Elsevier

Computer-Aided Design

Volume 46, January 2014, Pages 205-210
Computer-Aided Design

Technical note
How the Beast really moves: Cayley analysis of mechanism realization spaces using CayMos

https://doi.org/10.1016/j.cad.2013.08.033Get rights and content

Highlights

  • Bijective representation of connected components as curves in a minimal ambient dimension.

  • Defining and finding the “distance” between two different connected components.

  • Analysis and visualization of realization spaces for well-studied mechanisms.

Abstract

For a common class of 2D mechanisms called 1-dof tree decomposable linkages, the following fundamental problems have remained open: (a) How to canonically represent (and visualize) the connected components in the Euclidean realization space. (b) How to efficiently find two realizations representing the shortest “distance” between two connected components. (c) How to classify and efficiently find all the connected components, and the path(s) of continuous motion between two realizations in the same connected component, with or without restricting the realization type (sometimes called orientation type).

For a subclass of 1-dof tree-decomposable linkages that includes many commonly studied 1-dof linkages, we solve these problems by representing a connected component of the Euclidean realization space as a curve in a carefully chosen Cayley (non-edge distance) parameter space; and proving that the representation is bijective. We also show that the above set of Cayley parameters is canonical for all generic linkages with the same underlying graph, and can be found efficiently. We add an implementation of these theoretical and algorithmic results into the new software CayMos, and give (to the best of our knowledge) the first complete analysis, visualization and new observations about the realization spaces of many commonly studied 1-dof linkages such as the amusing and well-known Strandbeest, Cardioid, Limacon and other linkages.

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 (G,l̄), is a graph G=(V,E) with fixed length bars as edges, i.e.  l̄:ER. A 2D Euclidean realization or configuration G(p) of (G,l̄) is an assignment of points p:VR2 to the vertices of G satisfying the bar lengths in l̄, 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 O(|V|) 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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