Mobile Robot Localization: A Set-Membership Approach

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Interval Analysis The Global Localization Problem The Proposed Method Experimental Results

Mobile Robot Localization: A Set-Membership Approach

Rémy G UYONNEAU - Sébastien L AGRANGE - Laurent H ARDOUIN - Philippe L UCIDARME

University of Angers - LISA

January 24 2013

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Introduction

◦ Robot localization is an important issue of mobile robotics

◦ The robotics challenge called CAROTTE

¹

◦ The Simultaneous Localization And Mapping (SLAM) and the global localization problems

◦ In this presentation a set membership approach will be considered to deal with the global localization problem

1. CArtographie par ROboT d’un TErritoire (Robot Land Mapping) organized by the

french ANR (National Research Agency) and the DGA (french army)

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Summary

1 Interval Analysis

2 The Global Localization Problem

3 The Proposed Method

4 Experimental Results

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Summary

1 Interval Analysis Interval Analysis

Constraint Satisfaction Problem Q-Relaxed Intersection

2 The Global Localization Problem

3 The Proposed Method

4 Experimental Results

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Interval Analysis

Definitions

An Interval Vector

An interval vector, or a box [ p ] is defined as a closed subset of R ⁿ

[ p ] = ([x],[y],· · · ) = ([x, x],[y,y],· · · ) ⊂ R ⁿ

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Interval Analysis

Interval Arithmetic

Definition

Any real number elementary operators such as +, −, ×, ÷ and functions such as exp , sin , sqr , sqrt , can be easily extended to intervals

Example

Be [x] and [y] two intervals, we define

→ [x] + [y] = [x + y,x + y]

→ [x] × [y] = [min(xy, xy,xy,xy), max(xy,xy,xy,xy)]

Be f ∈ {cos,sin, exp,tan, log, sqrt, sqr} , we define

→ f ([x]) = {f (x) with x ∈ [x]}

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Constraint Satisfaction Problem

Definitions

Constraint Satisfaction Problem (CSP)

A CSP is defined by three sets. A set of variables V ^{, a set of}

domains D for those variables and a set of constraints C ^connecting

the variables together

CSP :







V ⁼ ^{x 1 ,x 2 ,· · · , x n } D ⁼ ^{[x 1 ],[x ₂ ], · · · , [x n ]}

C ⁼ ^{c 1 ,c ₂ ,· · · , c m }







Constraint propagation

This kind of problem can be solved using constraint propagation.

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

-20 0 20 40 60 80 100

-10 -8 -6 -4 -2 0 2 4 6 8 10

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

20 40 60 80 100

x ₂ = x ² ₁

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

-20 0 20 40 60 80 100

-10 -8 -6 -4 -2 0 2 4 6 8 10

x ₂ = − 2x 1 + 1

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

20 40 60 80 100

x ₁ = (1 − x ₂ )/2

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

-20 0 20 40 60 80 100

-10 -8 -6 -4 -2 0 2 4 6 8 10

|x ₁ | = √

x ₂

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

20 40 60 80 100

|x ₁ | = √

x ₂

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

-20 0 20 40 60 80 100

-10 -8 -6 -4 -2 0 2 4 6 8 10

|x ₁ | = √

x ₂

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

20 40 60 80 100

x ₂ = − 2x 1 + 1

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

-20 0 20 40 60 80 100

-10 -8 -6 -4 -2 0 2 4 6 8 10

|x ₁ | = √

x ₂

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Example

Example of CSP

Let x ₁ and x ₂ be two variables with [x ₁ ] = [−10,10] and [x 2 ] = [−20, 100] there domains. We consider the constraints

c ₁ : x ₂ = x ² ₁ and c ₂ : x ₂ = − 2x 1 + 1

20 40 60 80 100

The result of the

constraint propagation

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the

[ x i ] excepted q at most

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the [ x i ] excepted q at most

We consider five interval

vectors : [ x ₁ ],· · · , [ x ₅ ]

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the [ x i ] excepted q at most

{0}

T

i=1,···,5

[ x i ]

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the [ x i ] excepted q at most

{1}

T

i=1,···,5

[ x i ]

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the [ x i ] excepted q at most

{2}

T

i=1,···,5

[ x i ]

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Q-Relaxed Intersection

Definition

Let be m interval vectors [ x ₁ ], · · · , [ x m ] of R ⁿ . The q-relaxed intersection

{q}

T ([ x i ]) is defined as all the x ∈ R ⁿ that are in all of the [ x i ] excepted q at most

{3}

T

i=1,···,5

[ x i ]

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Summary

1 Interval Analysis

2 The Global Localization Problem The Robot

The Environment The Objective

3 The Proposed Method

4 Experimental Results

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The Robot

The considered robot

We consider a mobile wheeled robot with a LIDAR

^a

sensor. Its pose is defined by p = (x,y,θ) , with (x,y) its localization and θ its orientation

a. Light Detection And Ranging

The measurements The sensor provides a set of measurements :

= { d = (d ,d )} , i = 1, · · · ,n

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The Robot

Robot

Obstacle

The robot pose

Robot

Obstacle

A measurement d i

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The Environment

The map

The known environment E ∈ R ² is discretized with a resolution δ x , δ y

and this lead to a grid G composed of n × m cells (i,j) . At each cell (i, j) is associated g _i,j ∈ {0, 1}

g _i,j =

( 1 if there is an obstacle in the cell (i,j)

0 else

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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The Objective

Hypotheses

◦ Bounded error context

◦ Outliers can be considered

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Summary

1 Interval Analysis

2 The Global Localization Problem

3 The Proposed Method The Measurement CSP The Localization Algorithm

4 Experimental Results

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The Measurement CSP

The Context

Let [ p ] = ([x],[y],[θ]) be an initial domain that encloses the robot pose

(x, y,θ) and D = { d i },i = 1,· · · , n a set of n telemeter measurements

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The Measurement CSP

The Measurements Coordinates

The coordinates (w i

_x

, w i

_y

) of an obstacle in the map are defined by w i

_x

w i

_y

=

cos(θ) sin(θ)

−sin(θ) cos(θ)

d i

_x

d i

_y

+

x y

Robot ^Obstacle

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The Measurement CSP

The Robot-Measurement Distance

|| d i || ² = (x − w _i

_x

) ² + (y − w _i

_y

) ²

Robot ^Obstacle

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The Measurement CSP

The Measurement-Measurement Distance

|| d i − d j || ² = (w i

x

− w _j

_x

) ² + (w i

y

− w _j

_y

) ²

Robot ^Obstacle

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The Measurement CSP

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The Measurement CSP

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The Measurement CSP

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The Measurement CSP

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The Measurement CSP

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The Measurement CSP

Mobile Robot Localization: A Set-Membership Approach