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Technical Brief

Efficient Fitting of the Feed Correction Polynomial for Real-Time Spline Interpolation

[+] Author and Article Information
Kaan Erkorkmaz

Associate Professor
Precision Controls Laboratory,
Department of Mechanical and Mechatronics Engineering,
University of Waterloo,
200 University Avenue West,
Waterloo, ON N2L 3G1, Canada
e-mail: kaane@uwaterloo.ca

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received March 16, 2014; final manuscript received March 29, 2015; published online July 8, 2015. Assoc. Editor: Xiaoping Qian.

J. Manuf. Sci. Eng 137(4), 044501 (Aug 01, 2015) (8 pages) Paper No: MANU-14-1119; doi: 10.1115/1.4030300 History: Received March 16, 2014; Revised March 29, 2015; Online July 08, 2015

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.

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Figures

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Fig. 2

Three commonly used spline interpolation methods [3]: (a) natural interpolation, (b) interpolation using Taylor series expansion, and (c) interpolation with feedrate correction polynomial

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Fig. 3

Parameterization of the feed correction polynomial using data generated from the numerical integration of the arc length (s) over the spline parameter (u) [3]

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Fig. 4

Implementation comparing the new parameterization method for the feed correction polynomial with the earlier results reported in Ref. [3]. (a) 88-Segment toolpath, (b) natural interpolation, (c) first order Taylor, (d) feed correction polynomial (earlier solution, using matrix calculations [3]), and (e) feed correction polynomial (computed with the proposed solution).

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Fig. 7

Implementation of the proposed feed correction method on an airfoil profile. (a) 48-Segment toolpath, (b) natural interpolation, and (c) feed correction polynomial (computed with the proposed solution).

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Fig. 5

Comparison between seventh order feed correction polynomial and fifth order variations (i.e., justification of using seventh order): (a) seventh order feed correction polynomial parameterized considering u–s data, as well as zeroth, first, and second order boundary conditions (proposed), (b) fifth order feed correction polynomial parameterized considering u–s data and only zeroth and first order boundary conditions, and (c) fifth order feed correction polynomial parameterized considering only zeroth, first, and second order boundary conditions

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Fig. 6

(a) Computational load and (b) average computational time comparison between the earlier solution [3] and proposed new method for computing the feed correction polynomial for one segment. The reported results are based on the 88-segment toolpath in Fig. 4(a), which was split, on average, to 440 divisions per segment (= Mk) during the computation of the segment arc-lengths and feed correction polynomials.

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