Applying this operator successively to ξ(t), we obtain the recursive formulae
etc., where primes indicate derivatives with respect to ξ. The parametric speed
σ and its derivatives, required in (29.5), can be expressed recursively as follows
etc. If the feedrate V is given as a function of a physically significant variable,
such as arc length, curvature, or time, derivatives with respect to that variable
must be transformed into derivatives with respect to ξ for use in (29.5).
In practice, the custom has typically been to retain only the linear term in
(29.3), with no attempt to estimate the truncation error, which may become
significant whenever the curvature is high and/or the parametric speed is low.
Improving the accuracy of this scheme by incorporating higher–order terms
may incur significant computational cost, since the coefficients of the quadratic
and subsequent terms become increasingly complicated in the case of variable
feedrates. Yang and Kong [475] first proposed a variable–feedrate interpolator,
based on a Taylor series truncated after the quadratic term. However, as noted
in [191], their expansion contains an erroneous quadratic term (valid only for a
constant feedrate, although a dependence on the curve parameter is explicitly
indicated). This error was subsequently repeated by Yeh and Hsu [477].
A systematic derivation of the correct Taylor coefficients, up to third order
in (29.3), was presented in [193] for the cases of constant feedrate and feedrates
dependent on time, arc length, or curvature. For a constant feedrate, we simply
set V
= V
= 0 in (29.5). For a time–dependent feedrate, the recursive forms
are most convenient for implementation. The case of a feedrate dependent on
arc length or curvature is more involved — details may be found in [193].
Taylor series interpolators inevitably incur truncation errors, through the
omission of higher–order terms in the series (29.3). Since each reference–point