On the degenerated cases of architecturally singular planar parallel manipulators

In Theorem 3 of A. Karger, Architecturally singular non-planar parallel manipulators, Mechanism and Machine Theory, Vol. 43, No.3, pp. 335-346, 2008 all types of architecturally singular parallel manipulators with four collinear anchor points are listed. As the manipulator is assumed to be architecturally singular in the cited theorem the given conditions for each entry of this list are sufficient for classification. However, we prove that in contrast to the items 1 to 10, the degenerated planar cases (items 11 and 12) do not contain conditions which are sufficient for an architecturally singular manipulator design. We propose sufficient conditions for these two cases in order to decide whether a given manipulator (with 4 points collinear) is architecturally singular or not by checking only the list instead of calculating the determinant of the Jacobian in all poses. Moreover we give a geometric interpretation of the degenerated planar cases.