The purpose of this paper is to define functional realizations of the Khizhnik–Zamolochikov–Bernard (KZB) connection on the bundle of conformal blocks over the moduli space of curves.

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.

A simple and efficient predictor-corrector method uses finite difference theory to predict the parametric changes required to generate the desired curvilinear distances along the trajectory, and then correct the erorrs arising from this prediction. Polynomial approximation methods successfully approximate joint speeds and accelerations rather than require a closed-form inverse Jacobian solution.

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