In spline toolpath interpolation, a crucial point is solving the mapping between the spline parameter (u) and actual arc length (s) accurately, so that the toolpath is traveled without undesirable fluctuations or discontinuities in the feedrate profile. To achieve this, various techniques have been proposed in literature, including Taylor series interpolation, iterative numerical methods, and approximating the mapping between u and s with a feed correction polynomial. This paper presents a new way to parameterize the seventh order feed correction polynomial, which was introduced by Erkorkmaz and Altintas (2005, “Quintic Spline Interpolation With Minimal Feed Fluctuation,” ASME J. Manuf. Sci. Eng., 127(2), pp. 339–349). The proposed technique has a closed-form solution that can be efficiently implemented in real-time, rather than having to construct and solve a linear equation system with 14 unknowns for each spline segment. In this paper, the new solution is derived step by step, and simulation case studies are presented which demonstrate that the new method accurately parameterizes the feed correction polynomial in approximately 43% less computational time, compared to applying the former solution of Erkorkmaz and Altintas (2005, “Quintic Spline Interpolation With Minimal Feed Fluctuation,” ASME J. Manuf. Sci. Eng., 127(2), pp. 339–349). This is because matrix multiplication operations and a dedicated linear equation solver, which are cumbersome to implement inside a real-time computer numerical controller (CNC), are avoided in the new solution.

References

References
1.
Li
,
Z.-L.
, and
Zhu
,
L.-M.
,
2014
, “
Envelope Surface Modeling and Tool Path Optimization for Five-Axis Flank Milling Considering Cutter Runout
,”
ASME J. Manuf. Sci. Eng.
,
136
(
4
), p.
041021
.10.1115/1.4027415
2.
Pan
,
Y.
,
Zhou
,
C.
,
Chen
,
Y.
, and
Partanen
,
J.
,
2014
, “
Multitool and Multi-Axis Computer Numerically Controlled Accumulation for Fabricating Conformal Features on Curved Surfaces
,”
ASME J. Manuf. Sci. Eng.
,
136
(
3
), p.
031007
.10.1115/1.4026898
3.
Erkorkmaz
,
K.
, and
Altintas
,
Y.
,
2005
, “
Quintic Spline Interpolation With Minimal Feed Fluctuation
,”
ASME J. Manuf. Sci. Eng.
,
127
(
2
), pp.
339
349
.10.1115/1.1830493
4.
Wang
,
F.-C.
, and
Yang
,
D. C. H.
,
1993
, “
Nearly Arc-Length Parameterized Quintic-Spline Interpolation for Precision Machining
,”
Comput. Aided Des.
,
25
(
5
), pp.
281
288
.10.1016/0010-4485(93)90085-3
5.
Wang
,
F.-C.
,
Wright
,
P. K.
,
Barsky
,
B. A.
, and
Yang
,
D. C. H.
,
1999
, “
Approximately Arc-Length Parameterized C3 Quintic Interpolatory Splines
,”
ASME J. Mech. Des.
,
121
(
3
), pp.
430
439
.10.1115/1.2829479
6.
Farouki
,
R. T.
, and
Shah
,
S.
,
1996
, “
Real-Time CNC Interpolators for Pythagorean-Hodograph Curves
,”
Comput. Aided Geom. Des.
,
13
(
7
), pp.
583
600
.10.1016/0167-8396(95)00047-X
7.
Farouki
,
R. T.
,
Al-Kandari
,
M.
, and
Sakkalis
,
T.
,
2002
, “
Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves
,”
Adv. Comput. Math.
,
17
(
4
), pp.
369
383
.10.1023/A:1016280811626
8.
Huang
,
J.-T.
, and
Yang
,
D. C. H.
,
1992
, “
Precision Command Generation for Computer Controlled Machines
,”
Precision Machining: Technology and Machine Development and Improvement
, Vol. PED 58,
ASME
,
New York
, pp.
89
104
.
9.
Shpitalni
,
M.
,
Koren
,
Y.
, and
Lo
,
C.-C.
,
1994
, “
Realtime Curve Interpolators
,”
Comput. Aided Des.
,
26
(
11
), pp.
832
838
.10.1016/0010-4485(94)90097-3
10.
Otsuki
,
T.
,
Kozai
,
H.
, and
Wakinotani
,
Y.
,
1998
, “
Free-Form Curve Interpolation Method and Apparatus
,” U.S. Patent No. 5,815,401.
11.
Lin
,
R.-S.
,
2000
, “
Real-Time Surface Interpolator for 3-D Parametric Surface Machining on 3-Axis Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
40
(
10
), pp.
1513
1526
.10.1016/S0890-6955(00)00002-X
12.
Heng
,
M.
, and
Erkorkmaz
,
K.
,
2010
, “
Design of a NURBS Interpolator With Minimal Feed Fluctuation and Continuous Feed Modulation Capability
,”
Int. J. Mach. Tools Manuf.
,
50
(
3
), pp.
281
293
.10.1016/j.ijmachtools.2009.11.005
13.
Yuen
,
A.
,
Zhang
,
K.
, and
Altintas
,
Y.
,
2013
, “
Smooth Trajectory Generation for Five-Axis Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
71
, pp.
11
19
.10.1016/j.ijmachtools.2013.04.002
14.
Lei
,
W. T.
,
Sung
,
M. P.
,
Lin
,
L. Y.
, and
Huang
,
J. J.
,
2007
, “
Fast Real-Time NURBS Path Interpolation for CNC Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
47
(
10
), pp.
1530
1541
.10.1016/j.ijmachtools.2006.11.011
15.
Cheng
,
C. W.
,
Tsai
,
M. C.
, and
Maciejowski
,
J.
,
2006
, “
Feedrate Control for Non-Uniform Rational B-Spline Motion Command Generation
,”
Proc. Inst. Mech. Eng., Part B
,
220
(
11
), pp.
1855
1861
.10.1243/09544054JEM401
16.
Press
,
W. H.
,
Flannery
,
B. P.
,
Teukolsky
,
S. A.
, and
Vetterling
,
W. T.
,
1992
,
Numerical Recipes in C: The Art of Scientific Computing
,
2nd ed.
,
Cambridge University Press
,
New York
.
17.
MathWorks
, “
MATLAB R2014b Documentation
,” MathWorks, Inc., Natick, MA, http://www.mathworks.com/help/matlab/ref/mldivide.html
18.
Altintas
,
Y.
,
2000
,
Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design
,
Cambridge University Press
,
Cambridge, UK
.
You do not currently have access to this content.