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Research Papers

Automated Generation of Linkage Loop Equations for Planar One Degree-of-Freedom Linkages, Demonstrated up to 8-Bar

[+] Author and Article Information
Brian E. Parrish

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

J. Michael McCarthy

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

David Eppstein

Information and Computer Sciences,
Department of Computer Science,
University of California,
Irvine, CA 92697
e-mail: eppstein@ics.uci.edu

1Corresponding author.

Manuscript received September 26, 2014; final manuscript received November 24, 2014; published online December 31, 2014. Assoc. Editor: Thomas Sugar.

J. Mechanisms Robotics 7(1), 011006 (Feb 01, 2015) (8 pages) Paper No: JMR-14-1260; doi: 10.1115/1.4029306 History: Received September 26, 2014; Revised November 24, 2014; Online December 31, 2014

In this paper, we present an algorithm that automatically creates the linkage loop equations for planar one degree of freedom, 1DOF, linkages of any topology with revolute joints, demonstrated up to 8 bar. The algorithm derives the linkage loop equations from the linkage adjacency graph by establishing a rooted cycle basis through a single common edge. Divergent and convergent loops are identified and used to establish the fixed angles of the ternary and higher links. Results demonstrate the automated generation of the linkage loop equations for the nine unique 6-bar linkages with ground-connected inputs that can be constructed from the five distinct 6-bar mechanisms, Watt I–II and Stephenson I–III. Results also automatically produced the loop equations for all 153 unique linkages with a ground-connected input that can be constructed from the 71 distinct 8-bar mechanisms. The resulting loop equations enable the automatic derivation of the Dixon determinant for linkage kinematic analysis of the position of every possible assembly configuration. The loop equations also enable the automatic derivation of the Jacobian for singularity evaluation and tracking of a particular assembly configuration over the desired range of input angles. The methodology provides the foundation for the automated configuration analysis of every topology and every assembly configuration of 1DOF linkages with revolute joints up to 8 bar. The methodology also provides a foundation for automated configuration analysis of 10-bar and higher linkages.

FIGURES IN THIS ARTICLE
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Copyright © 2015 by ASME
Topics: Linkages , Cycles
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Figures

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Fig. 1

Example Watt I linkage reaching a task position

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Fig. 2

Double butterfly 8-bar adjacency matrix, graph, and linkage sketch

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Fig. 3

Example 8-bar adjacency matrix and adjacency graph

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Fig. 4

The first level shortest path

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Fig. 5

A second level shortest path

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Fig. 6

Two second level shortest paths

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Fig. 7

A third level shortest path

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Fig. 8

Another third level shortest path

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Fig. 9

The four unique cycles

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Fig. 11

Double butterfly unique mechanisms

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Fig. 12

Double butterfly unique linkages

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Fig. 13

Naming convention, example 8-bar

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