Passive modular groups with eight elements from inverse structural modeling of planar linkages

Abstract

Bi-mobile mechanisms commonly used in robotics are obtained as structural solutions from plane linkages with five degrees of freedom, in which the basis and effector are nominated and two active kinematic pairs of the system are positioned. The basis and the effector are determined by inverse structural modeling, the inverse model consisting exclusively of passive modular groups. It is obtained by connecting the effector with plane motion dependent on two independent parameters through a virtual kinematic pair at the basis canceling the degree of mobility.  The placement of active pairs is achieved by direct structural modeling, the direct model consisting of active and, eventually, passive modular groups. A mechanism performing with precision and accuracy of motion must have the inverse model and the direct model consisting of a minimum number of modular groups, as appropriate passive or active ones.The optimal inverse model is always a Baranov truss, from which the elimination of the basis results a single passive modular group. Bi-mobile mechanisms with three independent contours are obtained from the 40 kinematic chains with five degrees of freedom and three independent contours mentioned in the literature [Appendix 1]. They have a number of 8 mobile elements and 11 lower kinematic pairs, and the optimal inverse models, characterized by 9 elements and 12 kinematic pairs being Baranov trusses. The number of passive modular groups with 8 elements obtained from a Baranov truss with nine elements is C₉⁸ = C₉¹ = 9, from which indistinct solutions due to symmetrical elements are eliminated. Thus, there are 188 structures, of which 95 ones are found in the solutions of the inverse models mentioned and are specially marked.