Architecturally singular non-planar parallel manipulators

In this contribution we present the complete list of all non-planar architecture singular parallel manipulators (Stewart-Gough type manipulators). The main result is that they are in a certain sense trivial, because any such parallel manipulator has four collinear points in the platform or in the base. This follows that they are not useful for practical purposes. We see that all of them must satisfy certain assembly conditions (the space of joint parameters is degenerated, we cannot prescribe freely the length of legs). All non-planar architecture singular parallel manipulators have self-motions at every possible configuration. To every non-planar manipulator we can add infinitely many legs without disturbing its geometrical properties and self-motions, for planar platform and base it is not true. Inspection of the list of non-planar architecture singular parallel manipulators shows that they are invariant with respect to projective transformations in both fixed and moving spaces.