Cayley Configuration Spaces of a Common Class of Mechanisms in Two Dimensions

Abstract

We study Cayley configuration spaces of a class of 1 degree-of-freedom linkages (graphs with specified edge lengths), obtained by dropping an edge from a tree-decomposable graph. The class includes well-known mechanisms based on the four-bar, as well as strandbeest, cardioid, limacon etc. The Cayley configuration space is the set of intervals of attainable lengths for a \emph{base} nonedge (e.g. the dropped edge) over the linkage's 2 dimensional realizations. We require \emph{quadratic radical solvability (QRS)} (an extension of ruler-and-compass-realizability) of the interval endpoints, and tree-decomposability guarantees efficient, ruler-and-compass construction of the linkage realization, given the Cayley configuration. Due to these restrictions of Kempe universality, this class of \emph{low Cayley complexity (LCC)} graphs is common in mechanical computer aided design and kinematics. Our main contributions are the following. (1) We show that the definition of LCC is robust, and depends only on the graph, no matter the choice of base nonedge whose addition ensures tree-decomposability. (2) We give an efficient algorithmic characterization of LCC graphs (3) We show (graph) planarity is equivalent to LCC for a natural subclass of 1-degree-of-freedom tree-decomposable graphs. Counterexamples show impossibility of such finite forbidden minor characterizations when the above subclass is enlarged. (4) We give an easily testable definition of genericity of LCC linkages (i.e. with underlying LCC graphs) based on their edge lengths. (5) For generic LCC linkages, we give an algorithm to find both paths of continuous motion (provided they exist) between two distinct realizations, in time linear in a discrete measure of the length of the path. Nontrivial generalizations of these results to non-LCC, 1-degree-of-freedom tree-decomposable linkages. Several accessible open problems are posed.