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

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.

► The common link-pair and common Bennett-linkage methods are used in construction.
► Two constructive forms of double-subtractive-Goldberg 6R linkage are formed.
► The bifurcation behaviours of the linkages are analysed.
► Two non-constructive forms of double-subtractive-Goldberg 6R linkage are found.
► The resultant 6R linkage is related to Bricard linkages after isomerisation.