Desingularization of nonredundant serial manipulator trajectories using Puiseux series

It is shown that smooth spatial paths for nonredundant serial robots can always be smoothly reparametrized in the vicinity of kinematic singularities with finite root multiplicity using a Puiseux series. This reparametrization, based on the algebraic structure of the manipulator kinematics, is formed using the νth root of the path parameter distance to the singularity, where ν is some integer not exceeding the root multiplicity of the singularity. By contrast, self-motion singularities, which have infinite root multiplicity, do not possess such a reparametrization. However, smooth motions directly through such singularities are 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