Probabilistic Fundamentals in Robotics Pr Pr obabilistic Mode ls - - PDF document

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Probabilistic Fundamentals in Robotics

Pr

• babilistic Mode ls of Mobile Robots

Pr

• babilistic Mode ls of Mobile Robots

Robot Motion Robot Motion

Basilio Bona DAUIN – Politecnico di Torino

Course Outline

• Basic mathematical framework
• Probabilistic models of mobile robots
• Mobile robot localization problem
• Robotic mapping
• Probabilistic planning and control

Reference textbook Thrun, Burgard, Fox, “Probabilistic Robotics”, MIT Press, 2006 http://www.probabilistic‐robotics.org/

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Probabilistic models of mobile robots

• Robot motion

– Kinematics Velocity motion model – Velocity motion model – Odometry motion model

• Robot perception

– Maps – Beam model of laser range finders Correlation based measurement models – Correlation‐based measurement models – Feature‐based measurement models

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Introduction

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Kinematic states

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• rientation

( ) y t ( ) t q ( ) x t

Probabilistic kinematics

State (pose or location) control From Wikipedia: Odometry is the use of data from the movement of actuators to estimate change in position

• ver time.

Odometry is used by some robots to estimate their position relative to a starting location. The method is sensitive to errors due to the integration of velocity measurements over time In applications, controls are sometimes provided by rover odometry The method is sensitive to errors due to the integration of velocity measurements over time to give position estimates. Rapid and accurate data collection, equipment calibration, and processing are required in most cases for odometry to be used effectively.

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Example

y y

darker points show higher probabilities of being there

x x

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the orientation is not shown, but contributes to the uncertainty of the final location

Motion models

Velocity model: the simplest one, assumes that the control is given as a velocity command to the motors; velocity remain constant in the sampling interval [t‐1 t) sampling interval [t 1, t) Acceleration model: is slightly more complicated, assuming a constant acceleration motion, i.e., a linearly increasing velocity Odometric model: assumes the accessibility to odometric Odometric model: assumes the accessibility to odometric information, usually provided by wheel sensors, but often also by

• ther means (i.e., visual odometry)

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Motion models

• Odometric models are usually more accurate than

velocity models, but odometry is available only after the motion command has been executed, while velocity commands are available before performing the actual motion

• Odometric models are good for estimation, while velocity

models are better suited for path planning

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Velocity motion model

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Velocity motion model: noise‐free

( ) ( )

t t t t

x y q = x

T T t

r

t

q

( )

t t t

v w = u

T c

y

t

y

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90

t

q -

 c

x

t

x

Velocity motion model: noise‐free

t

x q D

t t t

v r w =

1 t-

x q D is negative c

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q D is negative

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Exact velocity model

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Velocity models

Exact Euler

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Runge‐Kutta

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Velocity models

Exact Euler R K tt

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Runge‐Kutta

Odometry errors

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Error noise

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Velocity model with error noise

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Velocity motion model algorithm

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Example

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Odometry motion model

• Odometry is obtained integrating sensor reading from wheel

encoders, or from other sources (e.g., visual odometry)

• Odometry provides the information of the relative motion of

y p the robot.

• Odometry is more accurate than velocity
• Odometry measurements are available only AFTER a control

has been supplied to the robot, then they should be better considered as measurements, … but usually the are included , y as control signals ut

• For this reason odometry cannot be used for planning and

control

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Odometry motion model

• Odometry model considers the

motion in the time interval motion in the time interval

• 1. A first rotation
• 2. A translation
• 3. A second rotation

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Each of them is noisy

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Odometry model

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Example

• Repeated application of the sensor model for short

movements

• Typical banana‐shaped distributions obtained for 2D‐

yp p projection of 3D posterior

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Example

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Sampling

• One can use normal (Gaussian) distributions or triangular

distributions for describing uncertainty and for sampling

Normal distribution Triangular distribution Normal distribution Triangular distribution

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How to Sample from Normal or Triangular Distributions?

• Sampling from a normal distribution
• Sampling from a triangular distribution

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Normally distributed samples

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Triangular distributed samples

103 samples 104 samples

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106 samples 105 samples

Sample odometry motion model

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Sample normal distribution

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Example

Start

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Motion and maps

• In many cases we have a map m that represents the

environment where the robot moves

• Occupacy maps distinguish free (traversable) from

p y p g ( )

• ccupied space: robot pose shall be always in free space
• A motion model that takes into consideration a map

computes

• If the map m carries information relevant to pose

Map‐based motion model

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• If the map m carries information relevant to pose

estimation

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• Approximation

Map‐free estimate Consistency on the pose with the map

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This is the result of checking model consistency at the final pose, instead of verifying it on the entire path to the goal

Thank you. Any question?

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