外文翻译--五轴数控加工的刀具路径规划与动力学仿真 英文版

Science China Press and Springer-Verlag Berlin Heidelberg 2010 Review Mechanical Engineering SPECIAL TOPIC: Huazhong University of Science and Technology October 2010 Vol.55 No.30: 34083418 doi: 10.1007/s11434-010-3247-7 Tool path generation and simulation of dynamic cutting process for five-axis NC machining DING Han1*, BI QingZhen2, ZHU LiMin2& XIONG YouLun1 1 State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China； 2 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China Received October 9, 2009； accepted December 29, 2009 Five-axis NC machining provides a valid and efficient way to manufacture the mechanical parts with complex shapes, which are widely used in aerospace, energy and national defense industries. Its technology innovations have attracted much attention in re-cent years. In this paper, the state-of-the-art techniques for five-axis machining process planning are summarized and the chal-lenging problems are analyzed from the perspectives of tool path generation, integrated geometric/mechanistic simulation and machining stability analysis. The recent progresses in accessibility-based tool orientation optimization, cutter location (CL) plan-ning for line contact and three-order point contact machining, shape control of cutter envelope surface and milling stability pre-diction are introduced in detail. Finally, the emerging trends and future challenges are briefly discussed. five-axis machining, tool path generation, integrated geometric/mechanistic simulation, dynamics simulation Citation: Ding H, Bi Q Z, Zhu L M, et al. Tool path generation and simulation of dynamic cutting process for five-axis NC machining. Chinese Sci Bull, 2010, 55: 34083418, doi: 10.1007/s11434-010-3247-7 In conventional three-axis NC machining only the transla-tion motions of the cutter are permitted while the cutter ori-entation is allowed to change in a five-axis machine tool because of the two additional rotational axes. The advan-tages of five-axis NC machining mainly depend on the con-trol of tool orientations: (1) The collision between the part and the cutter can be avoided by selecting the accessible tool orientation, which provides the ability to machine the complicated shapes such as aerospace impeller, turbo blade and marine propeller. (2) A large machining strip width can be obtained if the tool orientation is properly planed so that the tool tip geometry matches the part geometry well. Also, the highly efficient flank milling can be applied to machine aerospace impeller by using a five-axis machine tool. (3) The cutting conditions can be improved in five-axis ma-chining. For example, it is possible to shorten the tool overhang length if the tool orientation is optimized. Deter-mining the safe and shortest tool length is very helpful when *Corresponding author (email: ) the surface is machined in a confined space, in which only the small-diameter cutters can be used. The cutting area of a cutter, which affects the cutting force, cutter wear and ma-chined surface quality can also be controlled by changing the cutter orientation. Besides the above advantages, there exist several chal-lenging problems in five-axis machining. Since the tool orientation is adjustable, it is hard to image the complicated spatial motion of the tool. Thus, it is much more difficult to generate the collision-free and high efficient tool paths, which limits its wide application. Furthermore, the cutting force prediction and dynamics simulation are more complex because the involved cutting parameters are time-varying during the machining process. Current works about five- axis machining fall into three categories 1: tool path gen-eration, integrated geometric/mechanistic simulation and dynamics simulation, as shown in Figure 1. Tool path gen-eration is the process to plan the cutter trajectory relative to the part based on the part model, machining method and tolerance requirement. The cutter trajectory affects greatly DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 3409 the cutting efficiency and quality. It is also the foundation of integrated geometric/mechanistic simulation, which de-pends on the cutting geometry and cutting force modeling techniques. The cutting geometry reflects the meshing state between the cutter and the workpiece during the material removing process. By integrating the cutting geometry and cutting force models, the transient cutting force can be pre-dicted. The cutting force then can be applied to dynamics simulation, feedrate scheduling, and prediction and com-pensation of deformation. The goal of dynamics simulation is to predict the cutting stability and the machined surface profile based on the cutting force and the dynamics charac-teristics of the machine tool-cutter-fixture system. Dynam-ics simulation is helpful to optimize the cutting parameters and the tool path. The literatures on five-axis NC machining are enormous. A lot of related commercial systems have been developed, such as the general-purpose CAM softwares UG and CATIA, the special CAM software Max-AB for machining impeller and Turbosoft for machining blade, and the dy-namics simulation software CutterPro. European Commis-sion supported a project about flank milling optimization that is called “Flamingo”. Because of the obvious advan-tages of flank milling in cutting efficiency and surface qual-ity, a number of famous companies (SNECMA, Rolls Royce, Dassault Systmes) and a university (Hannover) participated in this project. The researches on five-axis high-efficiency and high-precision machining have also been carried out in some famous companies, such as United Technologies, Pratt & Whitney and Concepts NREC. Do-mestic researchers have developed some CAM systems such as KM, 5BDM and Dynacut, but the fundamental re-searches and industrial applications of five-axis machining are still in the primary level. Current commercial CAM systems provide a lot of strategies for tool path generation and simulation of dy-namic cutting process. However, the performances in intel-ligence, usability and computation efficiency still need to be improved. For instance, the selection of the strategy for cut-ter orientation optimization depends on the skill of the pro-grammer； it is difficult to automatically generate the opti-mum tool orientations that consider simultaneously all the Figure 1 Three challenging problems in five-axis NC machining. objectives required by the practical cutting process, such as collision avoidance, large effective cutting width, globally cutter orientation smoothness and shorter tool length. Also, most of the existing works about dynamics simulation aim to three-axis machining. Models and algorithms applicable to five-axis machining need to be explored. 1 Tool path generation Tool path generation is the most important technology in NC programming. The critical problem in five-axis ma-chining is to plan cutter orientations. Theoretically, the tool orientation can be any point on the Gauss Sphere. In fact, the feasible tool orientations are only a limited area on the Gauss Sphere because of the constraints of global collision avoidance and machine joint angle limits. To improve ma-chining efficiency and quality, the tool orientation of each cutter location (CL) data should be optimized by consider-ing the important factors related to a practical cutting proc-ess. The factors consist of geometrical constraints, kine-matic constraints, dynamic characteristics and physical fac-tors. How to take into account these factors is the most challenging issue in the research of tool path generation. 1.1 Collision avoidance Collision avoidance must be first considered in the process of tool path generation. There are mainly two kinds of ideas to avoid interference: (1) First generating and then adjusting cutter orientation to avoid collision. (2) Access-based tool path generation. With the former idea, cutter orientations are first planned according to some strategies. A collision detection method is then used to detect the collision be-tween the tool and the parts. If collision occurs, the tool orientations must be changed as shown in Figure 2. With the latter idea, the cutter orientations are generated directly in the accessibility cones as shown in Figure 3. The research about the first idea focuses on the algo-rithms to improve the collision detection efficiency and ad-just cutter orientations to avoid collision. In practical appli-cations, tool paths are usually composed of thousands to hundred thousands of tool positions. The collision detection often requires large computation time and resource. There-fore lots of algorithms have been proposed to improve the computation efficiency of collision detection 2,3. When machining a complex shape, the detection and adjustment processes usually repeat several times. Collision avoidance is of first concern. It is difficult to consider other factors affect-ing the cutting process when adjusting cutter orientations. The access-based tool path generation method consists of two steps. Collision-free cutter orientations at every cutter contact (CC) point are first computed. The set of colli-sion-free cutter orientations is called accessibility cone. The cutter orientations are then generated in the accessibility 3410 DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 Figure 2 Detecting and adjusting cutter orientation to avoid collision 2. (a) Collision detection； (b) adjust cutter orientation. Figure 3 Access-based collision-free tool path generation. (a) Accessi-bility cone； (b) collision-free tool path. cones. The most obvious merit of this method is that the iterative process of adjusting cutter orientations can almost be avoided. Based on the accessibility cone, the manufac-turability can be directly determined. Furthermore, the cut-ter orientation optimization can be carried out in the colli-sion-free space. Other objectives such as cutting forces and velocity smoothness may also be considered. The problem with this idea is the difficulty in efficiently computing ac-cessibility cones. Usually computing accessibilities will cost large computation time because complex shape may consist of hundreds of thousands of polygonal meshes. Some algo-rithms were proposed to improve computation efficiency such as the C-space (Configuration Space) methods 4,5 and visibility-based methods 610. Though C-space is an elegant concept to deal with collision avoidance, the free C-space cannot be explicitly and efficiently computed. Wang et al. 5 showed that the elapsed time to compute an accessibility cone for a part composed of only 10000 trian-gles would be 1190.33 min. Furthermore, the algorithm did not consider the collision of the tool holder. A cutter can be abstracted as a light ray that emits from the CL point if its radius is ignored. Then the problem of collision avoidance is transformed into that of visibility. We 68 described cutters visibility cone using the concept of C-space and proposed three strategies to accelerate the computation speed using the hidden-surface removal techniques in com-puter graphics. The manufacturability of a complex surface was also analyzed based on the visibility cone. However, the conventional visibility is only the necessary condition of accessibility because a milling tool usually consists of sev-eral cylindrical shapes with finite radii. The real accessible directions cannot be directly obtained from the visibility cone, and secondary collision checking and avoidance strat-egies are still needed 9. The accessibility will be equal to the visibility if both the machined surface and the interfer-ence checking surface are replaced by their offset surfaces 10. However, the offset surface is usually not easy to ob-tain and the collision avoidance of the tool holder cannot be guaranteed. Furthermore, the method only applies to ball- end cutters and cannot be extended to other types of cutters. We 11,12 proposed a high-efficient algorithm to compute the accessibility cone using graphics hardware. The algo-rithm has almost linear time complexity and applies to both flat-end and torus-end cutters. Generally, the CL point can be specified by the CC point, outward normal direction of the machined surface and cutter orientation. If the viewing direction is opposite to the cutter orientation, the global ac-cessibility of the cutter is then equal to the complete visi-bilities of the involved cylinders and cones. This equiva-lence provides an efficient method for detecting the acces-sibility of the milling cutter by using the occlusion query function of the graphics hardware. The computation effi-ciencies of the three algorithms are compared in Table 1. It is found that the computation time of our algorithm is less than 2% of that in 9 even though both the number of tri-angles and the number of cutter orientations are greater than 10 times of those in 9. The average computation time for one cutter orientation at one contact point is less than 2 of that in 9. The average computation time is also much less than that in 3 even though the number of inputted triangles is much greater than that in 3. 1.2 Cutting efficiency Nowadays, ball-end cutters are widely employed for five-axis NC machining. The major advantages of ball-end milling are that it applies to almost any surface and it is Table 1 The comparison of computation time Inputted models Method Computation platform Triangle Cutter center point Cutter orientations Computation time Average computa-tion time Ref. 9 SGI work station, Dual CPU 250M 10665 1500 80 51.63 m 2.58102s Ref. 3 CPU 2.4G, RAM 512M 12600 50000 1 61.61s 1.23103s Our method 12 CPU 2.4G, RAM 512M 139754 2000 1026 60.53 s 2.95105s DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 3411 relatively easy to generate the tool path. From the manufac-turers point of view, however, the main disadvantage of ball-end milling is that it is very time consuming. It may require more finish passes and each pass removes only a small amount of material. Compared with ball-end cutter, non-ball-end cutter possesses more complex geometry, and exhibits different “effective cutting profiles” at different locations. Thus, it is possible to position the cutter so that its “effective cutting profile” well matches the design surface, which results in a great improvement of the machining strip width. Hence, increasing attention has been drawn onto the problem of tool path optimization for milling complex sur-faces with non-ball-end cutters. In five-axis machining, the machined surface is formed by the swept envelope of the cutter surface. The true ma-chining errors are the deviations between the design surface and the cutter envelope surface. It is well known that the shape of the cutter envelope surface cannot be completely determined unless all the cutter positions are given 13,14. Due to the difficulty and complexity in locally modeling the cutter envelope surface, most works adopted the approxi-mate or simplified models, which formulate the problem of optimal cutter positioning as that of approximating the cut-ter surface to the design surface in the neighborhood of the current CC point 15. These optimization models do not characterize the real machining process. Also, they only apply to certain surfaces or cutters. Only a few works have addressed the cutter positioning problem from the perspective of local approximation of cutter envelope surface to design surface 1517. For a flat-end or disk cutter, Wang et al. 15 and Rao et al. 16 developed the third- and second-order approximate models of the cutter envelope surface, respectively. However, for such a cutter, its envelope surface is swept by the cutting circle, which is not a rotary surface. Therefore, the two me-thods cannot be applied to other types of rotary cutters. Re-cently, Gong et al. 17 developed a mathematical model that describes the second-order approximation of the enve-lope surface of a general rotary cutter in the neighborhood of the CC point, and then proposed a cutter positioning strategy that makes the cutter envelope surface have a con-tact of second-order with the design surface at the CC point. However, theoretically speaking, a third-order contact be-tween the cutter envelope surface and the design surface could be achieved by adjusting the cutter orientation. This means that the cutter location planning based on the sec-ond-order model does not take full advantage of the effi-ciency and power that the five-axis machining offers. The above models are not compatible with each other. Also, the optimal CL is determined by solving two equations derived from the second- and third-order contact conditions. Due to the constraints of machine joint angle limits, global colli-sion avoidance and tool path smoothness, maybe there is no feasible solution to this system of equations. In our recent works 18,19, the geometric properties of a pair of line contact surfaces were investigated. Then, based on the observation that the cutter envelope surface contacts with the cutter surface and the design surface along the characteristic curve and cutter contact (CC) path, respec-tively, a mathematical model describing the third-order ap-proximation of the cutter envelope surface according to just one given cutter location (CL) was developed. It was shown that at the CC point both the normal curvature of the normal section of the cutter envelope surface and its derivative with respect to the arc length of the normal section could be de-termined by those of the cutter surface and the design sur-face. This model characterizes the intrinsic relationship among the cutter surface, the cutter envelope surface and the design surface in the vicinity of the CC point. On this basis, a tool positioning strategy was proposed for effi-ciently machining free-form surfaces with non-ball-end cutters. The optimal CL was obtained by adjusting the in-clination and tilt angles of the cutter until its envelope sur-face and the design surface had the third-order contact at the CC point, which resulted in a wide machining strip. The strategy can handle the constraints of joint angle limits, global collision avoidance and tool path smoothness in a nature way, and applies to general rotary cutters and com-plex surfaces. Numerical examples demonstrated that the third-order point contact approach could improve the ma-chining strip width greatly as compared with the recently reported second-order one. A comparison of the machining strip widths using different CLs for the five-axis machining of a helical surface with a toroidal cutter is summarized in Table 2. The values of the tool parameters chosen for simu-lation are: radius of the torus R=10 mm, and radius of the corner r=2.5 mm. Compared with the point milling, the flank milling can increase the material removal rate, lower the cutting forces, eliminate necessary hand finish and ensure improved com-ponent accuracy. It offers a better choice for machining slender surfaces. Lartigue et al. 20 proposed an approach to globally optimize the tool path for flank milling. The basic idea is to deform the tool axis trajectory surface so that the tool envelope surface fits the design surface ac-cording to the least-squares criterion. To simplify the com-putation, an approximate distance measure was employed. For a cylindrical cutter, Gong et al. 21 presented the error propagation principle, and transformed the problem into that of least-squares (LS) approximation of the axis trajectory surface to the offset surface of the design surface. In these two works, not the local geometric error, but the geometric Table 2 Comparison of the machining strip widths for different CLs Tolerance (mm) Ball-end cutter (R = 5.5 mm) Toroidal cutter (Second order contact) Toroidal cutter (Third order contact) = 0.005 0.69 2.48 5.28 = 0.01 0.98 3.12 6.14 3412 DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 error between the envelope surface of the cutter and the design surface, was of the first concern. Thus it was called the global optimization method. Although the LS method was easy for implementation and efficient in computation, it could not incorporate readily the non-over- cut constraint required by semi-finish milling, and more importantly, it did not conform to the minimum zone crite- rion recommended by ANSI and ISO standards for toler- ance evaluation. Fur-thermore, the geometric deviation of the machined surface from the nominal one was not clearly defined and the influ-ence of the deformation of the tool axis trajectory surface on the change of this deviation was not quantitatively analyzed. In our studies 22,23, the maximum orthogonal distance from the point on the design surface to the tool envelope surface was introduced to characterize the geometric error of the machined surface. The first-order gradient and sec-ond-order Hessian matrix of the distance function about the control parameters governing the form of the axis trajectory surface were derived. On this basis, the complete principle, model and algorithm for global tool path optimization for five-axis flank milling using cylindrical cutters were devel-oped from the perspective of surface approximation follow-ing the minimum zone criterion, and applied to flank mill-ing of non-developable ruled surface. The geometrical pre-cision was improved about 30% compared with the existing algorithms. Another advantage is that our method can easily deal with the overcut-free constraints. The comparison re-sults are listed in Table 3. In our model, the envelope sur-face of the cutter was of no concern due to the fact that the envelope surface of a cylindrical cutter is the offset surface of the tool axis trajectory surface. Therefore, the approach was only applicable to cylindrical cutters, and could not be generalized to optimize the tool paths of other types of ro-tary cutters. Cylindrical cutters can satisfy most of the demands for flank milling. However, when an application requires flank milling within a confined space, a conical cutter may prove to be more suitable because it has a smaller tip and a stronger shank in comparison with a cylindrical cutter which has a small diameter. Recently, increasing attention has been drawn onto the problem of using a conical cutter for flank milling. In our recent works 24,25, based on the observation that conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter was repre-sented as a sphere-swept surface. Then, an approach was presented to efficiently compute the signed distance between Table 3 The comparison of flank milling tool path generation algorithm RRD 27 MBM 28 Gong et al. 21 Our method 22 Maximum undercut (mm) 0.220 0.264 0.093(0.228) 0.068(0.228) Maximum overcut (mm) 0.220 0.211 0.119(0.172) 0.067(0.172) a point in space and the swept surface without constructing the swept surface itself. The first order differential incre-ment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory sur-face was derived. By using the distance function, the tool path optimizations for semi-finished and finished millings with conical cutters are formulated as two constrained opti-mization problems in a unified framework. The sequential approximation algorithm along with a hierarchical algo-rithmic structure is developed to solve them. The proposed theories and methods apply to general rotary cutters. Here, an example of tool path optimization for flank milling of a blade of an impeller with a conical cutter is reported. The blade was defined by two directrices, which were both B-spline curves of Order 3. The cutter parameters were: the bottom radius was 6.25 mm, height 30 mm, and taper angle 10. And 50100 points were sampled from the design sur-face. A smooth axis trajectory surface was generated by using Chious method 26. The maximum undercut and overcut were 0.0896 mm and 0.0239 mm, respectively. Af-ter optimization of the axis trajectory surface, the maximum undercut and overcut reduced to 0.0062 mm and 0.0061 mm, respectively. It was seen that the global tool path optimiza-tion approach improved the machining accuracy greatly. 1.3 Cutting process condition optimization The cutting process conditions such as the smoothness of the tool path and the rigidity of the whole cutting system are paid more attention in high speed machining. The tool path smoothness and tool overhang length affect the dynamics characteristics of five-axis NC machining. The cutter orien-tations would also influence the valid cutting parameters such as cutting speed and cutting area, and hence the cutting force and surface quality. Therefore, the cutting process conditions should be taken into account when planning the tool path. (i) Cutter orientation smoothness. The drastic change of the tool orientation must be avoided in a practical five-axis cutting process 29,30. Generally, the measure of cutter orientation smoothness can be defined in the machine tool coordinate system, the workpiece coordinate system and the process coordinate system. The three measures reflect the rotational motion of the machine tool, the transition of cut-ter orientations relative to the workpiece and the change of cutting conditions, respectively. The measure defined in the machine tool coordinate sys-tem is the commonly used one in current research works. Kersting et al. 31 proposed an interesting algorithm to smooth cutter orientations in the free C-space. Castagnetti et al. 29 defined a new measure in the machine tool coordi-nate system to improve the cutting efficiency and the even-ness of the rotational motions. Their results showed that the cutting time could be greatly shortened by optimizing cut-ting orientations. We 11,12 proposed a model to globally DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 3413 smooth cutter orientations in the feasible space. The con-straint on the angle between two orientations at two neighboring CC points was considered in the model. The global smoothness problem was formulated as a discrete minimization problem and solved by the shortest path algo-rithm in graph theory. The research works about smoothing cutter orientations in the workpiece coordinate system focus on the NURBS tool path generation. The researchers in Dassult Systmes company described a new tool path format composed of two NURBS curves 32. The distance between the two curves is constant and the smoothness of the cutter orientations is guaranteed. The format has been supported by the 840D NC system of Siemens company. We 33 proposed a new method to generate this kind of tool path for five-axis ma-chining based on the “point-line” kinematics. From the viewpoint of kinematics, a point-line is the abstract of a tool. By using the screw theory in kinematics, the mapping from the space of point-line in Euclidean three-dimensional space into the hyperplane in dual quaternion space was con-structed. The problem of point-line motion design was con-verted to that of projective Bzier or B-spline image curve design in the hyperplane of dual quaternion. The resulting point trajectory is a NURBS curve in Euclidean three- dimensional space and orientation curve is a NURBS curve on unit sphere. Such two NURBS curves, named dou-ble-NURBS curves, can describe the tool path in NC ma-chining. The measure defined in the process coordinate system indicates the change of cutting conditions. Optimizing cutter orientations according to this measure is helpful to smooth cutting force. Ozturk et al. 34 gave the relationship be-tween the cutter orientation and the cutting force and showed that the cutting force acting on a ball-end cutter depended greatly on the cutter orientation. We 30,35 pro-posed an algorithm to globally smooth cutter orientations based on a mesh-based model. The measures defined in the three coordinate systems were comprehensively considered. The approach has two advantages: (1) The cutter orienta-tions are smoothed along both the feed direction and the pick-feed direction； (2) only the accessibility cones of mesh points are required to compute and computation efficiency is improved. Simulations showed that the global smoothing of cutter orientations was helpful to improve cutting effi-ciency, evenness of feed velocity and smoothness of cutting force. (ii) Shorter tool length. The use of shorter cutters with-out collision is a key advantage of five-axis machining be-cause the magnitude of tool deflection and the stability of the cutting process are greatly affected by the slenderness ratio of the cutter. In the existing works, the shortest colli-sion-free cutter length is generally considered in the five-axis machining simulation process. For example, the minimal tool overhang length can be calculated by the simulation software such as Vericut. With this method, the shortest safe tool length (SSTL) can only be determined according to a predefined tool path. However, the SSTL along a tool path is essentially determined by the tool ori-entations in machining of a complex shape. Therefore, the SSTL should be considered in the process of the tool path generation. The tool length is usually ignored in the existing tool path generation algorithms. Morimoto et al. 10 proposed a novel algorithm to shorten the overhang length for 3+2 axis machining with a ball-end cutter by properly selecting cutter orientations. In this work, the offset surfaces of the ma-chined surface and the interference checking surface must be constructed, which is not an easy task. Furthermore, the estimated safe tool length is too conservative. We 36 pro-posed a GPU- based algorithm to compute the SSTL along a cutter orientation at a CL point based on the GPU-based accessibility detection method and developed an efficient method to generate the SSTL for 3+2 axis machining proc-ess. In succession, we 37,38 proposed a novel algorithm to determine the SSTL for 5-axis NC machining with a short ball-end cutter by optimizing the tool orientations under the constraints of global collision avoidance and tool orientation smoothness. The optimization problem was formulated as a constrained combinatorial optimization problem and solved by a dynamic programming technique. It would generate concurrently the SSTL and collision-free tool path. 2 Integrated geometric/mechanistic simulation As a foundation of physical simulation, the dynamic cutting force simulation plays an important role in feed-rate sched-uling, spindle speed optimization, chatter prediction, adap-tive control of machining process, monitoring of tool wear and broken, prediction of surface topographic, error analysis and compensation, and so on. The dynamic cutting force in the material removal process is usually predicted based on the instantaneous cutting conditions which mainly consist of the cutting geometry and the cutting force coefficients. The cutting force coefficients are usually determined by an ex-perimental calibration 39,40. So, modeling swept volume of cutter and cutter-workpiece engagement becomes a pri-mary work. 2.1 Integration of geometry simulation and cutting force prediction The computation of the envelope surface of cutter is critical for the modeling of swept volume. Numerical methods are usually used, including the Jacobian rank-deficiency method, swept-envelope differential equation algorithm, implicit modeling and Minkowski sum method 41. The high-order differential or transcendental equations are gen-erally needed to be solved in the numerical methods, which require great computation costs. Chiou et al. 42,43 re-3414 DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 ported two explicit (closed-form) expressions of the swept profile of a generalized APT cutter undergoing a five-axis tool motion. The one derived in 42 is directly related to the kinematical model of the machine tool due to the im-proper representation of the velocity of the cutter, and the machine-configu- ration independent one developed in 43 requires to define a instantaneous auxiliary frame at the CC point. Du et al. 44,45 simplified the derivations and com-putations in 42 by introducing a instantaneous auxiliary frame and rigid-body velocity. In our studies 13,14, two methods were proposed to analytically compute the swept envelope surfaces of the rotary tools. The first one was based on the observation that many surfaces of revolution can be treated as a canal surface, i.e. the envelope surface of an one-parameter family of spheres. The analytical expres-sions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spa-tial motions were derived by using the envelope theory of sphere congruence. For the toroidal cutter, two methods for determining the effective patch of the envelope surface were proposed. With the present model, it was shown that the swept surfaces of a torus and a cylinder can be easily con-structed without complicated calculations, and that the mini-mum distance between the swept surface and a simple surface and the signed distance between the swept surface and a point in space could be easily computed without construct-ing the swept surface itself. The second one was based on the tangency condition in envelope theory and the body velocity representation in spatial kinematics. No additional moving frames or local frames are required, and the computational formulas are independent of the types of the machines. The modeling of cutter-workpiece engagement is the foundation for the cutting force simulation in five-axis milling. The usual method used so far can be classified into three types. The first is the solid geometry method, used by Altintas et al. 46 to identify the instantaneous cutter- workpiece intersection and chip load distribution in the ACIS solid modeling environment. The second is the ana-lytical method. Elbestaw et al. 47,48 employed a NURBS curve to represent the cutting edge profile and then deter-mined the instantaneous in-cut segments and chip load by computing the intersection between the NURBS curve and the locally defined surface. The third is the discrete geome-try method. Jerard et al. 40 used the extended Z-buffer model to represent the workpiece. The instantaneous contact area and chip load were obtained by computing the intersec-tions of the cutter envelope with Z-buffer elements. 2.2 Feed-rate optimization based on cutting force model On the basis of the integration of geometry simulation and cutting force prediction for the five axis milling process, the feed-rate can be optimized according to the predicted cut-ting force. Nowadays, the feed-rate optimization algorithms in the commercial softwares are mostly based on the vol-ume (or material removal rate) analysis. In this common method, the feed-rate is set in inverse ratio to the instanta-neous removed material. The main drawbacks of this me-thod lie in two aspects. Firstly, to a certain degree, the ma-terial removal rate is in the ratio of the magnitude of the instantaneous cutting force, but it cannot predict the direc-tion of the force. Secondly, it is difficult to keep the magni-tude of the instantaneous cutting force at a near-constant level. To overcome these problems, Elbestawi et al. 47,48 proposed a feed-rate optimization method for five axis ma-chining via the cutting force model. Lazoglu et al. 49 com-pleted a comparative study of the force-based feed-rate scheduling strategies. In our work 50, we proposed a feed-rate optimization method for five axis flank milling considering the constraint on the cutting force. Based on the cubic interpolation technique, the optimization model is established by using the time series assigned to the corre-sponding cutter locations as the design variable, the sum of the total time series as the objective function, the machine kinematical performance indices (i.e., velocity, acceleration and jerk of each axis) as the constraint functions, and the maximum magnitude of the instantaneous cutting force as the constraint of the whole milling process. The feed-rate can be calculated via the optimization model. This method is applicable to rough milling of free-form surfaces and semi-finish milling of ruled surfaces or ruled surface-like free-form surfaces. 3 Dynamics simulation The dynamics simulation for five-axis machining is used to obtain the time-varying status data of the cutting process, which are the foundation of the cutting process optimization. The essential work of dynamics simulation includes dy-namics modeling, cutting process stability analysis and cut-ting parameter optimization. 3.1 Dynamics modeling There are three kinds of cutter-workpiece dynamics models: (1) The coupled vibration model of the cutter and the work-piece. The model is often employed in the machining of the thin-wall workpiece. Ratchev et al. 51 proposed a coupled vibration model of the thin-wall workpiece and the cutter based on the FEM. Kovecses et al. 52 studied the analyti-cal vibration model of the thin-wall workpiece. However, the workpiece vibration model and the coupled vibration model for the machining of the sheet workpiece are usually ignored in the existing works. (2) The contact dynamics model of the workpiece and the fixture. Hu et al. 53 ana-lyzed the dynamic stability of the fixture based on the lumped parameter model that comes from the flexible mul-tibody dynamics theory. Kapoor et al. 54 studied the con- DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 3415 tact dynamics model that considers the friction between the fixture and the workpiece. The contact rigidity matrix was identified by the experimental method 55. (3) Melkote et al. 56 investigated the relationship between the dynamic fixture stability and the dynamic actions between the cutter and the workpiece (the time-varying characteristics of the workpiece inertia, rigidity and natural frequency in the process of material removing). 3.2 Stability analysis for machining process Up to now, based on the structural dynamics of the “cut-ter-workpiece” system, considerable attention has been de-voted to stability analysis (or chatter analysis) in the field of dynamic analysis of the milling process. The regenerative chatter and mode coupling chatter are the two main kinds of self-excited chatter. And the former one occurs earlier than the later one in most cases. Altintas and Budak 57 pre-sented an analytical method (ZOA method) for predicting milling stability lobes based on the mean of the Fourier se-ries of the dynamic milling coefficients. This method is ef-ficient and fast, but it cannot predict the existence of the additional stability regions and period doubling bifurcations in the case of low immersion milling. Recently, Altintas and his co-workers 58 explored the multi-frequency solution of chatter stability, which can predict stability lobe diagram accurately in low radial immersion milling. Bayly et al. 59 proposed the temporal finite element analysis (TFEA) for milling stability prediction. This method is efficient and accurate for small times in the cut, but not quite suitable in full and near-full immersion cases. Insperger and Stpn 60 developed the semi-discretization (SD) method. The key point of this method is that only the time-delay term of the dynamical system is discretized while the time domain terms are all unchanged. The multi-frequency and SD me-thods take into account the effect of higher harmonics mostly due to multiple mode excitation or highly interrupted cutting. However, their computational efficiencies are not high 61. In our work 62, a full-discretization (FD) method for prediction of milling stability was introduced. The response of the system of the dynamic milling process considering the regenerative effect is calculated via a direct integration scheme with the help of discretizing the time period. Then, the Duhamel term of the response is solved using FD method, i.e., the involved system state, time-pe- riodic and time delay items are simultaneously approxi-mated by means of linear interpolation. After obtaining the discrete map of the state transition on one time interval, a closed form expression for the transition matrix of the sys-tem is constructed. The milling stability is then predicted based on Floquet theory. Compared with SD method, FD method has much higher computational efficiency without loss of any numerical precision. For one degree of the free-dom milling model, the computation time of the proposed method can be reduced nearly 75%； and for two degrees of freedom milling model, the computation time can be re-duced about 60%. A summary of the methods for stability analysis is listed in Table 4. 3.3 Cutting parameters optimization The existing research on the chatter-free cutting parameters optimization concentrates on the three-axis machining. Bu-dak et al. 63 developed a method to compute the optimum axial and radial depth. In their studies, the optimization ob-ject was to maximize the material removal rate and the con-straint condition was to guarantee the chatter-free machin-ing. Altintas et al. 64 proposed a method to optimize the spindle speed and feed rate based on the milling process simulation and stability prediction. The existing works on the stability prediction and cutting parameter optimization are based on the deterministic parameter model. However, the deterministic model cannot account for the parameter uncertainties in a practical milling process. Usually there are physical and geometrical uncertain parameters in the cutter-workpiece structures. The former includes the elastic module and Poisson ratio, and the later consists of the workpiece thickness and other geometry dimensions. As a result, the chatter-free machining cannot be guaranteed. In the previous studies 65, uncertainties in the milling proc-ess were handled from the perspective of feedback control. The uncertainties in the cutting process were accommodated using a control system, and complex controllers were de-signed to compensate for the known process effects and accounted for the force-feed nonlinearity inherent in the cutting operations. To our best knowledge, the milling process dynamics taking account of the uncertain parame-ters are addressed little. We presented a robust optimization model for selecting cutting parameters in five-axis milling 6668. The interval algebra was introduced to characterizeTable 4 Comparison of methods for stability analysis Scope of application Methods Low radial immersion High radial immersion Computational load ZOA method 57 No Yes Lowest Frequency domain Multi-frequency method 58 Yes Yes High TFEA method 59 Yes No Low Semi-discretization method 60 Yes Yes High Time domain The proposed method 62 Yes Yes Low 3416 DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 the parameters uncertainties. The upper and lower bounds of the stability lobe diagram and cutter dynamics response were obtained using the sensitivity analysis technique. In comparison with the deterministic model, the stability of the milling process was guaranteed by adopting the results de-rived from the robust optimization model. 4 Future work The five-axis NC machining provides a valid and efficient way to manufacture the mechanical parts with complex shapes, which are widely used in aerospace, energy and national defense industries. Its technology innovations have attracted much attention in recent years. The involved fun-damental theories and key technologies have become the major topics of many national projects. Some emerging trends for future studies in this field are as follows: (i) The development of the full dynamics model for simulating the five-axis cutting process. Dynamics simu-lation is the foundation of high-efficiency and high-precise NC machining. The cutting system is composed of the ma-chine tool, the cutter and the fixture. Most of the existing works focus on the individual sub-systems, however, the full dynamics model should be considered. For example, the coupling between the large overall rigid motion and the vibration of the milling cutter is usually ignored in the pre-vious studies. However, the cutter feed-rate is time-varying during the five-axis machining process because of the addi-tional rotational motion. According to the multibody dy-namics theory, the elastic deformation of the cutter is af-fected by the acceleration of its large overall rigid motion. Therefore, the relationship between the cutting parameters and the machined surface quality should be investigated based on a full dynamics model taking into account this coupling effect. (ii) The development of the intelligent closed-loop manu-facturing methodology that integrates design, machining and measurement. Due to the time-varying cutting condi-tions and a lot of uncertain factors in the five-axis machin-ing, open-loop machining sometimes cannot satisfy the high quality requirements in both geometry precision and physi-cal performance. The closed-loop machining is a valid way to solve this problem and will be an important research area. The closed-loop machining consists of three fundamental steps: (1) Planning and simulating the cutting process； (2) Measuring the machined surface and analyzing the meas-ured data. (3) Evaluating the machined surface quality and re-designing the nominal surface. The challenging issues include the efficient online measurement to obtain the data about the geometry precision and physical performance, the re-design of the nominal surface based on the requirement of the physical performance, the accurate computation of the material volume to be removed in the compensation machining process and the process planning that considers the dynamics characteristics and physical constraints of the cutting process. (iii) The development of the multiphysic model for simu-lating the surface forming process. Higher surface quality is more and more desired by high-performance parts. The surface quality is affected by the cutting force, the cutting heat, the cutting deformation, and so on. The multiphysic modeling of these coupling effects is the foundation of cut-ting process control and cutting parameter optimization. The existing multiphysic models mainly apply to turning and the three-axis milling processes. The multiphysic simulation of the five-axis milling process becomes a challenging prob-lem because of the time-varying cutting conditions. The key issues in multiphysic simulation include the quantitive de-scription, prediction and control of the physic field, the mapping between the cutting parameters and the physical performance of the machined surface, and the new cutting process optimization approaches. This work was supported by the National Basic Research Program of China (2005CB724103) and the National Natural Science Foundation of China (50835004). 1 Ding H, Xiong Y L. Computational manufacturing. Prog Nat Sci, 2002, 12: 816 2 Wang Q H, Li J R, Zhou R R. Graphics-assisted approach to rapid collision detection for multi-axis machining. Int J Adv Manuf Tech, 2006, 30: 853863 3 Ilushin O, Elber G, Halperin D, et al. Precise global collision detec-tion in multi-axis NC-machining. Comput Aided Design, 2005, 37: 909920 4 Morishige K, Takeuchi Y, Kase K. Tool path generation using C-space for 5-axis control machining. J Manuf Sci E-T ASME, 1999, 121: 144149 5 Wang J, Roberts C A, Danielson S. Local and global accessibility evaluation with tool geometry. Comput Aided Des Appl, 2007, 4: 1929 6 Yin Z P, Ding H, Xiong Y L. Visibility theory and algorithms with application to manufacturing processes. Int J Prod Res, 2000, 38: 28912909 7 Yin Z P, Ding H, Xiong Y L. Accessibility analysis in manufacturing processes using visibility cones. Sci China Ser E-Tech Sci, 2002, 45: 4757 8 Yang W Y, Ding H, Xiong Y L. Manufacturability analysis for a sculptured surface using visibility cone computation. Int J Adv Manuf Tech, 1999, 15: 317321 9 Balasubramaniam M, Laxmiprasad P, Sarma S, et al. Generating 5-axis NC roughing paths directly from a tessellated representation. Comput Aided Design, 2000, 32: 261277 10 Morimoto K, Inui M. A GPU based algorithm for determining the optimal cutting direction in deep mold machining. In: ISAM - IEEE Int Symp Assem Manuf, Michigan USA, 2007, 203208 11 Ding H, Bi Q Z, Wang Y H, et al. Collision-free and orienta-tion-smooth tool path generation method for five-axis NC machining. Chinese Patent, Grant No. ZL200710045183.9, 2007-08-23 12 Bi Q Z, Wang Y H, Ding H. A GPU-based algorithm for generating collisionfree and orientation-smooth five-axis finishing tool paths of a ball-end cutter. Int J Prod Res, 2010, 48: 11051124 13 Zhu L M, Zheng G, Ding H. Formulating the swept envelope of ro-tary cutter undergoing general spatial motion for multi-axis NC ma-chining. Int J Mach Tool Manu, 2009, 49: 199202 14 Zhu L M, Zhang X M, Zheng G, et al. Analytical expression of the DING Han, et al. Chinese Sci Bull October (2010) Vol.55 No.30 3417 swept surface of a Rotary cutter using the envelope theory of sphere congruence. J Manuf Sci E-T ASME, 2009, 131: 4456 15 Wang X C, Wu X T, Li Y B. Second order curvature catering-a new approach in manufacture of sculptured surfaces. Acad J XJTU, 1992, 26: 5158 16 Rao A, Sarma R. On local gouging in five-axis sculptured surface machining using flat-end tools. Comput Aided Design, 2000, 32: 409420 17 Gong H, Cao L X, Liu J. Second order approximation of tool enve-lope surface for 5-axis machining with single point contact. Comput Aided Design, 2008, 40: 604615 18 Zhu L M, Ding H, Xiong Y L. Third Order Point Contact Approach for Five-Axis Sculptured Surface Machining Using Non-ball End Tools -Part I: Third Order Approximation of Tool Envelope Surface. Sci China: Tech Sci, 2010, 53: 19041912 19 Zhu L M, Ding H, Xiong Y L. Third order point contact approach for five-axis sculptured surface machining using non-ball end tools-Part II: Tool positioning strategy. Sci China: Tech Sci, 2010, 53: 2190 2197 20 Lartigue C, Duc E, Affouard A. Tool path deformation in 5-axis flank milling using envelope surface. Comput Aided Design, 2003, 35: 375382 21 Gong H, Cao L X, Liu J. Improved positioning of cylindrical cutter for flank milling ruled surfaces. Comput Aided Design, 2005, 37: 12051213. 22 Ding H, Zhu LM. Global optimization of tool path for five-axis flank milling with a cylindrical cutter. Sci China Ser E: Tech Sci, 2009, 52: 24492459 23 Ding Y, Zhu L M, Ding H. Semidefinite programming for Chebyshev fitting of spatial straight line with applications to cutter location planning and tolerance evaluation. Precis Eng, 2007, 31: 364368 24 Zhu L M, Zheng G, Ding H. Global optimization of tool path for five-axis flank milling with a conical cutter. Comput Aided Design, 2010, doi: 10.1016/j.cad.2010.06.005 25 Zhang X M, Zhu L M, Zheng G, et al. Tool path optimisation for flank milling ruled surface based on the distance function. Int J Prod Res, doi: 10.1080/00207540902993019 26 Chiou C J. Accurate tool position for five-axis ruled surface machin-ing by swept envelope approach. Comput Aided Design, 2004, 36: 967974 27 Redonnet J M, Rubio W, Dessein G. Side milling of ruled surfaces: Optimum positioning of the milling cutter and calculation of inter-ference. Int J Adv Manuf Tech, 1998, 14: 459465 28 Menzel C, Bedi S, Mann S. Triple tangent flank milling of ruled sur-faces. Comput Aided Design, 2004, 36: 289296 29 Castagnetti C, Duc E, Ray P. The Domain of Admissible Orientation concept: A new method for five-axis tool path optimisation. Comput Aided Design, 2008, 40: 938950 30 Bi Q Z, Wang Y H, Zhu L M, et al. Wholly smoothing cutter orienta-tions for five-axis NC machining based on cutter contact point mesh. Sci China: Tech Sci, 2010, 53: 12941303 31 Kersting P, Zabel A. Optimizing NC-tool paths for simultaneous five-axis milling based on multi-population multi-objective evolu-tionary algorithms. Adv Eng Softw, 2009, 40: 452463 32 Langeron J M, Duc E, Lartigue