FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES A.ABRAM AND R.GHRIST Also “Robot navigation functions on manifolds with boundary” – Daniel Koditschek

FINDING TOPOLOGY IN A FACTORY: CONFIGURATION

SPACES

A.ABRAM AND R.GHRIST

Also “Robot navigation functions on manifolds with boundary” – Daniel Koditschek and Elon Rimon.

2

Motivation: Consider an automated factory with a cadre of Robots.

Figure1: Two Robots finding their way from start points to destination.

FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

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Configuration Spaces: We define .

What looks like? - .

( ) ( )2 2 2 2 2N

N

C = ´ ´ ×××́ - D¡ ¡ ¡ ¡ ¡14444444444244444444443

( )2NC ¡

( ) ( ){ }21 2, , , : for some i j

N

N i jx x x x xD ××× Î = ¹@ ¡

( )2 2C ¡ 3 1S´¡


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Why homeomorphic to ?

homeomorphic to .

Figure2: may be represented as

}

1 2 1 2, , ,

r

a a b bx x x xæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

r

( )2 2C ¡

rr

( )2 2C ¡

3 1S´¡

ab

( )( )2 2 0,0´ -¡ ¡

( )( )2 2 0,0´ -¡ ¡ 3 1S´¡

( )( ) ( )2 1 10,0 0, S S- ¥ ´ ´¡ ; ; ¡


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How does configuration space helps us with robot motion planning problem?

Safe control scheme using vector field on configuration space.

Figure3: A Vector Field in Configuration space translates to robot motion.


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Navigation function:

It can be shown that all initial conditions away from a set of zero measure are successfully brought to by .

( )( )( )

2 Ploaron with minimum at

3 Morseon

4 Admissibleon

O

dq ÎF F

F

F

( )

Let bea compact connected analytic manifold with boundary.

A map : 0,1 is a if it is:

1 Analyticon

navigation functionf é ù® ê úë û

F

F

F .

dq fÑ


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Remark: Global attracting equilibrium state is topologically impossible.

Thereisnosmoothnondegeneratevectorfield, ,

onthe freespace, ,with 0obsticles,

whichistransverseon ,suchthat theflow

inducedby admitsaglobalasymptotic

stableequilibriumstate.

f

M

x f

>

¶

= -&

F

F


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One particular solution: Koditschek and Rimon.

Composition of repulsive and attractive potentials.

Figure4: “Attractive” and “repulsive” potentials produce navigation function.


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Koditschek and Rimon in more detail[1]:

Sphere World (for )( )2 2

0 0 0,r d qqb = -

( )2 2 ,i i i

r d qqb = -

( ) ( )iq qb b=Õ

2¡

- Sphere World Boundary ( ) ( )( )

2

,k

k goalq d qqg =

( )k qg

b

- Obstacle

Repulsive Attractive

( ) ( )( )

( ) ( )

2

1 12

2

,

,

goalkk

k k

goal

d qqq q

d qq q

gf x s

bb

æ öç ÷ç ÷= =ç ÷ç ÷è ø é ù+ê ú

ê úë û

o oTotal: … →


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Koditschek and Rimon in more detail[2]:K=3 K=4 K=6

Figure5: Koditschek and Rimon Navigation function.


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Navigation properties are invariant under deformation.

So this solution is valid for any manifold to which Sphere World is deformable.

let : 0,1 beanavigationfunctionon ,

: analiticdiffeomorphism.

Then ,

isanavigationfunctionon .

h

h

f

ff

é ù® ê úë û®%@ o

M M

F M

F


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Robots moving about a collection of tracks embedded in the floor.

Figure6: “Robots on an graph”.


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Example: ( )2 Y CW complex.C -

Figure7: Realization of .

( )2 YC


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It is hard to visualize even simple conf. spaces.

Discretized configuration space

Figure8: Even Simple graph leads to complicated configuration spaces.

( )cellsin whichclosure intersectsthediagonal

ND G = G´ G´ ×××́ G- D

D = G D

%

%


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Discretized configuration space

We can think of this as imposing the restriction that any path between two robots must be at least one full edge apart.

Figure9: Excluded Configurations [left] Closure of edge [center] Remaining Configurations [right].

( )cellsin whichclosure intersectsthediagonal

ND G = G´ G´ ×××́ G- D

D = G D

%

%

closure


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Example: 0-cells 1-cells 2-cells

( )2 YD

Figure10: Realization of .

( )2 YD

´3 2´ ´3 2 2

0


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Using same strategy it is easy to apprehend those spaces. Some interesting results appear.

Those are rather surprising results:

Figure11: homeomorphic to closed orientable manifold g = 6.

( )25

D K

( )2

5 3,3

let beconnected(uncolored)graphwithout loops.If is

homeomorphic to closed 2 dimensional manifold, then

D

K or K

G G

- G=

( )( )25

# # # 30 60 20D K faces edges verticesc = - + = - +


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How close are those discretized spaces to original ones?

( ) ( )

( )

Theorem 1: For any 1 and any graph with at least vertices

deformation retracts to if and only if:

1 each path between distinct vertices of valance not two

passes throught at least 1 edges, an

N N

N N

C D

N

> G

G G

-

( )d

2 each path from a vertex to itself that cannot be shrunk

to apoint in passes through at least 1 edges.NG +


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How this works -

Figure11: Graph that does not comply [upper] graph that complies [lower]

with the theorem.


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Another powerful result:

( )Theorem 2: Given a graph having verticesof

valence greater then two the space deforamtion

retracts to subcomplex of dimension at most .

N

V

C

V

G

G


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How this works[1] - Consider tree . Theorem 2 insuresthat

deformationretractsto 1 dimensional subcomplex

that isagraph. We can determine topological features

of the graph by EulerCaracteristic. Using adoubleinduction

ar

k kk prong r r-

-

-

( )

( )( )

( )

gument on and onecanprove that has

homotopy type of graph with P distinct loops joined

together like petals on daisy. Where

2 !P= 1+ 2 1

1

Nk

N k C r

N kNk N k

k

+ -- - +

-


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How this works[2] -

P = 5

Figure12: Topological structure (homology class) of 5-prone tree.


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Thank You!

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FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES A.ABRAM AND R.GHRIST Also “Robot navigation functions on manifolds with boundary” – Daniel Koditschek