COMPUTER AIDED

GEOMETRIC DESIGN

ELSEVIER Computer Aided Geometric Design 15 (1998) 567-584

Intersection approach to multi-point machining of sculptured surfaces

A n d r e w W a r k e n t i n * , F a t h y I sma i l , S a n j e e v Bed i

Automation and Control Group, Department of Mechanical Engineering. Universit3 of Waterloo. Waterloo N2L 3Gl Ontario. Canada

Received December 1996: revised July 1997

Abstract

Five-axis machining of sculptured surfaces is gaining wider acceptance in many industries. This paper examines the potential of a new tool positioning strategy where the desired surface is machined at more than one point of contact between the desired surface and the cutting tool. The method is called Multi-Point Machining or simply MPM. Intersection theory is used to gain insight into the multi-point contact problem as well as to solve for these contact points. The approach is demonstrated numerically for a sphere, a quadratic surface and a cubic B6zier surface. Experimental verification is also conducted for the spherical and quadratic surfaces. © 1998 Elsevier Science B.V.

Keywords: Five-axis; Machining; Surface; Multi-point

1. Introduct ion

The application of 5-axis NC machining has increased dramatically in the past ten years (Altan et al., 1993). Many companies are switching to 5-axis technology in order to reduce manufacturing cost, increase accuracy, and decrease machining time.

However, the full benefit of 5-axis technology has yet to be realized in the area of sculptured surface machining. These surfaces are encountered in many objects such as turbine blades, automobile parts, aircraft components, and especially in molds and dies. In most countries, the mold and die industry consists primarily of small companies, typically with fewer than 100 employees. In the US approximately 15 000 die shops represent an annual sales volume of about $20 billion (Altan et al., 1993).

* Corresponding author. E-maih awarkent@ravi.uwaterloo.ca.

0167-8396/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0167-8396(97)00039-3

568 A. Warkentin et aL / Computer Aided Geometric Design 15 (1998) 567--584

ball nose end mill

I

.~>1~ ssfeed ! i

/ s , /

d /

desired surface''-,..,,,,[/,, /

^ -,~p.--tool path >

~ scallop

Fig. 1. Scallops left behind when machining with a ball nose end mill.

Sculptured surfaces are generally produced in three stages; roughing, finishing, and benchwork. Roughing cuts are used to remove most of the material from a workpiece. The conventional method of finish machining a sculptured surface consists of removing material from a roughed out workpiece with a ball-nosed end mill. The resulting surface is left with a large number of scallops as shown in Fig. 1. Benchwork consisting of grinding and polishing is used to remove these scallops. The time spent on finishing and benchwork is dependent on the size of these scallops. The scallops size is related to the crossfeed; reducing the crossfeed decreases the scallop size at the expense of increased machining time. A recent survey by LeBlond Makino of Mason, Ohio (Technology Trends, 1994), stated that a small mold will typically require 57 hours of roughing, 127 hours of finishing, and 86 hours of grinding and polishing. Over 78% of the total machining time is spent on finishing, grinding and polishing. Clearly, there is a need for faster machining techniques that produce smaller scallops, and hence require little or ultimately no benchwork.

For faster finishing operations, researchers have turned to 5-axis machining for a solution. A number of tool positioning strategies have been developed to improve the placement of different types of tools relative to the design surface such that the material left behind for subsequent polishing is minimized. The most commonly used tools in the machining of molds and dies are shown in Fig. 2; they are: the ball nose cutter with radius, r, the flat bottom end mill of radius R, and the toroidal cutter which is characterized by two radii, the radius of the insert r, and the radial distance between the center of the tool and the center of the insert, R. As the tools rotate the cutting surfaces of the above cutters are a sphere, a cylinder, and a toms corresponding to the ball nose, the flat and the toroidal cutters respectively. A review of tool positioning techniques most relevant to the current work is presented next.

In 1992 Jensen and Anderson (1993) proposed a method for calculating an optimal tool angle based on the local curvature of the cutter contact point. A flat end mill is placed on the surface such that the feed direction lines up with the direction of minimum

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curvature on the surface. The tool is inclined such that the effective radius of the tool at the cutter location equals the minimum radius of curvature of the surface. This procedure matches the local geometry of the tool with that of the surface as closely as possible without gouging the surface in the neighborhood of the cutter contact point. The authors also noted that the profile of a torus is a 4th order curve while the profile of a fiat end mill is only a 2nd order curve. Therefore, a better match between the surface and a toroidal end mill should be possible. Jensen et al. (1993) extended this work to the toroidal end mill and developed a numerical procedure for calculating crossfeeds. Rao et al. (1996a) developed a similar technique that they called the Principal Axis Method, PAM, and successfully used it to machine a surface patch. Using simulation and cutting tests, Rao et al. (1996b) investigated the principal limitation of this technique which is the difficulty in lining up the feed direction with the direction of minimum curvature. Unless this condition is met, scallops are larger than expected and gouging may occur. Kruth and Klewais (1994) used curvature matching as a first approximation for their tool inclination calculation. The authors recognized the importance of the workpiece global geometry and not just the local curvature at a point. Even with curvature matching, gouging may still occur when the surface curvature is changing radically. They checked to see if any portion of the cutting tool was penetrating the desired surface by numerically approximating the distance between the tool and the surface. The tool inclination angle was altered based on the location and depth of gouging.

All of the above tool positioning strategies attempt to maximize the metal removal by considering the local geometry of a single point on the design surface and a single point on the tool. Warkentin et al. (1995) proposed a tool positioning strategy called multi-point machining, MPM, which matches the geometry of the tool to the geometry of the surface by positioning the tool in a manner that maximizes the number of contacl points between the surface and the tool. The authors were able to machine spherical surfaces with virtually no scallops in a fraction of the time required by conventional machining techniques. They fully exploited the special features of the spherical surfaces. but they did not demonstrate the technique on a general surface. In the present paper a general technique for finding multi-point contact at a tool position is developed. The tool-surface interface is modeled as an intersection problem between two surfaces. The nature of multi-point contact is examined by considering the contact between a tool and two surfaces, namely, a spherical cavity and a more general quadratic concave parametric surface. These surfaces have been machined to verify the numerical formulations and as such to demonstrate the potential of the multi-point machining concept. Finally, MPM is used to machine a cubic B~zier surface as a further confirmation of the applicability of the method.

2. Tool surface contact

When ball nose, flat, or toroidal cutters rotate about their axes, their cutting edges sweep spherical, cylindrical or toroidal cutting surfaces respectively, as shown in Fig. 2. The geometry of the cutting surfaces is used to position the tools rather than the geometry of the tool itself. When a ball-nose end mill is placed on a typical surface, it contacts the

570 A. Warkentin et al. / Computer Aided Geometric Design 15 (I998) 567-584

I

i

I

ball nose end mill Cutting surface is

a sphere

I

I

i I

I

i

toroidal cutter Cutting surface

is a torus

I

I

I

i I

I

flat end mill Cutting surface

is a cylinder

Fig. 2. Commonly used cutting tools.

z,taxls

i

torus parameters

• z,taxls

tool in multi-I:~int contact with a surface.

i ; Z,taxis

tp°s nt

U-~-JfCCl

section through first cutter contact point

Fig. 3. Multi-point contact between a toroidal cutter and a surface.

surface at a single point such that the surface normal at the cutter contact point is collinear with the sphere 's radius. Most NC software packages use this idea for positioning of ball- nose end mills. The center of the tool is simply offset a distance equal to the tool 's radius along the surface normal. A multi-point contact between a ball-nosed cutter and a surface is not possible, provided that the local radius of curvature of the surface is greater than the tool 's radius. Since there is only one point of contact under most situations, this tool will not be considered any further in this study.

A flat end mill can be considered to be a toroidal cutter with an infinitesimal insert radius. Accordingly flat and toroidal cutters will be considered together. If a torus or the flat bottom of a cyl inder is dropped onto a surface it would stabilize in a position where

A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584 571

it contacts the surface at more than one point. In Fig. 3 the tool is shown with three cutter contact points, eel, cc2, cc3. At each point of contact, the surface normal, n~, and the tool normal, nt, are collinear. These normal vectors intersect the tool axis because of the symmetry of the toms.

The cutting surface of a toroidal tool can be defined implicitly by the equation:

(X2 + y2 + z 2 +/~2 _ r2)2 _ 4R2(x 2 + y2) : 0

or parametrically as:

T ( ~ , O ) = Ty(O,O) = | ( R + rcos (O))s in (O) . Tz (0, O) L r sin(0)

(2.1)

(2.2)

The radii, R and r and the angles, 0 and 0 are shown in Fig. 3. Most CAD/CAM packages use parametric descriptions for surfaces. This practice will be continued here and accordingly the surface will be defined in terms of the parameters u and v as:

(2.3)

In order to calculate multiple points of contact between the tool and the surface, the geometry of the entire surface underneath the tool must be considered. This fact makes the multi-point problem much more difficult to solve than the single point contact solution. In order for a point of contact to exist, the tool must be placed tangentially on the desired surface. In other words, corresponding points on the tool and surface must share the same location, and their normal vectors must be collinear. Equations to model these conditions can be formulated using the parametric representation of the tool and surface. The cutting surface of the a toroidal tool is defined parametrically by Eq. (2.2) and the parametric equation of the surface is given by Eq. (2.3). The coordinate systems of the tool and the surface can be related using a matrix transformation. The tool is positioned relative to the surface by translating the tool equation along a vector (X, Y, Z). The desired tool orientation is achieved by a rotation of A and a rotation of B about two orthogonal axes. The equation of the transformed toms is shown below.

Ts(0,0)] T ( O , O , X , Y , Z , A , B ) = [ T r a n s f o r m a t i o n ( X , Y , Z , A , B ) ] Tu(O,O ) .

%(o o) (2.4)

Transformation(X, Y, Z, A, B) is a composite transformation including the translation and rotation that describes the position and orientation of the tool in the workpiece coordinate system. Together, the position and orientation of a tool is referred to as a tool configuration. The surface normal is expressed as:

~S(u,v) × OS(u,v) n~(u, v) = au a~ as(~,v) as( .... ) (2.5)

572 A. Warkenfin et al. / Computer Aided Geometric Design 15 (1998) 567-584

112

n~

intersection

1

tangency

1

no intersection

Fig. 4. The relationship between tangency and intersection.

and the tool normal is expressed as:

OT(O,O,X,Y,Z,A,B) × OT(~,O,X,Y,Z,A,B) a~ ao (2.6) nt(O,O,X,Y,Z,A,B)---- I aT(O'O'X'YZ'A'B)aO × aT(O,O,X,Y,Z,A,B)aO [ "

For tangential contact, points on the tool and the surfaces must share the same location in space and have the same normal direction. Therefore, all points of tangential contact between a surface and a tool must satisfy the following set of vector equations.

T(¢, O, X, Y, Z, A, t3) - S(u, v) = 0, (2.7)

nt(0, O, X, Y, Z, A, B) - ns(U, v) = 0. (2.8)

Each vector equation consists of three components. The result is a set of 6 non- linear equations in the 9 unknowns: X, Y, Z, A, B, 0, 0, u, v. A given transformation, T(X, Y, Z, A, t3), will result in a set of surface parameters, (u, v), and a corresponding set of tool parameters (0, 0). The solution set may contain a single point solution, a multi-point solution, or a curve of intersection solution, but the vast majority of tool configurations will produce no solution. If Eq. (2.7) is considered by itself, additional solutions will become possible. These additional solutions consist of tool-surface inter- sections. These intersections provide some insight into the tool configurations that might produce multiple tangential contact points.

The above concept can be illustrated by examining the intersection of two curves shown in Fig. 4. In this figure, two curves are gradually moved apart along a common normal. Just before the two curves separate, the two curves are tangent to each other; tangency is the boundary case between intersection and no intersection. The singular nature of tangent points makes them difficult if not impossible to locate. The generation of intersection curves is also a difficult problem but relatively easier to solve. Markot and Magedson (1989) and Krieziz (1990) showed that when a curve of intersection forms a closed loop, there is at least one pair of points within the loop that share a common normal. These points are referred to as characteristic points. As the area of the loop approaches zero, the location of the characteristic point approaches the location of a tangent point. This means that the center of a small intersection loop can be used as a good approximation to a tangent point. The area of this loop provides an indication of the accuracy of that approximation. The next section describes a technique used to examine the nature of contact between the tool and the surface based on intersection.

A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584 573

3. Solution by intersection

In the previous section, equations were used to model tangency between a tool and a surface. Eq. (12.7) is the equation of intersection between the surfaces and Eq. (2.8) is the equation of normality between the two surface. These equations are difficult to solve simultaneously. However, it is possible to solve for the intersection between two surfaces. Tangency solutions are a subset of all the possible intersection solutions. Theoretically Eq. (2.7) could be solved for intersection solutions but it is not in the most convenient form to do so. The tool parameters 0, and O, and the surface parameters u, and c, describe the same set of intersection points. The Cartesian coordinates of these points can be determined by substitution into Eqs. (2.2) and (2.3). However, since we have the luxury of being able to define the tool implicitly using Eq. (2.1), this unnecessary duplication can be avoided by substituting the parametric equation of the surface into

Z'

tool coordinate system Y'

z,

~ i ~ surface coordinate system

Y2 Fig. 5. Tool and surface coordinate systems.

,•X2, Y

intersectior

/ f I

/ surface, S' I .J B

~l ) toroidal cutting surface ns

' \lto'. "- / \

section B-B

Fig. 6. Generation of an intersection curve,

574 A. Warkentin et aL / Computer Aided Geometric Design 15 (1998) 567-584

the implicit definition of the tool. This will give a single set of parameters defining the intersection point. Unfortunately, the implicit definition of the tool does not allow the use of convenient matrix operations for translations and rotations. This limitation can be overcome be performing the required transformations on the surface instead of the tool.

In order to find an intersection, the surface, S, must be transformed into the tool coordinate system as shown in Fig. 5. The surface is translated so that the point, P0(u, v), is located at the center of the toms. The surface is rotated so that surface normal, ns, is lined up with the z-axis of the tool, and the directions of minimum and maximum curvatures, A1, A2, are aligned with the x and y axes respectively. The alignment of these axes is accomplished by rotating the surface by 71 about the z-axis and by "72 about the z-axis. In this orientation, the surface equation is given by:

x

0 0 [ COS("/I) C0S(72 ) s i n ( " / 2 ) [ sin~71)

- sin(72) C0S(72)

v) - p (uo, vo) l ~(~, v) p~(uo, ~o) I " ~z(~,~) pz(~o, vo) J

sin/ !] cos(7~)

0

(3.1)

Having now expressed the workpiece surface in the tool coordinates, the intersection curves between the tool and the workpiece can be calculated. These intersections can subsequently be changed by rotating the surface about the tool y-axis by the angle /3 and around the x-axis by the angle a, followed by a translation along the tool axis of a distance d as illustrated in Fig. 6. This exercise leads to changing the surface equation form S' into S" according to Eq. (3.2).

[i ° s"(u, v) = cos(a) sin(a)

sin,o,°]rC°o ][ ] s(/3) 0 sin(/3) s~(u, v) 1 o 4 ( ~ , ~ )

cos(a) L - sin(/3) o cos(/3) s'~(u,v)

(3.2)

The surface equation, S ' ( u , v), is then substituted into Eq. (2.1) to form the equation of intersection in the (u, v) plane.

(S~12 + -y'q"2 + -~.q"2 + R 2 _ r2)2_4R2{S,,2~ x + S'u p2) = 0 . (3.3)

Tangency points can be determined by choosing values of (a,/3, d) such that the inter- section loops are small.

The procedure described above for calculating the intersection loops was implemented using the symbolic computation package MAPLE. The minimization process of these loops was accomplished by trial and error.

A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584 5 7 5

4. Examples of multi-point machining technique

In this section the technique is applied to three example surfaces consisting of a sphere, a second order parametric and a cubic Brzier surface. The intersection procedure was used to demonstrate the arrangement and number of contact points for the sphere and the second order parametric surface. Based on these results a multi-point algorithm has been developed to machine concave surfaces. The details of this algorithm will be published shortly.

4.1. Example 1: Multi-point machining of a sphere

Results from the intersection analysis were compared with those obtained using the vector based multi-point solution method developed by Warkentin et al. (1995) and summarized in the following paragraph.

In (Warkentin et al., 1995) the multi-point machining was explained using the "drop the coin" concept. If a coin is placed in a spherical surface, the coin will touch the surface tangentially at every point on its circumference. As the coin slides along the surface, the surface is generated along the entire circumference of the coin. If the coin is now replaced by the bottom of a milling cutter, the surface will be generated along a circle of contact between the tool and the surface. Fig. 7 shows a cross-section through the axis of a toroidal cutter placed in multi-point contact with a spherical cavity. This figure shows that the surface normals at eel, cc2, and P0, are collinear with the insert radii and the tool axis, respectively. When these conditions are met, the center of the sphere forms right angled triangles with points CCl and ec2 which can be used to calculate the distance, d, from the surface to the tool center, tpo~ along the sphere's radius, R~. This result, was derived in (Warkentin et al., 1995) as:

d = ~ / ( / ~ s - T') 2 - R 2 - -/~s- (4.1)

For the intersection calculations, a sphere centered at the tool's origin is defined below.

S(u, v) = = v . (4.2)

The point of interest, P0(O, O) = (0, O, - R , ) , was selected to be the bottom point of the sphere. In this case, only a translation of R.~ along the tool axis was needed to align the normal and principal directions at P0 with the tool coordinate system at the tool center, tpos. Intersection curves were calculated as the sphere was translated by - d , along the tool axis. For this special case S"(u , v) is given below.

+

sz (u, v) R~ - d

The resulting equation of intersection is:

- - - - - - -}- - - r 2 j - - 4 R 2 ( U 2 + V 2 ) = 0 . (4.4)

576 A. Warkentin et aL / Computer Aided Geometric Design 15 (1998) 567-584

cal c e

cc~ Po

Fig. 7. Cross-section of a toroidal cutter in tangential contact with a spherical cavity.

!nterseclicn t Y (mrs) intersection 1 0 ~

~ x Imm) ~°1

(a) d = 0.0 (mm) (b) d =-5.0 (mm) (c) d : -5.464 (ram)

Fig. 8. Intersection loops between a toms and a sphere.

Fig. 8 shows some intersection curves for different values of d projected onto the xy plane. The values of R, r, R~ in this example, are 7.9375, 4.7625 and 50.0 mm respec- tively. In Fig. 8(a), the point of interest, P0, is at the tool center, tpos, and the surface normal at P0 is collinear the tool axis. The two circles are the intersection curves pro- duced by the inner and outer surfaces of the toms. As the tool moves out of the surface, these circles move closer together as seen in part (b). In part (c) both circles are coinci- dent. The curves of intersection have become a line of tangency. The exact value of d calculated from the equations derived in (Warkentin et al., 1995) is - 5 . 4 6 4 mm. The d value calculated by intersection is - 5 . 4 6 4 mm which agrees with the exact answer.

The proposed technique was used to machine a spherical cavity on a JO 'TECH 5-axis CNC milling machine. The machined part was compared to a similar part machined with the traditional 3-axis technique using a ball nose cutter. The test sphere had a radius, R~, of 50 mm and a depth of 20 mm. The cutting conditions for the 5-axis machining were: spindle speed = 20,000 rpm, feed = 0.0325 mm/tooth, 0.5 mm depth of cut, 6.5 mm crossfeed, using a 2 flute 10 mm diameter end mill. The tool path was generated by offsetting the tool a distance, d, from the test sphere. The value of d was calculated

k. Warkentin et al. /Computer Aided Geometric Design 15 (1998) 56~584 577

using the equations in (Warkentin et al., 1995). Since a flat end mill was used instead of a toroidal cutter, the insert radius was set to zero in all the equations. A photograph of the resulting spherical cavity is shown in Fig. 9. For comparison, the same size dish was machined using an 8 mm ball nosed cutter with the same speed and feed. The crossfeed was 0.44 mm at the rim of the dish. A photograph of the resulting surface is shown in Fig. 10. Measurements using a Talyrond machine determined that the maximum scallop height for the multi-point contact was 2.5 /~m, and 30 l~m for the single point contact. In addition, the machining time was also a tenfold reduction in machining time. The results lkw the ball nose cutter would have been about 5% better for the single point contact if a 10 mm ball nosed cutter had been used instead of the 8 mm ball nosed cutter.

4.2. Example 2: Multi-point machining o f a second order surface

The previous example demonstrated how multi-point contact may be achieved when the surface under the tool is spherical. In this example, the method is demonstrated on the second order parametric surface given by:

S( , t ,v ) = ~',; = - 1 3 1 . 2 5 u + 28.125,, 2 . (4.5) S~ F(~, ~,)

where F ( u , t,) - 5.859375(u2v 2 + ~t2v) - 3.9025v2u + 76.171875t, 2 + 6.727430556v e 27.34375'u~,- 50.78125u + 24.95659722v + 12.0569722. Every point on this surface is elliptic. An examination of possible multi-point contact solutions for this surface should provide some insight into the nature of the multi-point contact between ellip- tic surfaces and the toroidal cutter. The intersection equation was determined by trans- forming Eq. (4.5) into the tool coordinate system and substituting it into Eq. (3.3) as described earlier. Due to their length, the resulting equations will not be reproduced here. The resulting intersection curves in a - e space were transformed into Cartesian coordinates by back substitution into Eq. (4.5). During this investigation, the point of interest on this surface will be P0(0.5, 0.5). The tool parameters are: /? - 7.9375 mm and r -= 4.7625 mm.

Fig. 11 shows some of the curves of intersection projected onto the :r-9 plane while finding the tangency points when the tool is aligned with the surface normal. The curves of intersection form two elliptic shapes. As the tool is lifted out of the surface these curves approach one another tangentially at opposite points on the .r axis. The resulting curves look like two half moons facing each other. Just before the tool leaves the surface, the curves of intersection consist of two small loops. The centers of these loops are assumed to be the tangent points.

Intersection loops were also calculated for tilt ct angles of 0 °, 2 °, 4 °, 6 °, 7.6 °, 15 "~, 20 ~ . 30 ° and corresponding/3 angles of -0 .115 °, -0 .1 °. 0.08 °, 0.07 °, 0.07 °. See Fig. 6 for the definitions of the rotation angles ct and ,3. The resulting loops have all been plotted in the z - y plane as shown in Fig. 12. The loops are labeled a, b, c, d, c, f , .q. h which corresponds to, (, = 0 °, 2 °, 4 °, 6 °, 7.6 °. 15 °, 20 ° and 30 ° respectively. For the cases of tilt angles corresponding to (z. b. c, and d there were two intersection loops as indicated

578 A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584

Fig. 9. Spherical dish machined with a 10 mm end mill on a 5-axis milling machine.

Fig. 10. Spherical dish machined with an 8.0 mm ball-nosed cutter on a 3-axis milling machine.

, ,y (ram)

10-

• , ID

x (mm)

(a) d = 0.0 (ram)

' ,, (ram) 10 +

) -10

)

x (ram)

(b) d = -4.0 (tara)

10 t y (mm)

1 x(mm) -10 -10 { ~1'0 x(mm)

(c) d = -5.0 (ram) (d) d = -5.28 (mm)

Fig. 11. Loops of intersection between the toms and the parametric surface when the tilt angle is zero.

A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584 579

b, 0 a'C) ,

I I I -10 - -6 -4

y (turn)

6 -

4 -

2 -

-2 0 2

2

4 6 8 10 x (ram)

Fig. 12. Loops of intersection between the toms and the parametric surface at different tilt angles.

by the subscript on the letters. The ordinate axis, 9, is aligned with the direction of minimum curvature and the abscissa axis, .c, is aligned with the direction of maximum curvature. The center point of a small loop was assumed to be the coordinate of a point of contact. A loop was considered small enough when a change in the third decimal place of d would cause the loop to disappear. The half circle was drawn on the graph to emphasize the symmetry of the contact points. Between ~ values of 0 ° and 7.6 °, the solutions occur in pairs symmetrically about the axis of minimum curvature. Beyond an c~. of 7.6 ° the solutions occur only on the direction of minimum curvature. Loop ~ is the approximate location at which the multi-point solution becomes a single point solution. All single point machining takes place at tilt angles at or beyond point c.

The present example demonstrates that a toroidal tool can be placed on a general concave surface such that two points of contact exist between the surface and the tool. Also, the contact points are arranged symmetrically about the direction of minimum curvature on the surface. This insight has resulted in an algorithm that can calculate 2- point contact tool positions automatically given the desired separation between the contact points. A more complete description of the implementation of the 2-point algorithm will be published at a later date.

The 2-point algorithm was used to machine the test surface described by Eq. (4.5). A tool path was generated for a contact point separation, u,, of 10.0 mm. No attempt was made to optimize the cutter contact separation. The resulting tool configuration file along with milling machine kinematics, and workpiece setup was used to generate machine readable NC code used to position the tool relative to the workpiece. This code was used to machine the test surface on a FADAL VMC 4020 5-axis tilt-rotary table milling machine. The cutting conditions used were: spindle speed = 4000 rpm, feed = 0.05 mm/tooth, 1.0 mm depth of cut, using a cutter of the dimensions /{ = 7.9375 mm and r = 4.7625 mm.

Fig. 13 shows the machined workpiece and a close up of the surface. The close up of the surface shows a section of two adjacent tool paths. Each tool path can be recognized by its feed marks. In between the tool paths a faint jagged line can be seen. In single point machining you would expect the largest surface deviation to occur at this line. In

580 A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584

Fig. 13. Photographs of the parametric surface machined with the multi-point method, (a) the machined workpiece, (b) a close up of the surface.

multi-point machining, the surface deviation is at a minimum near this line because the cutter contact points are located along this line. The maximum surface deviation occurs at the midpoint between the two contact points. The jaggedness of the line is due to smearing caused by the small depths of cut at points where the adjacent cutter contact points almost overlap. Since large surface deviations don't occur at the intersection be- tween two tool paths the multi-point workpiece has extremely smooth appearance. This feature may be useful in applications that require high surface smoothness but do not necessarily require high surface accuracy,

A. Warkentin et al. /Computer Aided Geometric Design 15 (1998) 567-584

0.04

I desired surface.N, g 000. I , , " ' , , , ] [t~ [ . ~ / - - - seperation between

I, ~'~ t '~ , f ~ INk ~"4k, |,.,.L | - - ~ contact points

goost s,,re,,u ace" 0.0 20.0 40.0 50.0 60.0 80.0

distance (ram)

Fig. 14. Measured surface deviations at cross section ,~ . . . . 62.6 mm.

Table 1 Control points for B6zier surface. All values in mm

- 6 0 30 3O 6O

60 60* 0 0 45

30 25 2O I 0 55

30 6O 5 5 3O

- 6 0 45 5 -10 5

*z value at a: = - 6 0 and ,q = 60.

58/

In addition to the smooth appearance of the surface produced using MPM, a quanti- tative assessment of the deviation from the desired surface was also conducted. These measurements were performed by scanning the machined surface using a coordinate mea- suring machine, CMM, One such scan, close to the midsection of the workpiece surface, is shown in Fig. 14. Measurements were taken every 0.1 mm along this section. An off- set of approximately 25 #m from the design surface can be observed. This error can be attributed to the positioning error associated with each of the 5-axes and possible errors in the setup. The minimum deviations from the desired surface occur near the contact points and the maximum deviations occur in between these points. Even with the offset, the surface is within the tolerance of ±75 t~m typically assigned to a sculptured surface. If the surface fluctuations are considered without the offset, the average peak-to-peak variation for the entire surface is 25 /ml; well within tolerance.

The same surface was machined by Rao et al. (1996b) using a 10 mm ball nosed end mill with a 0.4 mm crossfeed. The average scallop height peak-to-peak was 0.013 mm. Although the average scallop height was about half that of the multi-point workpiece the crossfeed was 25 times smaller which translates into 25 times as much machining time.

5. Example 3: Multi-point machining of a cubic B6zier surface

Cubic surfaces are commonly used in the mold industry because of their design flex- ibility. For this reason, and for the sake of completeness we present an example of

582 A. Warkentin et al. / Computer Aided Geometric Design 15 (1998) 567-584

(-60,-60, 0)

Fig. 15. Cubic B6zier surface.

machining such a surface. In this example, the arrangement of contact points is similar to that in Example 2, and therefore will not be repeated here. The presentation will be conducted using simulation only, since the experimental verification has already been conducted for two surfaces. The surface in the present example is a cubic B6zier surface defined using the 16 control point listed in Table 1. The surface is shown in Fig. 15.

A tool path for this surface was generated in the same manner as in the previous example. Metal removal was simulated using the "mow the grass" technique advanced by Jerard et al. (1989). The following parameters were used for the simulation: tool dimensions R = 7.9375 mm, r = 4.7625 ram, crossfeed = 8 mm and cutter contact separation w = 6.4 mm. The results of the simulation are shown in Fig. 16. This graph shows the deviation between the surface and the tool at every tool position along the tool path. These deviations were computed along the surface normal. To emphasize the fluctuations between the design surface and the machined surface, the magnitude of these deviations are plotted on the xy plane. The simulation predicts a maximum scallop height of 17 #m at one corner of the surface. The average surface deviation is only 2 #m. For comparison, the same tool path was used to simulate 3-axis machining with a ball nosed end mill of the same size as the toroidal cutter. The resulting maximum and average surface deviations were 1378 ~m and 364 ,am respectively; two orders of magnitude higher then those obtained from MPM.

Fig. 16 clearly illustrates the two types of scallops produced by multi-point machining. The large rounded scallops are produced between the cutter contact points underneath the tool. These scallops are highly dependent on the surface curvature as illustrated by the large 17 #m scallop produced at the region of the surface with the highest surface curvature. It should be possible to control this phenomenon by varying the cutter contact separation distance along the tool path. The sharper scallops are produced between ad- jacent tool positions in the same manner as single point machining. These scallops can be controlled by varying the crossfeed.

A. Warkentin et al. / Computer Aided Geometric" Design 15 (I998) 567-584

scal lop left be tween cutter contact ,9o~nts

scal lop left be tweep 1001 DOSItlOP S

583

Fig. 16. Surface deviation produced when machining the Bezier surface.

Concluding remarks

This paper has demonstrated that multiple points of contact do exist between certain types of tools and surfaces. This fact can be used to decrease the time required to machine surfaces using a 5-axis arrangement. Future work in multi-point machining will concentrate on improving the algorithms used to calculate the tool configurations and to examine the effect of feed direction, cutter contact point separation, and crossfeed on surface deviation and machining time.

Acknowledgments

The authors would like to thank IRDI and McMaster University for the use of their equipment and the Natural Sciences and Engineering Research Council of Canada for financial support.

References

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