Locally optimal cutting positions for 5-axis sculptured surface machining

COMPUTER-AIDED DESIGN Computer-Aided Design 35 (2003) 69±81

www.elsevier.com/locate/cad

Locally optimal cutting positions for 5-axis sculptured surface machining Joung-Hahn Yoon a,*, Helmut Pottmann a, Yuan-Shin Lee b a

Institute of Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10/113, A-1040 Wien, Austria b Department of Industrial Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA Received 29 December 2000; revised 20 September 2001; accepted 24 September 2001

Abstract The paper presents a local condition for collision-free 5-axis milling of sculptured surfaces. We consider cutter positions, which guarantee local gouging avoidance. This can replace concepts such as `cutting pro®le' or `effective cutting shape' which are of approximate nature. Based on second order approximations of the machined strip width, we also present locally optimal cutting positions for cutting directions. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Local millability; Dupin indicatrix; Machined strip width

1. Introduction Sculptured (or free-form) surfaces are commonly used in industry. Corresponding design modules are included in all important CAD systems. There is an extensive literature on sculptured surfaces. Thus, these surfaces are very well understood and ef®cient algorithms are available for their design [6]. This picture somewhat changes when we turn to manufacturing. Manufacturing of parts (e.g. dies and molds) which exhibit sculptured surfaces is often based on numerically controlled (NC) milling. Current NC machining software, particularly 5-axis machining software, is incorporating a rather undesirable high amount of user interactivity or does not fully exploit the ¯exibility of 5-axis machining. This is related to problems such as motion speci®cation and collision avoidance, whose algorithmic treatment is quite complex. Currently, NC machined parts are lacking a high quality surface ®nish. Therefore, NC machining is followed by a manual grinding and polishing procedure. This causes increased costs and, in particular, the CAD models are no longer valid within the desired accuracy. Thus, improvement of current NC software is highly desirable from the point of view of industry. Partially, the problems with 5-axis machining are based * Corresponding author. Tel.: 143-1-58801-11311; fax: 143-1-5880111399. E-mail addresses: [email protected] (J.-H. Yoon), [email protected] (H. Pottmann), [email protected] (Y.-S. Lee).

on a lack of fundamental research on the mathematical and computational side of the problem area. Even for 3-axis machining, there is a need for theoretical research, as recently been pointed out by Choi et al. [2], but for 5-axis machining it is much more severe. A survey of mathematical fundamentals on NC machining of sculptured surfaces is given in books by Marciniak [13] and Radzevich [16]. They described basic concepts for tool positioning, locally gouge-free machining and NC simulation and veri®cation. For a recent survey on the latter topic, see Ref. [1]. A continuation of the local geometric approach to methods for tool selection, motion planning and local interference checking in the case of 5-axis machining has been performed by Lee [8±10,12], by Jensen and Anderson [7] (`curvature matched machining') and by Choi et al. [3]. There, basic differential geometry and analytic geometry serve for the development of algorithms for NC machining. It is important to note that these papers use some rough approximations, such as the `effective cutting shape' in order to determine a locally optimal cutter position. This may lead to unwanted collisions and has to be improved for machining high quality surfaces. In investigating collision-free machining of sculptured surfaces, one has to distinguish between local and global gouging effects. Local gouging means that in all arbitrary small neighborhoods of the current touching point the milling tool cuts into the ®nal surface of the workpiece, thereby destroying it. It is well known that this situation can be completely characterized by the curvatures of the two surfaces involved, thus giving an easily evaluated criterion of local millability [13].

0010-4485/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0010-448 5(01)00176-2

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More complicated are global collisions occurring in some distance from the touching point. For 3-axis machining it has been shown that if some conditions on the shape of the workpiece hold (which are easier to check for than global collisions), the absence of local gouging implies the complete absence of collisions at all [5,15]. We may expect theorems of this type for 5-axis machining. It has been shown that if we lead a cutter such that all axis positions pass through a ®xed point and if all points of the workpiece surface can be seen from this ®xed point (star-shaped workpiece), then local millability implies global millability [14]. But this theorem cannot guarantee fully to exploit the ¯exibility of 5-axis machining. Recently, Wallner and Pottmann presented a global millability theorem for general workpieces [18]. They investigated several possible con®guration manifolds of tool positions relative to a workpiece under different aspects; the degree of freedom of the motion of the tool, the correspondence between the contact point and the tool position, and the presence or absence of unwanted collisions between tool and workpiece. In the present paper, we study locally optimal cutting positions. A new method is presented to ®nd the locally millable cutting positions by using the Dupin indicatrices of cutter surface and designed surface at the contact point. Moreover, we compute a rating of all possible relative cutter positions at a ®xed contact point with respect to the obtained local surface quality in some given cutting direction. The machined surface quality is measured by the machined strip width, for which we present a second order approximation. This result is an extension of recent work by Rao and Sarma [17]. But unlike Rao and Sarma, we do not need to compute a parameterization of the swept surface of the moving cutter in order to derive its second order behavior at the contact point of the cutter. This can be done in a simpler geometric way using concepts of classical constructive differential geometry. The new methods presented in this paper can overcome the weakness of the current methods such as `cutting pro®les' and `effective cutting shape' methods and fully exploit the possibility of ®nding the locally optimal cutting positions for sculptured surface machining. Illustrative examples are also provided in this paper. 2. Local millability of smooth surfaces We describe the local millability test when restricted to smooth surfaces, which are C 2. Other surfaces will be considered later. Let X be a designed surface. X is the boundary of a solid, and we speak of the solid as of the interior of X and will call the ambient space the exterior of X. It is our goal that the object in the ambient space of X is cut by the cutter without undercutting to the solid of X. Let S be a cutter surface. The cutting tool is, geometrically, a convex body of rotational symmetry. The rotation of the cutter around its axis can be completely neglected from the geometric

point of view. The actual cutter consists of a shaft and an active part. The latter, while rotating, has a surface of revolution as its envelope. It is this surface which we consider as cutter surface. The convex cutter means that the line segment which joins any two point of S is contained in S . In most applications, this assumption is satis®ed. We further assume that S is strictly convex, which means that the line segments are contained in the interior except their two end points. So it does not contain parts of planes, cylinders or cones. This condition is not often ful®lled. However, there is always a sequence of strictly convex cutters, which converges to the actual cutter. Collision checking involves, in principle, only the measurement of distances, and not of derivatives. Thus, a suf®ciently close approximation will fail the test for millability if the original cutter has failed, and vice versa. While milling the surface X, the cutter surface S undergoes a motion such that the enveloping surface is just the given surface X. Hence, at any time instant, the current position of the cutter surface must be tangent to X at a point, the so-called cutter contact point. Although in practice both cutter and workpiece are moving, it is suf®cient for our considerations to view the relative motion and thus consider X as ®xed. In 3-axis machining, the cutter axis direction is ®xed in the system of the workpiece X. In 5-axis machining, however, the cutter axis can be located in any direction (within the cutter's workspace). For the mathematical representation we will use a local Cartesian …x; y; z† coordinate system, whose origin is at the cutter contact point and whose z-axis agrees with the common surface normal of cutter and design surface at the contact point p. Let S a(p) be the cutter position whose normalized axis vector is a and where the cutter contact point is p. Since a is a unit vector, it can be represented by two angles (f , u ). Here, f is the angle between a and the z-axis and u is the angle between the x-axis and the plane which is spanned by the cutter axis and the z-axis (see Fig. 1). De®nition 1. The surface X is said to be locally millable by the cutter S a(p) at a point p, if (i) there is a neighborhood U of p such that the solid bounded by X and S a(p) have no point in common in this neighborhood except p; and (ii) the two surfaces S a(p) and X do not have second order contact at p. Let us formulate this de®nition in the local coordinate system. This will also lead us to the motivation for the second part of the de®nition. Both surfaces X and S can be represented as graphs z ˆ f …x; y† and z ˆ s…x; y† of realvalued functions f and s, respectively. Since surfaces are tangent to the xy-plane at the origin, the function values and the ®rst order partial derivatives vanish at (0, 0) for f and s. If additionally the second order directional derivatives in

J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81

71

as svv ˆ vT Hs v;

…5†

and it is also well-known that Eq. (1) is equivalent to det…Hs 2 Hf † . 0

Fig. 1. Local coordinates at a contact point.

a tangent direction v satisfy s vv . fvv

for all v;

…1†

there is a neighborhood U of p such that s…x; y† $ f …x; y†

for all …x; y† [ U;

…2†

and the equality holds where (x, y) is at the origin 0. This means that above U, S a(p) does not interfere with X. If there is a direction vector v such that s vv , fvv ;

…3†

then S intersects the interior of the solid bounded by X in an arbitrarily small neighborhood. If s vv ˆ fvv we cannot tell the behavior from the second order in®nitesimal behavior. The case that for all v, svv $ fvv but not for all v svv . fvv almost never occurs, if we do not have line contact between the surface X and the cutting tool. This case splits into two subcases: First, we may have svv ˆ fvv for all v. Then, we have second order contact of design surface and cutter at the contact point. Here, a third order analysis would be necessary to decide whether there occurs local interference or not. For this reason, it is practical that the de®nition of local millability excludes second order contact at p. In the second subcase, we have svv ˆ fvv for only two vectors v, where v ˆ v1 ; 2v1 : Again, in this case a second order analysis is not suf®cient to decide on the local millability. It is exactly the situation where local millability becomes critical. It is a boundary case between millability and non-millability. Excluding this boundary case, we can test the local millability with only second order Taylor approximants. It is well known that svv (and analogously, fvv) can be expressed in terms of the Hessian ! sxx sxy ; …4† Hs ˆ sxy syy

and

sxx . fxx :

…6†

The surface X is locally millable by the cutter S a(p) at a point p, if Eq. (6) holds there. In a point where several C 2 surface patches meet, Eq. (6) has to be satis®ed for each of them. Glaeser, Wallner and Pottmann suggested an equivalent condition in terms of the Euclidean curvature indicatrices of X and S [5,15]. The indicatrices have the advantage that they are independent of the choice of a base plane. Let p [ X and let n be the surface normal of X in p. Fix any Cartesian coordinate system in the tangent plane t p whose origin is the point p. A surface tangent t(u ) is determined by its angle u with the x-axis. The plane spanned by n and a surface tangent t(u ) intersects x in a curve c(u ) which has a curvature radius r (u ) at p. This radius is given a negative sign if the curve is locally beneath t p and a positive sign if it is locally above t p. De®nition 2. The diagram which in polar coordinates has the equation p  …r; u† ˆ r…u†; u …7† whenever the square root is de®ned, is called the (signed) Euclidean indicatrix of curvature (Dupin indicatrix) ip of the surface X at the point p. We de®ne the interior of the indicatrix ip as the star-shaped (with respect to the origin) domain, whose boundary is ip. It may be the whole plane. The connection to our local coordinate system is as follows. The surface X has a second order Taylor approximant at p, zˆ

1 2

… f xx x2 1 2fxy xy 1 fyy y2 †:

…8†

It describes an elliptic or hyperbolic paraboloid if p is an elliptic or hyperbolic surface point (here, we have det Hf ± 0). For det Hf ˆ 0; we get either a parabolic cylinder or the plane z ˆ 0 ( fxx ˆ fxy ˆ fyy ˆ 0; there, p is a parabolic surface point or ¯at point, respectively). The signed Dupin indicatrix ip of X at p is obtained by intersecting the second order approximant with the plane z ˆ 1=2 and projecting it into the tangent plane z ˆ 0 (see Appendix). This yields as equation of the signed Dupin indicatrix ip : fxx x2 1 2fxy xy 1 fyy y2 ˆ 1:

…9†

The derivation of this result is the following: Since all ®rst order directional derivatives of f vanish, the second order directional derivative fvv is equal to the corresponding signed normal curvature for any tangent direction v. It is easy to see that the signed Dupin indicatrix ip is either

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Fig. 2. Local millability condition: cutter indicatrix iq contained in the interior of the design surface indicatrix ip. Fig. 3. Local gouging not shown by the cutting pro®les (left) and the effective cutter shape (right).

an ellipse, a hyperbola, a pair of parallel lines or void. The indicatrix iq of the cutter S is always an ellipse, since the cutter is positioned above t p and strictly convex. In detail, see Appendix. With the explanations and de®nitions given above, a noncritical situation of local millability can be characterized as follows. Proposition 1. A surface X is locally millable by the cutter S a …p† at a point p, if at the common contact point p the indicatrix iq of S is contained in the interior of the indicatrix ip of X (see Fig. 2). Note that local millability is critical if the Dupin indicatrix iq is inside ip, but touches ip in two diametral points. These two points lie on the tangent, along which the normal curvatures of X and S agree. Remark 1. Local millability is completely independent of a possible instantaneous cutting direction. However, the literature discussing local gouging uses the cutting direction. Two concepts are used, both being just rough approximations. One is the cutting pro®le, which is the intersection of the cutter with the cutting direction's perpendicular plane through the contact point. In the most recent article on the mathematical treatment of the local cutting situation [9], cutting pro®les are taken in the cutting direction and orthogonal to it in order to avoid `rear gouging and side gouging'. It is taken care that the curvature radii of these pro®les do not exceed the corresponding curvature radii of the designed surface. As is seen from Fig. 3 (left), this is not suf®cient to guarantee local millability. This agrees also with observations at test objects by Jensen (private communication). Another concept present in the literature (e.g. [11,12]) is that of the effective cutting shape. This is the silhouette of the cutter for orthogonal projection in the cutting direction. The corresponding curvature radius at the contact point (`effective cutting radius') is then compared with the curvature radius of the design surface in the direction normal to the cutting direction. The effective cutting radius is easily deter-

mined in a geometric way, since the Dupin indicatrix of the projection cylinder is a pair of parallel tangents of iS . Again, as shown from Fig. 3 (right), the effective cutting shape is not the appropriate tool to avoid local gouging. The problems of the cutting pro®le and the effective cutting shape are caused by testing only in one or two normal sections. It must be tested in all normal sections, in fact, three would be suf®cient (see Remark 2). As the above statement shows, we can test local millability with only second order Taylor approximants. Another interpretation of local millability is based on the difference function z ˆ s…x; y† 2 f …x; y†: It is the graph of the deviation between the two surfaces measured in direction of the contact normal. We have non-critical local millability if this graph is an elliptic paraboloidÐ or in other wordsÐif the Dupin indicatrix of the graph of s±f at the contact point is an ellipse. We will now exploit this in more detail. We choose an even more adapted coordinate system by requiring that the x-axis is de®ned by the maximum principal direction of the designed surface and the y-axis is de®ned via its minimum principal direction. Let Kmax and Kmin be the maximum and minimum of principal curvatures of the designed surface X at p, respectively. Thus, the designed surface is X : f …x; y† ˆ

1 2

…K max x2 1 Kmin y2 † 1 …p†;

…10†

where ( p ) denote terms of at least third order which are irrelevant now. Let kmax and kmin be the maximum and minimum of principal curvatures of the cutter surface S at p, respectively. Moreover, u shall be the angle between the maximum principal directions of the surfaces where 2p=2 # u # p=2 (see Fig. 4). Since the cutter is represented by

S : s…x; y† ˆ

1 2

…k max x2 1 kmin y2 † 1 …p†;

…11†

J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81

73

Theorem 1. Let Kmax and Kmin be the maximum and minimum principal curvatures of the designed surface X at p, respectively. And let kmax and kmin be the maximum and minimum principal curvatures of the cutter surface S at p, respectively. Denote by u the angle between the maximum principal directions of the two surfaces. Then, the surface X is locally millable by the cutter S at point p, if k max 1 kmin 2 …Kmax 1 Kmin † . 0

and

2…kmax 2 Kmax †…kmin 2 Kmin † 1sin2 u…kmax 2 kmin †…Kmax 2 Kmin † , 0;

Fig. 4. Cutter rotated by u .

when u ˆ 0; it is represented by

S u : su …x; y† ˆ

1 2 kmax …cos

1 ˆ

1 2

hold.

ux 1 sin uy†

1 2 kmin …2sin

2 2

ux 1 cos uy† 1 …p†

…kmax cos2 u 1 kmin sin2 u†x2

1…kmax 2 kmin †sin u cos uxy 1 12 …kmax sin2 u 1 kmin cos2 u†y2 1 …p†: …12† Then the indicatrix of su ±f is given by: …kmax cos2 u 1 kmin sin2 u 2 Kmax †x2 12…kmax 2 kmin †sin u cos uxy …13†

For local millability, we have to check that Eq. (13) is an ellipse. Since the center of Eq. (13) is the origin, there is an easy method checking the local millability. We de®ne a, b and c as follows: a ˆ kmax cos2 u 1 kmin sin2 u 2 Kmax ; and

…14†

c ˆ kmax sin2 u 1 kmin cos2 u 2 Kmin : Then local millability is characterized by: b2 2 ac , 0

and

c . 0:

Remark 3. To avoid possible misunderstanding we would like to point out the following. Local millability means that there is a neighborhood of the contact point p such that the solid bounded by a cutter and a surface do not have a point in common in this neighborhood except p. Then it guarantees the existence of such a neighborhood. But we cannot say anything about the size of such a neighborhood only with local geometry. So the local millability theorem, Theorem 1, cannot be used to detect rear gouging or side gouging. These types of gouging are considered as global gouging. 3. Locally optimal cutting positions

1…kmax sin2 u 1 kmin cos2 u 2 Kmin †y2 ˆ 1:

b ˆ …kmax 2 kmin †sin u cos u

(16)

…15†

This is the same condition as b2 2 ac , 0 and a . 0: For the sake of convenience, the condition b2 2 ac , 0 and a 1 c . 0 is the same as Eq. (15). Remark 2. By the considerations above, local millability holds if the indicatrix of su ±f is an ellipse. The general form of the indicatrix is ax2 1 2bxy 1 by2 ˆ 1; since it is a quadratic curve and its center is the origin. If three coef®cients a, b and c are determined, we can check the local millability. That is, the local millability can be tested in only three normal sections. Then three coef®cients are determined.

In 5-axis machining, there are many cutting positions which are locally millable at a point. If we have a rating of all possible relative cutter positions at a ®xed contact point with respect to the obtained local surface quality in some given cutting direction, among these positions we later choose the one with the best ®t and which does not cause interference. Now we rate the cutter positions with respect to the `machined strip width'. Let Xh be the offset surface to the surface X at a distance h . 0: That is, Xh is the surface whose points are represented by x 1 hn…x† where x [ X and n(x) is the outer unit normal vector at x. Let the part surface X be machined within the tolerance h, that is, the material inside the surface X may not be removed and the material outside the surface Xh must be removed. If the cutter S mills X at p, then the exact machined region is bounded by the curve of points in X such that points Xh which are equidistant to them de®ne an intersection curve between the cutter surface S and the offset surface Xh. (cf. Fig. 5). The de®nition of an exact machined region, though strict, is inconvenient. In order to obtain its shape, we have to calculate an intersection curve between the tool surface S and the offset surface Xh. The problem may be substantially simpli®ed if one uses a ®xed direction, de®ned by the unit normal vector n(p), to measure the distance between the surface. The boundary of the

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J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81

Fig. 7. Toroidal-end mill. Fig. 5. Comparison of machined region's de®nitions; the machined region r and the exact machined region re.

approximate machined region consists of those points in X for which the points in the approximate offset X 0h ˆ x 1 hn…p† are on the tool surface S . By de®nition, a machined region is an orthogonal projection of the approximate region into the xy-plane. The difference between the exact machined region and the machined region is negligible in nearly all practical situations. The approximate offset X 0h is located inside the region between X and Xh. Then we can say that the approximate machined region to any tolerance h lies inside the exact machined region to h. So we are on the safe side working with the approximate version. (cf. [13]). Consider a curve of cutter contact points. At consecutive contact points there are consecutive machined regions. A machined strip is de®ned as an envelope of machined regions [13]. This is not quite precise, since each machined region lies in the tangent plane of the corresponding contact point and we would ®rst have to project back to the surface X before we can form this envelope. In the de®nition of Marciniak [13], a machined strip width measured in the (tangential) cutting direction u is the orthogonal projection of the machined region on the straight line orthogonal to u (see Fig. 6). We will at ®rst work with this de®nition and

later in Section 4 present a more precise version via the surface swept by the moving cutter. Suppose the cutter position is the same as in Section 2. Let I…x; y† be the left hand side of Eq. (13). The indicatrix of s±f is represented by I…x; y† ˆ 1: Then the machined region of this situation is represented by: I…x; y† ˆ 2h;

…17†

in the xy-plane. If the conditions Eq. (15) are satis®ed and the surfaces are represented by the second order Taylor approximants, the machined region is an ellipse. We see that it is just a scaled version of the Dupin indicatrix of the difference function s±f. The cutting tools commonly used in 5-axis milling are end mill cutters. There are ball-end mill, toroidal-end mill and ¯at-end mill. Fig. 7 shows a toroidal-end mill which is commonly used for 5-axis sculptured surface machining. A ball-end mill and a ¯at-end mill are the extreme cases of a toroidal-end mill. So we consider only a toroidal-end mill. When the cutter axis is inclined by the angle f where 0 # f # p=2 (cf. Fig. 1), the maximum kmax and minimum kmin of principal curvatures are as following (cf. [4,13] and Appendix): kmax ˆ

1 a

and

kmin ˆ

sin f : b 1 a sin f

…18†

The maximum principal direction of the cutter is the direction of the meridian (intersection with a plane through the rotational axis). First, we may assume that the maximum principal curvature of the cutter is greater than the principal curvatures of the designed surface, since otherwise it cannot be locally millable. The following inequalities are equivalent to Eq. (16) in the case of the toroidal-end mill, these are obtained by substituting Eq. (18) in Eq. (16)

a{2 2 a…Kmax 1 Kmin †}sin f 1 b{1 2 a…Kmax 1 Kmin †} . 0;

Fig. 6. Machined strip width.

b…Kmax 2 Kmin † 2 sin u 2 …1 2 aKmin †sin f 1 bKmin , 0: 1 2 aKmax …19†

J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81

Put j ˆ sin u and h ˆ sin f: Then we have local millability if the cutter location is in the intersection of the following regions,

h.

b…Kmax 2 Kmin † bKmin h. j2 1 ; …1 2 aKmax †…1 2 aKmin † 1 2 aKmin

…20†

where 21 # j # 1 and 0 # h # 1: Since k max . Kmax by Theorem 1, the minimum of the second formula's right hand side is greater than the ®rst formula's right side. So we can check only the second inequality for local millability. Proposition 2. A surface X is locally millable by the cutter S f ,u , if k max . Kmax and

h.

b…Kmax 2 Kmin † bKmin j2 1 ; …1 2 aKmax †…1 2 aKmin † 1 2 aKmin

…21†

holds where j ˆ sin u and h ˆ sin f:

b…K max 2 Kmin † bKmin j2 1 : …1 2 aKmax †…1 2 aKmin † 1 2 aKmin

…22†

Since the coef®cient of j 2 is always positive, all cutter positions are locally millable when K max , 0 and no cutter position is locally millable when Kmin . max{kmin } ˆ 1=…a 1 b†: Suppose that X is locally millable. The machined region is an ellipse. The lengths of its major and minor axes are s p h…a 1 c 1 4b2 1 …a 2 c†2 † Al ˆ 2 b2 1 ac

and …23†

s p h…a 1 c 2 4b2 1 …a 2 c†2 † As ˆ ; 2 b2 1 ac

where h is a tolerance and a, b, c are de®ned in Eq. (14). The angles between x-axis and the major and minor axes are arctan

p 2…a 2 c† 2 …a 2 c†2 1 4b2 2b

p 2…a 2 c† 1 …a 2 c†2 1 4b2 ; arctan 2b

b…Kmax 2Kmin †j 2 2ah 4a 2 h2 …12aKmax †…12aKmin †2 2 {22a…Kmax 1Kmin †}1 D D4

2bh 4abh2 {12a…Kmax 1Kmin †}2 2 D D4 1 : 2ah 4a 2 h2 …12aKmax †…12aKmin †2 2 {22a…Kmax 1Kmin †}1 D4 D

bKmin …12aKmax †1

…25†

In the auxiliary (j , h )-plane these positions are represented by a parabola. For a critical situation, the machined region is a pair of parallel lines, since we have agreement of f and s up to second order in some direction. Hence, we then have D ˆ 1: In fact, Eq. (25) coincides with Eq. (22) for D ˆ 1: For a ®xed A l ˆ D; the cutter positions are represented by a parabola. The vertices of these parabolas are on the h -axis and the value of the h -coordinate of the vertex decreases as D ! 1: The coef®cient of j 2 (curvature at the parabola's vertex) also decreases as D ! 1 and is positive in the locally millable region. In fact, there exist pairs of D and h such that the coef®cient of j 2 is negative. In this case 1 2 aK max 2 1 2 aKmin 2 D ,h, D : 2a 2a

The critical situation for local millability occurs for

hˆ

D. Substituting Eq. (14) and D to a, b, c and Al, respectively, we get the following: hˆ

2b{1 2 a…K max 1 Kmin †} ; a{2 2 a…Kmax 1 Kmin †}

and …24†

respectively. In a locally millable region, let us study all cutter positions for which Al equals some given constant

75

…26†

Since h must be small and D as big as possible, however, the above inequalities do not make sense in practical use. In the cases of a meaningful value h, the contour parabolas with respect to ®xed values of D are distributed as shown in Fig. 8. Let C be an angle between the cutting direction and the angle of the smallest axis of the machined region where 2p=2 # C # p=2: Then the machined strip width is q …27† 2 A l2 cos2 C 1 A s2 sin2 C: See Fig. 9. Let W be the half of the machined strip width. Rearranging W by a, b, c of Eq. (27), for a given cutting direction C , W is represented by:  s p 2 2 2 h{a 1 c 1 4b 1 …a 2 c† …1 2 2 sin C†} Wˆ : …28† 2 b 2 1 ac Now we may put j ˆ sin u; h ˆ sin f and z ˆ sin C: For a ®xed W, Eq. (28) is an algebraic surface in the auxiliary (j , h , z )-space. The surface may not be included fully in the locally millable region. For a ®xed cutting direction C , the graph of Eq. (28) with a ®xed W is similar as that of Eq. (25). However, this graph is not a parabola. Examples of the algebraic surfaces are in Figs. 10 and 11. These two examples are of the radii of cutter as followings; a ˆ 0:1 and b ˆ 0:9: The left surfaces of these ®gures are the boundaries of locally millable regions. The right three surfaces are isosurfaces to W ˆ 0:5; 1 and 2 with the allowance tolerance h ˆ 0:1: Fig. 10 shows the case of

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Fig. 8. Isolines of the lengths of the major axis of the machined region, D ˆ 0:5; 0:67; 0:83; 1; ¼; 1 for Kmax ˆ 1; Kmin ˆ 0:1; a ˆ 0:1; b ˆ 0:9 and h ˆ 0:1:

designed surface whose principal curvatures are 1 and 0.1. Fig. 11 shows for K max ˆ 0 and Kmin ˆ 20:5: With increasing values of W, the corresponding isosurfaces of W are getting closer to the boundary of the locally millable region. When the cutting direction is closer to the minor axis of the second order approximant of the machined region, the quality of machining is better. The machining quality could still be improved by using higher order approximants rather than second order approximants. The above C is dependent of the rotation angle u . It is necessary that Eq. (28) is changed to a cutting direction independent of a semi-axis of the machined region. Let Q be an angle between x-axis and a cutting direction which is presented by a vector in the xy-plane of local coordinates where 2p=2 # Q # p=2: Then p 2…a 2 c† 1 …a 2 c† 2 1 4b2 : …29† Q ˆ C 1 arctan 2b So we obtain the following equation: s 2h…a cos2 Q 1 2b sin Q cos Q 1 c sin 2 Q Wˆ : 2 b 2 1 ac

Fig. 9. Machined strip width for the cutting direction C .

After choosing the value of W ˆ W0 whose error with respect to the Taylor approximant is suf®ciently small and which is as big as possible, we can get the optimal cutting positions which are between the boundary of locally millable region and the isosurface of W0. If the cutting direction Q ˆ Q 0 was given, we can choose the cutting positions which are between the boundary of locally millable region and the isosurface of W on the plane z ˆ sin Q 0 : If the cutting position u ˆ u0 and f ˆ f0 was given, we can choose the cutting directions Q 0 # Q # Q 1 such that Q 0 and Q 1 are two solutions of W…u 0 ; f0 ; Q† ˆ W 0 : In the case of the ball-end mill, the principal curvatures of cutter are always constant 1/a , since b ˆ 0: The inclination

…30†

Fig. 10. Isosurfaces of the machined strip widths W ˆ 0:5; 1 and 2 for Kmax ˆ 1; Kmin ˆ 0:1; a ˆ 0:1; b ˆ 0:9 and h ˆ 0:1:

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77

Fig. 12. Machined strip width and indicatrices of design surface, cutter, swept surface.

Fig. 11. Isosurfaces of the machined strip widths W ˆ 0:5; 1 and 2 for Kmax ˆ 0; Kmin ˆ 20:5; a ˆ 0:1; b ˆ 0:9 and h ˆ 0:1:

angle and the rotation angle do not make sense. It is locally millable if the maximum principal curvature of the designed surface is less than 1/a . Now we consider the case of the ¯at-end mill. The ¯at-end mill is not smooth, i.e. its actual cutting part is not C 1 smooth but only C 0 continuous. Even if we consider the ¯at-end mill as the limit of a toroidal-end mill, its maximum principal curvature is in®nity. So, we cannot represent it via the Taylor approximant. As mentioned before, a suf®ciently close approximation will fail the test for millability if the original cutter has failed, and vice versa. We can consider the ¯at-end mill as a toroidal-end mill such that a is a suf®ciently small positive number and b is equal to the radius of the ¯at-end mill. As a tends to 0, its indicatrix can be considered as a line segment. Its direction is given by the minimum principal direction and its length is twice the reciprocal of the minimum principal curvature. Thus, we may interpret both a ball cutter and a ¯at-end mill as a special or limit case of a toroidal end mill. Remark 4. Recently, Rao and Sarma computed the conditions of local millability for a ¯at-end mill with a certain swept surface [17]. The swept surface is de®ned by the envelope of the cutter surface positions to a curve of cutter contact points on the design surface. For a ¯at-end mill, this does not require an envelope computation, since the cutting circles (bottom circles of cutter positions) form a surface. Rao and Sarma gave a parametric representation of this surface, from which they derive the curvature and the second order estimate for the machined strip width. We can do this for more general cutters without a parametric representation of the envelope of cutter positions to a given cutter contact curve. Our derivation is shown in

Fig. 12. The indicatrix iS of the swept surface S of the moving cutter has double tangencies with the indicatrices of design surface X and cutter surface S . Moreover, it has to pass through the point which is on the indicatrix of the design surface in the current tangential cutting direction u. This information is suf®cient to compute the indicatrix iS of the swept surface. An exact machined strip de®nition would require the intersection of the swept surface S with the offset of X at some distance h. Again we simplify by measuring distances in direction of the surface normal at the current cutter contact point and by considering the second order approximants only. Then the second order approximant of the simpli®ed machined strip is bounded by a pair of parallel lines in the tangent plane. These lines are parallel to the cutting direction and lie at distance 2W to the cutter contact point, where W is the machined strip width de®nition from Eq. (30). Thus, we have found a much more geometric motivation for the de®nition of W than the earlier one due to Marciniak [13]. 4. The case of non-smooth surfaces In practical applications, the design surface is often just C 1, but composed of patches of higher smoothness. Furthermore, the design surface may have edges and vertices. We will therefore extend our investigation to surfaces which are C 0 and composed of C 2 patches. Joining the patches, a non-C 2 part can occur and it shall consist of piecewise C 2 curves. First, suppose a contact point p is on the C 1 part where k patches meet. Thus, there are k different curvature behaviors at p. Since there exists a unique tangent plane at p, the local coordinates of p are well-de®ned. The concept of an indicatrix is that the curvature behaviors of normal sections are represented on the tangent plane. Since the non-C 2 part is decomposed into piecewise C 2 curves, the indicatrix of p is divided into k parts such that each boundary point belongs to a one-sided normal section. If the indicatrix of the cutter is included in the indicatrix of the design surface, its cutter position is in the locally millable region of the cutter. For a direction u in the tangent plane, the element of indicatrix of

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Fig. 14. Indicatrix iS and substitute iX in the case of an edge with tangent t. Fig. 13. Interior of indicatrix iX in the C 1 case.

p which is located in a C 1 part where k patches meet is a point of the indicatrix of a patch which generates surface in the direction u . Fig. 13 shows an example of the interior of indicatrix at C 1 part. Let us study edges of the designed surface, which shall be curves e, where two patches meet with different tangent planes. There are two cases, one is a ridge and the other is a ravine. In the case of a ravine, it is impossible to mill by a smooth cutter. So we consider only the case of a ridge. Let p be a point in a ridge e. A ridge has in each of its points p a wedge Tp of admissible tangent planes t p. For t p in Tp, we can test the local millability similar to the smooth case. In terms of curvatures of e and cutter this can be expressed as follows [4]: denote the osculating plane of e at p by E and the radius of curvature of e by R. The edge tangent at p will by denoted by t. For all admissible tangent planes t p, choose the local coordinate system in t p such that the y-axis coincides with t. Let c ˆ /…E; t p †: Draw the indicatrix of the cutter and the two points (in polar coordinates) p  P 2 ˆ 2P1 ; …31† P1 R=…sin c†; f ; as in Fig. 14. If P1 and P2 are outside of the indicatrix of the cutter, the cutter position is locally millable. Its computation is easy. For a ®xed tangent plane t p, the locally millable cutting positions are represented by the rotation and inclination angles with respect to the local coordinate system. And a tangent plane is represented by a normal vector in S 2. So we have locally millable cutting positions at a point in a ridge which are represented by four angles (coordinates) and which are all locally millable cutting positions of each tp [ Tp : The vertex analysis is similar to the case of a ridge. A vertex p also has a wedge Tp of admissible tangent planes t p. For each t p, we can consider the indicatrix of designed surface. The indicatrices are similar as those of ridge points. 5. Implementation and examples The proposed method has been implemented and tested

for several examples of part surfaces. Fig. 15 shows a surface with a monkey saddle point which is described by z ˆ 0:01{…x 2 5†3 2 3…x 2 5†…y 2 5†2 } in the global coordinate system. The center point of the monkey saddle is (5, 5, 0). We use a toroidal-end mill with radius R ˆ …a 1 b† ˆ 0:5 and corner radius a ˆ 0:125: The machining tolerance h is set to be 0.001. The cutter positions (u , f ) are ®xed by (908, 5.58). The rotation angle u ˆ 908 means that the maximum principal direction of surface and the minimum principal direction of cutter coincide. The tilt angle f ˆ 5:58 is chosen such that it guarantees local millability at every surface point. The gray colors on the surface describe the lengths of major axes of the second order approximated machined regions. Fig. 16 shows the distribution of the optimal tilt angle for ®xed rotation angle u ˆ 908: The optimal tilt angles are obtained with Proposition 2 by:  sin f ˆ max

 b…Kmax 2 Kmin † bKmin sin2 u 1 1 e; 0 ; …1 2 aKmax †…1 2 aKmin † 1 2 aKmin

…32† where e is a suf®ciently small positive number. The contours are isolines for the tilt angle. For given tool positions, we can obtain machined strip widths with respect to cutting directions. The maximum of machined strip width occur if a cutting direction coincides with the minor axis of the second order approximant of the machined region. The vector ®eld visualized the optimal cutting directions. In this case, the machined strip widths are between 1.7 and 1.8. When the cutter positions and the cutting directions are optimal, the distribution of machined strip width is comparatively even. Fig. 17 shows an example of a chair seat. A ¯at-end mill with radius r ˆ 0:5 is used for testing. For a ¯at-end mill it is easier to compute the optimal cutter position for a given cutting direction. The optimal cutter position for the ¯at-end mill is that the rotation angle u is the same as the angle indicating the cutting direction and the tilt angle f is chosen such that the indicatrix of cutter contacts to that of design surface. The top of Fig. 18 shows the distribution of machined strip widths for given cutting directions which are ®xed by the minimum of principal direction of surface. The vector ®eld illustrates given cutting direction on surface. The tool positions are obtained by the proposed

J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81

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1

0.8

1 0

0.6

10

5

1

0.5

0

2

0

8

3

0. 5 1

6

5

0.4

0

6

5

4

4

2

4

2

6 0

4

4

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7

2

2

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6

8

0

Fig. 15. Example part with a monkey saddle point and machined strip width for a toroidal-end mill and its tool position (908, 5.58).

Fig. 17. Example part of a chair seat.

method. The bottom of Fig. 18 shows the distribution of machined strip widths for given cutting directions which are ®xed globally from the front of chair to the back. If the normal curvature in the perpendicular direction of cutting direction is positive, the width is suf®ciently large. 6. Conclusions and future research We presented a geometric method of locally optimal cutting positions for 5-axis sculptured surface machining. The method includes ®nding locally millable cutter

5

5

4

3

4 4

3 4

2 3

1 5 1

2

4

3

2

4

3

4

4 4

5

5

3

Fig. 16. Optimal tilt angle of cutter for given cutting direction ˆ minimum principal direction of surface and given rotation angle ˆ 908.

Fig. 18. Distributions of the machined strip width for given cutting directions; the minimum principal directions of surface (top) and an globally ®xed directions (bottom).

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positions by considering local geometries ( ˆ principal curvatures and directions) of designed and cutter surfaces and updating locally optimal cutting positions by considering their machined strip widths. Based on second order approximations of machined strip width, we could ®nd locally optimal cutting positions for cutting directions if we checked only the local geometries of two surfaces and cutter dimensions ( ˆ radii of cutter). We plan to construct a method ®nding all globally millable cutting positions. So far as we considered only geometric aspects of machining, we also plan to investigate mechanical constraints for the relationship between cutting direction and orientation. Combining these to the result of this paper, we can choose the optimal cutting position and direction. It allows to perform the optimized tool path planning in geometric and mechanical sense. Acknowledgements This work has been supported by grant No. P13938-MAT of the Austrian Science Foundation and by the Korean Research Foundation Grant (KRF-99-D007).

kmax cos2 u 1 kmin sin2 u:

…A2†

A point p is called elliptic if kmax kmin . 0; hyperbolic if kmax kmin , 0; ¯at if kmax ˆ kmin ˆ 0; and parabolic in the remaining cases. For the curvature radius r (u ) in the u -direction, the diagram which in polar coordinates has the equation …r; u† ˆ

p  r… u† ; u ;

…A3†

whenever the square root is de®ned, is called the signed Dupin indicatrix ip of the surface X at the point p. That is, for every tangent direction u , it corresponds to the square root of normal curvature radius when it is non-negative. As seen in Section 2, the surface X is represented by Taylor expansion at p as following: zˆ

1 2

… fxx x2 1 2fxy xy 1 fyy y2 † 1 O…3†:

…A4†

The normal curvature of X at p in the u direction is

Appendix A. Dupin indicatrix Let a : I ! R2 be a unit speed curve, so ia 0 …s†i ˆ 1 for each s [ I: Then T…s† ˆ a 0 …s† is called the unit tangent vector at a (s). The unit normal vector N(s) is obtained from T(s) by rotating anticlockwise through p/2. Since T(s) has constant length 1, i.e. a (s) is a unit speed curve, its derivative T 0 …s† ˆ a 00 …s† measures the way the curve is turning. Since T 0 …s† is perpendicular to T(s), there is a real number k (s) such that T 0 …s† ˆ k…s†N…s†:

corresponding to principal curvatures. Then if k…U† ˆ cos uEmax 1 sin uEmin ; the normal curvature of X in the U direction is

…A1†

We call k (s) the curvature of a at s and the reciprocal of curvature. The circle whose center is a…s† 1 N…s†=k…s† and radius is 1=uk…s†u is called the osculating circle at s. We call r…s† ˆ 1=k…s† the radius of curvature of a at s. Let X: D ! R3 be a regular surface and p be a point in X. Then there are the tangent plane tp X and the normal vector N(p) at p. For every tangent vector T, it can be represented by the angle, say u , between a ®xed tangent vector and T. If P(u ) is the plane determined by u and the normal vector N(p), then P cuts from X near p a curve c(u ) called the normal section of X in the u direction. The curvature k (u ) of a planar curve c(u ) is called the normal curvature of X at p in the u direction. The maximum and the minimum values k max, k min of normal curvature of X at p are called the principal curvatures of X at p. The directions in which these extreme values occur are called principal directions of X at p. Unit vector in these directions are called principal vectors of X at p. A point p is umbilic if kmax ˆ kmin : If p is a nonumbilic point, there are exactly two principal directions, and these are orthogonal. Let Emax and Emin be principal vectors

k…u† ˆ cos2 ufxx x2 1 2 cos u sin ufxy xy 1 sin2 ufyy y2 : …A5† The equation of Dupin indicatrix in Cartesian coordinates is obtained from p r ˆ 1= k…u†;

…A6†

substituting x and y to r cos u and r sin u; respectively. Then the Dupin indicatrix of X at p is ip ˆ fxx x2 1 2fxy xy 1 fyy y2 ˆ 1:

…A7†

The signed Dupin indicatrix ip of X at p is obtained by intersecting the second order approximant with the plane z ˆ 1=2 and projecting it into the tangent plane z ˆ 0: Since the equation of indicatrix is a quadratic function of x and y, the indicatrix is a quadratic curve. The signed Dupin indicatrix ip is an ellipse if p is an elliptic point and X is locally above the tangent plane t p. It is a hyperbola if p is a hyperbolic surface point, a pair of parallel lines if p is a parabolic point and X locally above t p. It is void if p is parabolic or elliptic and X locally beneath t p. References [1] Blackmore D, Leu MC, Wang LP, Jiang H. Swept volume: a retrospective and prospective view. Neural, Parallel Scient Comput 1997;5:81±102. [2] Choi BK, Kim DH, Jerard RB. C-space approach to tool-path generation for die and mold machining. Comp Aided Des 1997;29:657±69.

J.-H. Yoon et al. / Computer-Aided Design 35 (2003) 69±81 [3] Choi BK, Park JW, Jun CS. Cutter location data optimization in 5-axis surface machining. Comp Aided Des 1993;25:377±86. [4] do Carmo MD. Differential geometry of curves and surfaces. Englewood Cliffs: Prentice-Hall, 1976. [5] Glaeser G, Wallner J, Pottmann H. Collision-free-3-axis milling and selection of cutting tools. Comp Aided Des 1999;31:225±32. [6] Hoschek J, Lasser D. Fundamentals of computer aided geometric design. Wellesley: A.K. Peters, 1993. [7] Jensen CG, Anderson DC. Accurate tool placement and orientation for ®nish surface machining. In: Dunn WW, editor. Proceedings of the Symposium of Concurrent Engineering. ASME Winter Annual Meeting, 1992. [8] Lee Y-S. Admissible tool orientation control of gouging avoidance for 5-axis complex surface machining. Comp Aided Des 1997;29: 507±21. [9] Lee Y-S. Mathematical modeling using different end-mills and tool placement problems for 4- and 5-axis NC complex surface machining. Int J Prod Res 1998;36:785±814. [10] Lee Y-S. Non-isoparametric tool path planning by maching strip evaluation for 5-axis sculptured surface machining. Comp Aided Des 1998;30:559±70. [11] Lee Y-S, Chang TC. 2-phase approach to global tool interference avoidance in 5-axis machining. Comp Aided Des 1995;27:715±29. [12] Lee Y-S, Chang TC. Automatic cutter selection for 5-axis sculptured machining. Int J Prod Res 1996;34:111±35. [13] Marciniak K. Geometric modelling for numerically controlled machining. New York: Oxford University Press, 1991. [14] Pottmann H, Ravani B. Singularities of motions constrained by contacting surfaces. Mech Mach Theory 2000;35:963±84. [15] Pottmann H, Wallner J, Glaeser G, Ravani B. Geometric criteria for gouge-free three-axis milling of sculptured surfaces. ASME J Mech Des 1999;121:241±8. [16] Radzevich SP. Multi-axis NC machining of sculptured surface part surfaces. Kiev: Vishcha Shcola Publishers, 1991. [17] Rao A, Sarma R. On local gouging in ®ve-axis sculptured machining using ¯at-end tools. Comp Aided Des 2000;32:409±20. [18] Wallner J, Pottmann H. On the geometry of sculptured surface machining. Curve Surf Des 1999;1234:417±32.

81 Joung-Hahn Yoon is a post-doctoral researcher of the mathematics department at the University of Technology in Vienna, Austria. He received his PhD (1999), MS (1993) and BS (1991) degrees from the department of mathematics at Seoul National University, Korea. His research interests include differential geometry and its applications to CAD/CAM.

Helmut Pottmann is a professor of geometry at the University of Technology in Vienna, Austria. His research interests include classical geometry and its applications, especially to Computer Aided Geometric Design, kinematics, and the relations between geometry, numerical analysis and approximation theory. A list of his recent publications can be found at http://www.geometrie.tuwien.ac.at/pottmann.

Yuan-Shin Lee is an associate professor of industrial engineering at North Carolina State University, USA. He received his PhD (1993) and MS (1990) degrees from Purdue University, USA, both in industrial engineering, and his BS degree from National Taiwan University, Taiwan, in mechanical engineering. His research interests include 3 and 5-axis sculptured surface manufacturing, CAD/CAM integration, computer-aided process planning, and computational geometry for design and manufacturing.