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Research Papers

Piecewise B-Spline Tool Paths With the Arc-Length Parameter and Their Application on High Feed, Accurate CNC Milling of Free-Form Profiles

[+] Author and Article Information
Zezhong C. Chen

Department of Mechanical and Industrial Engineering,  Concordia University, 1455 De Maisonneuve West, Montreal, QC H3G 1M8, Canadazcchen@encs.concordia.ca

Maqsood A. Khan

Department of Industrial and Manufacturing Engineering,  NED University of Engineering and Technology,Karachi 75270, Pakistanmaqcan@yahoo.ca

J. Manuf. Sci. Eng 134(3), 031007 (May 07, 2012) (13 pages) doi:10.1115/1.4006551 History: Received January 31, 2011; Revised February 14, 2012; Published May 04, 2012; Online May 07, 2012

To conduct B-spline curve machining, first, B-spline tool paths with feed rates are planned; and second, the B-spline interpolator generates tool trajectories in real-time based on the paths fed into the computer numerically controlled (CNC) controller. Currently, the paths are often planned geometrically with a nonarc-length parameter. Literally, the interpolator can process B-spline paths with the arc-length parameter well, while it sometimes is challenged to work with the nonarc-length parameterized B-spline paths. As a consequence, it is difficult to ensure high accuracy of the tool trajectories in B-spline machining in terms of their corresponding paths; especially, if the feed is very high, smooth tool kinematics cannot be well maintained. To root out these problems, a new type of tool path—piecewise B-spline tool paths with the arc-length parameter—is first proposed in this work. Given a B-spline path with a nonarc-length parameter, it is accurately converted into a B-spline path with an arc-length parameter before sending it into the CNC controller. Furthermore, if the prescribed feed rate is very high and the arc-length parameterized B-spline path is disqualified, it is split into pieces represented with distinct arc-length parameterized B-spline paths in different feed rates. The main advantage of these piecewise paths is that they can eliminate the problems encountered by the existing B-spline interpolator with input of nonarc-length parameterized B-spline paths. Therefore, the piecewise arc-length parameterized B-spline paths are a genuine solution to high feed-and-accuracy B-spline machining.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Illustration of the real-time B-spline interpolation on a B-spline path with a nonarc-length parameter u

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Figure 2

Illustration for real-time B-spline interpolation of a B-spline path with the arc-length parameter; (a) with a normal feed rate, and (b) with a very high feed rate

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Figure 3

The flowchart of the B-spline conversion method (in the shaded blocks) and the approach to piecewise arc-length parameterized B-spline paths (in the plain blocks)

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Figure 4

(a) A B-spline path with a nonarc-length parameter and its control polygon, and (b) the decomposed eleven Bezier curve segments and their control polygons

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Figure 5

A Bezier curve with the control points of its control polygon, the chord and the hatched convex hull

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Figure 6

The B-spline path with a nonarc-length parameter in Fig. 4 is decomposed for sample cutter locations based on the prescribed tolerances: (a) 1 mm and (b) 0.01 mm

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Figure 7

A quadratic B-spline curve with a knot vector, in which two knots are very close, and the nine control points of the curve

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Figure 8

Illustration of the given B-spline path with a nonarc-length parameter and the converted B-spline path with the arc-length parameter

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Figure 9

For the example in Fig. 4, (a) the path error plot and (b) the parameterization error plot of the converted B-spline path with the arc-length parameter and 27 control points

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Figure 10

For the example in Fig. 4, (a) the path error plot and (b) the parameterization error plot of the converted B-spline path with the arc-length parameter and 50 control points

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Figure 11

For the example in Fig. 4, (a) a B-spline path with the arc-length parameter and (b) the cutter trajectory error plot

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Figure 12

For the example in Fig. 4, (a) the piecewise B-spline paths with the arc-length parameter and (b) the cutter trajectory error plot

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Figure 13

A given B-spline path with a nonarc-length parameter

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Figure 14

The results in converting the given nonarc-length parameterized B-spline path to an arc-length parameterized B-spline path; (a) sample cutter locations of the given path and (b) the converted arc-length parameterized B-spline path

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Figure 15

The path and parameterization accuracy of the converted arc-length parameterized B-spline path; (a) the path error curve and (b) the parameterization error curve of the path

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Figure 16

The tool trajectory accuracy and the feed rate of using the converted arc-length parameterized B-spline path in simulation; (a) the cutter trajectory error curve and (b) the feed rate error curve of the path

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Figure 17

Illustration of the seven arc-length parameterized B-spline paths with different feed rates

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Figure 18

Illustration of the accuracy and the different feed rates of the piecewise arc-length parameterized B-spline paths; (a) the cutter trajectory error curve and (b) the feed rate curve of the paths.

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Figure 19

Cutting results of using the given and the calculated paths: (a) the cut using the nonarc-length parameterized B-spline path with a feed rate of 400 mm/s and (b) the cut using the piecewise arc-length parameterized B-spline paths with different feed rates

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Figure 20

The zoom-in pictures of the cuts at the turns in regions 1 and 2 of Fig. 1 for comparison

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