Exact Taylor series coefficients for variable-feedrate CNC curve interpolators

Recent papers [Yang DCH, Kong T. Computer Aided Design 1994;26:225-34; Yeh S-S, Hsu P-L. Computer Aided Design 1999;31:349-57] formulate real-time CNC interpolators for variable feedrates along parametric curves. These interpolators employ truncated Taylor series to compute successive reference-point parameter values, but in both papers an erroneous coefficient for the highest (quadratic) term is cited. The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain-rule differentiation. Compact recursive formulae are presented here to compute the correct coefficients, up to the cubic term, in cases where the feedrate depends on: (i) elapsed time; (ii) curve are length; or (iii) local path curvature. The local and cumulative effects of truncation errors on the accuracy of such interpolator schemes are also assessed, and compared with the essentially exact interpolators for Pythagorean-hodograph curves. (C) 2001 Elsevier Science Ltd. All rights reserved.