A polynomial equation for a coupler curve of the double butterfly linkage

This paper presents a closed form polynomial equation for the path of a point fixed in the coupler links of the single degree-of-freedom eight-bar linkage commonly referred to as the double butterfly linkage. The revolute joint that connects the two Coupler links of this planar linkage is a special point on the two links and is chosen to be the coupler point. A systematic approach is presented to obtain the coupler curve equation, which expresses the Cartesian coordinates of the coupler point as a function of the link dimensions only; i.e., the equation is independent of the angular joint displacements of the linkage. From this systematic approach, the polynomial equation describing the coupler curve is shown to be, at most, forty-eighth order. This equation is believed to be an original contribution to the literature on coupler curves of a planar eight-bar linkage. The authors hope that this work will result in the eight-bar linkage playing a more prominent role in modern machinery.