Probabilistic Robotics
Introduction
2
Robot Environment Interaction
•
A robot can, at least, hypothetically, keep a
record of all past sensor measurements and
control actions. Such a collection is referred to as
data.
•
Two different streams of data
•
Environment measurement data
•
Control data
•
corresponds to the change of state in the time interval
(t

1;t]
3
Robot Environment Interaction
•
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)
4
Robot Environment Interaction
•
For state variable
•
If the state variable is complete
•
This is an example of Conditional independence
(CI).
5
Robot Environment Interaction
•
For measurement data
•
If the state variable is complete
•
This is another example of Conditional
independence (CI).
6
Robot Environment Interaction
State transition probability
measurement probability
7
Robot Environment Interaction
•
The state transition probability and the
measurement probability together describes the
dynamic stochastic system of the robot and its
environment.
•
See Figure 2.2.
8
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?
9
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.
predication
10
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.
11
Simple Example of Bayes Filter
Algorithm
•
Suppose a robot obtains measurement
z
•
What is
P(openz)?
12
Causal vs. Diagnostic Reasoning
•
P(openz)
is
diagnostic
.
•
P(zopen)
is
causal
.
•
Often
causal
knowledge is easier to
obtain.
•
Bayes rule allows us to use causal
knowledge:
count frequencies!
13
Example
•
P(zopen) = 0.6
P(z
open) = 0.3
•
P(open) = P(
open) = 0.5
•
z
raise
s
the probability that the door is open
.
14
Combining Evidence
•
Suppose our robot obtains another
observation
z
2
.
•
How can we integrate this new
information?
•
More generally, how can we estimate
P(x z
1
...z
n
)
?
15
Recursive Bayesian Updating
Markov assumption
:
z
n
is
independent
of
z
1
,...,z
n

1
if
we know
x.
16
Example: Second Measurement
•
P(z
2
open) = 0.5
P(z
2

open) = 0.6
•
P(openz
1
)=2/3
•
z
2
lowers the probability that the door is open
.
17
Example
•
The previous examples seems only concern
with measurement. What about control data
(or motion, action)?
•
How does control data play its role?
18
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
?
19
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
.
20
Modeling Actions
•
To incorporate the outcome of an
action
u
into the current “belief”, we
use the conditional pdf
P(xu,x’)
•
This term specifies the pdf that
executing
u
changes the state
from
x’ to x
.
21
Example: Closing the door
22
State Transition (probability
distribution)
P(xu,x’)
for
u
= “close door”:
If the door is open, the action “close
door” succeeds in 90% of all cases.
23
Integrating the Outcome of Actions
Continuous case:
Discrete case:
What’s going
on here?
24
Example: The Resulting Belief
25
Bayes Filters: Framework
•
Given:
•
Stream of observations
z
and action data
u:
•
Sensor model
P(zx).
•
Action model
P(xu,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
:
State transition probability
measurement probability
New terms
26
Bayes Filters: The Algorithm
•
Algorithm Bayes_filter ( )
•
for all do
•
•
endfor
•
return
Action model
Sensor model
27
Bayes Filters
Bayes
z
= observation
u
= action
x
= state
Markov
Markov
Total prob.
Markov
What is it?
Action model
Sensor model
recursion
28
Bayes Filters: An Example
•
Page 28

31.
29
Markov Assumption
(the Complete State Assumption)
Underlying Assumptions
•
Static world
•
Independent noise
•
Perfect model, no approximation errors
30
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
do
11.
12.
13.
Return
Bel’(x)
31
Bayes Filters are Familiar!
•
Kalman filters
•
Particle filters
•
Hidden Markov models
•
Dynamic Bayesian networks
•
Partially Observable Markov Decision
Processes (POMDPs)
32
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.
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