Smoothing of Time-Optimal Feedrates for Cartesian CNC Machines

Abstract

Minimum-time traversal of curved paths by Cartesian CNC machines, subject to prescribed bounds on the magnitude of acceleration along each axis, usually involves a “bang-bang” control strategy in which the acceleration bound is realized by one or another of the machine axes at each instant during the motion. For a path specified by a polynomial parametric curve and prescribed acceleration bounds, the time-optimal feedrate may be expressed in terms of a C0 piecewise-rational function of the curve parameter. This function entails sudden changes in either the identity of the limiting axis, or the sign of acceleration on a single limiting axis, incurring demands for instantaneous changes of motor torque that may not be physically realizable. A scheme is proposed herein to generate smoothed C1 (slightly sub-optimal) feedrate functions, that incur only finite rates of change of motor torque and remain consistent with the axis acceleration bounds. An implementation on a 3-axis CNC mill driven by an open-architecture software controller is used to illustrate this scheme.