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Simultaneous optimization of tool path and shape for five-axis flank milling
Computer-Aided Design 44 (2012) 1229–1234
Contents lists available at SciVerse ScienceDirect
Computer-Aided Design journal homepage: www.elsevier.com/locate/cad
Technical note
Simultaneous optimization of tool path and shape for five-axis flank milling LiMin Zhu a,∗ , Han Ding a , YouLun Xiong b a
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China
b
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, PR China
article
info
Article history: Received 22 May 2011 Accepted 1 June 2012 Keywords: Rotary cutter Flank milling Tool path Tool shape Distance function
abstract By representing the swept envelope of a generic rotary cutter as a sphere-swept surface, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization of the tool path and shape for five-axis flank milling. If the tool path is fixed, a novel tool shape optimization method is obtained. If the tool shape is fixed, a tool path optimization method applicable to any rotary cutter is obtained. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Numerical examples are given to confirm its validity. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Flank milling is performed by employing the side of a cutter to touch the desired surface. Compared with point milling, flank milling has its unique advantages. It can increase the material removal rate, lower the cutting forces, eliminate necessary hand finish, ensure improved component accuracy and result in longer tool life. Thus it offers a good choice for machining slender parts, like turbine blades and impellers. Recently, increasing attention was drawn onto the problem of optimum positioning of the cutter for flank milling. In most of the works, a tool path is represented by a huge data set of cutter locations (CLs), and each CL is generated to eliminate or reduce the local interference between the cutter and workpiece. However, the machined surface is formed by the swept envelope of the cutter surface. The true machining errors are the deviations between the design surface and cutter envelope surface. It is well known that the shape of the cutter envelope surface can not be completely determined unless all the CLs are obtained. So, only a few works addressed the cutter positioning problem from the perspective of approximation of the cutter envelope surface to design surface. Chiou [1] described a swept envelope-based method for tool positioning. The initial cutter positions were located to contact with two directrices. Then the swept profile of the cutter was calculated based on the cutter motion. Finally the cutter locations were adjusted to reduce the machining errors by comparing the swept profile with the designed ruled surface. By considering the envelope surface, Senatore et al. [2] analyzed the performance of
∗
Corresponding author. Tel.: +86 2134206545. E-mail address: [email protected] (L.M. Zhu).
0010-4485/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2012.06.003
an improved positioning method for flank milling of ruled surfaces with cylindrical cutters. Lartigue et al. [3] proposed a method to deform the two curves that define the tool axis trajectory so that the tool envelope surface fitted the design surface as much as possible. The geometric deviation between the two surfaces was evaluated by the sum of the squared distances of the points on the design surface to the envelope surface. To simplify the computation, an approximate distance measure was employed. Pechard et al. [4] stated that control of the trajectory smoothness was as essential as control of the geometrical deviations, and developed a method that aims at minimizing the geometrical deviations between the tool envelope surface and design surface while preserving the trajectory smoothness. For cylindrical cutters, Gong et al. [5] presented the error propagation principle, and formulated the problem of tool path optimization as that of least squares (LS) fitting of the tool envelope surface to the point cloud on the offset surface of the design surface. Later, Gong and Wang [6] extended this idea to deal with generic rotary cutters, and determined the optimum CL by LS fitting of a spatial line to a series of post-processed point data. Although the LS method was easy for implementation and efficient in computation, it did not conform to the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation [7,8], which requires the maximum norm of the error vector be minimized. More importantly, the geometric deviation of the machined surface from the nominal one was not clearly defined and the influence of the change of the tool axis trajectory on the change of this deviation was not quantitatively analyzed. Based on the differential properties of the signed point-to-surface distance function [9,10], we developed the model and algorithm for tool path planning from the perspective of surface approximation following the minimum zone criterion. The envelope surface of
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L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
a
b
Fig. 1. Geometry of a generic rotary cutter.
the cutter was of no concern in the model due to the fact that the envelope surface of a cylindrical cutter is the offset surface of the tool axis trajectory surface. Later, the approach was extended to handle conical cutters by representing the cutter envelope surface as a sphere-swept surface [11]. All the above methods focused on optimizing the tool positions with the tool shape being fixed. Chaves-Jacob et al. [12] proposed a novel approach that optimizes the tool shape for a given trajectory–surface pair to reduce the interferences. It offers a new degree of freedom to improve the machining accuracy. Monies et al. [13] introduced an algorithm to calculate the error between the conical cutter and the workpiece so one can determine the optimal cutter dimensions (cone radius and angle). This is a special case of tool shape optimization. In this note, we will extend our previous works [10,11] to develop a method that can optimize simultaneously the tool path and shape for five-axis flank milling. If the tool path is fixed, it becomes a method of tool shape optimization after implementation of a commercial CAM software to position the tool. If the tool shape is fixed, it becomes a tool path optimization method applicable to any rotary cutters. It also applies to the problems of cutter dimension optimization [13] and flank millable surface design [14]. The remainder of this paper is organized as follows. In Sections 2 and 3, the swept envelope of a rotary milling cutter is modeled as a sphere-swept surface. In Section 4, the properties of the signed distance between the cutter swept envelope and a point are presented. In Section 5, simultaneous optimization of the tool path and shape for five-axis flank milling is formulated as a minimax problem. Numerical examples are given in Section 6, and conclusions are summarized in Section 7.
its control points can be viewed as the shape parameters. Then, we have x′ (a) tan γ (a) = ′ z ( a) cos γ (a) =
z ′ (a) x′ (a)2
+
z ′ (a)2
c (u; a) = [0,
=
0,
0,
z (a)]T ,
a ∈ [a0 , a1 ]
where u = [u1 , . . . , um ] ∈ R denotes the collection of the shape parameters of the generatrix. If the generatrix is a B-spline curve, m
r (a)
.
0,
0,
r (a) sin γ (a) + z (a)]T x(a)x′ (a) + z (a)z ′ (a)
T
z ′ ( a)
x(a) x′ (a)2 + z ′ (a)2
(1)
r (u; a) =
z ′ (a)
.
(2)
Remark. To represent a surface of revolution as a canal surface, the planar generatrix should be smooth enough, and its radius of curvature should be large enough. For the practical rotary cutters used for flank milling, these conditions are always satisfied. 3. Representation of cutter swept envelope as sphere-swept surface As illustrated in Fig. 2, the tool motion is usually represented by two guiding curves C1 (t ) and C2 (t ). They determine the following tool axis trajectory surface Saxis (v ; l, t ) = C1 (t ) + l
g (u; a) = [x(a),
x(a)
This cutter surface can be treated as a canal surface, i.e. the envelope surface of a one-parameter family of spheres. The center c and radius r of any sphere in the family can be expressed as smooth functions of the parameter a, i.e.
2. Representation of a cutter surface as a canal surface A rotary cutter is enclosed by a surface of revolution. As shown in Fig. 1, the cutter surface is generated by rotating a generatrix on the x–z plane around the z-axis. The planar generatrix can be parametrically expressed as
=
C2 (t ) − C1 (t )
∥C2 (t ) − C1 (t )∥
,
t ∈ [t0 , t1 ]
(3)
where v = [v1 , . . . , vn ] ∈ Rn denotes the collection of the shape parameters of the two curves. Normally, C1 (t ) gives the position of the tool tip. The unit vector A(t ) =
C2 (t ) − C1 (t ) ∥C2 (t ) − C1 (t )∥
describes the orientation of the tool axis.
(4)
L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
1231
Fig. 2. Tool axis trajectory surface defined by two guiding curves.
As stated above, the cutter surface can be modeled as the envelope surface of a one-parameter family of spheres. Therefore, a two-parameter family of spheres would be generated if the cutter undergoes a one-parameter spatial motion. The center S and radius R of any sphere in the family can be expressed as the following smooth functions of the parameters a and t S ( w ; a, t ) = C1 ( t ) +
x(a)x′ (a) + z (a)z ′ (a) C2 (t ) − C1 (t ) z ′ (a)
∥C2 (t ) − C1 (t )∥
x(a) x′ (a)2 + z ′ (a)2
(5)
R(w ; a, t ) = r (u; a) =
z ′ (a)
,
(a, t ) ∈ [a0 , a1 ] × [t0 , t1 ]
(6)
where w = [w1 , . . . , wm+n ] = [u, v ] denotes the collection of the shape parameters of both the tool surface and the tool axis trajectory surface. The cutter swept surface is identically the envelope surface of this two-parameter family of spheres. In this sense, it is a bivariate sphere-swept surface. Although it can be analytically expressed, the parametric expression is complicated [15]. 4. Geometric computation without constructing the swept surface For NC machining simulation and verification, distance computation between the cutter swept surface and the part model surface is of great importance. Because a complex surface can always be approximated by a point cloud, here the computation of the signed distance between the cutter swept surface and a point is of concern. A point can be viewed as a sphere with a very small radius. According to the work on computing the minimum distance between two sphere-swept surfaces [16], the point to surface distance can be computed by minimizing the distance between the point and a sphere in the sphere congruence, as shown in Fig. 3. Hence, we have dp,Senvelope (w ) = min(∥p − S (w ; a, t )∥ − R(w ; a, t )). (a,t )
(7)
The minimum value is obtained by solving the following system of equations
Sa (w ; a, t )·(p − S (w ; a, t )) + R a ( w ; a, t ) = 0 ∥p − S (w ; a, t )∥ St (w ; a, t )·(p − S (w ; a, t )) + Rt (w ; a, t ) = 0. ∥p − S (w ; a, t )∥
Fig. 3. Point to envelope surface distance.
Although the signed distance must be computed by an iterative approach, its first order differential increment with respect to the differential deformations of both the tool surface and the tool axis trajectory surface can be explicitly computed, which is shown in the following proposition. Proposition. If the signed distance dp,S envelope (w ) is well-defined, its first order differential increment has the form
1dp,S envelope (w ) m +n S (w ; a, t ) − p Swi (w ; a, t )· ≈ − Rwi (w ; a, t ) · 1wi . ∥S (w; a, t ) − p∥ i =1 The proof can be completed in a way similar to the proof of proposition 1 in [11], in which only the deformation of the tool trajectory surface is considered. The proposition characterizes quantitatively the change of the distance between the tool envelops surface and a point under the changes of both the tool trajectory and shape. 5. Model for simultaneous optimization of tool path and shape From the geometric viewpoint, the envelope surface of the cutter can be treated as the machined surface. Obviously, we hope that the machined surface Senvelope approximates to the designed surface Sdesign as close as possible. For a dense set of data points {pi ∈ R3 , 1 ≤ i ≤ k1 } sampled from Sdesign , this leads to the following minimax problem Min–max
The signed distance between the point and the swept envelope surface is positive if the point lies on the exterior of the swept volume, and negative if the point lies in the interior of the swept volume.
max |dpi ,Senvelope (w )|.
1≤i≤k1
By introducing one extra variable ξ , which represents the maximum deviation between the design surface and the tool envelope surface, problem Min–max can be reformulated as the following differentiable constrained optimization problem min
(w ,ξ )∈Rm+n+1
NLP s.t.
(8)
min
w ∈Rm+n
ξ
−ξ ≤ dpi ,Senvelope (w ) ≤ ξ ,
1 ≤ i ≤ k1 .
As with any nonlinear optimization problem, a good initial solution is needed. If the machined part has no profile error, i.e., the design surface is the same as the cutter envelope surface, the solution to problem NLP is the same as that to the LS approximation problem defined as LS
min
w ∈Rm+n
k1 [dpi ,Senvelope (w )]2 . i=1
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L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
a
b
Fig. 4. Geometry of a barrel cutter.
a Fig. 6. Convergence of the minimax algorithm.
side of the cutter. One generates the machined surface, and the other could be used to check whether global gouging occurs. For a dense set of data points {qi ∈ R3 , i = 1, . . . , k2 } sampled from the adjacent surfaces, the global gouging could be avoided by imposing the following constraints CON
b
dqi ,Senvelope (w ) ≥ 0,
i = 1, . . . , k2 .
6. Numerical examples In order to demonstrate the validity of the proposed method, we give a simulation of simultaneous optimization of the tool path and shape for five-axis flank milling of the concave surface expressed in Eq. (9) with a barrel cutter.
(60 + 50 · cos ϕ) · cos θ F (u, v) = (60 + 50 · cos ϕ) · sin θ , 10 + 50 · sin ϕ + 20 · θ π θ ∈ 0, . 4
Fig. 5. Convergence of the LS algorithm.
In practical situations, the real machined part has a profile error to the order of microns, so the two solutions would be very close. Since problem LS can be solved more easily, its solution can serve as an initial estimate to problem NLP. A number of classical numerical optimization algorithms can be applied to problems NLP and LS because the derivatives of the involved objective functions and constraint functions are all available. In the above model, both the tool path and shape are optimized to meet the accuracy requirement of the machined surface. For some engineering applications, however, the tool shape could not be changed arbitrarily because global gouging might occur. Take the impeller blades for example, the cutter must be ensured not to interference with the adjacent blades. In this case, a constraint of global gouging avoidance needs be included in the optimization model. There are two separate envelope surfaces swept by the
9π 7π ϕ ∈ − ,− , 16
16
(9)
As shown in Fig. 4, R0 and dc are the parameters that determine the cutter shape. Their initial values are chosen as 40 and 16, respectively. Eight CLs are determined using Chiou’s method [1], and an initial axis trajectory surface is generated by interpolating eight pairs of points on these cutter axes with two B-spline curves of order 3. 50 × 100 points are sampled from the design surface. Two optimum tool paths along with the optimum tool shapes are determined with the LS and Minmax optimization algorithms presented in [9] with trivial modifications of the computer codes. The convergence processes of the two algorithms are depicted in Figs. 5 and 6, respectively. When LS optimization is applied, the maximum undercut increases from 0 to 0.0069 and the maximum overcut reduces from 1.0411 to 0.0174. When Minimax optimization is applied, the maximum undercut reduces to 0.0023 and the maximum overcut reduces to 0.0040. The distributions of the geometric errors before and after optimization are illustrated in Figs. 7–9. It is observed that the present optimization approach improves the machining accuracy greatly.
L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
Fig. 7. Distribution of the geometric errors before optimization.
1233
Fig. 10. Geometric model of the impeller.
Fig. 11. Distribution of the geometric errors.
Fig. 8. Distribution of the geometric errors after LS optimization.
model of the impeller is shown in Fig. 10. The cutter dimension parameters are the bottom cutter radius and the half taper angle. The blades of the given impeller are non-ruled surfaces. If a ruled surface is constructed by connecting the boundaries of a blade, there is a maximum error of 0.2878 mm between the approximated ruled surface and the original blade surface. Therefore, the cutter shape needs to be optimized. The initial bottom radius and half taper angle of the conical cutter are chosen as 5 mm and 10°, respectively. With the cutter geometry being fixed, its path is optimized using the approach presented in [11], and the resulting maximum undercut and overcut are 0.0599 mm and 0.0568 mm, respectively. Taking the above tool geometry and tool path as an initial solution, the approach that simultaneously optimizes the tool path and shape is applied. The optimal cutter bottom radius and half taper angle are obtained as 3.7407 mm and 4.8863°, respectively. The maximum undercut and overcut reduce to 0.0118 mm and 0.0113 mm, respectively. It is seen that optimization of both the tool trajectory and shape improves the machining accuracy greatly. In practical machining, a near-optimal cutter with bottom radius 4 mm and half taper angle 5° is chosen from the cutter series. Accordingly, the maximum undercut and overcut become 0.0169 mm and 0.0166 mm, respectively. The distribution of the geometric errors is depicted in Fig. 11. The real machined impeller is shown in Fig. 12.
Fig. 9. Distribution of the geometric errors after minimax optimization.
7. Conclusions In practical applications, it is rather expensive to customize a cutter with the specific shape. The present approach could be utilized to optimize the dimension or several specific shape parameters of the cutter. To show this, an example of flank milling impeller blades with a conical cutter is given. The geometric
In this note, the swept envelope of a generic rotary cutter is represented as a sphere-swept surface. Then, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization
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L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
Fig. 12. Real machined impeller.
of the tool path and shape for five-axis flank milling. If the tool path is fixed, it becomes a method of tool shape optimization for a given trajectory–surface pair. If the tool shape is fixed, it becomes a tool path optimization method applicable to any rotary cutters. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Acknowledgments This work was partially supported by the National Key Basic Research Program under grant No. 2011CB706804, the National Natural Science Foundation of China under grant No. 50835004, and the Science & Technology Commission of Shanghai Municipality under grant No. 10JC1408000. References [1] Chiou CJ. Accurate tool position for five-axis ruled surface machining by swept envelope approach. Computer-Aided Design 2004;36(10):967–74. [2] Senatore J, Monies F, Redonnet J-M, Rubio W. Analysis of improved positioning in five-axis ruled surface milling using envelope surface. Computer-Aided Design 2005;37:989–98. [3] Lartigue C, Duc E, Affouard A. Tool path deformation in 5-axis flank milling using envelope surface. Computer-Aided Design 2003;35(4):375–82. [4] Chaves-Jacob J, Poulachon G, Duc E. Geometrical deviations versus smoothness in 5-axis high-speed flank milling. International Journal of Machine Tools and Manufacture 2009;41(12):918–29.
[5] Gong H, Cao LX, Liu J. Improved positioning of cylindrical cutter for flank milling ruled surfaces. Computer-Aided Design 2005;37(12):1205–13. [6] Gong H, Wang N. Optimize tool paths of flank milling with generic cutters based on approximation using the tool envelope surface. Computer-Aided Design 2010;41(12):981–9. [7] International organization for standardization. ISO/R 1101. Technical drawings-geometrical tolerancing. Geneva: ISO; 1983. [8] American Society of Mechanical Engineers. ANSI Standard Y14.5. Dimensioning and tolerancing. New York: ASME; 1994. [9] Zhu LM, Zhang XM, Ding H. Geometry of signed point-to-surface distance function and its application to surface approximation. Transactions of ASME, Journal of Computing & Information Science in Engineering 2010;10(4): 041003. [10] Ding H, Zhu LM. Global optimization of tool path for five-axis flank milling with a cylindrical cutter. Science in China—Series E 2009;52(8):2449–59. [11] Zhu LM, Zheng G, Ding H, Xiong YL. Global optimization of tool path for fiveaxis flank milling with a conical cutter. Computer-Aided Design 2010;42(10): 903–10. [12] Chaves-Jacob J, Poulachon G, Duc E. New approach to 5-axis flank milling of free-form surfaces: computation of adapted tool shape. Computer-Aided Design 2009;49(6):454–61. [13] Monies F, Felices JN, Rubio W, et al. Five-axis NC milling of ruled surfaces: optimal geometry of a conical tool. International Journal of Production Research 2002;40(12):2901–22. [14] Li CG. Surface design for flank milling. Ph.D. Thesis, University of Waterloo; 2007. [15] Zhu LM, Zhang XM, Zheng G, Ding H. Analytical expression of the swept surface of a rotary cutter using the envelope theory of sphere congruence. Transactions of ASME, Journal of Manufacturing Science & Engineering 2009; 131(4):041017. [16] Lee K, Seong JK, Kim KJ, et al. Minimum distance between two sphere-swept surfaces. Computer-Aided Design 2007;39:452–9.
Contents lists available at SciVerse ScienceDirect
Computer-Aided Design journal homepage: www.elsevier.com/locate/cad
Technical note
Simultaneous optimization of tool path and shape for five-axis flank milling LiMin Zhu a,∗ , Han Ding a , YouLun Xiong b a
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China
b
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, PR China
article
info
Article history: Received 22 May 2011 Accepted 1 June 2012 Keywords: Rotary cutter Flank milling Tool path Tool shape Distance function
abstract By representing the swept envelope of a generic rotary cutter as a sphere-swept surface, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization of the tool path and shape for five-axis flank milling. If the tool path is fixed, a novel tool shape optimization method is obtained. If the tool shape is fixed, a tool path optimization method applicable to any rotary cutter is obtained. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Numerical examples are given to confirm its validity. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Flank milling is performed by employing the side of a cutter to touch the desired surface. Compared with point milling, flank milling has its unique advantages. It can increase the material removal rate, lower the cutting forces, eliminate necessary hand finish, ensure improved component accuracy and result in longer tool life. Thus it offers a good choice for machining slender parts, like turbine blades and impellers. Recently, increasing attention was drawn onto the problem of optimum positioning of the cutter for flank milling. In most of the works, a tool path is represented by a huge data set of cutter locations (CLs), and each CL is generated to eliminate or reduce the local interference between the cutter and workpiece. However, the machined surface is formed by the swept envelope of the cutter surface. The true machining errors are the deviations between the design surface and cutter envelope surface. It is well known that the shape of the cutter envelope surface can not be completely determined unless all the CLs are obtained. So, only a few works addressed the cutter positioning problem from the perspective of approximation of the cutter envelope surface to design surface. Chiou [1] described a swept envelope-based method for tool positioning. The initial cutter positions were located to contact with two directrices. Then the swept profile of the cutter was calculated based on the cutter motion. Finally the cutter locations were adjusted to reduce the machining errors by comparing the swept profile with the designed ruled surface. By considering the envelope surface, Senatore et al. [2] analyzed the performance of
∗
Corresponding author. Tel.: +86 2134206545. E-mail address: [email protected] (L.M. Zhu).
0010-4485/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2012.06.003
an improved positioning method for flank milling of ruled surfaces with cylindrical cutters. Lartigue et al. [3] proposed a method to deform the two curves that define the tool axis trajectory so that the tool envelope surface fitted the design surface as much as possible. The geometric deviation between the two surfaces was evaluated by the sum of the squared distances of the points on the design surface to the envelope surface. To simplify the computation, an approximate distance measure was employed. Pechard et al. [4] stated that control of the trajectory smoothness was as essential as control of the geometrical deviations, and developed a method that aims at minimizing the geometrical deviations between the tool envelope surface and design surface while preserving the trajectory smoothness. For cylindrical cutters, Gong et al. [5] presented the error propagation principle, and formulated the problem of tool path optimization as that of least squares (LS) fitting of the tool envelope surface to the point cloud on the offset surface of the design surface. Later, Gong and Wang [6] extended this idea to deal with generic rotary cutters, and determined the optimum CL by LS fitting of a spatial line to a series of post-processed point data. Although the LS method was easy for implementation and efficient in computation, it did not conform to the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation [7,8], which requires the maximum norm of the error vector be minimized. More importantly, the geometric deviation of the machined surface from the nominal one was not clearly defined and the influence of the change of the tool axis trajectory on the change of this deviation was not quantitatively analyzed. Based on the differential properties of the signed point-to-surface distance function [9,10], we developed the model and algorithm for tool path planning from the perspective of surface approximation following the minimum zone criterion. The envelope surface of
1230
L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
a
b
Fig. 1. Geometry of a generic rotary cutter.
the cutter was of no concern in the model due to the fact that the envelope surface of a cylindrical cutter is the offset surface of the tool axis trajectory surface. Later, the approach was extended to handle conical cutters by representing the cutter envelope surface as a sphere-swept surface [11]. All the above methods focused on optimizing the tool positions with the tool shape being fixed. Chaves-Jacob et al. [12] proposed a novel approach that optimizes the tool shape for a given trajectory–surface pair to reduce the interferences. It offers a new degree of freedom to improve the machining accuracy. Monies et al. [13] introduced an algorithm to calculate the error between the conical cutter and the workpiece so one can determine the optimal cutter dimensions (cone radius and angle). This is a special case of tool shape optimization. In this note, we will extend our previous works [10,11] to develop a method that can optimize simultaneously the tool path and shape for five-axis flank milling. If the tool path is fixed, it becomes a method of tool shape optimization after implementation of a commercial CAM software to position the tool. If the tool shape is fixed, it becomes a tool path optimization method applicable to any rotary cutters. It also applies to the problems of cutter dimension optimization [13] and flank millable surface design [14]. The remainder of this paper is organized as follows. In Sections 2 and 3, the swept envelope of a rotary milling cutter is modeled as a sphere-swept surface. In Section 4, the properties of the signed distance between the cutter swept envelope and a point are presented. In Section 5, simultaneous optimization of the tool path and shape for five-axis flank milling is formulated as a minimax problem. Numerical examples are given in Section 6, and conclusions are summarized in Section 7.
its control points can be viewed as the shape parameters. Then, we have x′ (a) tan γ (a) = ′ z ( a) cos γ (a) =
z ′ (a) x′ (a)2
+
z ′ (a)2
c (u; a) = [0,
=
0,
0,
z (a)]T ,
a ∈ [a0 , a1 ]
where u = [u1 , . . . , um ] ∈ R denotes the collection of the shape parameters of the generatrix. If the generatrix is a B-spline curve, m
r (a)
.
0,
0,
r (a) sin γ (a) + z (a)]T x(a)x′ (a) + z (a)z ′ (a)
T
z ′ ( a)
x(a) x′ (a)2 + z ′ (a)2
(1)
r (u; a) =
z ′ (a)
.
(2)
Remark. To represent a surface of revolution as a canal surface, the planar generatrix should be smooth enough, and its radius of curvature should be large enough. For the practical rotary cutters used for flank milling, these conditions are always satisfied. 3. Representation of cutter swept envelope as sphere-swept surface As illustrated in Fig. 2, the tool motion is usually represented by two guiding curves C1 (t ) and C2 (t ). They determine the following tool axis trajectory surface Saxis (v ; l, t ) = C1 (t ) + l
g (u; a) = [x(a),
x(a)
This cutter surface can be treated as a canal surface, i.e. the envelope surface of a one-parameter family of spheres. The center c and radius r of any sphere in the family can be expressed as smooth functions of the parameter a, i.e.
2. Representation of a cutter surface as a canal surface A rotary cutter is enclosed by a surface of revolution. As shown in Fig. 1, the cutter surface is generated by rotating a generatrix on the x–z plane around the z-axis. The planar generatrix can be parametrically expressed as
=
C2 (t ) − C1 (t )
∥C2 (t ) − C1 (t )∥
,
t ∈ [t0 , t1 ]
(3)
where v = [v1 , . . . , vn ] ∈ Rn denotes the collection of the shape parameters of the two curves. Normally, C1 (t ) gives the position of the tool tip. The unit vector A(t ) =
C2 (t ) − C1 (t ) ∥C2 (t ) − C1 (t )∥
describes the orientation of the tool axis.
(4)
L.M. Zhu, et al. / Computer-Aided Design 44 (2012) 1229–1234
1231
Fig. 2. Tool axis trajectory surface defined by two guiding curves.
As stated above, the cutter surface can be modeled as the envelope surface of a one-parameter family of spheres. Therefore, a two-parameter family of spheres would be generated if the cutter undergoes a one-parameter spatial motion. The center S and radius R of any sphere in the family can be expressed as the following smooth functions of the parameters a and t S ( w ; a, t ) = C1 ( t ) +
x(a)x′ (a) + z (a)z ′ (a) C2 (t ) − C1 (t ) z ′ (a)
∥C2 (t ) − C1 (t )∥
x(a) x′ (a)2 + z ′ (a)2
(5)
R(w ; a, t ) = r (u; a) =
z ′ (a)
,
(a, t ) ∈ [a0 , a1 ] × [t0 , t1 ]
(6)
where w = [w1 , . . . , wm+n ] = [u, v ] denotes the collection of the shape parameters of both the tool surface and the tool axis trajectory surface. The cutter swept surface is identically the envelope surface of this two-parameter family of spheres. In this sense, it is a bivariate sphere-swept surface. Although it can be analytically expressed, the parametric expression is complicated [15]. 4. Geometric computation without constructing the swept surface For NC machining simulation and verification, distance computation between the cutter swept surface and the part model surface is of great importance. Because a complex surface can always be approximated by a point cloud, here the computation of the signed distance between the cutter swept surface and a point is of concern. A point can be viewed as a sphere with a very small radius. According to the work on computing the minimum distance between two sphere-swept surfaces [16], the point to surface distance can be computed by minimizing the distance between the point and a sphere in the sphere congruence, as shown in Fig. 3. Hence, we have dp,Senvelope (w ) = min(∥p − S (w ; a, t )∥ − R(w ; a, t )). (a,t )
(7)
The minimum value is obtained by solving the following system of equations
Sa (w ; a, t )·(p − S (w ; a, t )) + R a ( w ; a, t ) = 0 ∥p − S (w ; a, t )∥ St (w ; a, t )·(p − S (w ; a, t )) + Rt (w ; a, t ) = 0. ∥p − S (w ; a, t )∥
Fig. 3. Point to envelope surface distance.
Although the signed distance must be computed by an iterative approach, its first order differential increment with respect to the differential deformations of both the tool surface and the tool axis trajectory surface can be explicitly computed, which is shown in the following proposition. Proposition. If the signed distance dp,S envelope (w ) is well-defined, its first order differential increment has the form
1dp,S envelope (w ) m +n S (w ; a, t ) − p Swi (w ; a, t )· ≈ − Rwi (w ; a, t ) · 1wi . ∥S (w; a, t ) − p∥ i =1 The proof can be completed in a way similar to the proof of proposition 1 in [11], in which only the deformation of the tool trajectory surface is considered. The proposition characterizes quantitatively the change of the distance between the tool envelops surface and a point under the changes of both the tool trajectory and shape. 5. Model for simultaneous optimization of tool path and shape From the geometric viewpoint, the envelope surface of the cutter can be treated as the machined surface. Obviously, we hope that the machined surface Senvelope approximates to the designed surface Sdesign as close as possible. For a dense set of data points {pi ∈ R3 , 1 ≤ i ≤ k1 } sampled from Sdesign , this leads to the following minimax problem Min–max
The signed distance between the point and the swept envelope surface is positive if the point lies on the exterior of the swept volume, and negative if the point lies in the interior of the swept volume.
max |dpi ,Senvelope (w )|.
1≤i≤k1
By introducing one extra variable ξ , which represents the maximum deviation between the design surface and the tool envelope surface, problem Min–max can be reformulated as the following differentiable constrained optimization problem min
(w ,ξ )∈Rm+n+1
NLP s.t.
(8)
min
w ∈Rm+n
ξ
−ξ ≤ dpi ,Senvelope (w ) ≤ ξ ,
1 ≤ i ≤ k1 .
As with any nonlinear optimization problem, a good initial solution is needed. If the machined part has no profile error, i.e., the design surface is the same as the cutter envelope surface, the solution to problem NLP is the same as that to the LS approximation problem defined as LS
min
w ∈Rm+n
k1 [dpi ,Senvelope (w )]2 . i=1
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a
b
Fig. 4. Geometry of a barrel cutter.
a Fig. 6. Convergence of the minimax algorithm.
side of the cutter. One generates the machined surface, and the other could be used to check whether global gouging occurs. For a dense set of data points {qi ∈ R3 , i = 1, . . . , k2 } sampled from the adjacent surfaces, the global gouging could be avoided by imposing the following constraints CON
b
dqi ,Senvelope (w ) ≥ 0,
i = 1, . . . , k2 .
6. Numerical examples In order to demonstrate the validity of the proposed method, we give a simulation of simultaneous optimization of the tool path and shape for five-axis flank milling of the concave surface expressed in Eq. (9) with a barrel cutter.
(60 + 50 · cos ϕ) · cos θ F (u, v) = (60 + 50 · cos ϕ) · sin θ , 10 + 50 · sin ϕ + 20 · θ π θ ∈ 0, . 4
Fig. 5. Convergence of the LS algorithm.
In practical situations, the real machined part has a profile error to the order of microns, so the two solutions would be very close. Since problem LS can be solved more easily, its solution can serve as an initial estimate to problem NLP. A number of classical numerical optimization algorithms can be applied to problems NLP and LS because the derivatives of the involved objective functions and constraint functions are all available. In the above model, both the tool path and shape are optimized to meet the accuracy requirement of the machined surface. For some engineering applications, however, the tool shape could not be changed arbitrarily because global gouging might occur. Take the impeller blades for example, the cutter must be ensured not to interference with the adjacent blades. In this case, a constraint of global gouging avoidance needs be included in the optimization model. There are two separate envelope surfaces swept by the
9π 7π ϕ ∈ − ,− , 16
16
(9)
As shown in Fig. 4, R0 and dc are the parameters that determine the cutter shape. Their initial values are chosen as 40 and 16, respectively. Eight CLs are determined using Chiou’s method [1], and an initial axis trajectory surface is generated by interpolating eight pairs of points on these cutter axes with two B-spline curves of order 3. 50 × 100 points are sampled from the design surface. Two optimum tool paths along with the optimum tool shapes are determined with the LS and Minmax optimization algorithms presented in [9] with trivial modifications of the computer codes. The convergence processes of the two algorithms are depicted in Figs. 5 and 6, respectively. When LS optimization is applied, the maximum undercut increases from 0 to 0.0069 and the maximum overcut reduces from 1.0411 to 0.0174. When Minimax optimization is applied, the maximum undercut reduces to 0.0023 and the maximum overcut reduces to 0.0040. The distributions of the geometric errors before and after optimization are illustrated in Figs. 7–9. It is observed that the present optimization approach improves the machining accuracy greatly.
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Fig. 7. Distribution of the geometric errors before optimization.
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Fig. 10. Geometric model of the impeller.
Fig. 11. Distribution of the geometric errors.
Fig. 8. Distribution of the geometric errors after LS optimization.
model of the impeller is shown in Fig. 10. The cutter dimension parameters are the bottom cutter radius and the half taper angle. The blades of the given impeller are non-ruled surfaces. If a ruled surface is constructed by connecting the boundaries of a blade, there is a maximum error of 0.2878 mm between the approximated ruled surface and the original blade surface. Therefore, the cutter shape needs to be optimized. The initial bottom radius and half taper angle of the conical cutter are chosen as 5 mm and 10°, respectively. With the cutter geometry being fixed, its path is optimized using the approach presented in [11], and the resulting maximum undercut and overcut are 0.0599 mm and 0.0568 mm, respectively. Taking the above tool geometry and tool path as an initial solution, the approach that simultaneously optimizes the tool path and shape is applied. The optimal cutter bottom radius and half taper angle are obtained as 3.7407 mm and 4.8863°, respectively. The maximum undercut and overcut reduce to 0.0118 mm and 0.0113 mm, respectively. It is seen that optimization of both the tool trajectory and shape improves the machining accuracy greatly. In practical machining, a near-optimal cutter with bottom radius 4 mm and half taper angle 5° is chosen from the cutter series. Accordingly, the maximum undercut and overcut become 0.0169 mm and 0.0166 mm, respectively. The distribution of the geometric errors is depicted in Fig. 11. The real machined impeller is shown in Fig. 12.
Fig. 9. Distribution of the geometric errors after minimax optimization.
7. Conclusions In practical applications, it is rather expensive to customize a cutter with the specific shape. The present approach could be utilized to optimize the dimension or several specific shape parameters of the cutter. To show this, an example of flank milling impeller blades with a conical cutter is given. The geometric
In this note, the swept envelope of a generic rotary cutter is represented as a sphere-swept surface. Then, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization
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Fig. 12. Real machined impeller.
of the tool path and shape for five-axis flank milling. If the tool path is fixed, it becomes a method of tool shape optimization for a given trajectory–surface pair. If the tool shape is fixed, it becomes a tool path optimization method applicable to any rotary cutters. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Acknowledgments This work was partially supported by the National Key Basic Research Program under grant No. 2011CB706804, the National Natural Science Foundation of China under grant No. 50835004, and the Science & Technology Commission of Shanghai Municipality under grant No. 10JC1408000. References [1] Chiou CJ. Accurate tool position for five-axis ruled surface machining by swept envelope approach. Computer-Aided Design 2004;36(10):967–74. [2] Senatore J, Monies F, Redonnet J-M, Rubio W. Analysis of improved positioning in five-axis ruled surface milling using envelope surface. Computer-Aided Design 2005;37:989–98. [3] Lartigue C, Duc E, Affouard A. Tool path deformation in 5-axis flank milling using envelope surface. Computer-Aided Design 2003;35(4):375–82. [4] Chaves-Jacob J, Poulachon G, Duc E. Geometrical deviations versus smoothness in 5-axis high-speed flank milling. International Journal of Machine Tools and Manufacture 2009;41(12):918–29.
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