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Research Papers

Computing the Branches, Singularity Trace, and Critical Points of Single Degree-of-Freedom, Closed-Loop Linkages

[+] Author and Article Information
David H. Myszka

Department of Mechanical
and Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka1@udayton.edu

Andrew P. Murray

Department of Mechanical
and Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: amurray1@udayton.edu

Charles W. Wampler

General Motors R&D Center,
Warren, MI 48090
e-mail: charles.w.wampler@gm.com

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 7, 2013; final manuscript received September 6, 2013; published online December 27, 2013. Assoc. Editor: J. M. Selig.

J. Mechanisms Robotics 6(1), 011006 (Dec 27, 2013) (10 pages) Paper No: JMR-13-1036; doi: 10.1115/1.4025752 History: Received February 07, 2013; Revised September 06, 2013

This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.

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References

Erdman, A.Sandor, G., and Kota, S., 2001, Mechanism Design: Analysis and Synthesis, Vol. 1, No. 4/e, Prentice–Hall, Englewood Cliffs, NJ.
Kovacs, P., and Mommel, G., 1993, “On the Tangent-Half-Angle Substitution,” Computational Kinematics, J.Angeles, ed., Kluwer Academic Publishers, Norwell, MA, pp. 27–40.
PortaJ., Ros, L., Creemers, T., and Thomas, F., 2007, “Box Approximations of Planar Linkage Configuration Spaces,” ASME J. Mech. Des., 129(4), pp. 393–405. [CrossRef]
Wampler, C. W., 1999, “Solving the Kinematics of Planar Mechanisms,” ASME J. Mech. Des., 121(3), pp. 387–391. [CrossRef]
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]
Wampler, C. W., 1996, “Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics From a New Perspective,” Proceedings of the ASME Design Technical Conference, Paper No. 96-DETC/MECH-1210.
Nielsen, J., and Roth, B., 1999, “Solving the Input/Output Problem for Planar Mechanisms,” ASME J. Mech. Des., 121(2), pp. 206–211. [CrossRef]
Wampler, C. W., and Sommese, A. J., 2011, “Numerical Algebraic Geometry and Algebraic Kinematics,” Acta Numer., pp. 469–567. [CrossRef]
Chase, T., and Mirth, J., 1993, “Circuits and Branches of Single-Degree-of-Freedom Planar Linkages,” ASME J. Mech. Des., 115(2), pp. 223–230. [CrossRef]
Foster, D., and Cipra, R., 1998, “Assembly Configurations and Branches of Planar Single-Input Dyadic Mechanisms,” ASME J. Mech. Des., 120(3), pp. 381–386. [CrossRef]
Foster, D., and Cipra, R., 2002, “An Automatic Method for Finding the Assembly Configurations of Planar Non-Single-Input Dyadic Mechanisms,” ASME J. Mech. Des., 124(3), pp. 58–67. [CrossRef]
Mirth, J., and Chase, T., 1993, “Circuit Analysis of Watt Chain Six-Bar Mechanisms,” ASME J. Mech. Des., 115(2), pp. 214–222. [CrossRef]
Davis, H., and Chase, T., 1994, “Circuit Analysis of Stephenson Chain Six-Bar Mechanisms,” Proceedings of the ASME Design Technical Conferences, DE-Vol. 70, pp. 349–358.
Wantanabe, K., and Katoh, H., 2004, “Identification of Motion Domains of Planar Six-Link Mechanisms of the Stephenson-Type,” Mech. Mach. Theory, 39, pp. 1081–1099. [CrossRef]
Ting, K. L., and Dou, X., 1996, “Classification and Branch Identification of Stephenson Six-Bar Chains,” Mech. Mach. Theory, 31(3), pp. 283–295. [CrossRef]
Litvin, F., and Tan, J., 1989, “Singularities in Motion and Displacement Functions of Constrained Mechanical Systems,” Int. J. Robot. Res., 8(2), pp. 30–43. [CrossRef]
Gosselin, C., and Angeles, J., 1990, “Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290. [CrossRef]
Murray, A., Turner, M., and Martin, D., 2008, “Synthesizing Single DOF Linkages via Transition Linkage Identification,” ASME J. Mech. Des., 130(2), p. 022301. [CrossRef]
Innocenti, C., and Parenti-Castelli, V., 1998, “Singularity-Free Evolution From One Configuration to Another in Serial and Fully Parallel Manipulators,” ASME J. Mech. Des., 120(1), pp. 73–99. [CrossRef]
McAree, P. R., and Daniel, R. W., 1999, “An Explanation of Never-Special Assembly Changing Motions for 3-3 Parallel Manipulators,” Int. J. Robot. Res., 18(6), pp. 556–574. [CrossRef]
Wenger, P., and Chablat, D., 1998, “Workspace and Assembly-Modes in Fully Parallel Manipulators: A Descriptive Study,” Advances in Robot Kinematics, Kluwer Academic Publishers, Norwell, MA, pp. 117–126.
Wenger, P., Chablat, D., and Zein, M., 2007, “Degeneracy Study of the Forward Kinematics of Planar 3-RPR Parallel Manipulators,” ASME J. Mech. Des., 129(12), pp. 1265–1268. [CrossRef]
Zein, M., Wenger, P., and Chablat, D., 2006, “Singular Curves and Cusp Points in the Joint Space of 3-RPR Parallel Manipulators,” Proceedings of IEEE International Conference on Robotics and Automation.
Zein, M., Wenger, P., and Chablat, D., 2008, “Non-Singular Assembly Mode Changing Motions for 3-RPR Parallel Manipulators,” Adv. Rob. Kinematics, 43(4), pp. 480–490.
Macho, E., Altuzarra, O., Pinto, C., and Hernandez, A., 2008, “Transitions Between Multiple Solutions of the Direct Kinematic Problem,” Advances in Robot Kinematics, Springer, The Netherlands, Part 5, pp. 301–310.
MerletJ.-P., 2007, “A Formal-Numerical Approach for Robust In-Workspace Singularity Detection,” IEEE Trans. Rob., 23(3), pp. 393–402. [CrossRef]
BohigasO., Manubens, M., and Ros, L., 2013, “Singularities of Non-Redundant Manipulators: A Short Account and a Method for Their Computation in the Planar Case,” Mech. Mach. Theory, 68(1), pp. 1–17. [CrossRef]
Lu, Y., Bates, D. J., Sommese, A. J., and Wampler, C. W., 2007, “Finding all Real Points of a Complex Curve,” Proceedings of Midwest Algebra, Geometry and Its Interactions Conference, Contemporary Mathematics, AMS, Vol. 448, pp. 183–205.
Besana, G. M., Di Rocco, S., Hauenstein, J. D., Sommese, A. J., and Wampler, C. W., 2013, “Cell Decomposition of Almost Smooth Real Algebraic Surfaces, Numerical Algorithms” (in press).
Bates, D. J., Hauenstein, J. D., Sommese, A. J., and Wampler, C. W., Bertini, “Software for Numerical Algebraic Geometry,” Available at http://www.nd.edu/∼sommese/bertini
Zhou, H., and CheungE. H. M., 2004, “Adjustable Four-Bar Linkages for Multi-Phase Motion Generation,” Mech. Mach. Theory, 39(3), pp. 261–279. [CrossRef]
Naik, D. P., and Amarnath, C., 1989, “Synthesis of Adjustable Four Bar Function Generators through Five Bar Loop Closure Equations,” Mech. Mach. Theory, 24(6), pp. 523–526. [CrossRef]
Lin, L., MyszkaD., Murray, A., and Wampler, C., 2013, “Using the Singularity Trace to Understand Linkage Motion Characteristics,” Proceedings of the ASME International Design Engineering Technical Conferences, Paper No. DETC2013-13244.
Sommese, A. J., and Wampler, C. W., 2005, Numerical Solution of Systems of Polynomials Arising In Engineering and Science, World Scientific Press, Singapore.
Allgower, E. L., and Georg, K., 1997, “Numerical Path Following,” Handbook of Numerical Analysis, Vol. V, North-Holland, Amsterdam.
PennockG., and Kassner, D., 1992, “Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-Type Robot,” ASME J. Mech. Des., 114(1), pp. 87–95. [CrossRef]

Figures

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Fig. 1

A four-bar mechanism

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Fig. 2

The motion curve for a four-bar mechanism as expressed in Eq. (7) or Eq. (8)

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Fig. 3

Projections of the motion curve from Fig. 2. Singularities, shown with circular markers, occur where tangents have no x component. (In this case, x = θ2.)

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Fig. 4

Stephenson III linkage position vector loop

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Fig. 5

Projection of the Stephenson III critical curve. Circular markers designate the critical points. Regions of equal GIs and circuits are identified.

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Fig. 6

A Stephenson III (a7 = 11.0) having a net-zero actuation that places the linkage in an alternate GI

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Fig. 7

A cusp on a singularity trace in (a) with the associated a motion curves, above the cusp in (b) and below the cusp in (c)

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