Using the Singularity Trace to Understand Linkage Motion Characteristics

This paper provides examples of a method used to analyze the motion characteristics of single-degree-of-freedom, closed-loop linkages under study a designated input angle and one or two design parameters. The method involves the construction of a singularity trace, which is a plot that reveals changes in the number of geometric inversions, singularities, and changes in the number of branches as a design parameter is varied. This paper applies the method to Watt II, Stephenson III and double butterfly linkages. For the latter two linkages, instances where the input angle is able to rotate more than one revolution between singularities have been identified. This characteristic demonstrates a net-zero, singularity free, activation sequence that places the mechanism into a different geometric inversion. Additional observations from the examples are given. Instances are shown where the singularity trace for the Watt II linkage includes multiple coincident projections of the singularity curve. Cases are shown where subtle changes to two design parameters of a Stephenson III linkage drastically alters the motion. Additionally, isolated critical points are found to exist for the double butterfly, where the linkage becomes a structure and looses the freedom to move.Copyright © 2013 by ASME and General Motors

[1]  F. Park,et al.  Singularity Analysis of Closed Kinematic Chains , 1999 .

[2]  T. Chase,et al.  Circuits and Branches of Single-Degree-of-Freedom Planar Linkages , 1990 .

[3]  David H. Myszka,et al.  Mechanism Branches, Turning Curves, and Critical Points , 2012 .

[4]  Raymond J. Cipra,et al.  Assembly Configurations and Branches of Planar Single-Input Dyadic Mechanisms , 1998 .

[5]  Charles W. Wampler,et al.  ISOTROPIC COORDINATES , CIRCULARITY , AND BEZOUT NUMBERS : PLANAR KINEMATICS FROM A NEW PERSPECTIVE , 1996 .

[6]  David H. Myszka,et al.  Singularity Analysis of an Extensible Kinematic Architecture: Assur Class N, Order N−1 , 2009 .

[7]  Henry P. Davis,et al.  Circuit analysis of Stephenson chain six-bar mechanisms , 1994 .

[8]  Charles W. Wampler,et al.  Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation , 2001 .

[9]  Bernard Roth,et al.  Solving the Input/Output Problem for Planar Mechanisms , 1999 .

[10]  Damien Chablat,et al.  Non-Singular Assembly-mode Changing Motions for 3-RPR Parallel Manipulators , 2008, ArXiv.

[11]  J. Schmiedeler,et al.  USING A SINGULARITY LOCUS TO EXHIBIT THE NUMBER OF GEOMETRIC INVERSIONS, TRANSITIONS AND CIRCUITS OF A LINKAGE , 2010 .

[12]  E. Allgower,et al.  Numerical path following , 1997 .

[13]  R. W. Daniel,et al.  An Explanation of Never-Special Assembly Changing Motions for 3–3 Parallel Manipulators , 1999, Int. J. Robotics Res..

[14]  C. Innocenti,et al.  Singularity-Free Evolution From One Configuration to Another in Serial and Fully-Parallel Manipulators , 1998 .

[15]  Andrew P. Murray,et al.  Synthesizing Single DOF Linkages Via Transition Linkage Identification , 2008 .

[16]  Thomas R. Chase,et al.  Circuit analysis of watt chain six-bar mechanisms , 1990 .

[17]  Hiroaki Katoh,et al.  Identification of motion domains of planar six-link mechanisms of the Stephenson-type , 2004 .

[18]  Faydor L. Litvin,et al.  Singularities in Motion and Displacement Functions of Constrained Mechanical Systems , 1989, Int. J. Robotics Res..

[19]  Charles W. Wampler,et al.  Solving the Kinematics of Planar Mechanisms , 1999 .

[20]  Damien Chablat,et al.  Singular curves and cusp points in the joint space of 3-RPR parallel manipulators , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

[21]  Charles Pinto,et al.  Transitions between Multiple Solutions of the Direct Kinematic Problem , 2008 .

[22]  Raymond J. Cipra,et al.  An Automatic Method for Finding the Assembly Configurations of Planar Non-Single-Input-Dyadic Mechanisms , 2002 .

[23]  Arthur G. Erdman,et al.  Mechanism Design : Analysis and Synthesis , 1984 .

[24]  Clément Gosselin,et al.  Singularity analysis of closed-loop kinematic chains , 1990, IEEE Trans. Robotics Autom..

[25]  Kwun-Lon Ting,et al.  Classification and branch identification of Stephenson six-bar chains , 1996 .

[26]  Damien Chablat,et al.  Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators , 2007, ArXiv.

[27]  Charles W. Wampler,et al.  Finding All Real Points of a Complex Curve , 2006 .

[28]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .

[29]  Damien Chablat,et al.  Workspace and Assembly modes in Fully-Parallel Manipulators : A Descriptive Study , 2007, ArXiv.

[30]  Jonathan D. Hauenstein,et al.  Cell decomposition of almost smooth real algebraic surfaces , 2013, Numerical Algorithms.

[31]  Andrew J. Sommese,et al.  Numerical algebraic geometry and algebraic kinematics , 2011, Acta Numerica.