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Review SPECIAL TOPIC: Huazhong University of Science and Technology Mechanical Engineering

October 2010 Vol.55 No.30: 3408−3418 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 2

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China; 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 recent years. In this paper, the state-of-the-art techniques for five-axis machining process planning are summarized and the challenging 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) planning for line contact and three-order point contact machining, shape control of cutter envelope surface and milling stability prediction 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: 3408−3418, doi: 10.1007/s11434-010-3247-7

In conventional three-axis NC machining only the translation motions of the cutter are permitted while the cutter orientation is allowed to change in a five-axis machine tool because of the two additional rotational axes. The advantages of five-axis NC machining mainly depend on the control 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 machining. For example, it is possible to shorten the tool overhang length if the tool orientation is optimized. Determining the safe and shortest tool length is very helpful when *Corresponding author (email: [email protected])

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

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 machined surface quality can also be controlled by changing the cutter orientation. Besides the above advantages, there exist several challenging 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 fiveaxis machining fall into three categories [1]: tool path generation, integrated geometric/mechanistic simulation and dynamics simulation, as shown in Figure 1. Tool path generation 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 csb.scichina.com

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the cutting efficiency and quality. It is also the foundation of integrated geometric/mechanistic simulation, which depends 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 predicted. The cutting force then can be applied to dynamics simulation, feedrate scheduling, and prediction and compensation 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 characteristics of the machine tool-cutter-fixture system. Dynamics 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 dynamics simulation software CutterPro. European Commission supported a project about flank milling optimization that is called “Flamingo”. Because of the obvious advantages of flank milling in cutting efficiency and surface quality, a number of famous companies (SNECMA, Rolls Royce, Dassault Systèmes) 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. Domestic researchers have developed some CAM systems such as KM, 5BDM and Dynacut, but the fundamental researches 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 dynamic cutting process. However, the performances in intelligence, usability and computation efficiency still need to be improved. For instance, the selection of the strategy for cutter orientation optimization depends on the skill of the programmer; it is difficult to automatically generate the optimum tool orientations that consider simultaneously all the

Figure 1

Three challenging problems in five-axis NC machining.

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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 machining 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 machining efficiency and quality, the tool orientation of each cutter location (CL) data should be optimized by considering the important factors related to a practical cutting process. The factors consist of geometrical constraints, kinematic constraints, dynamic characteristics and physical factors. 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 between 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 algorithms to improve the collision detection efficiency and adjust cutter orientations to avoid collision. In practical applications, tool paths are usually composed of thousands to hundred thousands of tool positions. The collision detection often requires large computation time and resource. Therefore 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 affecting 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 collision-free cutter orientations is called accessibility cone. The cutter orientations are then generated in the accessibility

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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) Accessibility 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 manufacturability can be directly determined. Furthermore, the cutter orientation optimization can be carried out in the collision-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 accessibility cones. Usually computing accessibilities will cost large computation time because complex shape may consist of hundreds of thousands of polygonal meshes. Some algorithms were proposed to improve computation efficiency such as the C-space (Configuration Space) methods [4,5] and visibility-based methods [6−10]. 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 triangles 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 Table 1

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radius is ignored. Then the problem of collision avoidance is transformed into that of visibility. We [6−8] described cutter’s visibility cone using the concept of C-space and proposed three strategies to accelerate the computation speed using the hidden-surface removal techniques in computer 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 several cylindrical shapes with finite radii. The real accessible directions cannot be directly obtained from the visibility cone, and secondary collision checking and avoidance strategies are still needed [9]. The accessibility will be equal to the visibility if both the machined surface and the interference checking surface are replaced by their offset surfaces [10]. However, the offset surface is usually not easy to obtain and the collision avoidance of the tool holder cannot be guaranteed. Furthermore, the method only applies to ballend 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 algorithm 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 accessibility of the cutter is then equal to the complete visibilities of the involved cylinders and cones. This equivalence provides an efficient method for detecting the accessibility of the milling cutter by using the occlusion query function of the graphics hardware. The computation efficiencies 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 triangles 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

The comparison of computation time

Method

Computation platform

Ref. [9] Ref. [3] Our method [12]

SGI work station, Dual CPU 250M CPU 2.4G, RAM 512M CPU 2.4G, RAM 512M

Triangle 10665 12600 139754

Inputted models Cutter center point Cutter orientations 1500 80 50000 1 2000 1026

Computation Average computatime tion time 51.63 m 61.61s 60.53 s

2.58×10−2 s 1.23×10−3 s 2.95×10−5 s

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relatively easy to generate the tool path. From the manufacturer’s 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 surfaces with non-ball-end cutters. In five-axis machining, 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 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 approximate or simplified models, which formulate the problem of optimal cutter positioning as that of approximating the cutter 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 [15−17]. 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 methods cannot be applied to other types of rotary cutters. Recently, Gong et al. [17] developed a mathematical model that describes the second-order approximation of the envelope 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 contact of second-order with the design surface at the CC point. However, theoretically speaking, a third-order contact between 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 second-order model does not take full advantage of the efficiency 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 collision 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

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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, respectively, a mathematical model describing the third-order approximation 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 determined by those of the cutter surface and the design surface. 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 efficiently machining free-form surfaces with non-ball-end cutters. The optimal CL was obtained by adjusting the inclination and tilt angles of the cutter until its envelope surface 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 complex surfaces. Numerical examples demonstrated that the third-order point contact approach could improve the machining 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 simulation 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 component 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 according to the least-squares criterion. To simplify the computation, 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)

δ = 0.005 δ = 0.01

0.69 0.98

Toroidal cutter (Second order contact) 2.48 3.12

Toroidal cutter (Third order contact) 5.28 6.14

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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. Furthermore, the geometric deviation of the machined surface from the nominal one was not clearly defined and the influence 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 second-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 developed from the perspective of surface approximation following the minimum zone criterion, and applied to flank milling of non-developable ruled surface. The geometrical precision 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 results are listed in Table 3. In our model, the envelope surface 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 rotary 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 represented 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)

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a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface 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 optimization problems in a unified framework. The sequential approximation algorithm along with a hierarchical algorithmic 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 50×100 points were sampled from the design surface. A smooth axis trajectory surface was generated by using Chiou’s method [26]. The maximum undercut and overcut were 0.0896 mm and 0.0239 mm, respectively. After 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 optimization 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 orientations 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 cutter orientations relative to the workpiece and the change of cutting conditions, respectively. The measure defined in the machine tool coordinate system 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 coordinate system to improve the cutting efficiency and the evenness of the rotational motions. Their results showed that the cutting time could be greatly shortened by optimizing cutting orientations. We [11,12] proposed a model to globally

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smooth cutter orientations in the feasible space. The constraint 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 algorithm 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 Systèmes 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 machining 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 constructed. The problem of point-line motion design was converted to that of projective Bézier or B-spline image curve design in the hyperplane of dual quaternion. The resulting point trajectory is a NURBS curve in Euclidean threedimensional space and orientation curve is a NURBS curve on unit sphere. Such two NURBS curves, named double-NURBS curves, can describe the tool path in NC machining. 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 between 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] proposed 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 orientations 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 efficiency, evenness of feed velocity and smoothness of cutting force. (ii) Shorter tool length. The use of shorter cutters without collision is a key advantage of five-axis machining because 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 collision-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

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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 orientations 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 machined 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] proposed 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 process. 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 scheduling, spindle speed optimization, chatter prediction, adaptive 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 experimental calibration [39,40]. So, modeling swept volume of cutter and cutter-workpiece engagement becomes a primary 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 generally needed to be solved in the numerical methods, which require great computation costs. Chiou et al. [42,43] re-

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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 improper 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 computations 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 expressions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spatial 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 constructed without complicated calculations, and that the minimum 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 constructing 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 cutterworkpiece intersection and chip load distribution in the ACIS solid modeling environment. The second is the analytical method. Elbestaw et al. [47,48] employed a NURBS curve to represent the cutting edge profile and then determined 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 geometry 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 intersections 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 cutting force. Nowadays, the feed-rate optimization algorithms

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in the commercial softwares are mostly based on the volume (or material removal rate) analysis. In this common method, the feed-rate is set in inverse ratio to the instantaneous removed material. The main drawbacks of this method lie in two aspects. Firstly, to a certain degree, the material removal rate is in the ratio of the magnitude of the instantaneous cutting force, but it cannot predict the direction of the force. Secondly, it is difficult to keep the magnitude 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 machining via the cutting force model. Lazoglu et al. [49] completed 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 corresponding 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 dynamics modeling, cutting process stability analysis and cutting 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 workpiece. 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 analytical 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] analyzed the dynamic stability of the fixture based on the lumped parameter model that comes from the flexible multibody dynamics theory. Kapoor et al. [54] studied the con-

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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 “cutter-workpiece” system, considerable attention has been devoted 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] presented an analytical method (ZOA method) for predicting milling stability lobes based on the mean of the Fourier series of the dynamic milling coefficients. This method is efficient 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 Stépán [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 methods 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-peTable 4

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riodic and time delay items are simultaneously approximated 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 system 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 freedom 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 reduced 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. Budak et al. [63] developed a method to compute the optimum axial and radial depth. In their studies, the optimization object was to maximize the material removal rate and the constraint condition was to guarantee the chatter-free machining. 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 process were handled from the perspective of feedback control. The uncertainties in the cutting process were accommodated using a control system, and complex controllers were designed 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 parameters are addressed little. We presented a robust optimization model for selecting cutting parameters in five-axis milling [66−68]. The interval algebra was introduced to characterize

Comparison of methods for stability analysis Methods

Frequency domain Time domain

ZOA method [57] Multi-frequency method [58] TFEA method [59] Semi-discretization method [60] The proposed method [62]

Scope of application Low radial immersion High radial immersion No Yes Yes Yes Yes No Yes Yes Yes Yes

Computational load Lowest High Low High Low

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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 derived 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 fundamental 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 simulation is the foundation of high-efficiency and high-precise NC machining. The cutting system is composed of the machine 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 previous studies. However, the cutter feed-rate is time-varying during the five-axis machining process because of the additional rotational motion. According to the multibody dynamics theory, the elastic deformation of the cutter is affected 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 manufacturing methodology that integrates design, machining and measurement. Due to the time-varying cutting conditions and a lot of uncertain factors in the five-axis machining, open-loop machining sometimes cannot satisfy the high quality requirements in both geometry precision and physical 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 measured 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

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the dynamics characteristics and physical constraints of the cutting process. (iii) The development of the multiphysic model for simulating 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 cutting 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 problem because of the time-varying cutting conditions. The key issues in multiphysic simulation include the quantitive description, 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 2

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