Time-Optimal feedrates along Curved Paths for Cartesian CNC Machines with prescribed Bounds on Axis velocities and Accelerations

The authors consider the problem of specifying the feedrate variation along a curved path, that yields the minimum traversal time for a multi-axis CNC machine subject to given bounds on the feasible velocity and acceleration along each axis. The torque-speed characteristics of the axis drive motors are first discussed, and are used to determine constraints on the velocity and acceleration for each axis. For a path specified by a polynomial parametric curve r(ξ), it is shown that (the square of) the time-optimal feedrate can be determined as a piecewise-rational function of the curve parameter ξ, with break-points that correspond to the roots of certain polynomials. In general, each feedrate segment is characterized by the saturation of either velocity or acceleration on one machine axis at each instant throughout the motion. The feedrate function admits a real-time interpolator algorithm that can drive the machine from the analytic curve description, eliminating the need for piecewise-linear/circular G code approximations. The theoretical and computational aspects of such time-optimal feedrates are presented, along with experimental results from implementation on a three-axis CNC mill driven by an open-architecture software controller. Compared to prior time-optimal feedrate algorithms (based on acceleration bounds only), the new algorithms give physically valid feedrates at high speeds, where the motor voltage ratings become a limiting factor.