Desingularization of nonredundant serial manipulator trajectories using Puiseux series

It is shown that smooth spatial paths for non-redundant serial robots can always be smoothly reparameterized in the vicinity of kinematic singularities with finite root multiplicity using a Puiseux series (i.e., fractional power series). This reparameterization, based on the algebraic structure of the manipulator kinematics, is formed using the -th root of the distance of the path parameter to the singularity, where is some integer not exceeding the root multiplicity of the singularity. For the very common case where a singularity has multiplicity two, we can always take = 2, so that the reparameterization is formed from the square root of the path parameter’s distance to the singularity. By contrast, self-motion singularities (such as the wrist singularity), which have infinite root multiplicity, do not possess such a reparameterization. However, smooth motions directly throughsuch singularitiesare generally possible because approaching path solutions are unaffected by them. These results help quantify how joint derivatives blow up at singularities, and can readily be used to generate singularity-robust motions along any path, with no spatial error. This is illustrated with an example involving the PUMA manipulator.