Kinematic Synthesis of Planar and Spatial Linkages Using Gradients and Superposition of Scalar Fields

Abstract

It is well known that the solutions to the three-position motion generation problem for the synthesis of the planar four-bar linkages results in a set of center-point and circle point circles. The four-position solution set yields a cubic curve, and the five-position solution yields a set of points. There are a variety of methods including graphical and algebraic, for generating center-point and circle-point circles for the 3-position, and Burmester curves for the 4-position planar motion generation problem. Dyads are synthesized based on these solution sets. In general, it is difficult or impossible to generalize these methods to 3D mechanisms. This paper begins by briefly reviewing the geometric constructions used in classical Burmester theory. Then, it presents a method using a scalar field to find the center-point and circle-point circles by finding the points where these fields are equal. The developed method uses Euclid’s chord angle principle to generate the solutions. This principle is then generalized to 3D to form the cone angle principle. This principle is used in a manner directly analogous to the planar case to synthesize the S-S dyad. Similar gradient and superposition methods using scalar fields based on the equations for the spherical R-R dyad, and the spatial C-C dyad are used to generate the spatial equivalent to the center-point circles and Burmester curves. Although these results have been known for some time, the method for generating these solution sets is new for spatial synthesis.