On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators

Abstract

In this paper, the singularity loci of general three-degree-of-freedom planar parallel manipulators are studied and a graphical representation of these loci in the manipulator's workspace is obtained. The algorithm used here is based on the determination of the roots of the determinant of the manipulator's Jacobian matrix. As mentioned elsewhere, two different types of singularities can occur when parallel manipulators are actuated. Both types are considered here and it is shown that one of the two types leads to a trivial description while the second one is more challenging. On the other hand, architectural singularities are not considered since they are assumed to be eliminated from the outset by a proper choice of the kinematic parameters. Indeed, for the type of manipulator studied here, architectural singularities are very easy to predict and were studied in detail elsewhere. Analytical expressions describing the singularity locus of a planar parallel manipulator are obtained here. Moreover, it is shown that, for a given orientation of the platform, the singularity locus in the plane of motion is a quadratic form, i.e., either a hyperbola, a parabola or an ellipse. Examples illustrating these results are given. For each of these examples, the corresponding singularity locus is graphically superimposed on the manipulator's workspace. This feature has been included in a package developed for the CAD of parallel manipulators. Cases of manipulators for which the singularity locus is located outside of the workspace for certain orientations of the platform are presented. Additionally, three-dimensional representions of the singularity loci are given. The graphical representation of the singularity loci is a very powerful design tool which can be of great help, especially in the context of parallel manipulators.