7,768 research outputs found
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
Deconstructing Approximate Offsets
We consider the offset-deconstruction problem: Given a polygonal shape Q with
n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance,
as the Minkowski sum of another polygonal shape P with a disk of fixed radius?
If it does, we also seek a preferably simple-looking solution P; then, P's
offset constitutes an accurate, vertex-reduced, and smoothened approximation of
Q. We give an O(n log n)-time exact decision algorithm that handles any
polygonal shape, assuming the real-RAM model of computation. A variant of the
algorithm, which we have implemented using CGAL, is based on rational
arithmetic and answers the same deconstruction problem up to an uncertainty
parameter \delta; its running time additionally depends on \delta. If the input
shape is found to be approximable, this algorithm also computes an approximate
solution for the problem. It also allows us to solve parameter-optimization
problems induced by the offset-deconstruction problem. For convex shapes, the
complexity of the exact decision algorithm drops to O(n), which is also the
time required to compute a solution P with at most one more vertex than a
vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011,
submitted to DC
Multi-contact Walking Pattern Generation based on Model Preview Control of 3D COM Accelerations
We present a multi-contact walking pattern generator based on preview-control
of the 3D acceleration of the center of mass (COM). A key point in the design
of our algorithm is the calculation of contact-stability constraints. Thanks to
a mathematical observation on the algebraic nature of the frictional wrench
cone, we show that the 3D volume of feasible COM accelerations is a always a
downward-pointing cone. We reduce its computation to a convex hull of (dual) 2D
points, for which optimal O(n log n) algorithms are readily available. This
reformulation brings a significant speedup compared to previous methods, which
allows us to compute time-varying contact-stability criteria fast enough for
the control loop. Next, we propose a conservative trajectory-wide
contact-stability criterion, which can be derived from COM-acceleration volumes
at marginal cost and directly applied in a model-predictive controller. We
finally implement this pipeline and exemplify it with the HRP-4 humanoid model
in multi-contact dynamically walking scenarios
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