New methods have been developed to control a mechanism's realtime Cartesian motion along spatially complex curves such as Non-Uniform Rational B-splines (NURBS). The methods dynamically map the critical trajectory parameters between parameter space, Cartesian space, and joint space. Trajectory models that relate Cartesian tool speeds and accelerations to joint speeds and accelerations have been generalized so that they can be applied to most classes of robots and CNC mechanisms.