07-sensor-models

• Again, independence between
different components is assumed.
23
Example
Likelihood field Map m P(z|x,m)
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San Jose Tech Museum
Occupancy grid map
Likelihood field
2
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.
0.3 0.25 0.2 0.15 0.1 0.05 0 0 40 50 60 70
10 20 30 20 30 40 10 50 60 0 70
19
Influence of Angle to Obstacle
"sonar-3"
0.25 0.2 0.15 0.1 0.05 0 0 40 50 60 70
10 20 30 20 30 40 10 50 60 0 70
20
Summary Beam-based Model
• Assumes independence between beams.
• Justification? • Overconfident!
• Models physical causes for measurements.
1. Beams reflected by obstacles 2. Beams reflected by persons / caused by crosstalk 3. Random measurements 4. Maximum range measurements
6
Proximity Measurement
3
Beam-based Sensor Model
• Scan z consists of K measurements.
z = {z1 , z 2 ,..., z K }
• Individual measurements are
P ( z | x , m ) = ∏ P ( z k | x , m)
k =1 K
independent given the robot position.
4
Beam-based Sensor Model
P ( z | x , m ) = ∏ P ( z k | x , m)
k =1
K
5
Typical Measurement Errors of an Range Measurements
How can we determine the model parameters?
10
Raw Sensor Data
Measured distances for expected distance of 300 cm.
Sonar
Laser
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Approximation
• Maximize log likelihood of the data P( z | zexp ) • Search space of n-1 parameters.
28
Additional Models of Proximity Sensors
• Map matching (sonar,laser): generate
small, local maps from sensor data and match local maps against global model. by scan endpoints, match scan into this map.
• • • • Sonar (time of flight) Radar (phase and frequency) Laser range-finders (triangulation, tof, phase) Infrared (intensity)
• Visual sensors: Cameras • Satellite-based sensors: GPS
~0.01 sec
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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?
• Measurement can be caused by …
• • • • • • • • a known obstacle. cross-talk. an unexpected obstacle (people, furniture, …). missing all obstacles (total reflection, glass, …). in measuring distance to known obstacle. in position of known obstacles. in position of additional obstacles. whether obstacle is missed.
• • • • Hill climbing Gradient descent Genetic algorithms …
• Deterministically compute the n-th
parameter to satisfy normalization constraint.
12
Approximation Results
7
• Noise is due to uncertainty …
Beameasurement noise Unexpected obstacles
0
zexp
zmax
0
zexp
zmax
Phit ( z | x, m) = η
1 e 2πb
1 ( z − z exp ) − 2 b
Introduction to Mobile Robotics
Probabilistic Sensor Models
Wolfram Burgard Cyrill Stachniss Giorgio Grisetti
SA-1
Sensors for Mobile Robots
• Contact sensors: Bumpers • Internal sensors
• 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.
1 z max
Pmax ( z | x, m) = η
1 z small
9
Resulting Mixture Density
α hit α unexp P ( z | x , m) = α max α rand
T
Phit ( z | x, m) Punexp ( z | x, m) ⋅ Pmax ( z | x, m) P ( z | x , m) rand
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Scan-based Model
• Beam-based model is …
• 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 Matching
• Extract likelihood field from scan and
use it to match different scan.
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Scan Matching
• Extract likelihood field from first scan
and use it to match second scan.
17
Influence of Angle to Obstacle
"sonar-1"
0.3 0.25 0.2 0.15 0.1 0.05 0 0 40 50 60 70
10 20 30 20 30 40 10 50 60 0 70
18
Influence of Angle to Obstacle