Unification of planar linkage synthesis through kinematic mapping

A general treatment which includes special cases of the five poses planar four-bar synthesis problem is investigated. There are 0, 2, or 4 real solutions to this problem formulated with five simultaneous equations. This method exposes solutions containing RR, PR, or RP dyads. The kinematic mapping is used to represent planar constraint equations in a three-dimensional projective image space because it offers a comprehensive way to solve problems in kinematic synthesis. The motion of a planar rigid body under a RR constraint can be mapped to points on a skew one-sheet hyperboloid with circles as generators which are generated by intersecting the hyperboloid with planes parallel to X3 = 0 in the image space. Similarly, a RP or PR dyad generates points on a unique hyperbolic paraboloid ruled by lines when intersected with planes parallel to X3 = 0. The five-pose four-bar linkage synthesis problem leads to five second-order constraint equations. By solving the constraint equations for RR dyads or four constraint equations for a RP or a PR dyad, four-bar linkage design parameters can be computed. In this thesis, examples which produce two RR dyads, four RR dyads and one RR dyad with one RP dyad will be analyzed. Furthermore, an approximate solution, with kinematic mapping, is given to the planar elliptical trammel synthesis problem