Time-Aware Non-Uniform Rational Basis Spline (T-NURBS)

Explicitly adopting time as a parameter, the defi-nition of non-uniform rational basis spline profile known short as NURBS is revisited and updated. This results in another NURBS definition that is aware of time, and not only the geometry. Also, it can jointly exist as-is at the CAD-CAM side, and at the motion numeric controller side without resorting to segmentation, curve fitting, and interpolation techniques usually used when extracting motion information from the standard geometric NURBS profiles. This gives rise to the notion of 'what you see is what you get' when the proposed NURBS definition is used. First, working at the jerk signal level and by using quadratic polynomials with time as the independent variable, quintic polynomials are obtained at the position level and are smoothly glued together to form the needed basis functions that facilitate introducing time-aware splines. Similarly, the trigonometric sine function is used to define another set of time-aware basis functions. Second, and as with standard NURBS, the herein-defined time-aware splines are extended and put into the rational polynomial form such that the proposed time-aware NURBS structure is revealed. Despite being normalized, the time signature used to define the basis functions persists once velocity, acceleration, and jerk profiles are obtained. At the level of the coefficient, these kinematical quantities are neatly written using vector notation that- with the aid of a developed algorithm- reduces the computation burden at the motion numeric controller side during real-time execution. This results in a smooth motion with reduced feed rate variation while adhering to any imposed kinematical constraints. The usefulness, and simplicity of the proposed approach is mainly demonstrated through numeric simulation where the proposed concept of 'what you see is what you get' is verified.