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Technical Brief

A Fourier Descriptor-Based Approach to Design Space Decomposition for Planar Motion Approximation

[+] Author and Article Information
Xiangyun Li

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China
e-mail: xiangyun.app@gmail.com

Jun Wu

Department of Quality Control,
DMG MORI Manufacturing USA, Inc.,
Davis, CA 95618

Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794

1Corresponding author.

Manuscript received December 25, 2015; final manuscript received April 18, 2016; published online September 6, 2016. Assoc. Editor: J. M. Selig.

J. Mechanisms Robotics 8(6), 064501 (Sep 06, 2016) (5 pages) Paper No: JMR-15-1349; doi: 10.1115/1.4033528 History: Received December 25, 2015; Revised April 18, 2016

In an earlier work, we have combined a curve fitting scheme with a type of shape descriptor, Fourier descriptor (FD), to develop a unified method to the synthesis of planar four-bar linkages for generation of both open and closed paths. In this paper, we aim to extend the approach to the synthesis of planar four-bar linkages for motion generation in an FD-based motion fitting scheme. Using FDs, a given motion is represented by two finite harmonic series, one for translational component of the motion and the other for rotational component. It is shown that there is a simple linear relationship between harmonic content of the rotational component and that of the translational component for a planar four-bar coupler motion. Furthermore, it is shown that the rotational component of the given motion identifies a subset of design parameters of a four-bar linkage including link ratios, while the translational component determines the rest of the design parameters such as locations of the fixed pivots. This leads naturally to a decomposed design space for four-bar mechanism synthesis for approximate motion generation.

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Figures

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Fig. 1

A rigid body with M as its moving frame and F as the fixed frame. Point (x, y) and angle θ represent the location and the orientation of M, respectively.

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Fig. 2

A four-bar mechanism

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Fig. 3

Fourier descriptors of a four-bar linkage at kω where k=−5,−4,−3,−2,−1,2,3,4,5

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Fig. 4

Four-bar linkage with ground link AD, crank AB, coupler link BC, and output link CD. AB=b,BC=c,CD=d,AD=a,BD=e.

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Fig. 5

The graph of task motion and synthesized motion in image space. The fitting error I1 is 3.7184×10−4 and I2 is 5.5131×10−5.

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Fig. 6

The graph of task motion and synthesized motion in image space. The fitting error I1 is 0.5244 and I2 is 5.6873×10−4.

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