Equivariant Motion Planning Hellen Colman Wright College, Chicago
Topological Robotics A new discipline at the intersection of topology, engineering and computer science that 1. studies pure topological problems inspired by robotics and 2. uses topological ideas and algebraic topology tools to solve problems of robotics.
Motion Planning Problem (MPP) Robot: A mechanical system capable of moving autonomously. Physical space: The real world X where the robot can move. MPP: Given an initial position A and a fjnal position B , fjnd a path in X that moves the robot from A to B .
Motion Planning Problem (MPP) Robot: A mechanical system capable of moving autonomously. Physical space: The real world X where the robot can move. MPP: Given an initial position A and a fjnal position B , fjnd a path in X that moves the robot from A to B .
Several robots
Confjguration Space A confjguration is a specifjc state of a system; and the confjguration space is the collection of all possible confjgurations for a given system. Example If a point robot moves in a physical space X, then the state of the system = position of the robot confjguration space C 1 ( X ) is just X.
Confjguration Space - two robots Example If two robots move in a physical space X, then the confjguration state of the system = combined position of both robots space C 2 ( X ) = X × X − ∆
Motion Planning Algorithm (MPA) A MPA is a function that assigns to each pair of confjgurations A Defjnition and B , a continuous motion α from A to B . Let PX be the space of paths in X and ev : PX → X × X be the evaluation map, ev ( α ) = ( α (0) , α (1)) . A MPA is a section s : X × X → PX of ev , i.e. ev ◦ s = id .
Does this section exist? related to the stability of robot behavior is about its continuity. Theorem (Farber) and only if the space X is contractible. ▶ If the space is connected, yes. Otherwise, there is no MPA. ▶ But even when the section exists, a fundamental question A continuous motion planning algorithm s : X × X → PX exists if
Does this section exist? related to the stability of robot behavior is about its continuity. Theorem (Farber) and only if the space X is contractible. ▶ If the space is connected, yes. Otherwise, there is no MPA. ▶ But even when the section exists, a fundamental question A continuous motion planning algorithm s : X × X → PX exists if
Farber’s Topological Complexity Defjnition Topological complexity is a homotopy invariant. The topological complexity TC ( X ) is the least integer k such that X × X may be covered by k open sets { U 1 , . . . , U k } , on each of which there is a continuous section s i : U i → PX such that ev ◦ s i = i U i : U i ֒ → X × X . If no such integer exists then we set TC ( X ) = ∞ .
Domains of continuity: MPA: and B . between A and B . TC ( S 1 ) = 2 1. U = { ( x , y ) ∈ S 1 × S 1 | x is not antipodal to y } 2. V = { ( x , y ) ∈ S 1 × S 1 | x is not equal to y } 1. s 1 : U → PX such that s 1 ( A , B ) = shortest path between A 2. s 2 : V → PX such that s 2 ( A , B ) = counterclockwise path
TC as a sectional category Defjnition (1930) least integer k such that X may be covered by k open sets Defjnition (1960) least integer k such that B may be covered by k open sets The Lusternik-Schnirelmann category of a space X , cat ( X ) , is the { U 1 , . . . , U k } , each of which is contractible in X . The sectional category of a fjbration p : E → B , secat ( p ) , is the { U 1 , . . . , U k } on each of which there exists a map s : U i → E such that ps = i U i : U i ֒ → B . We have that TC ( X ) = secat ( ev : PX → X × X )
A group G acting on the space X All maps continuous. Translation groupoid G ⋉ X with objects ( G ⋉ X ) 0 = X arrows ( G ⋉ X ) 1 = G × X Equivariant map φ ⋉ f : G ⋉ X → K ⋉ Y f : X → Y , φ : G → K f ( gx ) = φ ( g ) f ( x )
A group G acting on the space X All maps continuous. Translation groupoid G ⋉ X with objects ( G ⋉ X ) 0 = X arrows ( G ⋉ X ) 1 = G × X Equivariant map φ ⋉ f : G ⋉ X → K ⋉ Y f : X → Y , φ : G → K f ( gx ) = φ ( g ) f ( x )
Notions of equivalences for group actions 2. Morita equivalence: 1. Natural equivalence: � φ ⋉ f G ⋉ X K ⋉ Y ⇄ ψ ⋉ h with ( ψ ⋉ h ) ◦ ( φ ⋉ f ) ∼ = id G ⋉ X and ( φ ⋉ f ) ◦ ( ψ ⋉ h ) ∼ = id K ⋉ Y where ∼ = means equivalent by a natural transformation. ψ ⋉ σ φ ⋉ ϵ � K ⋉ Y . G ⋉ X J ⋉ Z with ψ ⋉ σ and φ ⋉ ϵ essential equivalences.
� � � � � Y X � 2. (fully faithful) the following diagram is a pullback: Essential Equivalence φ ⋉ ϵ : G ⋉ X → K ⋉ Y 1. (essentially surjective) ϕ ′ ◦ π is an open surjection: ϕ ′ X × Y ( K × Y ) π K × Y p 2 ϵ � Y φ × ϵ � K × Y G × X ( p 2 ,ϕ ′ ) ( p 2 ,ϕ ) ϵ × ϵ � Y × Y X × X G × X = { (( k , y ) , ( x , x ′ )) | y = ϵ ( x ) , ky = ϵ ( x ′ ) } .
� � � � � Y X � 2. (fully faithful) the following diagram is a pullback: Essential Equivalence φ ⋉ ϵ : G ⋉ X → K ⋉ Y 1. (essentially surjective) ϕ ′ ◦ π is an open surjection: ϕ ′ X × Y ( K × Y ) π K × Y p 2 ϵ � Y φ × ϵ � K × Y G × X ( p 2 ,ϕ ) ( p 2 ,ϕ ′ ) ϵ × ϵ � Y × Y X × X G × X = { (( k , y ) , ( x , x ′ )) | y = ϵ ( x ) , ky = ϵ ( x ′ ) } .
� � � � � Y X � 2. (fully faithful) the following diagram is a pullback: Essential Equivalence φ ⋉ ϵ : G ⋉ X → K ⋉ Y 1. (essentially surjective) ϕ ′ ◦ π is an open surjection: ϕ ′ X × Y ( K × Y ) π K × Y p 2 ϵ � Y φ × ϵ � K × Y G × X ( p 2 ,ϕ ) ( p 2 ,ϕ ′ ) ϵ × ϵ � Y × Y X × X G × X = { (( k , y ) , ( x , x ′ )) | y = ϵ ( x ) , ky = ϵ ( x ′ ) } .
� Y An ee has to reach to all orbits and there is a bijection induced by � � 2. (fully faithful) the following diagram is a pullback: X � � � Essential Equivalence φ ⋉ ϵ : G ⋉ X → K ⋉ Y 1. (essentially surjective) ϕ ′ ◦ π is an open surjection: ϕ ′ X × Y ( K × Y ) π K × Y p 2 ϵ � Y φ × ϵ � K × Y G × X ( p 2 ,ϕ ) ( p 2 ,ϕ ′ ) ϵ × ϵ � Y × Y X × X G × X = { (( k , y ) , ( x , x ′ )) | y = ϵ ( x ) , ky = ϵ ( x ′ ) } . φ : { g ∈ G | x ′ = gx } → { k ∈ K | ϵ ( x ′ ) = k ϵ ( x ) } .
Any notion relevant to the geometric object defjned by the action, � should be invariant under Morita equivalence. Morita Equivalence ∼ Two actions G × X → X and K × Y → Y are Morita equivalent if there is a third action J × Z → Z and two essential equivalences ψ ⋉ σ φ ⋉ ϵ � G ⋉ X J ⋉ Z K ⋉ Y . We write G ⋉ X ∼ K ⋉ Y .
Examples 1. Let G be a topological group, then 2. If H is a subgroup of G acting on X , then e ⋉ X ∼ G ⋉ ( G × X ) H ⋉ X ∼ G ⋉ ( G × H X ) where [ gh , x ] = [ g , hx ] .
band M . Example Z 2 ⋉ I ∼ ϵ S 1 ⋉ M There is an essential equivalence between the mirror action of Z 2 on the interval I = ( − 1 , 1) and the action of S 1 on the Moebius ϵ
Examples 1. If G acts freely on X, then G ⋉ X ∼ e ⋉ X / G 2. If H ⊴ G acts freely on X, then G ⋉ X ∼ G / H ⋉ X / H
refmection. Example ( Z 2 × Z 2 ) ⋉ S 1 ∼ Z 2 ⋉ S 1 There is an essential equivalence between the action of Z 2 × Z 2 on the circle by rotation+refmection and the action of Z 2 on S 1 by just Z 2 × Z 2 = { e , ρ, σ, ρσ } acting on S 1 Z 2 × Z 2 / <ρ> = < σ > = Z 2 acting on S 1 / <ρ> = S 1
Pronk-Scull characterization Any essential equivalence is a composite of maps as below: 1. (quotient map) G ⋉ X → G / K ⋉ X / K where K ⊴ G and K acts freely on X . 2. (inclusion map) K ⋉ Z → H ⋉ ( H × K Z ) where K ≤ H acting on Z and H × K Z = H × Z / ∼ with [ hk , z ] ∼ [ h , kz ] for any k ∈ K .
Equivariant LS-category X i U integer k such that X may be covered by k invariant open sets The equivariant category of a G -space X , cat G ( X ) , is the least { U 1 , . . . , U k } , each of which is G -compressible into a single orbit. That is, inclusion map i : U → X is G -homotopic to a G -map c : U → X with c ( U ) ⊆ orb G ( z ) for some z ∈ X . − → ❆ ✁ ✕ ✁ ❆ ✁ ❆ ❆ ❯ ✁ orb G ( z )
Examples
Examples
Examples
Examples
A i X U Equivariant Clapp-Puppe A -category Let A be a class of G -invariant subsets of X . The equivariant A -category, A cat G ( X ) , is the least integer k such that X may be covered by k G -invariant open sets { U 1 , . . . , U k } , each G -compressible into some space A ∈ A . − → ❆ ✁ ✕ ✁ ❆ ✁ ❆ ❆ ❯ ✁
X A U i Equivariant Clapp-Puppe A -category Let A be a class of G -invariant subsets of X . The equivariant A -category, A cat G ( X ) , is the least integer k such that X may be covered by k G -invariant open sets { U 1 , . . . , U k } , each G -compressible into some space A ∈ A . − → ❆ ✁ ✕ ✁ ❆ ✁ ❆ ❆ ❯ ✁ In particular, A cat G ( X ) = cat G ( X ) when A = orbits.
E The G-sectional category (Colman-Grant) U B The equivariant sectional category of a G -map p : E → B , secat G ( p ) , is the least integer k such that B may be covered by k invariant open sets { U 1 , . . . , U k } on each of which there exists a G -map s : U i → E such that ps ≃ G i U i : U i ֒ → B . � ✒ � ❄ ֒ →
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