This paper describes a mechanism design methodology that draws plane curves which have finite Fourier series parameterizations, known as trigonometric curves. We present three ways to use the coefficients of this parameterization to construct a mechanical system that draws the curve. One uses Scotch yoke mechanisms for each of the terms in the coordinate trigonometric functions, which are then added using a belt or cable drive. The second approach uses two-coupled serial chains obtained from the coordinate trigonometric functions. The third approach combines the coordinate trigonometric functions to define a single-coupled serial chain that draws the plane curve. This work is a version of Kempe's universality theorem that demonstrates that every plane trigonometric curve has a linkage which draws the curve. Several examples illustrate the method including the use of boundary points and the discrete Fourier transform to define the trigonometric curve.
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April 2017
Technical Briefs
Design of Mechanisms to Draw Trigonometric Plane Curves
Yang Liu,
Yang Liu
Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: liuy14@uci.edu
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: liuy14@uci.edu
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J. Michael McCarthy
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
Fellow ASME
Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: jmmccart@uci.edu
Search for other works by this author on:
Yang Liu
Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: liuy14@uci.edu
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
Fellow ASME
Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: jmmccart@uci.edu
Manuscript received October 20, 2016; final manuscript received January 13, 2017; published online March 9, 2017. Assoc. Editor: Venkat Krovi.
J. Mechanisms Robotics. Apr 2017, 9(2): 024503 (8 pages)
Published Online: March 9, 2017
Article history
Received:
October 20, 2016
Revised:
January 13, 2017
Citation
Liu, Y., and Michael McCarthy, J. (March 9, 2017). "Design of Mechanisms to Draw Trigonometric Plane Curves." ASME. J. Mechanisms Robotics. April 2017; 9(2): 024503. https://doi.org/10.1115/1.4035882
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