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BeamSensorModels
PieterAbbeelUCBerkeleyEECS
ManyslidesadaptedfromThrun,BurgardandFox,ProbabilisAcRoboAcs
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n
ThecentraltaskistodetermineP(z|x),i.e.,theprobabilityofameasurementzgiventhattherobotisatposiAonx.
n Ques0on:WheredotheprobabiliAescomefrom?
n Approach:Let’strytoexplainameasurement.
ProximitySensors
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Beam-basedSensorModeln ScanzconsistsofKmeasurements.
n IndividualmeasurementsareindependentgiventherobotposiAon.
},...,,{ 21 Kzzzz =
∏=
=K
kk mxzPmxzP
1
),|(),|(
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Beam-basedSensorModel
∏=
=K
kk mxzPmxzP
1
),|(),|(
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TypicalMeasurementErrorsinRangeMeasurements
1. Beamsreflectedbyobstacles
2. Beamsreflectedbypersons/causedbycrosstalk
3. Randommeasurements
4. Maximumrangemeasurements
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Beam-basedProximityModel
Measurement noise
zexp zmax 0
bzz
hit ebmxzP
2exp )(
21
21),|(
−−
=π
η⎭⎬⎫
⎩⎨⎧ <
=−
otherwisezz
mxzPz
0e
),|( expunexpλλη
Unexpected obstacles
zexp zmax 0
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Beam-basedProximityModel
Random measurement Max range
max
1),|(z
mxzPrand η=smallz
mxzP 1),|(max η=
zexp zmax 0 zexp zmax 0
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ResulAngMixtureDensity
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
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=
),|(),|(),|(),|(
),|(
rand
max
unexp
hit
rand
max
unexp
hit
mxzPmxzPmxzPmxzP
mxzP
T
α
α
α
α
How can we determine the model parameters?
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RawSensorData
Measured distances for expected distance of 300 cm.
Sonar Laser
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ApproximaAonn Maximizeloglikelihoodofthedata
n Searchspaceofn-1parameters.n Hillclimbing
n Gradientdescent
n GeneAcalgorithms
n …
n
DeterminisAcallycomputethen-thparametertosaAsfynormalizaAonconstraint.
)|( expzzP
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ApproximaAonResults
Sonar
Laser
300cm 400cm
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ApproximaAonResults
Laser
Sonar
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"sonar-0"
0 10 20 30 40 50 60 70 01020
30405060
700
0.050.10.150.20.25
InfluenceofAngletoObstacle
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"sonar-1"
0 10 20 30 40 50 60 70 01020
30405060
700
0.050.10.150.20.250.3
InfluenceofAngletoObstacle
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"sonar-2"
0 10 20 30 40 50 60 70 01020
30405060
700
0.050.10.150.20.250.3
InfluenceofAngletoObstacle
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"sonar-3"
0 10 20 30 40 50 60 70 01020
30405060
700
0.050.10.150.20.25
InfluenceofAngletoObstacle
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n Assumesindependencebetweenbeams.
n JusAficaAon?
n Overconfident!
n Modelsphysicalcausesformeasurements.
n MixtureofdensiAesforthesecauses.
n Assumesindependencebetweencauses.Problem?
n ImplementaAon
n Learnparametersbasedonrealdata.
n
Differentmodelsshouldbelearnedfordifferentanglesatwhichthesensorbeamhitstheobstacle.
n Determineexpecteddistancesbyray-tracing.
n Expecteddistancescanbepre-processed.
SummaryBeam-basedModel
