Transcript CS225B Robot Programming Laboratory

Probabilistic Models of Sensing and Movement
Move to probability models of sensing and movement
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Project 2 is about complex behavior using sensing
Sensor interpretation is difficult – simple interpretation in this section
Artifacts [goal-directed motion] and reactive behaviors
Lectures
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Probabilistic sensor models
Probabilistic representation of uncertain movement
Particle filter implementation
Project
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PF for motion model
Markov localization with PF
Stretch – feature-based localization
Slides thanks to Steffen Gutmann
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Probabilistic Sensors Models
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.
p ( x | z )  p ( z | x) p ( x)
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Beam-based Sensor Model
Scan z consists of K measurements.
z  {z1 , z2 ,...,zK }
Individual measurements are independent given the robot
position.
K
P( z | x, m)   P( zk | x, m)
k 1
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Beam-based Sensor Model
K
P( z | x, m)   P( zk | x, m)
k 1
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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
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Proximity Measurement
Measurement can be caused by …
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a known obstacle.
cross-talk.
an unexpected obstacle (people, furniture, …).
missing all obstacles (total reflection, glass, …).
Noise is due to uncertainty …
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in measuring distance to known obstacle.
in position of known obstacles.
in position of additional obstacles.
whether obstacle is missed.
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Beam-based Proximity Model
Measurement noise
0
zexp
1
Phit ( z | x, m) 
e
2b
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Unexpected obstacles
zmax
1 ( z  zexp )

2
b
2
0
zexp
  e  z
Punexp ( z | x, m)  
 0
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zmax
z  zexp 

otherwise
Beam-based Proximity Model
Random measurement
zexp
0
Prand ( z | x, m) 
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zmax
1
zmax
Max range
0
zexp
zmax
0 if z  zmax
Pmax ( z | x, m)  
1 if z  zmax
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Resulting Mixture Density
  hit 
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 unexp 
P ( z | x, m)  
 max 
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 
 rand 
T
 Phit ( z | x, m) 
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 Punexp ( z | x, m) 
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Pmax ( z | x, m) 
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 P ( z | x, m) 
 rand
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How can we determine the model parameters?
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Raw Sensor Data
Measured distances for expected distance of 300 cm.
Sonar
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Laser
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Approximation
Search for -parameters that maximize the
(log) likelihood of the data
P( z | zexp )
Search space of n-1 parameters.
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Hill climbing
Gradient descent
Genetic algorithms
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Deterministically compute the n-th parameter to satisfy normalization constraint.
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Approximation Results
Laser
Sonar
400cm
300cm
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Example
z
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P(z|x,m)
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Summary Beam-based Model
Assumes independence between beams.
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Justification?
Overconfident!
Models physical causes for measurements.
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Mixture of densities for these causes.
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.
Determine expected distances by ray-tracing.
Expected distances can be pre-processed.
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Scan-based Model
Beam-based model is …
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not smooth for small obstacles and at edges.
not very efficient.
Idea: Instead of following along the beam, just check the end
point.
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Scan-based Model
Probability is a mixture of …
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a Gaussian distribution with mean at distance to closest obstacle,
a uniform distribution for random measurements, and
a small uniform distribution for max range measurements.
Again, independence between different components is
assumed.
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Example
Likelihood field
Map m
P(z|x,m)
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San Jose Tech Museum
Occupancy grid map
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Likelihood field
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Properties of Scan-based Model
Highly efficient, uses 2D tables only.
Smooth w.r.t. to small changes in robot position.
Allows gradient descent, scan matching.
Ignores physical properties of beams.
Will it work for ultrasound sensors?
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