Path-Tracking through Singularities

Given a mapping F from Rn to Rm with $n\geq m,$ a basic computational problem that has applications in robotics is to solve for paths x(t) in the domain of F given a target path z(t) in the range of F so that F(x(t))=z(t). Results are given for the existence and characterization of solution paths through singularities of F, with emphasis on folds and cusps. Reparametrizations using Puiseux series are derived in cases where x(t) is not differentiable.