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.
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.