New methods have been developed to control a mechanism's realtimeCartesian motion along spatially complex curves such as Non-Uniform RationalB-splines (NURBS). The methods dynamically map the critical trajectoryparameters between parameter space, Cartesian space, and joint space. Trajectorymodels that relate Cartesian tool speeds and accelerations to joint speeds andaccelerations have been generalized so that they can be applied to most classesof robots and CNC mechanisms.

A simple and efficient predictor-correctormethod uses finite difference theory to predict the parametric changes requiredto generate the desired curvilinear distances along the trajectory, and thencorrect the erorrs arising from this prediction. Polynomial approximationmethods successfully approximate joint speeds and accelerations rather thanrequire a closed-form inverse Jacobian solution.

The numericalalgorithms prove to be time bounded (fixed number of computational steps), andthe generated trajectories are smooth and continuous. Both simulation andphysical experiments using an Open-Architecture Controller demonstrate thefeasibility and usefulness of the developed trajectory generation algorithms andmethods. The methods can be conducted at trajectory rates greater than 100 Hz,depending on mechanism complexity.