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Research Papers

Design of a Linkage System to Write in Cursive

[+] Author and Article Information
Yang Liu

Robotics and Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: liuy14@uci.edu

J. Michael McCarthy

Professor
Fellow ASME
Robotics and Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: jmmccart@uci.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received June 16, 2017; final manuscript received June 26, 2017; published online July 20, 2017. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 17(3), 031015 (Jul 20, 2017) (8 pages) Paper No: JCISE-17-1120; doi: 10.1115/1.4037229 History: Received June 16, 2017; Revised June 26, 2017

This paper presents a design methodology for a system of linkages that can trace planar Bezier curves that represent cursive handwriting of the alphabet and Chinese characters. This paper shows that the standard degree n Bezier curve can be reparameterized so that it takes the form of a trigonometric curve that can be drawn by a one degree-of-freedom coupled serial chain consisting of 2n links. A series of cubic Bezier curves that define a handwritten name yields a series of six-link coupled serial chains that trace these curves. We then show how to simplify this system using cubic trigonometric Bezier curves to obtain a series of four-link serial chains that approximate the system of Bezier curves. The result is a methodology for the design of a mechanical system that draws complex plane curves such as the cursive alphabet and Chinese characters.

FIGURES IN THIS ARTICLE
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Copyright © 2017 by ASME
Topics: Linkages , Chain , Design
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References

Figures

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Fig. 1

The six-link coupled serial chain is constructed so that each sequence of two links, Lk and Mk, rotates in opposite directions at a multiple k of the base joint rate. The phase angles, ψk and ηk, define the initial configuration of the trigonometric Bezier curve.

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Fig. 2

Cubic trigonometric Bezier curve with λ = −0.5 in dashed compared to the cubic Bezier curve in solid and cubic trigonometric Bezier curve with λ = 0.5 in dashed compared to the cubic Bezier curve in solid

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Fig. 3

The cursive letters that define Yang are defined by eight cubic Bezier curves listed in Table 2

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Fig. 4

Comparison of curve 1 in Yang drawn by a six-link chain (solid) and by a four-link chain (dashed)

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Fig. 5

An example showing the vectors used to calculate the ground pivots Gj

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Fig. 6

The curves generated by six-link serial chains and the curves generated by corrected four-link serial chains

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Fig. 7

The comparison of curves generated by six-link and corrected four-link serial chains

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Fig. 8

The eight four-link coupled serial chains that draw the Bezier curves defining the cursive Yang

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Fig. 9

The linkage system that can draw Yang in cursive. These chains are driven by the same input.

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Fig. 10

Script form of the Chinese character long or dragon

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Fig. 11

Nine Bezier curves used to define the script form of the Chinese character long. The control points are listed in Table 6.

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Fig. 12

The cubic trigonometric Bezier curves with λ = −0.5 (dashed) fit the cubic Bezier curves (solid), and the cubic trigonometric Bezier curves with λ = 0.01 (dashed) vary from the cubic Bezier curves (solid)

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Fig. 13

The nine four-link coupled serial chains that draw the Bezier curves defining Chinese character long

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Fig. 14

The linkage system that draws the script form of the Chinese character long

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