Anno accademico 2006/2007. Davide Migliore
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1 Roboica Anno accademico 2006/2007 Davide Migliore
2 Today Eercise session: An Off-side roblem Robo Vision Task Measuring NBA layers erformance robabilisic Roboics Inroducion The Bayesian Filer 27/0/2006 Corso di Roboica 06/07 2/5
3 Eercise Session 27/0/2006 Corso di Roboica 06/07 3/5
4 . An Off-side roblem Find vanishing poin of he field-boom direcion b b a a 27/0/2006 Corso di Roboica 06/07 4/5
5 N.B. Wha is Cross-Raio? D case: T 2 ' H 2 2 Cross i j i de i 2 j j 2 27/0/2006 Corso di Roboica 06/07 5/5
6 . An Off-side roblem a c b d a and b: ca+b/2: d : abscissae of he endpoins of a segmen abscissa of segmen midpoin poin a he infinie along he segmen direcion CR a c b c a d b d a c b c a b c d Harmonic 4-uple abcd a b and a b are image of symmeric segmens same image of he midpoin c same vanishing poin d 27/0/2006 Corso di Roboica 06/07 6/5
7 . An Off-side roblem Solve { for CR CR a' c' a' b' c' d' b' c' a'' c' a' d' b' d' a' ' b'' c' d' b'' c' a'' d' b'' d' c d Sysem of wo linear equaions in c d and c +d Two degree equaion whose soluions are c and d among he wo soluions he one for d is he value eernal o he range [a b ] 27/0/2006 Corso di Roboica 06/07 7/5
![2. Robo Vision Task Daa: The line a and b are parallel a' [-0] and a'' [-2-] b' [0] and b'' [.5-0.](/docs-images/92/109917335/images/8-1.jpg "5] [ 6 ] The camera mari is: [ 0 0 2 0 2/2 2/2 0 2/2 2/2 ] a'' a' b' b'' Find angle o follow he poin!")
8 2. Robo Vision Task Daa: The line a and b are parallel a' [-0] and a'' [-2-] b' [0] and b'' [.5-0.5] [ 6 ] The camera mari is: [ /2 2/2 0 2/2 2/2 ] a'' a' b' b'' Find angle o follow he poin! 27/0/2006 Corso di Roboica 06/07 8/5
9 2. Robo Vision Task The vanishing poin V [0] In homogeneus represenaion [0] V Vanishing poin Now remember and we wan find X X Ignoring erinsic parameers a'' a' b' b'' M [ 0 2/2 M [ 0 2/2 2/2] ] 0 2/2 2/2 0 2/2 2/2 27/0/2006 Corso di Roboica 06/07 9/5
10 2. Robo Vision Task The direcion of he robo is: 0 0 dm V [ ][ 0 ] [ 0 0 2/2 2/ /2 2/2 2] [ 0 0 ] The poin in h-coordinae is T [ 6 ] dm T [ /2 2/2 The angle is 60 ][ 6 ] [ /2 2/2 2] [ 3/2 0 /2 ] 27/0/2006 Corso di Roboica 06/07 0/5
11 2. Measuring NBA layers Esimae he jump heigh of Nae Robinson Hp weak perspecive effec 27/0/2006 Corso di Roboica 06/07 /5
12 2. Measuring NBA layers 27/0/2006 Corso di Roboica 06/07 2/5
13 robabilisic Roboics 27/0/2006 Corso di Roboica 06/07 3/5
14 robabilisic Roboics Key idea: Eplici represenaion of uncerainy using he calculus of probabiliy heory ercepion sae esimaion Acion uiliy opimiaion Quaniies such a sensor measuremens conrolsand saes of a robo and is environmen are all modeled as random variables 27/0/2006 Corso di Roboica 06/07 4/5
15 Aioms of robabiliy Theory ra denoes probabiliy ha proposiion A is rue. 0 r A r True r False 0 r A B r A + r B r A B 27/0/2006 Corso di Roboica 06/07 5/5
16 A Closer Look a Aiom 3 r A B r A + r B r A B True A A B B B 27/0/2006 Corso di Roboica 06/07 6/5
17 Using he Aioms r A A r True r A r A + r A r A A r A + r A r False r A + r A 0 r A 27/0/2006 Corso di Roboica 06/07 7/5
18 Discree Random Variables X denoes a random variable. X can ake on a counable number of values in { 2 n }. Xi or i is he probabiliy ha he random variable X akes on value i. is called probabiliy mass funcion.. E.g. Room /0/2006 Corso di Roboica 06/07 8/5
19 Coninuous Random Variables X akes on values in he coninuum. px or p is a probabiliy densiy funcion DF. r a b p d a p E.g. b 27/0/2006 Corso di Roboica 06/07 9/5
20 Coninuous Random Variables A common densiy funcion in ha of he onedimensional normal disribuion wih mean µ and variance σ 2 The DF of a normal disribuion is given by he Gaussian funcion: p 2 2 /2 ep{ 2 2 } 2 Abbreviae as N ; µσ 2 27/0/2006 Corso di Roboica 06/07 20/5
21 Coninuous Random Variables Ofen will be a muli-dimensional vecor so he normal disribuions are called mulivariae wih DF: p de 2 /2 ep{ 2 T } Where µ is he mean vecor and Σ a posiive semidefinie and symmeric mari called covariance mari 27/0/2006 Corso di Roboica 06/07 2/5
22 Join and Condiional robabiliy X and Yy y If X and Y are independen hen y y y is he probabiliy of given y y X Yy y y / y if y >0 y y y If X and Y are independen hen y 27/0/2006 Corso di Roboica 06/07 22/5
23 Law of Toal robabiliy Marginals Discree case y y y y y Coninuous case p d p p y dy p p y p y dy 27/0/2006 Corso di Roboica 06/07 23/5
24 Bayes Formula y y y y y y y likelihood prior evidence If is a quaniy ha we would like o infer from y he p will be referred o as prior probabiliy disribuion and y is called he daa The probabiliy py is called he poserior probabiliy disribuion over X 27/0/2006 Corso di Roboica 06/07 24/5
25 27/0/ /0/2006 Corso di Roboica 06/07 Corso di Roboica 06/ /5 /5 Normaliaion y y y y y y η η y y y y y au : au au : η η Algorihm:
26 Condiioning Law of oal probabiliy: d d y y y d 27/0/2006 Corso di Roboica 06/07 26/5
27 Bayes Rule wih Background Knowledge y y y 27/0/2006 Corso di Roboica 06/07 27/5
28 Condiioning Toal probabiliy: d d y y d 27/0/2006 Corso di Roboica 06/07 28/5
29 Condiional Independence y y equivalen o and y y y 27/0/2006 Corso di Roboica 06/07 29/5
30 Condiional Independence Condiional independence does no imply absolue independence p y p p p y p p y The converse is general unrue: p y p p y p y p p 27/0/2006 Corso di Roboica 06/07 30/5
31 The Epecaion of a RV The epecaion of a random variable X is given by E[ X ] p E [ X ] p d The epecaion is a linear funcion of a random variable E [ax b]ae [ X ] b The covariance of X is obained as Cov[ X ]E [ X E [ X ]] 2 E [ X 2 ] E [ X ] 2 27/0/2006 Corso di Roboica 06/07 3/5
32 Simple Eample of Sae Esimaion Suppose a robo obains measuremen Wha is open? 27/0/2006 Corso di Roboica 06/07 32/5
33 Causal vs. Diagnosic Reasoning open is diagnosic. open is causal. Ofen causal knowledge is easier o obain. Bayes rule allows us o use causal knowledge: coun frequencies! open open open 27/0/2006 Corso di Roboica 06/07 33/5
34 Eample open 0.6 open 0.3 open open 0.5 open open open open p open + open p open open raises he probabiliy ha he door is open. 27/0/2006 Corso di Roboica 06/07 34/5
35 Combining Evidence Suppose our robo obains anoher observaion 2. How can we inegrae his new informaion? More generally how can we esimae... n? 27/0/2006 Corso di Roboica 06/07 35/5
36 27/0/ /0/2006 Corso di Roboica 06/07 Corso di Roboica 06/ /5 /5 Recursive Bayesian Updaing n n n n n n Markov assumpion: n is independen of... n- if we know n i i n n n n n n n n η η
37 Eample: Second Measuremen 2 open open 0.6 open 2/3 open 2 2 open 2 open open open + open 2 open lowers he probabiliy ha he door is open. 27/0/2006 Corso di Roboica 06/07 37/5
38 A Typical ifall Two possible locaions and p2 d p d p d Number of inegraions 27/0/2006 Corso di Roboica 06/07 38/5
39 Acions Ofen he world is dynamic since acions carried ou by he robo acions carried ou by oher agens or jus he ime passing by change he world. How can we incorporae such acions? 27/0/2006 Corso di Roboica 06/07 39/5
40 Typical Acions The robo urns is wheels o move The robo uses is manipulaor o grasp an objec lans grow over ime Acions are never carried ou wih absolue cerainy. In conras o measuremens acions generally increase he uncerainy. 27/0/2006 Corso di Roboica 06/07 40/5
41 Modeling Acions To incorporae he oucome of an acion u ino he curren belief we use he condiional pdf u This erm specifies he pdf ha eecuing u changes he sae from o. 27/0/2006 Corso di Roboica 06/07 4/5
42 Eample: Closing he door 27/0/2006 Corso di Roboica 06/07 42/5
43 Sae Transiions u for u close door : open closed 0 If he door is open he acion close door succeeds in 90% of all cases. 27/0/2006 Corso di Roboica 06/07 43/5
44 Inegraing he Oucome of Coninuous case: Acions u u ' ' d' Discree case: u u ' ' 27/0/2006 Corso di Roboica 06/07 44/5
45 Eample: The Resuling Belief closed u closed u ' ' closed u open open + closed u closed closed open u open u ' ' open u open open + open u closed closed closed 3 8 u 6 27/0/2006 Corso di Roboica 06/07 45/5
46 Given: Bayes Filers: Framework Sream of observaions and acion daa u: Sensor model. Acion model u. rior probabiliy of he sysem sae. Waned: d { u u Esimae of he sae X of a dynamical sysem. The poserior of he sae is also called Belief: } Bel u u 27/0/2006 Corso di Roboica 06/07 46/5
47 Markov Assumpion p u p 0: : : p u p u : : : Underlying Assumpions Saic world Independen noise erfec model no approimaion errors 27/0/2006 Corso di Roboica 06/07 47/5
48 27/0/ /0/2006 Corso di Roboica 06/07 Corso di Roboica 06/ /5 /5 d Bel u η Bayes Filers Bayes Filers u u u u η Bayes observaion u acion sae u u Bel Markov u u η Markov d u u u η d u u u u η Toal prob. Markov d u u η Measuremen updae
49 Bel Bayes Filer Algorihm η u Bel d. Algorihm Bayes_filer Beld : 2. η0 3. If d is a percepual daa iem hen 4. For all do For all do Else if d is an acion daa iem u hen 0. For all do. 2. Reurn Bel Bel ' Bel η η + Bel' Bel' η Bel' Bel' u ' Bel ' d' 27/0/2006 Corso di Roboica 06/07 49/5
50 Bayes Filers are Familiar! Bel η u Bel d Kalman filers aricle filers Hidden Markov models Dynamic Bayesian neworks arially Observable Markov Decision rocesses OMDs 27/0/2006 Corso di Roboica 06/07 50/5
51 Summary Bayes rule allows us o compue probabiliies ha are hard o assess oherwise. Under he Markov assumpion recursive Bayesian updaing can be used o efficienly combine evidence. Bayes filers are a probabilisic ool for esimaing he sae of dynamic sysems. 27/0/2006 Corso di Roboica 06/07 5/5
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