Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox,...
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Transcript of Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox,...
![Page 1: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/1.jpg)
Beam Sensor Models
Pieter AbbeelUC Berkeley EECS
Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
![Page 2: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/2.jpg)
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Proximity Sensors
The central task is to determine P(z|x), i.e., the probability of a measurement z given that the robot is at position x.
Question: Where do the probabilities come from?
Approach: Let’s try to explain a measurement.
![Page 3: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/3.jpg)
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Beam-based Sensor Model Scan z consists of K measurements.
Individual measurements are independent given the robot position.
},...,,{ 21 Kzzzz
K
kk mxzPmxzP
1
),|(),|(
![Page 4: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/4.jpg)
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Beam-based Sensor Model
K
kk mxzPmxzP
1
),|(),|(
![Page 5: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/5.jpg)
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Typical Measurement Errors of an Range Measurements
1. Beams reflected by obstacles
2. Beams reflected by persons / caused by crosstalk
3. Random measurements
4. Maximum range measurements
![Page 6: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/6.jpg)
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Proximity Measurement
Measurement can be caused by … a known obstacle. cross-talk. an unexpected obstacle (people, furniture, …). missing all obstacles (total reflection, glass,
…). Noise is due to uncertainty …
in measuring distance to known obstacle. in position of known obstacles. in position of additional obstacles. whether obstacle is missed.
![Page 7: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/7.jpg)
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Beam-based Proximity Model
Measurement noise
zexp zmax0
b
zz
hit eb
mxzP
2exp )(
2
1
2
1),|(
otherwise
zzmxzP
z
0
e),|( exp
unexp
Unexpected obstacles
zexp zmax0
![Page 8: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/8.jpg)
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Beam-based Proximity Model
Random measurement Max range
max
1),|(
zmxzPrand
smallzmxzP
1),|(max
zexp zmax0zexp zmax0
![Page 9: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/9.jpg)
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Resulting Mixture Density
),|(
),|(
),|(
),|(
),|(
rand
max
unexp
hit
rand
max
unexp
hit
mxzP
mxzP
mxzP
mxzP
mxzP
T
How can we determine the model parameters?
![Page 10: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/10.jpg)
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Raw Sensor Data
Measured distances for expected distance of 300 cm.
Sonar Laser
![Page 11: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/11.jpg)
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Approximation Maximize log likelihood of the data
Search space of n-1 parameters. Hill climbing Gradient descent Genetic algorithms …
Deterministically compute the n-th parameter to satisfy normalization constraint.
)|( expzzP
![Page 12: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/12.jpg)
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Approximation Results
Sonar
Laser
300cm 400cm
![Page 13: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/13.jpg)
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Example
z P(z|x,m)
![Page 14: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/14.jpg)
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Approximation Results
Laser
Sonar
![Page 15: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/15.jpg)
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"sonar-0"
0 10 20 30 40 50 60 70 010203040506070
00.050.10.150.20.25
Influence of Angle to Obstacle
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"sonar-1"
0 10 20 30 40 50 60 70 010203040506070
00.050.10.150.20.250.3
Influence of Angle to Obstacle
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"sonar-2"
0 10 20 30 40 50 60 70 010203040506070
00.050.10.150.20.250.3
Influence of Angle to Obstacle
![Page 18: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/18.jpg)
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"sonar-3"
0 10 20 30 40 50 60 70 010203040506070
00.050.10.150.20.25
Influence of Angle to Obstacle
![Page 19: Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read.](https://reader036.fdocuments.us/reader036/viewer/2022062320/56649cf75503460f949c80b5/html5/thumbnails/19.jpg)
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Summary Beam-based Model Assumes independence between beams.
Justification? Overconfident!
Models physical causes for measurements. Mixture of densities for these causes. Assumes independence between causes. Problem?
Implementation Learn parameters based on real data. Different models should be learned for different angles
at which the sensor beam hits the obstacle. Determine expected distances by ray-tracing. Expected distances can be pre-processed.