Set-Membership Approach to the Kidnapped Robot Problem

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Set-Membership Approach to the Kidnapped Robot Problem

Benoît Desrochers, Simon Lacroix and Luc Jaulin IROS 2015, Hamburg.

Presentation available at https://youtu.be/oFvl0_NQpuc

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1 OMNE

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We consider the bounded-error estimation problem f_i (p₎ _∈ _[y_i_]

where

[y_i] ⊂ R ^{is the} ith collected interval data,

p _∈ Rⁿ is the parameter vector to be estimated, p^∗ is the true value for p_.

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The set of all p consistent with the ith measurement y_i is P_i ⁼ ^f_i⁻¹ ^([y_i^])^.

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Inlier or outlier ?

[y_i] is an inlier if p^∗ _∈ P_i^. It is an outlier if p^∗ _∈_/ P_i^.

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OMNE (Outlier Minimal Number Estimator) [Walter and Lahanier, 1987]

λ(p_{) =} _card_{i _| p _∈_/ P_i^} q^∗ = min

p λ(p₎ P ⁼ ^λ⁻¹ ^(q^∗⁾ ^.

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2 Set formulation

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q-relaxed intersection

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OMNE corresponds to

P_i ⁼ ^f_i⁻¹ ^([y_i^]) P^{q} ⁼

{q}

P_i

q^∗ = min q | P^{q} ⁼ ^∅ P ⁼ P^{q^∗^}^.

Outer GOMNE solves the problem with intervals and a local search.

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3 Illustrative example

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Range only localization with 3 landmarks using interval analysis.

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P^{0}^,P^{1}^, P^{2}

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4 DEM

The map is described by M ^{= (x}_i^{, y}_i^{, z}_i⁾ ^{, i} ^≥ ^1.

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Example of digital map

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Aerial orthoimage built from UAV

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Velodyne Lidar

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Original range data

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Down sampling

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Extraction of vertical shapes

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5 Localization

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p = (x, y, ψ) P_i ⁼ ^f_i⁻¹ ⁽M⁾ P^{q} ⁼

{q}

P_i

q^∗ = min q | P^{q} ⁼ ^∅ P ⁼ P^{q^∗^}^.

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Contribution of the paper: We are able to compute the global minimum q^∗.

We used a MonteCarlo search to speed up the calculus.

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