| dc.description.abstract | 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. | |
