Modeling the surface swept by a generalized cutter ... - Semantic Scholar
PII: s0010-4485(97)00033-x
Computer-Afded Design. Vol 30, No. 8 pp. 587-594, 1998 ‘T’ 1998 Elsevier Science Ltd. All rights resewed Printed in Great Britam 0010.4485/98/$19.00+0.00
ELSEVIER
Modeling the surface swept by a generalized cutter for NC verification Yun C. ChungtT,
Jung W. Park*, Hayong Shins and Byoung K. Choit” a z-direction vector model as shown in Figure I is used widely 6. When cutting simulation is performed on a workpiece represented by a collection of z-direction vectors (ZDV), it is required to compute the intersection point between a ZDV and the cutter swept volume ‘. In 3-axis NC machining, the bottom of the cutter-swept volume for a linear motion NC-code block forms a cutter-swept surface (CSS) of single-valued type. If the CSS is represented in a single-valued form, z = f(x, y), intersection points between the CSS and ZDVs are easily obtained. Presented in this article is a procedure for constructing a single-valued model of the CSS for a generalized cutter referred to as automatically programmed tool (APT)-like cutter *. An approximation method for evaluating the CSS for flat endmills was first proposed by Anderson ‘, and a more comprehensive treatment of the subject was provided by Jerard et al.‘. The basic idea of these methods is to represent the CSS as a union of simple surface primitives, like circular cylinder, sphere, or elliptic cylinderg9”. The main drawback of this approach is that no simple surface primitives exist for the CSS of a generalized cutter. For example, the CSS of a rounded (or filleted) endmill renders no simple surface primitives. As a result, a search method is used in evaluating the CSS of a rounded endmill’. Other related research results include the method proposed by Wang” for obtaining a parametric surface model of 5-axis cutter-swept volume. machining,
Presented in this article is a procedure for representing the cutterswept surface (CSS) of a generalized cutter in a single-valued form, ; =,f(x, v). The key idea is that the z-value of the CSS at a 2D point (x, y) is expressed as the sum of 1) the z-value at a point on the silhouette curve of the cutter bottom surface and 2) the incremental z-value along the cutter movement direction. Thus, the main part of the modeling method is to obtain the silhouette curvr equations. which becomes a root finding problem for a quartic polynomial (when the cutter bottom surface contains a toroidal
surface). The proposed method not only renders a single-valued representation for the CSS of a generalized cutter (which was not possible with the existing methods) including rounded endmill but also results in a computational scheme that is faster than the existing schemes for ball- and flat endmills. Science Ltd. All rights reserved. Keywords: cutter path verification, sweeps, depth buffer
silhouettes,
0 1998 Elsevier
single valued,
INTRODUCTION Sculptured surface machining requires a careful checking of NC-data for possible overcutting (gouging) or undercutting. and cutting simulation has become a standard means fol checking or verifying NC cutter paths. As a result, a considerable amou:t60f attention was paid to the subject of cutting simulation NC verification via cutting simulation is a critical issue in die-cavity machining. Since die surfaces are visible from above, mostly 3-axis NC machines are employed in die cavity machining. In the cutting simulation of 3-axis NC
SINGLE-VALUED MODELING OF APT-LIKE CIJTTER In this section, the cutter bottom surface (CBS) of a generalized cutter is modeled as a single-valued surface of form : = f(x, y). This CBS model is utilized later in obtaining a single-valued model of its cutter swept surface (CSS). Shown in Figure 2a is the geometry of an APT-like cutter defined by six parameters (d, r, f, CY,6, h). * In this article, for notational convenience. the same cutter geometry is defined by an other set of six parameters (f, r, a, 1,. m, n), as depicted in Figure 2b. Obviously, in order to have a valid cutter geometry, the following restrictions on the parameters are required:
*To whom correspondence should be addressed. Fax: +82 42 8611692: E-mail: [email protected] WAM Laboratory, Department of Industrial Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373. I Kusong-dong, Yusong-gu, Taejon, Korea iSchool of Mechanical Engineering, Yeungnam University, Gyoung-san. Korea BChrysler Corporation (Chrysler Technology Center), Auburn Hills. Ml. USA ll Present address: CAD/CAM Research Institute of Cubictek Co., Seoul. Korea. Pqwr Keceired: 22 Mrrrcl~ 19%. Ampted: I I Jurw 1097
C1) (2) (3) (4)
587
All parameter values are non-negative. b must be positive when u is positive. m must be positive when rz is positive. b/a must be greater than n/m if both a and n are positive.
Surface swept by a generalized cutter: Y. C. Chung et al.
(4 Figure 3
(b)
(4
Special cases of the APT-like cutter
From the results of eqns (1) and (2) the single-valued expression of the CBS for the APT-like cutter in its standard position (shown in Figure 4c) is expressed as z =CBSA&, Figure 1
Saxis cutting simulation
with z-direction
vectors
(n/m)
Shown in Figure 3 are special cases of the APT-like cutter which are widely used in die-cavity machining. Alternatively, these special cases become primitive elements of the generalized cutter geometry. The parameter values for the rounded endmill shown in Figure 3c aref= m > 0, r > 0, and a = b = n = 0. The flat endmill of Figure 3a and the ball endmill of Figure 3b are special cases of the rounded endmill (r = 0 and f = m = 0, respectively). The cone endmill of Figure 3d is defined by a > 0, b > 0, and f = r=m=n=O.
z = CBSRG, y 1f, r)
0,
for t R)}
and (Ju’+(d-v)2
N(u, v).D = 0 >R)}
(8)
where R is the radius of the cutter and d is the projected distance of the cutter movement. A point falling inside the CSS domain is classified into one of the three regions as follows: RegionA : v i w Region B:
(11)
wO (A7)
where, 8 = cos - 1(RI&Q3 Further, the coefficients of the quadratic expressed in terms of _v, as follows:
eqn (A3) are
c, = Cl: - 4uz + 44’,)/2,
for D 5 0 or 1~1< IsI r
uf - 4~2 + 4y,)/2,
for D < 0 and jc[/ 2 Is/ r
(y , + d-)/2,
for D 5 0
(y,-Lm)/2
forD>O
C?=
(As)
In eqn (A8), the plus/minus signs are assigned such that only the valid root is selected satisfying the following constraints for a rounded endmill max(f,
u)stSf+r.
Finally, using the results of eqns (A8), and (15) is expressed as (A9)
Yun-Ghan Chung recutfly joined Cuhictek Co., a CAD/CAM softwnre company in Korea. He received II BS from h’unynny lJniver.sity in 1989, and an MS and o Ph.D. from KAIST, in 1991 nnd 1996, all in industrial Pngineerin,q. His research interests include sur&ce and solid modelroolparh ,qenerarion, CAD/CAM ing, vysrem and computer ,gruphic.v.
Jung W. Park is u ,full-rime lecturer qf mechanical engineering. He received 0 BS in physics,from Seoul National Universi@, Karen, in 1987, und an MS and a Ph.D. from KAIST, in 1991 and 1995, all in industrial engineering. He has worked for Chrysler Corporution (Chrysler Technology Center), USA, from mid-1995 through early 1997 as CAD/CAM software system developer. He joined Yeungnam University in early 1997. His research interests are in surface modeling, multi-axis NC machining, nnd offline progrcrmming.
1
i
593
Surface swept by a generalized cutter: Y. C. Chung et al.
HLI.LYIIZ~ Shin is n post doctorul resenrcher at the CAD/CAM/CAE R&D department oj Chyler Corporation. He received a BS ,fiom Seoul National Universiy irl 1985 and an MS and Ph.D. in industrial engineering from Korea Advanced Institute qf Science and Technology His m&r research interests are in the area oj’geometric modeling and tool path generation. His e-mail address is hs.?O@ch~.~ler. corn
594
Byoung K. Choi is LI professor of mamfacturing systems engineering in the Department qf Industrial Engineering ot Korea Advanced Institute of Science and Technology since he joined KAIST in 1983. He received a BS from Seoul Nutional Universizy, an MS ,from KtUST, and a Ph.D. from Purdue University. all in industritrl engineering. His research interests are in the same areu of sculptured sw$ace modeling, die-cavity muchining, CAPP, .vwtrm modeling and simulation, and virtual mixup.facturing. He can he reached via e-muil crt hkchoi@he;ier.kai.st.ac.kr
Computer-Afded Design. Vol 30, No. 8 pp. 587-594, 1998 ‘T’ 1998 Elsevier Science Ltd. All rights resewed Printed in Great Britam 0010.4485/98/$19.00+0.00
ELSEVIER
Modeling the surface swept by a generalized cutter for NC verification Yun C. ChungtT,
Jung W. Park*, Hayong Shins and Byoung K. Choit” a z-direction vector model as shown in Figure I is used widely 6. When cutting simulation is performed on a workpiece represented by a collection of z-direction vectors (ZDV), it is required to compute the intersection point between a ZDV and the cutter swept volume ‘. In 3-axis NC machining, the bottom of the cutter-swept volume for a linear motion NC-code block forms a cutter-swept surface (CSS) of single-valued type. If the CSS is represented in a single-valued form, z = f(x, y), intersection points between the CSS and ZDVs are easily obtained. Presented in this article is a procedure for constructing a single-valued model of the CSS for a generalized cutter referred to as automatically programmed tool (APT)-like cutter *. An approximation method for evaluating the CSS for flat endmills was first proposed by Anderson ‘, and a more comprehensive treatment of the subject was provided by Jerard et al.‘. The basic idea of these methods is to represent the CSS as a union of simple surface primitives, like circular cylinder, sphere, or elliptic cylinderg9”. The main drawback of this approach is that no simple surface primitives exist for the CSS of a generalized cutter. For example, the CSS of a rounded (or filleted) endmill renders no simple surface primitives. As a result, a search method is used in evaluating the CSS of a rounded endmill’. Other related research results include the method proposed by Wang” for obtaining a parametric surface model of 5-axis cutter-swept volume. machining,
Presented in this article is a procedure for representing the cutterswept surface (CSS) of a generalized cutter in a single-valued form, ; =,f(x, v). The key idea is that the z-value of the CSS at a 2D point (x, y) is expressed as the sum of 1) the z-value at a point on the silhouette curve of the cutter bottom surface and 2) the incremental z-value along the cutter movement direction. Thus, the main part of the modeling method is to obtain the silhouette curvr equations. which becomes a root finding problem for a quartic polynomial (when the cutter bottom surface contains a toroidal
surface). The proposed method not only renders a single-valued representation for the CSS of a generalized cutter (which was not possible with the existing methods) including rounded endmill but also results in a computational scheme that is faster than the existing schemes for ball- and flat endmills. Science Ltd. All rights reserved. Keywords: cutter path verification, sweeps, depth buffer
silhouettes,
0 1998 Elsevier
single valued,
INTRODUCTION Sculptured surface machining requires a careful checking of NC-data for possible overcutting (gouging) or undercutting. and cutting simulation has become a standard means fol checking or verifying NC cutter paths. As a result, a considerable amou:t60f attention was paid to the subject of cutting simulation NC verification via cutting simulation is a critical issue in die-cavity machining. Since die surfaces are visible from above, mostly 3-axis NC machines are employed in die cavity machining. In the cutting simulation of 3-axis NC
SINGLE-VALUED MODELING OF APT-LIKE CIJTTER In this section, the cutter bottom surface (CBS) of a generalized cutter is modeled as a single-valued surface of form : = f(x, y). This CBS model is utilized later in obtaining a single-valued model of its cutter swept surface (CSS). Shown in Figure 2a is the geometry of an APT-like cutter defined by six parameters (d, r, f, CY,6, h). * In this article, for notational convenience. the same cutter geometry is defined by an other set of six parameters (f, r, a, 1,. m, n), as depicted in Figure 2b. Obviously, in order to have a valid cutter geometry, the following restrictions on the parameters are required:
*To whom correspondence should be addressed. Fax: +82 42 8611692: E-mail: [email protected] WAM Laboratory, Department of Industrial Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373. I Kusong-dong, Yusong-gu, Taejon, Korea iSchool of Mechanical Engineering, Yeungnam University, Gyoung-san. Korea BChrysler Corporation (Chrysler Technology Center), Auburn Hills. Ml. USA ll Present address: CAD/CAM Research Institute of Cubictek Co., Seoul. Korea. Pqwr Keceired: 22 Mrrrcl~ 19%. Ampted: I I Jurw 1097
C1) (2) (3) (4)
587
All parameter values are non-negative. b must be positive when u is positive. m must be positive when rz is positive. b/a must be greater than n/m if both a and n are positive.
Surface swept by a generalized cutter: Y. C. Chung et al.
(4 Figure 3
(b)
(4
Special cases of the APT-like cutter
From the results of eqns (1) and (2) the single-valued expression of the CBS for the APT-like cutter in its standard position (shown in Figure 4c) is expressed as z =CBSA&, Figure 1
Saxis cutting simulation
with z-direction
vectors
(n/m)
Shown in Figure 3 are special cases of the APT-like cutter which are widely used in die-cavity machining. Alternatively, these special cases become primitive elements of the generalized cutter geometry. The parameter values for the rounded endmill shown in Figure 3c aref= m > 0, r > 0, and a = b = n = 0. The flat endmill of Figure 3a and the ball endmill of Figure 3b are special cases of the rounded endmill (r = 0 and f = m = 0, respectively). The cone endmill of Figure 3d is defined by a > 0, b > 0, and f = r=m=n=O.
z = CBSRG, y 1f, r)
0,
for t R)}
and (Ju’+(d-v)2
N(u, v).D = 0 >R)}
(8)
where R is the radius of the cutter and d is the projected distance of the cutter movement. A point falling inside the CSS domain is classified into one of the three regions as follows: RegionA : v i w Region B:
(11)
wO (A7)
where, 8 = cos - 1(RI&Q3 Further, the coefficients of the quadratic expressed in terms of _v, as follows:
eqn (A3) are
c, = Cl: - 4uz + 44’,)/2,
for D 5 0 or 1~1< IsI r
uf - 4~2 + 4y,)/2,
for D < 0 and jc[/ 2 Is/ r
(y , + d-)/2,
for D 5 0
(y,-Lm)/2
forD>O
C?=
(As)
In eqn (A8), the plus/minus signs are assigned such that only the valid root is selected satisfying the following constraints for a rounded endmill max(f,
u)stSf+r.
Finally, using the results of eqns (A8), and (15) is expressed as (A9)
Yun-Ghan Chung recutfly joined Cuhictek Co., a CAD/CAM softwnre company in Korea. He received II BS from h’unynny lJniver.sity in 1989, and an MS and o Ph.D. from KAIST, in 1991 nnd 1996, all in industrial Pngineerin,q. His research interests include sur&ce and solid modelroolparh ,qenerarion, CAD/CAM ing, vysrem and computer ,gruphic.v.
Jung W. Park is u ,full-rime lecturer qf mechanical engineering. He received 0 BS in physics,from Seoul National Universi@, Karen, in 1987, und an MS and a Ph.D. from KAIST, in 1991 and 1995, all in industrial engineering. He has worked for Chrysler Corporution (Chrysler Technology Center), USA, from mid-1995 through early 1997 as CAD/CAM software system developer. He joined Yeungnam University in early 1997. His research interests are in surface modeling, multi-axis NC machining, nnd offline progrcrmming.
1
i
593
Surface swept by a generalized cutter: Y. C. Chung et al.
HLI.LYIIZ~ Shin is n post doctorul resenrcher at the CAD/CAM/CAE R&D department oj Chyler Corporation. He received a BS ,fiom Seoul National Universiy irl 1985 and an MS and Ph.D. in industrial engineering from Korea Advanced Institute qf Science and Technology His m&r research interests are in the area oj’geometric modeling and tool path generation. His e-mail address is hs.?O@ch~.~ler. corn
594
Byoung K. Choi is LI professor of mamfacturing systems engineering in the Department qf Industrial Engineering ot Korea Advanced Institute of Science and Technology since he joined KAIST in 1983. He received a BS from Seoul Nutional Universizy, an MS ,from KtUST, and a Ph.D. from Purdue University. all in industritrl engineering. His research interests are in the same areu of sculptured sw$ace modeling, die-cavity muchining, CAPP, .vwtrm modeling and simulation, and virtual mixup.facturing. He can he reached via e-muil crt hkchoi@he;ier.kai.st.ac.kr
