A Graph-Theory-Based Method for Topological and Dimensional Representation of Planar Mechanisms as a Computational Tool for Engineering Design

Abstract

In this paper, a graph-theory-based approach for representing planar mechanisms is presented, the Santiago Portilla method (SPM). From the corresponding adjacency matrix, SPM generates an extended matrix containing the complete characterization of a planar mechanism, including all the information about both topology and geometry. This matrix representation can be used for the optimal design of mechanisms, allowing simultaneously the topological and dimensional synthesis by means of computational tools such as the metaheuristic algorithms. A case study corresponding to the design of a fixed-linear-trajectory tracker mechanism is included in order to test the efficiency of the proposed approach. It was carried out by addressing the design as an optimization problem and solving it with the differential evolution algorithm, representing the individuals in its population by the matrix form generated by the SPM. The results of the case study show that the SPM and its matrix representation constitute a useful and flexible tool for the solution of the real engineering problems involving the design of planar mechanisms.

An Innovative Optimization Design Procedure for Mechatronic Systems with a Multi-Criteria Formulation

“…A specific research line has been addressing the development of topology applications. For example, in [19] with a topology synthesis procedure to provide a complete atlas, in [20] with a description of topology and kinematics of mechanisms to address topology synthesis and particularization of a solution automatically, and in [21] with simultaneous shape and topology optimization for planar mechanisms.…”

A new synthesis method for planar Rubik’s cube mechanism

“…Crossley 20 introduced graph theory into mechanism for the first time to represent the topological structure of kinematic chain and presented a simple algorithm to solve Gruebler's theory for constrained kinematic chains. Santiago-Valentín et al 21 presented a novel Santiago-Portilla Method of representing any type of planar mechanisms, independently of the number of elements or the link type. Slaboch et al 22 first introduced a method of matrix representations contained the five topological characteristics, including the connectivity of the links and joints, the type of joints, the kinematic orientation of the joints, the fixed links, and the degrees of freedom of the mechanism.…”