A NEW GEOMETRIC DERIVATION OF THE PLANE-SYMMETRIC RPRP LINKAGE OF DELASSUS

摘要

A plane symmetric RPRP linkage with special geometry is regarded as a special one of Delassus four-bar HPHP paradoxical linkages. Based on the theorem that any subset of translations is normal or invariant by conjugation in the group of a spatial translation, we introduce a new geometric derivation of the plane-symmetric RPRP paradoxical linkage of Delassus through a doubly ruled surface of revolute hyperboloid of one sheet by means of two special RPR open kinematic chains. An explicit vector calculation is done for verification. Two 2-DoF translation parallel mechanisms that are congruent by any given translation produce the same translational motion of their end effectors. From the plane symmetry of the hyperboloid, a novel parallel generator of hyperboloidal translation is obtained and a special case with coinciding R axes is considered. Then we geometrically derive a plane symmetric RPRP mechanism with 1-DoF finite paradoxical mobility.