This paper introduces some basic concepts and new approaches regarding the kinematic structure and functional analysis of mechanisms. The theory and approach are illustrated on shaft couplings involving pode joints. Kinematic structure of pode joints is given and some new concepts, such as multiple contacting points and effective and idle degrees-of-freedom, are introduced. A systematic approach which includes a modified graph representation and a modified degree-of-freedom equation is presented. Using this approach the mobility analysis of a class of difficult and complex mechanisms can be treated. Several specific examples are presented to illustrate the basic theory.

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