General formulation of the singularity locus for a 3-dof regional manipulator

The analysis of singularities is a central aspect in the design of robotic manipulators. Such analyses are usually based on the use of geometric parameters like DH parameters. However, the manipulator kinematics is naturally described using the concept of screws and twists, associated to Lie groups and algebras. These give rise to general and coordinate-invariant singularity conditions on the manipulator geometry. In this setting no restrictions are imposed onto the type of joints, as it is the case when using DH parameters. In this paper a single closed-form equation is presented that gives a complete description of the singularity locus of an arbitrary regional manipulator in terms of two joint variables and all design parameters, expressed by joint screw coordinates, together with the coordinates for the wrist centre. Some examples are reported, and it is shown that the expression can be used to analyse bifurcations in the singularity locus. The simple form of the condition should make it useful for practical design as well as for a deeper understanding of singularities.