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# Scallop cusps. (a) Ball-end milling is shown. A design surface (dark grey) and machined surface (light blue), consisting of several machined strips, are shown. (b) A zoom-in of the machined strips. The intersection of two neighbor strips introduces a sharp edge (cusp) and its distance from the design surface is known as the scallop height.

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We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet t...

## Contexts in source publication

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... strip width and the optimal machining direction. The research proposed here aims at the finishing stage of machining when the motion of the milling tool completes the desired shape of the workpiece. Typically, this is a very time demanding process because small-radii milling tools are used to eliminate or reduce the remaining scallop cusps, see Fig. 1. To optimize performance of this operation, various approaches varying tool orientation have been developed. The idea is to adapt the milling tool, usually a torus or a cylinder, such that the contact circle possesses higher order contact with the surface. This technique is known as curvature matched machining, see [1,21,22,9,25,5] and ...

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... main goal of this research is to detect large parts of free-form surfaces that are manufacturable by a single sweep of a rotary cutter. We emphasize here again that such a patch is scallop-free, cf. Fig. 1, because of the tan- gential contact between the two surfaces (the cutter and the material block), and therefore is desirable for CNC machining because it does not require any post-process ...

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... both sur- face of revolution Φ and its trajectory R, see Fig. 12 Check if the distance between Φ and Ω is bellow ε Return surface of revolution Φ opt and its motion R opt ...

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... an initialization aims at finding the best tangential mo- tion of a rigid line. Its trajectory, a ruled surface R ini , provides an initial guess for the optimization stage, explained later in Figure 11: Optimization settings. The initial ruled surface R(t, s), see (7), is uniformly discretized along the rulings (s-direction) with n = 7 samples (green dots). ...

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... 4.5. The reason is that by now, the surfaces of revolu- tion associated to l vary over time and the optimization seeks for congruent surfaces of revolution, see Fig. ...

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... we represent the two boundary curves a(t) and b(t) as cubic B-spline curves. Further, we equidistantly sample s ∈ [0, 1] and thus obtain trajectories R(t, s j ), j = 1, . . . n, of n uni- formly sampled points on l (see Fig. 11). We denote these trajectories by C j (t) and the vector of unknown distances by Four iterations of the optimization tested on an ex- act envelope surface are shown. In every iteration, the optimized milling tool (framed) and its motion (yellow) are displayed. The actual envelopes are color- coded by the absolute value of the one-sided ...

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... would result in only quadratic objective function (12), however, alternating optimizations often suffer with slow convergence, and therefore we optimize all unknowns simultaneously. If not stated differently in Section 5, we use the default values µ 1 = 1, µ 2 = µ 4 = 0.1, and µ 3 = 0.001. Four iterations of the optimiza- tion cycle are shown in Fig. ...

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... loop. As the ruled surface R is changed, the surface of revolution Ψ is also updated by recomputing its meridian. This is achieved by computing a planar envelope of a set of circles centered along l having d j s as radii. By default, 30 uniformly sampled points on l are used and the half-meridian is a cubic spline curve with ten control points. Fig. 13 shows four iterations of the half-meridian ...

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... this section, we present several examples of envelope de- tection. An example with an exact envelope as the input surface is shown in Fig. 14 top. The color-coding reflects the one-sided absolute error ε between the machined (Ω) and the designed (Φ) surfaces, i.e., considered over a discrete set of samples of the ruled surface R, see also Fig. 11 as a reference for this sampling. If not stated differently, the design surface Φ is normalized such that the di- agonal of its ...

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... this section, we present several examples of envelope de- tection. An example with an exact envelope as the input surface is shown in Fig. 14 top. The color-coding reflects the one-sided absolute error ε between the machined (Ω) and the designed (Φ) surfaces, i.e., considered over a discrete set of samples of the ruled surface R, see also Fig. 11 as a reference for this sampling. If not stated differently, the design surface Φ is normalized such that the di- agonal of its bounding box is one. Additionally in Fig. 14, an exact envelope was deformed by locally destroying the exact envelope property. The results show that our algorithm still de- tects a dominant sub-patch of the ...

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... the one-sided absolute error ε between the machined (Ω) and the designed (Φ) surfaces, i.e., considered over a discrete set of samples of the ruled surface R, see also Fig. 11 as a reference for this sampling. If not stated differently, the design surface Φ is normalized such that the di- agonal of its bounding box is one. Additionally in Fig. 14, an exact envelope was deformed by locally destroying the exact envelope property. The results show that our algorithm still de- tects a dominant sub-patch of the exact envelope, but optimizes the surface of revolution and its trajectory ...

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... results of another test conducted on the exact envelope are shown in Fig. 15 where a random noise was applied on the exact envelope. The optimized solution differs marginally (ε < 0.1%) from the one without noise, cf. Fig. 14 top, which shows a very good stability of our algorithm. Fig. 16 shows another reconstruction from distorted data where an exact, yet incomplete, envelope Φ is given as the in- put. Our ...

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... results of another test conducted on the exact envelope are shown in Fig. 15 where a random noise was applied on the exact envelope. The optimized solution differs marginally (ε < 0.1%) from the one without noise, cf. Fig. 14 top, which shows a very good stability of our algorithm. Fig. 16 shows another reconstruction from distorted data where an exact, yet incomplete, envelope Φ is given as the in- put. Our method recovers the exact solution within a very fine threshold on the distance error, see Eq. (13); ε < 0.1% of the bounding box of ...

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... results of another test conducted on the exact envelope are shown in Fig. 15 where a random noise was applied on the exact envelope. The optimized solution differs marginally (ε < 0.1%) from the one without noise, cf. Fig. 14 top, which shows a very good stability of our algorithm. Fig. 16 shows another reconstruction from distorted data where an exact, yet incomplete, envelope Φ is given as the in- put. Our method recovers the exact solution within a very fine threshold on the distance error, see Eq. (13); ε < 0.1% of the bounding box of ...

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... example testing our algorithm on industrial data is shown in Fig. 17. The depicted geometry is a reference surface that is used for benchmarking toolpath generation and material re- moval simulation algorithms in industrial settings. A single sweep approximation is compared with an approach using sev- eral patches. A surface-driven initialization was used and the direction of the rulings were set ...

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... example with industrial data is shown in Fig. 18. The initialization was surface-driven and the shape of the tool was constrained to be linear. Two envelopes of the optimal con- ical cutters that approximate the input geometry within a very fine threshold ε = 0.00045 are ...

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... this experiment, a surface-driven initialization was used, now requiring the axis direction to be roughly perpendicular to the dominant edge of the reference surface. The optimal conical cutters and their motions derived by our algorithm are displayed. A comparison of our two initialization strategies is shown in Fig. 21: the line-driven (Section 4.3) and the surface-driven (Section 4.4) initialization of the motion of the milling axis. An example of the approximation of a free-form blade using a conical envelope is shown in Fig. 22. The initialization was surface-driven; the result satisfies a very fine tolerance (ε < ...

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... with Φ. Since our objective was to approximate Φ with one, or several, large patches of the size ideally equal the designed surface, we did not consider the collision detection in our optimization framework. We con- duct collision detection as a post-process, by testing penetration of the half-axes, provided by some thickness, with Φ. Fig. 19 shows two results of our algorithm where one solution passed this test whilst the other ...

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We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet t...

We investigate a recently introduced methodology for 5-axis flank computer numerically controlled (CNC) machining, called double-flank milling. Certain geometries, such as curved teeth of spiral bevel gear, admit this approach where the milling tool has tangential contact with the material block on two sides, yielding a more efficient variant of fl...

## Citations

... Namely, the robot machining performance would be improved compared with the two-step framework. However, research about this is lacking due to the relatively high number of variables, the complexity of tool positioning and machining error evaluation in flank milling [30]. To address this problem, a direct planning approach for the flank milling toolpath of a 6-DoF robot is proposed in this paper. ...

Robotic flank milling has outstanding advantages in machining large-scale slender surfaces. Currently, the paths for this process are mainly generated by optimizing redundant robot degrees of freedom (DoFs) on the basis of conventional 5-axis flank milling paths. This two-step framework, however, does not enable optimal robot kinematic and dynamical performance compared to the direct generation of 6-DoF robot paths, limiting the machining efficiency and effectiveness. This paper presents an optimization method to directly generate a toolpath with six DoFs for robotic flank milling. Firstly, the kinematic model of the milling system and the representation of the 6-DoF toolpath are established. Then, the standard geometric error for flank milling that conforms to the geometric specification is defined, and an efficient algorithm based on conformal geometric algebra is proposed to solve it. On this basis, the toolpath optimization model with toolpath smoothness and robot stiffness as objective functions is established. A sequential quadratic programming algorithm is proposed to solve this highly non-linear problem based on the lexicographic order of arrays. The simulations and experiments demonstrate that the proposed method has better efficiency, robustness, and effectiveness compared with the existing methods. Due to the improvement of smoothness and stiffness, the productivity, accuracy, and finish of the machining are all enhanced.

... (1) Noninterference tool orientations; [8][9][10][11][12][13][14][15] (2) Cutter shape optimization; [6,7,[16][17][18][19][20] Determining the noninterference tool orientation in multi-axis machining is a precondition for designing a useable large-size cutter and noninterference machining. The existing research for the noninterference tool orientation is mainly carried out from two strategies. ...

... The first aspect is mainly aimed at side milling. Its purpose is to optimizing the cutter shape to make the rotary surface of the cutter matching the machined surface so that the machining error is reduced [16][17][18][19][20]. Zhu et al. [16,17] optimized the conical cutter shape for side milling by minimizing the distance between the envelope surface of the cutter and the machined surface. ...

... But due to the relatively fixed shape of the conical cutter, it only suits to optimize some surfaces with specific geometric features, such as the ruled surface. To solve this problem, some studies tried to use drum cutter and optimize its shape [18][19][20]. Bo et al. [18] improved machining accuracy by optimizing the similarity between the rotating generatrix of the drum cutter and the isocurve of the machined surface. Meng et al. [19] proposed a comprehensive drum cutter optimization method. ...

Selecting the appropriate cutter size and determining the noninterference tool orientation are the key steps in planning the four-axis machining of complex channel parts. In this paper, based on the invariance property of isometric transformation and the function of the minimum distance between surfaces, a nonlinear numerical optimization method is proposed to accurately solve the noninterference critical tool orientation and maximum taper angle of the conical cutter. Furthermore, the optimization method for the maximum taper angle of the conical cutter can handle the different cases, whether the tool orientation is preset or not. Firstly, the basic theory of the proposed optimization method for the conical cutter is given. Secondly, the optimization method for calculating the noninterference tool orientation is provided. Thirdly, the local optimization method for the taper angle of a conical cutter with the tool orientation preset, and the global optimization method for the maximum taper angle of the conical cutter are provided. The conical cutter solved by the local optimization method can reach critical state with a single constraint surface, but the conical cutter solved by the global optimization method can reach critical state with multiple constraint surfaces. Finally, the proposed method is verified by the simulation experiments. The results show that the provided method accurately calculates the result for the noninterference tool orientation of a given conical cutter and the max taper angle of the conical cutter. Furthermore, it proves that the collaborative optimization strategy of the tool orientation and cutter taper angle used in the global optimization method is much better for the max taper angle of the conical cutter.

... This subsequently broadens the application range of the high-order osculating method, but highly skilled programmers are then required due to the complexity of the relevant algorithms. Peng et al. [6,7] introduced a new method that approximates free-form surfaces by envelopes of the revolving surfaces' one-parameter motions. An automatic algorithm was also proposed to detect patches of free-form surfaces that can be approximated by rotational cone envelopes under a rigid body motion, thus considerably reducing the requirements for human intervention and execution times. ...

To demonstrate the closeness between the tool and nominal surfaces during the flank milling of centrifugal blades, discrete points on the conical cutter tool axis are projected in the nominal surface normal direction, and a mapping curve on the conical surface is obtained. On this basis, a novel toolpath planning method, under hub surface constraints, is proposed for flank milling centrifugal blades using a conical cutter based on the BUA method. This is a nonlinear optimisation problem, therefore the programming linearization method is provided. The effectiveness of the proposed BUA method is validated using a numerical example featuring a ruled surface and compared against the least square (LS) and two-point offset (DPO) methods. The influence of the conical cutter cone angle on the planning error is also analysed. The proposed BUA method is also applied to a centrifugal blade flank milling numerical example and its feasibility is verified.

... Another class of research considers curved tools for flank milling, either of a given shape [7,23], or custom-shaped [24,25,26,27,28]. These approaches do not look only for the motion of the tool as a variable, but optimize also the shape of the tool itself. ...

... In our contribution, we do not consider these physical issues and focus solely on the geometric approximation between the design surface and the milled conical envelope. In contrast to [20,25], we consider conical tools, but our computational framework supports fixed conical tools as well as those resulting from our optimization process. ...

Existing flank milling path-planning methods typically lead to tiny gaps or overlaps between neighboring paths, which causes artifacts and imperfections in the workpiece. We propose a new multi-strip path-planning method for 5-axis flank milling of free-form surfaces which targets G1 (tangent-plane) continuity of the neighboring strips along shared boundaries. While for some geometries one cannot achieve G1 continuity and high approximation quality at the same time, our optimization framework offers a good trade-off between machining accuracy in terms of distance error and the G1 connection of neighboring strips. We demonstrate our algorithm on synthetic free-form surfaces as well as on industrial benchmark datasets, showing that we are able to meet fine industrial tolerances and simultaneously significantly reduce the kink angle of adjacent strips, and consequently to improve the surface finish in terms of smoothness.

... Free-form surface parts with large size and thin walls are widely used in the aerospace industry to form aerodynamic shapes, such as large aircraft skin parts. Large skin parts are generally manufactured by CNC (computer numerical control) machine tools from the blank obtained by sheet metal forming, stretch forming, welding, etc. [1]. Due to the poor rigidity of the thin-walled parts, the low dimensional accuracy of the blank, and the deformation induced by clamping and self-gravity, the actual shape of the blank is inconsistent with the expected shape [2]. ...

Machining contours on a large thin-walled part is a difficult task due to inevitable deformation of the actual part. The deformation of the contour can significantly influence the rigidity of the thin-walled part; thus, it is of importance to minimize the deformation of the contour compared with the design contour. This paper proposes a novel method to adaptively calculate the optimal contour on the deformed surface based on optimized parallel projection. The geometry data of free-form surface is obtained by on-machine measurement (OMM), and then, the actual surface is reconstructed using the measurement data. The reconstructed surface is matched with the design surface, and then, the design contour was projected onto the reconstructed surface to obtain the locating contour. The iterative nearest point (ICP) algorithm is used to adjust the design contour to match the locating contour. Finally, the projection direction is optimized to minimize contour deformation, and the adjusted contour is projected onto the reconstructed surface along the optimal direction to obtain the new contour. The effectiveness of this method is verified by machining and measuring experiments.

... Because the machining cost is high, ordinary CNC machine tools are used by some manufacturers instead of the five-axis CNC machine tools that have very high costs. Therefore, it is necessary to analyze the relationship and transform the corresponding profile of the rotor in different cross-sections [12] [13] [14] [15] [16]. ...

... This subsequently broadens the application range of the high-order osculating method, but highly-skilled programmers are then required due to the complexity of the relevant algorithms. Peng et al. [6][7] introduced a new method that approximates free-form surfaces by envelopes of the revolving surfaces' one-parameter motions. An automatic algorithm was also proposed to detect patches of free-form surfaces that can be approximated by rotational cone envelopes under a rigid body motion, thus considerably reducing the requirements for human intervention and execution times. ...

... The one-sided Hausdorff distance of the characteristic curve relative to the design surface can be changed by varying these three parameters. Consequently, the mapping curve formula can be expressed as follows: (6) where, ( , , ) ...

To demonstrate the closeness between the tool and nominal surfaces during the flank milling of centrifugal blades, discrete points on the conical cutter tool axis are projected in the nominal surface normal direction, and a mapping curve on the conical surface is obtained. On this basis, a novel toolpath planning method, under hub surface constraints, is proposed for flank milling centrifugal blades using a conical cutter based on the BUA method. This is a nonlinear optimisation problem, therefore the programming linearisation method is provided. The effectiveness of the proposed BUA method is verified via numerical example of a ruled surface and compared with the least square (LS) and two-point offset (DPO) methods. The influence of the conical cutter cone angle on the planning error is also analysed. The proposed BUA method is also applied to a centrifugal blade flank milling numerical example and its feasibility is verified.

... To improve efficiency and quality of free-form surface machining using formed cutters is not a new idea. There are several relevant works that deal with five-axis CNC flank machining of complex surfaces using curved tools (Bo et al., 2016;Machchhar et al., 2017;Bo & Barton, 2019;Rajain et al., 2022). Although form cutters are more expensive than standardized straight cutters, to replace specialized machines with CNC machine tools is still valuable and cost effective in practice. ...

Computer numerical control (CNC) milling provides greater flexibility and universality for machining complex gears compared to dedicated gear manufacturing. A critical challenge in popularizing the use of 5-axis flank milling to spiral bevel gears is to achieve acceptable machining accuracy that ensures the meshing performance of the finished gears. Previous studies, which used approaches such as gear design modification, using multiple tool paths, and end milling, failed to resolve this issue. Thus, this paper proposes a computational scheme to improve the machining accuracy of 5-axis flank milling of spiral bevel gears by optimizing the tool path and cutter geometry. The scheme minimizes the geometric deviations between the machined surface and original design using heuristics and optimization algorithms. A simplified tooth contact analysis method was developed to quantitatively evaluate the contact path of the meshing gears. The simulation results of real gears show that the proposed scheme outperforms previous methods in reducing machining errors and further enhance the meshing performance by optimizing the design of form cutters. This work developed an effective approach for flexible and low-cost manufacturing of complex gears.

... Five-axis computer numerical control (CNC) machining is widely used to manufacture various complex parts in the field of molds, dies, and aeroengine. In computer-aided manufacturing (CAM) environments, the toolpaths are always splined [1][2][3], but they are mostly described by the consecutive small linear segments when fed into CNC. Due to the tangential discontinuity of the path, the tool often needs to stop at each corner point, which not only reduces the efficiency of machining but also harms the machining accuracy. ...

... To extend the local smoothing methods to 5-axis toolpaths, a typical method is conducted based on the tool position path and the orientation path on the unit sphere. Tulsyan and Altintas [12] generated the 3 C continuous toolpath by inserting B-spline curves at the corners of the tool position path and tool orientation path. In this approach, the control points of the tool orientation splines need to be evaluated by a Newton-Raphson iterative process. ...

... Thus, the polynomial is defined as follows to ensure the 3 G continuity of the toolpath. It should be noted that although all elements in this section are represented in conformal space using CGA, the 3 G continuity achieved is for curves in 6D Euclidean space with the help of geometric features. As proved in Appendix A, the path of TTP separated from the 6D space has the same continuity as long as any three adjacent TTP locations in the linear path are not co-linear, and the same conclusion is reached for the TAP path. ...

The chain of linear path segments is widely used in five-axis machining, but the tangential discontinuity of the path reduces machining efficiency and accuracy. This paper proposes an analytical G3 continuous corner smoothing method based on the conformal geometric algebra. Based on the locations of the tooltip point and a tool axis point, the toolpath is represented in the 6-dimensional Euclidean space. With the help of the representation of conformal transformations and circles in conformal geometric algebra, the path in the 6-dimensional space is smoothed by G3 continuous circle-based splines under the constraints of the maximum deviation error tolerance. The proposed approach can generate a smooth toolpath that passes through the discrete cutter locations given in the original linear segments analytically. The cycle time of machining is improved thanks to the small curvature and G3 continuity of the toolpath. The effectiveness and efficiency of the proposed method are validated by simulations and experiments.

... Figure 16 illustrates the proposed approach. Bo et al. [190] adapt the idea of the VF-based TPG to flank machining as follows. Given a free-form surface Φ , the algorithm seeks finite lines characterized by the local tangential movability along Φ . ...

... By default the VFPD methods in Table 1 are designed for point five-axis machining with perpendicular axis. However, we also mention the remarkable works of He et al. [189] and Bo et al. [190] on flank milling (see Subsect. 2.2. The algorithms designed and verified on four-or three-axis machines are indicated in the column "Additional features". ...

... The optimized toolpath usually provides a considerable reduction of machining time. The Fig. 17 Flank milling [190]. First row. ...

Toolpath generation (TPG) for multi-axis milling machines using vector fields (VF) and vector field analysis (VFA) is becoming increasingly popular in the manufacturing industry. Therefore, the paper presents a survey of algorithms and methods of TPG based on the vector fields of preferred directions (VFPD) for five-axis CNC machining. Two hundred relevant citations in top manufacturing and optimization journals during 1995–2021 have been presented and discussed. Additional 79 references in Appendices are related to a classification of five-axis machines, the theory and recent advances in the area of the vector and tensor fields.