Embed Size (px)
DESCRIPTION
Probabilistic Methods in Mobile Robotics. Stereo cameras. Sonar. Tactiles. Infra-red. Laser range-finder. Sonar. Bayes Formula. 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). - PowerPoint PPT Presentation
Citation preview
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
