Beam Sensor Models - Advanced Robotics - Lecture Slides, Slides of Robotics

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Beam Sensor Models

2 Proximity Sensors

n 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.

n Question : Where do the probabilities come from?

n Approach : Let s try to explain a measurement.

4

Beam-based Sensor Model

=

K k k

P z x m P z x m

1

5 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

7 Beam-based Proximity Model

Measurement noise

zexp z (^0) max b z z hit e b P z x m 2 ( exp ) 2 1 2

− − = π η ⎭

− otherwise z z P z x m z 0 e ( | , ) exp unexp λ η λ

Unexpected obstacles

zexp z (^0) max

8 Beam-based Proximity Model

Random measurement Max range

max

z P z x m rand

small z P z x m

max = η zexp z (^0) max zexp z (^0) max

10 Raw Sensor Data

Measured distances for expected distance of 300 cm.

Sonar Laser

11 Approximation

n Maximize log likelihood of the data

n Search space of n-1 parameters.

n Hill climbing

n Gradient descent

n Genetic algorithms

n …

n Deterministically compute the n-th parameter to

satisfy normalization constraint.

( | ) exp P z z

13 Example z P(z|x,m)

15 Approximation Results

Laser

Sonar

17 **"sonar-1" 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0

0.** Influence of Angle to Obstacle

18 **"sonar-2" 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0

0.** Influence of Angle to Obstacle

20 Summary Beam-based Model

n Assumes independence between beams.

n Justification? n Overconfident!

n Models physical causes for measurements.

n Mixture of densities for these causes. n Assumes independence between causes. Problem?

n Implementation

n Learn parameters based on real data. n Different models should be learned for different angles at which the sensor beam hits the obstacle. n Determine expected distances by ray-tracing. n Expected distances can be pre-processed.