High-speed cornering by CNC machines under prescribed bounds on axis accelerations and toolpath contour error

Original Article


To exactly execute a sharp corner in the toolpath, the feedrate of a CNC machine must instantaneously drop to zero at that point. This constraint is problematic in the context of high-speed machining, since it incurs very high deceleration/acceleration rates near sharp corners, which increase the total machining time, and may incur significant path deviations (contour errors) at these points. A strategy for negotiating sharp corners in high-speed machining is proposed herein, based upon a priori toolpath/feedrate modifications in their vicinity. Each corner is smoothed by replacing a subset of the path that contains it with a conic “splice” segment, deviating from the exact corner by no more than a prescribed tolerance ϵ, along which the square of the feedrate is specified as a Bernstein-form polynomial. The problem of determining the fastest traversal of the conic segments under known axis acceleration bounds can then be formulated as a constrained optimization problem, and by exploiting some well-known properties of Bernstein-form polynomials this can be approximated by a simple linear programming task. Some computed examples are presented to illustrate the implementation and performance of the high-speed cornering strategy.


CNC machine High-speed machining Feedrate Contour error Cornering Path modification Constrained optimization Linear programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dahlquist G, Björck A (1974) Numerical methods. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  2. 2.
    de Souza AF, Coelho RT (2007) Experimental investigation of feed rate limitations on high speed milling aimed at industrial applications. Int J Adv Manuf Technol 32:1104–1114CrossRefGoogle Scholar
  3. 3.
    Erkorkmaz K, Yeung C-H, Altintas Y (2006) Virtual CNC system. Part II. High speed contouring application. Int J Mach Tools Manuf 46:1124–1138CrossRefGoogle Scholar
  4. 4.
    Ernesto CA, Farouki RT (2010) Solution of inverse dynamics problems for contour error minimization in CNC machines. Int J Adv Manuf Technol 49:589–604CrossRefGoogle Scholar
  5. 5.
    Farin G (1997) Curves and surfaces for computer aided geometric design, 4th edn. Academic Press, San DiegoMATHGoogle Scholar
  6. 6.
    Farouki RT, Goodman TNT (1996) On the optimal stability of the Bernstein basis. Math Comput 65:1553–1566CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Farouki RT, Manni C, Sestini A (2001) Real-time CNC interpolators for Bézier conics. Comput Aided Geom Des 18:639–655CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Farouki RT, Neff CA (1990) On the numerical condition of Bernstein-Bézier subdivision processes. Math Comput 55:637–647MATHMathSciNetGoogle Scholar
  9. 9.
    Farouki RT, Rajan VT (1987) On the numerical condition of polynomials in Bernstein form. Comput Aided Geom Des 4:191–216CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Farouki RT, Rajan VT (1988) Algorithms for polynomials in Bernstein form. Comput Aided Geom Des 5:1–26CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Halkin H (1965) A generalization of LaSalle’s “bang-bang” principle. SIAM J Control 2:199–202MathSciNetGoogle Scholar
  12. 12.
    Hausdorff F (1957) Set theory (translated by JR Aumann et al). Chelsea, New YorkMATHGoogle Scholar
  13. 13.
    Imani BM, Jahanpour J (2008) High-speed contouring enhanced with PH curves. Int J Adv Manuf Technol 37:747–759CrossRefGoogle Scholar
  14. 14.
    Jahanpour J, Imani BM (2008) Real-time PH curve CNC interpolators for high speed cornering. Int J Adv Manuf Technol 39:302–316CrossRefGoogle Scholar
  15. 15.
    Komanduri R, Subramanian K, von Turkovich BF (eds) (1984) High speed machining, PED-vol 12. ASME, New YorkGoogle Scholar
  16. 16.
    LaSalle JP (1960) The time optimal control problem. In: Cesari L, LaSalle JP, Lefschetz S (eds) Contributions to the theory of nonlinear oscillations, vol 5. Princeton University PressGoogle Scholar
  17. 17.
    Lee ETY (1987) The rational Bézier representation for conics. In: Farin GE (ed) Geometric modeling: algorithms and new trends. SIAM, PhiladelphiaGoogle Scholar
  18. 18.
    Schultz H, Moriwaki T (1992) High-speed machining. Ann CIRP 41:637–643CrossRefGoogle Scholar
  19. 19.
    Smith S, Tlusty J (1997) Current trends in high-speed machining. ASME J Manuf Sci Eng 119:664–666CrossRefGoogle Scholar
  20. 20.
    Tlusty J (1993) High-speed machining. CIRP Ann 42:733–738CrossRefGoogle Scholar
  21. 21.
    Timar SD, Farouki RT, Smith TS, Boyadjieff CL (2005) Algorithms for time-optimal control of CNC machines along curved tool paths. Robot Comput-Integr Manuf 21:37–53CrossRefGoogle Scholar
  22. 22.
    Tsai Y-F, Farouki RT (2001) Algorithm 812: BPOLY: an object-oriented library of numerical algorithms for polynomials in Bernstein form. ACM Trans Math Softw 27:267–296CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaDavisUSA

Personalised recommendations