This paper presents the explicit mapping relations between topological structure and position and orientation characteristic (POC) of mechanism motion output. It deals with (1) the symbolic representation and the invariant property of the topological structure of the mechanism, (2) the matrix representation of POC of mechanism motion output, and (3) the POC equations of serial and parallel mechanisms and the corresponding symbolic operation rules. The symbolic operation involves simple mathematic tools and fewer operation rules and has clear geometrical meaning, so it is easy to use. The POC equations cannot only be used for structural analysis of the mechanism (such as determining POC of the relative motion between any two links of a mechanism and the rank of single-loop kinematic chain and calculating the full-cycle DOF of a mechanism, etc.) but can be used for structural synthesis of the mechanism as well (e.g., structural synthesis of the rank-degenerated serial mechanism, the over constrained single-loop mechanism, and the rank-degenerated parallel mechanism, etc.).
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