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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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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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