Simultaneous type and dimensional synthesis approach with expandable solution space for planar linkages

Classic Burmester theory states that up to 4 exact solutions can be found for dyad synthesis problem with five prescribed poses. Also, numerous approaches have been proposed to obtain the optimal approximated solution for more than five prescribed poses. However, many cases have shown that even five given poses could result in no exact solution, or for more than five poses, the mathematical optimal solution might not be suitable under practical conditions. In those situations, a designer usually wants to gradually lower the accuracy requirement so as to bring more approximated solutions into consideration. This paper proposed an N-pose motion synthesis approach with expandable solution space for planar linkages. Based on kinematic mapping theory, the optimal joint type and linkage dimensions for planar dyads or triads can be simultaneously obtained. The proposed work mainly focused on the situation that no exact solution exists for five given poses, or that mathematical optimal solution cannot satisfy practical requirements. In those two cases, our approach showed that the solution space can be expanded by introducing or gradually increasing error tolerance, and hereby we could gradually obtain more approximate solutions, which will then be utilized to determine the best-suited dyads subject to various practical constraints. Finally, four-bar linkages or parallel linkage systems can be constructed to approximate the N-pose given motion and satisfy practical constraints as well.