Series parallel linkages

Abstract

Abstract. We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.

Configuration Workspaces of Series-Parallel Mechanisms

“…If the constraints between the ground and end-effector form a seriesparallel graph [4], we say the mechanism is series-parallel. These definitions are consistent with those commonly found in literature [5,6]. Our method places no restriction on the type of constraints; examples include revolute or prismatic joints, irregularly shaped slots, and non-collision constraints.…”

“…In general, most approaches to solving workspaces of linkages can be addressed within the framework of series-parallel linkages. Series-parallel linkages were introduced by Cruickshank and McLaughlin [6], though it appears that some of the ideas involved have been around a bit earlier (albeit not in any formal presentation). For example, the ROMA algebraic constraint solver in Seybold's thesis uses a series-parallel reduction to simplify constraint solving in general assemblies [20].…”

The realizable extension problem and the weighted graph (K 3,3, l)

“…However, one recent development to this end, is a result contained in [10], and appears in [2], which gives realizability conditions for weighted graphs where the graph is contained in the class of series-parallel graphs. …”

“…Given a graph with preassigned edge lengths then a common problem is to determine if this weighted graph can be realized in E 2 . A graph G is a pair (V G , E G ) where V G , known as the vertex set of G, is a finite set, and E G , known as the edge set of G, is a multiset whose elements are elements of [V G ] 2 , the set of 2-element subsets of V G .…”

“…The fact that realizability conditions exist for the weighted graph (K 4 , k) appears to be something of a rarity as there does not appear to exist in the literature general realizability conditions, analogous to the (K 4 , k) case for other (non-trivial) weighted graphs. However, one recent development to this end, is a result contained in [10] (and will appear in [2]) which gives realizability conditions for weighted graphs where the graph is contained in the class of series-parallel graphs.…”