hyperbolic geometry and robotics

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hyperbolic geometry and robotics

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

indiana university dec 2004
collaborators a. abrams emory v. peterson
uiuc, s. lavalle uiuc
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configurations in robotics
there are numerous settings in robotics and
automation for which coordination is vital
zlatanov et al. u. laval
the key unifying idea in motion planning comes
from taking a topological point of view...
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reconfiguration spaces
trade degrees of freedom for dimension
system state or positions
point in configuration space C
motion planning repetitive actions centralized
controls kinematic constraints optimal planning
path space of C loop space of C vector fields on
C distributions on C geodesics on C
however, there are settings for which
discretized spaces are more naturalreconfigura
tion
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motivation metamorphic robots
Chirikjian, JHU Rus Vona, MIT Yim, PARC
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motivation self-assembly
through passive and active means
UWISC, Chem REU
G. Whitesides, Harvard
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motivation digital microfluidics
droplet can be manipulated on a grid via
electrowetting
(richard fair duke)
goal manipulate inputs, reactions, and products
lab-on-a-chip
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motivation group theory
group theory has been using discrete
configuration spaces to study groups for well
over 100 years
finitely presented group
cayley graph larger complexes
this is our perspective
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definitions and examples
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formal reconfigurable systems
domain graph G (lattice)
states labelings of vertices of G by an
alphabet A (Z2)
generator (support, trace two unordered local
states on support)
support subgraph of G trace subgraph of
support (where things move)
admissible the support matches a local state
support
trace
local states
a reconfigurable system is a set of states closed
under generators
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example 2-d hexagonal
Chirikjian et al.
support
trace
local states
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the state complex
idea build a cubical complex out of states,
generators...
generators fi i1..K commute sup(fm) n
tr(fn) 0 m ? n
each cube of dimension K corresponds to K
commutative moves
(undirected version of high-dimensional automata
pratt)
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example planar sliding system
tiles 2-d, square
local moves row, column slides
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example articulated robot arm
states length N chain in planar lattice
moves rotating end flipping corners
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example articulated robot arm
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example points on a graph G
states arrangements of N points on vertices
moves slide a point along a free edge
theorem a. abrams refinements converge in
homotopy type to the (smooth) configuration
space of the graph
this models microfluidic arrays, as well as
configuration spaces of robots on tracks...
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configuration spaces of graphs
a. abrams, 2000
1. determine the local structure
2. euler characteristic computation
vertices - edges faces -10
this is a closed surface of genus 6
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labeled graphs and permutohedra
states labeled vertices of a graph
generators exchange distinct labels on
neighboring vertices
5-gon with 5 labels genus 16
filled-in cayley graph of S5 via transpositions
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other examples...
algebra
diagram groups of semigroup presentations
farley
geometry
spaces of triangulations of polygons
automation
parallel assembly schemes gklavins
biology
spaces of phylogenetic trees billera-holmes-vogtm
ann
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notice
all of these surfaces we construct as state
complexes are of genus gt 1
can you construct a sphere?
can you construct interesting three-manifolds?
surprisingly, the answer to these topological
questions depends almost completely on the
geometry of these objects
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and now some geometry
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cat(0) geometry
let X be a space on which you can measure
distance via geodesics (shortest paths)
in X
d
d
in E2
X is cat(0) ? all triangles are no fatter than in
E2
X is nonpositively curved npc iff X is locally
cat(0) (cat(0) npc simply connected)
cartan, andronov, toponogov or comparare ab
triangulos
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gromovs link condition
link simplicial complex of incident cells
theorem gromov cube complex 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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gromovs link condition
link simplicial complex of incident cells
link
theorem gromov cube complex 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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theorem
AG all state complexes are nonpositively
curved.
link
proof simple application of the link condition
commutativity as defined is pairwise determined
corollary all higher homotopy groups vanish
fundamental group is torsion-free
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the realization problem
theorem GP let L be any flag simplicial
complex. There exists a reconfigurable system
whose state complex has all links L.
proof explicit construction
(cf. result of m. davis)
states labelings of V(L) by 0,1
generators change one label at a time
this yields interesting closed manifolds of all
dimensions
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the realization problem
sois anything not possible?
yes the following npc spaces cannot arise as a
subcomplex of a state complex
(cf. recent work of Haglund Wise on special
cube complexes)
are there any weaker restrictions...say...on p1 ?
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a nice embedding theorem
theorem GP the fundamental group of a state
complex with generators fk k1..K embeds
into the group with presentation fk1..K
fifjfjfi if fi and fj physically commute
these are artin right-angled groups, and are very
nice...
corollary fundamental groups of state complexes
are linear
the proof of the above theorem is very similar
to the recent results of crisp wiest, who prove
the result for the example of points on a
graph...
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some positive results
theorem GP any npc subcomplex of a product of
graphs can be realized as a state complex.
the converse is not, however, true
observation GP any finite simply connected
state complex is a subcomplex of some cube
(of sufficiently high dimension)
open problem characterize the set of all cube
complexes which arise as state complexes
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well ok then
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but what is this good for?
(classical) theorem geodesics are unique on a
cat(0) space
proof
assume two geodesics in X insert a middle point
build the comparison triangle in the euclidean
plane
cat(0) inequality implies that the two geodesics
are the same
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but what is this good for?
this is very useful for optimization problems
general example
given some state trajectory from a distributed,
ad hoc, or probabilistic path-planner
1. perform curve shortening on state complex
2. this must converge to the global
minimum obtainable by homotopy of the initial
path
npc gt no local minima at which to get stuck
do not have to compute the entire state complex
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optimal scheduling
let I1,,IN denote N closed intervals, denoting
the paths of N distinct robots in a
configuration space (includes positions,
orientations, states...)
coordination space
C I1 x x IN - O
O (open) obstacle set where, e.g., robots
collide
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optimal scheduling
for obstacle sets O defined by collisions,
the coordination space is cylindrical
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pareto optimality
goal optimal coordination/scheduling
but each robot/process has its own cost
function (e.g., elapsed time)
we could use...
average cost
maximal cost
nonlinear weighted cost?
use pareto-optimization
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pareto optimality
definition a path is pareto-optimal iff it is
minimal with respect to the partial order on cost
vectors.
cost vectors A(a1,,aN) B(b1,,bN)
A B ? ai bi for all i
incomparable if
ai lt bi and aj gt bj for some i, j
equivalent if
ai bi for all i
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pareto optimality
problem classify compute pareto optimal path
classes
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pareto optimality
problem classify compute pareto optimal path
classes
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example
cylindrical
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example
cylindrical
noncylindrical
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what is the difference?
(non)positive curvature
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curvature coordination
theorem glavalle cylindrical roadmap
coordination spaces are npc
proof the coordination space is a compact
hausdorff limit of npc state complexes obtained
by discretization
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pareto optimality
theorem glavalle the number of pareto optimal
classes is finite in the cylindrical case
proof
discretize system to a cubical state complex
deform pareto-optimal paths to a left-greedy
geodesic
left-greedy paths are unique up to homotopy
niblo-reeves in an npc cube complex
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pareto optimality
why doesnt this work in the case of positive
curvature?
all the steps work except one
? can approximate by cube paths
? left-greedy cube paths are pareto optima
? there is no longer a unique left-greedy path
exponential blow-up in paths
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open problems
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pursuit - evasion...
e.g., lion man problems
does the lion catch the man?
answers (simply connected)
dimension 2 lion wins
dimension 3 lion can lose
all known examples of lion losing involve
positive curvature...
conjecture on a cat(0) space, pursuit algorithms
exist and are fast
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shortest paths...
given points p, q in a workspace
what is the complexity of finding the shortest
path p ? q?
answers
dimension 2 polynomial
dimension 3 np hard canny-reif
all known proofs utilize obstacles which force
positive curvature...
conjecture on a cat(0) space, shortest path
algorithms are of polynomial complexity
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the last word
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a meta-theorem
dynamical systems hyperbolic dynamics 1960s
hyperbolic dynamics are the only class of
chaotic dynamics for which we have good
theorems...
topology hyperbolic manifolds 1970s
hyperbolic 3-manifolds are both interesting and
tractable the poincaré conjecture is
difficult because of elliptization...
group theory hyperbolic groups 1980s
hyperbolic groups form a class of groups which
are both interesting and have good algorithmic
properties...
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a meta-theorem
robotics hyperbolic configurations
there is a significant class of robotics settings
and problems for which a cat(0) geometry
is natural
this class is both fascinating and tractable
hard problems should experience a dramatic
drop in complexity