6. General Formula of Degrees of Freedom for Spatial Mechanisms

The general DOF formula which is of great importance to mechanism topology, kinematics and dynamics is introduced in this chapter. The content deals with: (1) The general DOF formula for PMs is proposed based on the topologically equivalent principle, POC equation of serial mechanisms in Chap. 4 and POC equation of PMs in Chap. 5. This DOF formula is independent of motion position of a PM and it is not necessary to establish the fixed coordinate system. Hence, it can guarantee that full-cycle DOF is obtained. (2) Based on the DOF formula, the criteria for determination of inactive pairs and the criteria for selection of driving pairs are proposed. (3) With replacement method for topological equivalent kinematic chains, DOF of general multi-loop spatial mechanisms can be calculated with this DOF formula very easily. (4) This DOF formula can be used for calculating DOF of PMs whose POC set contains non-independent element. (5) Based on the DOF formula, formulas for determining number of independent displacement equations, number of over-constraints and redundancy of mechanisms are derived. In brief , this DOF formula reveals the internal relations among topological structure, DOF and POC set of spatial mechanisms and will be used for topology design of PMs (refer to Chap. 9).