Introduction Presentation of the problem The proposed method Results
A Visibility Information for Multi-Robot Localization
Rémy G UYONNEAU - Sébastien L AGRANGE - Laurent H ARDOUIN
University of Angers - LISA/LARIS4
thOctober 2013
The problem
A team of robots q 1,0
q 3,0
q 5,0
q 4,0
q 2,0
Introduction Presentation of the problem The proposed method Results
The problem
An environment q 1,0
q 3,0
q 5,0
q 4,0
q 2,0
The problem
The robots are drifting
q 1,1 q 2,1
q 5,1
q 3,1
q 4,1
Introduction Presentation of the problem The proposed method Results
The problem
The robots are drifting q 1,2
q 2,2
q 5,2
q 4,2
q 3,2
The problem
q 1,2
q 2,2
q 5,2
q 4,2
q 3,2
Is it possible to avoid the drifting of the robots using
a boolean visibility information ?
Introduction Presentation of the problem The proposed method Results
Summary
1 Presentation of the problem
2 The proposed method
3 Results
The environment
• An environment E is a set of m obstacles : E = S m j=1 ε j
• An obstacle ε j is a connected subset of R 2
ε 1
ε 2 ε 3
ε 4
Introduction Presentation of the problem The proposed method Results The robots
The robots
• A team R is a set of robots
• A robot r i is characterized by the discrete time dynamic equation : q i,k+1 = f (q i,k ,u i,k )
• q i,k = (x i,k , θ i,k )
• Evaluation of u i,k using odometry
• Evaluation of θ i,k using a compass
• Measurements : the visibility between the robots
The visibility information
ε 1
ε 2 ε 3
ε 4 r 1
r 2
r 3
Introduction Presentation of the problem The proposed method Results The robots
The visibility information
ε 2
ε 1
ε 3
ε 4 r 1
r 2
r 3
(r 2 V r 3 ) E ≡ (r 3 V r 2 ) E
The visibility information
ε 1
ε 2 ε 3
ε 4 r 1
r 2
r 3
(r 1 V r 2 ) E ≡ (r 2 V r 1 ) E
Introduction Presentation of the problem The proposed method Results The robots
The visibility information
ε 2
ε 1
ε 3
ε 4 r 1
r 2
r 3
(r 1 V r 3 ) E ≡ (r 3 V r 1 ) E
The robots
• A team R is a set of robots
• A robot r i is characterized by the discrete time dynamic equation : q i,k+1 = f (q i,k ,u i,k )
• q i,k = (x i,k , θ i,k )
• Evaluation of u i,k using odometry
• Evaluation of θ i,k using a compass
• Measurements : the visibility between the robots
• z i,t = {0, 1, 1,· · · , 0}
Introduction Presentation of the problem The proposed method Results The approach
The approach
• Pose tracking problem
• Robot initial poses are known
• Bounded error context
• x i,0 ∈ [x i,0 ]
• u i,0 ∈ [u i,0 ]
• θ i,k ∈ [θ i,k ]
Set-membership drifting
ε 2
ε 1
ε 3
ε 4
x i,0
Introduction Presentation of the problem The proposed method Results The approach
Set-membership drifting
ε 2
ε 1
ε 3
ε 4
[x i,0 ]
Set-membership drifting
ε 2 ε 3
ε 4 [x i,0 ]
[x i,1 ]
[x i,1 ] = f([x i,0 ], [u i,0 ])
Introduction Presentation of the problem The proposed method Results The approach
Set-membership drifting
ε 1
ε 3
ε 4 [x i,0 ]
[x i,1 ]
[x i,2 ]
[x i,2 ] = f ([x i,1 ],[u i,1 ])
Set-membership drifting
ε 1
ε 3
ε 4 [x i,0 ]
[x i,1 ]
[x i,2 ]
Introduction Presentation of the problem The proposed method Results The objective
Objective
How to contract a box over a visibility (non-visibility) information ?
Objective
ε 3
ε 4 [x i,k ]
[x j,k ]
Introduction Presentation of the problem The proposed method Results The objective
Objective
ε 3
ε 4 [x i,k ]
[x j,k ]
(r i V r j ) E
Objective
(r i V r j ) E
ε 3
ε 4 [x i,k ]
[x j,k ]
Introduction Presentation of the problem The proposed method Results The objective
Objective
(r i V r j ) E
ε 3
ε 4 [x i,k ]
[x j,k ]
Objective
ε 3
ε 4 [x i,k ]
[x j,k ]
Difficult to evaluate with any obstacles
Introduction Presentation of the problem The proposed method Results (non-)Visibility contractors
Summary
1 Presentation of the problem
2 The proposed method
3 Results
(non-)Visibility contractors
• A box (interval vector) ≡ convex polygon
• Considering polygons as obstacles
→ Visible and non-visible spaces defined by line equations
ε 3
ε 4
[x i,k ]
Introduction Presentation of the problem The proposed method Results (non-)Visibility contractors
(non-)Visibility contractors
• A box (interval vector) ≡ convex polygon
• Considering polygons as obstacles
→ Visible and non-visible spaces defined by line equations
ε 3
ε 4
[x i,k ]
(non-)Visibility contractors
• A box (interval vector) ≡ convex polygon
• Considering polygons as obstacles
→ Visible and non-visible spaces defined by line equations
ε 3
ε 4
[x i,k ]
Introduction Presentation of the problem The proposed method Results (non-)Visibility contractors
The environment characterizations
ε 1
ε 2 ε 3
ε 4
An environment E
The environment characterizations
ε 1
ε 2 ε 3
ε 4 An inner characterization
E − ⊆ E
Introduction Presentation of the problem The proposed method Results (non-)Visibility contractors
The environment characterizations
ε 1
ε 2 ε 3
ε 4 An outer characterization
E + ⊇ E
Environment/characterizations
• (r i V r j ) E ⇒ (r i V r j ) E
−• (r i V r j ) E ⇒ (r i V r j ) E
+Introduction Presentation of the problem The proposed method Results Environment/characterizations
Environment/characterizations - example
ε 2
ε 1
ε 3
ε 4 x i,k
x j,k
Environment/characterizations - example
ε 1
ε 3
ε 4 x i,k
x j,k
(r i V r j ) E
Introduction Presentation of the problem The proposed method Results Environment/characterizations
Environment/characterizations - example
ε 1
ε 3
ε 4 x i,k
x j,k
(r i V r j ) E ⇒ (r i V r j ) E
−Environment/characterizations - example
ε 2 ε 3
ε 4 x i,k
x j,k
(r i V r j ) E
Introduction Presentation of the problem The proposed method Results Environment/characterizations
Environment/characterizations - example
ε 2 ε 3
x i,k x j,k
(r i V r j ) E ⇒ (r i V r j ) E
+Video
Introduction Presentation of the problem The proposed method Results Conclusion
