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A Task Driven Approach to Unified Synthesis of Planar Four-bar Linkages using Algebraic Fitting of a Pencil of G-manifolds

[+] Author and Article Information
Q. J. Ge

Computational Design Kinematics Lab Department of Mechanical Engineering Stony Brook University Stony Brook, New York, 11794-2300
qiaode.ge@stonybrook.edu

Anurag Purwar

Computational Design Kinematics Lab Department of Mechanical Engineering Stony Brook University Stony Brook, New York, 11794-2300
anurag.purwar@stonybrook.edu

Ping Zhao

Computational Design Kinematics Lab Department of Mechanical Engineering Stony Brook University Stony Brook, New York, 11794-2300
pzhao@ic.sunysb.edu

Shrinath Deshpande

Computational Design Kinematics Lab Department of Mechanical Engineering Stony Brook University Stony Brook, New York, 11794-2300
shrinath.deshpande@stonybrook.edu

1Corresponding author.

ASME doi:10.1115/1.4035528 History: Received December 17, 2012; Revised December 12, 2016

Abstract

This paper studies the problem of planar four-bar motion approximation 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. 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.

Copyright (c) 2016 by ASME
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