16,871 research outputs found
Computational topology with Regina: Algorithms, heuristics and implementations
Regina is a software package for studying 3-manifold triangulations and
normal surfaces. It includes a graphical user interface and Python bindings,
and also supports angle structures, census enumeration, combinatorial
recognition of triangulations, and high-level functions such as 3-sphere
recognition, unknot recognition and connected sum decomposition.
This paper brings 3-manifold topologists up-to-date with Regina as it appears
today, and documents for the first time in the literature some of the key
algorithms, heuristics and implementations that are central to Regina's
performance. These include the all-important simplification heuristics, key
choices of data structures and algorithms to alleviate bottlenecks in normal
surface enumeration, modern implementations of 3-sphere recognition and
connected sum decomposition, and more. We also give some historical background
for the project, including the key role played by Rubinstein in its genesis 15
years ago, and discuss current directions for future development.Comment: 29 pages, 10 figures; v2: minor revisions. To appear in "Geometry &
Topology Down Under", Contemporary Mathematics, AM
Topological Prismatoids and Small Simplicial Spheres of Large Diameter
We introduce topological prismatoids, a combinatorial abstraction of the
(geometric) prismatoids recently introduced by the second author to construct
counter-examples to the Hirsch conjecture. We show that the `strong -step
Theorem' that allows to construct such large-diameter polytopes from
`non--step' prismatoids still works at this combinatorial level. Then, using
metaheuristic methods on the flip graph, we construct four combinatorially
different non--step -dimensional topological prismatoids with
vertices. This implies the existence of -dimensional spheres with
vertices whose combinatorial diameter exceeds the Hirsch bound. These examples
are smaller that the previously known examples by Mani and Walkup in 1980 (
vertices, dimension ).
Our non-Hirsch spheres are shellable but we do not know whether they are
realizable as polytopes.Comment: 20 pages. Changes from v1 and v2: Reduced the part on shellability
and general improvement to accesibilit
Fast and Continuous Foothold Adaptation for Dynamic Locomotion through CNNs
Legged robots can outperform wheeled machines for most navigation tasks
across unknown and rough terrains. For such tasks, visual feedback is a
fundamental asset to provide robots with terrain-awareness. However, robust
dynamic locomotion on difficult terrains with real-time performance guarantees
remains a challenge. We present here a real-time, dynamic foothold adaptation
strategy based on visual feedback. Our method adjusts the landing position of
the feet in a fully reactive manner, using only on-board computers and sensors.
The correction is computed and executed continuously along the swing phase
trajectory of each leg. To efficiently adapt the landing position, we implement
a self-supervised foothold classifier based on a Convolutional Neural Network
(CNN). Our method results in an up to 200 times faster computation with respect
to the full-blown heuristics. Our goal is to react to visual stimuli from the
environment, bridging the gap between blind reactive locomotion and purely
vision-based planning strategies. We assess the performance of our method on
the dynamic quadruped robot HyQ, executing static and dynamic gaits (at speeds
up to 0.5 m/s) in both simulated and real scenarios; the benefit of safe
foothold adaptation is clearly demonstrated by the overall robot behavior.Comment: 9 pages, 11 figures. Accepted to RA-L + ICRA 2019, January 201
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