This contribution presents new considerations on the theory of type synthesis of fully parallel platforms. These considerations prove that the theory of type synthesis of fully parallel platforms can be dealt with by analyzing two types of fully parallel platforms, where the displacements of the moving platform generate a subgroup of the Euclidean group, SE(3), including in this type, 6DOF parallel platforms and lower mobility platforms or, more precisely, parallel platforms where the displacements of the moving platforms generate only a subset of the Euclidean group. The theory is based on an analysis of the subsets and subgroups of the Euclidean group, SE(3), and their intersections. The contribution shows that the different types of parallel platforms are determined by the intersections of the subgroups or subsets, of the Euclidean group, generated by the serial connecting chains or limbs of the parallel platform. From an analysis of the intersections of subgroups and subsets of the Euclidean group, this contribution presents three possibilities for the type synthesis of fully parallel platforms where the displacements of the moving platform generate a subgroup of the Euclidean group, and two possibilities for the type synthesis of fully parallel platforms where the displacements of the moving platforms generate only a subset of the Euclidean group. An example is provided for each one of these possibilities. Thus, once these possible types of synthesis are elucidated, the type synthesis of fully parallel platforms is just reduced to the synthesis of the serial connecting chains or limbs that generate the required subgroups or subsets of the Euclidean group.

This contribution presents a comprehensive theory for the type synthesis of fully parallel platforms. The theory deals with both types of platforms, 6 D.O.F. parallel platforms and lower mobility platforms. The theory is based on an analysis of the subsets and subgroups of the Euclidean group, SE(3). It is likely that the theory can be also developed based on an analysis of the subspaces and subalgebras of the Lie algebra, se(3), of the Euclidean group, SE(3).

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