Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds - Timar, Sebastian D; Farouki, Rida T (Robotics and computer-integrated manufacturing, 2008)
Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis CNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analytic function of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed-dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments that correspond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependent acceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values may be efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolator algorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of the time-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is a significant improvement over that proposed in [Farouki RT, Rajan VT. Algorithms for polynomials in Bernstein form. Comput Aided Geometric Des 1988;5:1-26], since the addition of motor voltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments.