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(xi), we show that the (square of the) time-optimal feedrate can be determined as a piecewise-rational function of the curve parameter xi, 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. (C) 2004 Elsevier Ltd. All rights reserved.

[1]  Rida T. Farouki,et al.  Real rational curves are not 'unit speed' , 1991, Comput. Aided Geom. Des..

[2]  Jui-Jen Chou,et al.  On the Generation of Coordinated Motion of Five-Axis CNC/CMM Machines , 1992 .

[3]  Donald R. Smith.  Variational methods in optimization , 1974 .

[4]  S. Smith,et al.  Current Trends in High-Speed Machining , 1997 .

[5]  Rida T. Farouki,et al.  Exact Taylor series coefficients for variable-feedrate CNC curve interpolators , 2001, Comput. Aided Des..

[6]  Rida T. Farouki,et al.  Algorithms for polynomials in Bernstein form , 1988, Comput. Aided Geom. Des..

[7]  Moshe Shpitalni,et al.  Realtime curve interpolators , 1994, Comput. Aided Des..

[8]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[9]  Jui-Jen Chou,et al.  Command Generation for Three-Axis CNC Machining , 1991 .

[10]  Rida T. Farouki,et al.  Performance analysis of CNC interpolators for time-dependent feedrates along PH curves , 2001, Comput. Aided Geom. Des..

[11]  Friedrich Pfeiffer,et al.  A concept for manipulator trajectory planning , 1987, IEEE J. Robotics Autom..

[12]  H. Halkin.  A Generalization of LaSalle’s “Bang-Bang” Principle , 1964 .

[13]  Zvi Shiller,et al.  On singular time-optimal control along specified paths , 1994, IEEE Trans. Robotics Autom..

[14]  J. Tlusty.  High-Speed Machining , 1993 .

[15]  S. Lefschetz.  Contributions to the theory of nonlinear oscillations , 1950 .

[16]  Z. Shiller,et al.  Computation of Path Constrained Time Optimal Motions With Dynamic Singularities , 1992 .

[17]  Syh-Shiuh Yeh,et al.  The speed-controlled interpolator for machining parametric curves , 1999, Comput. Aided Des..

[18]  J. Bobrow,et al.  Time-Optimal Control of Robotic Manipulators Along Specified Paths , 1985 .

[19]  Zvi Shiller,et al.  Robust computation of path constrained time optimal motions , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[20]  Daniel C. H. Yang,et al.  Parametric interpolator versus linear interpolator for precision CNC machining , 1994, Comput. Aided Des..

[21]  Jean-Jacques E. Slotine,et al.  Improving the Efficiency of Time-Optimal Path-Following Algorithms , 1988, 1988 American Control Conference.

[22]  Rida T. Farouki,et al.  Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves , 1998, Comput. Aided Des..

[23]  Jean-Jacques E. Slotine,et al.  Fast Algorithms for Near-Minimum-Time Control of Robot Manipulators , 1994, Int. J. Robotics Res..

[24]  Rong-Shine Lin,et al.  Real-Time Interpolators for Multi-Axis CNC Machine Tools , 1994 .

[25]  Rida T. Farouki,et al.  Real-time CNC interpolators for Pythagorean-hodograph curves , 1996, Comput. Aided Geom. Des..

[26]  Rida T. Farouki,et al.  On the numerical condition of polynomials in Bernstein form , 1987, Comput. Aided Geom. Des..

[27]  Kang G. Shin,et al.  Minimum-time control of robotic manipulators with geometric path constraints , 1985 .

[28]  Rida T. Farouki,et al.  G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines , 1999 .