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A Taste of Localization Problem Course Summary

Introduction to ROBOTICS

- Dr. John (Jizhong) Xiao
- Department of Electrical Engineering
- City College of New York
- jxiao_at_ccny.cuny.edu

Topics

- Brief Review (Robot Mapping)
- A Taste of Localization Problem
- Course Summary

Mapping/Localization

- Answering robotics big questions
- How to get a map of an environment with imperfect

sensors (Mapping) - How a robot can tell where it is on a map

(localization) - It is an on-going research
- It is the most difficult task for robot
- Even human will get lost in a building!

Review Use Sonar to Create Map

What should we conclude if this sonar reads 10

feet?

there isnt something here

there is something somewhere around here

10 feet

Local Map

unoccupied

no information

occupied

What is it a map of?

Several answers to this question have been tried

cell (x,y) is unoccupied

cell (x,y) is occupied

oxy

oxy

Its a map of occupied cells.

pre 83

What information should this map contain, given

that it is created with sonar ?

Each cell is either occupied or unoccupied --

this was the approach taken by the Stanford Cart.

What is it a map of ?

Several answers to this question have been tried

cell (x,y) is unoccupied

cell (x,y) is occupied

oxy

oxy

Its a map of occupied cells. Its a map of

probabilities p( o S1..i ) p( o

S1..i )

pre 83

The certainty that a cell is occupied, given the

sensor readings S1, S2, , Si

83 - 88

The certainty that a cell is unoccupied, given

the sensor readings S1, S2, , Si

The odds of an event are expressed relative to

the complement of that event.

Its a map of odds.

probabilities

evidence log2(odds)

p( o S1..i )

odds( o S1..i )

The odds that a cell is occupied, given the

sensor readings S1, S2, , Si

p( o S1..i )

Combining Evidence

- The key to making accurate maps is combining

lots of data.

Combining Evidence

- The key to making accurate maps is combining

lots of data.

p( o S2 ? S1 )

defn of odds

odds( o S2 ? S1)

p( o S2 ? S1 )

p( S2 ? S1 o ) p(o)

.

Bayes rule ()

p( S2 ? S1 o ) p(o)

p( S2 o ) p( S1 o ) p(o)

conditional independence of S1 and S2

.

p( S2 o ) p( S1 o ) p(o)

p( S2 o ) p( o S1 )

.

Bayes rule ()

p( S2 o ) p( o S1 )

previous odds

precomputed values

the sensor model

Update step multiplying the previous odds by a

precomputed weight.

Mapping Using Evidence Grids

Evidence Grids...

represent space as a collection of cells, each

with the odds (or probability) that it contains

an obstacle

evidence log2(odds)

Lab environment

likely free space

likely obstacle

lighter areas lower evidence of obstacles

being present

not sure

darker areas higher evidence of obstacles being

present

Mobot System Overview

Content

- Brief Review (Robot Mapping)
- A Taste of Localization Problem
- Course Summary

Whats the problem?

- WHERE AM I?
- But what does this mean, really?
- Reference frame is important
- Local/Relative Where am I vs. where I was?
- Global/Absolute Where am I relative to the world

frame? - Location can be specified in two ways
- Geometric Distances and angles
- Topological Connections among landmarks

Localization Absolute

- Proximity-To-Reference
- Landmarks/Beacons
- Angle-To-Reference
- Visual manual triangulation from physical points
- Distance-From-Reference
- Time of Flight
- RF GPS
- Acoustic
- Signal Fading
- EM Bird/3Space Tracker
- RF
- Acoustic

Triangulation

Land

Landmarks

Works great -- as long as there are reference

points!

Lines of Sight

Unique Target

Sea

Compass Triangulation

cutting-edge 12th century technology

Land

Landmarks

Lines of Sight

North

Unique Target

Sea

Localization Relative

- If you know your speed and direction, you can

calculate where you are relative to where you

were (integrate). - Speed and direction might, themselves, be

absolute (compass, speedometer), or integrated

(gyroscope, Accelerometer) - Relative measurements are usually more accurate

in the short term -- but suffer from accumulated

error in the long term - Most robotics work seems to focus on this.

Localization Methods

- Markov Localization
- Represent the robots belief by a probability

distribution over possible positions and uses

Bayes rule and convolution to update the belief

whenever the robot senses or moves - Monte-Carlo methods
- Kalman Filtering
- SLAM (simultaneous localization and mapping)
- .

Environment Representation

- Environment Representation
- Continuos Metric x,y,q
- Discrete Metric metric grid
- Discrete Topological topological grid

Continuous metric

Real environment

Discrete Topological

Metric grid

Environment Representation

- Continuous Metric, (x,y,q)
- Topological (landmark-based, state space

organized according to the topological structure

of the environment) - Grid-Based (the world is divided in cells of

fixed size resolution and precision of state

estimation are fixed beforehand) - The latter suffers from computational overhead

Probability Review

- Discrete Random Variables
- X denotes a random variable.
- X can take on a countable number of values in

x1, x2, , xn. - P(Xxi), or P(xi), is the probability that the

random variable X takes on value xi. - P( ) is called probability mass function.
- E.g.

.

Probability Review

- Continuous Random Variables
- X takes on values in the continuum.
- p(Xx), or p(x), is a probability density

function. - E.g.

p(x)

x

Probability Review

- Joint and Conditional Probability
- P(Xx and Yy) P(x,y)
- If X and Y are independent then P(x,y) P(x)

P(y) - P(x y) is the probability of x given y P(x

y) P(x,y) / P(y) P(x,y) P(x y) P(y) - If X and Y are independent then P(x y) P(x)

Law of Total Probability, Marginals

Discrete case

Continuous case

Probability Review

- Law of total probability

Conditional Independence

- equivalent to
- and

Bayes Formula

If y is a new sensor reading

Prior probability distribution

?

Posterior probability distribution

?

Generative model, characteristics of the sensor

?

?

Does not depend on x

Bayes Rule with Background Knowledge

Markov Localization

- What is Markov Localization ?
- Special case of probabilistic state estimation

applied to mobile robot localization - Initial Hypothesis
- Static Environment
- Markov assumption
- The robots location is the only state in the

environment which systematically affects sensor

readings - Further Hypothesis
- Dynamic Environment

Markov Localization

- Applying probability theory to robot localization
- Markov localization uses an explicit, discrete

representation for the probability of all

position in the state space. - This is usually done by representing the

environment by a grid or a topological graph with

a finite number of possible states (positions). - During each update, the probability for each

state (element) of the entire space is updated.

Markov Localization

- Instead of maintaining a single hypothesis as to

where the robot is, Markov localization maintains

a probability distribution over the space of all

such hypothesis - Uses a fine-grained and metric discretization of

the state space

Example

- Assume the robot position is one- dimensional

The robot is placed somewhere in the environment

but it is not told its location

The robot queries its sensors and finds out it is

next to a door

Example

The robot moves one meter forward. To account for

inherent noise in robot motion the new belief is

smoother

The robot queries its sensors and again it finds

itself next to a door

Basic Notation

Bel(Ltl ) Is the probability (density) that the

robot assigns to the possibility that its

location at time t is l

The belief is updated in response to two

different types of events sensor readings,

odometry data

Notation

- Goal

Markov assumption (or static world assumption)

Markov Localization

- Measurement

- Action

Update Phase

a

b

c

Update Phase

Recursive Bayesian Updating

Markov assumption zn is independent of

z1,...,zn-1 if we know x.

Example Closing the door

Example Second Measurement

- P(z2open) 0.5 P(z2?open) 0.6
- P(openz1)2/3

- z2 lowers the probability that the door is open.

Action Prediction Phase

- Often the world is dynamic since
- actions carried out by the robot,
- actions carried out by other agents,
- or just the time passing by
- change the world.
- How can we incorporate such actions?

Modeling Actions

- To incorporate the outcome of an action u into

the current belief, we use the conditional pdf - P(xu,x)
- This term specifies the pdf that executing u

changes the state from x to x.

State Transitions

- P(xu,x) for u close door
- If the door is open, the action close door

succeeds in 90 of all cases.

Integrating the Outcome of Actions

Continuous case Discrete case

Example The Resulting Belief

Bayes Filters Framework

- Given
- Stream of observations z and action data u
- Sensor model P(zx).
- Action model P(xu,x).
- Prior probability of the system state P(x).
- Wanted
- Estimate of the state X of a dynamical system.
- The posterior of the state is also called Belief

Markov Assumption

Measurement probability

?

State transition probability

?

- Markov Assumption
- past and future data are independent if one knows

the current state

- Underlying Assumptions
- Static world, Independent noise
- Perfect model, no approximation errors

Bayes Filters

z observation u action x state

Summary

- Measurement

- Action

Content

- Brief Review (Robot Mapping)
- A Taste of Localization Problem
- Course Summary

Mobile Robot

Mobile Robot Locomotion

Locomotion the process of causing a robot to move

- Tricycle

- Differential Drive

Swedish Wheel

- Synchronous Drive

- Omni-directional

- Ackerman Steering

Differential Drive

Property At each time instant, the left and

right wheels must follow a trajectory that moves

around the ICC at the same angular rate ?, i.e.,

- Kinematic equation

- Nonholonomic Constraint

Differential Drive

- Basic Motion Control

R Radius of rotation

- Straight motion
- R Infinity VR VL

- Rotational motion
- R 0 VR -VL

Tricycle

- Steering and power are provided through the front

wheel - control variables
- angular velocity of steering wheel ws(t)
- steering direction a(t)

d distance from the front wheel to the rear axle

Tricycle

Kinematics model in the world frame ---Posture

kinematics model

Synchronous Drive

- All the wheels turn in unison
- All wheels point in the same direction and turn

at the same rate - Two independent motors, one rolls all wheels

forward, one rotate them for turning - Control variables (independent)
- v(t), ?(t)

Ackerman Steering (Car Drive)

- The Ackerman Steering equation

Car-like Robot

Driving type Rear wheel drive, front wheel

steering

Rear wheel drive car model

forward velocity of the rear wheels

angular velocity of the steering wheels

non-holonomic constraint

l length between the front and rear wheels

Robot Sensing

- Collect information about the world
- Sensor - an electrical/mechanical/chemical device

that maps an environmental attribute to a

quantitative measurement - Each sensor is based on a transduction principle

- conversion of energy from one form to another - Extend ranges and modalities of Human Sensing

Gas Sensor

Gyro

Accelerometer

Metal Detector

Pendulum Resistive Tilt Sensors

Piezo Bend Sensor

Gieger-Muller Radiation Sensor

Pyroelectric Detector

UV Detector

Resistive Bend Sensors

CDS Cell Resistive Light Sensor

Digital Infrared Ranging

Pressure Switch

Miniature Polaroid Sensor

Limit Switch

Touch Switch

Mechanical Tilt Sensors

IR Sensor w/lens

IR Pin Diode

Thyristor

Magnetic Sensor

Polaroid Sensor Board

Hall Effect Magnetic Field Sensors

Magnetic Reed Switch

IR Reflection Sensor

IR Amplifier Sensor

IRDA Transceiver

IR Modulator Receiver

Radio Shack Remote Receiver

Solar Cell

Lite-On IR Remote Receiver

Compass

Compass

Piezo Ultrasonic Transducers

Sensors Used in Robot

- Resistive sensors
- bend sensors, potentiometer, resistive

photocells, ... - Tactile sensors contact switch, bumpers
- Infrared sensors
- Reflective, proximity, distance sensors
- Ultrasonic Distance Sensor
- Motor Encoder
- Inertial Sensors (measure the second derivatives

of position) - Accelerometer, Gyroscopes,
- Orientation Sensors Compass, Inclinometer
- Laser range sensors
- Vision, GPS,

Motion Planning

Path Planning Find a path connecting an initial

configuration to goal configuration without

collision with obstacles

- Configuration Space
- Motion Planning Methods
- Roadmap Approaches
- Cell Decomposition
- Potential Fields
- Bug Algorithms

Motion Planning

- Motion Planning Methodololgies
- Roadmap
- Cell Decomposition
- Potential Field
- Roadmap
- From Cfree a graph is defined (Roadmap)
- Ways to obtain the Roadmap
- Visibility graph
- Voronoi diagram
- Cell Decomposition
- The robot free space (Cfree) is decomposed

into simple regions (cells) - The path in between two poses of a cell can

be easily generated - Potential Field
- The robot is treated as a particle acting

under the influence of a potential field U, - where
- the attraction to the goal is modeled by

an additive field - obstacles are avoided by acting with a

repulsive force that yields a negative field

Global methods

Local methods

Full-knowledge motion planning

Cell decompositions

Roadmaps

visibility graph

exact free space represented via convex polygons

voronoi diagram

approximate free space represented via a quadtree

Potential field Method

- Usually assumes some knowledge at the global

level

The goal is known the obstacles sensed

Each contributes forces, and the robot follows

the resulting gradient.

Thank you!

This Thurday Final Exam Time Dec. 13,

630pm-830pm, Place SH-378 Coverage Mobile

Robot Close-book with 2 pages cheat sheet, but Do

Not Cheat