robert ghrist

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topology, geometry, robotics

• robert ghrist
• department of mathematics
• university of illinois
• urbana-champaign

aaron abrams --- math UGA steve lavalle --- cs
UIUC eric klavins --- ee UW dan koditschek
--- eecs UM
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topology, geometry, robotics
1. prefatory remarks

• 2. nontraditional c-spaces

3. curvature and optimization
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topology, geometry, robotics

• what topology is good for

new ways of looking at old problems
slick solutions to new problems
robustness!
is the topology interesting?
are the applications honest?
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topology, geometry robotics
needs configuration spaces for coordination

• automated guided vehicles
• koditschek

• assembly (self-)
• klavins

• metamorphic robots
• abrams

Millibots CMU
G. Whitesides, Harvard
M. Yim, PARC Labs
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configuration spaces
trade degrees of freedom for dimension
positions shape system state
point in configuration space C
motion planning repetitive actions centralized
controls kinematic constraints optimal
reconfiguration
path space of C loop space of C vector fields on
C distributions on C geodesics on C
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classical examples
distinct labeled points on R2
workspace W R2 - O (obstacles)
CN WN - D
D (xi) xi xj
braid group p1(CN(R2)/SN)
mechanical linkages
theorem kapovich-millson Any manifold can
arise as (components of) C-space of a planar
linkage.
zlatanov et al. U. Laval
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topology, geometry, robotics
1. prefatory remarks

• 2. nontraditional c-spaces

3. curvature and optimization
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nontraditionalc-spaces
1. configurations on graphs
2. coordination spaces
3. state complexes
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graph-constrained AGVs
with d. koditschek
tracks in floor electrified guidewires optical
paths.
collision avoidance is challenging
what is the topology of CN(G) GN - D ?
goal vector fields to realize goals
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topological results
theorems G,98 for any N and any graph G,

• CN(G ) is determined up to homotopy
• by its fundamental group
• (viz., it is a finite K(p,1))

2. CN(G ) deformation retracts to a complex of
dimension bounded above by essential
vertices of G
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braid groups of graphs
G radial K-prong tree GK
p1(CN(GK )) free group of rank 1(NK-2N-K1)(N
K-2)!/(K-1)!
A C2(K5 ) and C2(K3,3 )
p1(C2(K5 )) surface group of genus 6
p1(C2(K3,3 )) surface group of genus 4
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trees
G there is a (serre) fibration of CN(GK) over
a simplicial (K-1)-complex with contractible
fiber
example K3 unlabeled
this fibration behaves nicely under restricting
points to the boundary of the tree
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trees
UCN(tree) as a graph of spaces
split tree at midpoint
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4
3
2
1
0
0
1
2
3
4
5
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nontraditionalc-spaces
1. configurations on graphs
2. coordination spaces
3. state complexes
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roadmaps in a workspace
approximate W by a graph - roadmap
what to do for multiple objects?
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coordination spaces
w/lavalle,okane
given N robots in workspace W
construct N roadmap graphs G(G1,,GN)
coordination space
C(GW) G1 x x GN - O
O obstacle set (open) where, e.g., robots
collide in W
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coordination spaces
w/lavalle,okane
for obstacle sets O defined by collisions, the
coordination space is cylindrical
for each i, j
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nontraditionalc-spaces
1. configurations on graphs
2. coordination spaces
3. state complexes
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shape-planning metamorphic robots
1. different language / approach for each type
of system
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shape-planning metamorphic robots
1. different language / approach for each type
of system
2. early approach transition graph
chirikjian et al.
vertices shapes edges local moves
(cf. cayley graph for group)
3. recent approach distributed
controllers amato et al., rus et al., yim et
al.
advantages faster, smaller computations
problems ad hoc? not optimal?
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local reconfigurable systems
workspace W (lattice or graph)
states labelings of W with some alphabet A
generator pair of local states on some support
action translation of support into W
(admissible on state U if it matches a local
state)
F
U1
U0
a local reconfigurable system is a set of
generators and states closed under all
admissible actions
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example 2-d hexagonal
Chirikjian et al.
support
local states
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the state complex
idea transition graph ? 1-skeleton of a
cubical complex
each cube of dimension K corresponds to K
physically independent commutative moves
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example planar sliding system
tiles 2-d, square
local moves row, column slides
other examples 3-d, compression systems,
assembly sequences klavins
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example articulated robot arm
states length N curve in lattice
moves rotating end flipping corners
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example articulated robot arm
prop the state complex S is contractible for
any number of segments
proof use symbolic representation of states
actions transposing x?y changing last letter
induct on the length of the word
yyyxxxyxyxyxyy
state complex converges to a smooth
configuration space
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example points on a graph G
states arrangements of N points on vertices
moves move a point along a free edge
theorem refinements converge in homotopy type
to the c-space of the graph
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computing C2(K5 )
A. Abrams, 2001
1. determine the local structure
2. euler characteristic computation
vertices (5)(4) 20
local euler characteristic 1 3 3/2 1/2
euler characteristic 10 ? genus 6
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generalization labeled graphs
states labels on vertices
moves exchange distinct labels on neighboring
vertices
5-gon with 5 labels genus 16
filled-in cayley graph of S5 via transpositions
other examples hyperbolic 3-manifolds
(icosahedral link)
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advantages of state complex
1. topology

• cf. 1-skeleton of an N-cube N2N-1 edges
• convergence phenomena under refining lattice
• natural generalization of C-space to discrete
  setting

2. geometry

• geodesics optimize elapsed time
• universal geometric feature nonpositive
  curvature
• helps to prove convergence of algorithms

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topology, geometry, robotics
1. prefatory remarks

• 2. nontraditional c-spaces

3. curvature and optimization
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cat(0) geometry
let X be a metric space with local geodesics
in X
d
d
in E2
X is nonpositively curved npc iff X is
locally cat(0)
this approach to geometry has a rich history
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gromovs link condition
link simplicial complex of incident cells
X is npc ? link of each vertex is a flag complex
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gromovs link condition
link simplicial complex of incident cells
link
X is npc ? link of each vertex is a flag complex
if the edges look like a k-simplex, there really
is a k-simplex spanning them
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npc is prevalent
among manifolds
among groups
among robotic systems?
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npc is prevalent
theorems 1. A,S c-spaces of graphs are
npc 2. GL cylindrically deleted coordination
spaces are npc 3. AG state complexes are npc
proofs 1.2. filtrate by cubical complexes and
apply link condition 3. reinterpret link
condition in terms of actions of generators
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npc is prevalent
note you really need a local reconfigurable
system
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geodesic design
observations
1. geodesics on S optimize elapsed time
2. geodesics on S are unique up to homotopy
proof S has no positive curvature.
strategem
given a shape trajectory from a distributed
controller edge-path in S
1. perform curve shortening on S
2. this must converge to the global
minimum obtainable from the initial path
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geodesics are not discrete
want to use a geodesic among cube paths
a normal cube path niblo-reeves uses high
dimensional cells as early as possible
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curve-shortening algorithm
given edge-path in S
march along checking commutativity
multiple sweeps through path reduce length
you do not need to construct S.
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morals
classical configuration spaces are not the
complete story
nonpositive curvature and CAT(0) geometry are
useful for optimization algorithms