Toward Error-free Manufacturing of Fractals - CyberLeninka

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ScienceDirect Procedia CIRP 12 (2013) 43 – 48

8th CIRP Conference on Intelligent Computation in Manufacturing Engineering

Toward error-free manufacturing of fractals A.M.M. Sharif Ullah*, R. Omori, Y. Nagara, A. Kubo, J. Tamaki Kitami Institute of Technology, 165 Koen-cho, Kitami 090-8507, Japan * Corresponding author. Tel.: +81-157-26-9207; fax: +81-157-26-9207. E-mail address: [email protected].

Abstract Fractals are self-similar-scale-invariant entities and often used for modeling complex shapes including natural (e.g., plant, leaf, flower, animal, landscape, etc.) and aesthetically valuable ones (e.g., shapes used in jewelry, buildings, etc.). The purpose of this study is to provide more insights into the error-free manufacturing of fractals. In particular, Hilbert-type space-filling fractals and Iterative Function System based fractals (leaf-like shapes) are manufactured by using both additive (stereolithography) and subtractive (milling) manufacturing processes. The manufacturing errors are identified and the plausible solutions for overcoming the errors are outlined. The findings demonstrate that novel manufacturing process, tooling, and algorithms are needed for achieving error-free manufacturing of fractals. © 2013 The Authors. Published by Elsevier B.V. © 2012 The Published by Elsevier B.V. SelectionRoberto and/or peer-review under responsibility of Professor Roberto Teti. Selection andAuthors. peer review under responsibility of Professor Teti Keywords: Fractals; CAD; CAM; Stereolithography; End Milling; Error; Accuracy.

1. Introduction Fractals are self-similar-scale-invariant mathematical entities introduced in 1700s. These entities were brought into attention again in 1960s by Mandelbrot as a powerful means for modeling complex shapes [1-2]. Particularly, natural objects (plants, leafs, flowers, animals, landscape, etc.) can easily be modeled by using fractals [3-5]. In manufacturing, fractals have been found useful in improving material removal processes, designing manufacturing systems, and modeling surfaces [6-14]. In addition, manufacturing of physical model of fractals with the aid of additive manufacturing processes (e.g., stereolithography) has earned an attention, too [1517]. However, the accuracy of the manufactured fractals has not yet been reported. Not only the additive manufacturing processes, but also the subtractive manufacturing processes (e.g., end milling) can be used for manufacturing fractals. This possibility has not yet been investigated. The purpose of this study is to provide more insights into the error-free manufacturing of fractals. In particular, space-filling and IFS (Iterative Function System) based fractals are considered. The remainder of this article is

2212-8271 © 2013 The Authors. Published by Elsevier B.V. Selection and peer review under responsibility of Professor Roberto Teti doi:10.1016/j.procir.2013.09.009

organized, as follows: Section 2 describes the manufacturing of Hilbert-type space-filling fractals by using both stereolithography and end milling. Section 3 describes the manufacturing of IFS based fractals (leaflike shapes) by using end milling. Section 4 outlines some strategies for eliminating the manufacturing errors of fractals. Section 5 concludes this study. 2. Manufacturing Hilbert-Type Space-Filling Fractals Space-filling fractals were proposed in late 1800s [18-19]. One of the well-studied space-filling fractals is Hilbert-type fractals introduced by Hilbert in 1890s. The space-filling process of Hilbert-type fractals (summarized by Rose [18]) is described, as follows: Let I = {t | 0 t 1} be a unit interval and Q = {(x,y) | 0 x 1, 0 y 1} be a unit square. For a positive integer N {1,2,...} the interval I can be partitioned into 4N subintervals of length 4 -N. The square Q can also be partitioned into 4N sub-squares of side 2-N. While filling the square, a one-to-one correspondence among the subintervals and sub-squares is maintained by the following conditions: Adjacent Condition: Adjacent subintervals correspond to adjacent sub-squares with an

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A.M.M. Sharif Ullah et al. / Procedia CIRP 12 (2013) 43 – 48

edge in common. Nesting Condition: If at the N-th partition, the subinterval INk corresponds to a sub-square QNk, then at the (N+1)-st partition the 4 subintervals of INk correspond to the 4 sub-squares of QNk. Figure 1 illustrates the simplest Hilbert-type fractals for N = 1, 2, 3, and 4. Note that the space is gradually filled with the increase in N and no crossings have taken place (i.e., one-to-one correspondence is maintained).

(N = 1)

stereolithography requires a relatively longer time to manufacturing the fractals because a minimal amount of time is needed for the proper polymerization of the photosensitive resin used [20-21].

3D CAD models

(N = 2)

Stereolithography

(N = 3)

(N = 4)

Fig. 1. Hilbert-type space-filling fractals.

The Hilbert-type fractals shown in Fig. 1 can be manufactured by using both additive manufacturing process (e.g., stereolithography) and subtractive manufacturing process (e.g., milling). First, consider the additive manufacturing (stereolithography). Figure 2 shows the stereolithography based manufacturing of Hilbert-type fractals. As seen from Fig. 2, 3D CAD models are created and the data is used to run a stereolithography apparatus (SLA) available at Kitami Institute of Technology. Some of the physical models of Hilbert-type fractals are shown in Fig. 2. The models are made of polymers (Somos based photosensitive resin). The physical models seem accurate. However, it is observed that the models have a form error. Particularly, the corners are not sharp as it should be. To investigate the form error more systematically, the coordinates of the outer boundaries of the physical models are determined by using a laser microscope. Figure 3 shows one of the results. As seen from Fig. 3, the corners are found to be rounded. The radius of curvature of the rounded corners depends on the radius of curvature of the laser beam used in manufacturing process (stereolithography). Thus, this is an inherent error of this process and the elimination of this error is not possible by process parameter selection or so. Note that

Physical Models Fig. 2. Manufacturing Hilbert-type fractals by using stereolithography.

Fig. 3. A form error of physical model (stereolithography based) of Hilbert-type fractals.

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As such, the process produces a substantial amount of burden on the environment [22]. One of the alternatives is to use a subtractive manufacturing process such as end milling. To investigate this possibility a study has been carried out, as follows. Tool path

3. Manufacturing IFS Based Fractals Let ((xi,y , i) be the coordinates of a point in i-th iteration and ((xi+1,y , i+1) be the coordinates of a point in the next iteration. The following mapping holds between ( i+1,y (x , i+1) and ((xi,y , i). xi yi

1 1

a b xi c d yi

e f

(1)

In equation (1), a, b, c, d, d e, and f are all real numbers. In IFS, the values of (a, b, c, d, e, ff) are stochastically assigned from a predefined sets of values. For example, for modeling a fern leaf the following IFS is used [3-4]: T Tool Fig. 4. Physical model of Hilbert-type fractals made by using end milling.

Leftover

input x0 , y0 for i 1 n p random 0,1 if p 0,0.1 a, b, c, d , e, f 0,0,0,0.16,0,0 if p 0.1,0.18 a, b, c, d , e, f 0.2, 0.26,0.23,0.22,0,1.6 if p 0.18,0.26 a, b, c, d , e, f 0.15,0.28,0.26,0.24,0,0.44 if p 0.26 a, b, c, d , e, f 0.75,0.04, 0.04,0.85,0,1.6 xi 1 axi byi e yi 1 cxi dyi f

n = 1000

n = 2000

(2)

n = 5000

Fig. 5. Form error in fractals manufactured by end milling.

A desktop-type CNC machine tool (available at Kitami Institute of Technology) y is used to manufacture the Hilbert-type fractals shown in Fig. 1. The coordinates of the Hilbert-type fractals have been used as the tool path. Accordingly, NC programs have been generated for performing the end milling operations by using cutting tools having very small diameters (1 mm to 2 mm). Figure 4 shows one of the results. The manufactured fractals are examined carefully to ensure whether or not they are free from errors. One of the errors is a form error similar to that of the previous case. The radius of the tool rounds the corners of the manufactured fractals. Figure 5 shows a magnified view of the manufactured fractals demonstrating the existence of this error. The error lefts extra materials in some corners of the manufactured fractals (hereinafter referred to as leftover). r Thus, for f error-free manufacturing of fractals, it is necessary to remove the leftover. The error elimination process is described in Section 4.

Fig. 6. Modeling fern leaf by IFS (equation (2)).

The formulation in equation (2) ensures that the probability of (a, b, c, d, e, ff) = (0, 0, 0, 0.16, 0, 0) is equal to 0.1, (a, b, c, d, e, ff) = (0.2, -0.26, 0.23, 0.22, 0, 0.16) is equal to 0.08, (a, b, c, d, d e, ff) = (-0.15, 0.28, 0.26, 0.24, 0, 0.44) is equal to 0.08, and (a, b, c, d, e, ff) = (0.75, 0.04, -0.04, 0.85, 0, 1.6) is equal to 0.74. Figure 6 illustrates the models of the fern leaf for n = 1000, 2000, 5000. As seen from Fig. 6, the larger the value of n, the better the model. To model other similar objects, the formulation shown in equation (2) needs slight modification. This issue is beyond the scope of this study and, thus, not discussed further.

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To manufacture the physical model of the fern leaf shown in Fig. 6, end milling can be used. To perform end milling, a NC program is needed. The NC program can be generated either from the bitmap image data or from the coordinates (x ( i,y , i), i = 0,1,...,n.

of the options is to use a triangular cutting tool, as schematically illustrated in Fig. 9. As seen from Fig. 9, the triangular tool is able to enter into the region of leftover, which is not possible by using a circular tool.

Circular tool

Triangular tool

Physical Model

Bitmap data

NC Program

Fig. 9. Eliminating the leftover using a triangular tool.

Fig. 7. Bitmap data driven fern leaf manufacturing. Point based NC Program Physical Model

Figure 10 shows the parameters involved in the tool path generation of a triangular tool. Let P, Q, and R be the vertices of equilateral triangle and T be the center. The sides of the triangle are all l. The triangle is placed in square boundary ABCD. The vertices of the triangle should not cross ABCD. The vertex P always follows the line AB.

C

B

R

( i+1 ,y , i+11) (xi ,,yyi ) (x

P

T

t Fig. 8. Point driven fern leaf manufacturing.

Figures 7 and 8 show the examples of physical models wherein the NC programs have been generated from the bitmap image data and from the points representing the fern leaf (see Fig. 6 and equation (2)), respectively. The machining time has been almost the same for both models, which is substantially shorter than the previous case. As seen from Figs. 7 and 8, it is clear that the fern leaf has not been manufactured properly. This is due to the uneven distribution of the points that model the fern leaf. The region modeled by many points has severely been distorted (e.g., see the top-right segment of the physical models in Figs. 7-8). Even though the depth of cut and tool diameter were decreased, the degree of distortion was not disappeared. 4. Eliminating Errors This section outlines some strategies for eliminating the errors reported in the previous two sections. First, the form error called leftover is considered. The shape of the cutting tool creates this error. Therefore, other shapes of cutting tool can be considered to eliminate the error. One

N l Q

d=AB/2 y A

x

D

Fig. 10. Parameters of tool path generation of a triangular tool.

At a time t the coordinates of P(t), Q(t), t and R(t), and T t) are given as follows: T( Pt Qt Rt Tt

xA, d t x A l sin x A l sin Pt Qt 3

,d ,d Rt

t l cos t l cos

(3)

In equation (3), is the angle that PQ edge makes with AB edge, is the angle that PR edge makes with AB edge, d is the distance between points A and P (at t = 0, d = AB/2), and is the in-feed (mm/sec). Let N be the rotational speed of the tool (revolution per second (rps)). The angles and can be expressed in terms of N N, as follows:

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3

2 N Nt

3

(4)

2 N Nt

( 1,y (x , 1)

The coordinates of T( T t) is used for generating the NC program. The maximum distance travelled by P is d and by this time Q should just touch AB edge. Otherwise, a collision occurs. This yields the following relationship among the in-feed, size of the boundary, rotational speed, and time available for machining (ta). (5)

Inner tool path

Outer tool p path IFS based model of tree

A representative segment

Model for machining

Fig. 11. Remodeling of IFS generated fractals.

( 2,y (x , 2)

( c,y (x , c) Fig. 12. A mapping for creating points by rotation and translation.

input : xc , y c , x1 , y1 rotate x1 , y1 around xc , y c so that x2 xc xc x1 cos yc y 2 y c xc x1 sin yc extend x2 , y 2 to a distance r so that xc,2 x3 r xc d y c,2 y3 r yc d 1

x2 , y 2 y1 sin y1 cos x3 , y 3

(6)

5

0.5

()

The relationships shown in equation (5) indicate that a very low rotational speed, as well as a very slow infeed, is necessary for removing the leftover. Further investigations are needed in this regard, however. Recall the physical models of IFS based fractals as shown in Figs. 7 and 8. The main problem is the uneven distribution of the points that model the shape. This problem can be eliminated in many ways. One of the straightforward ways is to remodel the outline and inner segment by using some curves. Such remodeling is explained in Fig. 11. As seen from Fig. 11, for the sake of remodeling, it is important to identify a representative segment of the model. The remodeling is applied to the outer and inner portions of the representative segment using a mapping, as shown in Fig. 12. The relationships among the points used in the mapping are defined by equation (6). As seen from Fig. 12, the mapping needs two given points ((x ( c,y , c) and ((x1,y , 1)), an angle ( )), and a distance (r). r The point (x1,y , 1) can be rotated by an angle around ((xc,y , c) to generate a point (x2,y , 2). The other point, ((x3,y , 3), is generated on the line passing through ( c,y (x ,yc) and ((x2,y , 2) at a distance r from ((xc,y , c). Similar procedure can be applied recursively to generate points like ((x2,y ,y2) and (x3,y , 3). The values of and r can be redefined after each iteration. After performing the iteration as many times as needed, the points (x ( c,y , c) and ( 1,y (x , 1) of the first iteration, and the points playing the role of ((x3,y , 3) in each iteration are used to represent the remodeled shape. Figure 13 shows how r and are redefined after each iteration to create the outline.

( 3,y (x , 3) r

r (mm)

6Nd t a

1 6N

d

0 0

10 20 30 Iterations

0 -5 0

10 20 30 Iterations

Fig. 13. Angle of rotation and distance functions.

Using end milling (tool diameter 1 mm) the remodeled outlines of the fractals shown in Fig. 11 has been manufactured. The result is shown in Fig. 14. As seen from Fig. 14, the desired outline of the shape has been created properly and no distortions of the shape are observed, as it was observed in the previous case. In the next phase of the research, attempts will be made to removal materials from the inner portion of the remodeled fractals. This way the manufacturing errors of IFS based fractals can be eliminated. However, the presented procedure of remodeling puts an extra burden on the user because the user needs to define r and . To reduce the burden, the user should be assisted by a means so that s/he can easily indentify the

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required functions for determining r and . This issue remains open for further research.

Fig. 14. Machining of remodeled IFS based fractals.

5. Concluding Remarks The Computer-Aided Design (CAD) of fractals is a matured field. On the contrary, Computer-Aided Manufacturing (CAM) of fractals is still a new field. The authors have provided some insights into the error-free manufacturing of fractals either by using additive manufacturing or by using subtractive manufacturing. In particular, Hilbert-type space-filling fractals are manufactured by using both additive and subtractive manufacturing processes and IFS based fractals are manufactured by subtractive manufacturing. Further research is needed to fully materialize the presented strategies of eliminating the manufacturing errors of fractals. Acknowledgments The authors acknowledge the contributions of two former students of Kitami Institute of Technology Mr. Masaki Sakai and Mr. Tooru Ogura. References [1] Mandelbrot, B.B. ,1967. How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, Vol.156, No.3775, p.636. [2] Mandelbrot, B.B., 1982. The Fractal Geometry of Nature, San Francisco: W. H. Freeman. [3] Barnsley, M., 1993. Fractals Everywhere. San Diego: Academic Press. [4] Hutchinson, J. (1981). Fractals and self-similarity, Indiana University Mathematics Journal, Vol.30, No.5, pp.713-747.

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