Synthesis of a Spherical Cycloidal Mechanism

Abstract

Using rotational displacement matrices to describe spherical motions, closed-form synthesis equations are derived for two rotation matrix mechanisms. These equations are used to calculate the linkage parameters of a spherical cycloidal crank mechanism so that it will guide a rigid body through five finitely separated positions lying on the surface of a unit sphere. It is shown that the synthesis equation for calculating the coordinates of the fixed and moving points lying on the unit sphere is of fourth degree. This indicates that a maximum of four unique solutions of the mechanism are possible when five finitely separated positions are specified. A synthesis example is presented illustrating the proposed synthesis technique using a spherical cycloidal crank mechanism to guide a rigid body.