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Probabilistic Robotics Introduction Probabilities Bayes rule Bayes filters
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Probabilistic Robotics
Introduction
ProbabilitiesBayes rule
Bayes filters
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Probabilistic Robotics
Key idea: Explicit representation of uncertainty using the calculus of probability theory
• Perception = state estimation• Action = utility
optimization
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Pr(A) denotes probability that proposition A is true.
•
•
•
Axioms of Probability Theory
1)Pr(0 A
1)Pr( True
)Pr()Pr()Pr()Pr( BABABA
0)Pr( False
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A Closer Look at Axiom 3
B
BA A BTrue
)Pr()Pr()Pr()Pr( BABABA
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Using the Axioms
)Pr(1)Pr(
0)Pr()Pr(1
)Pr()Pr()Pr()Pr(
)Pr()Pr()Pr()Pr(
AA
AA
FalseAATrue
AAAAAA
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Discrete Random Variables
•X denotes a random variable.
•X can take on a countable number of values in {x1, x2, …, xn}.
•P(X=xi), or P(xi), is the probability that the random variable X takes on value xi.
•P( ) is called probability mass function.
.
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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
dxxpbax )()),(Pr(
x
p(x)
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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 yP(x | y) = P(x,y) / P(y)P(x,y) = P(x | y) P(y)
• If X and Y are independent thenP(x | y) = P(x)
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Law of Total Probability, Marginals
y
yxPxP ),()(
y
yPyxPxP )()|()(
x
xP 1)(
Discrete case
1)( dxxp
Continuous case
dyypyxpxp )()|()(
dyyxpxp ),()(
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Bayes Formula
evidence
prior likelihood
)(
)()|()(
)()|()()|(),(
yP
xPxyPyxP
xPxyPyPyxPyxP
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Normalization
)()|(
1)(
)()|()(
)()|()(
1
xPxyPyP
xPxyPyP
xPxyPyxP
x
yx
xyx
yx
yxPx
xPxyPx
|
|
|
aux)|(:
aux
1
)()|(aux:
Algorithm:
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Conditioning
• Law of total probability:
dzyzPzyxPyxP
dzzPzxPxP
dzzxPxP
)|(),|()(
)()|()(
),()(
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Bayes Rule with Background Knowledge
)|(
)|(),|(),|(
zyP
zxPzxyPzyxP
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Conditioning
• Total probability:
dzzPzyxPyxP
dzzPzxPxP
dzzxPxP
)(),|()(
)()|()(
),()(
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Conditional Independence
)|()|(),( zyPzxPzyxP
),|()( yzxPzxP
),|()( xzyPzyP
equivalent to
and
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Simple Example of State Estimation
• Suppose a robot obtains measurement z
• What is P(open|z)?
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Causal vs. Diagnostic Reasoning
•P(open|z) is diagnostic.
•P(z|open) is causal.
•Often causal knowledge is easier to obtain.
•Bayes rule allows us to use causal knowledge:
)()()|(
)|(zP
openPopenzPzopenP
count frequencies!
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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
openpopenzPopenpopenzP
openPopenzPzopenP
• z raises the probability that the door is open.
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Combining Evidence
•Suppose our robot obtains another observation z2.
•How can we integrate this new information?
•More generally, how can we estimateP(x| z1...zn )?
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Recursive Bayesian Updating
),,|(
),,|(),,,|(),,|(
11
11111
nn
nnnn
zzzP
zzxPzzxzPzzxP
Markov assumption: zn is independent of z1,...,zn-1 if we know x.
)()|(
),,|()|(
),,|(
),,|()|(),,|(
...1...1
11
11
111
xPxzP
zzxPxzP
zzzP
zzxPxzPzzxP
ni
in
nn
nn
nnn
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Example: Second Measurement
• P(z2|open) = 0.5 P(z2|open) = 0.6
• P(open|z1)=2/3
625.08
5
31
53
32
21
32
21
)|()|()|()|(
)|()|(),|(
1212
1212
zopenPopenzPzopenPopenzP
zopenPopenzPzzopenP
• z2 lowers the probability that the door is open.
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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?
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Typical Actions
• The robot turns its wheels to move
• The robot uses its manipulator to grasp an object
• Plants grow over time…
• Actions are never carried out with absolute certainty.
• In contrast to measurements, actions generally increase the uncertainty.
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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.
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Example: Closing the door
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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
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Integrating the Outcome of Actions
')'()',|()|( dxxPxuxPuxP
)'()',|()|( xPxuxPuxP
Continuous case:
Discrete case:
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Example: The Resulting Belief
)|(1161
83
10
85
101
)(),|(
)(),|(
)'()',|()|(
1615
83
11
85
109
)(),|(
)(),|(
)'()',|()|(
uclosedP
closedPcloseduopenP
openPopenuopenP
xPxuopenPuopenP
closedPcloseduclosedP
openPopenuclosedP
xPxuclosedPuclosedP
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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 tttt zuzuxPxBel
},,,{ 11 ttt zuzud
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Markov Assumption
Underlying Assumptions• Static world• Independent noise• Perfect model, no approximation errors
),|(),,|( 1:1:11:1 ttttttt uxxpuzxxp )|(),,|( :1:1:0 tttttt xzpuzxzp
32111 )(),|()|( ttttttt dxxBelxuxPxzP
Bayes Filters
),,,|(),,,,|( 1111 ttttt uzuxPuzuxzP Bayes
z = observationu = actionx = state
),,,|()( 11 tttt zuzuxPxBel
Markov ),,,|()|( 11 tttt uzuxPxzP
Markov11111 ),,,|(),|()|( tttttttt dxuzuxPxuxPxzP
1111
111
),,,|(
),,,,|()|(
ttt
ttttt
dxuzuxP
xuzuxPxzP
Total prob.
Markov111111 ),,,|(),|()|( tttttttt dxzzuxPxuxPxzP
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Bayes Filter Algorithm
1. Algorithm Bayes_filter( Bel(x),d ):2. 0
3. If d is a perceptual data item z then4. For all x do5. 6. 7. For all x do8.
9. Else if d is an action data item u then10. For all x do11.
12. Return Bel’(x)
)()|()(' xBelxzPxBel )(' xBel
)(')(' 1 xBelxBel
')'()',|()(' dxxBelxuxPxBel
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
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Bayes Filters are Familiar!
• Kalman filters
• Particle filters
• Hidden Markov models
• Dynamic Bayesian networks
• Partially Observable Markov Decision Processes (POMDPs)
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
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Example: State Representations for Robot Localization
Grid Based approaches
(Markov localization)
Particle Filters (Monte Carlolocalization)
Kalman Tracking
Discrete Representations
Continuous Representations
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Summary
•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.
