Abstract
Minimum-time traversal of curved paths by Cartesian CNC machines, subject to prescribed bounds on the magnitude of acceleration along each axis, usually involves a “bang-bang” control strategy in which the acceleration bound is realized by one or another of the machine axes at each instant during the motion. For a path specified by a polynomial parametric curve and prescribed acceleration bounds, the time-optimal feedrate may be expressed in terms of a C 0 piecewise-rational function of the curve parameter. This function entails sudden changes in either the identity of the limiting axis, or the sign of acceleration on a single limiting axis, incurring demands for instantaneous changes of motor torque that may not be physically realizable. A scheme is proposed herein to generate smoothed C 1 (slightly sub-optimal) feedrate functions, that incur only finite rates of change of motor torque and remain consistent with the axis acceleration bounds. An implementation on a 3-axis CNC mill driven by an open-architecture software controller is used to illustrate this scheme.
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References
EIA Standard RS–274–D, Interchangeable variable block data format for positioning, contouring, and contouring/positioning numerically controlled machines, Electronic Industries Association, Engineering Dept., Washington, D.C (1979)
Bobrow, J.E., Dubowsky, S., Gibson, J.S.: Time–optimal control of robotic manipulators along specified paths. International Journal of Robotics Research 4(3), 3–17 (1985)
Chou, J.–J., Yang, D.C.H.: Command generation for three–axis CNC machining. ASME Journal of Engineering for Industry 113, 305–310 (1991)
Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, 4th edn. Academic Press, San Diego (1997)
Farouki, R.T., Manjunathaiah, J., Nicholas, D., Yuan, G.–F., Jee, S.: Variable feedrate CNC interpolators for constant material removal rates along Pythagorean–hodograph curves. Computer Aided Design 30, 631–640 (1998)
Farouki, R.T., Shah, S.: Real–time CNC interpolators for Pythagorean–hodograph curves. Computer Aided Geometric Design 13, 583–600 (1996)
Farouki, R.T., Tsai, Y.–F.: Exact Taylor series coefficients for variable–feedrate CNC curve interpolators. Computer Aided Design 33, 155–165 (2001)
Florian, C.: An Introduction to the Theory of Equations. Macmillan Company, New York (1969)
Halkin, H.: A generalization of LaSalle’s “bang–bang” principle. SIAM Journal on Control 2, 199–202 (1965)
Huang, J.–T., Yang, D.C.H.: A generalized interpolator for command generation of parametric curves in computer–controlled machines. In: Proceedings, Japan/USA Symposium on Flexible Automation. vol. 1, pp. 393–399. ASME (1992)
Komanduri, R., Subramanian, K., von Turkovich, B.F. (eds.): High Speed Machining, PED, vol. 12. ASME, New York (1984)
LaSalle, J.P.: The time optimal control problem. In: Cesari, L., LaSalle, J.P., Lefschetz, S. (eds.) Contributions to the Theory of Nonlinear Oscillations, vol. 5, Princeton Univ. Press, Princeton (1960)
Pfeiffer, F., Johanni, R.: A concept for manipulator trajectory planning. IEEE Journal of Robotics and Automation RA–3(2), 115–123 (1987)
Shiller, Z., Lu, H.H.: Robust computation of path constrained time optimal motions, Proceedings. In: IEEE International Conference on Robotics and Automation, Cincinnati, OH, pp. 144–149 (1990)
Slotine, J.J.E., Yang, H.S.: Improving the efficiency of time–optimal path–following algorithms. IEEE Transaction on Robotics and Automation 5(1), 118–124 (1989)
Smith, S., Tlusty, J.: Current trends in high–speed machining. ASME Journal of Manufacturing Science and Engineering 119, 664–666 (1997)
Timar, S.D., Farouki, R.T., Boyadjieff, C.L.: Time-optimal feedrates along curved paths for Cartesian CNC machines with prescribed bounds on axis velocities and accelerations (2005) (preprint)
Timar, S.D., Farouki, R.T., Smith, T.S., Boyadjieff, C.L.: Algorithms for time-optimal control of CNC machines along curved tool paths. Robotics and Computer-Integrated Manufacturing 21(1), 37–53 (2005)
Tlusty, J.: High–speed machining. CIRP Annals. 42, 733–738 (1993)
Tsai, Y.–F., Farouki, R.T., Feldman, B.: Performance analysis of CNC interpolators for time–dependent feedrates along PH curves. Computer Aided Geometric Design 18, 245–265 (2001)
Uspensky, J.V.: Theory of Equations. McGraw-Hill, New York (1948)
Yang, D.C.H., Kong, T.: Parametric interpolator versus linear interpolator for precision CNC machining. Computer Aided Design 26, 225–234 (1994)
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Boyadjieff, C.L., Farouki, R.T., Timar, S.D. (2005). Smoothing of Time-Optimal Feedrates for Cartesian CNC Machines. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_6
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DOI: https://doi.org/10.1007/11537908_6
Publisher Name: Springer, Berlin, Heidelberg
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