Probabilistic Methods in Mobile Robotics

Probabilistic Methods inMobile Robotics

Stereo cameras

Infra-red

Sonar

Laser range-finder

Sonar

Tactiles

Bayes Formula

)(

)()|()|(

)()|()()|()(

Bp

ApABpBAp

ApABpBpBApBAp

A Simple Example: Estimating the state of a door

Suppose a robot obtaines measurement s What is p(Door=open|SensorMeasurement=s)? Short form: p(open|s)

Causal vs. Diagnostic Reasoning

We’re interested in p(open|s) (called diagnostic reasoning)

Often causal knowledge like p(s|open) is easier to obtain.

From causal to diagnostic:

Apply Bayes rule:

)()()|(

)|(sp

openpopenspsopenp

Normalization

)()()|(

)|(sp

openpopenspsopenp

)()()|(

)|(sp

openpopenspsopenp

)()|()()|(

)()()(

openpopenspopenpopensp

openspopenspsp

)()|()()|()()|(

)|(openpopenspopenpopensp

openpopenspsopenp

Example

p(s|open) = 0.6 p(s|open) = 0.3 p(open) = p(open) = 0.5

67.03

2

5.03.05.06.0

5.06.0)|(

)()|()()|(

)()|()|(

sopenp

openpopenspopenpopensp

openpopenspsopenp

s raises the probability, that the door is open.

Integrating a second Measurement ... New measurement s2

p(s2|open) = 0.5 p(s2|open) = 0.6

625.08

5

31

53

32

21

32

21

)|(

)|()|()|()|(

)|()|()|(

12

1212

1212

ssopenp

sopenpopenspsopenpopensp

sopenpopenspssopenp

s2 lowers the probability, that the door is open.

Where am I?

+

Mobile Robot Localization

Principle of Robot Localization

Lt: position of the robot at time t

Given:

Map and sensor model:

Motion model:

Initial state of the robot:

Data

Sensor information (sonar, laser range-finder, camera) oi

Odometry information ai

Markov Localization as State Estimation (1)

)|( lLoOP tt

)',|( 11 lLaAlLP ttt

)( 0LP

TTT aoaod ,...,, 121

Motion Model )',|( 11 lLaAlLP ttt

Model for Proximity Sensors

The sensor is reflected either by a known or by an unknown obstacle:

Laser sensor Sonar sensor

Motion:

Perception:

… is optimal under the Markov assumption

Kalman filters, Hidden Markov Models, DBN

Markov Localization as State Estimation (2)

)'()',|()( 1'

11 lLPlLaAlLPlLP tl

tttt

)()|()( 11 lLPlLoOPlLP tttt

Grid-based Markov LocalizationThree-dimensional grid over the sate space of the robot:

Localization Example (1)

Localization Example

Sample-based Density Representation

D. Fox, Univ. of Washington

Global Localization (sonar)

Example Run Sonar

Example Run Laser

Localization for AIBO robots

D. Fox, Univ. of Washington

Localization for AIBO robots

D. Fox, Univ. of Washington

Mobile Robot Mapping

Mapping the Allen Center: Raw Data

Mapping the Allen Center

Multi-robot Mapping

Robot A Robot B Robot C