2 Probabilistic Robot

8/2/2019 2 Probabilistic Robot

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City College of New York 1

Dr. Jizhong Xiao

Department of Electrical Engineering

City College of New York [email protected]

Probabilistic Robotics

Advanced Mobile Robotics




Robot Navigation

Fundamental problems to provide a mobile

robot with autonomous capabilities:

•Where am I going

•What’s the best way there?

•Where have I been? how to create an

environmental map with imperfect sensors?

• Where am I? how a robot can tell where

it is on a map?

• What if you’re lost and don’t have a map?

Mapping

Localization

Robot SLAM

Path Planning

Mission Planning




Representation of the Environment• Environment Representation

– Continuos Metric x,y,q

– Discrete Metricmetric grid

– Discrete Topological

topological grid




Localization, Where am I?

?

• Odometry, Dead Reckoning

• Localization base on external sensors,

beacons or landmarks

• Probabilistic Map Based Localization

Observation

Mapdata base

Prediction of Position(e.g. odometry)

P e r c e p

t i o n

Matching

Position Update(Estimation?)

raw sensor data or extracted features

predicted position

position

matched observations

YES

Encoder

P e r c e p t i o n




Localization Methods• Mathematic Background, Bayes Filters• Markov Localization:

– Central idea: represent the robot’s belief by a probability distributionover possible positions, and uses Bayes’ rule and convolution to updatethe belief whenever the robot senses or moves

– Markov Assumption: past and future data are independent if one knows thecurrent state

• Kalman Filtering – Central idea: posing localization problem as a sensor fusion problem

– Assumption: gaussian distribution function

• Particle Filtering – Central idea: Sample-based, nonparametric Filter – Monte-Carlo method

• SLAM (simultaneous localization and mapping)

• Multi-robot localization




Markov Localization• Applying probability theory to robot localization

• Markov localization uses anexplicit, discrete representation for the probability ofall position in the state space.

• This is usually done by representing the environment by agrid or a topological graph with a finite number of possible

states (positions).

• During each update, the probability for each state (element) of the entire space is updated.


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Markov Localization Example• Assume the robot position is one-

dimensionalThe robot is placed

somewhere in the

environment but it isnot told its location

The robot queries its

sensors and finds out it

is next to a door



Markov Localization Example

The robot moves one

meter forward. To

account for inherent

noise in robot motion the

new belief is smoother

The robot queries its

sensors and again it

finds itself next to adoor




• Falls in between model-based and behavior-based techniques – There are models, and sensor measurements, but they are

assumed to be incomplete and insufficient for control

– Statistics provides the mathematical glue to integratemodels and sensor measurements

• Basic Mathematics

– Probabilities – Bayes rule

– Bayes filters



• The next slides are provided by theauthors of the book "Probabilistic

Robotics“, you can download from thewebsite:

http://www.probabilistic-robotics.org/




Mathematic Background

ProbabilitiesBayes rule

Bayes filters




Key idea:Explicit representation of uncertainty usingthe calculus of probability theory

– Perception = state estimation

– Action = utility optimization



Pr(A) denotes probability that proposition A is true.

•

•

•

Axioms of Probability Theory

1)Pr(0 A

1)Pr( True

)Pr()Pr()Pr()Pr( B A B A B A

0)Pr( False



A Closer Look at Axiom 3

B

B A A BTrue

)Pr()Pr()Pr()Pr( B A B A B A



Using the Axioms

)Pr(1)Pr(

0)Pr()Pr(1

)Pr()Pr()Pr()Pr(

)Pr()Pr()Pr()Pr(

A A

A A

False A ATrue

A A A A A A



Discrete Random Variables

• X denotes a random variable.

• X can take on a countable number of values in {x1,x2, …, xn}.

• P(X=x i ), or P(x i ), is the probability that the randomvariable X takes on value x i.

• P( ) is called probability mass function.

• E.g. 02.0,08.0,2.0,7.0)( RoomP

.




Continuous Random Variables

• X takes on values in the continuum.

• p(X=x), or p(x), is a probability density function.

• E.g.

b

a

dx x pba x )()),(Pr(

x

p(x)




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)




Law of Total Probability, Marginals

y

y xP xP ),()(

y

yP y xP xP )()|()(

x

xP 1)(

Discrete case

1)( dx x p

Continuous case

dy y p y x p x p )()|()(

dy y x p x p ),()(




Bayes Formula

)( y x p

evidence

priorlikelihood

)(

)()|()(

yP

xP x yP y xP

)()|()()|(),( xP x yP yP y xP y xP

Posterior probability distribution

)( x y p

)( x p Prior probability distribution

If y is a new sensor reading

Generative model, characteristics of the sensor

)( y p

Does not depend on x




Normalization

)()|(

1)(

)()|()(

)()|(

)(

1

xP x yP yP

xP x yP yP

xP x yP

y xP

x

y x

x

y x

y x

y xP x

xP x yP x

|

|

|

aux)|(:

aux

1

)()|(aux:

Algorithm:




Bayes Rulewith Background Knowledge

)|(

)|(),|(),|(

z yP

z xP z x yP z y xP




Conditioning

• Total probability:

dz zP z y xP y xP

dz zP z xP xP

dz z xP xP

)(),|()(

)()|()(

),()(




Conditional Independence

)|()|(),( z yP z xP z y xP

),|()( y z xP z xP

),|()( x z yP z yP

equivalent to

and




Simple Example of State Estimation

• Suppose a robot obtains measurement z • What is P(open|z)?




Causal vs. Diagnostic Reasoning

• P(open|z) is diagnostic.

• P(z|open) is causal.

• Often causal knowledge is easier toobtain.

• Bayes rule allows us to use causalknowledge:

)(

)()|()|(

zP

openPopen zP zopenP

count frequencies!




Example

• P(z|open) = 0.6 P(z|open) = 0.3

• P(open) = P(open) = 0.5

67.03

2

5.03.05.06.0

5.06.0)|(

)()|()()|(

)()|(

)|(

zopenP

open popen zPopen popen zP

openPopen zP

zopenP

• z raises the probability that the door is open.




Combining Evidence

• Suppose our robot obtains anotherobservation z2.

• How can we integrate this newinformation?

• More generally, how can we estimate P(x| z1...zn )?




Recursive Bayesian Updating

),,|(),,|(),,,|(),,|(

11

11111

nn

nnnn

z z zP z z xP z z x zP z z xP

Markov assumption: zn is independent of z 1 ,...,z n-1 if we know x.

)()|(

),,|()|(

),,|(

),,|()|(),,|(

...1

...1

11

11

111

xP x zP

z z xP x zP

z z zP

z z xP x zP z z xP

ni

in

nn

nn

nnn




Example: Second Measurement

• P(z2|open) = 0.5 P(z2|open) = 0.6

• P(open|z1)=2/3

625.08

5

3

1

5

3

3

2

2

13

2

2

1

)|()|()|()|(

)|()|(

),|(1212

12

12

zopenPopen zP zopenPopen zP

zopenPopen zP

z zopenP

• z 2 lowers the probability that the door is open.




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 anobject

• Plants 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 actionu into the current “belief”, we use theconditional pdf

P(x|u,x’)

• This term specifies the pdf thatexecuting u changes the state from x’ to x .




Integrating the Outcome of Actions

')'()',|()|( dx xP xu xPu xP

)'()',|()|( xP xu xPu xP

Continuous case:

Discrete case:




Example: Closing the door




State Transitions

P(x|u,x’) for u = “close door”:

If the door is open, the action “close door”succeeds in 90% of all cases.

open closed0.1 1

0.9

0

P(open)=5/8 P(closed)=3/8




Example: The Resulting Belief

)|(1

16

1

8

3

1

0

8

5

10

1

)(),|(

)(),|(

)'()',|()|(

16

15

8

3

1

1

8

5

10

9

)(),|(

)(),|(

)'()',|()|(

uclosed P

closed Pclosed uopenP

openPopenuopenP

xP xuopenPuopenP

closed Pclosed uclosed P

openPopenuclosed P

xP xuclosed Puclosed P




Bayes Filters: Framework

• 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).

• Wanted:

– Estimate of the state X of a dynamical system.

– The posterior of the state is also called Belief:

),,,|()( 11 t t t t zu zu xP x Bel

},,,{ 11 t t t zu zud

M k A ti




Markov Assumption

Underlying Assumptions

• Static world, Independent noise

• Perfect model, no approximation errors

),|(),,|( 1:1:11:1 t t t t t t t u x x pu z x x p

)|(),,|( :1:1:0 t t t t t t x z pu z x z p

State transition probability

Measurement probability

Markov Assumption:

– past and future data are independent if one knows the current state

z = observation




)(),|()|(

t t t t t t t

dx x Bel xu xP x zP

Bayes Filters

),,,|(),,,,|( 1111 t t t t t u zu xPu zu x zP Bayes

u = action x = state

),,,|()( 11 t t t t zu zu xP x Bel

Markov ),,,|()|( 11 t t t t u zu xP x zP

Markov11111 ),,,|(),|()|(

t t t t t t t t dxu zu xP xu xP x zP

1111

111

),,,|(

),,,,|()|(

t t t

t t t t t

dxu zu xP

xu zu xP x zP

Total prob.

Markov111111 ),,,|(),|()|( t t t t t t t t dx z zu xP xu xP x zP




Bayes Filter Algorithm

1. Algorithm Bayes_filter( Bel(x),d ):

2. 0

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

10. For all x do11.

12. Return Bel’(x)

)()|()(' x Bel x zP x Bel

)(' x Bel

)(')(' 1 x Bel x Bel

')'()',|()(' dx x Bel xu xP x Bel

111 )(),|()|()( t t t t t t t t dx x Bel xu xP x zP x Bel




Bayes Filters are Familiar!

• Kalman filters

• Particle filters

• Hidden Markov models

• Dynamic Bayesian networks

• Partially Observable Markov Decision Processes(POMDPs)

111 )(),|()|()( t t t t t t t t dx x Bel xu xP x zP x Bel




Summary

• Bayes rule allows us to computeprobabilities that are hard to assessotherwise.

• Under the Markov assumption, recursiveBayesian updating can be used toefficiently combine evidence.

• Bayes filters are a probabilistic tool forestimating the state of dynamic systems.



Thank You

Homework 3: Exercises 1, 2 and 3 on pp36~37 of textbook,

Due date March 5

Next class: Feb 21, 2012

Homework 2: Due date: Feb 27

2 Probabilistic Robot

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