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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.).

Position and orientation characteristic (POC) equations for topological structure synthesis of serial and parallel mechanisms were proposed in a published paper by the authors. This paper will further prove the correctness and strictness of the theoretical foundation for POC equations and also be a reply to the reviewers of our follow-up papers. The main contents of this paper include: symbolic representation of mechanism topological structure and its invariance, velocity characteristic (VC) set and POC set of link and its invariance, one-to-one correspondence between elements of the VC set and POC set, POC equations for the serial mechanism and 10 corresponding "union" operation rules, and POC equations for the parallel mechanism and 14 corresponding "intersection" operation rules. In addition, the interrelations and differences among three methods (POC set based method, screw theory based method, and displacement subgroup based method) for mechanism topological structure design are concluded.

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This study presents a solution for the inverse kinematics problem in serial 6R manipulator. Using only seven equations-composed of Duffy's four kinematical equations containing three angles and three corresponding angles' identical equations-instead of the traditional 14 equations, the authors reduced the inverse kinematics problem in the general 6R manipulator to a univariate polynomial with a minimum degree based on the Groebner Base method. From that, they concluded that the maximum number of the solutions is 16, generally. Also, the mathematics mechanization method can be extended to solve other mechanism problems involving nonlinear equations symbolically.