Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming

[+] Author and Article Information
Edward C. Kinzel

School of Mechanical Engineering,  Purdue University, West Lafayette, IN 47907

James P. Schmiedeler

Department of Mechanical Engineering,  The Ohio State University, Columbus, OH 43210

Gordon R. Pennock

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

J. Mech. Des 128(5), 1070-1079 (Oct 25, 2005) (10 pages) doi:10.1115/1.2216735 History: Received April 20, 2005; Revised October 25, 2005

This paper presents an original approach to the kinematic synthesis of planar mechanisms for finitely separated positions. The technique, referred to here as geometric constraint programming, uses the sketching mode of commercial parametric computer-aided design software to create kinematic diagrams. The elements of these diagrams are parametrically related so that when a parameter is changed, the design is modified automatically. Geometric constraints are imposed graphically through a well-designed user interface, and numerical solvers integrated into the software solve the relevant systems of equations without the user explicitly formulating those equations. This allows robust algorithms for the kinematic synthesis of a wide variety of mechanisms to be “programmed” in a straightforward, intuitive manner. The results provided by geometric constraint programming exhibit the accuracy and repeatability achieved with analytical synthesis techniques, while simultaneously providing the geometric insight developed with graphical synthesis techniques. The key advantages of geometric constraint programming are that it is applicable to a broad range of kinematic synthesis problems, user friendly, and highly accessible. To demonstrate the utility of the technique, this paper applies geometric constraint programming to three examples of the kinematic synthesis of planar four-bar linkages: Motion generation for five finitely separated positions, path generation for nine finitely separated precision points, and function generation for four finitely separated positions.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Five finitely separated positions for motion generation

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Figure 2

Congruent triangles ΔAiCiDi

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Figure 3

The center point curve for positions 1–4

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Figure 4

Center points OA and OB and circle points A and B

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Figure 5

The synthesized four-bar linkage and the coupler curve

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Figure 6

Nine finitely separated precision points for path generation

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Figure 7

The construction to locate the center points and circle points

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Figure 8

The synthesized four-bar linkage and a portion of the coupler curve

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Figure 9

Four finitely separated positions for function generation

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Figure 10

The synthesized four-bar linkage in an intermediate position



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