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.

Copyright © 2015 by ASME
Topics: Linkages , Cycles
Your Session has timed out. Please sign back in to continue.


Tsai, L.-W., 2000, Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press, Boca Raton, FL.
Sunkari, R. P., and Schmidt, L. C., 2006, “Structural Synthesis of Planar Kinematic Chains by Adapting a McKay-Type Algorithm,” Mech. Mach. Theory, 41(9), pp. 1021–1030. [CrossRef]
McKay, B. D., 1998, “Isomorph-Free Exhaustive Generation,” J. Algorithms, 26(2), pp. 306–324. [CrossRef]
Ding, H., and Huang, Z., 2007, “The Establishment of the Canonical Perimeter Topological Graph of Kinematic Chains and Isomorphism Identification,” ASME J. Mech. Des., 129(9), pp. 915–923. [CrossRef]
Ding, H., Hou, F., Kecskeméthy, A., and Huang, Z., 2012, “Synthesis of the Whole Family of Planar 1-DOF Kinematic Chains and Creation of Their Atlas Database,” Mech. Mach. Theory, 47, pp. 1–15. [CrossRef]
Ding, H., Yang, W., Huang, P., and Kecskeméthy, A., 2013, “Automatic Structural Synthesis of Planar Multiple Joint Kinematic Chains,” ASME J. Mech. Des., 135(9), p. 091007. [CrossRef]
Tuttle, E. R., 1996, “Generation of Planar Kinematic Chains,” Mech. Mach. Theory, 31(6), pp. 729–748. [CrossRef]
Manolescu, N., 1973, “A Method Based on Baranov Trusses, and Using Graph Theory to Find the Set of Planar Jointed Kinematic Chain and Mechanisms,” Mech. Mach. Theory, 8(1), pp. 3–22. [CrossRef]
Verho, A., 1973, “An Extension of the Concept of the Group,” Mech. Mach. Theory, 8(2), pp. 249–256. [CrossRef]
Soh, G. S., and McCarthy, J. M., 2007, “Synthesis of Eight-Bar Linkages as Mechanically Constrained Parallel Robots,” 12th World Congress in Mechanism and Machine Science (IFToMM), Besancon, France, June 18–21. http://www.dmg-lib.org/dmglib/handler?docum=20412009
Perez, A., and McCarthy, J. M., 2005, “Clifford Algebra Exponentials and Planar Linkage Synthesis Equations,” ASME J. Mech. Des., 127(5), pp. 931–940. [CrossRef]
Soh, G. S., and McCarthy, J. M., 2008, “The Synthesis of Six-Bar Linkages as Constrained Planar 3R Chains,” Mech. Mach. Theory, 43(2), pp. 160–170. [CrossRef]
McCarthy, J. M., and Soh, G. S., 2010, Geometric Design of Linkages, Springer, New York.
Wampler, C. W., 2001, “Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex Plane Formulation,” ASME J. Mech. Des., 123(3), pp. 382–387. [CrossRef]
Dixon, A., 1909, “The Eliminant of Three Quantics in Two Independent Variables,” Proceedings of the London Mathematical Society , C. F. Hodgson & Son, London, UK, pp. 49–69.
Nielsen, J., and Roth, B., 1999, “Solving the Input/Output Problem for Planar Mechanisms,” ASME J. Mech. Des., 121(2), pp. 206–211. [CrossRef]
Dhingra, A. K., Almadi, A. N., and Kohli, D., 2001, “A Gröbner–Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms,” ASME J. Mech. Des., 122(4), pp. 431–438. [CrossRef]
Porta, J. M., Ros, L., and Thomas, F., 2009, “A Linear Relaxation Technique for the Position Analysis of Multiloop Linkages,” IEEE Trans. Rob., 25(2), pp. 225–239. [CrossRef]
Chase, T. R., and Mirth, J. A., 1993, “Circuits and Branches of Single-Degree-of-Freedom Planar Linkages,” ASME J. Mech. Des., 115(2), pp. 223–230. [CrossRef]
Myszka, D. H., Murray, A. P., and Wampler, C. W., 2012, “Mechanism Branches, Turning Curves and Critical Points,” ASME Paper No. DETC2012-70277. [CrossRef]
Kecskeméthy, A., Krupp, T., and Hiller, M., 1997, “Symbolic Processing of Multiloop Mechanism Dynamics Using Closed-Form Kinematics Solutions,” Multibody Syst. Dyn., 1(1), pp. 23–45. [CrossRef]
Whitney, H., 1932, “Non-Separable and Planar Graphs,” Trans. Am. Math. Soc., 34(2), pp. 339–362. [CrossRef]
Dijkstra, E. W., 1959, “A Note on Two Problems in Connexion With Graphs,” Numer. Math., 1(1), pp. 269–271. [CrossRef]
Silvester, J. R., 2000, “Determinants of Block Matrices,” Math. Gaz., 84(501), pp. 460–467. [CrossRef]


Grahic Jump Location
Fig. 1

Example Watt I linkage reaching a task position

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
Fig. 3

Example 8-bar adjacency matrix and adjacency graph

Grahic Jump Location
Fig. 4

The first level shortest path

Grahic Jump Location
Fig. 5

A second level shortest path

Grahic Jump Location
Fig. 6

Two second level shortest paths

Grahic Jump Location
Fig. 7

A third level shortest path

Grahic Jump Location
Fig. 8

Another third level shortest path

Grahic Jump Location
Fig. 9

The four unique cycles

Grahic Jump Location
Fig. 11

Double butterfly unique mechanisms

Grahic Jump Location
Fig. 12

Double butterfly unique linkages

Grahic Jump Location
Fig. 13

Naming convention, example 8-bar




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In