Singular set of four screws: an intrinsic solution

We study singular four-screw systems and the set of configurations of associated mechanisms. After some mathematical considerations, the obtention of a remarkable change of variables allows to simplify significantly the equations of the set of singularities and the one of configurations, all calculations being coordinate-free. In a second time, we analyse the converse problem. Intrinsic conditions about the screw system (X1, X2, X3, X4) in order that the associated mechanism moves are found. This intrinsic algebraic method underlines the symmetries of the problem and preserves the geometrical meaning of each relation. Its principle may be extended to any screw system.