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An improved method for scheduling the tool paths for three-axis surface machining
International Journal of Machine Tools & Manufacture 41 (2001) 133–147
An improved method for scheduling the tool paths for threeaxis surface machining Chih-Ching Lo a b
a,*
, Rong-Shine Lin
b
Department of Mechanical Engineering, Feng Chia University, Taichung 407, Taiwan, ROC Department of Mechanical Engineering, Chung Cheng University, Chiayi 621, Taiwan, ROC Received 16 December 1999; accepted 13 June 2000
Abstract This paper presents a new tool-path scheduling method to improve the accuracy and efficiency of threeaxis surface machining. The features of the proposed method include: (1) surface-error-based segmentation in the feed-forward direction; (2) consideration for the coupled effect of the segmentation in the feedforward and the path-interval directions; and (3) compensation for the tool location to control the surface error and the segmentation efficiency. In this paper, we consider a ball-end mill for surface machining. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Surface machining; Tool-path scheduling; Ball-end mill
1. Introduction In today’s manufacturing industries, the CAM system schedules the tool paths and feeds them to the CNC machine tool for on-line part machining. One of the most important CAM techniques is to generate the tool paths for machining of parametric or free-form surfaces that are widely utilized to express the shapes of complex parts. For machining of these parametric surfaces, threeaxis CNC milling machines adopting ball-end cutters are frequently utilized [1–3]. The tool-path scheduling for three-axis ball-end milling of a parametric surface includes: (1) segmentation in the path-interval or side-step direction; and (2) segmentation in the machining or feed-forward direction. The existing approaches to determining the tool-path intervals are the iso-parametric [4], the iso-planar [5,6], and the iso-scallop [7,8] methods. Basically, the path interval is computed based on the limit of the scallop height that is an important source for the * Corresponding author. Tel.: +886-445-17250, ext 3504; fax: +886-445-16545. E-mail addresses: [email protected] (C.-C. Lo), [email protected] (R.-S. Lin).
0890-6955/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 0 ) 0 0 0 5 0 - X
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surface machining error. With the determined path interval, a series of machining paths passing over the surface are scheduled. In current surface machining applications, each machining path in the feed-forward direction is further divided into a series of linear or spline segments that are then fed to the CNC machine tools. In this paper, we consider the case of linear segmentation since the linear motion command is the most popular in the CNC machining applications. Linear segmentation in the feed-forward direction is conducted according to the limit of the chordal deviation that is another source for surface machining error [7]. A basic procedure for the linear segmentation is first to compute a set of cutter-contact (CC) points along the machining path (that is located on the surface), and then to compute the corresponding cutter-location (CL) points through tool offsetting. Consequently, the continuously straight lines connecting these CL points are the linear motion commands for the surface machining. In the current approach, the segmentation in the feed-forward direction is independent of the segmentation in the path-interval direction. However, the surface error is a combination of the chordal error due to the feed-forward segmentation and the scallop error due to the path-interval segmentation. Accordingly, to achieve high efficiency and accuracy in tool-path scheduling, we should conduct the feed-forward segmentation with consideration of the effect of the path-interval segmentation. Moreover, the feed-forward segmentation should be conducted based on the surface error, rather than the chordal error considered in the traditional methods. In this paper, we present an improved method for tool-path scheduling, with which the surface error is controlled within the desired tolerance and the segmentation efficiency is maximized. With the proposed method that considers the coupled effects of the segmentation in the feedforward and the path-interval directions, we can improve the efficiency in the feed-forward segmentation (by reducing the number of desired linear segments) and the efficiency in the pathinterval segmentation (by reducing the total machining length). 2. Existing methods and problems As shown in Fig. 1, a series of machining paths over the surface are assigned to cut the surface that is denoted by S(u,v), where u and v are the surface parameters. These machining paths are
Fig. 1. The machining paths on a surface.
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the curves that the cutter-contact (CC) point passes through. The CC point denotes the location at which the cutter edge touches the surface. In practice, the motion commands fed to the CNC machine tools are not the CC paths, but the cutter-location (CL) paths that the tool center passes through. For a ball-end cutter, the tool offsetting that relates the CC point (C) to the CL point (L) can be formulated as L⫽C⫹r· sign(nˆ·zˆ)·nˆ,
(1)
where nˆ is the unit normal vector to the surface, and zˆ is the unit vector along the tool axis (that is set in the positive z-direction in this paper). The typical tool-path scheduling method is first to assign the CC paths one-by-one over the surface, and then to break each CC path into a series of linear segments. The existing methods to determine the CC paths include the iso-parametric, the iso-planar, and the iso-scallop methods. Basically, the path interval or the side step is computed based on the limit of the scallop height which is an important source of the surface error. A geometrical illustration for the scallop height (h) and the path interval (l) is shown in Fig. 2. For a given path interval, the scallop height can be calculated by [7, 9] l2[rs−r] , h⫽ 8rsr
(2)
where rs is the radius of curvature in the path-interval direction. In Eq. (2), rs is defined to be positive for a concave shape and negative for a convex shape. Note that when utilizing the isoparametric and the iso-planar methods, a conservative path interval is determined so that the maximum scallop height all over the CC path will not exceed an assigned limit (e.g. hⱕ0.01 mm). When utilizing the iso-scallop method, the scallop height is maintained constant along the CC path. The linear segmentation in the machining or feed-forward direction is conducted based on the limit of the chordal deviation that is another source of the surface machining error. For a given incremental length (d) in the feed-forward direction, the chordal error (d) can be calculated by d⫽
d2 , 8rc
(3)
where rc is the radius of curvature on the CC path, and d=|Ci+1⫺Ci|, where Ci+1 and Ci are two
Fig. 2. Path interval and scallop height.
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Fig. 3. Linear segmentation and chordal error.
consecutive CC points (please refer to Fig. 3). Based on Eq. (3), the existing tool-path planning methods break a CC path into a series of linear segments so that the chordal error will not exceed an assigned limit (e.g. d⬍0.01 mm). A drawback of the current methods for the feed-forward segmentation is that the chordal error does not have a strong relation with the surface error. As illustrated in Fig. 3, D is a point on the linear segment CiCi+1 and corresponds to the maximum deviation error (or chordal error) from the CC path. In practice, D may still be very close to the surface, and then corresponds to a very small surface error. Therefore, the current methods with chordal-error-based segmentation are not efficient approaches. However, as will be addressed in a later section, we can easily delete this problem by revising Eq. (3), for which rc is replaced by rf, where rf is the radius of curvature on the surface (rather than the CC path) in the feed-forward direction. Another drawback of the current methods is that an additional error component due to the tool offsetting is not considered. A typical result for feed-forward segmentation is shown in Fig. 4. The current segmentation procedure first takes a series of CC points along the CC path (based on the scheduled chordal error limitation, ds), then computes the correspondent CL points (based
Fig. 4. Additional error component due to the tool offsetting.
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on tool offsetting), and then connect these CL points with continuously linear segments. As can be seen in Fig. 4, if the tool follows the CL segmented path, the practical CC path will not follow the scheduled CC path. Consequently, the practical chordal error becomes dp. Note that dp depends on not only ds but also the surface curvature that is variable along the CC path. As will be addressed in a later section, we can delete this problem by modifying the tool offsetting algorithm, Eq. (1). The third and most important drawback for the current methods is that the feed-forward segmentation is independent of the path-interval segmentation. In practice, the resultant surface error is a combination of the segmentation in the path-interval and the feed-forward directions. In the following, we first investigate the combinative effects of the feed-forward and the path-interval segmentation, and then construct the basis for the method presented in a later section. As shown in Fig. 5, Ci and Ci+1 are two consecutive CC points for a specific linear segment. On the CC segment, Cm is the middle point that corresponds to the chordal error d. Along the machining path, Pi, Pi+1 and Pm are the peaks of the scallop corresponding to Ci, Ci+1 and Cm, respectively. Let e be a vector representing the deviation errors from the surface at the concerned points, Ci, Cm, Ci+1, Pi, Pm and Pi+1. Based on the geometrical illustration in Fig. 5, we have
Fig. 5. Surface errors due to the path-interval and feed-forward segmentation.
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e(Ci,Cm,Ci+1,Pi,Pm,Pi+1)⫽(0,⫺d,0,h,h⫺d,h),
(4a)
if the surface is convex in the feed-forward direction, and e(Ci,Cm,Ci+1,Pi,Pm,Pi+1)⫽(0,d,0,h,h⫹d,h),
(4b)
if the surface is concave in the feed-forward direction. For iso-parametric and iso-planar machining methods, h is not constant and hⱕhlimit, where hlimit is the assigned limit for the scallop height. In contrast, h=hlimit for the iso-scallop machining method. In these current methods, d is set as a constant (i.e. d=dlimit) for all linear segments. In the current machining planning methods, dlimit and hlimit are usually set as the same value, i.e. dlimit=hlimit=elimit, where elimit is the allowable tolerance for the surface machining. Consequently, for the convex case as referred to in Eq. (4a), the maximum deviation error is equal to ⫺elimit and is located at Cm. For the concave case as referred to in Eq. (4b), the maximum surface error is equal to h+elimit(ⱕ2elimit) and is located at Pm. As can be seen, the surface error is not well controlled because it may exceed the assigned tolerance, elimit. Besides, the range of the surface error is varying from one machining segment to another (note that ⫺elimitⱕeⱕh for the convex case and elimitⱕeⱕh+elimit for the concave case, where h is variable for non-isoscallop machining). As stated above, the maximum surface error caused by the traditional methods (when letting dlimit=hlimit=elimit) may be close to 2elimit, and we need to select conservative values for dlimit and hlimit so that the surface error is still smaller than elimit. This can be done by letting dlimit=hlimit=elimit/2. However, the segmentation efficiency in the feed-forward and the path-interval directions is degraded. In principle, (according to Eqs. (2) and (3), d⬀√d and l⬀√h), the feedforward incremental length (d) and the path-interval length (l) will become 1/√2 of the original value. Consequently, the total machining length may be increased by 40% (=√2⫺1) and the number of total linear segments may be increased by 100% (=√2×√2⫺1). In the following, an improved tool-path scheduling method is proposed to cope with the above problem. In the proposed method, the feed-forward segmentation is conducted with regard to the path-interval segmentation so that the surface error for every machining strip is always controlled within the range of [⫺elimit,elimit]. Moreover, we can improve the segmentation efficiency in both the feed-forward and the path-interval directions.
3. Proposed method 3.1. Surface-error-based segmentation As has been illustrated above, the feed-forward segmentation based on Eq. (3) is not efficient because the chordal error from the CC path is not directly related to the surface machining error. To eliminated this drawback, Eq. (3) can be modified by d⫽
d2 , 8rt
(5)
where rt is the radius of curvature on the surface (rather than the CC path) in the feed-forward direction. Consequently, the error d computed by Eq. (5) is a deviation error from the surface
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(rather than the CC path). Note that the scallop height is also an error component deviated from the surface. According to Eq. (5), the segment distance for a given deviation error d is: d=√8drt. 3.2. Modification of tool offsetting As has been illustrated by Fig. 4, through the tool offsetting, the practical CC path is different to the scheduled one. Consequently, the practical chordal error dp is different to ds, with which the CC path is segmented. This problem can be eliminated by modifying the tool offsetting algorithm as referred to in Eq. (1). As illustrated in Fig. 6, we can correct the offset CL point so that dp=ds. This can be done by replacing the tool offset radius by r+dr. Accordingly, Eq. (1) is modified as L⫽C⫹(r⫹dr)·sign(nˆ·zˆ)·nˆ,
(6)
where dr is the correction term for the tool radius. Based on the geometric illustration shown in Fig. 6, we can get dr dr⫽ rt−d
(7)
where rt is the radius of surface curvature in the feed-forward direction, and d(=dp=ds) is the assigned chordal deviation error. Substituting Eq. (7) into Eq. (6) yields rrt ·sign(nˆ·zˆ)·nˆ. L⫽C⫹ rt−d
(8)
3.3. Feed-forward segmentation with respect to path-interval segmentation The proposed method for feed-forward segmentation is conducted with regard to the pathinterval segmentation. The idea is to control the surface errors at the concerned points Ci, Cm,
Fig. 6. Modification of the tool offsetting.
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Ci+1, Pi, Pm, and Pi+1), among which the two extreme cases are just corresponding to e=⫺elimit and e=elimit. This is done by adopting a variable chordal error limit in the feed-forward segmentation and moving the CC points away from the surface. In other words, the two control variables are d and s, where s is the correcting amount for the CC points. In the following, the convex and the concave cases are discussed, respectively. For the convex case as referred to in Fig. 5(a) and Eq. (4a), the two extreme cases are ⫺d and h that correspond to Cm and Pi, respectively. If we move the CC point away from the surface by s, the two extreme cases will become
再
e(Cm)=−d+s e(Pi)=h+s
.
(9)
In order to make the two extreme cases equal to ⫺elimit and elimit, respectively, we can have
再
d=2elimit−h . s=elimit−h
(10)
For the concave case as referred to in Fig. 5(b) and Eq. (4b), the two extreme cases are 0 and h+d that correspond to Ci and Pm, respectively. If we move the CC point away from the surface by s, the two extreme cases will become
再
e(Ci)=s e(Pm)=h+d+s
.
(11)
In order to make the two extreme cases equal to ⫺elimit and elimit, respectively, we can have
再
d=2elimit−h s=−elimit
.
(12)
Based on Eq. (10) or Eq. (12), we can obtain the two variables d and s, and then use them to conduct the feed-forward segmentation and the CC point correction, respectively. Note that for both convex and concave cases, the chordal error limit (d) is set as (2elimit⫺h), where the scallop height h is usually varying along the machining path. Besides, the CC point correcting amount (s) can be merged in the tool offsetting algorithm, i.e. replacing the term (r+dr) in Eq. (6) by (r+dr+s). With the proposed method, the surface machining errors are always controlled within the range of [⫺elimit,elimit]. The problem addressed in the above section is removed. Because the chordal-error limit (d) required for the feed-forward segmentation is a function of h, the feed-forward segmentation depends on the result of the path-interval segmentation. If we let h⬍hlimit=elimit in the path-interval segmentation (this follows the conventional methods), then d=2elimit⫺h⬎elimit. Under this situation, the number of required linear segments is always less than that for the conventional methods (for which d=elimit/2). Therefore, the proposed method can provide an efficiency improvement in the feed-forward segmentation.
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As will be discussed in detail in the later section, another important situation introduced by the proposed method is that we can have: hlimit⬎elimit (for conventional methods, hlimit=elimit/2). Under this situation, the number of machining paths required to complete the surface can be reduced because we can utilize a larger path interval. Therefore, the proposed method can provide an efficiency improvement in the path-interval segmentation. 3.4. Determination of path interval As referred to in Eqs. (10) and (12), the two parameters d and s required for the feed-forward segmentation and the CC point correction are functions of elimit and h. The former (elimit) is the desired accuracy for the surface machining and is set as a constant for the tool path planning. The latter (h) depends on the sort of tool path scheduling. The existing methods, for which hⱕ hlimit=elimit is assigned in the determination of the tool path interval, have been proven inaccurate and non-efficient in the previous section. As referred to in Eqs. (10) and (12), d=2elimit⫺h. Geometrically, the only limitation is that the chordal error limit (d) must be greater than zero. Consequently, we have: hⱕhlimit⬍2elimit. This means that the proposed method can adopt a scallop height limit that is close to twice the accuracy limit, elimit. According to Eq. (2), we have: l⬀√h. Consequently, we can increase the path interval by 40% (=√2⫺1). This corresponds to a 40% reduction of the number of tool paths required to complete the surface. However, under this condition a small chordal-error limit (d) is needed in the feed-forward segmentation. Therefore, there is a compromise between the efficiency improvement in the feed-forward segmentation and the path-interval segmentation. Note that the efficiency in the feed-forward segmentation is related to the number of linear segments required to complete the surface. In practice, it is related to the size of the machining program, rather than the total distance (or the machining time) for the machining paths. In contrast, the efficiency of the path-interval segmentation is related to the total distance for the machining paths. It is obvious that the efficiencies in both segmentation directions depends on the selection of the scallop-height limit, hlimit. The effects of hlimit will be further evaluated in the later simulation examples. After the scallop-height limit (hlimit) is determined, we can conduct the path-interval segmentation based on the assigned machining way (iso-parametric, iso-planar, or iso-scallop machining). Note that along the machining path, h=hlimit for the iso-scallop method and hⱕhlimit for the isoparametric and iso-planer methods. 3.5. Computation of scallop height In the feed-forward segmentation, Eq. (10) or Eq. (12) can be applied for every CC segment. Before the evaluation of every CC point, the current value for the scallop height (h) must be calculated. For the iso-scallop method, the computation is not needed, because h=hlimit. For the iso-parametric and iso-planar methods, we can utilize Eq. (2). However, we need the conversion algorithms that convert the path-interval distance (l) to the parametric interval (⌬v, if u is the parameter along the machining path) for the parametric method or to the plane interval (⌬m) for the iso-planar method. The conversion algorithm for l and ⌬v can be obtained from the geometric illustration in Fig.
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Fig. 7. Geometric relationship between l and ⌬v.
7. In Fig. 7, t is the unit vector for the machining direction and n is the unit normal vector to the surface. Then, the unit vector for the path-interval direction is n×t. Based on the geometrical illustration in Fig. 7, we can obtain l⫽(n⫻t)·
∂S ⌬v. ∂v
(13)
The conversion algorithm for l and ⌬m can be obtained from the geometrical illustration in Fig. 8. In Fig. 8, m is the unit vector perpendicular to the parallel planes on which the machining paths are located. Based on the geometrical illustration in Fig. 8, we can obtain ⌬m . l⫽ m·(n×t)
(14)
Fig. 8.
Geometric relationship between l and ⌬m.
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3.6. Tool-path generation procedure Based on the algorithms stated above, the procedure for the tool-path generation is sketched below. The first step is to assign the surface tolerance limit elimit and the scallop height limit hlimit (for which the domain is: elimitⱕhlimit⬍2elimit). elimit is determined based on the requirement of the surface accuracy (e.g. elimit=0.01 mm). hlimit is determined based on the trade-off between the efficiencies of the path-interval segmentation and the feed-forward segmentation. If we want to minimize the number of total linear segments (or size of machining program) required to complete the surface, hlimit should be close to elimit. If we want to minimize the total machining length (or machining time) required to complete the surface, hlimit should be close to 2elimit. The second step is to schedule the machining paths based on the path-interval segmentation. The ways for path-interval segmentation include iso-parametric, iso-planar, and iso-scallop methods. For the iso-parametric method, we generate the CC paths by incrementing the parametric interval ⌬v (if u is the path variable). For the iso-planar method, we generate the CC paths by locating them on parallel vertical planes for which the plane normal and interval are m and ⌬m. For iso-parametric and iso-planar methods, the maximum value of the scallop height (h) on every CC path cannot exceed hlimit. For the iso-scallop method, the path-interval distance (l) is variable along a CC path so that h=hlimit. The detailed algorithms for the three path-interval segmentation methods can be found in existing literature or our previous research work [9,10]. The third step is to segment the CC path and correct the CC points along the path. In the generation of an iso-parametric (or the iso-planar) path, ⌬v (or ⌬m) is given. Based on Eq. (13) or Eq. (14), we can obtain l. Then, we can substitute l into Eq. (2) and obtain the current scallop height, h. Based on the calculated value of h, we can obtain the two parameters, d and s, and then conduct the feed-forward segmentation and the CC point correction, respectively. The final step is to conduct the tool offsetting so as to obtain the CL points along the tool path. Note that the tool offsetting is based on the improved algorithm as referred to Eq. (8). By connecting the CL points by a series of linear segments, we obtain the CL path that is to be fed to the CNC machine tool.
4. Examples In this section, examples for three-axis machining of a bicubic spline surface, which is one of the most popular parametric forms in current CAD/CAM applications, are demonstrated. The bicubic spline surface is expressed by
冦
x=[u3u2u1][Bx][v3v2v1]t y=[u3u2u1][By][v3v2v1]t, 0⬍u⬍1, 0⬍v⬍1, 3 2
3 2
z=[u u u1][Bz][v v v1]
t
(15)
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where
[Bx]⫽
冤
−5 −15
−15
45
30 −45
−10
0
15
0 −30
冥 冤
0
45
15 15
10
, [By]⫽
25 −15
−75
90 −45
60 −90
−10
15
冥 冤
0 −15 30
45 −45 15
45
−20
, [Bz]⫽
15 0
−5
90 −90 0
30
−75 10
90 0 −30 0 0
冥
.
45
A ball-end mill, for which r=10 mm, is utilized here. The surface tolerance (elimit) is required to be controlled within 0.01 mm. Iso-parametric, iso-planar, and iso-scallop methods are all considered for the path-interval segmentation. Typical results for the iso-parametric, iso-planar, and iso-scallop machining paths for the above bicubic surface are shown in Fig. 9. For the traditional segmentation method, we need to let h⬍hlimit=elimit/2 and d=elimit/2 so that the surface error will not exceed elimit (=0.01 mm). For the proposed method, we can select hlimit in the range of [elimit,2elimit] and adapt d along the machining path so that the segmentation efficiency can be well controlled. As stated in the above section, the segmentation efficiency of the proposed method in the feed-forward and the path-interval directions is a function of hlimit. In this paper, the efficiency of the feed-forward segmentation is represented by N, which is the number of total linear segments required for the surface machining. The efficiency of the pathinterval segmentation is represented by L, which is the total length for the surface machining paths. For all three (iso-parametric, iso-planar and iso-scallop) machining methods, the efficiency (N and L) of the proposed method (by letting hlimit=elimit, 1.3elimit, 1.5elimit, 1.7elimit, respectively) is evaluated and compared with the traditional method. The results are shown in Fig. 10. As compared with the traditional method, the proposed method provides a significant improvement in the surface machining efficiency. For the proposed method, there exists a compromise between the efficiencies of the feed-forward segmentation and the path-interval segmentation. As shown in Fig. 10, when utilizing a large hlimit, the total machining length (L) is reduced, but the number of linear segments (N) is increased. Note that as hlimit is close to 2elimit, N will go to infinity.
5. Conclusions An improved method for tool-path scheduling for three-axis surface machining has been presented. The concepts of the proposed method include: (1) the surface-error-based segmentation in the feed-forward direction; (2) consideration of the coupled effect of the segmentation in the feed-forward and the path-interval directions; and (3) compensation for the tool location to minimize the surface error and improve the segmentation efficiency. With the proposed method, we can control the surface error within the desired surface machining tolerance. Moreover, we can improve the efficiency in the feed-forward segmentation (by reducing the number of desired linear segments) and the efficiency in the path-interval segmentation (by reducing the total machining length). The compromise between the efficiencies in the feed-forward segmentation and the pathinterval segmentation is based on the selection of the scallop height limit, hlimit.
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Fig. 9. Machining paths for a bicubic surface.
Acknowledgement The author would like to acknowledge the financial support of the National Science Council, Taiwan ROC under grant number: NSC 88-2212-E-035-010.
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Fig. 10. The total machining path length (L) and the number of linear segments (N).
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An improved method for scheduling the tool paths for threeaxis surface machining Chih-Ching Lo a b
a,*
, Rong-Shine Lin
b
Department of Mechanical Engineering, Feng Chia University, Taichung 407, Taiwan, ROC Department of Mechanical Engineering, Chung Cheng University, Chiayi 621, Taiwan, ROC Received 16 December 1999; accepted 13 June 2000
Abstract This paper presents a new tool-path scheduling method to improve the accuracy and efficiency of threeaxis surface machining. The features of the proposed method include: (1) surface-error-based segmentation in the feed-forward direction; (2) consideration for the coupled effect of the segmentation in the feedforward and the path-interval directions; and (3) compensation for the tool location to control the surface error and the segmentation efficiency. In this paper, we consider a ball-end mill for surface machining. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Surface machining; Tool-path scheduling; Ball-end mill
1. Introduction In today’s manufacturing industries, the CAM system schedules the tool paths and feeds them to the CNC machine tool for on-line part machining. One of the most important CAM techniques is to generate the tool paths for machining of parametric or free-form surfaces that are widely utilized to express the shapes of complex parts. For machining of these parametric surfaces, threeaxis CNC milling machines adopting ball-end cutters are frequently utilized [1–3]. The tool-path scheduling for three-axis ball-end milling of a parametric surface includes: (1) segmentation in the path-interval or side-step direction; and (2) segmentation in the machining or feed-forward direction. The existing approaches to determining the tool-path intervals are the iso-parametric [4], the iso-planar [5,6], and the iso-scallop [7,8] methods. Basically, the path interval is computed based on the limit of the scallop height that is an important source for the * Corresponding author. Tel.: +886-445-17250, ext 3504; fax: +886-445-16545. E-mail addresses: [email protected] (C.-C. Lo), [email protected] (R.-S. Lin).
0890-6955/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 0 ) 0 0 0 5 0 - X
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surface machining error. With the determined path interval, a series of machining paths passing over the surface are scheduled. In current surface machining applications, each machining path in the feed-forward direction is further divided into a series of linear or spline segments that are then fed to the CNC machine tools. In this paper, we consider the case of linear segmentation since the linear motion command is the most popular in the CNC machining applications. Linear segmentation in the feed-forward direction is conducted according to the limit of the chordal deviation that is another source for surface machining error [7]. A basic procedure for the linear segmentation is first to compute a set of cutter-contact (CC) points along the machining path (that is located on the surface), and then to compute the corresponding cutter-location (CL) points through tool offsetting. Consequently, the continuously straight lines connecting these CL points are the linear motion commands for the surface machining. In the current approach, the segmentation in the feed-forward direction is independent of the segmentation in the path-interval direction. However, the surface error is a combination of the chordal error due to the feed-forward segmentation and the scallop error due to the path-interval segmentation. Accordingly, to achieve high efficiency and accuracy in tool-path scheduling, we should conduct the feed-forward segmentation with consideration of the effect of the path-interval segmentation. Moreover, the feed-forward segmentation should be conducted based on the surface error, rather than the chordal error considered in the traditional methods. In this paper, we present an improved method for tool-path scheduling, with which the surface error is controlled within the desired tolerance and the segmentation efficiency is maximized. With the proposed method that considers the coupled effects of the segmentation in the feedforward and the path-interval directions, we can improve the efficiency in the feed-forward segmentation (by reducing the number of desired linear segments) and the efficiency in the pathinterval segmentation (by reducing the total machining length). 2. Existing methods and problems As shown in Fig. 1, a series of machining paths over the surface are assigned to cut the surface that is denoted by S(u,v), where u and v are the surface parameters. These machining paths are
Fig. 1. The machining paths on a surface.
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the curves that the cutter-contact (CC) point passes through. The CC point denotes the location at which the cutter edge touches the surface. In practice, the motion commands fed to the CNC machine tools are not the CC paths, but the cutter-location (CL) paths that the tool center passes through. For a ball-end cutter, the tool offsetting that relates the CC point (C) to the CL point (L) can be formulated as L⫽C⫹r· sign(nˆ·zˆ)·nˆ,
(1)
where nˆ is the unit normal vector to the surface, and zˆ is the unit vector along the tool axis (that is set in the positive z-direction in this paper). The typical tool-path scheduling method is first to assign the CC paths one-by-one over the surface, and then to break each CC path into a series of linear segments. The existing methods to determine the CC paths include the iso-parametric, the iso-planar, and the iso-scallop methods. Basically, the path interval or the side step is computed based on the limit of the scallop height which is an important source of the surface error. A geometrical illustration for the scallop height (h) and the path interval (l) is shown in Fig. 2. For a given path interval, the scallop height can be calculated by [7, 9] l2[rs−r] , h⫽ 8rsr
(2)
where rs is the radius of curvature in the path-interval direction. In Eq. (2), rs is defined to be positive for a concave shape and negative for a convex shape. Note that when utilizing the isoparametric and the iso-planar methods, a conservative path interval is determined so that the maximum scallop height all over the CC path will not exceed an assigned limit (e.g. hⱕ0.01 mm). When utilizing the iso-scallop method, the scallop height is maintained constant along the CC path. The linear segmentation in the machining or feed-forward direction is conducted based on the limit of the chordal deviation that is another source of the surface machining error. For a given incremental length (d) in the feed-forward direction, the chordal error (d) can be calculated by d⫽
d2 , 8rc
(3)
where rc is the radius of curvature on the CC path, and d=|Ci+1⫺Ci|, where Ci+1 and Ci are two
Fig. 2. Path interval and scallop height.
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Fig. 3. Linear segmentation and chordal error.
consecutive CC points (please refer to Fig. 3). Based on Eq. (3), the existing tool-path planning methods break a CC path into a series of linear segments so that the chordal error will not exceed an assigned limit (e.g. d⬍0.01 mm). A drawback of the current methods for the feed-forward segmentation is that the chordal error does not have a strong relation with the surface error. As illustrated in Fig. 3, D is a point on the linear segment CiCi+1 and corresponds to the maximum deviation error (or chordal error) from the CC path. In practice, D may still be very close to the surface, and then corresponds to a very small surface error. Therefore, the current methods with chordal-error-based segmentation are not efficient approaches. However, as will be addressed in a later section, we can easily delete this problem by revising Eq. (3), for which rc is replaced by rf, where rf is the radius of curvature on the surface (rather than the CC path) in the feed-forward direction. Another drawback of the current methods is that an additional error component due to the tool offsetting is not considered. A typical result for feed-forward segmentation is shown in Fig. 4. The current segmentation procedure first takes a series of CC points along the CC path (based on the scheduled chordal error limitation, ds), then computes the correspondent CL points (based
Fig. 4. Additional error component due to the tool offsetting.
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on tool offsetting), and then connect these CL points with continuously linear segments. As can be seen in Fig. 4, if the tool follows the CL segmented path, the practical CC path will not follow the scheduled CC path. Consequently, the practical chordal error becomes dp. Note that dp depends on not only ds but also the surface curvature that is variable along the CC path. As will be addressed in a later section, we can delete this problem by modifying the tool offsetting algorithm, Eq. (1). The third and most important drawback for the current methods is that the feed-forward segmentation is independent of the path-interval segmentation. In practice, the resultant surface error is a combination of the segmentation in the path-interval and the feed-forward directions. In the following, we first investigate the combinative effects of the feed-forward and the path-interval segmentation, and then construct the basis for the method presented in a later section. As shown in Fig. 5, Ci and Ci+1 are two consecutive CC points for a specific linear segment. On the CC segment, Cm is the middle point that corresponds to the chordal error d. Along the machining path, Pi, Pi+1 and Pm are the peaks of the scallop corresponding to Ci, Ci+1 and Cm, respectively. Let e be a vector representing the deviation errors from the surface at the concerned points, Ci, Cm, Ci+1, Pi, Pm and Pi+1. Based on the geometrical illustration in Fig. 5, we have
Fig. 5. Surface errors due to the path-interval and feed-forward segmentation.
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e(Ci,Cm,Ci+1,Pi,Pm,Pi+1)⫽(0,⫺d,0,h,h⫺d,h),
(4a)
if the surface is convex in the feed-forward direction, and e(Ci,Cm,Ci+1,Pi,Pm,Pi+1)⫽(0,d,0,h,h⫹d,h),
(4b)
if the surface is concave in the feed-forward direction. For iso-parametric and iso-planar machining methods, h is not constant and hⱕhlimit, where hlimit is the assigned limit for the scallop height. In contrast, h=hlimit for the iso-scallop machining method. In these current methods, d is set as a constant (i.e. d=dlimit) for all linear segments. In the current machining planning methods, dlimit and hlimit are usually set as the same value, i.e. dlimit=hlimit=elimit, where elimit is the allowable tolerance for the surface machining. Consequently, for the convex case as referred to in Eq. (4a), the maximum deviation error is equal to ⫺elimit and is located at Cm. For the concave case as referred to in Eq. (4b), the maximum surface error is equal to h+elimit(ⱕ2elimit) and is located at Pm. As can be seen, the surface error is not well controlled because it may exceed the assigned tolerance, elimit. Besides, the range of the surface error is varying from one machining segment to another (note that ⫺elimitⱕeⱕh for the convex case and elimitⱕeⱕh+elimit for the concave case, where h is variable for non-isoscallop machining). As stated above, the maximum surface error caused by the traditional methods (when letting dlimit=hlimit=elimit) may be close to 2elimit, and we need to select conservative values for dlimit and hlimit so that the surface error is still smaller than elimit. This can be done by letting dlimit=hlimit=elimit/2. However, the segmentation efficiency in the feed-forward and the path-interval directions is degraded. In principle, (according to Eqs. (2) and (3), d⬀√d and l⬀√h), the feedforward incremental length (d) and the path-interval length (l) will become 1/√2 of the original value. Consequently, the total machining length may be increased by 40% (=√2⫺1) and the number of total linear segments may be increased by 100% (=√2×√2⫺1). In the following, an improved tool-path scheduling method is proposed to cope with the above problem. In the proposed method, the feed-forward segmentation is conducted with regard to the path-interval segmentation so that the surface error for every machining strip is always controlled within the range of [⫺elimit,elimit]. Moreover, we can improve the segmentation efficiency in both the feed-forward and the path-interval directions.
3. Proposed method 3.1. Surface-error-based segmentation As has been illustrated above, the feed-forward segmentation based on Eq. (3) is not efficient because the chordal error from the CC path is not directly related to the surface machining error. To eliminated this drawback, Eq. (3) can be modified by d⫽
d2 , 8rt
(5)
where rt is the radius of curvature on the surface (rather than the CC path) in the feed-forward direction. Consequently, the error d computed by Eq. (5) is a deviation error from the surface
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(rather than the CC path). Note that the scallop height is also an error component deviated from the surface. According to Eq. (5), the segment distance for a given deviation error d is: d=√8drt. 3.2. Modification of tool offsetting As has been illustrated by Fig. 4, through the tool offsetting, the practical CC path is different to the scheduled one. Consequently, the practical chordal error dp is different to ds, with which the CC path is segmented. This problem can be eliminated by modifying the tool offsetting algorithm as referred to in Eq. (1). As illustrated in Fig. 6, we can correct the offset CL point so that dp=ds. This can be done by replacing the tool offset radius by r+dr. Accordingly, Eq. (1) is modified as L⫽C⫹(r⫹dr)·sign(nˆ·zˆ)·nˆ,
(6)
where dr is the correction term for the tool radius. Based on the geometric illustration shown in Fig. 6, we can get dr dr⫽ rt−d
(7)
where rt is the radius of surface curvature in the feed-forward direction, and d(=dp=ds) is the assigned chordal deviation error. Substituting Eq. (7) into Eq. (6) yields rrt ·sign(nˆ·zˆ)·nˆ. L⫽C⫹ rt−d
(8)
3.3. Feed-forward segmentation with respect to path-interval segmentation The proposed method for feed-forward segmentation is conducted with regard to the pathinterval segmentation. The idea is to control the surface errors at the concerned points Ci, Cm,
Fig. 6. Modification of the tool offsetting.
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Ci+1, Pi, Pm, and Pi+1), among which the two extreme cases are just corresponding to e=⫺elimit and e=elimit. This is done by adopting a variable chordal error limit in the feed-forward segmentation and moving the CC points away from the surface. In other words, the two control variables are d and s, where s is the correcting amount for the CC points. In the following, the convex and the concave cases are discussed, respectively. For the convex case as referred to in Fig. 5(a) and Eq. (4a), the two extreme cases are ⫺d and h that correspond to Cm and Pi, respectively. If we move the CC point away from the surface by s, the two extreme cases will become
再
e(Cm)=−d+s e(Pi)=h+s
.
(9)
In order to make the two extreme cases equal to ⫺elimit and elimit, respectively, we can have
再
d=2elimit−h . s=elimit−h
(10)
For the concave case as referred to in Fig. 5(b) and Eq. (4b), the two extreme cases are 0 and h+d that correspond to Ci and Pm, respectively. If we move the CC point away from the surface by s, the two extreme cases will become
再
e(Ci)=s e(Pm)=h+d+s
.
(11)
In order to make the two extreme cases equal to ⫺elimit and elimit, respectively, we can have
再
d=2elimit−h s=−elimit
.
(12)
Based on Eq. (10) or Eq. (12), we can obtain the two variables d and s, and then use them to conduct the feed-forward segmentation and the CC point correction, respectively. Note that for both convex and concave cases, the chordal error limit (d) is set as (2elimit⫺h), where the scallop height h is usually varying along the machining path. Besides, the CC point correcting amount (s) can be merged in the tool offsetting algorithm, i.e. replacing the term (r+dr) in Eq. (6) by (r+dr+s). With the proposed method, the surface machining errors are always controlled within the range of [⫺elimit,elimit]. The problem addressed in the above section is removed. Because the chordal-error limit (d) required for the feed-forward segmentation is a function of h, the feed-forward segmentation depends on the result of the path-interval segmentation. If we let h⬍hlimit=elimit in the path-interval segmentation (this follows the conventional methods), then d=2elimit⫺h⬎elimit. Under this situation, the number of required linear segments is always less than that for the conventional methods (for which d=elimit/2). Therefore, the proposed method can provide an efficiency improvement in the feed-forward segmentation.
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As will be discussed in detail in the later section, another important situation introduced by the proposed method is that we can have: hlimit⬎elimit (for conventional methods, hlimit=elimit/2). Under this situation, the number of machining paths required to complete the surface can be reduced because we can utilize a larger path interval. Therefore, the proposed method can provide an efficiency improvement in the path-interval segmentation. 3.4. Determination of path interval As referred to in Eqs. (10) and (12), the two parameters d and s required for the feed-forward segmentation and the CC point correction are functions of elimit and h. The former (elimit) is the desired accuracy for the surface machining and is set as a constant for the tool path planning. The latter (h) depends on the sort of tool path scheduling. The existing methods, for which hⱕ hlimit=elimit is assigned in the determination of the tool path interval, have been proven inaccurate and non-efficient in the previous section. As referred to in Eqs. (10) and (12), d=2elimit⫺h. Geometrically, the only limitation is that the chordal error limit (d) must be greater than zero. Consequently, we have: hⱕhlimit⬍2elimit. This means that the proposed method can adopt a scallop height limit that is close to twice the accuracy limit, elimit. According to Eq. (2), we have: l⬀√h. Consequently, we can increase the path interval by 40% (=√2⫺1). This corresponds to a 40% reduction of the number of tool paths required to complete the surface. However, under this condition a small chordal-error limit (d) is needed in the feed-forward segmentation. Therefore, there is a compromise between the efficiency improvement in the feed-forward segmentation and the path-interval segmentation. Note that the efficiency in the feed-forward segmentation is related to the number of linear segments required to complete the surface. In practice, it is related to the size of the machining program, rather than the total distance (or the machining time) for the machining paths. In contrast, the efficiency of the path-interval segmentation is related to the total distance for the machining paths. It is obvious that the efficiencies in both segmentation directions depends on the selection of the scallop-height limit, hlimit. The effects of hlimit will be further evaluated in the later simulation examples. After the scallop-height limit (hlimit) is determined, we can conduct the path-interval segmentation based on the assigned machining way (iso-parametric, iso-planar, or iso-scallop machining). Note that along the machining path, h=hlimit for the iso-scallop method and hⱕhlimit for the isoparametric and iso-planer methods. 3.5. Computation of scallop height In the feed-forward segmentation, Eq. (10) or Eq. (12) can be applied for every CC segment. Before the evaluation of every CC point, the current value for the scallop height (h) must be calculated. For the iso-scallop method, the computation is not needed, because h=hlimit. For the iso-parametric and iso-planar methods, we can utilize Eq. (2). However, we need the conversion algorithms that convert the path-interval distance (l) to the parametric interval (⌬v, if u is the parameter along the machining path) for the parametric method or to the plane interval (⌬m) for the iso-planar method. The conversion algorithm for l and ⌬v can be obtained from the geometric illustration in Fig.
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Fig. 7. Geometric relationship between l and ⌬v.
7. In Fig. 7, t is the unit vector for the machining direction and n is the unit normal vector to the surface. Then, the unit vector for the path-interval direction is n×t. Based on the geometrical illustration in Fig. 7, we can obtain l⫽(n⫻t)·
∂S ⌬v. ∂v
(13)
The conversion algorithm for l and ⌬m can be obtained from the geometrical illustration in Fig. 8. In Fig. 8, m is the unit vector perpendicular to the parallel planes on which the machining paths are located. Based on the geometrical illustration in Fig. 8, we can obtain ⌬m . l⫽ m·(n×t)
(14)
Fig. 8.
Geometric relationship between l and ⌬m.
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3.6. Tool-path generation procedure Based on the algorithms stated above, the procedure for the tool-path generation is sketched below. The first step is to assign the surface tolerance limit elimit and the scallop height limit hlimit (for which the domain is: elimitⱕhlimit⬍2elimit). elimit is determined based on the requirement of the surface accuracy (e.g. elimit=0.01 mm). hlimit is determined based on the trade-off between the efficiencies of the path-interval segmentation and the feed-forward segmentation. If we want to minimize the number of total linear segments (or size of machining program) required to complete the surface, hlimit should be close to elimit. If we want to minimize the total machining length (or machining time) required to complete the surface, hlimit should be close to 2elimit. The second step is to schedule the machining paths based on the path-interval segmentation. The ways for path-interval segmentation include iso-parametric, iso-planar, and iso-scallop methods. For the iso-parametric method, we generate the CC paths by incrementing the parametric interval ⌬v (if u is the path variable). For the iso-planar method, we generate the CC paths by locating them on parallel vertical planes for which the plane normal and interval are m and ⌬m. For iso-parametric and iso-planar methods, the maximum value of the scallop height (h) on every CC path cannot exceed hlimit. For the iso-scallop method, the path-interval distance (l) is variable along a CC path so that h=hlimit. The detailed algorithms for the three path-interval segmentation methods can be found in existing literature or our previous research work [9,10]. The third step is to segment the CC path and correct the CC points along the path. In the generation of an iso-parametric (or the iso-planar) path, ⌬v (or ⌬m) is given. Based on Eq. (13) or Eq. (14), we can obtain l. Then, we can substitute l into Eq. (2) and obtain the current scallop height, h. Based on the calculated value of h, we can obtain the two parameters, d and s, and then conduct the feed-forward segmentation and the CC point correction, respectively. The final step is to conduct the tool offsetting so as to obtain the CL points along the tool path. Note that the tool offsetting is based on the improved algorithm as referred to Eq. (8). By connecting the CL points by a series of linear segments, we obtain the CL path that is to be fed to the CNC machine tool.
4. Examples In this section, examples for three-axis machining of a bicubic spline surface, which is one of the most popular parametric forms in current CAD/CAM applications, are demonstrated. The bicubic spline surface is expressed by
冦
x=[u3u2u1][Bx][v3v2v1]t y=[u3u2u1][By][v3v2v1]t, 0⬍u⬍1, 0⬍v⬍1, 3 2
3 2
z=[u u u1][Bz][v v v1]
t
(15)
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where
[Bx]⫽
冤
−5 −15
−15
45
30 −45
−10
0
15
0 −30
冥 冤
0
45
15 15
10
, [By]⫽
25 −15
−75
90 −45
60 −90
−10
15
冥 冤
0 −15 30
45 −45 15
45
−20
, [Bz]⫽
15 0
−5
90 −90 0
30
−75 10
90 0 −30 0 0
冥
.
45
A ball-end mill, for which r=10 mm, is utilized here. The surface tolerance (elimit) is required to be controlled within 0.01 mm. Iso-parametric, iso-planar, and iso-scallop methods are all considered for the path-interval segmentation. Typical results for the iso-parametric, iso-planar, and iso-scallop machining paths for the above bicubic surface are shown in Fig. 9. For the traditional segmentation method, we need to let h⬍hlimit=elimit/2 and d=elimit/2 so that the surface error will not exceed elimit (=0.01 mm). For the proposed method, we can select hlimit in the range of [elimit,2elimit] and adapt d along the machining path so that the segmentation efficiency can be well controlled. As stated in the above section, the segmentation efficiency of the proposed method in the feed-forward and the path-interval directions is a function of hlimit. In this paper, the efficiency of the feed-forward segmentation is represented by N, which is the number of total linear segments required for the surface machining. The efficiency of the pathinterval segmentation is represented by L, which is the total length for the surface machining paths. For all three (iso-parametric, iso-planar and iso-scallop) machining methods, the efficiency (N and L) of the proposed method (by letting hlimit=elimit, 1.3elimit, 1.5elimit, 1.7elimit, respectively) is evaluated and compared with the traditional method. The results are shown in Fig. 10. As compared with the traditional method, the proposed method provides a significant improvement in the surface machining efficiency. For the proposed method, there exists a compromise between the efficiencies of the feed-forward segmentation and the path-interval segmentation. As shown in Fig. 10, when utilizing a large hlimit, the total machining length (L) is reduced, but the number of linear segments (N) is increased. Note that as hlimit is close to 2elimit, N will go to infinity.
5. Conclusions An improved method for tool-path scheduling for three-axis surface machining has been presented. The concepts of the proposed method include: (1) the surface-error-based segmentation in the feed-forward direction; (2) consideration of the coupled effect of the segmentation in the feed-forward and the path-interval directions; and (3) compensation for the tool location to minimize the surface error and improve the segmentation efficiency. With the proposed method, we can control the surface error within the desired surface machining tolerance. Moreover, we can improve the efficiency in the feed-forward segmentation (by reducing the number of desired linear segments) and the efficiency in the path-interval segmentation (by reducing the total machining length). The compromise between the efficiencies in the feed-forward segmentation and the pathinterval segmentation is based on the selection of the scallop height limit, hlimit.
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Fig. 9. Machining paths for a bicubic surface.
Acknowledgement The author would like to acknowledge the financial support of the National Science Council, Taiwan ROC under grant number: NSC 88-2212-E-035-010.
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Fig. 10. The total machining path length (L) and the number of linear segments (N).
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