4/22/2008 1 It d ti t E b dd dS t Introduction to EmbeddedSystems EECS 124, UC Berkeley, Spring 2008 Lecture 23: Localization and Mapping Gabe Hoffmann Gabe Hoffmann Ph.D. Candidate, Aero/Astro Engineering Stanford University Overview • Statistical Models • Localization • Occupancy Grid Mapping • Simultaneous Localization and Mapping (SLAM) This lecture draws material from S. Thrun, W. Burgard, & D. Fox, Probabilistic Robotics, 2005
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I t d ti t E b dd d S tIntroduction to Embedded SystemsEECS 124, UC Berkeley, Spring 2008Lecture 23: Localization and Mapping
Gabe HoffmannGabe HoffmannPh.D. Candidate, Aero/Astro EngineeringStanford University
Overview
• Statistical Models
• Localization
• Occupancy Grid Mapping
• Simultaneous Localization and Mapping (SLAM)
This lecture draws material fromS. Thrun, W. Burgard, & D. Fox, Probabilistic Robotics, 2005
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Statistics Background
• Conditional Probability
• Bayes’ Rule
• Independence of two variables
Conditional Independence
Statistics Background (cont.)• Theorem of Total Probability
• Gaussian Probability Distributions
-5
0
5
-5
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50
0.05
0.1
x1
x2
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Markov Assumption
• Future and past independent given present
Motion Model
• Stochastic model of future state:
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• Velocity input model
Two Wheeled Robots
• Odometry “input” model
NoiseCommand
NoiseMeasurement
Two Wheeled Robots (cont.)
• Next state
• Noise model
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(Forward) Measurement Model
• Stochastic model of measurements:
Sensor Examples
• Bump Bearing Sensor Model
• GPS
• Camera
• Directional Antenna
• Ultrasonic Ranger
• LIDAR
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Forward vs. Inverse Sensor Models
• Forward ModelC f
Forward Measurement Model
– Common for localization
• Inverse Model– Useful when state less complicated than InverseMeasurement Model