtains measurement What is open? 7
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
Eample open 0.6 open 0.3 open open 0.5 open openopen open open open open p open + open p open open 0.6 0.5 0.6 0.5 + 0.3 0.5 2 3 0.67 raises the probabilit that the door is open. 9
Combining Evidence Suppose our robot obtains another observation 2. How can we integrate this new information? More generall how can we estimate... n? 20
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 Eample: Second Measurement 2 open 0.5 2 open 0.6 open 2/3 0.625 8 5 3 5 3 3 2 2 3 2 2 2 2 2 2 + + open open open open open open open 2 lowers the probabilit that the door is open.
A Tpical itfall Two possible locations and 2 0.99 2 0.09 0.07 0.9 p2 d p d 0.8 0.7 0.6 p d 0.5 0.4 0.3 0.2 0. 0 5 0 5 20 25 30 35 40 45 50 Number of integrations 23