Visibility Contractors: Application to Mobile Robot Localization

(1)

Application to Mobile Robot Localization

Rémy G UYONNEAU

J

OURNÉE

MEA

LISA/LARIS - University of Angers (France)

Laurent HARDOUIN Sébastien LAGRANGE

5th December 2013

Rémy Guyonneau - Journée MEA LISA/LARIS - University of Angers (France)

Visibility Contractors - Application to mobile robot localization 1 / 26

(2)

!

- Computer graphics - Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

(3)

! Visibility is studied and used in several fields - Computer graphics

- Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

r 1

r 2

(4)

!

- Computer graphics - Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

r 1

r 2

(5)

1 General Presentation

2 Visibility contractors

3 Multi-robot localization

4 Global localization

5 Conclusion

(6)

1 General Presentation 2 Visibility contractors

3 Multi-robot localization

4 Global localization

5 Conclusion

(7)

normal

General Presentation Visibility contractors Multi-robot localization Global localization Conclusion

Objectives

• Developing a contractor associated to the constraint x ₁ sees x ₂

[ x ₁ ]

[ x ₂ ]

(8)

• Developing a contractor associated to the constraint x ₁ does not see x ₂

[ x ₁ ]

[ x ₂ ]

(9)

normal

Visibility between two points

• ( x ₁ V x ₂ ) "

j

, Seg ( x ₁ ; x ₂ ) \ " j = ;

! "

j

: connected subset of R

ⁿ

^{, with} x

1

62 "

j

and x

2

62 "

j

x ₁

x ₂ x ₃

" j

(10)

normal

Visibility between two points

• ( x ₁ V x ₂ ) "

j

, Seg ( x ₁ ; x ₂ ) \ " j = ;

! "

j

: connected subset of R

ⁿ

^{, with} x

1

62 "

j

and x

2

62 "

j

! Reflexive relation : ( x

1

V x

1

)

"j

! Symmetric relation : ( x

1

V x

2

)

"j

( x

2

V x

1

)

"j

! Non-transitive relation : ( x

₁

V x

₂

)

"j

^ ( x

₂

V x

₃

)

"j

6) ( x

₁

V x

₃

)

"j

x ₁

x ₂ x ₃

" j

(11)

( 1 2 ) "

j

, ( 1 ; 2 ) \ " j = ;

! "

j

: connected subset of R

ⁿ

^{, with} x

1

62 "

j

and x

2

62 "

j

! Reflexive relation : ( x

1

V x

1

)

"j

! Symmetric relation : ( x

1

V x

2

)

"j

( x

2

V x

1

)

"j

! Non-transitive relation : ( x

₁

V x

₂

)

"j

^ ( x

₂

V x

₃

)

"j

6) ( x

₁

V x

₃

)

"j

!

( x

1

V x

3

)

"j

^c

= ( x

1

V x

3

)

"j

x ₁

x ₂ x ₃

" j

(12)

normal

Visibility spaces of a point

• E _"

_j

( x ) = f x _i 2 R ⁿ j ( x _i V x ) "

j

g

x E _"

_j

( x )

" j

(13)

• E _"

_j

( x ) = f x _i 2 R ⁿ j ( x _i V x ) "

j

g

• E Û _"

_j

( x ) = f x _i 2 R ⁿ j ( x _i V x ) _"

j

g

!

E

_"_j

( x )

^c

= E Û

_"_j

( x )

x

E Û _"

_j

( x )

" j

(14)

normal

Visibility spaces of a set

• E _"

_j

(X) = f x _i 2 R ⁿ j 8 x 2 X;( x _i V x ) _"

j

g

X " j

E _"

_j

(X)

(15)

normal

Visibility spaces of a set

• E _"

_j

(X) = f x _i 2 R ⁿ j 8 x 2 X;( x _i V x ) _"

j

g

• E Û _"

_j

(X) = f x _i 2 R ⁿ j 8 x 2 X;( x _i V x ) "

j

g

X " j

Û E _"

_j

(X)

(16)

• E _"

_j

(X) = f x _i 2 R j 8 x 2 X;( x _i V x ) _"

j

g

• E Û _"

_j

(X) = f x _i 2 R ⁿ j 8 x 2 X;( x _i V x ) "

j

g

• E e _"

_j

(X) = f x _i 2 R ⁿ j 9 x ₁ 2 X;9 x ₂ 2 X;( x _i V x ₁ ) _"

j

^ ( x _i V x ₂ ) _"

j

g

X " j

e E _"

_j

(X)

(17)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

" 1

" 2

" 3

(18)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

X

" 1

" 2

" 3

(19)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

X

" 1

" 2

" 3

E _"

₁

(X)

(20)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

X

" 1

" 2

" 3

E _"

₂

(X)

(21)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

X

" 1

" 2

" 3

E _"

₃

(X)

(22)

normal

Consideration of an environment

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

X

" 1

" 2

" 3

E _E (X)

(23)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

(24)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

E Û _"

₁

(X)

(25)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

E Û _"

₂

(X)

(26)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

Û E _"

₃

(X)

(27)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

S n

O

j = 1 E Û _"

_j

(X)

(28)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

2 ^E ^Û _E (X)

(29)

• E = ^S ⁿ _j ₌

^O

₁ " j

• E _E (X) = ^T ⁿ _j ₌

^O

₁ ^E _"

j

(X)

• E Û _E (X) ^S ⁿ _j ₌

^O

₁ ^E ^Û _"

j

(X)

X

" 1

" 2

" 3

E Û _E (X)

(30)

normal

Objectives

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

(31)

normal

Objectives

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

! x ₁ 2 [ x ₁ ] ^et x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) E ) x ₂ 62 Ê ^Û E ([ x ₁ ]) ând x ₁ 62 Ê ^Û E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(32)

normal

Objectives

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

! x ₁ 2 [ x ₁ ] ^et x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) E ) x ₂ 62 Ê ^Û E ([ x ₁ ]) ând x ₁ 62 Ê ^Û E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(33)

normal

Objectives

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

! x ₁ 2 [ x ₁ ] ^et x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) E ) x ₂ 62 Ê ^Û E ([ x ₁ ]) ând x ₁ 62 Ê ^Û E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(34)

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) E

(35)

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) E

! x ₁ 2 [ x ₁ ] ^and x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) _E ) x ₂ 62 Ê _E ([ x ₁ ]) ând x ₁ 62 Ê _E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(36)

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) E

! x ₁ 2 [ x ₁ ] ^and x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) _E ) x ₂ 62 Ê _E ([ x ₁ ]) ând x ₁ 62 Ê _E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(37)

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) _E

• Developing a contractor associated to the constraint ( x ₁ V x ₂ ) E

! x ₁ 2 [ x ₁ ] ^and x ₂ 2 [ x ₂ ]

! ( x ₁ V x ₂ ) _E ) x ₂ 62 Ê _E ([ x ₁ ]) ând x ₁ 62 Ê _E ([ x ₂ ])

[ x ₁ ]

[ x ₂ ]

(38)

1 General Presentation

2 Visibility contractors 3 Multi-robot localization

4 Global localization

5 Conclusion

(39)

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

x e ₁

_j

e ₂

_j

(40)

normal

Visibility of a point

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

E_"s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] = ; _

x

det ( x

i

e

1_j

j e

2_j

e

1_j

) > 0 _

^x

det ( x

i

e

1_j

j x e

1_j

) > 0 _

x

det(x

i

e

2_j

jx e

2_j

) < 0g

x e ₁

_j

e ₂

_j

(41)

normal

Visibility of a point

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

E_"s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] = ; _

x

det ( x

i

e

1_j

j e

2_j

e

1_j

) > 0 _

^x

det ( x

i

e

1_j

j x e

1_j

) > 0 _

x

det(x

i

e

2_j

jx e

2_j

) < 0g

x e ₁

_j

e ₂

_j

(42)

normal

Visibility of a point

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

E_"s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] = ; _

x

det ( x

i

e

1_j

j e

2_j

e

1_j

) > 0 _

^x

det ( x

i

e

1_j

j x e

1_j

) > 0 _

x

det(x

i

e

2_j

jx e

2_j

) < 0g

x e ₁

_j

e ₂

_j

(43)

normal

Visibility of a point

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

E_"s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] = ; _

x

det ( x

i

e

1_j

j e

2_j

e

1_j

) > 0 _

^x

det ( x

i

e

1_j

j x e

1_j

) > 0 _

x

det(x

i

e

2_j

jx e

2_j

) < 0g

x e ₁

_j

e ₂

_j

(44)

normal

Visibility of a point

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

E_"s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] = ; _

x

det ( x

i

e

1_j

j e

2_j

e

1_j

) > 0 _

^x

det ( x

i

e

1_j

j x e

1_j

) > 0 _

x

det(x

i

e

2_j

jx e

2_j

) < 0g

x e ₁

_j

e ₂

_j

(45)

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space

! Non-visible space Û

E_"^s

j

( x ) =f x

i

2 R

²

j

[ x

i

[ x ] \ [ e

1_j

[ e

2_j

] 6= ; ^

^x

det ( x

i

e

1_j

j e

2_j

e

1_j

) 0 ^

x

det ( x

i

e

1_j

j x e

1_j

) 0 ^

x

det ( x

i

e

2j

j x e

2j

) 0 g

x e ₁

_j

e ₂

_j

(46)

normal

Visibility of a segment

[ x ]

(47)

normal

Visibility of a segment

[ x ]

x ₁ x ₂

x ₃

x ₄

(48)

" _j = ( 1

j

; 2

j

)

e ₁

_j

e ₂

_j

x ₁

x ₂

(49)

normal

Visibility of a segment

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space - E _"

^s

j

( Seg ( x ₁ ; x ₂ ))

E_"s

j(Seg(x1;x2)) = fxi2R²j

(x1= x2) ^ x1det(x_i e₁_jje₂_j e₁_j) >0_

x1det(xi e1jjx1 e1j) >0^ x2det(xi e1jjx2 e1j) >0_ x1det(x_i e₂_jjx₁ e₂_j) <0^ x2det(x_i e₂_jjx₂ e₂_j) <0

_ (x1= x2) ^

e1det(x_i e₁_jjx₁ e₁_j) >0_ e1det(x_i e₁_jjx₂ e₁_j) <0

^ e2det(xi e2jjx1 e2j) >0_ e2det(xi e2jjx2 e2j) <0

_ [x_i[x₁[x₂] \ [e₁_j[e₂_j] = ;

g:

e ₁

_j

e ₂

_j

x ₁

x ₂

(50)

" _j = ( 1

j

; 2

j

)

! Visible space - E _"

^s

j

( Seg ( x ₁ ; x ₂ ))

! Non-visible space - E Û _"

^s

j

(Seg ( x ₁ ; x ₂ )) Û E _"

^s

j

( Seg ( x ₁ ; x ₂ )) = ^E ^Û _"

^s_j

( x ₁ )\ ^E ^Û _"

^s_j

( x ₂ )

e ₁

_j

e ₂

_j

x ₁

x ₂

(51)

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

e ₁

_j

e ₂

_j

[ x ]

(52)

normal

Visibility of a box

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space - E _"

^s

j

([ x ])

e ₁

_j

e ₂

_j

[ x ]

(53)

• With a segment as obstacle - " ^s _j = Seg ( e ₁

_j

; e ₂

_j

)

! Visible space - E _"

^s

j

([ x ])

! Non-visible space - E Û _"

^s

j

([ x ])

e ₁

_j

e ₂

_j

[ x ]

(54)

normal

With a convex polygon as obstacle

• Convex polygon : set of segments

! "

^pj

= S

ⁿPj k=1

"

^sk

e ₂

_j

e ₁

_j

[ x ]

e ₃

_j

(55)

normal

With a convex polygon as obstacle

• Convex polygon : set of segments

! "

^pj

= S

ⁿPj k=1

"

^sk

! ^E

_"^p

j

([ x ]) = T

ⁿPj k=1

E

_"^s

k

([ x ])

e ₂

_j

e ₁

_j

[ x ]

e ₃

_j

(56)

• Convex polygon : set of segments

! "

^pj

= S

ⁿPj k=1

"

^sk

! ^E

_"^p

j

([ x ]) = T

ⁿPj k=1

E

_"^s

k

([ x ])

! Û E

_"^p

j

([ x ]) = S

ⁿPj k=1

Û E

_"^s

k

([ x ])

e ₂

_j

e ₁

_j

[ x ]

e ₃

_j

(57)

1 General Presentation

2 Visibility contractors

3 Multi-robot localization

4 Global localization

5 Conclusion

(58)

normal

Problem presentation

• Team of robots

(59)

normal

Problem presentation

• Team of robots

• Initial poses known

(60)

normal

Problem presentation

• Team of robots

• Initial poses known

(61)

normal

Problem presentation

• Team of robots

• Initial poses known

(62)

normal

Problem presentation

• Team of robots

• Initial poses known

(63)

normal

Problem presentation

• Team of robots

• Initial poses known

! The robots are drifting

(64)

normal

Problem presentation

• Team of robots

• Initial poses known

• Known environment

(65)

• Team of robots

• Initial poses known

• Known environment

• Is it possible to avoid the drifting of the robot by using a boolean information : the visibility between the robots ?

(66)

normal

The robots

• Bounded error context

! q _i _; ₀ 2 [ q _i _; ₀ ]

(67)

normal

The robots

• Bounded error context

• Evaluation of the orientation by using a compass

! i ; k 2 [ i ; k ]

Compass CMPS10 Compass SEN12753P

(68)

normal

The robots

• Bounded error context

• Evaluation of the orientation by using a compass

• Inter-robot communication

! At each time step k each robot knows the position

estimation of all the robots

(69)

• Bounded error context

• Evaluation of the orientation by using a compass

• Inter-robot communication

• Boolean measurements

! r ₁ sees r ₂ , ( x ₁ V x ₂ ) _E

! r ₁ does not see r ₂ , ( x ₁ V x ₂ ) _E

! z _i _; _k = f 0 ; 1 ; ; 1 g

• 0 : the robot r

i

does not see the first robot

• 1 : the robot r

i

sees the second robot

• • 1 : the robot r

_i

sees the last robot

(70)

normal

Environment characterisations

• Environment E

! E = ^S ⁿ _j ₌

^O

₁ " j

! Sets of convex polygons

" 1 " 2

" 4

" 3

(71)

normal

Environment characterisations

• Environment E

• Inner characterisation E

! E E

" 1 " 2

" 4

" 3

" 1

" ₂ " 3

" 4

" ₅

" 6

" 7

" 8

(72)

normal

Environment characterisations

• Environment E

• Inner characterisation E

• Outer characterisation E ⁺

! E E ⁺

" ⁺ ₁ " ⁺ ₂

" ⁺ 3 " ⁺ 4

" ⁺ 5

" ⁺ 6

" ⁺ ₇

" 1 " 2

" 4

" 3

(73)

normal

Environment characterisations

• Environment E

• Inner characterisation E

• Outer characterisation E ⁺

• Environment/Characterisations

x

1

2 Û E

_E+

( x

2

) ^, x

1

62 ^E

_E+

( x

2

)

x

2

2 Û E

_E₊

( x

1

) ^, x

2

62 ^E

_E+

( x

1

)

(74)

normal

Environment characterisations

• Environment E

• Inner characterisation E

• Outer characterisation E ⁺

• Environment/Characterisations

! r ₁ sees r ₂ ) ( x ₁ V x ₂ ) _E

x

1

2 ^E

_E

( x

2

) ^, x

1

62 E Û

_E

( x

2

) x

2

2 ^E

_E

( x

1

) ^, x

2

62 E Û

_E

( x

1

)

"

¹

"

²

"

4

"

3

"

1

"

2

"

3

"

4

"

5

"

6

"

7

"

8

x

1

x

2

(75)

• Environment E

• Inner characterisation E

• Outer characterisation E ⁺

• Environment/Characterisations

! r ₁ sees r ₂ ) ( x ₁ V x ₂ ) _E

x

1

2 ^E

_E

( x

2

) ^, x

1

62 E Û

_E

( x

2

) x

2

2 ^E

_E

( x

1

) ^, x

2

62 E Û

_E

( x

1

)

! r ₁ does not see r ₂ ) ( x ₁ V x ₂ ) _E

⁺

x

1

2 Û E

_E+

( x

2

) ^, x

1

62 ^E

_E+

( x

2

) x

2

2 Û E

_E₊

( x

1

) ^, x

2

62 ^E

_E+

( x

1

)

"

⁺1

"

⁺2

"

⁺3

"

⁺4

"

⁺5

"

⁺6

"

⁺7

"

1

"

2

"

3

"

4

x

1

x

2

(76)

• Simulator

(77)

1 General Presentation

2 Visibility contractors

3 Multi-robot localization

4 Global localization 5 Conclusion

(78)

x 1

x ₂

(79)

x 1

x ₂

(80)

q ₁

q ₂

x 1

x ₂

Two poses are consistent with the constraints

(81)

q ₁

q ₂

x 1

x ₂

How to process the measurement intersection constraint ?

(82)

normal

Considering an original constraint

• Presentation of the constraint

(83)

normal

Considering an original constraint

• Presentation of the constraint

intersection !

! the measurement must not intersect the obstacle

(84)

normal

Considering an original constraint

• Presentation of the constraint

! the measurement must not intersect the obstacle

(85)

• Presentation of the constraint

• Constraint formalisation

(86)

normal

Considering an original constraint

• Presentation of the constraint

• Constraint formalisation

w

2_i

x

2

( 0 ; 0 )

obstacle

x

w

i

x

1

w

1i

! Visibility relations between the robot and all the detected obstacles

(87)

• Presentation of the constraint

• Constraint formalisation

w

2_i

x

2

( 0 ; 0 ) x

w

i

x

1

w

1i

! Visibility relations between the robot and all the detected obstacles

! 8" j 2 E;8 i ;( x V w _i ) "

j

(88)

normal

Environment characterization

• Contraction over a visibility information

• Inner characterisation

(89)

normal

Environment characterization

• Contraction over a visibility information

• Inner characterisation

[ w _i ]

" 1

" 2

(90)

normal

Environment characterization

• Contraction over a visibility information

• Inner characterisation

[ w _i ]

" 1

" 2

(91)

normal

Environment characterization

• Contraction over a visibility information

• Inner characterisation

[ w _i ]

" 1

" 2

(92)

• Contraction over a visibility information

• Inner characterisation

• Environment characterisation

" 1 " 2

" 4

" 3

" ₁ " 2

" 3

(93)

Considered environment

(94)

Inner characterisation

(95)

Considered measurements

(96)

Result without visibility

(97)

Result with visibility

(98)

! Original formalisation of the visibility information

! Optimal visibility contractors

! Two application to mobile robot localization Multi-robot localization

- Avoid the drifting of the robots

- Results are critical over the number of robots and the environment topology

- The contractors can be added to classical localization approach Original measurement intersection constraint

- Improvement of the localization results - Avoidance of symmetries

- Those contractors could be useful to other application

Visibility Contractors: Application to Mobile Robot Localization

Application to Mobile Robot Localization

Rémy G UYONNEAU

J

MEA

5th December 2013

!

- Computer graphics - Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

! Visibility is studied and used in several fields - Computer graphics

- Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

r 1

r 2

!

- Computer graphics - Telecommunication - Robotics...

! Usually associated to bearing or ranging data

! We consider the visibility as a boolean information - Application to mobile robot localization

r 1

r 2

1 General Presentation

2 Visibility contractors

3 Multi-robot localization

4 Global localization

5 Conclusion

1 General Presentation 2 Visibility contractors

3 Multi-robot localization

4 Global localization

5 Conclusion

normal

Objectives

• Developing a contractor associated to the constraint x 1 sees x 2

[ x 1 ]

[ x 2 ]

• Developing a contractor associated to the constraint x 1 does not see x 2

[ x 1 ]

[ x 2 ]

normal

Visibility between two points

• ( x 1 V x 2 ) "

, Seg ( x 1 ; x 2 ) \ " j = ;

! "

: connected subset of R

, with x

62 "

and x

62 "

x 1

x 2 x 3

" j

normal

Visibility between two points

• ( x 1 V x 2 ) "

, Seg ( x 1 ; x 2 ) \ " j = ;

! "

: connected subset of R

, with x

62 "

and x

62 "

! Reflexive relation : ( x

V x

)

! Symmetric relation : ( x

V x

)

( x

V x

)

! Non-transitive relation : ( x

V x

)

^ ( x

V x

)

6) ( x

V x

)

• Developing a contractor associated to the constraint x ₁ sees x ₂

[ x ₁ ]

[ x ₂ ]

• Developing a contractor associated to the constraint x ₁ does not see x ₂

[ x ₁ ]

[ x ₂ ]

• ( x ₁ V x ₂ ) "

, Seg ( x ₁ ; x ₂ ) \ " j = ;

^{, with} x

x ₁

x ₂ x ₃

• ( x ₁ V x ₂ ) "

, Seg ( x ₁ ; x ₂ ) \ " j = ;

^{, with} x

x ₁

x ₂ x ₃

^{, with} x

x ₁

x ₂ x ₃

• E _"

( x ) = f x _i 2 R ⁿ j ( x _i V x ) "

x E _"

• E _"

( x ) = f x _i 2 R ⁿ j ( x _i V x ) "

• E Û _"

( x ) = f x _i 2 R ⁿ j ( x _i V x ) _"

E Û _"

• E _"

(X) = f x _i 2 R ⁿ j 8 x 2 X;( x _i V x ) _"

E _"