| For on-line Cartesian trajectory control, a new procedure has been developed to construct a real-time Cartesian trajectory for spatial complex curves. The methods establish the relationships among parametric space, Cartesian space, joint space and time. Trajectory models calculate both tool speeds and accelerations as they relate to joint speeds and accelerations for controlling the mechanisms. Both simple trapezoidal velocity profile and a more complex S-curve velocity profile have been implemented for predicting the speed and distance changes. A simple and efficient predictor-corrector method using numerical finite difference theory has been developed to predict the parameter changes of complex curves with respect to distance changes, and then correct the errors arising from this prediction. Polynomial approximation methods successfully approximate joint speeds and accelerations rather than require closed form Jacobian solutions. The numerical algorithms prove to be time bounded (fixed number of steps), and the generated trajectories are smooth and continuous. Both simulation and physical experiments demonstrate the feasibility and usefulness of the developed trajectory generation algorithms and methods. The methods can be conducted at trajectory rates approaching 100 Hz, depending on mechanism type. |
