Algorithms for time-optimal control of CNC machines along curved tool paths

The problem of specifying the feedrate variation along a curved path, that yields minimum traversal time for a 3-axis CNC machine subject to constraints on the feasible acceleration along each axis, is addressed. In general, this time-optimal feedrate incurs “bang-bang control,” i.e., maximum acceleration/deceleration is demanded of at least one axis throughout the motion. For a path defined by a polynomial parametric curve r(ξ), we show that the (square of the) time-optimal feedrate can be determined as a piecewise-rational function of the curve parameter ξ, with break-points corresponding to the roots of certain polynomial equations. Furthermore, this type of feedrate function is amenable to a real-time interpolator algorithm that drives the machine directly from the analytic curve description, eliminating the need for linear/circular G code approximations. The theoretical and computational aspects of such time-optimal feedrate functions are presented, together with experimental results from their implementation on a 3-axis mill driven by an open-architecture software controller.