Robust Localisation Using Separators

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Robust Localisation Using Separators

COPROD’14

Würzburg, September 21, 2014 Luc Jaulin and Benoît Desrochers

ENSTA Bretagne, IHSEV, OSM, LabSTICC.

http://www.ensta-bretagne.fr/jaulin/

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1 Contractors

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C([x_]) _⊂ _[x_] (contractance) [x_] _⊂ _[y_] _{⇒ C}_([x_]) _{⊂ C}_([y_]) (monotonicity)

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Inclusion

C₁ ⊂ C₂ ⇔ ∀[x] ∈ IRⁿ_, _C₁_([x]) ⊂ C₂([x]).

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A set S _is _consistent _with _C _{(we write} S _{∼ C}_{) if} C([x]) ∩ S _{= [}x] ∩ S_.

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C is minimal if

S _{∼ C}

S _{∼ C}₁ ^{⇒ C ⊂ C}¹^.

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The negation ¬C of C is deﬁned by

¬C ([x]) = {x _∈ [x] | x ∈ C/ ([x_])}. It is not a box in general.

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2 Separators

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A separator S is pair of contractors Sⁱⁿ,S^out such that Sⁱⁿ([x]) ∪ S^out([x]) = [x] (complementarity).

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A set S _is _consistent _with _S _{(we write} S _{∼ S}_{), if} S _{∼ S}^out _and S _{∼ S}ⁱⁿ_.

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The remainder of S is

∂S([x]) = Sⁱⁿ([x]) ∩ S^out([x]).

∂S is a contractor, not a separator.

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We have

¬Sⁱⁿ([x]) ∪ ¬S^out([x]) ∪ ∂S([x]) = [x] . Moreover, they do not overlap.

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¬Sⁱⁿ([x]), ¬S^out([x]) and ∂S([x])

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Inclusion

S₁ ⊂ S₂ ⇔ S₁ⁱⁿ ⊂ S₂ⁱⁿ and S₁^out ⊂ S₂^out. Here ⊂ means more accurate.

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S is minimal if

S₁ ⊂ S ⇒ S₁ = S. i.e., if Sⁱⁿ and S^out are both minimal.

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3 Paver

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We want to compute X⁻_,X⁺ _{such that} X⁻ _⊂ X _⊂ X⁺_.

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Algorithm Paver(in: [x_],_S_{; out:} X⁻_, X⁺₎ 1 L := {[x_]}_;

2 Pull [x] from L;

3 [xⁱⁿ],[x^out] = S([x]);

4 Store [x_]_\[xⁱⁿ_] _into X⁻ and also into X⁺_; 5 [x] = [xⁱⁿ] ∩ [x^out];

6 If w([x]) < ε, then store [x] in X⁺_,

7 Else bisect [x] and push into L the two childs 8 If L = ∅, go to 2.

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For the implementation, the paving is represented by a binary tree.

The ith node of the tree contains two boxes: [xⁱⁿ](i) and [x^out](i).

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The binary tree is said to be minimal if for any node i₁ with brother i₂ and father j, we have









(i) [xⁱⁿ](i₁) = ∅, [x^out](i₁) = ∅

(ii) [xⁱⁿ_](j) _∩ _[x^out_{](j) =} _[xⁱⁿ_](i₁₎ _∩ _[x^out_](i₁₎

⊔ [xⁱⁿ](i₂) ∩ [x^out](i₂)

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4 Algebra

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Contractor algebra does not allow decreasing operations, i.e.,

∀i, C_i ⊂ C_i^′ ⇒ E (C₁,C₂, . . .) ⊂ E C₁^′ ,C₂^′ , . . . .

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The complementary C of a contractor C, the restriction C₁\C₂, etc. cannot be deﬁned.

Separators extend the operations allowed for contractors to non monotonic expressions.

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The complement of S = Sⁱⁿ,S^out is S = S^out,Sⁱⁿ .

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The exponentiation of S = Sⁱⁿ,S^out is deﬁned as S⁰ = {⊤,⊤}

S^k+1 = ¬S^k ^out ⊔ (Sⁱⁿ ◦ ∂S^k),¬S^k ⁱⁿ ⊔ (S^out ◦ ∂S^k) .

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Example.

S¹ = { ¬(S⁰)^out ⊔ Sⁱⁿ ◦ ( S⁰ ⁱⁿ ∩ S⁰ ^out),

¬(S⁰)ⁱⁿ ⊔ S^out ◦ ( S⁰ ⁱⁿ ∩ S⁰ ^out) }

= ¬⊤ ⊔ Sⁱⁿ ◦ (⊤ ∩ ⊤) ,¬⊤ ⊔ S^out ◦ (⊤ ∩ ⊤)

= Sⁱⁿ,S^out = S.

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If S_i = S_iⁱⁿ,S_i^out , i ≥ 1, are separators, we deﬁne

S₁ ∩ S₂ = S₁ⁱⁿ ∪ S₂ⁱⁿ,S₁ôut ∩ S₂ôut (intersection) S₁ ∪ S₂ = S₁ⁱⁿ ∩ S₂ⁱⁿ,S₁ôut ∪ S₂ôut (union)

{q}

S_i =







{m−q−1}

S_iⁱⁿ,

{q}

S_i^out





 (relaxed intersection)

S₁\S₂ = S₁ ∩ S₂. (diﬀerence)

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q-relaxed intersection

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Theorem. If S_i are subsets of Rⁿ_{, we have} (i) S₁ _∩ S₂ _∼ _S₁ _{∩ S}₂ (ii) S₁ _∪ S₂ _∼ _S₁ _{∪ S}₂

(iii) S_i _∼ _S_i

(iv) S_i _{∼ S}^k

i , k ≥ 0 (v)

{q}

S_i _∼

{q}

S_i (vi) S₁_\S₂ _∼ _S₁_\S₂_.

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5 Inversion of separators

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The inverse of Y _⊂ Rⁿ _by f : Rⁿ _→ R^m is deﬁned as X ₌ f⁻¹ (Y_{) =} _{x _| f (x) ∈ Y_}_.

f can be a translation, rotation, homothety, projection, . . ..

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We deﬁne

f⁻¹ _(S_Y) = f⁻¹_(S_Yⁱⁿ),f⁻¹_(S_Y^out) .

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Theorem. The separator f⁻¹ (SY) is a separator associ- ated with the set X ₌ f⁻¹ ₍Y_{), i.e.,}

f⁻¹ (Y₎ _∼ f⁻¹ (SY).

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Example. If

f ^x¹

x₂ = x₁ + 2x₂ x₁ − x₂ ,

a separator f⁻¹ _(S_Y) obtained by the following algorithm.

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Separator SX(in: [x_] _,_S_Y_{; out:} _{[xⁱⁿ_],_[x^out_]}) 1 [y₁] = [x₁] + 2[x₂];

2 [y₂] = [x₁] − [x₂];

3 [yⁱⁿ] = S_Yⁱⁿ ([y]) ; [yôut] = S_Yôut ([y]) ; 4 [xⁱⁿ_{] = [}x_{]; [}xôut_{] = [}x_];

5 [xⁱⁿ₁ ] = [xⁱⁿ₁ ] ∩ [y₂ⁱⁿ] + [xⁱⁿ₂ ] ; 6 [xⁱⁿ₂ ] = [xⁱⁿ₂ ] ∩ [xⁱⁿ₁ ] − [y₂ⁱⁿ] ;

7 [xôut₁ ] = [xôut₁ ] ∩ [y₂ôut] + [xôut₂ ] ; 8 [xôut₂ ] = [xôut₂ ] ∩ [xôut₁ ] − [y₂ôut] ; 9 [xⁱⁿ₁ ] = [xⁱⁿ₁ ] ∩ [y₁ⁱⁿ] − 2[xⁱⁿ₂ ];

10 [xⁱⁿ₂ ] = [xⁱⁿ₂ ] ∩ ₂¹ [y₁ⁱⁿ] − [xⁱⁿ₁ ] ; 11 [xôut₁ ] = [xôut₁ ] ∩ [y₁ôut] − 2[xôut₂ ];

12 [xôut₂ ] = [xôut₂ ] ∩ ₂¹ [y₁ôut] − [xôut₁ ] .

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A map M

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Rot(M₎

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Rot(M₎ _∪ M

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6 Atomic separators

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6.1 Equation-based separators

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If

X ₌ _{f ₍x₎ _≤ _0}_,

the pair Sⁱⁿ,S^out , where S^out: f (x₎ _≤ ₀ _and _Sⁱⁿ_: f (x) ≥ 0, is a separator for X_.

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6.2 Database-based separators

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Optimal separator using boundaries

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7 Path planning

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Wire loop game : a metal loop on a handle and a curved wire. The player holds the loop in one hand and attempts to guide it along the curved wire without touching.

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Wire loop game. Is it possible to perform to complete circular path ?

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The feasible conﬁguration space is

M ₌ _{(x₁_{, x}₂₎ _∈ _{[−π, π]} _| f₂ (x) ∈ Y _and f₃ (x) ∈/ Y_} where

f_ℓ₍x_{) = 4} ^cos^x¹

sinx₁ + ℓ cos (x₁ + x₂) sin (x₁ + x₂) . A separator for M _is

SM = f₂⁻¹ _(S_Y) ∩ f₃⁻¹ _S_Y .

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8 Robust localization

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beacons x_i y_i [d_i]

1 1 3 [1,2]

2 3 1 [2,3]

3 −1 −1 [3,4]

P^{q} ₌

{q}

p ∈ R² _| _(p₁ ₋ _x_i₎² _{+ (p}₂ ₋ _y_i₎² _∈ _[d_i_]

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

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9 With real data

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Robot equipped with a laser rangeﬁnder and a compass.

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143 distances collected by the rangeﬁnder ±10cm

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P^{16}

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References

L. Jaulin (2001). Path planning using intervals and graphs.

Reliable Computing, issue 1, volume 7, 1-15.

L. Jaulin and B. Desrochers (2014). Introduction to the Algebra of Separators with Application to Path Planning.

Engineering Applications of Artiﬁcial Intelligence volume 33, pp. 141-147