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Three-axis machining of compound surfaces using flat and filleted endmills
PII: SOOlO-4485(98)00021-9
Computer-Aided Design, Vol. 30. NO. 8, pp. 641-647, 1998 c 1998 Elsevw Science Ltd. All rights reserved Printed in Great Britain 001 O-4485/98/$1 9.00+0.00
Three-axis machining of compound surfaces using flat and filleted endmills Ji Seon Hwangt
and Tien-Chien Chang*
accommodate the flat and filleted endmills. It is a robust method for cutter-path generation, yet there are some intrinsic limitations in the Z-map surface-representation scheme, as stated in Ref.‘. Tang et al.” used the offset-surface approach. In their study, instead of using an approximate surface representation, cutter paths are computed directly from the original part surfaces by intersecting their offset surfaces as well as the offsets of surface boundaries with section planes. It is, however, expected that a great deal of computational effort is needed for the offset-surface intersection, which may be unstable as the surface complexity increases. The method presented in this paper consists of two major steps: a part surface is first tessellated into the triangular polyhedron; cutter paths are then computed from the polyhedron surface. Part surfaces in this study are compound surfaces that include trimmed surface elements. It is unnecessary to discuss the surface-triangulation step in detail since there are a number of available techniques”-‘4 and some of them are for trimmed surfaces1’.14. Therefore the following discussions are focused more on how cutter locations are determined from the triangulated part surface. The proposed method is, in fact, the extended work of Ref.6 where only the ball-endmill case was given. The formulae for computing the cutter-path interval are also presented.
Flat and filleted endmills are less frequently used than ball-end cutters in 3-axis sculptured surface machining. However, they improve cutting efficiency to a great degree in some applications. such as machining smooth part surfaces or rough cutting. Presented in this paper is a method to generate cutter paths to make effective use of these cutter types. A part surface is first approximated into a triangular polyhedron. Cutter paths are then generated from the tessellated surface model. The robust method makes it possible to machine any compound sculptured surfaces regardless of their complexity. An efficient algorithm is used for calculating cutterlocation data. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: compound spherical cutters
surfaces, cutter path generation,
non-
In 3-axis sculptured surface machining, ball endmills are more widely used than the other types of tools like flat or filleted endmills. It is easy to locate a ball-end cutter on a curved surface due to its simple geometric shape. The cutter also produces uniform cusps on the part surface, which makes it convenient to grind or polish the surface in the next operation. Flat and filleted endmills, however, offer an important merit if they are used properly. It is the cutting efficiency. Parts can always be cut by the periphery of such cutters at the maximum speed. but not with the ball endmill’,2. A suitable application of them can be the rough cutting or finishing of smooth surfaces that do not have radical forms. Presented in this paper is a method to generate tool paths of the flat and filleted endmills in the 3-axis machining of compound sculptured surfaces. A set of surface elements representing a part shape to cut is called a compound surface. Several approaches have been addressed to generate cutter paths for the compound surface machining’-‘“. Duncan and Mair’s ‘Polyhedral machining’ ’ was an early work on this problem, in which triangulated surface models were first used in the calculation of cutter-location (CL) data. The proposed method of this paper employs the same idea but to extend it to the application of non-spherical cutters. The Z-map approacha,’ can easily
CUTTER-PATH PLANNING Many types of cutter-path topology are available in compound surface machining. One of them is called Cartesian path that is generated using parallel section planes6. Others are the ‘curve-referenced’ paths for which reference curves are used to determine the path topology. They include the ‘normal-to-curve’, ‘parallel-to-curve’, and so on 15. Cartesian tool path is actually a special case of the normal-to-curve type in which the reference curve is a straight line. Although the method of this paper is not limited to Cartesian machining, it is mainly focused on the generation of Cartesian tool paths that are used more often than the other types. An issue arises when the flat and filleted endmills are used in the rough cutting of sculptured surfaces. Since such cutters, particularly the tip-inserted ones, usually do not have the cutting force at their bottom, it should always be the upward cutting, as shown in Figure la. While it is an important jssue in practice, it is not difficult to solve technically.
TDepartment of Stamping Tool Manufacturing, Hyundai Motor Company. 700 Yangjung-dong, Ulsan 6X3-791, South Korea $School of Industrial Engineering, Purdue University. West Lafayette. IN 47907. USA Paper Rrt,rived: 6 Augu.vt 1996. Rr~~isrd: 14 F&-uurv 199X. Axepted: 5 Man,h 1998
641
Three-axis
machining
of compound
surfaces: J. S. Hwang and T.-C. Chang
a
Figure 1
Cutter-path planning; (a) upward cutting, (b) part-thickness control
In the figure, the upward-cutting tool path, depicted by a heavy line, can easily be obtained from the original by examining the z-coordinates of CL data. Another issue in the rough cutting is the part-thickness control. In any machining process, a uniform thickness of material needs to remain on the part surface during the roughing so that it can be removed in the following finish cutting. To control part thickness, an offset cutter can simply be used. In Figure lb, a virtual filleted endmill is applied to generating cutter paths, instead of the flat endmill used in actual machining. The virtual cutter is the offset of the flat endmill by part thickness. The flat-end cutter paths can then be produced, raising the z-components of the generated CL data by the part-thickness value.
CUTTER-LOCATION ALGORITHM
DATA COMPUTATION
Given a part surface, Cartesian tool paths are generated by first defining a series of vertical section planes and then computing cutter locations at a set of tool axes (vertical lines) on each plane6. Assuming that the part surface is already triangulated, an important part of the process is determining cutter locations from the polyhedron surface whose faces are triangles. This section presents the algorithm to compute the interference-free cutter location at a vertical line. For a clear discussion, some terms are defined first. A cutter-location (CL) point of a flat or a filleted endmill is the centre point of the cutter-bottom circle, depicted by e in Figure 2. A cutter-contact (CC) point is the point at which the cutter touches the part surface. The point r, in the figure is a CC point. When a (interference-free) CL point has been computed at a vertical tool axis passing through a 2D point (x,, ye), there is a region on the part surface in which the CC point always exists. This is called the CC region. It is the tool-shadow area projected in the tool-axis direction. When the part surface is a triangular polyhedron, the CC point is located on the facets that are overlapped with the CC region. Therefore, the interference-free CL point can be obtained by computing the CL point (the z-value of the CL point, actually) with every facet under the tool shadow and then choosing the highest one6. Figure 2 shows the filleted-endmill case where the inner area of the heavy circle is the CC region for computing the interference-free CL point e. The triangles overlapped with a CC region can be collected by checking the 2D distance between the tool axis and triangles. The operation can be speeded up by pregrouping or indexing facets with integer numbers (it is
642
b
called ‘bucketing’ in Ref. lh). Given a tool axis, the following procedure produces an interference-free CL point. (1) With all the vertex points within the CC region, compute the CL points and keep the highest one. (2) For each triangle overlapped with the CC region, do the following. (2.1) If all three vertices of the triangle are on or lower than the current cutter-bottom plane, skip to the next loop. (2.2) Compute the new CL point with the infinite plane containing the triangle. If the new CL point is equal to or lower than the current, skip to the next loop. (2.3) Obtain the CC point on the infinite plane. If the CC point is located within the triangular bound, update the current CL point with the new one and skip to the next loop. (2.4) Identify an edge of the facet that may gouge the current cutter. If the edge does not exist, skip to the next loop. (2.5) Obtain the CC point on the infinite line containing the edge. If the CC point is not located on the edge segment, skip to the next loop. (2.6) Compute the new CL point with the edge. If the new point is higher than the current, update the current CL point with the new one. The main idea of the above procedure is to reduce computation time. The initial CL point is computed from all the vertex points in the CC region. The cutter at this
+u
JLL!
Z
L
rc
ie
v
L Figure 2
+X
CL point. CC point and CC region
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang initial point can interfere only with faces and edges of triangles but not with vertices. The CL point is then updated (raised) obtaining the new CL points computed from faces and edges. During the process, if any triangle is located below the current cutter-bottom plane, it is rejected immediately from further computation. Because the CL-point calculation is simpler with vertex points than with faces and edges, the rejection can reduce the total computation time. The overall procedure that lifts up the cutter until it is cleared from all the faces and edges is similar to that of Ref.h. The difference is that the flat and filleted endmills are used in this study. The next section gives the analytic formulae to compute CL and CC points from vertices, triangle faces, and edges for both the flat and filleted endmills.
CL-POINT COMPUTATION EDGE, AND FACE
FROM VERTEX,
This section presents the CL-point computation methods from the vertex point, edge segment. and triangular facet of the polyhedron surface. Both the flat- and filleted-endmill cases are given. Without loss of generality, it is assumed that the vertical tool axis passes through the origin point of the locally defined coordinate system. Thus the aim is to compute the :-component of the CL point e = (0, 0, ze).
where n = (u, b, c) is the unit normal vector of the plane and its direction points outwards from the part surface. The zvalue of the CL point e = (0. 0, z,) is then given by . -- _ dlc + Rltan 8 .>e (3) where :, = - d/c is the z-value of the intersection point of the plane with the tool axis (ei in the figure), and 0 = sin-I((,) is the angle between the normal vector and the horizontal plane. Since the tool-axis vector in 3-axis machining is u= (0, 0, l), the CC point on the infinite plane is given in vector notation as rc = e + (R tan 0)~ - (Rlcos 6)n
Case 3: from the edge An edge is defined by two end points, r, = (xl, yl, zl) and r2 x (.rz, y2, c2), and expressed by q
r(J) = (I -pPl
(5)
+pr?
When the edge and a flat endmill of a radius R are projected onto the xy-plane, the intersection of them generates the following quadratic equation: + 2(X, (X2- _Y,)
((.X2-.X-l ): + (y2 - .v[)?)$
Flat-endmill
(4)
The test if the CC point is within the facet’s boundary can be performed on the xv-plane by projecting the facet and the CC point onto the plane. The details of the test can be found in Ref.‘.
case +v,(?.2-?.,))~+~~f+~:-R2=0
Case 1: from the vertex Let r = (x, y, :) be a vertex point and R be the radius of a flat endmill. The :-value of the CL point is ze = :
Figure 4 shows the 2D view. If the roots of the equation are imaginary, the cutter cannot touch the edge. When they are real, the :-component of the CL point e is given by
(I)
if (x’ +J’)“’ otherwise.
5 R. The
tool
cannot
touch
the
r
point,
max((l -p,)z~
+PI:Z,
(I -p&,
if p, E [O, ] andp, (1 -p,);,
Case 2: from the facet As shown in Figure 3, a triangular facet is defined by three vertex points r,, r2 and r3, and a flat endmill of a radius R touches the infinite plane containing the facet. The plane equation can be obtained from the three points and expressed by
(6)
i,, = (
+P~zz),
E [0, I]
+plQ.
if p, E LO, 1 and pZ @ LO,1I (I - p&,
I
ifp,
(7)
+p92. ~[0,11andp2E[0,1]
where p, and p2 are the real roots. If p, @ [O,l] and p2 $?T [O,l], the CC point is located outside the edge-segment limits.
Filleted-endmill
case
Case 1: from the vertex Let the body radius of a filleted-endmill be R and its comer radius be Y. The :-component of the CL point e = (0, 0, ie) computed from a vertex point r = (x, y, z) is ify
..>
i 0
Ze =
I- L - r + (r2 - (q - R + Y)‘)“‘,
(8)
rl
r2
1 L_,X
Figure
3
Flat-end cutter location
r2
e
x
R
rc
rl
I
with a facet
w Y
be r3i
Y
if R-r
Figure
4
Flat-end cutter location
with an edge
643
Three-axis
machining
of compound
surfaces: J. S. Hwang and T.-C. Chang
where q = (x’ + y*)“’ is the horizontal distance between the point and the tool axis. The cutter cannot touch the point if q > R. Case 2: from the facet As in the flat-endmill case, the infinite plane of the triangular facet is expressed by eqn (2), and the angle between the facet normal vector and the horizontal plane is 8. The z-value of the CL point e is then z, = - dlc - r + (R - r)/tan 8 + r/sin 9
(9)
and the CC point is computed by r, = e + (Y+ (R - r)tan %)u - (r + (R - r)lcos %)n
(10)
Case 3: from the edge The two end points r, and r2 of an edge segment are first rotated about the tool axis so that they are located on the vertical plane that is parallel to the yz-plane. The rotated points are given by r3 = (.x3, y3, z3) and r4 = (x4, y4, zJ) as shown in Figure 5. To make the computation simple, the filleted tool is offset inwards by the cutter-comer radius r and the line segment is offset to be a cylinder of a radius r. Then the problem is to compute the location of a virtual flat-end cutter of a radius R - r, touching the cylinder, as shown in the figure. When the (virtual) flat-end cutter moves down until it touches the cylinder. a CC point, depicted by rt in the figure, is defined. The observation of the geometry in the top view gives an idea that, at the CC point rt, the (flat-end) cutter-bottom circle contacts with an ellipse externally. The ellipse is the intersection of the cutter-bottom plane and the cylinder. It is depicted by a dotted line in Figure 5. To compute the CC point on the q-plane, the following equations are needed: x’++(&r)’ (x -x&?
+ (4‘- J&t2
the line segment. The horizontal distance from the edge line to the tool axis is xd = x3 = x4. If xd > R, the tool cannot touch the line. Now the equations have only three unknown variables, x, v and yd, and combining them results in an implicit equation, ,f(x) = (r2 - i).u’(x - xd)’ + t”(R-r)‘(x
(12) The root of the equation can be computed employing the well-
known Newton-Raphson search. The average number of iterations was five in the implementation test, which is more efficient than directly solving the polynomial equation of degree four. Note that the external contact point can always be found by setting its bounds, xd(R - r)lR 5 x 5 min(xd, R - r), which can easily be proved by considering the case where the ellipse is the circle of a radius r. The middle value of the bounds can be used for the initial value for the iteration. Once the x-component x, of the (virtual) CC point r, is found from the search, yt is calculated using the first one of eqn (1 l), and given by yI = sign(z, - z3) ((R - r)’ - XT) “I
Jo = sign(z, - zi) ( I_Y, I + r sin d( I - (xt - xd)‘/r2)“‘) (14) Let the equation of the transformed be expressed as
(15)
wheref= -(z4-zJ), g=y4-y3,andh=y~zJ-yJz3.The i.--value of the CL point e is then given by ze = - @, + h)lg - (glf)(y,
d
(16)
DETERMINATION
While the tool paths (CL paths) that were generated using vertical planes are planar, the CC paths (the sequences of CC points) are not. Thus, it is difficult to estimate the cutting error and the step length precisely. The exact computation is only possible when the envelope surface of the moving cutter is compared with the part surface, which requires much computation. A simple way is approximating it into a 2D problemh. Since the CL path is planar, a circle is defined locally in between the two consecutive CL points using the tangent at the CC point (for the first two points of the path) or the previous CL point (for those in the middle of the path). Then the cutting error is estimated computing the chordal deviation and so is the step length.
PATH-INTERVAL
644
- yt) - r
where the term z, = - @, + h)lg is the z-value at the real CC point r,, and - (g/‘)(y, - y,) is the z-directional distance from r, to r,. If y3 9 yc 5 yJ, the CC point is located on the line segment.
STEP-LENGTH
cutter location with an edge
edge line on the yz-plane
(11)
The first two equations are those for the circle and ellipse, respectively. The third one is for providing the tangent condition of two geometric entities. The longer radius of the ellipse is given by t= Jr/sin 41, where 4 = tan ~ ‘((Q - zj)/(y4 -F~)) is the angle of the slope of
Filleted-end
(13)
where the function sign(z) is defined as sign(z) = 1, when : P 1, and sign(z) = - I, otherwise. The y-value of the actual CC point r, can then be obtained by introducin 5 Xl and yt into another ellipse equation, (x - x$/r + (y - y,)‘>(r sin $)* = 1. The new ellipse is the xy-plane projection of the circle on the cylinder passing through r,. The farthest yc from the x-axis is taken (since it must be the centre of the external ellipse) and expressed by
,fifgz+h=O
= 1
y/x = (r’(y - yd))l(t2(x - xd))
Figure 5
- xd)? - r’x’ = 0
COMPUTATION
Path-interval computation is another important issue in sculptured surface machining. In Cartesian machining, it is
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang
b
a Figure 6
Path-intenTal computation for the Hat endmill: (a) 3D view, (b) 2D view
to determine the interval of two consecutive section planes. The exact computation of the path interval is again difficult for sculptured surfaces. Thus, the tangent-plane approach I7 is employed in this study. in which the part surface is assumed to be locally planar. The following sections give a detailed description of the path-interval computation for the flat and filleted endmills.
Flat-endmill
case
In Figure 6a, a fat endmill of a radius R cuts a plane surface whose unit normal vector is n. The cutter moves in the direction of the horizontal vector a, and the normal vector of the section plane is b. Both are the mutually perpendicular unit vectors. The problem here is to compute the path interval 6, given the scallop height S. A plane is created by offsetting the part surface by s. The intersection of the offset plane and the cutter produces an area, which is depicted by the shaded region in the figure. Let us call it the undercut region. The meaning of the region is that, under this region, the height of remained material on the part surface is less than or equal to S. Therefore it is easily conceivable that the track of the 2D undercut region along a path must touch the one in the next path to make the maximum height of uncut material equal to s. The idea is shown in Figure 6b. As can be seen in the figure, the path interval can actually be obtained by computing the width of the 2D undercut region in the direction of b. Let nxvbe the projected vector of the plane normal vector n = (x,, yi, z,) onto the horizontal plane. The width of the undercut region in the direction of n,. is given bq 1= min(s/sin (Y, 2R), where CY= cos ~ ‘(z,) is the slope angle of the plane surface. When the horizontal vector h is given by h = (n X n,)lln X n,, 1,and the angle between a and h is /3 = cos-‘(a.h), the path interval is computed by ifI=2Rorif/3?y 2R,
expressed by exact formulae. Because of the toroidal cutter edge, the boundary of the 2D undercut region is not an analytic curve. Remember that, in the flat-cutter case, the boundary curve consists of a line and a circular arc that make it possible to induce the closed-form solution. To compute the width of the 2D boundary curve for the filletedendmill case, piecewise linear approximation is applied. Let R be the cutter-body radius, r be the cutter-edge radius, and s be the scallop height. The vectors a, b, n, nAY,and h are defined as same as in the flat-endmill case. To sample points on the curve, two extreme points in the direction of n, are first computed as follows. When a local coordinate system is defined at the cutter-bottom centre in which the x-axis is parallel to the vector h as shown in Figure 7. the 2D frontend point v of the curve is given by v=&,
2m sin /3,
(0, R - r + u cos a + (r - s)sin 01.
Filleted-endmill
ifs
and (r - .s)Iu > tan CY
1
(18)
where LL = (2rs - s~)“~ and Q!= cos- ’ (z,), the slope angle of the part plane. The equation of the offset plane (the dotted line on the yz-plane in Figure 7) is ;: = (tan CX)~+ k
(19)
where k = r( 1 - cos a) - (R - r + r sin a)tan CY+ s/cos 01. Then the rear-end point w is computed by w = (x,, .v,) ’ (0, R - r - u cos CY+ (r - s)sin CY),if k 5 -n ( k ((R - r)’ -(k/tan
if /3<-yandl<2R
(0, r - R - w cos
/2)“2 and y = sin ~ ‘(m/R).
+r(l
case
The above method can also be applied to the filleted endmill. In this case. however, the path interval cannot be
CX)‘)“*, -k/tan
a), if
-III
<
(17, where m = (2Rl-
or if s < r and (r - s)/u 5 tan (Y
=
if /3 2 y and 1~ R
1 R+msinp-(R--l)cosp,
if s 2 r
(0, R),
and R < I < 2R 6=
YV)
, (0,
--R),
01 +
(r - Lj)sin a),
if n 5 k < n
+cosa)
if k?n+r(l+cosa) (20)
where
II = (R - r)tan 01.
L’= s - 2(R - r)sin o(,
and 645
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang w = (2~ - v2)“‘. Note that if the offset plane intersects with the cutter-bottom circle (in other words, if the boundary of the undercut region includes a line segment on the cutterbottom plane). two points at the end of the intersection line are computed as the rear end of the curve. Once two extreme points are obtained, the middle points in between the two points are sampled by the following method. Let us assume that p = (xP, yP, z,,) is a sampling point on the curve, where _v,, holding the condition ?‘$+s_vp ‘?‘>. is already known. The z-value !,, of the point can be computed by eqn (19), and the radius of the circle on the toroidal edge that passes through the point is given by
if zP 2 r
R, P=
if :,,
1 R-r+(2~,,-~~)“‘, The x-component is then .$ = _c (p’ - ,$)“I Figure
7
Path-interval
computation
for the filleted endmill
(22)
Therefore if a set of y-values are first chosen, then the Xvalues of the sampling points can be computed by the above process. Finally, the path interval is obtained by calculating the width of the point set in the direction of the section-plane normal vector b. The accuracy of the solution depends on
(b)
Figure 8
646
Implementation
test; (a) compound
surface, (b) triangular
(21)
polyhedron,
(c) cutter paths, (d) cutting simulation
Three-axis
machining
the number of sampling points so that further study is necessary to determine the number properly. In the implementation test, 12 points were sampled to approximate the curve.
of compound
6.
7. 8.
IMPLEMENTATION
TEST
The method presented in the paper was implemented on an IBM RS/6000-3BT computer using the C language. The compound surface model, shown in Figure #a, consists of three trimmed surfaces and the size is 50 X 40 mm’. The triangular polyhedron was generated from the surface with the sampling tolerance of 0.02 mm, producing 8097 facets. Figure 8b shows the polyhedron. The cutter paths were then produced from the polyhedron, as shown in Figure 8~. A filleted endmill was used. Its body radius was 3 mm and its corner radius was 1 mm. The scallop height was 0.05 mm. It took 0.7 s for triangulation and 17.9 s for cutter-path generation. The number of CL points was 4240. Figure 8~1 shows the snapshot of the cutting simulation of the generated cutter paths.
9.
IO. I I. 12. 13. II. IS.
16.
CONCLUSIONS In this paper, a cutter-path generation method is presented to use flat and filleted endmills in sculptured surface machining. The CL point is computed in an efficient way in that only a portion of the facets in a CC region are involved in the computation, using the rejection criterion. The paper also provides solutions to compute the path interval for both the flat and filleted cutters, in terms of the scallop height. Combined with the technique for ball-end cutters in Ref. ‘, the proposed methodology offers a robust and versatile tool for machining compound sculptured surfaces.
17.
surfaces: J. S. Hwang and T.-C. Chang
Hwang, J. S., Interference-free tool-path generation m the NC machining of parametric compound surfaces. Cornput.-Aidrd Des., 1992. 24( 12), 667-676. Lal. .I. Y. and Wang, D. J.. A strategy for finish cutting path generation of compound surfaces. Cnmput. Indust.. 1994. 25(2). 1X9-209. Kondo, T.. Kishinami, T. and Saito. K.. Machining system based on scanning data (in Japanese). J. .Jr~ptrn SW. Precision Eng.. 1988, 54(h). lO70- 1075. Choi. B. K.. Chung. Y. C., Park, J. W. and Kim, D. H., Unified CAMsystem architecture for die and mould manufacturing. Cornput.-AidcJd Dry., 1994, 260). 235-243. Tang, K.. Cheng, C. C. and Dayan, Y.. Offsetting surface boundaries and 3-axis gouge-free surface machimng. Compltt:Aided De.v.. 1995. 27( 12). 915m~927. Filip. D.. Magedson, R. and Markot. R.. Surface algorithms using hounds on derivatives. Con~prt.Aided G~om Des.. 1986. 3(4). 295-3 I I. Drysdalc, R. L.. Jerard. R. B., Schaudt. B. and Hauck, K., Discrete Gmulation of NC machining. Algorithmiccl. 1989, 4(l), X-60. Piegl. L. A. and Richard, A. M., Tessellating trimmed NURBS surlaces. Conrput..Aitfetf Des.. 1995. 27( I ). 6-76. Hamann, B. and Tsai, P.-Y.. A tessellation algorithm for the representatlon of trimmed NURBS surfaces with arbitrary trimming curves. (‘omprrt. -Aided Des., 1996, 28(6/7), 46 I-472. Klaus. R. and Schramm, P., Numerically-controlled milling of CAD surface data. In Hagen, H. and Rollers, D. (Eds.). Geometric Modelirr,y: Methods N& Applications. Springer-Verlag. Berlin. 199 I. pp. 213-275. Li. S. X. and Jerard. R. B.. S-axis machining of sculptured surfaces with a Rat-end cutter. Cornput.-Aided Des.. 1994. 26(3). 165- 178. Choi, B. K.. Park. J. W. and Jun. C. S., Cutter-location data optimization in 5.aui\ surface machining. Comtw-Aided De.\. 1993, 25(6), 377-386.
Ji Sean Hwmg i.s (I senior research rrqinrer at the Stamping Tool MunufucLlcturing Department of Hyundai Motor Compnn~. South Korea He received an MS from Korea Adwnced lnstitutr oj’Sciencr trnd Technology in 1987 and a Ph.D. ,j+om Purdue University in 1997, both in Industrial Engineering. His wwurch interests include CAD/CAM. .xxlplured wrj&e muchining. geometric modeling, cmd productiorl phoning arxl control.
REFERENCES I.
Vickers, G. W. and Quan, K. W., Ball-mills versus end-mills for curved surface machining. Trcrns. ASME, J. En,?. Indusf.. 1989, 111(I ). 22-x. 2. Bell. C.. Landi. B. and Sabin. M., The programming and use of numerical control to machine sculptured surfaces. Proc. 14th Machine Tool Design and Research Conf., 1973, pp. 233-238. 3. Duncan, .I. P. and Mair, S. G., Slulptured Surfaces in Engineering and Medicine. Cambridge University Press, Cambridge, MA, 1983. 4. Bobrow. J. E., NC machine tool path generation from CSG part representations. Cornput..Aided Des.. 1985, 17(2), 69-76. 5. Choi. B. K. and Jun, C. S., Ball-end cutter interference avoidance in NC machining of sculptured surfaces. Cornput:Aided Des., 1989, 21(6), 37 I-378.
Dr Tien-Chirn Chcrrr~is mprojessor in the School of’ Industrinl Engineering clt Putdue lJnil,er.sity His research interest is in thr urw of marmfacturing automtrtion. Hiv current research act&ties include cuuorncued proce.ts planning. .wdptured surjizce machining, and virrunl manufacturing. Dr Chang is the author and u-author of several hooks cmd articles on automated prw’ss pltrnr@ and computer-aided mnnlrfuc.trrrirly.
647
Computer-Aided Design, Vol. 30. NO. 8, pp. 641-647, 1998 c 1998 Elsevw Science Ltd. All rights reserved Printed in Great Britain 001 O-4485/98/$1 9.00+0.00
Three-axis machining of compound surfaces using flat and filleted endmills Ji Seon Hwangt
and Tien-Chien Chang*
accommodate the flat and filleted endmills. It is a robust method for cutter-path generation, yet there are some intrinsic limitations in the Z-map surface-representation scheme, as stated in Ref.‘. Tang et al.” used the offset-surface approach. In their study, instead of using an approximate surface representation, cutter paths are computed directly from the original part surfaces by intersecting their offset surfaces as well as the offsets of surface boundaries with section planes. It is, however, expected that a great deal of computational effort is needed for the offset-surface intersection, which may be unstable as the surface complexity increases. The method presented in this paper consists of two major steps: a part surface is first tessellated into the triangular polyhedron; cutter paths are then computed from the polyhedron surface. Part surfaces in this study are compound surfaces that include trimmed surface elements. It is unnecessary to discuss the surface-triangulation step in detail since there are a number of available techniques”-‘4 and some of them are for trimmed surfaces1’.14. Therefore the following discussions are focused more on how cutter locations are determined from the triangulated part surface. The proposed method is, in fact, the extended work of Ref.6 where only the ball-endmill case was given. The formulae for computing the cutter-path interval are also presented.
Flat and filleted endmills are less frequently used than ball-end cutters in 3-axis sculptured surface machining. However, they improve cutting efficiency to a great degree in some applications. such as machining smooth part surfaces or rough cutting. Presented in this paper is a method to generate cutter paths to make effective use of these cutter types. A part surface is first approximated into a triangular polyhedron. Cutter paths are then generated from the tessellated surface model. The robust method makes it possible to machine any compound sculptured surfaces regardless of their complexity. An efficient algorithm is used for calculating cutterlocation data. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: compound spherical cutters
surfaces, cutter path generation,
non-
In 3-axis sculptured surface machining, ball endmills are more widely used than the other types of tools like flat or filleted endmills. It is easy to locate a ball-end cutter on a curved surface due to its simple geometric shape. The cutter also produces uniform cusps on the part surface, which makes it convenient to grind or polish the surface in the next operation. Flat and filleted endmills, however, offer an important merit if they are used properly. It is the cutting efficiency. Parts can always be cut by the periphery of such cutters at the maximum speed. but not with the ball endmill’,2. A suitable application of them can be the rough cutting or finishing of smooth surfaces that do not have radical forms. Presented in this paper is a method to generate tool paths of the flat and filleted endmills in the 3-axis machining of compound sculptured surfaces. A set of surface elements representing a part shape to cut is called a compound surface. Several approaches have been addressed to generate cutter paths for the compound surface machining’-‘“. Duncan and Mair’s ‘Polyhedral machining’ ’ was an early work on this problem, in which triangulated surface models were first used in the calculation of cutter-location (CL) data. The proposed method of this paper employs the same idea but to extend it to the application of non-spherical cutters. The Z-map approacha,’ can easily
CUTTER-PATH PLANNING Many types of cutter-path topology are available in compound surface machining. One of them is called Cartesian path that is generated using parallel section planes6. Others are the ‘curve-referenced’ paths for which reference curves are used to determine the path topology. They include the ‘normal-to-curve’, ‘parallel-to-curve’, and so on 15. Cartesian tool path is actually a special case of the normal-to-curve type in which the reference curve is a straight line. Although the method of this paper is not limited to Cartesian machining, it is mainly focused on the generation of Cartesian tool paths that are used more often than the other types. An issue arises when the flat and filleted endmills are used in the rough cutting of sculptured surfaces. Since such cutters, particularly the tip-inserted ones, usually do not have the cutting force at their bottom, it should always be the upward cutting, as shown in Figure la. While it is an important jssue in practice, it is not difficult to solve technically.
TDepartment of Stamping Tool Manufacturing, Hyundai Motor Company. 700 Yangjung-dong, Ulsan 6X3-791, South Korea $School of Industrial Engineering, Purdue University. West Lafayette. IN 47907. USA Paper Rrt,rived: 6 Augu.vt 1996. Rr~~isrd: 14 F&-uurv 199X. Axepted: 5 Man,h 1998
641
Three-axis
machining
of compound
surfaces: J. S. Hwang and T.-C. Chang
a
Figure 1
Cutter-path planning; (a) upward cutting, (b) part-thickness control
In the figure, the upward-cutting tool path, depicted by a heavy line, can easily be obtained from the original by examining the z-coordinates of CL data. Another issue in the rough cutting is the part-thickness control. In any machining process, a uniform thickness of material needs to remain on the part surface during the roughing so that it can be removed in the following finish cutting. To control part thickness, an offset cutter can simply be used. In Figure lb, a virtual filleted endmill is applied to generating cutter paths, instead of the flat endmill used in actual machining. The virtual cutter is the offset of the flat endmill by part thickness. The flat-end cutter paths can then be produced, raising the z-components of the generated CL data by the part-thickness value.
CUTTER-LOCATION ALGORITHM
DATA COMPUTATION
Given a part surface, Cartesian tool paths are generated by first defining a series of vertical section planes and then computing cutter locations at a set of tool axes (vertical lines) on each plane6. Assuming that the part surface is already triangulated, an important part of the process is determining cutter locations from the polyhedron surface whose faces are triangles. This section presents the algorithm to compute the interference-free cutter location at a vertical line. For a clear discussion, some terms are defined first. A cutter-location (CL) point of a flat or a filleted endmill is the centre point of the cutter-bottom circle, depicted by e in Figure 2. A cutter-contact (CC) point is the point at which the cutter touches the part surface. The point r, in the figure is a CC point. When a (interference-free) CL point has been computed at a vertical tool axis passing through a 2D point (x,, ye), there is a region on the part surface in which the CC point always exists. This is called the CC region. It is the tool-shadow area projected in the tool-axis direction. When the part surface is a triangular polyhedron, the CC point is located on the facets that are overlapped with the CC region. Therefore, the interference-free CL point can be obtained by computing the CL point (the z-value of the CL point, actually) with every facet under the tool shadow and then choosing the highest one6. Figure 2 shows the filleted-endmill case where the inner area of the heavy circle is the CC region for computing the interference-free CL point e. The triangles overlapped with a CC region can be collected by checking the 2D distance between the tool axis and triangles. The operation can be speeded up by pregrouping or indexing facets with integer numbers (it is
642
b
called ‘bucketing’ in Ref. lh). Given a tool axis, the following procedure produces an interference-free CL point. (1) With all the vertex points within the CC region, compute the CL points and keep the highest one. (2) For each triangle overlapped with the CC region, do the following. (2.1) If all three vertices of the triangle are on or lower than the current cutter-bottom plane, skip to the next loop. (2.2) Compute the new CL point with the infinite plane containing the triangle. If the new CL point is equal to or lower than the current, skip to the next loop. (2.3) Obtain the CC point on the infinite plane. If the CC point is located within the triangular bound, update the current CL point with the new one and skip to the next loop. (2.4) Identify an edge of the facet that may gouge the current cutter. If the edge does not exist, skip to the next loop. (2.5) Obtain the CC point on the infinite line containing the edge. If the CC point is not located on the edge segment, skip to the next loop. (2.6) Compute the new CL point with the edge. If the new point is higher than the current, update the current CL point with the new one. The main idea of the above procedure is to reduce computation time. The initial CL point is computed from all the vertex points in the CC region. The cutter at this
+u
JLL!
Z
L
rc
ie
v
L Figure 2
+X
CL point. CC point and CC region
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang initial point can interfere only with faces and edges of triangles but not with vertices. The CL point is then updated (raised) obtaining the new CL points computed from faces and edges. During the process, if any triangle is located below the current cutter-bottom plane, it is rejected immediately from further computation. Because the CL-point calculation is simpler with vertex points than with faces and edges, the rejection can reduce the total computation time. The overall procedure that lifts up the cutter until it is cleared from all the faces and edges is similar to that of Ref.h. The difference is that the flat and filleted endmills are used in this study. The next section gives the analytic formulae to compute CL and CC points from vertices, triangle faces, and edges for both the flat and filleted endmills.
CL-POINT COMPUTATION EDGE, AND FACE
FROM VERTEX,
This section presents the CL-point computation methods from the vertex point, edge segment. and triangular facet of the polyhedron surface. Both the flat- and filleted-endmill cases are given. Without loss of generality, it is assumed that the vertical tool axis passes through the origin point of the locally defined coordinate system. Thus the aim is to compute the :-component of the CL point e = (0, 0, ze).
where n = (u, b, c) is the unit normal vector of the plane and its direction points outwards from the part surface. The zvalue of the CL point e = (0. 0, z,) is then given by . -- _ dlc + Rltan 8 .>e (3) where :, = - d/c is the z-value of the intersection point of the plane with the tool axis (ei in the figure), and 0 = sin-I((,) is the angle between the normal vector and the horizontal plane. Since the tool-axis vector in 3-axis machining is u= (0, 0, l), the CC point on the infinite plane is given in vector notation as rc = e + (R tan 0)~ - (Rlcos 6)n
Case 3: from the edge An edge is defined by two end points, r, = (xl, yl, zl) and r2 x (.rz, y2, c2), and expressed by q
r(J) = (I -pPl
(5)
+pr?
When the edge and a flat endmill of a radius R are projected onto the xy-plane, the intersection of them generates the following quadratic equation: + 2(X, (X2- _Y,)
((.X2-.X-l ): + (y2 - .v[)?)$
Flat-endmill
(4)
The test if the CC point is within the facet’s boundary can be performed on the xv-plane by projecting the facet and the CC point onto the plane. The details of the test can be found in Ref.‘.
case +v,(?.2-?.,))~+~~f+~:-R2=0
Case 1: from the vertex Let r = (x, y, :) be a vertex point and R be the radius of a flat endmill. The :-value of the CL point is ze = :
Figure 4 shows the 2D view. If the roots of the equation are imaginary, the cutter cannot touch the edge. When they are real, the :-component of the CL point e is given by
(I)
if (x’ +J’)“’ otherwise.
5 R. The
tool
cannot
touch
the
r
point,
max((l -p,)z~
+PI:Z,
(I -p&,
if p, E [O, ] andp, (1 -p,);,
Case 2: from the facet As shown in Figure 3, a triangular facet is defined by three vertex points r,, r2 and r3, and a flat endmill of a radius R touches the infinite plane containing the facet. The plane equation can be obtained from the three points and expressed by
(6)
i,, = (
+P~zz),
E [0, I]
+plQ.
if p, E LO, 1 and pZ @ LO,1I (I - p&,
I
ifp,
(7)
+p92. ~[0,11andp2E[0,1]
where p, and p2 are the real roots. If p, @ [O,l] and p2 $?T [O,l], the CC point is located outside the edge-segment limits.
Filleted-endmill
case
Case 1: from the vertex Let the body radius of a filleted-endmill be R and its comer radius be Y. The :-component of the CL point e = (0, 0, ie) computed from a vertex point r = (x, y, z) is ify
..>
i 0
Ze =
I- L - r + (r2 - (q - R + Y)‘)“‘,
(8)
rl
r2
1 L_,X
Figure
3
Flat-end cutter location
r2
e
x
R
rc
rl
I
with a facet
w Y
be r3i
Y
if R-r
Figure
4
Flat-end cutter location
with an edge
643
Three-axis
machining
of compound
surfaces: J. S. Hwang and T.-C. Chang
where q = (x’ + y*)“’ is the horizontal distance between the point and the tool axis. The cutter cannot touch the point if q > R. Case 2: from the facet As in the flat-endmill case, the infinite plane of the triangular facet is expressed by eqn (2), and the angle between the facet normal vector and the horizontal plane is 8. The z-value of the CL point e is then z, = - dlc - r + (R - r)/tan 8 + r/sin 9
(9)
and the CC point is computed by r, = e + (Y+ (R - r)tan %)u - (r + (R - r)lcos %)n
(10)
Case 3: from the edge The two end points r, and r2 of an edge segment are first rotated about the tool axis so that they are located on the vertical plane that is parallel to the yz-plane. The rotated points are given by r3 = (.x3, y3, z3) and r4 = (x4, y4, zJ) as shown in Figure 5. To make the computation simple, the filleted tool is offset inwards by the cutter-comer radius r and the line segment is offset to be a cylinder of a radius r. Then the problem is to compute the location of a virtual flat-end cutter of a radius R - r, touching the cylinder, as shown in the figure. When the (virtual) flat-end cutter moves down until it touches the cylinder. a CC point, depicted by rt in the figure, is defined. The observation of the geometry in the top view gives an idea that, at the CC point rt, the (flat-end) cutter-bottom circle contacts with an ellipse externally. The ellipse is the intersection of the cutter-bottom plane and the cylinder. It is depicted by a dotted line in Figure 5. To compute the CC point on the q-plane, the following equations are needed: x’++(&r)’ (x -x&?
+ (4‘- J&t2
the line segment. The horizontal distance from the edge line to the tool axis is xd = x3 = x4. If xd > R, the tool cannot touch the line. Now the equations have only three unknown variables, x, v and yd, and combining them results in an implicit equation, ,f(x) = (r2 - i).u’(x - xd)’ + t”(R-r)‘(x
(12) The root of the equation can be computed employing the well-
known Newton-Raphson search. The average number of iterations was five in the implementation test, which is more efficient than directly solving the polynomial equation of degree four. Note that the external contact point can always be found by setting its bounds, xd(R - r)lR 5 x 5 min(xd, R - r), which can easily be proved by considering the case where the ellipse is the circle of a radius r. The middle value of the bounds can be used for the initial value for the iteration. Once the x-component x, of the (virtual) CC point r, is found from the search, yt is calculated using the first one of eqn (1 l), and given by yI = sign(z, - z3) ((R - r)’ - XT) “I
Jo = sign(z, - zi) ( I_Y, I + r sin d( I - (xt - xd)‘/r2)“‘) (14) Let the equation of the transformed be expressed as
(15)
wheref= -(z4-zJ), g=y4-y3,andh=y~zJ-yJz3.The i.--value of the CL point e is then given by ze = - @, + h)lg - (glf)(y,
d
(16)
DETERMINATION
While the tool paths (CL paths) that were generated using vertical planes are planar, the CC paths (the sequences of CC points) are not. Thus, it is difficult to estimate the cutting error and the step length precisely. The exact computation is only possible when the envelope surface of the moving cutter is compared with the part surface, which requires much computation. A simple way is approximating it into a 2D problemh. Since the CL path is planar, a circle is defined locally in between the two consecutive CL points using the tangent at the CC point (for the first two points of the path) or the previous CL point (for those in the middle of the path). Then the cutting error is estimated computing the chordal deviation and so is the step length.
PATH-INTERVAL
644
- yt) - r
where the term z, = - @, + h)lg is the z-value at the real CC point r,, and - (g/‘)(y, - y,) is the z-directional distance from r, to r,. If y3 9 yc 5 yJ, the CC point is located on the line segment.
STEP-LENGTH
cutter location with an edge
edge line on the yz-plane
(11)
The first two equations are those for the circle and ellipse, respectively. The third one is for providing the tangent condition of two geometric entities. The longer radius of the ellipse is given by t= Jr/sin 41, where 4 = tan ~ ‘((Q - zj)/(y4 -F~)) is the angle of the slope of
Filleted-end
(13)
where the function sign(z) is defined as sign(z) = 1, when : P 1, and sign(z) = - I, otherwise. The y-value of the actual CC point r, can then be obtained by introducin 5 Xl and yt into another ellipse equation, (x - x$/r + (y - y,)‘>(r sin $)* = 1. The new ellipse is the xy-plane projection of the circle on the cylinder passing through r,. The farthest yc from the x-axis is taken (since it must be the centre of the external ellipse) and expressed by
,fifgz+h=O
= 1
y/x = (r’(y - yd))l(t2(x - xd))
Figure 5
- xd)? - r’x’ = 0
COMPUTATION
Path-interval computation is another important issue in sculptured surface machining. In Cartesian machining, it is
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang
b
a Figure 6
Path-intenTal computation for the Hat endmill: (a) 3D view, (b) 2D view
to determine the interval of two consecutive section planes. The exact computation of the path interval is again difficult for sculptured surfaces. Thus, the tangent-plane approach I7 is employed in this study. in which the part surface is assumed to be locally planar. The following sections give a detailed description of the path-interval computation for the flat and filleted endmills.
Flat-endmill
case
In Figure 6a, a fat endmill of a radius R cuts a plane surface whose unit normal vector is n. The cutter moves in the direction of the horizontal vector a, and the normal vector of the section plane is b. Both are the mutually perpendicular unit vectors. The problem here is to compute the path interval 6, given the scallop height S. A plane is created by offsetting the part surface by s. The intersection of the offset plane and the cutter produces an area, which is depicted by the shaded region in the figure. Let us call it the undercut region. The meaning of the region is that, under this region, the height of remained material on the part surface is less than or equal to S. Therefore it is easily conceivable that the track of the 2D undercut region along a path must touch the one in the next path to make the maximum height of uncut material equal to s. The idea is shown in Figure 6b. As can be seen in the figure, the path interval can actually be obtained by computing the width of the 2D undercut region in the direction of b. Let nxvbe the projected vector of the plane normal vector n = (x,, yi, z,) onto the horizontal plane. The width of the undercut region in the direction of n,. is given bq 1= min(s/sin (Y, 2R), where CY= cos ~ ‘(z,) is the slope angle of the plane surface. When the horizontal vector h is given by h = (n X n,)lln X n,, 1,and the angle between a and h is /3 = cos-‘(a.h), the path interval is computed by ifI=2Rorif/3?y 2R,
expressed by exact formulae. Because of the toroidal cutter edge, the boundary of the 2D undercut region is not an analytic curve. Remember that, in the flat-cutter case, the boundary curve consists of a line and a circular arc that make it possible to induce the closed-form solution. To compute the width of the 2D boundary curve for the filletedendmill case, piecewise linear approximation is applied. Let R be the cutter-body radius, r be the cutter-edge radius, and s be the scallop height. The vectors a, b, n, nAY,and h are defined as same as in the flat-endmill case. To sample points on the curve, two extreme points in the direction of n, are first computed as follows. When a local coordinate system is defined at the cutter-bottom centre in which the x-axis is parallel to the vector h as shown in Figure 7. the 2D frontend point v of the curve is given by v=&,
2m sin /3,
(0, R - r + u cos a + (r - s)sin 01.
Filleted-endmill
ifs
and (r - .s)Iu > tan CY
1
(18)
where LL = (2rs - s~)“~ and Q!= cos- ’ (z,), the slope angle of the part plane. The equation of the offset plane (the dotted line on the yz-plane in Figure 7) is ;: = (tan CX)~+ k
(19)
where k = r( 1 - cos a) - (R - r + r sin a)tan CY+ s/cos 01. Then the rear-end point w is computed by w = (x,, .v,) ’ (0, R - r - u cos CY+ (r - s)sin CY),if k 5 -n ( k ((R - r)’ -(k/tan
if /3<-yandl<2R
(0, r - R - w cos
/2)“2 and y = sin ~ ‘(m/R).
+r(l
case
The above method can also be applied to the filleted endmill. In this case. however, the path interval cannot be
CX)‘)“*, -k/tan
a), if
-III
<
(17, where m = (2Rl-
or if s < r and (r - s)/u 5 tan (Y
=
if /3 2 y and 1~ R
1 R+msinp-(R--l)cosp,
if s 2 r
(0, R),
and R < I < 2R 6=
YV)
, (0,
--R),
01 +
(r - Lj)sin a),
if n 5 k < n
+cosa)
if k?n+r(l+cosa) (20)
where
II = (R - r)tan 01.
L’= s - 2(R - r)sin o(,
and 645
Three-axis machining of compound surfaces: J. S. Hwang and T.-C. Chang w = (2~ - v2)“‘. Note that if the offset plane intersects with the cutter-bottom circle (in other words, if the boundary of the undercut region includes a line segment on the cutterbottom plane). two points at the end of the intersection line are computed as the rear end of the curve. Once two extreme points are obtained, the middle points in between the two points are sampled by the following method. Let us assume that p = (xP, yP, z,,) is a sampling point on the curve, where _v,, holding the condition ?‘$+s_vp ‘?‘>. is already known. The z-value !,, of the point can be computed by eqn (19), and the radius of the circle on the toroidal edge that passes through the point is given by
if zP 2 r
R, P=
if :,,
1 R-r+(2~,,-~~)“‘, The x-component is then .$ = _c (p’ - ,$)“I Figure
7
Path-interval
computation
for the filleted endmill
(22)
Therefore if a set of y-values are first chosen, then the Xvalues of the sampling points can be computed by the above process. Finally, the path interval is obtained by calculating the width of the point set in the direction of the section-plane normal vector b. The accuracy of the solution depends on
(b)
Figure 8
646
Implementation
test; (a) compound
surface, (b) triangular
(21)
polyhedron,
(c) cutter paths, (d) cutting simulation
Three-axis
machining
the number of sampling points so that further study is necessary to determine the number properly. In the implementation test, 12 points were sampled to approximate the curve.
of compound
6.
7. 8.
IMPLEMENTATION
TEST
The method presented in the paper was implemented on an IBM RS/6000-3BT computer using the C language. The compound surface model, shown in Figure #a, consists of three trimmed surfaces and the size is 50 X 40 mm’. The triangular polyhedron was generated from the surface with the sampling tolerance of 0.02 mm, producing 8097 facets. Figure 8b shows the polyhedron. The cutter paths were then produced from the polyhedron, as shown in Figure 8~. A filleted endmill was used. Its body radius was 3 mm and its corner radius was 1 mm. The scallop height was 0.05 mm. It took 0.7 s for triangulation and 17.9 s for cutter-path generation. The number of CL points was 4240. Figure 8~1 shows the snapshot of the cutting simulation of the generated cutter paths.
9.
IO. I I. 12. 13. II. IS.
16.
CONCLUSIONS In this paper, a cutter-path generation method is presented to use flat and filleted endmills in sculptured surface machining. The CL point is computed in an efficient way in that only a portion of the facets in a CC region are involved in the computation, using the rejection criterion. The paper also provides solutions to compute the path interval for both the flat and filleted cutters, in terms of the scallop height. Combined with the technique for ball-end cutters in Ref. ‘, the proposed methodology offers a robust and versatile tool for machining compound sculptured surfaces.
17.
surfaces: J. S. Hwang and T.-C. Chang
Hwang, J. S., Interference-free tool-path generation m the NC machining of parametric compound surfaces. Cornput.-Aidrd Des., 1992. 24( 12), 667-676. Lal. .I. Y. and Wang, D. J.. A strategy for finish cutting path generation of compound surfaces. Cnmput. Indust.. 1994. 25(2). 1X9-209. Kondo, T.. Kishinami, T. and Saito. K.. Machining system based on scanning data (in Japanese). J. .Jr~ptrn SW. Precision Eng.. 1988, 54(h). lO70- 1075. Choi. B. K.. Chung. Y. C., Park, J. W. and Kim, D. H., Unified CAMsystem architecture for die and mould manufacturing. Cornput.-AidcJd Dry., 1994, 260). 235-243. Tang, K.. Cheng, C. C. and Dayan, Y.. Offsetting surface boundaries and 3-axis gouge-free surface machimng. Compltt:Aided De.v.. 1995. 27( 12). 915m~927. Filip. D.. Magedson, R. and Markot. R.. Surface algorithms using hounds on derivatives. Con~prt.Aided G~om Des.. 1986. 3(4). 295-3 I I. Drysdalc, R. L.. Jerard. R. B., Schaudt. B. and Hauck, K., Discrete Gmulation of NC machining. Algorithmiccl. 1989, 4(l), X-60. Piegl. L. A. and Richard, A. M., Tessellating trimmed NURBS surlaces. Conrput..Aitfetf Des.. 1995. 27( I ). 6-76. Hamann, B. and Tsai, P.-Y.. A tessellation algorithm for the representatlon of trimmed NURBS surfaces with arbitrary trimming curves. (‘omprrt. -Aided Des., 1996, 28(6/7), 46 I-472. Klaus. R. and Schramm, P., Numerically-controlled milling of CAD surface data. In Hagen, H. and Rollers, D. (Eds.). Geometric Modelirr,y: Methods N& Applications. Springer-Verlag. Berlin. 199 I. pp. 213-275. Li. S. X. and Jerard. R. B.. S-axis machining of sculptured surfaces with a Rat-end cutter. Cornput.-Aided Des.. 1994. 26(3). 165- 178. Choi, B. K.. Park. J. W. and Jun. C. S., Cutter-location data optimization in 5.aui\ surface machining. Comtw-Aided De.\. 1993, 25(6), 377-386.
Ji Sean Hwmg i.s (I senior research rrqinrer at the Stamping Tool MunufucLlcturing Department of Hyundai Motor Compnn~. South Korea He received an MS from Korea Adwnced lnstitutr oj’Sciencr trnd Technology in 1987 and a Ph.D. ,j+om Purdue University in 1997, both in Industrial Engineering. His wwurch interests include CAD/CAM. .xxlplured wrj&e muchining. geometric modeling, cmd productiorl phoning arxl control.
REFERENCES I.
Vickers, G. W. and Quan, K. W., Ball-mills versus end-mills for curved surface machining. Trcrns. ASME, J. En,?. Indusf.. 1989, 111(I ). 22-x. 2. Bell. C.. Landi. B. and Sabin. M., The programming and use of numerical control to machine sculptured surfaces. Proc. 14th Machine Tool Design and Research Conf., 1973, pp. 233-238. 3. Duncan, .I. P. and Mair, S. G., Slulptured Surfaces in Engineering and Medicine. Cambridge University Press, Cambridge, MA, 1983. 4. Bobrow. J. E., NC machine tool path generation from CSG part representations. Cornput..Aided Des.. 1985, 17(2), 69-76. 5. Choi. B. K. and Jun, C. S., Ball-end cutter interference avoidance in NC machining of sculptured surfaces. Cornput:Aided Des., 1989, 21(6), 37 I-378.
Dr Tien-Chirn Chcrrr~is mprojessor in the School of’ Industrinl Engineering clt Putdue lJnil,er.sity His research interest is in thr urw of marmfacturing automtrtion. Hiv current research act&ties include cuuorncued proce.ts planning. .wdptured surjizce machining, and virrunl manufacturing. Dr Chang is the author and u-author of several hooks cmd articles on automated prw’ss pltrnr@ and computer-aided mnnlrfuc.trrrirly.
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