Proximity Sensors The central task is to determine P(z|x) , i.e., the - - PowerPoint PPT Presentation
Beam Sensor Models Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, ProbabilisAc RoboAcs Proximity Sensors The central task is to determine P(z|x) , i.e., the probability of a measurement z n given that the robot
Pieter Abbeel UC Berkeley EECS
Many slides adapted from Thrun, Burgard and Fox, ProbabilisAc RoboAcs
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The central task is to determine P(z|x), i.e., the probability of a measurement z given that the robot is at posiAon x.
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Ques0on: Where do the probabiliAes come from?
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Approach: Let’s try to explain a measurement.
n Scan z consists of K measurements. n Individual measurements are independent given the robot
2 1 K
=
K k k
1
=
K k k
1
1. Beams reflected by
2. Beams reflected by persons / caused by crosstalk 3. Random measurements 4. Maximum range measurements
Measurement noise
zexp zmax
b z z hit
e b m x z P
2 exp)
( 2 1
2 1 ) , | (
− −
= π η ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ < =
−
z z m x z P
z
e ) , | (
exp unexp λ
λ η Unexpected obstacles
zexp zmax
Random measurement Max range
max
1 ) , | ( z m x z P
rand
η =
small
z m x z P 1 ) , | (
max
η =
zexp zmax zexp zmax
⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ) , | ( ) , | ( ) , | ( ) , | ( ) , | (
rand max unexp hit rand max unexp hit
m x z P m x z P m x z P m x z P m x z P
T
α α α α
How can we determine the model parameters?
Measured distances for expected distance of 300 cm. Sonar Laser
n Maximize log likelihood of the data n Search space of n-1 parameters.
n Hill climbing n Gradient descent n GeneAc algorithms n …
n DeterminisAcally compute the n-th parameter to saAsfy normalizaAon
exp
Sonar Laser
300cm 400cm
Laser Sonar
"sonar-0" 10 20 30 40 50 60 70 0 10203040506070 0.05 0.1 0.15 0.2 0.25
"sonar-1" 10 20 30 40 50 60 70 0 10203040506070 0.05 0.1 0.15 0.2 0.25 0.3
"sonar-2" 10 20 30 40 50 60 70 0 10203040506070 0.05 0.1 0.15 0.2 0.25 0.3
"sonar-3" 10 20 30 40 50 60 70 0 10203040506070 0.05 0.1 0.15 0.2 0.25
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Assumes independence between beams.
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JusAficaAon?
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Overconfident!
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Models physical causes for measurements.
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Mixture of densiAes for these causes.
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Assumes independence between causes. Problem?
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ImplementaAon
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Learn parameters based on real data.
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Different models should be learned for different angles at which the sensor beam hits the obstacle.
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Determine expected distances by ray-tracing.
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Expected distances can be pre-processed.