A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

Abstract

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

A Task-Driven Approach to Optimal Synthesis of Planar Four-Bar Linkages for Extended Burmester Problem

“…In this section, we review a unified form of the kinematic constraints of four types of dyads (RR, PR, RP, and PP) in the image space; see Ref. [10] for details. A point X or line L on the coupler of a four-bar linkage can be geometrically constrained in one of the following four ways: (1) for an RR dyad, the point is constrained to be on a circle with center and radius given as homogeneous coordinates (a 0 , a 1 , a 2 , a 3 ); (2) for a PR dyad, the point is constrained to be on a fixed line having coordinates (L 1 , L 2 , L 3 ); (3) for an RP dyad, a moving line (l 1 , l 2 , l 3 ) is constrained to be tangent to a circle (a 1 , a 2 , a 3 ); and (4) for PP dyad, a point on line (L 1 , L 2 , L 3 ) is constrained to move along another line (2a 1 , 2a 2 , a 3 ).…”

“…This paper is a continuation of our research [10][11][12], wherein we have presented a task-driven approach to simultaneous type and dimensional synthesis of planar dyads for the motion generation problem. A four-bar linkage is constructed as combination of any two of the synthesized dyads.…”

“…A four-bar linkage is constructed as combination of any two of the synthesized dyads. This dyadic construction simplifies the synthesis process and renders the method as modular building block for synthesis of mechanisms with more links such as six-bar mechanisms [10]. By using the concepts of kinematic mapping [13,14] and planar quaternions [15,16], we obtained a unified form of kinematic constraints of the planar dyads and created an algorithm for unified type and dimensional synthesis of planar four-bar linkages.…”

“…The original contribution of this paper is in the reformulation of our framework [10][11][12] in a general way to extend the Burmester problem by accommodating a variety of geometric constraints. In addition, the new formulation solves problems which our previous approach could not solve.…”

Enabling Machine Design Innovation among Freshman Mechanical Engineering Students

“…The input task is a planar motion given as a set of discrete positions and orienta- tions (referred to as a pose, hereafter) and the app computes type-and dimensions of synthesized planar four-bar linkages, where their coupler interpolates through the given poses either exactly or approximately while minimizing an algebraic fitting error. The algorithm used in the app is an implementation of our simultaneous type and dimensional synthesis approach presented in Ge and Purwar, 28 Ge et al 29,30 In essence, the algorithm extracts the geometric constraints (circular, fixedline or line-tangent-to-a-circle) implicit in a given motion and matches them with corresponding mechanical dyad types enumerated earlier. In the process, the dimensions of the dyads are also computed.…”