robert ghrist

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optimal coordination on roadmaps

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

jason okane steve lavalle
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optimal coordination on roadmaps
1. coordination spaces

• 2. pareto optimality

3. curvature
4. computing optima
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coordin- cooper-
in warehouse/agv systems
in swarming systems
in metamorphic robot systems
assume
N distinct robots
N distinct cost functions
common workspace W
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planning
achieve goals safely, optimally
each robot has configuration space
do path-planning
but
full c-spaces can be unwieldy
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roadmap approximation
approximate Ci by a graph - roadmap
single - multiple objects
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coordination spaces
given N robots in workspace W
construct N independent roadmaps G(G1,,GN)
coordination space
C(G) G1 x x GN - O
O obstacle set (open)
cf. lozano-peres erdmann et al.
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example graph c-space
agvs with track constraints
GG1GN O (xi) xi xj
tracks guidewires optical paths
coordination problem
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example cylindrical
for obstacle sets O defined by pairwise
collisions, X is cylindrical
for each i, j
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optimality
goal optimal coordination
but each robot 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 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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why pareto?
lemma any optimum for any monotone
scalarization of the cost functions
is in fact a pareto optimum.
precomputing the pareto optima is good
set of all possibilities
changing priorities / costs
hopefully, this is a small set of optima
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example
cylindrical
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example
noncylindrical
cylindrical
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what is the difference?
(non)positive curvature
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nonpositive curvature
let X be a geodesic metric space
in X
d
d
in E2
X is nonpositively curved npc iff
small triangles in X arent fatter
than triangles in the plane
this approach to geometry has a rich history
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what is the difference?
npc
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example cube
the surface of a flat cube has positive
curvature at a corner
a solid flat cube does not
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gromovs link condition
link simplicial complex of incident cells
X is npc ? link of each vertex
cube complex is a flag complex
if the edges look like a k-simplex, there
really is a k-simplex spanning them
no triangles
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why is this good?
curvature for spaces which are not smooth
meta-theorem algorithmic problems are
intractable
positive
quadratic
nonpositive
for groups of
curvature
linear
negative
lemma geodesics on an npc space are unique
(up to homotopy, i.e. deformation)
goal repeat this for pareto optima
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curvature coordination
key lemma all cylindrical roadmap
coordination spaces are npc.
proof let Gi(k) denote the graph obtained
by bisecting each edge of Gi(k-1).
let C(k) be the largest closed subcomplex of
Gi(k) xx Gi(k) which does not intersect O.
this is called the stage-k discretization of C
(GW).
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curvature coordination
key lemma all cylindrical roadmap
coordination spaces are npc.
proof steps
C(k) satisfies the link condition for
each k and is thus npc
C is the hausdorff limit of C(k)
a compact hausdorff limit of npc spaces
is npc
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cubical complexes/paths
npc cubical complexes have several nice
group-theoretic properties
cube paths
sequence of cubes touching at corners
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left-greed is good
like a geodesic among cube paths
star(Ci) n Ci1 vi
a left-greedy cube path niblo-reeves uses
high dimensional cells as early as possible
theorem normal cube paths on an npc cube
complex are unique up to homotopy
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finiteness theorem
theorem for C cylindrical, there is a
finite number of pareto optimal classes
proof
given a pareto-optimal path in C
approximate it by a cube path in C(n)
uses the fact that C is npc
deform it to the (unique!) normal cube path
verify that cost functions have not changed
use compactness to go from discrete - finite
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finiteness theorem
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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finiteness theorem
positive curvature is not the only obstruction
need 1-d roadmaps for each agent
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finiteness theorem
positive curvature is not the only obstruction
need 1-d roadmaps for each agent
(discretizing this space yields positive
curvature)
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computing optima
problem given a path, compute its
pareto-optimal representative
key step
generalize left-greedy paths to smooth category
these are pareto-optimal
these are unique up to homotopy via npc
uses the fact that C is npc
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smooth left greedy paths
to optimize a given path
primary steps
propagate at top speed (slope 1 in each plane)
re-evaluate when crossing critical hyperplanes
homotopy class determines critical crossings
linear programming along multiple obstacles
end result is the unique left-greedy path
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a monotone example
m obstacle complexity p input path complexity

p output path complexity
time O(p m log(mp) pm½ log m)
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computing optima
challenge find all homotopy classes
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main point
nonpositive curvature is useful for
bounds and for algorithms
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open question
can we blunt the dimension curse?
maybe yes? npc geometry ? quadratic
bounds in geometric group theory
maybe no? there can exist a factorial
number of p.o. path classes
cf shortest path problem in 3-d
np hard canny-reif but this relies
on positive curvature