(a) a constant feedrate: V0,
(b) a function of the elapsed time: V (t),
(c) a function of the curve arc length: V (s),
(d) a function of the curve parameter: V (ξ),
(e) a function of the local curvature: V (κ).
Case (a) is the simplest — it amounts to computing a sequence of points along
the curve, spaced uniformly by the arc–length increment ∆s0 = V0∆t.Some
authors have employed case (d) as a means of specifying variable feedrate along
a curve [93,94,253,475] — but this is a rather “artificial” approach, since the
curve parameter ξ is unrelated to its intrinsic geometry (a re–parameterization
of the curve yields a different physical realization of the feedrate variation).
Cases (c) and (e) are more “natural” ways to prescribe a variable feedrate,
based on the curve geometry. For general polynomial or rational curves they
are computationally more difficult, but the special algebraic structure of PH
curves allows an analytic reduction of the interpolation integral (for cases of
practical interest) in their implementation. Case (b) is useful in specifying the
acceleration/deceleration phases of a motion that begins and ends at rest.
For each of the above modes (a)–(e) of feedrate specification, finding the
desired sequence of parameter values ξ0,ξ1,...,ξN amounts to integrating the
differential equation
dξ
dt
= V
σ
(29.1)
from t = 0 (corresponding to ξ0 =0)to t = ∆t, 2∆t,...,N∆t,where N is the
smallest integer such that ξN > 1. When V is known — directly, or indirectly
through s or κ — as a function of ξ, this may be cast in integral form as
ξkV
dξ = k∆t , k =1, 2,...,N, (29.2)
where the unknown reference–point parameter values ξ0,ξ1,...,ξN appear as
upper limits of integration. For the formulation of real–time interpolators, the
advantage of PH curves over general B´ ezier/B–spline curves derives from the
feasibility of a closed–form reduction of the “interpolation integral” in (29.2),
for a variety of useful feedrate functions V of the form (a)–(e) above.
29.2 Taylor Series Interpolators
Real–time interpolators for general B´ ezier/B–spline curves r(ξ) typically rely
on Taylor series expansions to approximate the parameter values of successive
reference points [93, 94, 253, 411, 475]. The reference–point parameter values
correspond to a discrete sampling ξk = ξ(k∆t) of the function ξ(t) describing
the variation of the curve parameter ξ with time t, according to the prescribed
feedrate. We may expand ξ(t) in a Taylor series about tk = k∆t to obtain
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