Enumeration of kinematic chains and mechanisms

Abstract

A problem still unsolved in kinematics is the enumeration of a complete list of kinematic chains and mechanisms without isomorphisms and without degenerate chains that operate in any screw system. In this paper, a method for the enumeration of kinematic chains without isomorphisms and degenerate chains for all screw systems and a new method of enumeration of kinematic chain inversions (i.e. mechanisms) based on group theory techniques are presented. New concepts of the group theory are introduced and a new method of enumeration of inversions is presented. Kinematic inversions are related to the symmetries of the chain which can be identified analysing the corresponding graph. The symmetry of a graph can be identified for the group of automorphisms of the graph and its orbits provides sets of vertices (links) that are in the same equivalence classes, i.e. they have the same properties of symmetry. The main definitions of group theory and examples of application of the new method of enumeration of inversions are presented. New results are obtained and divided in two classes: original results in non-planar screw systems (λ = 3) and results in agreement with a previously published list for planar kinematic chains and planar mechanisms (inversions). Two tables (1 and 3) are provided, which are up-to-date lists of kinematic chains and mechanisms for several screw systems.

Fractionation in planar kinematic chains: Reconciling enumeration contradictions

“…Simoni et al [6] present a variation of Sunkari and Schmidt's method for enumeration of kinematic chains with mobility 1 ≤ M ≤ 6 for several screw systems [6].…”

“…The methods of Sunkari and Schmidt [4] and Simoni et al [6] only present results for non-fractionated kinematic chains.…”

“…Eq. (1) is well known in mechanism and machine theory and it is often used in the structural synthesis of parallel robots [2][3][4][5][6]. However, it fails in some known cases [1].…”

“…Section 5 presents the results obtained by the new method. The table of planar kinematic chains presented in Simoni et al [6] is then presented for comparison of the results with those of other methods and to solve the contradictions found in literature. Simoni et al [6] consider only enumeration of nonfractionated kinematic chains for several screw systems.…”

“…The table of planar kinematic chains presented in Simoni et al [6] is then presented for comparison of the results with those of other methods and to solve the contradictions found in literature. Simoni et al [6] consider only enumeration of nonfractionated kinematic chains for several screw systems. The results of this paper are compared with those found in the literature for the enumeration of planar kinematic chains carried out in three ways; only non-fractionated, only fractionated and general (fractionated plus non-fractionated) kinematic chains that satisfy the mobility criteria.…”