The City College of New York 1 Dr. John (Jizhong) Xiao Department of Electrical Engineering City...

The City College of New York

1

Dr. John (Jizhong) Xiao

Department of Electrical Engineering

City College of New York

[email protected]

Mobile Robot Mapping

Introduction to ROBOTICS


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Topics• Brief Review: Motion Planning

• Mobile Robot Mapping– Using sonar to create maps– Bayes Rule– Evidence Grids


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Review: Motion Planning

– Configuration Space

– Motion Planning Methods• Roadmap Approaches• Cell Decomposition• Potential Fields• Bug Algorithms

Motion Planning: Find a path connecting an initial configuration to goal configuration without collision with obstacles


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Review: 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


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Full-knowledge motion planning

approximate free space

represented via a quadtree

Cell decompositionsRoadmaps

exact free space

represented via convex polygons

visibility graph

voronoi diagram


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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.


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Mobot System Overview

Abstraction level

Motor Modeling: what voltage should I set now ?

Control (PID): what voltage should I set over time ?

Kinematics: if I move this motor somehow, what happens in other coordinate systems ?

Motion Planning: Given a known world and a cooperative mechanism, how do I get there from here ?

Bug Algorithms: Given an unknowable world but a known goal and local

sensing, how can I get there from here?

Mapping: Given sensors, how do I create a useful map?

low-level

high-level

Localization: Given sensors and a map, where am I ?Vision: If my sensors are eyes, what do I do?


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Mobile Robot Mapping

• 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)– Getting from place to place under uncertain

conditions– Using vision effectively


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Sonar sensing

Why is sonar sensing limited to between ~12 in.

and ~25 feet ?

“The sponge”

Polaroid sonar emitter/receivers

sonar timeline

0

a “chirp” is emitted into the environment

75s

typically when reverberations from the initial chirp have stopped

.5sthe transducer goes into “receiving” mode and awaits a signal...

after a short time, the signal will be too weak to be detected

Sonar (sound navigation and ranging): range sensing using acoustic (i.e., sound) signal

Blanking time Attenuation


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Sonar effects

resolution: time / space

(d) Specular reflections cause walls to disappear

(e) Open corners produce a weak spherical wavefront

(f) Closed corners measure to the corner itself because of multiple reflections --> sonar ray tracing

(a) Sonar providing an accurate range measurement

(b-c) Lateral resolution is not very precise; the closest object in the beam’s cone provides the response


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Sonar modelinginitial time response

spatial response

Sonar amplitude profile: 3-D cone with side lobes

blanking time

accumulated responses

cone width


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Sonar Modelingresponse model (Kuc &

Siegel)

sonar reading

obstacle

c = speed of sound

a = diameter of sonar element (transducer)

t = time

z = orthogonal distance

= angle of environment surface

• Models the response, hR,with

• Then, add noise to the model to obtain a probability:

p( S | o )

chance that the sonar reading is S, given an obstacle at

location o

z =

S

o


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Using sonar to create maps

What should we conclude if this sonar reads 10 feet?

10 feet

what would a local map look like?


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10 feet

there is something somewhere around here

there isn’t something here

Local Map

unoccupied

occupied


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10 feet

there is something somewhere around here

there isn’t something here

Local Mapunoccupied

occupied

or ...

no information


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What should we conclude if this sonar reads 10 feet...

10 feet

and how do we add the information that the next sonar reading (as the robot moves) reads 10 feet, too?

10 feet


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Combining sensor readings

• The key to making accurate maps is combining lots of data.

• But combining these numbers means we have to know what they are !

What should our map contain ?

• small cells

• each represents a bit of the robot’s environment

• larger values => obstacle

• smaller values => free

what is in each cell of this sonar model / map ?


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What is it a map of?

Several answers to this question have been tried:

It’s a map of occupied cells. oxy oxy cell (x,y) is occupied

cell (x,y) is unoccupied

Each cell is either occupied or unoccupied -- this was the approach taken by the Stanford Cart.

pre ‘83

What information should this map contain, given that it is created with sonar ?


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It’s a map of occupied cells.

It’s a map of probabilities: p( o | S1..i )

p( o | S1..i )

The certainty that a cell is occupied, given the sensor readings S1, S2, …, Si

The certainty that a cell is unoccupied, given the sensor readings S1, S2, …, Si

oxy oxy cell (x,y) is occupied


• maintaining related values separately

• initialize all certainty values to zero

‘83 - ‘88

pre ‘83

What is it a map of ?

• contradictory information will lead to both values near 1 • combining them takes some work...


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It’s a map of occupied cells.

It’s a map of probabilities: p( o | S1..i )

p( o | S1..i )

It’s a map of odds.

The certainty that a cell is occupied, given the sensor readings S1, S2, …, Si

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.

odds( o | S1..i ) = p( o | S1..i )p( o | S1..i )

The odds that a cell is occupied, given the sensor readings S1, S2, …,

Si

oxy oxy cell (x,y) is occupied


‘83 - ‘88

pre ‘83

What is it a map of ?

probabilities

evidence = log2(odds)


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An example map

units: feetEvidence grid of a tree-lined outdoor path

lighter areas: lower odds of obstacles being present

darker areas: higher odds of obstacles being presenthow to combine them?


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Combining probabilities

How to combine two sets of probabilities into a single map ?


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Conditional probability

Some intuition...

p( o | S ) = The probability of event o, given event S .The probability that a certain cell o is occupied, given that the robot sees the sensor reading S .

p( S | o ) = The probability of event S, given event o .The probability that the robot sees the sensor reading S , given that a certain cell o is occupied.

• What is really meant by conditional probability ?

• How are these two probabilities related?

p( o | S ) = p(S | o) ??


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Bayes Rule

p( o S ) = p( o | S ) p( S )

- Joint probabilities/Conditional probabilities


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Bayes Rule

- Bayes rule relates conditional probabilities

p( o | S ) = p( S | o ) p( o )

p( S )Bayes rule

p( o S ) = p( o | S ) p( S )



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Bayes Rule

- Bayes rule relates conditional probabilities

p( o | S ) = p( S | o ) p( o )

- So, what does this say about

p( S )Bayes rule

odds( o | S2 S1 ) ?

p( o S ) = p( o | S ) p( S )


Can we update easily ?


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Combining evidence

So, how do we combine evidence to create a map?

What we want --

odds( o | S2 S1) the new value of a cell in the map after the sonar reading S2

What we know --

odds( o | S1) the old value of a cell in the map (before sonar reading S2)

p( Si | o ) & p( Si | o ) the probabilities that a certain obstacle causes the sonar reading Si


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Combining evidence

odds( o | S2 S1) = p( o | S2 S1 )

p( o | S2 S1 )


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Combining evidence

odds( o | S2 S1) = p( o | S2 S1 )

p( o | S2 S1 )

. = p( S2 S1 | o ) p(o)

p( S2 S1 | o ) p(o)

def’n of odds


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

Combining evidence

odds( o | S2 S1) = p( o | S2 S1 )

p( o | S2 S1 )

. =

. =

p( S2 S1 | o ) p(o)

p( S2 S1 | o ) p(o)

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

def’n of odds

Bayes’ rule (+)


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

Combining evidence

odds( o | S2 S1) = p( o | S2 S1 )

p( o | S2 S1 )

. =

. =

. =

p( S2 S1 | o ) p(o)

p( S2 S1 | o ) p(o)

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

def’n of odds

Bayes’ rule (+)

conditional independence of S1 and

S2

p( S2 | o ) p( o | S1 )p( S2 | o ) p( o | S1 ) Bayes’ rule

(+)


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

Combining evidence

odds( o | S2 S1) = p( o | S2 S1 )

p( o | S2 S1 )

. =

. =

. =

p( S2 S1 | o ) p(o)

p( S2 S1 | o ) p(o)

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

Update step = multiplying the previous odds by a precomputed weight.

def’n of odds

Bayes’ rule (+)

conditional independence of S1 and

S2

p( S2 | o ) p( o | S1 )p( S2 | o ) p( o | S1 ) Bayes’ rule

(+)

previous oddsprecomputed valuesthe sensor model


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Evidence Grids...

Mapping Using Evidence grids


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represent space as a collection of cells, each with the odds (or probability) that it contains an obstacle

Lab environment

not surelikely obstacle

likely free space

Evidence Grids...

evidence = log2(odds)


lighter areas: lower odds of obstacles being present

darker areas: higher odds of obstacles being present


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Lab environment


likely free space

• The relative locations of the robot within the map are assumed known.• It is important that the robot odometry is correct

Evidence Grids...



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hallway with some open doors

known map and estimated evidence grid

lab spaceCMU -- Hans

Moravec



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Motion Planning in Evidence Grids

Reduces to a search problem within the graph of cells, and the graph is created on the fly.

Search for a minimum-cost path, depending on

•length

•probability of collision


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Path Planning in Evidence Grids



cost = K stepij + collision penalty

•length


(i,j) path (i,j) path

stepij = length of traversing cell (i,j)

What should this be?


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Path Planning in Evidence Grids



cost = K stepij + -log( 1-p )

•length


(i,j) path (i,j) path

Suppose p is the chance of colliding with an obstacle in cell (i,j).

stepij = length of traversing cell (i,j)


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Robot Mapping


Lab environment


likely free space

• The relative locations of the robot within the map are assumed known.• It is important that the robot odometry is correct

• Equally plausible to consider the converse problem...

Evidence Grids...

Given a map of the environment, how do I determine where I am?

“Robot localization problem”


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Localization via mapping

Key idea: use what we already know how to do -- mapping

• Map the area around the robot

• Match the result to the known map of the world


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Localization via mapping

Key idea: use what we already know how to do -- mapping

• Map the area around the robot

• Match the result to the known map of the world


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Thank you!Homework 9 posted on the web

Next class: Localization

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The City College of New York 1 Dr. John (Jizhong) Xiao Department of Electrical Engineering City...

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