Cayley configuration spaces of 2D mechanisms, Part I: extreme points, continuous motion paths and minimal representations

Describing configuration spaces of 1-degree-of-freedom (1-dof) linkages in 2D is a difficult problem with a long history. We use the Cayley configuration space, a set of intervals of possible distance-values for an independent non-edge. We study the following complexity measures (associated with the linkage’s underlying graph): (a) Cayley size, i.e., the number of intervals, (b) Cayley computational complexity of computing the interval endpoints, as a function of the number of intervals, (c) Cayley (algebraic) complexity of describing the interval endpoints. In both parts of this paper, we restrict ourselves to 1-dof linkages obtained by dropping a bar from minimally rigid, tree-decomposable linkages, widely used in engineering and CAD, because they are quadratically-radically solvable (QRS, also called ruler-and-compass realizable). In Part I of this paper, we give an algorithm to determine the interval endpoints of a Cayley configuration space of a 1-dof tree-decomposable linkage, by characterizing the Cartesian realizations corresponding to these endpoints. We then focus on graphs with low Cayley (algebraic) complexity. For corresponding generic linkages (as defined here), we show how to find a path of continuous motion (provided one exists) between two given realizations, in time linear in a natural measure of the length of the path. We show that the number of such paths is at most two. In addition, we consider Measures (a) and (b) above for graphs with low Cayley complexity, and give a natural, minimal realization type, i.e. a minimal set of local orientations, whose specification guarantees Cayley size of 1 and O(|V |) Cayley computational complexity. As a corollary, given two realizations of a linkage with the same minimal realization type, a continuous motion path between them is guaranteed, which maintains the same minimal realization type. Specifying fewer local orientations results in a superpolynomial blow-up of both the Cayley size and computational complexity, provided P is different from NP. Preprint submitted to Journal of Symbolic Computation November 12, 2012