Real-Time NURBS Interpolator for Distributed Motion Control

Abstract

Off-line toolpath planning for machine tools inevitably leads to suboptimal use of the given equipment and, even more important, excludes the possibility to accommodate for unforeseen or unmodelled effects. Online path planning however requires the availability of real-time interpolators which are both numerically efficient and deterministic and which do not depend on global information on the path to interpolate. Continuity of the interpolant up to higher derivatives is highly desirable. This paper presents an interpolation scheme that meets all these requirements. The algorithm generates a third-order, C 2-continuous Non Uniform Rational B-Spline curve from a series of exactly interpolated position and velocity setpoints. This curve is at all instants completely determined from the first up to the last available setpoint. It is shown that a new setpoint affects only the last control points and the end of the knot vector of the interpolant. A graphical interpretation of the algorithm and a discussion on numerical issues and start conditions are presented.