Beam Sensor Models
Pieter Abbeel
UC Berkeley EECS
Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
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Proximity Sensors
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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 position x.
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Question: Where do the probabilities come from?
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Approach: Let s try to explain a measurement.
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Page 1!
Beam-based Sensor Model
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Scan z consists of K measurements.
z = {z1, z2 ,..., zK }
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Individual measurements are independent given the robot
position.
K
P ( z | x, m) = ∏ P ( z k | x, m)
k =1
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Beam-based Sensor Model
K
P ( z | x, m) = ∏ P ( z k | x, m)
k =1
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Page 2!
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
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Proximity Measurement
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Measurement can be caused by …
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a known obstacle.
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cross-talk.
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an unexpected obstacle (people, furniture, …).
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missing all obstacles (total reflection, glass, …).
Noise is due to uncertainty …
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in measuring distance to known obstacle.
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in position of known obstacles.
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in position of additional obstacles.
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whether obstacle is missed.
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Page 3!
Beam-based Proximity Model
Measurement noise
0
zexp
Phit ( z | x, m) = η
Unexpected obstacles
zmax
1 ( z − zexp )
b
−
1
e 2
2πb
0
2
zexp
zmax
⎧η λ e −λz
Punexp ( z | x, m) = ⎨
⎩ 0
z < zexp ⎫
⎬
otherwise⎭
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Beam-based Proximity Model
Random measurement
0
zexp
Prand ( z | x, m) = η
zmax
Max range
0
1
zexp
Pmax ( z | x, m) = η
z max
zmax
1
z small
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Page 4!
Resulting Mixture Density
⎛ α hit ⎞
⎜
⎟
⎜α unexp ⎟
P( z | x, m) = ⎜
α ⎟
⎜ max ⎟
⎜ α ⎟
⎝ rand ⎠
T
⎛ Phit ( z | x, m) ⎞
⎜
⎟
⎜ Punexp ( z | x, m) ⎟
⋅ ⎜
P ( z | x, m) ⎟
⎜ max
⎟
⎜ P ( z | x, m) ⎟
⎝ rand
⎠
How can we determine the model parameters?
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Raw Sensor Data
Measured distances for expected distance of 300 cm.
Sonar
Laser
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Page 5!
Approximation
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Maximize log likelihood of the data
P( z | zexp )
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Search space of n-1 parameters.
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Hill climbing
Gradient descent
Genetic algorithms
…
Deterministically compute the n-th parameter to
satisfy normalization constraint.
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Approximation Results
Laser
400cm
300cm
Sonar
Page 6!
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Example
z
P(z|x,m)
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Approximation Results
Laser
Sonar
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Page 7!
Influence of Angle to Obstacle
"sonar-0"
0.25
0.2
0.15
0.1
0.05
0
0
10 20
30
20
30 40
10
50 60
70 0
40
50
60
70
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Influence of Angle to Obstacle
"sonar-1"
0.3
0.25
0.2
0.15
0.1
0.05
0
0
10 20
30
20
30 40
10
50 60
70 0
40
50
60
70
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Page 8!
Influence of Angle to Obstacle
"sonar-2"
0.3
0.25
0.2
0.15
0.1
0.05
0
0
10 20
30
20
30 40
10
50 60
70 0
40
50
60
70
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Influence of Angle to Obstacle
"sonar-3"
0.25
0.2
0.15
0.1
0.05
0
0
10 20
30
20
30 40
10
50 60
70 0
40
50
60
70
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Page 9!
Summary Beam-based Model
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Assumes independence between beams.
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Justification?
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Overconfident!
Models physical causes for measurements.
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Mixture of densities for these causes.
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Assumes independence between causes. Problem?
Implementation
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Learn parameters based on real data.
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.
Page 10!
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