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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 and Elon Rimon.
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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´¡
FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat
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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- ¥ ´ ´¡ ; ; ¡
FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat
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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!