Topological Robotics - Rational Homotopy...

Topological RoboticsMinimal Navigational Complexity

FAC. SC. AIN CHOCK, CASABLANCA

5 JANVIER 2013

Professeur Agrégé-Docteur en MathMaster 1 en Sc de l’éducation, Univ. [email protected]

My Ismail Mamouni, CRMEF Rabat

My Ismail Mamouni (CRMEF, Rabat) Rational Homotopy Seminar Topological Robotics 1 / 24

Summary

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1 Recalling

2 When TC=1

3 When TC=cat


Recalling

Homotopy between functions

Two continuous maps f0, f1 : X −→ Y are homotopic if there is acontinuous map

H : X × [0,1] −→ Y

(called Homotopy) such that

f0(x) = H(x ,0), f1(x) = H(x ,1), for allx ∈ X

Then we write f0 ≃ f1.

Definition


Recalling

Homotopy between functions

A homotopy H : S1 × [0,1] −→ X between loops f0, f1 : S1 −→ X

Example


Recalling

Homotopy between spaces

DefinitionTwo spaces X ,Y are homotopy equivalent, written X ≃ Y , ifthere are continuous maps f : X −→ Y and g : Y −→ X suchthat

f ◦ g ≃ IdY and g ◦ f ≃ IdX


Recalling

Homotopy=continuous deformation


Recalling

Contractible Space

DefinitionA topological X space is called contractible, when

X ≃ {pt}


Recalling

Contractible Space

Rn is contractible, but Rn \ D is not


Recalling

Lusternik-Schnirelmann category

DefinitionFor any topological space, cat(X ) is the smallest n for which thereis an open covering {U0, . . . ,Un} by (n + 1) open sets, each ofwhich is contractible in X .

Poland (1899-1981) Russia (1905-1938)


Recalling

Lusternik-Schnirelmann category

☞ It was, as originally defined for the case of X a manifold, thelower bound for the number of critical points that a real-valuedfunction on X could possess.

☞ This should be compared with the result in Morse theory thatshows that the sum of the Betti numbers is a lower bound forthe number of critical points of a Morse function.

Interpretation


Recalling

Robotic

Motion Planning Algorithm☞ It takes as input a pair (A,B) ∈ X × X and output a path

γ : [0;1] −→ X from A to B

☞ Such algorithm is always possible when X is path-connected.

☞ That means, that the map

π : PX := X [0,1] −→ X × Xγ 7−→ (γ(0), γ(1))

π is surjective when X is path-connected, and has a section,called motion planning algorithm

s : X × X −→ PX

such that s ◦ π = id , i.e. s.π(A,B) = (A,B).


Recalling

Topological Robotics

TopologyIf we equip the path space PX with compact-open topology, thenwe can talk about nearby paths.


Recalling

Topological Robotics

☞ The question will be : Does there exist a continuous motionplanning in X ?

☞ Equivalently, is it possible to construct a motion planning in theconfiguration space X so that the continuous path s(A, B) in X,which describes the movement of the system from the initialconfiguration A to the final configuration B, depends continu-ously on the pair of points (A, B) ?

☞ Absence of continuity will result in the instability of behav-ior : there will exist arbitrarily close pairs (A,B) and (A′

,B′) ofinitial-desired configurations such that the corresponding pathss(A,B) and s(A′

,B′) are not close.

Stability of navigation


Recalling

The first result

A continuous motion planning s : X × X −→PX exists if and only if the configuration spaceX is contractible.

M. Farber (2003)


Recalling

Topological Complexity

Definition (M. Farber 2003)TC(X) is the smallest number n for which there is an open cover{U0, . . . ,Un} of X × X by (n + 1) open sets, for each of whichthere is a local section si : Ui −→ PX .


Recalling

Navigational Complexity

Homotopy Invariance : M. Farber (2003)TC(X) depends only on the homotopy type of X.

☞ This property of homotopy invariance allows us to simplify theconfiguration space X without changing the topological com-plexity, and hence, we may predict the character of instabilitiesof the behavior of the robot

☞ the homotopy invariant TC(X) is a new interesting tool whichmeasures the “navigational complexity” of X.

Interpretation


Recalling

TC : first results

M. Farber (2003)☞ TC(X ) = 0 iff X contractible

☞ TC(S2n+1) = 1

☞ TC(S2n) = 2

☞ cat(X ) ≤ TC(X ) ≤ cat(X × X )

☞ cat(X ) = TC(X ) when X is a topological group


When TC=1

Main result

M. Grant, G. Lupton, J. Oprea (2012)Let X be a path-connected CW complex of finite type. then

TC(X ) = 1 ⇐⇒ X ≃ S2n+1

Mark Grant Gregory Lupton John OpreaNottingham, UK Cleveland, USA Cleveland, USA


When TC=1

Detailed results

M. Grant, G. Lupton, J. Oprea (2012)Let X be a path-connected CW-complex of finite type, such thatTC(X ) = 1. Then

☞ X ≃ S2n+1 with 2n + 1 ≥ 3 when X is simply connected

☞ X ≃ S1 when π1(X ) 6= {e}.

M. Grant, G. Lupton, J. Oprea (2012)Let X be a non-contractible co-H-space, path-connected CWcomplex of finite type. then

TC(X ) = 1 or 2


When TC=cat

G. Lupton , J. Scherer (2011)Let X be a connected, CW H-space. Then

TC(X ) = cat(X )

Gregory Lupton Jerome SchererCleveland, USA Lausanne, Switzerland


When TC=cat

H-space

DefinitionH-space is a topological space X equipped with a continuousproduct

µ : X × X −→ X

with an identity homotopy element e such that µ(e, .) ≃ µ(.,e) ≃idX .

topological group is an H-space

coH-spaceis defined as the same from a comultiplication, namely, a map

µ : X −→ X ∧ X


When TC=cat

H-space : History

The Hin H-space was suggested by Jean-Pierre Serre in recognition ofthe influence exerted on the subject by Heinz Hopf a

a. J. R. Hubbuck. "A Short History of H-spaces", History of topology, 1999,pages 747-755

Jean Pierre Serre Heinz HopfFrance (1926- ) Germany (1894-1971)


When TC=cat

Sketch of proofs

Follow on the


When TC=cat

That’s all talks


When TC=cat

References

M. Farber, Invitation to Topological Robotics, EMS (2008).

M. Farber, Topology of robot motion planning , in : Morse TheoreticMethods in Nonlinear Analysis and in Symplectic Topology (P.Biran et al (eds.)) (2006), 185–230.

J.-C. Latombe, Robot Motion Planning, Kluwer (1991).

M. Grant, G. Lupton, J. Oprea, Spaces of Topological ComplexityOne, mathArXiv :1207.4725. (2012).

G. Lupton, J. Scherer, Topological complexity of H-spaces,mathArXiV : 1106.3399 (2011), to appear in Proc. Amer. Math.Soc.


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