Integration and Implementation

Abstract

This chapter presents an overall integration of the proposed methodologies throughout this dissertation. The primary objective of the integrated system is to construct the software tools for planning and conducting the manufacture of hybrid freeform surfaces using the multiple-axis ultraprecision machining process.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter presents an overall integration of the proposed methodologies throughout this dissertation. The primary objective of the integrated system is to construct the software tools for planning and conducting the manufacture of hybrid freeform surfaces using the multiple-axis ultraprecision machining process.

Section 7.1 presents the overall view of the developed integrated system and controller configuration, and also discusses the optimization of tool geometry in the integrated system for avoiding the tool interference onto the machined surface. Sections 7.2 and 7.3 present the two case studies to validate the creditability of the developed integrated system. Lastly, the concluding remarks are discussed in Sect. 7.4.

7.1 Integrated CAD/CAM System

The screenshot image of the main menu for the developed SolidWorks-API user-interface in the integrated CAD/CAM system is shown in Fig. 7.1. It can be seen that there are two different cutting processes, namely automated Guilloche machining technique (AGMT) and diamond turning processes. The user selects one of the cutting processes for the manufacturing of hybrid freeform surfaces. The details of two developed sub-systems and the configuration requirements of the controllers are discussed in the next sections. In addition, a tool optimization process and geometrical splitting of hybrid freeform surface also embedded into the system and its details are discussed in Sects. 7.1.3 and 7.1.4, respectively.

Fig. 7.1

Screenshot image for main menu of user-interface in the integrated system for the selection of a cutting process. Some content of this chapter has been reproduced with permission from [3, 9]

7.1.1 Integrated Sub-system for AGMT Process

The screenshot image of the developed user-interface for the AGMT process in a sub-system of the integrated system is illustrated in Fig. 7.2. First of all, the user has to select the type of Fresnel lens arrays to be machined and its layout. Then, the user shall input the lens dimensions, namely radius/apothem (depending on types of Fresnel lens) and pitch distance, before selecting the surfaces to be machined. Next, the user has to define all machining and tool parameters, and targeted accuracy errors for evaluating the critical machining parameters and profile errors by the developed integrated system. These critical parameters would generate at the end of evaluation process and display in the user-interface form. Then, the user would decide to keep or modify these critical values before post-processing the cutting points as NC-codes for the fabrication of Fresnel lens array.

Fig. 7.2

Screenshot image for user-interface of sub-system to generate the Guilloche tool trajectory for AGMT process

Furthermore, there is another option for exporting the detailed descriptions of the calculated critical parameters into a data file for the user to understand the results clearly. The developed sub-system for AGMT process would be validated with a case study. The details are presented and discussed in Sect. 7.2.

7.1.2 Integrated Sub-system for Diamond Turning Process

The screenshot image of the developed user-interface for diamond turning process in another sub-system of the integrated system is shown in Fig. 7.3. Similarly, the user has to select a type of diamond turning processes and its cutting strategy, before selecting the surfaces to be machined. Next, the user has to define all machining and tool parameters, and targeted accuracy errors for evaluating the critical machining parameters and profile errors by the developed integrated system. These critical parameters would generate at the end of evaluation process and display in the user-interface form. Then, the user would decide to keep or modify these critical values before post-processing the cutting points as NC-codes for the fabrication of freeform surface. Lastly, the user may export the detailed descriptions of the calculated critical parameters into a data file for the analytical studies. The developed sub-system for diamond turning process would be validated with a case study. The details are presented and discussed in the later sections.

Fig. 7.3

User interface of SolidWorks-API to generate the spiral tool trajectory for hybrid FTS/SSS process. Figure reproduced from [3]

7.1.3 Optimization of Tool Geometry

A tool geometry optimization process has been developed and implemented for not only machining freeform surfaces efficiently, but also preventing the tool to overcut or damage the workpiece. The design of a diamond tool is strongly influenced by the curvatures of freeform surface. Hence, it is important to consider the tool geometries, namely rake (γ), clearance (α) and included (β) angles in the selection/designing a new diamond tool. Although there are few developed mathematical models [1, 2] to optimize the tool geometry, they are limited to two-dimensional surface curvatures along the cutting and feed directions but not for three-dimensional surface curvatures. Thus, the methodologies have been proposed to come the above limitations by obtaining these critical tool geometrical angles directly from CAD system.

Figure 7.4 illustrates that there are three types of cutting interference to be taken into an account when selecting or designing a new diamond tool. Firstly, a rake interference occurs between tool rake face and the surface curvature when the tool rakeγ _tool is highly negative. This can be overcome by selecting γ _tool which is not more the critical rake γ _cr . Secondly, a flank interference occurs between tool flank faces and the surface curvature due to insufficient tool flank clearance α _tool . Hence, α _tool should be larger than critical front clearance α _cr so that the flank faces are free of any interference. Lastly, a side interference is also necessary to be avoided to prevent overcutting of side cutting edges onto the surface curvature. This can be overcome when the tool included angle β _tool is lesser than the critical included angle β _cr . These critical tool geometrical angles can be defined with the following conditions:

• (i) cutting direction’s boundary conditions:

  $$ \begin{array}{*{20}l} {\gamma_{i} = \pi - \phi_{i} - C,} \hfill & {{\text{if}} \; \phi_{i} \ge 0} \hfill \\ {\alpha_{i} = \left| {\phi_{i} } \right| + C,} \hfill & {{\text{if}} \; \phi_{i} < 0} \hfill \\ {\gamma_{cr} \le \,\arg \,\hbox{max} \left( {\gamma_{i} } \right)} \hfill & {} \hfill \\ {\alpha_{cr} \ge \,\arg \,\hbox{max} \left( {\alpha_{i} } \right)} \hfill & {} \hfill \\ \end{array} $$

  (7.1)

• (ii) feed direction’s boundary conditions:

  $$ \begin{array}{*{20}l} {\beta_{1} = 90^{ \circ } - \varepsilon_{i} - C,} \hfill & {{\text{if}}\,\varepsilon_{i} \ge \, 0} \hfill \\ {\beta_{2} = 90^{ \circ } - \left| {\varepsilon_{i} } \right| - C,} \hfill & {{\text{if}}\,\varepsilon_{i} < 0} \hfill \\ {\beta_{i} = \beta_{1} + \beta_{2} } \hfill & {} \hfill \\ {\beta_{cr} < \arg \,\hbox{max} \left( {\beta_{i} } \right)} \hfill & {} \hfill \\ \end{array} $$

  (7.2)

where ϕ and ε are the slopes of the freeform curvature along the cutting and feed directions, respectively, and C is a clearance angle to prevent tool faces overcutting/interfering against the machined surface.

Fig. 7.4

Types of cutting interference and critical tool angles

Lastly, the immediate availability of standard tool sizes is also an important factor in designing these critical angles, especially α _cr . Although a diamond tool is custom-made with respect to the critical tool geometrical angles, it would be not only costly but also required a long lead time. Furthermore, the maximum α _tool for the manufacturers to produce is 30° or lesser and may not be able to meet the requirement of α _cr . Hence, these drive the needs for alternative solution of designing diamond tool geometry at a shortest time with an economical cost [3]. The front clearance issues can be overcome by titling the tool holder, as illustrated in Fig. 7.5, and both α and γ would be also adjusted accordingly. Thus, the critical tilted angle λ _cr can be defined as:

$$ \lambda_{\text{cr}} = \alpha_{cr} - \alpha_{tool} $$

(7.3)

The details for determining the critical tool geometrical angles will be elaborated with case studies in the later sections.

Fig. 7.5

Schematic diagram for titling a tool holder

7.1.4 Geometrical Splitting of Hybrid Freeform Surface

A hybrid freeform surface may come in a hybrid form of several freeform surfaces, which is possible to manufacture by employing a hybrid FTS/SSS diamond turning process. Although the layered tool trajectory methodology has been developed in the previous studies (Chap. 4), this methodology is only designed for micro prism but not for any freeform feature. Hence, it is necessary to have a proper approach to split up the hybrid freeform surface into two types of features, which would be machined by FTS and SSS processes, simultaneously. In this study, the splitting up process can be done by based on their geometrical frequency properties using the fast Fourier transform (FFT) and Hilbert transform (HT) approaches.

FFT defines the generated tool trajectories into a different domain with several distinguished properties for detailed analysis [4], and the discrete Fourier transform of the generated tool trajectory signals dZ is given by:

$$ \begin{aligned} dZ\left( k \right) = & \sum\limits_{j = 1}^{N} {\partial z\left( j \right)} \omega_{N}^{{\left( {j - 1} \right)\left( {k - 1} \right)}} \\ \omega_{N} = & e^{{{{\left( { - 2\pi \sqrt { - 1} } \right)} \mathord{\left/ {\vphantom {{\left( { - 2\pi \sqrt { - 1} } \right)} N}} \right. \kern-0pt} N}}} \\ \end{aligned} $$

(7.4)

where ∂z is the incremental adjustment of z-values between two corresponding tool trajectory points.

Due to the fact that a tool trajectory from HCAA cutting strategy has multiple frequencies, it would be difficult to implement the FFT approach. Thus, Hilbert transform approach is necessary to overcome this multiple frequencies problem. HT approach has been widely employed as an important tool in different branches of science and technology, from complex analysis and optics, to circuit theory and control science [5, 6]. HT not only behaves similarly to FFT but also is a linear operator, making it useful for analyzing non-stationary signals by expressing frequency as a rate of change in phase, so that the frequency can vary with time. Hence, HT is often introduced as a convolution between f(x) and −1/(πx) [7]:

$$ H\left( {f\left( x \right)} \right) = - \frac{1}{\pi x} * f\left( x \right) = - \frac{1}{\pi }\int\limits_{ - \infty }^{\infty } {f\left( {x^{{\prime }} } \right)\frac{{dx^{{\prime }} }}{{x^{{\prime }} - x}}} $$

(7.5)

If x(n) is a causal and absolutely summable real sequence with a discrete time Fourier transform X(e ^jω), then Hilbert transform can rewritten [8]:

$$ \begin{aligned} X_{\text{im}} \left( {e^{j\omega } } \right) & = - \frac{1}{2\pi }\int\limits_{ - \pi }^{\pi } {X_{{re}} \left( {e^{j\omega } } \right)\coth \left( {\frac{\omega - \varphi }{2}} \right)} d\varphi \\ X_{{re}} \left( {e^{j\omega } } \right) & = x\left( 0 \right) + \frac{1}{2\pi }\int\limits_{ - \pi }^{\pi } {X_{\text{im}} \left( {e^{j\omega } } \right)\coth \left( {\frac{\omega - \varphi }{2}} \right)} d\varphi \\ \end{aligned} $$

(7.6)

where X _re (e ^jω) and X _re (e ^jω) are the real and imaginary parts of X(e ^jω).

The details for determining the geometrical splitting of hybrid freeform surface based on their discrete time frequencies will be elaborated with a case study in the later sections.

7.2 Case Study 1: Hexagonal Fresnel Lens Array Using AGMT Process

This case study presents the machining of hexagonal Fresnel array using AGMT process [9]. Fresnel lens array is one of these hybrid freeform surfaces, which may have facet groove orientations arranged in a rectangular or hexagonal layout to form a polygonal Fresnel lens [10,11,12]. Notwithstanding the fact that a radius of Fresnel profile in such layouts could be the same as that of circular one (Fig. 7.6), the facet grooves are parallel to an apothem. It is possible to manufacture such arrangements with extrusion process and followed by assembly process to form together within the congruent triangles having a common vertex at the centre of the polygon. However, this approach leads to the misalignment of assembled profiles which may impact the overall optical performance. Hence, this misalignment issue could be eliminated by employing the developed automated Guilloche machining technique (AGMT) with a linear tool motion, as demonstrated in Fig. 7.7.

Fig. 7.6

Fresnel lens designs; a cross-sectional profile of Fresnel zone, b-d circular, square and hexagonal types, respectively

Fig. 7.7

Schematic diagram of machining hexagonal Fresnel lens array; a calculation of tool control points, b Guilloche tool trajectory for one rotation from Steps (i) to (vii)

7.2.1 Experimental Validations

In this case study, the developed automated Guilloche machining technique was employed for machining an array of hexagonal Fresnel lenses, as illustrated in Fig. 7.8. Table 7.1 describes the machining parameters which are the inputs for the developed user-interface of the sub-system. The surface generation for the Guilloche tool trajectory was conducted by implementing the methodologies into the developed integrated system.

Fig. 7.8

A CAD model for hexagonal Fresnel lens array; a in slanted view, and b half-sectional view

Table 7.1 Input parameters for fabrication of hexagonal Fresnel lens array

Figure 7.9a and b show that the cutting points were successfully mapped onto the surface of central and offset hexagonal Fresnel lenses, respectively. These cutting points has been post-processed into the Guilloche tool trajectories, as illustrated in Fig. 7.9c and d, for the respectively lenses.

Fig. 7.9

Successful generation of Guilloche tool trajectory points using the developed integrated system; Cutting points were mapped on central and offset Fresnel lenses in (a–b) respectively, and (c–d) the simulated Guilloche tool trajectories were conducted successfully on the respectively lenses

At the same time, the critical machining parameters for the generated cutting points have also been pre-evaluated by the developed integrated system, as demonstrated in Fig. 7.10, before the tool trajectory points are further post-processed into NC-codes. It can be seen that the critical feedrates Δρ _cr for roughing and finishing processes are 0.01050 and 0.00475 mm, respectively. The critical angular pitching for roughing and finishing processes Δt _cr are 1.801° and 0.810°, respectively. Furthermore, the detailed information of these critical data are replicated from the developed API system and discussed in the next sections.

Fig. 7.10

Screenshot image for the output results of calculated critical parameters for optimal AGMT process by the developed integrated system

7.2.1.1 Critical Machining Parameters for AGMT Process

Figure 7.11 shows the critical results of feedrate Δρ _cr for roughing and finishing processes, which are reproduced from the developed system. The selection of feedrates for roughing and finishing processes should not be more than Δρ _cr of 0.01050 and 0.00475 mm, respectively. Otherwise, it would not be able to meet the requirements of E _ρ (as given in previous Table 7.1).

Fig. 7.11

Critical results of roughing and finishing feedrates with given E _ρ , reproduced from output data from developed system

On the other hand as illustrated from Fig. 7.12, the reproduced critical values of angular pitching Δt _cr based on a given r _{c, max} of 4.05 mm are 1.801° and 0.810° for roughing and finishing processes, respectively. Hence, the selection of angular pitching should not be more than Δt _cr in order to fulfill the requirement of targeted h _tol (refer to Table 7.1).

Fig. 7.12

Calculation of critical pitch angular for roughing and finishing processes; a r _{c, max} of an array, and b critical Δt _cr

7.2.1.2 Cutting Experiments and Results

The automated Guilloche machining technique, AGMT [9], has been employed to perform the cutting experiments. Table 7.2 describes the cutting conditions have been selected based the above approaches to meet the targeted requirements. Figure 7.13 illustrates the photographic images for the successful fabrication of an array of hexagonal Fresnel lenses. These machined Fresnel lenses were measured using an Olympus LEXT OLS4000 3D Measuring Laser Microscope with a confocal optical system. 3D measured profile data were further post-processed using MATLAB for surface characterization.

Table 7.2 Selected cutting conditions for hexagonal Fresnel lens array

Fig. 7.13

Photographic images of a machined hexagonal Fresnel lens array; a full view and b an enlarged view for the selected zone (dashed box). Figure reproduced from [9]

Figure 7.14a and e show the 3D contour measurements of the central and offset hexagonal Fresnel lenses, respectively. As illustrated in Fig. 7.14b–d and f–h, the measured E _ρ for the selected zones of the central and offset lenses, respectively, are able to achieve lesser than the required E _ρ of 0.1 µm. On the other hand, Fig. 7.15a and e present the selected areas of 3D contour measurements of the central and offset hexagonal Fresnel lenses, respectively, for the evaluation of sagittal errors h _err . From Fig. 7.15b–d and f–g, the measured h _err of respectively areas are also found to be lesser than the required h _tol of 0.1 µm.

Fig. 7.14

Cutting residual error of machined hexagonal Fresnel lenses

Fig. 7.15

Measured Sagitta errors of machined hexagonal Fresnel lenses

Thus, these results further validate that the selected machining parameter, the feedrate Δρ, for cutting the hexagonal Fresnel lens array is achieving the cutting residual error E _ρ lesser than 0.1 μm. On top of these, these Sagitta error h _err results also further validate that the selected second parameter Δt for machining hexagonal Fresnel lens array are also achieving the accuracies lesser than the h _tol of 0.1 μm.

Lastly, these experimental results have validated the credibility of the proposed Guilloche machining technique to fabricate accurate hexagonal Fresnel lens array in a single process. In additions, this can be only achieved with the implementation of the proposed approaches, namely cutting residual error and Sagitta error analyses.

7.3 Case Study 2: Multiple-Compound Eye Surface Design-B

7.3.1 Experimental Validations

In this case study, a multiple-compound eye surface, as illustrated in Fig. 7.16, has been machined by hybrid fast tool/slow slide servo (FTS/SSS) diamond turning process. Table 7.3 describes the selected experimental conditions for the cutting experiments. A rapidly solidified aluminum alloy RSA-6061 has been employed as a workpiece material for the cutting experiments. RSA-6061 is an ultra-fine grain aluminum alloy AA-6061, which is widely used for making optical inserts owning to its excellent nanometric surface finishing after diamond turning process. Furthermore, its fine microstructures by a superfast solidification in the melt-spinning process gives excellent mechanical and physical properties [13, 14].

Fig. 7.16

CAD model for multiple-compound eye Design-B, a 3D view, b top view, and c slanted view

Table 7.3 Cutting conditions for case study 2

At the same time, two different types of diamond tool geometries have also been employed for roughing and finishing processes. In the roughing process, a diamond tool having a nose radius of 0.2 mm, flank clearance of 7° and wedge angle of 60° is employed with the cutting feedrate of 0.005 mm/rev and depth of cuts of 10 µm. As for the finishing process, a diamond tool with nose radius of 10 µm, flank clearance of 30° and wedge angle of 30° is employed with the cutting feedrate of 0.001 mm/rev and depth of cuts of 5 µm.

Figure 7.17 presents a screenshot image of user-interface with the generated critical cutting parameters and tool geometrical angles by the developed system. It can be seen that the critical cutting parameters, namely constant angle Δθ, constant arc-length ΔS and transition radius r _trans , have been pre-defined for both roughing and finishing cutting experiments to achieve accurate freeform contours. Furthermore, the critical tool geometrical angles, namely γ _cr , α _cr and β _cr are also been evaluated to avoid any unnecessary tool interferences. The analytical details of these critical parameters can be extracted from the output files of the system and are presented in the next sections.

Fig. 7.17

Screenshot image for the calculated critical parameters by the developed integrated system. Figure reproduced from [3]

7.3.1.1 Critical Cutting Parameters for HCAA Cutting Strategy

Figure 7.18 illustrates a replication of output data from the developed system for the critical cutting parameters in the hybrid constant-arc and constant-angle (HCAA) cutting strategy. It can be seen that the constant angle cutting strategy has been selected for the outer and inner regions, and the middle region’s cutting strategy is the constant arc. The transition radii, r _trans1 and r _trans2 between these regions are located at 3.9865 and 0.2782 mm, respectively. In the constant arc cutting strategy, the critical value of ΔS for the finishing and roughing cuts, as illustrated in Fig. 7.18a, should not exceed 0.0175 and 0.035 mm, respectively. The calculated PV _err values of middle region for the finishing and roughing cuts are 0.9675 µm (<1.0 µm) and 1.9785 µm (<2.0 µm), respectively, which are also lesser than the targeted PV _err . On the other hand, the critical Δθ for both finishing and roughing cuts, as shown in Fig. 7.18c and d, are the same value of 180°. The calculated PV _err values of outer and inner regions are found to be near zero for both finishing and roughing cuts.

Fig. 7.18

Replication of output data for the details of critical cutting parameters from developed system. Figure reproduced from [3]

In summary, the cutting process begins with a constant-angle cutting strategy from outermost radius until the tool reaches r _trans1 of 3.9865 mm. Then, the cutting process continues with constant-arc cutting strategy for the middle region. When the tool reaches the next r _trans2 of 0.2782 mm, the cutting process reverts back to constant-angle cutting strategy for the inner region. This entire cycle shall be repeated for each cut until the whole cutting process completes.

7.3.1.2 Critical Tool Geometrical Angles

Figure 7.19 represents the replication of the critical tool geometrical angles from output data of the developed system. The critical values of tool rake γ _cr and front clearance α _cr angles are 70° and 20°, as illustrated in Fig. 7.19a and b, respectively, and the critical included angle β _cr, as shown in Fig. 7.19c, is about 135°. With the respect to above results, it can be seen that Tool #2 would be free of tool interferences but not for Tool #1. There would be a frank interference in the case of Tool #1 as α _tool of 5° is lesser than the critical one (α _cr  = 20°). Thus, Tool #1 is required to be tilted at λ _cr of 15°, as illustrated in Fig. 7.20, overcoming the frank interference with an additional spacer.

Fig. 7.19

Replication of output data for the details of critical tool angles from developed system. Figure reproduced from [3]

Fig. 7.20

Schematic setup for tilting Tool #1

7.3.1.3 Geometrical Splitting for Hybrid FTS/SSS Process

Figure 7.21 replicates the critical geometrical frequencies from output data of the developed system. It demonstrates that the generated tool trajectories for hybrid freeform surface has two notable frequencies of 6.098 and 47.03 Hz, as illustrated in Fig. 7.21a. Hence, the generated tool trajectory can be further split into two types of trajectories, namely low-order and high-order, as shown in Fig. 7.21b and c, respectively. Thus, the low-order tool trajectory having a frequency property of 6.245 Hz and a sag height of 0.3 mm would be handled by SSS process. On the other hand, the other high-order one having a frequency property of 46.48 Hz with a sag height of 0.02 mm would be handled by FTS process.

Fig. 7.21

Geometrical splitting of freeform features; a–c 3D profiles and d–f calculated frequency spectrums for the original, low-order (SSS) and high-order (FTS) tool trajectories, respectively

7.3.2 Cutting Experiments and Results

Several cutting experiments have been performed by employing a hybrid fast tool/slow slide servo (FTS/SSS) diamond turning process. hybrid constant-angle and constant-arc (HCAA) cutting strategy. The cutting conditions have been chosen based on the above results to fabricate the multiple-compound eye surface. Figure 7.22 shows that the generated spiral tool trajectory points have been successfully overlayed on the CAD surface by the developed SolidWorks-API system. These tool trajectory points were further post-processed into NC codes for the cutting process

Fig. 7.22

A successful generation of tool trajectories for HCAA cutting strategy, which are mapped onto the surface of the CAD model. Figure reproduced from [3]

Figure 7.23 illustrates the photographic images for the successful machining of the multiple compound eye surface. The machined surface were measured using a 3D measuring laser microscope with a confocal optical system (Olympus LEXT OLS4000). 3D measured surface data were further characterized using MATLAB software for the surface error.

Fig. 7.23

Photographic images of fabricated multiple compound eye surface; a fabricated workpiece mounted on a chuck, and b top slanted view. Figure reproduced from [3]

Figures 7.24 illustrates the replicated of 3D contour measurements for the central and offset compound eye surfaces. Figure 7.24b–c and d–e show that the measured contour errors for the selected regions of the central and offset compound eye surfaces, respectively, were found to be lesser than the targeted PV _tol of 1.0 µm. Hence, these results are validating the proper selection of critical cutting parameters, ΔS and Δθ, for the machining of the freeform surface using hybrid constant-arc and constant-angle cutting strategy.

Fig. 7.24

Contour error measurements of the central compound eye surface

In additions, no tool interference/rubbing marks are detected on the machined surface. Therefore, these experimental results have validated the credibility of the proposed methodologies, hybrid Hilbert and fast Fourier transformations HT/FFT, and tool geometrical optimization, for incorporating into the integrated SoildWorks-API system to fabricate hybrid freeform surface accurately.

7.4 Concluding Remarks

In the present study, an integrated CAD/CAM system has been developed to fabricate complex freeform surface accurately using multiple-axis diamond turning processes. The proposed methodologies not only replaces the needs of expensive specialized CAM software, but also provides an attractive solution with an integration of Visual Basic application programming interface (API) into SolidWorks for accurate and optimized surface generation of hybrid freeform surfaces. The conclusions are been drawn as follows:

• i. Two case studies explaining the implementation of the developed integrated system is presented. The profile accuracy requirements for the hybrid freeform surface have been met. The contour evaluation has demonstrated that the profile errors were lesser than the targeted requirements.

• ii. This can be only achieved with the implementation of several analytical approaches for accurate freeform surfaces. These approaches evaluate analytically for the cutting residual, Sagitta and cutting linearization errors.

• iii. The tool interference/rubbing marks are also eliminated with an aid of tool geometrical optimization approach.

• iv. In addition, the hybrid approach of Hilbert transformation (HT) and fast Fourier transformation (FFT) performs a proper segregation of freeform features in order to be fabricated by FTS and SSS processes simultaneously.

This study provides an essential contribution towards the improvement of CAD/CAM supports for multiple-axis ultraprecision machining of complex hybrid freeform surfaces.