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research-article

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages via Kinematic Mapping

[+] Author and Article Information
Xiangyun Li

Southwest Jiaotong University Chengdu, China
xiangyun.app@gmail.com

Ping Zhao

Hefei University of Technology Hefei, China
ping.zhao@hfut.edu.cn

Anurag Purwar

Stony Brook University Stony Brook, USA
anurag.purwar@stonybrook.edu

Qiaode Jeffrey Ge

Stony Brook University Stony Brook, USA
qiaode.ge@stonybrook.edu

1Corresponding author.

ASME doi:10.1115/1.4038305 History: Received April 28, 2017; Revised October 17, 2017

Abstract

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, a pencil of general manifolds that best fit the task image points in the least squares sense can be found using Singular Value Decomposition, and the singular vectors associated with the smallest singular values are used to form the null space solution; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constrains and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

Copyright (c) 2017 by ASME
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