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Abstract
Toolpath generation techniques have become increasingly critical in ultra-precision diamond turning for optical microstructures due to the dramatically enhanced geometrical complexity of the machined region. However, the conventionally used spiral toolpath is required for interpolation from the structural models, leading to random instability of the feeding axis and additional profile error between the toolpath and designed structures, which means an enlarged effect on the machining quality in ultra-precision machining. In this paper, a rotary-coordinate and shuttling-element cutting strategy based on integrated geometrical modelling and spiral toolpath generation is presented for ultra-precision turning of optical microstructures. Using the innovative rotary-coordinate and shuttling-element cutting method, the point clouds for the micro-structured modelling can be scattered along the spiral shape which can be directly fitted as the final toolpath. A series of simulation and cutting experiments have been carried out to realize the effectiveness of this method, and it is found that the preparation time in diamond turning can be significantly reduced along with ameliorating the machining quality.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultra-precision diamond turning is prone to obtain structures with high machining accuracy and excellent surface finish because of its cutting feature of continuity and consistency as well as the improving techniques of machine tools [1–6]. With the cutting-edge technologies of linear driving and spindle servo system used in this kind of machining, the ultra-precision machine can provide the primary motion and realize the feeding motion which is defined as C-axis [7–9]. This new pattern of feeding motion has broken through the limitation of conventional general turning work which can only satisfy the machining for surfaces with rotational symmetry [10–13]. As a result, a state-of-the-art turning formation method based on a servosystem is increasingly applied in ultra-precision machining of complicated microstructures of moulds for optical components with geometrical features of noncircular sectional elements. In particular, the diamond turning by slow tool servo (STS) system is widely utilized for optical moulds with high-amplitude and low-frequency microstructures due to its larger stroke of the linear axis of the machine tool [14,15]. By contrast, fast tool servo (FTS) turning can realize a high-frequency reciprocating Z-axis-direction feeding motion [16,17].
However, with the dramatically enhanced complexity of the structural geometry of optical microstructure, the performance of machine tools and the parameters of cutting tools as well as the high machining accuracy and surface finish of machined pieces have been facing unavoidable challenges [18–20]. Especially, the precise toolpath is much needed to be promoted to express the complicated micro-structured geometry as well as to be adapted to the dynamic performance of machine tools during the challenging manufacturing conditions, which is considered as the foundation of the ultra-precision diamond turning process, i.e., an optimized toolpath generation method is currently much needed to meet the more stringent requirements.
There have been multi-aspect studies on toolpath generation techniques for ultra-precision diamond turning, including the research on toolpath shape design [21–24], toolpath parameter planning [25–27], and adaptive toolpath for processing parameters [28–31]. Regarding toolpath shape design, Ao et al. [21] investigated an interpolation method to connect the Archimedean spiral toolpath and the vacant path in non-cutting regions to ensure the continuity of global machining trajectories. Focusing on multi-scale optical structures, Yang et al. [22] created a hybrid toolpath generation method to combine different processing modes. Based on the space Archimedean spiral, Gong et al. [23] innovated a quasi-revolutionary method to modify the local shape of spiral toolpaths. Similar to the previous methods, Huang et al. [24] modified the conventional spiral toolpath for the special double free face by alternately linking the practical cutting toolpath and the vacant path. From these studies above, the Archimedean spiral-shaped toolpath is generally used for ultra-precision diamond turning to ensure the stability of the X-axis of the machine tool. To be specific, keeping the X-axial feeding motion as a uniform linear motion during the machining process as much as possible is a basic requirement for optimizing machining quality. When it comes to basic toolpath generation and surface modelling, there have been precise and high-level mathematical expressions of typical surfaces. For example, Wu et al. [32] proposed two excellent design methods for the first time facing the ultra-efficient aspherical surface lenses in two-dimensional geometry. However, due to the improved complexity of micro-structured surfaces, the typical mathematical expressions cannot be directly used for optical micro-structured surfaces with multiple local geometrical features. As a result, the scattered point clouds data are increasingly considered for this situation.
A new problem occurs because of the specific cutting motion mode of each axis in diamond turning. The spiral toolpath is difficult to adapt to the scattered point clouds with the quadrilateral meshing method used for micro-structured models. This issue has been overlooked by previous research work. The conventional method using the interpolation in the X-axis direction breaks the continuity and stability of its feeding motion due to the imprecise spiral shape.
As for toolpath parameter planning, Guo et al. [25] presented a typical toolpath optimization method to obtain the uniform spiral toolpath using a modified hybrid method to integrate the constant-polar-angle and constant-arc sampling methods. Meanwhile, the interpolation error was controlled by determining the distance between toolpath points. Similarly, to control the sampling distance along the X-axial feeding direction and cutting-motion direction independently, Wang et al. [26] invented an adaptive toolpath generation method considering interpolation error and scallop-height error. In addition, Li et al. [27] developed a FTS-assisted toolpath design with compensation for ultra-precision turning. Considering the processing parameters, Mishra et al. [28] carried out a series of experiments which took into consideration the effect of process parameters including the feeds of the X-axis and C-axis, cutting depth, spindle speed, etc. in STS turning for freeform surface and eventually found the optimal combination of parameters. Facing an interference-free toolpath, You et al. [29] proposed a creative location-point-drive method which is especially adapted to hard-and-brittle material machining using non-zero-rake turning tools with comprehensive consideration of the feeding motion stability for the X-axis of the turning machine. Due to this innovation, the surface finish obtained could be improved by 44.81% which was of valuable guidance for ultra-precision diamond turning.
According to the investigations above, the toolpath parameters are planned by various interpolation methods to control the profile error and guarantee the continuity of machining motion. However, the relationship between toolpath parameters and processing parameters has not been fully analyzed and connected. Besides, the surface modelling and toolpath generation processes are operated independently, leading to calculation errors between the generated toolpath and designed machining surface. Hence, the machining quality for micro-structured surfaces cannot be further improved in ultra-precision turning.
In addition, global toolpath generation regulation is generally used in the conventional methods, which also increases the difficulty of toolpath design for typical optical surfaces [33,34]. To solve this problem, the complicated micro-structured surface studied in this paper can be seen as a composition array of multiple optical elements. Therefore, the sub-regional processing method [22,35–36] can be creatively introduced to this area. Its essence means segmenting the global surface into sub-regions to simplify the complicated geometrical features, and the invented toolpath generation method is easier to be applied to sub-regions with specific elements.
Hence, in this paper, a rotary-coordinate and shuttling-element cutting strategy based on integration with geometrical modelling for ultra-precision diamond turning of optical microstructures is presented. Firstly, the specific feeding-motion features are analyzed to explore the relationship between toolpath geometry and process parameters. Hence, the rotary-coordinate and shuttling-element (RCSE) strategy is created to deal with the key issue, wherein the necessity and feasibility of this strategy are evaluated. The optical microstructure is modelled with comprehensive consideration of the dynamic characteristics in diamond turning as well as the toolpath parameters. As a result, the point clouds for microstructure modelling could be scattered along the spiral-geometry mesh which would be directly extracted as the practical cutting contact points (CCP) without interpolation. During this process, the global optical micro-structured surface is segmented into different sub-regions with optical elements, and a sub-regional toolpath is generated and stitched for the final global spiral toolpath. This method can simplify the process of modelling and toolpath generation before machining operations in ultra-precision diamond turning and eliminate the errors between the designed machining surface and practical toolpath. Hence, the feeding motion stability of the X-axis can be ensured. Finally, the machining quality is significantly promoted.
2. Rotary-coordinate and shuttling-element (RCSE) cutting strategy
2.1 Necessity and feasibility analysis of the RCSE cutting strategy
The distinctive characteristic of linear-axis and rotary-axis feeding motions in slow tool servo (STS) diamond turning has restricted the shape of the toolpath during the generation process, i.e., the spiral toolpath shape is considered as the most applicable in this manufacturing situation. However, the specific toolpath shape is leading to new issues in machined region modelling and toolpath generation. In this section, the influence of the characteristics of multi-axis feeding motion in STS diamond turning for toolpath generation is analyzed, based on which the necessity of the proposed RCSE cutting strategy is discussed, following the detailed process of the creative method. This is also the key idea in this paper.
The influence of specific feeding motion characteristics for toolpath generation is shown in the diagram in Fig. 1. The precise motion of the C-axis, utilizing the main spindle servo system techniques, can be synchronized with the machine tool's linear axes to manufacture freeform surfaces with non-rotational optical microstructures. During the optical component manufacturing process, rapid reciprocating feeding motions of the X-axis or Z-axis are necessary to create optical microstructures on the workpiece surface. According to the machine tool's general assembly construction, the X-axis is connected to the heavier headstock and the Z-axis is connected to the lighter tool rest. Thus, the intense reciprocating motion of the X-axis can destabilize the spindle speed motion, leading to significant machining errors in the sensitive direction. This should be minimized in the turning process. Hence, the mode with constant rotation speed of the C-axis (spindle speed), single-direction feeding motion of the X-axis and reciprocating feeding motion of the Z-axis is usually taken by the conventional toolpath generation methods in the turning process. This specific feeding motion mode can only be realized by the Archimedean spiral shape of the toolpath.
However, the new issues of Archimedean spiral toolpath generation are found and analyzed. When this type of toolpath shape is specially adapted to the ultra-precision turning process designed, a realistic technical problem in operation should be considered. With the increasing enhancement of the complexity of the geometry of the micro-structured surface, the global machined region with multiple local geometrical features can hardly be directly expressed by mathematical equations. As a result, in practical processes, the geometrical features of machined optical micro-structured surfaces are generally expressed by point cloud. However, the existing point cloud model, no matter whether based on rectangular, triangular, or other shaped mesh, can hardly directly form the basic geometry of the Archimedean spiral shaped toolpath. Indeed, the interpolation by grid mesh is always needed as shown in Fig. 2(a) which leads to additional profile errors. This is because the nature of interpolation is to obtain an approximation of the true shape by fitting the practical profile with known points. The fitting precision depends on the grid density and interpolation techniques. Besides, the mesh grid is scattered and is inherently made up of lines and points at fixed intervals, which limits the types of shapes that it can precisely represent. An Archimedean spiral is a continuous curve, and its expression on the generally used mesh grid is surely an approximation. According to Fig. 2(a), it is obvious that the toolpath of conventional methods is not an exact spiral shape and exhibits significant profile errors although it is generated by the point cloud in higher density. In addition, the practical CCPs should be interpolated between the data point of the point cloud model to generate the final spiral-shaped toolpath during the toolpath planning process. That situation leads to two issues as follows:
- (i) The toolpath obtained by the interpolated process is difficult to be strictly spiral geometrically, which means that the X-axis is driven with random reciprocating motion, influencing the stability of its feeding motion.
- (ii) The value of the coordinate in the Z-axis direction by interpolation also results in error compared with that of the designed surface, generating profile error between the designed toolpath and the practical machined region surface.
Fig. 2. Schematic diagram of different strategies of toolpath generation. (a) Conventional method with interpolation by a grid mesh; (b) Innovative method with direct spiral toolpath generation.
2.2 Investigation of the RCSE-strategy toolpath generation method in STS diamond turning
Based on the analysis above, it is necessary to promote the toolpath generation method in the specific situation. Particularly, whether it is possible to directly generate a point cloud model where the data points are scattered along the spiral-geometry grid, which precisely expresses the complex geometrical features of the machined surface, and how to realize it, became considerations in this study. In that way, the toolpath can be fitted by the selected sample points form the spiral-base point cloud model without interpolation, as shown in Fig. 2(b). Thus, the problems mentioned above can be solved. This is the critical investigation and the most valuable aspect of this paper.
The global geometrical features of the microstructures of the moulds used for optical microlenses are of great complexity generally, while the local optical elements could have a regular distribution, i.e., in the form of micro-scale element arrays. As shown in Fig. 3, the concave elements are radially distributed, and each group of elements in circular patterns (simplified to the ‘group’ in the following part) are all arranged in a certain loop, which is the precondition for the feasibility of the RCSE strategy. In this situation, the global microstructure is segmented into local micro elements (simplified to the ‘element’ in the following part). Hence, the sub-regional toolpath can be designed for each element, and finally the global toolpath can be stitched, by which the issue of toolpath generation for complicated microstructures can be simplified. Meanwhile, the spiral topology of the toolpath is suitable for ultra-precision diamond turning. As a result, the toolpath can be streamlined and continuous in the whole machining area. Accordingly, the interruption or sudden change of parameters should be avoided in case of sudden change of kinematic parameters of the machine tool or interference of the cutting tool during the turning process, which means the deterioration of machining quality and even unrealizable operation of the cutting motion of each axis. This means that, how to combine the local toolpath planning with global processing issues and integrate sub-regional toolpath naturally into a continuous spiral toolpath should be comprehensively investigated.
Therefore, a novel toolpath generation method based on RCSE is presented. Specifically, using rotary coordinates allows the coordinate system to rotate suitably. This corresponds with the generation of a continuous spiral toolpath which moves from one element to the adjacent one, effectively keeping the element in a comparatively fixed position. Such an arrangement makes the machined surface like a swivel plate. Additionally, the shuttling element refers to the periodic motion of the CCP moving back and forth through the element, synchronized with the rotation of the coordinate system. The detailed process of the RCSE strategy for toolpath generation is discussed as follows:
Firstly, the rotating coordinate process in the RCSE strategy is introduced. The range of polar angle $\Delta \theta $ of an element corresponds with the rotation angle of the plane local coordinate system (PLCS) each time, leading the global micro-structured surface region to rotate intermittently as a ratchet. Hence, a rotation of coordinates $S({u\ast ,v\ast } )$ from the initial one $S({u^{\prime},v^{\prime}} )$ can be expressed as,
As shown in Fig. 4, the machined region with local optical elements can be segmented into different sub-regions along the circular direction. From the perspective of the XOY plane of the machined surface, each time the PLCS of the original sub-region is rotated anticlockwise it becomes the PLCS of the second sub-region. Consequently, the original machined surface appears to rotate clockwise. This results in the second sub-region aligning with the position of the original sub-region. Thus, the second sub-region can be seen as the new original element. This process is repeated until all elements of the group are operated. The procedure is characterized as rotating coordinates, akin to the motion of a ratchet wheel.
Specifically, it can be considered that the sub-regional toolpath remains in a stable position, while the rotation of the coordinate system and the offset of the element means that the toolpath has been switched from the current element to the next element in practice. The rotation angle of the coordinate system is transformed into the generated point cloud matrix as the critical positioning information, i.e., the global toolpath has only one independent variable, the polar angle $\theta $ along the circumferential direction.
Secondly, the shuttling elements process in the RCSE strategy is presented. This involves a shuttle-like reciprocating motion of the element, in conjunction with the rotation of the coordinate system, as shown in Fig. 5. A spiral toolpath is generated in a loop passing through each element in a certain direction. For instance, the relative position between the helical path and different elements (congruent or similar in shape) is changed from the periphery to the centre as shown in Fig. 4. However, when the mentioned method of rotary coordinate system is carried out, the element is offset accordingly by a certain distance relative to the opposite direction of the toolpath generating direction.
where d is the spacing distance between adjacent loops of the toolpath on the XOY plane which determines the side step of space for the toolpath.Considering the transformation from the polar coordinate for the spiral toolpath to the PLCS, the practical coordinate values $({x\ast ,y\ast } )$ of the sub-regional toolpath for the nth element can be determined as,
As shown in Fig. 5, taking elements I, II and III of group ${\textrm{N}_x}$ as examples, the shuttling-element process can be demonstrated. To begin with, the CCPs of the sub-regional toolpath in element I, with initial CCP ${\textrm{A}_1}$ and stop CCP ${\textrm{B}_1}$, shown as points with diagonal stripes, are generated orderly, whose detailed generation method can be seen in the following Section 2.3. Hence, the spiral toolpath is continuously generated and turned into the sub-region of element II, where the dotted points distinguish the CCPs in different sub-regions. However, according to the rotary coordinates process, the CCPs of element II can still be generated on the original position of the element I (this element is defined as the basic element) with initial CCP ${\textrm{A}_2}$ and stop CCP ${\textrm{B}_2}$, illustrated as a dark outline, and practically the CCPs are automatically transferred to the corresponding sub-region of element II as a light outline, which are precisely stitched with the CCPs in element I. Similarly, the CCPs in element III are still generated in the basic element with initial CCP ${\textrm{A}_3}$ and stop CCP ${\textrm{B}_3}$, filled with the outlined diamond grid and then transferred into a related practical sub-region where CCPs are shown as points with a light outline. As a result, the generated CCPs of different sub-regions are shuttling through the basic element, back and forth, just like a shuttle, which is the reason why the shuttling element process is so named. Hence, it can be considered that the creative RCSE strategy for sub-regional toolpath generation can dramatically simplify the calculation process of the algorithm, and in particular, the geometrical features of basic elements can be directly copied, transferred and expressed along the direction of the spiral array on the global machined region, i.e., the local issues are utilized to solve the global problem.
When it comes to the shuttling element process between different groups, as shown in Fig. 6, the spiral toolpath in a single loop can hardly cover the complete profile of an element. As a result, the toolpath passes through different positions of a certain element multiple times based on several adjacent loops. The shuttling element not only means that the toolpath is generated between adjacent elements, but also means that the toolpath is generated between different positions of a certain element. It is assumed that the number of elements in the first group along the circumferential direction is ${\textrm{N}_1}$, and there would be ${\textrm{L}_1}$ loops of the spiral toolpath to completely cover the profile of this group; the number of elements in the second group is ${\textrm{N}_2}$, and there are ${\textrm{L}_2}$ loops of the spiral toolpath to completely cover the profile of this group. During the processing, the element should shuttle back and forth combined with the rotation of the coordinate system, i.e., when generating the toolpath along the actual first group, the element is initially offset in a specific direction (e.g., if the toolpath is generated from the periphery to the centre, the element is offset from the interior to the exterior). After offsetting ${\textrm{N}_1}$ times, the toolpath for the first loop is generated. Upon entering the next loop, the element continues to be offset in the original direction, starting from a different position of the initial element and then follows the unidirectional rotation of the coordinate axis sequentially. It requires ${\textrm{L}_1}{\textrm{N}_1}$ times the offset of the element for the toolpath to cover the global profile of a group. Next, when the toolpath planning is related to the second group, the offset element naturally returns to the initial position similar to the initial offset element in the first group (i.e., the innermost position of an offset element), and then the element is offset from the interior to the exterior ${\textrm{L}_2}{\textrm{N}_2}$ times iteratively to complete the toolpath generation of the second group. In this process, the element is reciprocating as a moving shuttle, which is also the reason for the definition of the shuttling element.
2.3 Sub-regional toolpath generation basic elements
In this section, the detailed operations of the sub-regional toolpath generation method in a single element are presented. According to the previous rotary-shuttling method, when the local toolpath of a single element is generated, the coordinate system as well as the offset element are kept in a constant position. As shown in Fig. 6, the sub-regional spiral toolpath is from A to B. In this study, the Archimedes spiral is utilized, which can ensure the stability of the feeding motion of the X-axis under a certain spindle speed of the machine tool. As a result, it is widely used in ultra-precision diamond turning. When the sub-regional toolpath is being generated as shown in Fig. 6 along the direction from A to B, the CCPs are scattered by the iso-polar-angle criterion, i.e., the difference in polar value of adjacent CCPs is set as equal $\alpha $. Hence, the CCP $({x(\theta ),y(\theta )} )$ in the nth sub-region is determined as,
During this process, it is necessary to determine the certain cut-in and cut-out points on the special optical structure of the element for the toolpath, namely ${\textrm{P}_i}$ and ${\textrm{P}_o}$. In this case, the special optical structure is the concave spherical surface in the element, which needs to be distinguished from the plane. When the toolpath is generated on the plane, the corresponding Z-axis coordinate value is 0. After the tool is cutting in the optical structure, the corresponding Z-axis coordinate value needs to be determined using the equation for a concave spherical surface. Furthermore, if the designed optical structure is not a regular shape (e.g., spherical surface, ellipsoid surface, square or zig-zag shape, etc.), it can also be geometrically expressed in the form of discrete point cloud fitting. Based on the analysis above, the Pi and Po can be expressed by,
As mentioned above, when a sub-regional toolpath for an element is generated, the coordinate system is rotated as a ratchet wheel, and the CCPs are generated as a shuttle, where the sub-regional toolpath should be stitched precisely between different sub-regions when the RCSE strategy is applied. As shown in Fig. 5, the sub-regional toolpath of element II seems to be returned to a basic element, and the CCPs are discretely generated along the spiral path shuttling from ${\textrm{A}_2}$ to ${\textrm{B}_2}$. In fact, since the position information through the polar angle of the C-axis direction has been recorded on each CCP, the practical CCP of adjacent elements are precisely connected end to end, and stitching and integrating is performed for a continuous spiral toolpath of the global machining region. However, what should be highlighted is that the initial CCP of the sub-regional toolpath of element II, which is shown as a specific point on ${\textrm{A}_2}$, is just the stop CCP ${\textrm{B}_1}$ of the sub-regional toolpath of element I, where the two points are located in the arc of the equal polar radius of ${R_{{B_1}}}$. Indeed, point ${\textrm{A}_2}$ and point ${\textrm{B}_1}$ are of the same CCP, on account of the transformation of adjacent sub-regions in the RCSE process. Obviously, only in this way can the interference of different sub-regional CCPs be avoided at the endpoint, which is critical to ensure the continuity of the global toolpath. Besides, the successive rotation of the coordinate system and the synergistical matching of the element offset are the key factors to lead the regional toolpath to be precisely stitched. Finally, a spiral topological expression of the surface geometric features of the designed microstructure is achieved.
3. Analysis
To further evaluate the capability of the toolpath generated by the invented RCSE strategy, a typical micro-structured surface with a circular array was designed, with five groups for which there were 36 elements on each, as shown in Fig. 2. The optical element had the plane radius of 400 µm and largest depth of 2 µm. The global machined region had a diameter of 9.5 mm.
To compare the different characteristics of the conventional method (i.e., the practical CCPs are interpolated between the data point of the point cloud model based on rectangular mesh to generate the final spiral toolpath) and proposed RCSE method, the toolpaths of micro-structured surface were respectively generated by the two methods with the equal spiral spacing distance of 5 µm and isometric polar angle of 0.25°, as shown in Fig. 7(a)-(d).
Fig. 7. Modelling and toolpath generation by different methods. (a) Conventional surface modelling; (b) Conventional toolpath generation; (c) Proposed surface modelling with point cloud of CCPs; (d) Proposed toolpath generation.
3.1 Toolpath geometry analysis
The expansion of the toolpath along the C-axis direction was expressed on the C-X coordinate plane. As shown in Fig. 8(a), regarding the toolpath generated by the conventional method, the polar-radius values along the X-axis of the machined surface plane have great instability, which means the dramatic reciprocating motion of X-axis of the machine tool when it is feeding along the generated toolpath. According to the demonstration in Section 2.1, that severely affects the dynamic performance of the machine tool as well as the final machining quality. In contrast, the polar-radius values of the toolpath generated by the proposed method as shown in Fig. 8(b) were stable, forming a smooth descending line on the C-X coordinate plane. Obviously, when the X-axis of the machine tool is feeding along the generated toolpath, it maintains uniform linear motion to lead the dynamic capability of the machine tool to be fully utilized. It can be evaluated that the toolpath of the proposed RCSE strategy makes it easier to obtain better machining quality.
3.2 Analysis of toolpath kinematic parameters
Based on the expansion of the toolpath in Section 3.1, the kinematic parameters (i.e., velocity and acceleration) of the X-axis of the machine tool when the cutting tool is feeding along the practical toolpath during the machining process were determined and are illustrated in Fig. 9(a)-(d). It can be obviously seen that the kinematic parameters of the toolpath generated in the conventional method are of high value and rapidly change while those of the toolpath produced by the proposed method are stable (velocity) and close to zero (acceleration). In detail, the obtained values of kinematic parameters were collected, and their maximum values are listed in Table 1. Certainly, the kinematic parameters were simulated by the software, MATLAB, and are considered as the predicted values to evaluate the toolpath capability. In practice, the feeding velocity and acceleration of the X-axis in machining cannot reach an overlarge value due to being limited by the dynamic performance of the machine tool.
Fig. 9. Kinematic parameters of the X-axis. (a) Velocity of the X-axis under the conventional method; (b) Velocity of the X-axis under the proposed method; (c) Acceleration of the X-axis under the conventional method; (d) Acceleration of the X-axis under the proposed method.
Table 1. Analysis results of different processes of toolpath generation.
Hence, the significant difference between the conventional method and proposed RCSE strategy was proved, i.e., the RCSE method can effectively reduce the rapid change of X-axis velocity as well as the resolute value of its acceleration, which ameliorates the feeding stability of the X-axis during the diamond turning process, critically ensuring the machining quality. In addition, the attached average value of velocity can also reflect the promoted practical feeding-motion capability of the X-axis by the RCSE-generated toolpath to some extent.
3.3 Operational time comparison
The operational time of modelling and toolpath generation was recorded by the software, MATLAB, which is also listed in Table 1. As a result, the total operation time of modelling and toolpath generation of the conventional method was 14,543 seconds, of which the time for machined micro-structured surface modelling was 4,734 s, and the time of the following toolpath generation was 9,809 seconds. This reflects the time-consuming process of the conventional method, especially the interpolation of CCPs between the point cloud data. In contrast, the operation time of the investigated RCSE-strategy method was 1,791 seconds, including the modelling time and toolpath generation time. Thanks to the integration of modelling and toolpath generation by the RCSE strategy, both the modelling process and toolpath generation process are simplified, and the operation time can be dramatically reduced. Therefore, the presented method can also enhance the efficiency of the global machining process.
4. Experimental work
Comparative experiments involving both the conventional toolpath generation method and RCSE-strategy toolpath generation method were carried out to further confirm the practical performance of the proposed method. It is important to illustrate that in this study, the cutting location point (CLP), which practically determines the motion of the cutting tool during the machining process, was offset along the Z-axis of the machine tool [37]. This approach prevents any disruption to the optimized feeding-motion stability of the machine tool’s X-axis. In particular, the surface finish and roughness, form error as well as the machining following error, etc. were measured, calculated, and analyzed to compare the machining quality of different methods.
The STS turning experimental setup can be seen in Fig. 10(a). The 19 mm diameter workpiece, using material of RSA-905 aluminium alloy for its superior mechanical and material properties [38], was fixed on a fixture (aluminium alloy 6061) to be vacuum chucked by the spindle box of the machine tool. There were two cylindrical diamond cutting tools used for the rough and finish processes, where the nominal value of the nose radius was 0.1 mm, and the rake angle and the clearance angle were, respectively, 0° and 15°. For controlling variables, the processing parameters were equally set as 42 rpm spindle speed, a 1 µm depth of cut and 5 µm/rev feed rate. The obtained control workpiece ${\textrm{W}_1}$ and test workpiece ${\textrm{W}_2}$ can be seen in Fig. 10(b).
4.1 Surface finish
To begin with, the surface finish of the obtained workpiece was observed for a basic evaluation. As shown in Fig. 11, the workpieces were observed under the microscope magnification of 5x (Fig. 11(a)-(b)) for global distribution of the machined structures, and the microscope magnification of 20x (Fig. 11(c)-(d)) for a display of the surface finish of a single element. Evidently, the global machined quality of the surface finish of the RCSE strategy is much better than that of the conventional method, as there were significant machined marks along the toolpath and uneven machining effectiveness of elements in different sub-regions in ${\textrm{W}_1}$, while there was a smoother surface finish and uniform machining quality in ${\textrm{W}_2}$. Hence, it is proved that the RCSE strategy can promote the surface finish as well as the processing stability because of the ameliorated feeding motion capability of the machine tool.
Fig. 11. Observations of workpieces at different magnifications. (a) ${\textrm{W}_1}$ at 5x; (b) ${\textrm{W}_2}$ at 5x; (c) ${\textrm{W}_1}$ at 20x; (d) ${\textrm{W}_2}$ at 20x.
4.2 Surface roughness
Aiming to demonstrate the surface finish in detail, the values of surface roughness were measured by the optical profiling system (New Zygo NexView). The ${S_a}$ values of surface roughness were collected and analyzed first. Fifteen elements of each workpiece were randomly selected for measurement with the moderate scanning field of 0.214 × 0.214 $m{m^2}$, where there are three elements chosen in each group of the global machined micro-structured surface. To illustrate, the morphology of one of the selected elements of each workpiece is shown in Fig. 12. Furthermore, the analyzed results of the obtained ${S_a}$ values (areal average roughness, i.e., the average height of all measured points in the areal measurement) are shown in Table 2.
Fig. 12. Morphologies of representative samples for ${S_a}$. (a) Element of ${\textrm{W}_1}$; (b) Element of ${\textrm{W}_2}$.
Table 2. Analysis results of areal average roughness (${S_a}$).
Based on the results above, it can be significantly found that the ${\textrm{W}_2}$ by the RCSE strategy has improved machining quality according to its dramatic decrease in ${S_a}$ value of surface roughness. Specifically, the average value (AVG) of ${S_a}$ of ${\textrm{W}_2}$ has been reduced by 60.62% compared with ${\textrm{W}_1}$ as processed by the conventional method. Furthermore, the standard deviation (SD) of ${S_a}$ of ${\textrm{W}_2}$ is also obviously lower (91.68% decrease) than that of ${\textrm{W}_1}$, which means a higher consistency of global surface finishing by the RCSE strategy in light of the evenness of roughness values in different sub-regions. The characteristic of consistency is considered as a critical index for evaluating the machining quality because of the vital effect of fully utilizing the existing machine tool and obtaining the optimal processing precision [12,33].
Meanwhile, the ${S_z}$ values (the sum of the largest peak height value and the largest pit depth value within the defined area) of the maximum height of the surface roughness are also analyzed in this section. To investigate various elements at different cutting speeds, two elements from the high-speed area, mid-speed area and low-speed area of the global machined micro-structured surface were selected. Thus, a total of six elements of each workpiece were randomly chosen for measurement with the moderate scanning field of 0.214 × 0.214 mm2. To illustrate, the morphology of one of the selected elements of each workpiece is shown in Fig. 13. Furthermore, the analyzed results of the obtained ${S_a}$ values are shown in Table 3.
Fig. 13. Morphologies of representative samples for ${S_z}$. (a) Element of ${\textrm{W}_1}$; (b) Element of ${\textrm{W}_2}$.
Table 3. Analysis results of the maximum height of surface roughness value (${S_z}$).
Based on the results above, it can be significantly found that the ${\textrm{W}_2}$ by the RCSE strategy has optimized machining accuracy in light of its dramatic decrease in ${S_z}$ value of surface roughness. Specifically, the average value (AVG) of ${S_a}$ of ${\textrm{W}_2}$ has been reduced by 61.15% compared with ${\textrm{W}_1}$ processed by the conventional method, showing the more precise profile of the local measured area of the element. Besides, the standard deviation (SD) of ${S_z}$ of ${\textrm{W}_2}$ is also decreased and achieves a higher consistency of global machining quality by the RCSE strategy. However, the machining accuracy should be further proved by the profile error values which are calculated in the following section.
4.3 Following error and profile error
To further confirm the machining quality, the values of following error in machining were collected and the profile errors (or named as form errors) of the obtained workpieces were measured by the Optical Profiling System (New Zygo NexView) and calculated by the software, MATLAB. According to the theoretical explanation in Section 2, the X-axial following error during the machining process is focused on. By recording and collecting the related values, the maximum value of following error was 652 nm during the machining process of ${\textrm{W}_1}$, while this value in the machining process of ${\textrm{W}_2}$ was only 16 nm. The following error of the X-axis is almost eliminated by the RCSE strategy as the feeding motion stability of the X-axis is greatly ensured. This is the critical condition for guaranteeing the machining quality in the STS turning process. At the same time, the largest following error of the Z-axis of ${\textrm{W}_1}$ and ${\textrm{W}_2}$ during machining were, respectively, 266 nm and 226 nm, which is not a significant difference because of the similar amplitude and frequency of the reciprocating feeding motion of the Z-axis. For clarity, the values of the following error are summarized in Table 4. The following error is related to the profile error of the machined surface, as the larger value means a deterioration of the dynamic capability of the machine tool during the machining process. Thus, it can be predicted that ${\textrm{W}_2}$ has better capability in regard to the profile error.
Table 4. Maximum following error of different axes in machining.
To prove the above, the original values of two elements from the high-speed area, mid-speed area and low-speed area of the global machined micro-structured surface were selected respectively. Thus, a total of six elements of each workpiece were randomly chosen for calculation of the profile error of the surface of a whole sub-region. To illustrate, the morphology and profile of one of the selected elements of each workpiece are shown in Fig. 14. The profile error was measured along two slice directions, i.e., the cutting direction and radial direction which is the same as X-axial direction in machining. Furthermore, the average values of the calculated profile error along the two directions are shown in Table 5.
Fig. 14. Morphologies and profiles of representative samples. (a) Element of ${\textrm{W}_1}$; (b) Element of ${\textrm{W}_2}$.
Table 5. Profile error (AVG) in different directions.
Based on the results above, there are significant differences between different toolpath generation methods as well as different measured slice directions. In particular, the profile errors of ${\textrm{W}_2}$ were less than that of ${\textrm{W}_1}$ in various directions, where the profile error along the cutting direction of ${\textrm{W}_2}$ was 0.3822 µm, which decreased by 32.33% compared with ${\textrm{W}_1}$, while the profile error along the radial direction of ${\textrm{W}_2}$ was 0.2570 µm, which decreased by 58.38% compared with ${\textrm{W}_1}$. Furthermore, there was also a difference in the value changing of two directions. Specifically, as for ${\textrm{W}_1}$, the profile error along the radial direction was larger than that in the cutting direction, while regarding ${\textrm{W}_2}$, it was the inverse in that the profile error in the radial direction was the smaller one. This phenomenon reflects the effectiveness of the RCSE strategy that the feeding motion of the X-axis of the machine tool was dramatically optimized, which was focused on in this investigation. As found in Section 2, the reciprocating feeding motion of the X-axis has a stronger impact on the global machining quality. Due to the elimination of the reciprocating motion of the X-axis by the RCSE-strategy toolpath generation method, this issue has been well solved. Hence, both the following error and profile error of the X-axis can be restrained, leading to much better machining quality along the radial direction (X-axial direction), which manifests as a larger decline in profile error in this direction.
5. Conclusion
In this paper, the characteristics of feeding motion of different axes in STS ultra-precision diamond turning and its influence on toolpath parameters have been analyzed and the sub-regional RCSE-strategy toolpath generation method has been proposed. The main findings are summarized as follows:
- (1) The characteristics of feeding motion in STS diamond turning have been theoretically analyzed and the critical issue of mismatching between the Archimedean spiral toolpath and the fundamental modelling meshing shape has been found.
- (2) The rotary-coordinate process and shuttling-element process have been investigated. By transforming the cutting motion mode of the toolpath, coordinates and elements, the process of modelling and toolpath generation in ultra-precision diamond turning has been integrated and ameliorated.
- (3) An algorithm has been built to obtain and evaluate the practical toolpath. According to the calculated kinematic parameters, the RCSE strategy can dramatically promote the continuity and smoothness of the global toolpath.
- (4) An experiment has been carried out, which can prove the effectiveness of the innovative method compared with the conventional modelling and toolpath generation method. The measured surface finish, roughness and profile error can support the view that the proposed method can significantly ensure the stability of the X-axis of machine tools and enhance the machining quality.
The presented study can provide an innovative method to directly generate toolpaths without pre-modelling, which can simplify the process of machining process planning. Furthermore, the RCSE strategy has provided a creative and practical theory to solve the problem regarding dynamic performance and machining quality in ultra-precision turning.
Funding
PhD Studentship, Research Committee of The Hong Kong Polytechnic University (RHM7); Research and Innovation Office of The Hong Kong Polytechnic University (BBX2, BBX5).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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