687 research outputs found
The free splitting complex of a free group I: Hyperbolicity
We prove that the free splitting complex of a finite rank free group, also
known as Hatcher's sphere complex, is hyperbolic.Comment: 85 pages, including index and glossary. Many figures in the free
splitting complex, in the guise of commutative diagrams of maps between free
splittings. Changes from Version 1 to Version 2: Corrected an error in the
proof of Proposition 6.5, Step 2. Revamped the theory of free splitting units
in Section
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
Polynomial continuation in the design of deployable structures
Polynomial continuation, a branch of numerical continuation, has been applied
to several primary problems in kinematic geometry. The objective of
the research presented in this document was to explore the possible extensions
of the application of polynomial continuation, especially in the field
of deployable structure design. The power of polynomial continuation as a
design tool lies in its ability to find all solutions of a system of polynomial
equations (even positive dimensional solution sets). A linkage design problem
posed in polynomial form can be made to yield every possible feasible
outcome, many of which may never otherwise have been found.
Methods of polynomial continuation based design are illustrated here by way
of various examples. In particular, the types of deployable structures which
form planar rings, or frames, in their deployed configurations are used as
design cases. Polynomial continuation is shown to be a powerful component
of an equation-based design process.
A polyhedral homotopy method, particularly suited to solving problems in
kinematics, was synthesised from several researchers’ published continuation
techniques, and augmented with modern, freely available mathematical
computing algorithms. Special adaptations were made in the areas of level-k
subface identification, lifting value balancing, and path-following. Techniques
of forming closure/compatibility equations by direct use of symmetry,
or by use of transfer matrices to enforce loop closure, were developed as appropriate
for each example.
The geometry of a plane symmetric (rectangular) 6R foldable frame was examined
and classified in terms of Denavit-Hartenberg Parameters. Its design
parameters were then grouped into feasible and non-feasible regions, before
continuation was used as a design tool; generating the design parameters
required to build a foldable frame which meets certain configurational specifications.
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Two further deployable ring/frame classes were then used as design cases:
(a) rings which form (planar) regular polygons when deployed, and (b) rings
which are doubly plane symmetric and planar when deployed. The governing
equations used in the continuation design process are based on symmetry
compatibility and transfer matrices respectively.
Finally, the 6, 7 and 8-link versions of N-loops were subjected to a witness
set analysis, illustrating the way in which continuation can reveal the nature
of the mobility of an unknown linkage.
Key features of the results are that polynomial continuation was able to provide
complete sets of feasible options to a number of practical design problems,
and also to reveal the nature of the mobility of a real overconstrained
linkage
Towards printable robotics: Origami-inspired planar fabrication of three-dimensional mechanisms
This work presents a technique which allows the application of 2-D fabrication methods to build 3-D robotic systems. The ability to print robots introduces a fast and low-cost fabrication method to modern, real-world robotic applications. To this end, we employ laser-engraved origami patterns to build a new class of robotic systems for mobility and manipulation. Origami is suitable for printable robotics as it uses only a flat sheet as the base structure for building complicated functional shapes, which can be utilized as robot bodies. An arbitrarily complex folding pattern can be used to yield an array of functionalities, in the form of actuated hinges or active spring elements. For actuation, we use compact NiTi coil actuators placed on the body to move parts of the structure on-demand. We demonstrate, as a proof-of-concept case study, the end-to-end fabrication and assembly of a simple mobile robot that can undergo worm-like peristaltic locomotion.United States. Defense Advanced Research Projects Agency (Grant W911NF-08-C-0060)United States. Defense Advanced Research Projects Agency (Grant W911NF-08-1-0228
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