OPTIMAL CURVE FITTING FOR CNC MACHINING

Abstract

In general, CNC machines can only perform linear, circular and sometimes parabolic interpolations and hence, cannot generate higher-order and other functional curves used to define the objects to be manufactured by machining. Consequently, several segments of lower-order curves have to be used to approximate the given curve subject to a given tolerance. Overspecification of the number of segments than necessary to ensure that the tolerance is not exceeded can lead to lengthy programs and long processing time to compute and produce the curve segments. In this thesis, the approximation is posed as an optimization problem with the objective of obtaining an optimal number of composite lower-order curves to approximate the given curve segment. The lower-order curves employed are linear, circular and parabolic curves. Fast algorithms have been developed and used to determine the optimal numbers of lower-order curve segments for cubic Bezier, cubic polynomial, trigonometric, cycloidal and exponential curves. The algorithm for the single cubic Bezier curve has also been extended to composite (two or more) Bezier curves. The composite lines, circular arcs and parabolic curves used to approximate the various curves are compared and discussed. NC codes corresponding to the composite lower order curves have also been generated to machine the respective curved parts and the machining times and surface finish conditions are compared.