Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization via Kinematic Mapping

It has been well established that kinematic mapping theory could be applied in mechanism synthesis area, where discrete motion approximation problem could be converted to surface fitting problem of a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach for mixed exact and approximate motion realization problem for three types of planar dyad chains (RR, RP, PR) is presented. For all three types of dyads, N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, while up to four prescribed poses could be imposed as pose-constraint equations to this surface fitting system such that they could be exactly realized. The surface fitting problem is converted to a linear system with two quadratic constraint equations, which could be solved by null space analysis technique. On the other hand, the given exact poses are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.