Singularity analysis of 5-RPUR parallel mechanisms (3T2R)

Singularities of parallel mechanisms are related to the regularity of certain Jacobian matrices whose rank deficiency makes the mechanisms lose their inherent rigidity. This paper presents the results of a detailed investigation of the singular configurations of 5-degree-of-freedom parallel mechanisms performing all three translations and two independent rotations. The general architecture of the mechanism—arising from the type synthesis of symmetrical 5-degree-of-freedom parallel mechanisms—is comprised of a mobile platform attached to a base through five identical revolute–prismatic–universal revolute-jointed serial kinematic chains. From the results of screw theory, it is demonstrated that the Jacobian matrix of the above mechanism is constituted by six Plücker lines, a special case of Grassmann coordinates. This channels us to use the so-called Grassmann line geometry instead of applying a classical linear algebra approach. Grassmann line geometry can be regarded as a classification of the degeneration of the Plücker line set. Moreover, six simplified designs are proposed for which their singular configurations can be predicted by means of the Grassmann line geometry. The principles of this study can also be applied to other types of symmetrical 5-degree-of-freedom parallel mechanisms.