Kinematic-mapping based solution to various mixed-exact-and-approximated problem in planar motion synthesis

The design of mechanisms that lead a rigid-body through a set of prescribed discrete poses is usually referred to as "motion synthesis". In practical motion synthesis cases, aside of realizing a set of given poses, various types of geometric constraint conditions could also require to be satisfied, e.g. defining the coordinates of the center/circle points of dyad linkages, setting the ground line/coupler line for four-bar linkages, realization of additional task positions, etc. Some of these constraint conditions require to be realized exactly while others might allow approximation. To solve this mixed-exact-and-approximated problem, this paper proposed a kinematic-mapping-based approach, which builds on the previous work of the realization of an arbitrary number of approximated poses as well as up to four exact poses. We now have found that the aforementioned various types of constraint conditions could be converted to each other through a general linear constraint equation. Thus, those "approximated conditions" could be uniformly converted to several prescribed discrete poses so as to be formulated as a general approximated motion synthesis problem, which is actually a general quadratic surface fitting problem in kinematic-mapping space, while up to four "exact conditions" could be imposed as linear constraint equations to this surface fitting system such that they could be exactly realized. Through null-space analysis technique, both type and dimensions of the resulting optimal dyad linkages could be determined by the solution of this surface-fitting problem with constraints. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators.