Multiple linkage forms and bifurcation behaviours of the double-subtractive-Goldberg 6R linkage

Abstract In this paper, a particular type of double-subtractive-Goldberg 6R linkage is obtained by combining two subtractive Goldberg 5R linkages on the commonly shared ‘roof-links’ through the common link-pair method and common Bennett-linkage method. Two distinct linkage forms are obtained with the identical geometry conditions, yet different closure equations. Bifurcation behaviours of these two forms are analysed, leading to the discovery of two more linkage forms of this linkage, which cannot be constructed with Bennett linkages or Goldberg linkages directly. From the construction process, this 6R linkage belongs to the Bennett-based linkages. But about the bifurcation behaviours, it is closely related to the line-symmetric Bricard linkage because of its hidden symmetric property. Therefore, it could play an important role in exploring the relationship between the Bennett-based linkages and the Bricard linkages.

[1]  Xianwen Kong,et al.  Type synthesis of single-DOF single-loop mechanisms with two operation modes , 2009, 2009 ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots.

[2]  G. T. Bennett,et al.  The Skew Isogram Mechanism , 1914 .

[3]  Yan Chen,et al.  A spatial 6R linkage derived from subtractive Goldberg 5R linkages , 2011 .

[4]  Yan Chen,et al.  The original double-Goldberg 6R linkage and its bifurcation analysis , 2011 .

[5]  Chaoyang Song,et al.  A family of mixed double-Goldberg 6R linkages , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Andreas Müller,et al.  Generic mobility of rigid body mechanisms , 2009 .

[7]  Zhong You,et al.  Threefold-symmetric Bricard linkages for deployable structures , 2005 .

[8]  J.Eddie Baker,et al.  On the generation of new linkages from Bennett loops , 1981 .

[9]  Kenneth J. Waldron.  Hybrid overconstrained linkages , 1968 .

[10]  J. Denavit,et al.  A kinematic notation for lower pair mechanisms based on matrices , 1955 .

[11]  J.Eddie Baker,et al.  A comparative survey of the bennett-based, 6-revolute kinematic loops , 1993 .

[12]  W W Gan,et al.  Numerical Approach to the Kinematic Analysis of Deployable Structures Forming a Closed Loop , 2006 .

[13]  Zhong You,et al.  Spatial 6R linkages based on the combination of two Goldberg 5R linkages , 2007 .

[14]  Sergio Pellegrino,et al.  Computation of kinematic paths and bifurcation points , 2000 .

[15]  S. Guest,et al.  A symmetry analysis of mechanisms in rotating rings of tetrahedra , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  R. Bricard.  Mémoire sur la théorie de l'octaèdre articulé , 1897 .

[17]  Patrick W. Fowler,et al.  A symmetry-extended mobility rule , 2005 .

[18]  K Wohlhart.  Merging two general goldberg 5R linkages to obtain a new 6R space mechanism , 1991 .

[19]  J. Dai,et al.  Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds , 1998 .

[20]  R. Bricard.  Leçons de cinématique , 1926 .

[21]  J.Eddie Baker,et al.  An analysis of the Bricard linkages , 1980 .

[22]  S. Pellegrino.  Structural computations with the singular value decomposition of the equilibrium matrix , 1993 .

[23]  F. E. Myard.  Contribution à la géométrie des systèmes articulés , 1931 .

[24]  A. Mu¨ller.  Geometric Characterization of the Configuration Space of Rigid Body Mechanisms in Regular and Singular Points , 2005 .

[25]  Yan Chen,et al.  The line-symmetric octahedral Bricard linkage and its structural closure , 2010 .

[26]  Zhong You,et al.  An Extended Myard Linkage and its Derived 6R Linkage , 2008 .

[27]  Woon Huei Chai,et al.  Bifurcation of a special line and plane symmetric Bricard linkage , 2011 .