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1
Robot Environment Interaction
x
• Environment perception provides information about the environment’s state, and it tends to increase the robot’s knowledge.
• Motion (control date), on the other hand, tends to induce a loss of knowledge due to noise (uncertainty).
• The evolution of state and measurements is governed by probabilistic laws. (Probabilistic Robotics)
2
Robot Environment Interaction
x
• For state variable
• If the state variable is complete
• This is an example of Conditional independence (CI).
)|( :1,1:1,1:0 tttt uzxxp
)|()|( ,1:1,1:1,1:0 ttttttt uxxpuzxxp
3
Robot Environment Interaction
• For measurement data
• If the state variable is complete
• This is another example of Conditional independence (CI).
)|( :1,1:1,:0 tttt uzxzp
)|()|( :1,1:1,:0 tttttt xzpuzxzp
4
Robot Environment Interaction
x
)|()|( ,1:1,1:1,1:0 ttttttt uxxpuzxxp
State transition probability
)|()|( :1,1:1,:0 tttttt xzpuzxzp
measurement probability
5
Robot Environment Interaction
x
• The state transition probability and the measurement probability together describes the dynamic stochastic system of the robot and its environment.
• See Figure 2.2.
6
Robot Environment Interaction
• Besides measurement, control, etc, another key concept in probabilistic robotics is that of a belief.
• A belief reflects the robot’s internal knowledge about the state of the environment, because the state of the environment, to the robot, is unobservable.
• How belief is probabilistically represented in probabilistic robotics?
7
Robot Environment Interaction
• The belief of a robot is represented in the form of conditional probability distribution (CPD) as:
• Sometimes, the following CPD is also of interest.
)|()( :1,:1 tttt uzxpxbel
)|()( :1,1:1 tttt uzxpxbel
predication
8
Bayes Filter-The single most important algorithm in the book
• It calculates the belief distribution bel from measurement and control date.
• It is a recursive algorithm. It is the basis of all other algorithms in the book.
9
Simple Example of Bayes Filter Algorithm
• Suppose a robot obtains measurement z
• What is P(open|z)?
10
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!
11
Example
• P(z|open) = 0.6 P(z|open) = 0.3
• P(open) = P(open) = 0.5
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5.06.0)|(
)()|()()|(
)()|()|(
zopenP
openpopenzPopenpopenzP
openPopenzPzopenP
• z raises the probability that the door is open.
12
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 )?
13
Recursive Bayesian Updating
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),...,|(),...,,|(),...,|(
11
11111
nn
nnnn
zzzP
zzxPzzxzPzzxP
Markov assumption: zn is independent of z1,...,zn-1 if we know x.
)()|(
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...1...1
11
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in
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What’s going
on here?
14
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.
15
Example
•The previous examples seems only concern with measurement. What about control data (or motion, action)?
•How does control data play its role?
16
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?
17
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.
18
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.
19
Example: Closing the door
20
State Transition (probability distribution)
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
21
Integrating the Outcome of Actions
')'()',|()|( dxxPxuxPuxP
)'()',|()|( xPxuxPuxP
Continuous case:
Discrete case:
What’s going on here?
22
Example: The Resulting Belief
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)'()',|()|(
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uclosedP
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23
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 called Belief:
),...,,|()( 11 tttt zuzuxPxBel
},....,,{ 11 ttt zuzud
State transition probability
measurement probabilityNew terms
24
Bayes Filters: The Algorithm
• Algorithm Bayes_filter ( )
• for all do
•
• endfor
• return
111^ )(),|()( tttttt dxxbelxuxpxbel
ttt zuxbel ,),( 1
)( txbel
)()|( )( ^tttt xbelxzpxbel
tx Action model
Sensor model
25111 )(),|()|( ttttttt dxxBelxuxPxzP
Bayes Filters
)...,,|(),,|( :11:1:1:1 ttttttt uzxPuzxzP Bayes
z = observationu = actionx = state
),|()( :1:1 tttt zuxPxBel
Markov ),|()|( :11:1 ttttt uzxPxzP
Markov1:11:111 )...,,|(),|()|( ttttttttt dxuzxPxuxPxzP
1:11:11
:11:11
),|(
)...,,,|()|(
tttt
tttttt
dxuzxP
uzxxPxzPTotal prob.
Markov111111 )...,,,|(),|()|( tttttttt dxzzuxPxuxPxzP
What is it?
Action modelSensor modelrecursion
26
Bayes Filters: An Example
•Page 28-31.
27
Markov Assumption (the Complete State Assumption)
Underlying Assumptions• Static world• Independent noise• Perfect model, no approximation errors
In practice, Bayes filters have been found to be surprisely robust to violations of Markov assumption.
),|(),,|( 1:1:11:1 ttttttt uxxpuzxxp )|(),,|( :1:1:0 tttttt xzpuzxzp
28
Representations and Computation• There exist quite a variety of techniques and
algorithms that are all derived from the Bayes filter.
• Each such technique relies on different assumptions regarding the measurement and state transition probabilities and the initial belief.
• These assumptions then give rise to different types of posterior distributions, and
• The algorithms for computing them have different computational characteristics.
• Exact techniques only exist for highly specialized cases,
• In general, many require approximate.
29
Bayes Filters are Familiar!
• Kalman filters
• Particle filters
• Hidden Markov models
• Dynamic Bayesian networks
• Partially Observable Markov Decision Processes (POMDPs)
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
30
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.
31
Mobile Robot Localization
• Mobile robot localization is the problem of determining the pose of a robot relative to a given map of the environment. Because,
• Unfortunately, the pose of a robot can not be sensed directly, at least for now. The pose has to be inferred from data.
• A single sensor measurement is enough?
• The importance of localization in robotics.
• Mobile robot localization can be seen as a problem of coordinate transformation. One point of view.
32
Mobile Robot Localization
•Localization techniques have been developed for a broad set of map representations. • Feature based maps, location based
maps, occupancy grid maps, etc. (what exactly are they?) (See figure 7.2)
• (You can probably guess What is the mapping problem?)
•Remember, in localization problem, the map is given, known, available.
• Is it hard? Not really, because,
33
Mobile Robot Localization
•Most localization algorithms are variants of Bayes filter algorithm.
•However, different representation of maps, sensor models, motion model, etc lead to different variant.
•Here is the agenda.
34
Mobile Robot Localization
•We want to know different kinds of maps.
•We want to know different kinds of localization problems.
•We want to know different localization problems.
35
Mobile Robot Localization
Different kinds of maps.
36
Mobile Robot Localization – A Taxonomy
•Different kinds of Localization problems.
•Through 4 dimensions
• Local versus Global (initial knowledge)
• Static versus Dynamic (environment)
• Passive versus active (control of robots)
• Single robot or multi-robot
