Probabilistic Robotics

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2 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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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.