Robust real-time NURBS path interpolators

In a review of the real-time non-uniform rational B-splines (NURBS) path interpolation method in CNC controllers, it was found that none of the NURBS interpolators described in the literature has the necessary robustness against an extreme knot distribution. The problems begin with the calculation of the total length of the NURBS path: most interpolators handle knots as a global curve parameter and may deliver incorrect results if the knots are of an extreme distribution. Further, Taylor's-expansion-based NURBS interpolators may overlook the path portion of extremely small knot spans. To solve these problems, methods for improving the robustness of Taylor's-expansion-based NURBS interpolators are proposed in this study, and an improved robust fast NURBS path interpolator is described. These new methods process NURBS data on the basis of knot spans: the adaptive quadrature method is applied to each knot span, and the calculated lengths of all knot spans are summed to build the total path length. Further, the inverse length functions are also generated based on the knot spans, and a control mechanism is introduced to prove the validity of the resulting inverse length functions. Experimental results have proved the effectiveness of the proposed robust fast NURBS path interpolator.