Probabilistic Robotics

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(open|z)?

12

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:

count frequencies!

13

Example

•
P(z|open) = 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(open|z
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(x|u,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(x|u,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(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
:

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.