Application of Linear and Nonlinear Graphs in Structural Synthesis of Kinematic Chains

Abstract

Using graph theory and Polya’s theory of counting, the present paper performs structural synthesis and analysis of planar and three-dimensional kinematic chains. The Section 2 of the paper develops a mathematical model that permits one to perform structural analysis and synthesis of planar kinematic chains with kinematic elements such as revolute pairs, cam pairs, springs, belt-pulley, piston-cylinder, and gears. The theory developed is applied to enumerate eight-link kinematic chains with these kinematic elements. The Section 3 of the paper develops a mathematical model that permits one to perform structural analysis and synthesis of multi-loop spatial kinematic chains with higher and lower kinematic pairs. The theory developed is applied to enumerate all possible two-loop kinematic chains with or without general constraints.

On the number synthesis of kinematic chains

“…In 1973, Huang and Soni [26] performed structural synthesis and analysis of planar KCs by applying graph theory and Polya's theory. The proposed method enumerated 2-loop KCs, and the vertex-edge incidence matrix was used to test isomorphism.…”

“…In the 1960s, many researchers [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] demonstrated the feasibility of synthesizing the kinematic structure of mechanisms with closed chains by means of graph theory.…”

A Formalism for the Analysis and Design of Modular Kinematic Structures

“…During the past three decades, the kinematic structure of mechanisms has been investigated with the aid of linear graphs. This has provided a powerful tool in the early or conceptual stages of the design by making it possible to conduct a systematic search for the kinematic structures that satisfy given motion requirements (Dobrjanskyj and Freudenstein 1967;Davies 1968;Huang and Soni 1973;Freudenstein and Maki 1979). By way of comparison, other approaches are, more or less, based on a similar philosophy (Malik and Kerr 1992;Pikon and Wojnarowski 1995).…”

Identification of Cospectral Simple Jointed Kinematic Chains using Characteristic Polynomial of Degree Matrix and Structural Matrix

“…To address the problem of isomorphism in a very simple yet efficient way, graph theory plays a very important role. A. H. Soni et.al [1] used graph theory in structural synthesis of kinematic chains. Uicker et.al [2] discussed a method to address the kinematic equivalence of chains where the equality of identifying numbers is used to detect the kinematic equivalency.…”