Interval Analysis for Global Localization Problem …...Interval Analysis for Global Localization Problem using Range Sensors R emy GUYONNEAU, S ebastien LAGRANGE, Laurent HARDOUIN

The Global Localization ContextThe Proposed Method

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Interval Analysis for Global Localization Problemusing Range Sensors

Remy GUYONNEAU, Sebastien LAGRANGE, LaurentHARDOUIN and Philippe LUCIDARME.

Laboratoire d’Ingenierie des Systemes Automatises (LISA),University of Angers,

France.

June 13, 2011

R. Guyonneau, S. Lagrange, L. Hardouin, P. Lucidarme Interval Analysis for Global Localization Problem using Range Sensors 1


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Introduction

F Robot localization is an important issue of mobile robotics.

F Robotics challenge CAROTTE1

F Simultaneous Localization And Mapping (SLAM) and GlobalLocalization problems.

F In this presentation a set membership approach will beconsidered to deal with the global localization problem.

1CArtographie par ROboT d’un TErritoire (Robot Land Mapping) organizedby the french ANR (National Research Agency).



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Outline

1 The Global Localization Context

2 The Proposed MethodThe Static localizationThe Global Localization Algorithm

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The RobotThe EnvironmentThe Objective

The Global Localization Context



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The Robot

The considered robot

We consider a mobile wheeled robot with a LIDARa sensor. Its

pose is defined by p =

xyθ

, with (x , y) its localization and θ

its orientation.

aLight Detection And Ranging

The measurements

The sensor provides a set of nmeasurements:

D =

d0 = (d0x , d0y )...

dn = (dnx , dny )



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The Robot

Robot Obstacle

Figure 1: The robot’s pose.

Robot Obstacle

Figure 2: A measurement di .



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The Environment

The map

The known environment E ∈ R2 is discretized with a resolution δx ,δy and this lead to a grid G composed of n ×m cells (i , j). Ateach cell (i , j) is associated gi ,j ∈ {0, 1}:

gi ,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 are considered,

Figure 3: Measurements from thesensor.



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The Objective

Figure 4: We have a grid map.

Hypotheses






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The Objective


Hypotheses






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The Objective


Hypotheses



Figure 6: Initial domain.



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The Objective



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The Objective



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The Objective



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The Objective



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The Objective

The robot is localized

All the measurements fit withthe map.



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The Objective

The robot is localized

Except for one outlier.



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The Static localizationThe Global Localization Algorithm

The Proposed Method



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The Static localization

The Context

Let [p] =

[x ][y ][θ]

be an initial domain that encloses the robot’s

pose p =

xyθ

and D =

d0...dn

a set of n measurements.



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The Robot-Measurement Distance

||di ||2 = (x − wix )2 + (y − wiy )2.

Robot Obstacle



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The Measurement-Measurement Distance

||di − dj ||2 = (wix − wjx )2 + (wiy − wjy )2.

Robot Obstacle



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The Measurements Coordinates

The coordinates (wix ,wiy ) of an obstacle in the map are defined by:(wix

wiy

)=

(cos(θ) sin(θ)−sin(θ) cos(θ)

)(dixdiy

)+

(xy

)

Robot Obstacle



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The map constraint.

We define cG the constraintwhich says that themeasurement has to beconsistent with the map.



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The map contractor.

We define CG the contractorof the constraint cG.



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Considered Constraint Satisfaction Problem

V = {x , y , θ,di = (dix , diy )T ,wix ,wiy with i = 1, · · · , n},

D = {[x ] = [−∞,+∞], [y ] = [−∞,+∞], [θ] = [0, 2π],[wix ] = [−∞,+∞], [wiy ] = [−∞,+∞],[dix ] =obtained from the sensor,[diy ] =obtained from the sensor}.

C =

cwix: dix cos(θ) + diy sin(θ) + x

cwiy: −dix sin(θ) + diy cos(θ) + y

cdi : ||di ||2 = (x − wix )2 + (y − wiy )2

cdi,j : ||di − dj ||2 = (wix − wjx )2 + (wiy − wjy )2

cG : to be consistent with the map (using CG)



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The Global Localization Algorithm



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Assumption

We assume u = (ur , ul) the knowledge of the moving of the robot,with ur the speed of the right wheel and ul the speed of the leftwheel.



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Prediction Equation

f (pk ,uk) = pk+1 =

xk+1

yk+1

θk+1

=

xk +ukr +ukl

2 cos(θk)

yk +ukr +ukl

2 sin(θk)

θk +ukr−ukl

2

.

Retro-propagation Equation

f −1(pk+1,uk) = pk =

xk+1 −ukr +ukl

2 cos(θk+1 −ukr−ukl

2 )

yk+1 −ukr +ukl

2 sin(θk+1 −ukr−ukl

2 )

θk+1 −ukr−ukl

2

.



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Three Constraints

To be consistent with a data set and the map: The staticlocalization.

Retro-propagation : pk−1 = f −1(pk ,uk−1),

Prediction : pk+1 = f (pk ,uk),

with pk−1 ∈ Pk−1,pk ∈ Pk ,pk+1 ∈ Pk+1,uk−1 ∈ [uk−1] anduk ∈ [uk ].



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Demonstration

Simulation parameters

I A 1000× 1000 cells occupancy grid map (10× 10 meters).

I Each measurement has a bounded error of 10 centimetres anda max range dmax = 3 meters.




Demonstration



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Conclusion

F In this presentation we have seen that interval analysis couldbe used to solve the global localization problem.

F This method uses a discrete map so the time processingdepends of the size of the grid.

F The simulation results are promising and this method can beefficient in a real context.



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ConclusionFuture work

F Experimental tests after the CAROTTE challenge.

F The using of topological maps.



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Questions

??R. Guyonneau, S. Lagrange, L. Hardouin, P. Lucidarme Interval Analysis for Global Localization Problem using Range Sensors 30

Documents

Interval Analysis for Global Localization Problem …...Interval Analysis for Global Localization Problem using Range Sensors R emy GUYONNEAU, S ebastien LAGRANGE, Laurent HARDOUIN