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Machining accuracy improvement in five-axis flank milling of ruled surfaces
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 48 (2008) 914–921 www.elsevier.com/locate/ijmactool
Machining accuracy improvement in five-axis flank milling of ruled surfaces C.H. Chua, W.N. Huanga, Y.Y. Hsub, a
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan, ROC b Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan, ROC Received 21 May 2007; received in revised form 26 October 2007; accepted 30 October 2007 Available online 1 January 2008
Abstract The aim of this study is to develop a new adjustment method for improving machining accuracy of tool path in five-axis flank milling of ruled surfaces. This method considers interpolation sampling time of the five-axis machine tools controller in NC tool path planning. The actual interpolation position and orientation between G01 commands are estimated with the first differential approximation of Taylor expansion. The tool swept volume is modeled using the envelope surface and compared with the design surface to determine the deviation, which corresponds to the machining error induced by the linear interpolation. We propose a feedrate adjustment rule that automatically controls the tool motion at feedrate-sensitive corners based on a bisection method, thus limiting the maximum machining errors and improving the machining accuracy. Experimental cuts are conducted on different ruled surfaces to verify the effectiveness of the proposed method. The result shows that it can enhance the machining quality in five-axis flank milling in both simulation and practical operation. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ruled surface; Five-axis flank milling; Machining accuracy; Interpolation; Envelope surface
1. Introduction Five-axis machining has received much attention in both industry and the research community since 1980s. With two extra rotational degrees of freedom, it offers numerous advantages over three-axis machining, such as higher production rate and fewer setups, and thus is widely used in the manufacture of complex geometries like turbine blade, impeller, and molds. There are two different milling methods in five-axis CNC machining. In point milling, the cutting edges near the end of a tool perform the material removal. In contrast, the side face of a cutter does the machining in flank milling. Users can select an appropriate one according to the workpiece geometry, surface finish, machining time, and cost [1]. Five-axis flank milling is considered more suitable for the part consisting of ruled surfaces. Many studies have focused on improving the tool path planning of the ruled Corresponding author. Tel.: +886 3 5186498; fax: +886 3 5186521.
E-mail address: [email protected] (Y.Y. Hsu). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.10.023
surface machining. Several methods are now available, which automatically adjusts the tool position and orientation for producing good surface quality [2–4]. However, these studies did not consider the effect of the interpolation in NC controller. In particular, when CNC machine tools advance towards high-speed production with a feedrate of 40 m/min, an acceleration of 2 g [5], an interpolation sampling time of approximately 1 ms, and a singleinterpolation straight length of 0.667 mm, overlarge chord errors may occur in the interpolated tool motion and thus deteriorate the ‘‘actual’’ machined quality. This problem is particularly serious along the tool path with a small curvature radius, known as feedrate-sensitive corners. Five-axis flank milling uses the cutter side to perform the machining, and the chord error commonly used in end milling obviously cannot reflect the machining error [6,7]. Therefore, sophisticated geometries have been employed to describe the five-axis tool motion. Among them, the envelope surface was proposed to characterize the trajectory of tool sweeping in this space and the resultant part geometry after the cut. In addition, different error
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
estimation methods were used to define machining errors and establish the machining errors of five-axis flank milling, e.g. the distances from sampling points on the envelope surface to the design surface [8]. Generally speaking, there are three different types of parametric interpolator designs: uniform, speed-controlled, and adaptive feedrate interpolators [9]. Bedi et al. [10] reported that the next interpolation position can be obtained from the fixed curve parameter increment, but it cannot control feedrate because the calculation of interpolation position is time-independent. Shpitalni et al. [11] as well as Yeh and Hsu [12] used first- and second-order numerical approximations, respectively, to reduce the speed fluctuation caused by a uniform interpolator. To obtain a uniform feedrate reference, Wang and Wright [13] proposed a quintic parameter curve for suitable interpolation by re-parameterization of the curve with consideration of the approximate arc length. All these methods were attempted to fix the interpolated feedrate without taking the machining accuracy into account. Yong and Narayanaswami [14] summarized past research and proposed an off-line method that discovers feedrate-sensitive corners in the machining tool path and adjusts the feedrate in the areas. The above literature review shows that little research has been concerned with the machining error in flank milling induced by the interpolation in the NC controller. Therefore, this paper aims to establish a novel adjustment mechanism for the tool path in five-axis flank milling of ruled surfaces. This mechanism considers the interpolation capacity, specifically the sampling time interval, of fiveaxis machine tools in the NC part programming using CAD/CAM. The cutter feedrate is automatically adjusted by estimating the machining errors using an envelope surface to improve the machining accuracy around feedrate-sensitive corners. Experimental cuts are conducted with the adjusted tool path to demonstrate the improvement of the machined surface quality. The result verifies the feasibility of the proposed method as an effective offline path planning tool in five-axis flank milling. 2. Error estimation model
A ruled surface is defined by two boundary curves. The ruled lines, known as rulings, are constructed by connecting two points on the curves at the same parameter value u, as shown in Fig. 1. Such a surface can be expressed as [7] Sðu; vÞ ¼ ð1 vÞC 0 ðuÞ þ vC 1 ðuÞ
Fig. 1. Machining tool path of ruled surfaces.
be readily obtained in this case. N0(ui) vector is determined as the cross-product of the tangent vector T0(ui) at point C0(ui) and the corresponding ruled line vector R(ui). The end-point position of the tool central line TC0(ui) is calculated by shifting a distance of one tool radius along the N0(ui) vector. The other end-point TC1(ui) is determined in the same way. The tool orientation vector is given by Oðui Þ ¼ TC 1 ðui Þ TC 0 ðui Þ,
ðu; vÞ 2 ½0; 12 ,
(1)
where u is the curve parameter of C0(u) and C1(u) curves, v is the parameter of the ruled lines. C0(u) and C1(u) are the two boundary curves of the ruled surface. A simple way of generating the tool motion (position and orientation) in flank milling of a ruled surface is to guide the tool to follow the rulings. The tool locations can
(2)
where ui is the curve parameter at the ith position, TC1(ui) and TC0(ui) are the top and bottom end-point position of the tool central line at ui position, respectively. Generally speaking, tool interference occurs along the tool path generated in this manner [15] unless the three vectors N0(ui), N1(ui), and R(ui) remain in a plane, i.e. the ruling surface becomes developable. Given a curved path C, a CNC controller calculates the parameter ui of interpolation points on the curves by using the first-order derivative approximation of the Taylor expansion [12]: ui ¼ ui1
2.1. Ruled surfaces
915
V i1 Dt þ EðDtÞ2 , jdCðui1 Þ=duj
(3)
where ui is the curve parameter of the ith sampling time, Dt is the sampling time of the controller, Vi1 is the feedrate of the tool to travel from C(ui1) to C(ui) and E(Dt)2 is the first-order approximate error. 2.2. Machining error estimation In order to estimate the machining error, a swept surface is generated for two consecutive tool locations using the envelope surface. Fig. 2 shows the construction process of the envelope surface for a straight end milling cutter. Compared with three-axis machining, five-axis machining allows synchronous motion on five axes, which
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
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Fig. 2. Swept envelope surface.
include not only the translation movement between two points but also the change of tool orientation specified by the rotation axes. Thus, the complete information of any tool location includes six parameters (X, Y, Z, yA, yB, yC). The straight end milling tool profile Tm in the machine coordinate system (Xm, Ym, Zm) can be defined as [6] T m ðu; k; LÞ ¼ TðuÞ þ R cosðkÞ X t þ R sinðkÞ Y t þ L Z t , (4) where L is the height along the tool axis calculated from point P and T(u) represents the tool path with u as the curve parameter. R is the tool radius, kA[0, 2p], and (Xt, Yt, Zt) is the tool coordinate system. The vector O(ui) ¼ (Ox Oy Oz) along the tool axis with respect to the machine coordinate system (Xm, Ym, Zm) in BC-type fiveaxis CNC machining (such as DMG60T/DECKEL MAHO with B-axis and C-axis rotation axis) can be written as ½ Oðui Þ T ¼ RotðZ; hC Þ Rotðy; hB Þ ½ 0
0
1 T
¼ ½ cosðyC Þ sinðyB Þ sinðyC Þ sinðyB Þ
cosðyB Þ T , (5)
where Rot(Z,hC) ¼ [cos(yC) sin(yC) 0, sin(yC) cos(yC) 0; 0 0 1] and Rot(Y,hB) ¼ [cos(yB) 0 sin(yB); 0 1 0; sin(yB) 0 cos(yB)] are rotation matrices. _ i Þ of the tool axis can be The instantaneous velocity Oðu obtained by the cross-product of the angular velocity x of the tool axis and the unit vector of tool orientation: _ i Þ ¼ x Oðui Þ. Oðu
(6)
The instantaneous angular velocity x can be expressed as x ¼ h_ C þ RotðZ; hC Þ h_ B
(7)
Fig. 3. Machining error of ruled surfaces.
The velocity of the any point Q on the tool surface can be thus written as V Q ¼ V P þ x PQ,
(8)
where VP is the tool velocity at the point P. The normal vector N(Q) at point Q on the tool surface in the machine coordinate system is written as NðQÞ ¼ cosðkÞ X t þ sinðkÞ Y t ,
(9)
where Q is defined as the grazing point when VQ N(Q) ¼ 0. Swept curves can be constructed from the grazing points of discrete sampling positions on the tool path. They further construct the envelope surface through standardized geometric operation like lofting. It is necessary to distinguish the estimation of machining error between end milling and flank milling. Chord error is usually used to characterize the former. In contrast, a volumetric measure is more appropriate to estimate the error in flank milling, as the tool side surface contacts the work part in this case (see Fig. 3), i.e. the machining deviation cannot be precisely quantified only using chord error. Therefore, we construct the envelope surface to approximate the swept surface of the tool during spatial motion and use the result as a datum for comparing with the design surface.
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
The estimation of the surface error is the deviation e along the normal vector of the envelope surface E(u, v) to the ruled surface S(u, v) [8]. It is difficult to directly estimate the deviation amount on the free-form representation of a surface. Thus, we calculate it on discrete points on the machined surface. In other words, a set of sample points Pe are generated with the same parameter increment on E(u, v). A line follows the surface normal at each point and intersects with the design surface S(u, v) at some point Ps. The distance between the two points is considered the machining error at Ps. The error between the two surfaces is defined as the maximum deviation among all the distances.
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3. Accuracy control rules The parameter interpolation of feedrate-error control considers the limitation of the sampling capability of NC controller. The machining error of the ruled surface can be adaptively controlled by adjusting the machining feedrate. The adjustment procedure is shown in Fig. 4. It is applied to every G1 tool motion with a given feedrate. The tool position and tool axis are estimated at discrete locations according to the linear Taylor expansion described in Eq. (3). The envelope surface generated by the tool moving from the start position to the interpolation location ui is constructed and the resultant machining error is calculated.
Fig. 4. Feedrate-error control scheme.
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The feedrate needs to be adjusted when the error is greater than the specified tolerance; otherwise, the interpolation position ui and the initial feedrate is recorded. The same procedure then begins with the next interpolation position. The feedrate adjustment is conducted by a bisection method. Each time, the initial velocity Vc is reduced by half until either the error is within the tolerance, or the adjusted feedrate reaches a lower bound specified by the user, which is also determined by the acceleration capability of the NC controller. The reduced feedrate that fulfills the machining tolerance will override Vc and becomes a new interpolation point of the tool motion. The adjustment continues until the interpolation arrives at the end of the G1 command. Other regulation schemes (e.g. gold section method) are applicable, although the bisection method works quite well in our experimentation. 4. Validation and discussion In order to validate the proposed method, we designed a ruled surface, surface_A, consisting of two cubic Be´zier curves (Fig. 5). Surface_A is a developable surface that possesses a nice property of interference-free situation in five-axis flank machining. Such a design allows precise estimation of the machining error induced by the linear
interpolation rather than the tool interference. The feedrate-sensitive corner of surface_A occurs around the curve segment corresponding to a parameter range u ¼ 0.7–0.8. The surface geometry mimics a turbine blade with a portion of large curvature. A total of 100 points are sampled with the same parameter interval within the surface area between the two interpolation tool positions. A five-axis CNC machine tool DMG60T (DECKEL MAHO) with a HeidenhainTM controller iTNC530 is used for the cutting experiment. The sampling time of the controller is 0.0018 s; the tool is a + 6 mm straight-end HSS milling cutter. The cutting parameters include a feedrate at 100 mm/s, a spindle speed of 6000 rpm, and the maximum machining error of 0.005 mm specified by the user. In order to reduce the influence of tool deflection on the machining error, the part is made of phenolic resin, a special epoxy formed by compressing resin powders. This wood-like material is relatively soft compared to the cutting tool but remains integral during the cutting. Fig. 6 shows the finished part. The origin of the workpiece coordinate system is located at the top lefthand corners. In order to minimize the measure error, the cut with the machining accuracy control and the uncontrolled cut are performed on the same part during the same operation and setup. Thus, there are two machined
Fig. 5. Test surfaces A and B.
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
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leaving a uniform layer of stock material on the machined surfaces. The cavity in the middle was produced by tool retraction at the end of the finish cut. Zeiss UMC 850 CMM, which has an accuracy of 73 mm, is used to measure the machining errors in both cuts. Fig. 7 shows the distribution of the machining errors. The result indicates that the errors induced by the adjusted tool paths can be effectively reduced and in most regions, they can be maintained within the given allowance 0.005 mm. In order to illustrate the reproducibility of the method, a second set of experiment is conducted with different surface geometry (surface_B). The cutting condition remains the same as the first one. The part design and the measurement setup are also similar. Fig. 8 shows the finish part. The distribution of the machining error on the machined surface is shown in Fig. 8. The result indicates a significant improvement of the machining precision. The effect is distinct around the surface region with large curvature, i.e. the feedrate-sensitive area. It corresponds to the curve range u ¼ 0.1–0.2 in this case.
5. Conclusions
Fig. 6. Finished parts.
surfaces in the part. The start and end positions of each machined surface are marked in the top half figure. Note that the finish cut is of our major concern. A rough-cut operation has been conducted prior to the experimentation,
In practice, the tool path planning of multi-axis machining is accomplished using CAD/CAM systems, due to its complexity. However, the capability of the machine tools and the resultant machining accuracy are not considered in these systems. To overcome this problem, this study investigates the influence of the sampling time in NC controller on five-axis flank milling of ruled surfaces. Specifically, we are focused on the machining errors induced by linear interpolation between cutter locations. They deteriorate the machining quality particularly in the feedrate-sensitive corners. An error estimation method was proposed that constructs the envelope surface of the real tool motion and compares it with the design surface. The feedrate was adaptively adjusted using a bisection scheme. As a result, the interpolated tool positions as well as orientations produce better surface quality. The effectiveness of this work has been successfully verified in two cutting tests. The measured result of both machined parts by a CMM indicates that the machining error is significantly reduced in the feedrate-sensitive areas on the machined surfaces. The improvement is especially distinct around the surface region with large curvature. Moreover, the adjustment is not noticeable in the developable regions. This work provides an off-line tool for improving the machining precision in five-axis flank milling of ruled surfaces by considering the machine tool capability. It leads to several potential topics in future work. First, an acceleration control scheme may be required due to the varied feedrates in the tool motion. The scheme will involve real-time scheduling issues in NC controller. In addition, the cutting load should be taken into account in the feedrate modification. The machining error might be further reduced by doing so.
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Fig. 7. Error-measuring result of surface_A.
Fig. 8. Error-measuring result of surface_B.
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References [1] H.K. To¨nshoff, C. Gey, N. Rackow, Flank milling optimization—the Flamingo Project, Air and Space Europe 3 (2001) 60–63. [2] S. Bedi, S. Mann, C. Menzel, Flank milling with flat end milling cutters, Computer Aided Design 35 (2003) 293–300. [3] D.M. Tsay, M.J. Her, Accurate 5-axis machining of twisted ruled surfaces, ASME Journal of Manufacturing Science and Engineering 123 (2001) 731–738. [4] D.M. Tsay, H.C. Chen, M.J. Her, A study on five-axis flank machining of centrifugal compressor impellers, ASME Journal of Engineering for Gas Turbines and Power 124 (2002) 177–181. [5] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation, International Journal of Machine Tools & Manufacture 41 (2001) 1323–1345. [6] K. Weinert, S. Du, P. Damm, M. Stautner, Swept volume generation for the simulation of machining processes, International Journal of Machine Tools & Manufacture 44 (2004) 617–628. [7] J.C.J. Chiou, Accuracy tool position for five-axis ruled surface machining by swept envelope approach, Computer Aided Design 36 (2004) 967–974.
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[8] C. Li, S. Mann, S. Bedi, Error measurements for flank milling, Computer Aided Design 37 (2005) 1459–1468. [9] J.J. Chou, D. Yan, Command generation for three-axis CNC machining, ASME Journal of Engineering Industry 113 (1991) 305–310. [10] S. Bedi, I. Ali, N. Quan, Advanced interpolation techniques for CNC machines, ASME Journal of Engineering Industry 115 (1993) 329–336. [11] M. Shpitalni, Y. Koren, C.C. Lo, Real-time curve interpolators, Computer Aided Design 26 (11) (1994) 832–838. [12] S.S. Yeh, P.L. Hsu, The speed-controlled interpolator for machining parametric curves, Computer Aided Design 31 (1999) 349–357. [13] F.C. Wang, P.K. Wright, Open architecture controllers for machine tools. Part 2. A real time quintic interpolator, ASME Journal Manufacture Science and Engineering 120 (2) (1998) 425–432. [14] T. Yong, R. Narayanaswami, A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining, Computer Aided Design 35 (2003) 1249–1259. [15] C.H. Chu, J.T. Chen, Tool path planning for five-axis flank milling with developable surface approximation, International Journal of Advanced Manufacturing Technology 29 (7–8) (2006) 707–713.
International Journal of Machine Tools & Manufacture 48 (2008) 914–921 www.elsevier.com/locate/ijmactool
Machining accuracy improvement in five-axis flank milling of ruled surfaces C.H. Chua, W.N. Huanga, Y.Y. Hsub, a
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan, ROC b Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan, ROC Received 21 May 2007; received in revised form 26 October 2007; accepted 30 October 2007 Available online 1 January 2008
Abstract The aim of this study is to develop a new adjustment method for improving machining accuracy of tool path in five-axis flank milling of ruled surfaces. This method considers interpolation sampling time of the five-axis machine tools controller in NC tool path planning. The actual interpolation position and orientation between G01 commands are estimated with the first differential approximation of Taylor expansion. The tool swept volume is modeled using the envelope surface and compared with the design surface to determine the deviation, which corresponds to the machining error induced by the linear interpolation. We propose a feedrate adjustment rule that automatically controls the tool motion at feedrate-sensitive corners based on a bisection method, thus limiting the maximum machining errors and improving the machining accuracy. Experimental cuts are conducted on different ruled surfaces to verify the effectiveness of the proposed method. The result shows that it can enhance the machining quality in five-axis flank milling in both simulation and practical operation. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ruled surface; Five-axis flank milling; Machining accuracy; Interpolation; Envelope surface
1. Introduction Five-axis machining has received much attention in both industry and the research community since 1980s. With two extra rotational degrees of freedom, it offers numerous advantages over three-axis machining, such as higher production rate and fewer setups, and thus is widely used in the manufacture of complex geometries like turbine blade, impeller, and molds. There are two different milling methods in five-axis CNC machining. In point milling, the cutting edges near the end of a tool perform the material removal. In contrast, the side face of a cutter does the machining in flank milling. Users can select an appropriate one according to the workpiece geometry, surface finish, machining time, and cost [1]. Five-axis flank milling is considered more suitable for the part consisting of ruled surfaces. Many studies have focused on improving the tool path planning of the ruled Corresponding author. Tel.: +886 3 5186498; fax: +886 3 5186521.
E-mail address: [email protected] (Y.Y. Hsu). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.10.023
surface machining. Several methods are now available, which automatically adjusts the tool position and orientation for producing good surface quality [2–4]. However, these studies did not consider the effect of the interpolation in NC controller. In particular, when CNC machine tools advance towards high-speed production with a feedrate of 40 m/min, an acceleration of 2 g [5], an interpolation sampling time of approximately 1 ms, and a singleinterpolation straight length of 0.667 mm, overlarge chord errors may occur in the interpolated tool motion and thus deteriorate the ‘‘actual’’ machined quality. This problem is particularly serious along the tool path with a small curvature radius, known as feedrate-sensitive corners. Five-axis flank milling uses the cutter side to perform the machining, and the chord error commonly used in end milling obviously cannot reflect the machining error [6,7]. Therefore, sophisticated geometries have been employed to describe the five-axis tool motion. Among them, the envelope surface was proposed to characterize the trajectory of tool sweeping in this space and the resultant part geometry after the cut. In addition, different error
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
estimation methods were used to define machining errors and establish the machining errors of five-axis flank milling, e.g. the distances from sampling points on the envelope surface to the design surface [8]. Generally speaking, there are three different types of parametric interpolator designs: uniform, speed-controlled, and adaptive feedrate interpolators [9]. Bedi et al. [10] reported that the next interpolation position can be obtained from the fixed curve parameter increment, but it cannot control feedrate because the calculation of interpolation position is time-independent. Shpitalni et al. [11] as well as Yeh and Hsu [12] used first- and second-order numerical approximations, respectively, to reduce the speed fluctuation caused by a uniform interpolator. To obtain a uniform feedrate reference, Wang and Wright [13] proposed a quintic parameter curve for suitable interpolation by re-parameterization of the curve with consideration of the approximate arc length. All these methods were attempted to fix the interpolated feedrate without taking the machining accuracy into account. Yong and Narayanaswami [14] summarized past research and proposed an off-line method that discovers feedrate-sensitive corners in the machining tool path and adjusts the feedrate in the areas. The above literature review shows that little research has been concerned with the machining error in flank milling induced by the interpolation in the NC controller. Therefore, this paper aims to establish a novel adjustment mechanism for the tool path in five-axis flank milling of ruled surfaces. This mechanism considers the interpolation capacity, specifically the sampling time interval, of fiveaxis machine tools in the NC part programming using CAD/CAM. The cutter feedrate is automatically adjusted by estimating the machining errors using an envelope surface to improve the machining accuracy around feedrate-sensitive corners. Experimental cuts are conducted with the adjusted tool path to demonstrate the improvement of the machined surface quality. The result verifies the feasibility of the proposed method as an effective offline path planning tool in five-axis flank milling. 2. Error estimation model
A ruled surface is defined by two boundary curves. The ruled lines, known as rulings, are constructed by connecting two points on the curves at the same parameter value u, as shown in Fig. 1. Such a surface can be expressed as [7] Sðu; vÞ ¼ ð1 vÞC 0 ðuÞ þ vC 1 ðuÞ
Fig. 1. Machining tool path of ruled surfaces.
be readily obtained in this case. N0(ui) vector is determined as the cross-product of the tangent vector T0(ui) at point C0(ui) and the corresponding ruled line vector R(ui). The end-point position of the tool central line TC0(ui) is calculated by shifting a distance of one tool radius along the N0(ui) vector. The other end-point TC1(ui) is determined in the same way. The tool orientation vector is given by Oðui Þ ¼ TC 1 ðui Þ TC 0 ðui Þ,
ðu; vÞ 2 ½0; 12 ,
(1)
where u is the curve parameter of C0(u) and C1(u) curves, v is the parameter of the ruled lines. C0(u) and C1(u) are the two boundary curves of the ruled surface. A simple way of generating the tool motion (position and orientation) in flank milling of a ruled surface is to guide the tool to follow the rulings. The tool locations can
(2)
where ui is the curve parameter at the ith position, TC1(ui) and TC0(ui) are the top and bottom end-point position of the tool central line at ui position, respectively. Generally speaking, tool interference occurs along the tool path generated in this manner [15] unless the three vectors N0(ui), N1(ui), and R(ui) remain in a plane, i.e. the ruling surface becomes developable. Given a curved path C, a CNC controller calculates the parameter ui of interpolation points on the curves by using the first-order derivative approximation of the Taylor expansion [12]: ui ¼ ui1
2.1. Ruled surfaces
915
V i1 Dt þ EðDtÞ2 , jdCðui1 Þ=duj
(3)
where ui is the curve parameter of the ith sampling time, Dt is the sampling time of the controller, Vi1 is the feedrate of the tool to travel from C(ui1) to C(ui) and E(Dt)2 is the first-order approximate error. 2.2. Machining error estimation In order to estimate the machining error, a swept surface is generated for two consecutive tool locations using the envelope surface. Fig. 2 shows the construction process of the envelope surface for a straight end milling cutter. Compared with three-axis machining, five-axis machining allows synchronous motion on five axes, which
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
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Fig. 2. Swept envelope surface.
include not only the translation movement between two points but also the change of tool orientation specified by the rotation axes. Thus, the complete information of any tool location includes six parameters (X, Y, Z, yA, yB, yC). The straight end milling tool profile Tm in the machine coordinate system (Xm, Ym, Zm) can be defined as [6] T m ðu; k; LÞ ¼ TðuÞ þ R cosðkÞ X t þ R sinðkÞ Y t þ L Z t , (4) where L is the height along the tool axis calculated from point P and T(u) represents the tool path with u as the curve parameter. R is the tool radius, kA[0, 2p], and (Xt, Yt, Zt) is the tool coordinate system. The vector O(ui) ¼ (Ox Oy Oz) along the tool axis with respect to the machine coordinate system (Xm, Ym, Zm) in BC-type fiveaxis CNC machining (such as DMG60T/DECKEL MAHO with B-axis and C-axis rotation axis) can be written as ½ Oðui Þ T ¼ RotðZ; hC Þ Rotðy; hB Þ ½ 0
0
1 T
¼ ½ cosðyC Þ sinðyB Þ sinðyC Þ sinðyB Þ
cosðyB Þ T , (5)
where Rot(Z,hC) ¼ [cos(yC) sin(yC) 0, sin(yC) cos(yC) 0; 0 0 1] and Rot(Y,hB) ¼ [cos(yB) 0 sin(yB); 0 1 0; sin(yB) 0 cos(yB)] are rotation matrices. _ i Þ of the tool axis can be The instantaneous velocity Oðu obtained by the cross-product of the angular velocity x of the tool axis and the unit vector of tool orientation: _ i Þ ¼ x Oðui Þ. Oðu
(6)
The instantaneous angular velocity x can be expressed as x ¼ h_ C þ RotðZ; hC Þ h_ B
(7)
Fig. 3. Machining error of ruled surfaces.
The velocity of the any point Q on the tool surface can be thus written as V Q ¼ V P þ x PQ,
(8)
where VP is the tool velocity at the point P. The normal vector N(Q) at point Q on the tool surface in the machine coordinate system is written as NðQÞ ¼ cosðkÞ X t þ sinðkÞ Y t ,
(9)
where Q is defined as the grazing point when VQ N(Q) ¼ 0. Swept curves can be constructed from the grazing points of discrete sampling positions on the tool path. They further construct the envelope surface through standardized geometric operation like lofting. It is necessary to distinguish the estimation of machining error between end milling and flank milling. Chord error is usually used to characterize the former. In contrast, a volumetric measure is more appropriate to estimate the error in flank milling, as the tool side surface contacts the work part in this case (see Fig. 3), i.e. the machining deviation cannot be precisely quantified only using chord error. Therefore, we construct the envelope surface to approximate the swept surface of the tool during spatial motion and use the result as a datum for comparing with the design surface.
ARTICLE IN PRESS C.H. Chu et al. / International Journal of Machine Tools & Manufacture 48 (2008) 914–921
The estimation of the surface error is the deviation e along the normal vector of the envelope surface E(u, v) to the ruled surface S(u, v) [8]. It is difficult to directly estimate the deviation amount on the free-form representation of a surface. Thus, we calculate it on discrete points on the machined surface. In other words, a set of sample points Pe are generated with the same parameter increment on E(u, v). A line follows the surface normal at each point and intersects with the design surface S(u, v) at some point Ps. The distance between the two points is considered the machining error at Ps. The error between the two surfaces is defined as the maximum deviation among all the distances.
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3. Accuracy control rules The parameter interpolation of feedrate-error control considers the limitation of the sampling capability of NC controller. The machining error of the ruled surface can be adaptively controlled by adjusting the machining feedrate. The adjustment procedure is shown in Fig. 4. It is applied to every G1 tool motion with a given feedrate. The tool position and tool axis are estimated at discrete locations according to the linear Taylor expansion described in Eq. (3). The envelope surface generated by the tool moving from the start position to the interpolation location ui is constructed and the resultant machining error is calculated.
Fig. 4. Feedrate-error control scheme.
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The feedrate needs to be adjusted when the error is greater than the specified tolerance; otherwise, the interpolation position ui and the initial feedrate is recorded. The same procedure then begins with the next interpolation position. The feedrate adjustment is conducted by a bisection method. Each time, the initial velocity Vc is reduced by half until either the error is within the tolerance, or the adjusted feedrate reaches a lower bound specified by the user, which is also determined by the acceleration capability of the NC controller. The reduced feedrate that fulfills the machining tolerance will override Vc and becomes a new interpolation point of the tool motion. The adjustment continues until the interpolation arrives at the end of the G1 command. Other regulation schemes (e.g. gold section method) are applicable, although the bisection method works quite well in our experimentation. 4. Validation and discussion In order to validate the proposed method, we designed a ruled surface, surface_A, consisting of two cubic Be´zier curves (Fig. 5). Surface_A is a developable surface that possesses a nice property of interference-free situation in five-axis flank machining. Such a design allows precise estimation of the machining error induced by the linear
interpolation rather than the tool interference. The feedrate-sensitive corner of surface_A occurs around the curve segment corresponding to a parameter range u ¼ 0.7–0.8. The surface geometry mimics a turbine blade with a portion of large curvature. A total of 100 points are sampled with the same parameter interval within the surface area between the two interpolation tool positions. A five-axis CNC machine tool DMG60T (DECKEL MAHO) with a HeidenhainTM controller iTNC530 is used for the cutting experiment. The sampling time of the controller is 0.0018 s; the tool is a + 6 mm straight-end HSS milling cutter. The cutting parameters include a feedrate at 100 mm/s, a spindle speed of 6000 rpm, and the maximum machining error of 0.005 mm specified by the user. In order to reduce the influence of tool deflection on the machining error, the part is made of phenolic resin, a special epoxy formed by compressing resin powders. This wood-like material is relatively soft compared to the cutting tool but remains integral during the cutting. Fig. 6 shows the finished part. The origin of the workpiece coordinate system is located at the top lefthand corners. In order to minimize the measure error, the cut with the machining accuracy control and the uncontrolled cut are performed on the same part during the same operation and setup. Thus, there are two machined
Fig. 5. Test surfaces A and B.
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leaving a uniform layer of stock material on the machined surfaces. The cavity in the middle was produced by tool retraction at the end of the finish cut. Zeiss UMC 850 CMM, which has an accuracy of 73 mm, is used to measure the machining errors in both cuts. Fig. 7 shows the distribution of the machining errors. The result indicates that the errors induced by the adjusted tool paths can be effectively reduced and in most regions, they can be maintained within the given allowance 0.005 mm. In order to illustrate the reproducibility of the method, a second set of experiment is conducted with different surface geometry (surface_B). The cutting condition remains the same as the first one. The part design and the measurement setup are also similar. Fig. 8 shows the finish part. The distribution of the machining error on the machined surface is shown in Fig. 8. The result indicates a significant improvement of the machining precision. The effect is distinct around the surface region with large curvature, i.e. the feedrate-sensitive area. It corresponds to the curve range u ¼ 0.1–0.2 in this case.
5. Conclusions
Fig. 6. Finished parts.
surfaces in the part. The start and end positions of each machined surface are marked in the top half figure. Note that the finish cut is of our major concern. A rough-cut operation has been conducted prior to the experimentation,
In practice, the tool path planning of multi-axis machining is accomplished using CAD/CAM systems, due to its complexity. However, the capability of the machine tools and the resultant machining accuracy are not considered in these systems. To overcome this problem, this study investigates the influence of the sampling time in NC controller on five-axis flank milling of ruled surfaces. Specifically, we are focused on the machining errors induced by linear interpolation between cutter locations. They deteriorate the machining quality particularly in the feedrate-sensitive corners. An error estimation method was proposed that constructs the envelope surface of the real tool motion and compares it with the design surface. The feedrate was adaptively adjusted using a bisection scheme. As a result, the interpolated tool positions as well as orientations produce better surface quality. The effectiveness of this work has been successfully verified in two cutting tests. The measured result of both machined parts by a CMM indicates that the machining error is significantly reduced in the feedrate-sensitive areas on the machined surfaces. The improvement is especially distinct around the surface region with large curvature. Moreover, the adjustment is not noticeable in the developable regions. This work provides an off-line tool for improving the machining precision in five-axis flank milling of ruled surfaces by considering the machine tool capability. It leads to several potential topics in future work. First, an acceleration control scheme may be required due to the varied feedrates in the tool motion. The scheme will involve real-time scheduling issues in NC controller. In addition, the cutting load should be taken into account in the feedrate modification. The machining error might be further reduced by doing so.
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Fig. 7. Error-measuring result of surface_A.
Fig. 8. Error-measuring result of surface_B.
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