Approximation of planar offset curves with globally bounded error for B-spline NC tool paths

Abstract

The topic of representing the offset of a 2D B-spline curve in the same form has been a topic of research for a long time, and it has many industrial applications, especially in NC tool path generation for pocketing. For B-spline tool paths, it is often required that the tool paths have fewer control points, lower base function degree, and the approximation error is guaranteed within a given tolerance over the entire curve. However, the existing methods utilising global error control approximate the offsets of 2D free-form curves with high function degree and many control points. Although these offsets are useful in computer-aided design, they are inappropriate for the use of CNC machining. To address the problems in order to generate high quality B-spline tool paths, this original work formulates an error function of the offset approximation and then constructs a NURBS curve to globally bound the errors. By checking the maximum coefficient of the bounding curve, the upper bound of all the approximated offset errors is found and the errors can be reduced by segmenting the curve at appropriate knot values. The proposed new approach is more efficient, and the resulting offset approximations are more accurate, with fewer control points and lower function degree. It is useful to generate B-spline tool paths for CNC pocketing, which have the potential for other applications in industry.