Technical noteHow the Beast really moves: Cayley analysis of mechanism realization spaces using CayMos
Section snippets
Introduction and background
A key underlying barrier to understanding underconstrained geometric constraint systems is the classical problem of representing and efficiently finding the Euclidean realization spaces of 1-degree-of-freedom linkages, or mechanisms in 2D. A linkage , is a graph with fixed length bars as edges, i.e. . A 2D Euclidean realization or configuration of is an assignment of points to the vertices of satisfying the bar lengths in , modulo Euclidean motions.
Contributions
In this paper, we address the following questions for 1-dof tree-decomposable linkages with low Cayley complexity:
- (a)
How to canonically represent the connected components of the realization space as curves in an ambient space.
Theorem 3 gives such a bijective representation by characterizing a canonical set of non-edges, and proving that adding those non-edges results in global rigidity. Consequently, we obtain an ambient dimension in which each connected component bijectively corresponds to a
Canonical bijective representation of the connected components of the realization space as curves in an ambient space
In this section, we prove our main theorem, Theorem 3: for a 1-dof tree-decomposable linkage with low Cayley complexity, there is a bijective correspondence between the Cartesian realization space and complete Cayley distance vectors. The latter is a distance vector on base non-edges, defined before Lemma 2. This yields a canonical representation of the realization space. To define the complete Cayley distance vector, we need a theorem called the Four-cycle Theorem from [13] which gives
Conclusion
In this paper, we use CayMos to solve the problems (a)–(c) proposed in the abstract, for the natural and commonly occurring class of 1-dof tree-decomposable linkages with low Cayley complexity. Our future objective is to minimize the dimension of a complete Cayley vector. It has been shown in [12] that the minimum dimension is 2 for 1-path graphs, and a minimal complete Cayley vector has been shown for general 1-dof graphs of low Cayley complexity, assuming globally rigid clusters.
Some other
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