Direct Position Analysis of a Large Family of Spherical and Planar Parallel Manipulators with Four Loops

The direct position analysis (DPA) of a manipulator is the computation of the end-effector poses (positions and orientations) compatible with assigned values of the actuated-joint variables. Assigning the actuated-joint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator one-to-one correspond to the assembly modes of the structure that is generated by locking the actuated-joint variables of that manipulator. Determining the assembly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topology which are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links and four binary links) connected through twelve revolute pairs to form four closed loops.