# 数控机床英文参考文献翻译

be chosen as a function of the most unfavorable conditions which can
be met. If they are constantly adjusted to the real conditions and to
the intended goal, there will be a resulting improvement in the output.
This method is designated by the term “adaptive control.”
Pierre B´ ezier, Numerical Control: Mathematics and Applications [38]
29.1 Digital Motion Control
Multi–axis computer numerical control (CNC) machines use digital closed–
loop feedback controllers to drive the individual machine axes so as to execute
a given path at a speciﬁed (constant or variable) speed or feedrate. The digital
sampling frequency of such controllers is typically f = 1024 Hz. Within each
sampling interval ∆t =1/f ≈ 0.001 second, the controller must compare the
commanded machine position, computed from the speciﬁed path and feedrate,
with the actual machine position, measured by encoders on the machine axes,
in order to appropriately accelerate or decelerate the axis drive motors. Such
“closed–loop” control is essential for accurate path traversals at the speciﬁed
speeds under varying machine loads, external disturbances, etc.
Our interest here is in a particular component of the control algorithm, the
real–time interpolator. The task of this function is to compute, from the given
path geometry and speed variation, a stream of reference points with which
the measured machine positions may be compared. This must be performed in
real time, at the sampling frequency f (i.e., each reference–point computation
should consume only a fraction of the sampling interval ∆t). The interpolator
algorithm must be accurate, to ensure faithful realization of the desired path
and feedrate variation; it must be eﬃcient, to permit real–time execution; and
it should be versatile in terms of accommodating a variety of path geometries
620 29 Real–time CNC Interpolators
and speed variations. Although a seemingly simple and modest component of
the overall control algorithm, in practice the real–time interpolator may often
be the limiting factor in the actual CNC machine performance本文来自六~维\论|文/网，毕业论文 www.lwfree.cn 加7位QQ324'9114找源文.
CNC machines have traditionally relied on crude and data–intensive path
descriptions — piecewise–linear/circular “G code” approximations [2] — due
to the diﬃculty of formulating real–time interpolators that can accurately and
eﬃciently compute reference points on free–form curves, traversed at varying
feedrates. Any desired approximation accuracy can, in principle, be achieved
by resorting to suﬃciently small G code segments, but the discrete nature of
such path descriptions compromises the ability of the interpolator to sustain
smooth feedrate performance, especially at high speeds [445].
Several authors [93, 94, 253, 306, 409, 411, 475, 477] have recently proposed
CNC interpolators for the standard (B´ ezier/B–spline) curves of CAD systems.
However, the impossibility of exact arc length computation for such curves (see
§16.1) makes these interpolators inherently approximate — even for constant
feedrates — and frequently no attempt is made to estimate the approximation
error, which may be signiﬁcant for curves with strong curvature variations or
uneven parameterization. By contrast, PH curves admit analytic reduction of
the interpolation integral, yielding real–time interpolators that are essentially
exact and remarkably versatile in terms of the repertoire of feedrate variations
(with time, arc length, or curvature) they can accommodate.
Accurate feedrate performance becomes especially important in high–speed
machining [285,421,441] where one requires extreme rates of feed acceleration
and deceleration, and maintenance of very high feedrates. Moreover, failure of
the interpolator to properly maintain the commanded feedrates may induce
tool “chatter” or breakage through an inappropriate relationship between the
spindle rotation speed and the path traversal rate. Reliable interpolators for
time–dependent feedrates [445] are invaluable in this context.
Given a parametric curve1 r(ξ), the variables that concern us in the context
of real–time interpolator algorithms (with suitable physical dimensions)are:
• t = time (sec.),
• s = curve arc length (mm),
• ξ = curve parameter (dimensionless),
• σ =ds/dξ = “parametric speed” (mm),
• V =ds/dt = feedrate along curve (mm sec.
−1),
• κ = curvature (mm−1).
For a given sampling interval ∆t, the function of the real–time interpolator is
to compute a sequence reference points on the curve (identiﬁed by parameter
values ξ0,ξ1,...,ξN) that correspond to a discrete sampling (at the instants
0,∆t,...,N∆t) of a smooth traversal of the curve at the prescribed feedrate.
The feedrate might be speciﬁed in number of diﬀerent ways, such asHenceforth we use ξ as the curve parameter, since we reserve t for time. 3017

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