5. Position and Orientation Characteristics Equation for Parallel Mechanisms

Position and orientation characteristics (POC) equation for parallel mechanisms (PMs) is introduced in this chapter. The content deals with: (1) Based on velocity composition principle and topological structure invariance of PMs (Chap. 2), the unit vector set of moving platform velocity is “intersection” of unit vector sets of the end link velocity of its each branch, which depends only on the topological structure (excluding singular positions) of a mechanism (Chap. 3). (2) Since the unit vector set of velocity could be rewritten as the form of velocity characteristic (Chap. 3), the velocity characteristics (VC) equation and its operation rules for PMs are derived. (3) Based on one-to-one correspondence between elements of the POC set and elements of the VC set (excluding singular positions), the POC equation for PMs and the “intersection” operation rules of POC sets (twelve linear rules and two nonlinear criteria) are obtained. However, determination of the POC set always involves DOF calculation and inactive pair judgment (refer to Chap. 6). (4) The POC equation is independent of motion position (excluding singular positions), and it is not necessary to establish the fixed coordinate system. (5) Complex branches can be replaced by their topologically equivalent SOC branches in order to simplify POC set calculation of PMs. (6) This POC equation could be used for determining POC set of PMs when its topological structure is known (Sect. 5.5) , and can be used in structure synthesis, i.e. determining topological structure of a PM when its POC set and DOF are known (refer to Chap. 9). (7) This POC equation will be used for building general DOF formula for spatial mechanisms (refer to Chap. 6).