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Abstract
Mid-spatial-frequency (MSF) errors seriously damages the imaging performance of optical components. Path pattern is an important factor that affects the generation of MSF waviness in polishing. This paper proposes a versatile pseudo random tree-shaped path (RTSP) generation method imitating the growth rules of tree branch in nature, which can efficiently generate continuous, uniformly distributed and multi-directional paths on a specified surface. Furthermore, the effectiveness of RTSP to suppress MSF waviness is verified by numerical simulation and experimental results. Finally, the RTSP is applied to the polishing of Ti-6Al-4V titanium alloy mirror. Results indicate that grinding marks have been significantly removed, while no obvious MSF waviness is introduced.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of modern optical technology, ultra-precision optical components have been widely used in high-power laser systems, astronomical observation navigation and ultraviolet lithography systems [1,2]. The improvement of optical system performance, in turn put forward higher requirements on the stringent power spectral density (PSD) specification of optical components in full frequency regime, in terms of low-spatial, mid-spatial, and high-spatial frequencies [3]. At present, low-spatial frequency errors, that is, the surface form accuracy, can be convergent through the deterministic removal iteration of computer controlled optical surfacing (CCOS) technology [4]. The high-spatial frequency errors, corresponding to surface roughness, can be successfully reduced by smoothing technologies such as pitch polishing [5], float polishing [6] and chemical mechanical polishing (CMP) [7]. The mid-spatial frequency (MSF) errors (also called waviness) is usually undesirable but inevitable in optics fabrication, which may be caused by many factors, such as tool marks left by previous grinding or single-point diamond turning, uneven tool wear during polishing, changes of polishing slurry concentration, cut-off frequency limit of removal function and polishing path patterns, etc. [8]. The existence of MSF waviness will cause great damage on the performance of optical systems, even endanger their stable and safe operation. For example, the periodic waviness can cause small-angle scattering in the point spread function (PSF), produce speckle and reduce the image sharpness [9,10]. Up to now, MSF waviness suppression in polishing is still faces various challenges. Traditional manual polishing is time-consuming, laborious, harsh environment and has poor process consistency. The automatic polishing based on industrial robots or machine tools has greater advantages over manual methods in terms of improving productivity, enhancing processing quality and reducing manufacturing costs, and is being widely used. MSF waviness control is significantly important in automatic polishing and higher certainty for MSF waviness suppression is required. Once an unacceptable MSF waviness is introduced in the current polishing stage, the subsequent use of polishing tools with different cutting-off frequencies, and process parameters optimization such as polishing spacing are often required to gradually eliminate the MSF waviness caused by the previous process. This process is complexity, time consuming, which increases the risk of accidentally generating irreversible machining errors. Therefore, it is very necessary to maximize the suppression of MSF waviness in each polishing.
To effectively suppress or reduce MSF waviness in automatic polishing, a variety of approaches have been proposed in previous literature from different perspectives, which can be classified into three categories, namely, tool optimization approach, parameter optimization approach and path optimization approach. As a well-known example of tool optimization, lap tools made of pitch are prone to suppress MSF waviness effectively [11]. Further, Kim et al. [12] proposed to use rigid conformal polishing lap filled with non-Newtonian fluid to achieve the reduction of MSF waviness for large-sized workpiece. To adapt to sub-aperture deterministic polishing and improve MSF waviness removal efficiency, Zhu et al. [13] recently developed an improved bonnet tool filled with viscoelastic fluid, and its mechanism and effectiveness in reducing MSF waviness was revealed and verified. As an example of parameter optimization approach, Huang et al. [14] investigated the process parameters in bonnet polishing on MSF textures, so as to the effectively remove irregular surface ripples and reduce MSF errors by optimizing polishing parameters. Traditional polishing paths such as raster path, spiral path and Hilbert path are widely used in automatic polishing due to its uniformity and characteristics of strong traversal. However, this kind of path has an obvious regular pattern, and cannot imitate the randomness of manual polishing, which brings challenges to the suppression of MSF waviness in automatic polishing [10].
As another approach, path optimization approach have widely used in suppressing MSF waviness, such as adaptive paths and pseudo random paths. For example, Rososhansky et al. [15] first applied contact mechanics to the planning of polishing path to guide the adjustment of polishing path spacing. Han et al. [16] further enriched the physical uniform coverage path generation theory for complex surface polishing, including scanning and spiral paths. The experimental results show that the optimized path can achieve more uniform material removal, and the polishing waviness can be effectively suppressed. In addition, Han et al. [17] established a regional form error adaptive path planning method based on analytical convolution model, which can optimize the dynamic constraints of machine tools and suppress MSF waviness in deterministic polishing simultaneously. However, these paths still have obvious regularity and are regarded as mitigation methods of MSF waviness suppression. Pseudo random paths with multi-directions are proved to be effective and convenient in suppressing MSF waviness in automatic polishing [18]. Tam et al. [19] proposed a method to construct Peano paths for achieving uniform coverage of aspherical surfaces and verified the processing capability of the path to suppress MSF waviness. Dunn and Walker [20] firstly presented a pseudo-random tool path for using with CNC sub-aperture polishing techniques and demonstrated its feasibility for restraining MSF waviness. Dong et al. [21] proposed a random fractal-like path that exhibits high randomness, boundary adaption and step-length arbitrariness, which was successfully used to manufacture optical elements with strict low surface waviness requirement. Takizawa et al. [22] developed a novel circular pseudo-random path, which can optimize the smoothness of the feed rate of the polishing tool and further suppress the surface ripple. On this basis, Beaucamp et al. [10] proposed an extension of the circular-random path to aspheric and freeform surfaces, experiment results show that the random path can reduce the amplitude of MSF errors and enhance relative intensity of PSF images. In summary, paths with multi-directionality and randomness have great significance for the suppression of MSF waviness and are suitable for automatic polishing application.
However, existing random path planning methods often have complex algorithms without fully disclosed, low efficiency and poor versatility in automatic polishing. As an important research progress in this field, this paper proposes a random tree-shaped path (RTSP) imitating the topological rules of tree branch growth in nature. This idea can quickly and efficiently generate a continuous, uniformly distributed and multi-directional smooth path on the workpiece surface with arbitrary boundaries. In addition, it can efficiently generate random polishing paths with 4 or 6 directions or even any direction, which can meet the requirements of robot automatic polishing.
The rest of the paper is organized as follows. Section 2 introduces the basic idea of RTSP and describes its generation algorithm in detail. Section 3 details the experimental setup, design, and surface metrology. Section 4 gives the simulation results, which preliminarily certify the effectiveness of the proposed path in suppressing MSF waviness. Section 5 verifies the correctness of the simulation results by bonnet polishing experiments. Section 6 conducts polishing tests on a planar mirror made of titanium alloy, which demonstrates the application potential of the proposed path in automatic high quality polishing, followed by some important conclusions in Section 7.
2. Random tree-shaped path generation algorithm
2.1 Basic idea
A random tree-shaped path imitating the growth rules of branches [23] is proposed for polishing, as shown in Fig. 1. Firstly, branches are randomly and uniformly generated one by one in a two-dimensional weightless confined space to obtain a trunk covering the entire region. Then, according to the non-closed trunk generated above, the continuous disjoint and uniformly distributed random paths can be obtained through unilateral boundary extraction. Finally, the smooth path suitable for polishing is obtained by post-processing.
2.2 Algorithm
This section presents the detailed generation algorithm for RTSP, and the overall flow chart is shown in Fig. 2. This algorithm mainly includes four modules, that is, parameter input module, tree branch generation module, RTSP generation module and path post-processing module. Parameter input module is used to generate uniformly distributed control points according to the specified process parameters including workpiece boundary, path type, and path spacing. In tree branch generation module, the random tree branch is grown in fully covered region with control points. The boundaries of the branch curves are extracted in RTSP generation module to obtain the initial RTSP. Finally, the random path is smoothed by post-processing module, and the RTSP suitable for polishing is obtained. The algorithm is programmed and implemented based on MATLAB environment. This detailed algorithm can be described as following four steps.
Step 1: Specific process parameter inputs for generating control points.
Input process parameters include workpiece boundaries, path spacing and path type. Workpiece boundaries is used for constraining the layout range of the control points. Path spacing p is selected according to bonnet radius and surface quality requirement and also used for determining the interval of adjacent control points, which is set as 2 times of specified path spacing. Path type determines the randomness of the path direction. Thus, the control points ${P_1},{P_2}, \ldots {P_n}$, can be generated, as shown in Fig. 3(a), which is stored into the U set.
Fig. 3. Generation of random branch curve. (a) Layout of control points; (b) Generation of a trunk and branch;(c) 4-direction and (d) 6-direction branch curve.
Step 2: Generation of random branch curve.
Randomly select a point ${P_r} = random\{U \}$ as the starting point, and the set $U^{\prime}$ is a replica set from the set U with the element ${P_r}$ removed. Search for points ${P_{r1}},{P_{r2}}, \ldots {P_{rn}}$ whose distance from ${P_r}$ is pixel pitch, and store them in the set ${U_p}$. Judge whether the intersection of $U^{\prime}$ and ${U_p}$ is an empty set. If it is not empty, randomly select a point $P_r^\mathrm{^{\prime}}$ in the intersection of $U^{\prime}$ and ${U_p}$, connect ${P_r}$ and $P_r^\mathrm{^{\prime}}$ for branch growth display, take $P_r^\mathrm{^{\prime}}$ as the new ${P_r}$, and store the coordinate of old ${P_r}$ in the Q queue in sequence. If intersection of $U^{\prime}$ and ${U_p}$ is empty, the generation of truck is completed, as shown the orange curve in Fig. 3(b). Then judge whether $U^{\prime}$ is an empty set, that is, whether all control points are connected with branch. If $U^{\prime}$ is not empty, search for a point in set Q as the starting point, and connect the unconnected points in the control points, which can be analogized to the growing process of bypass branches (yellow curve) on the tree trunk as shown in Fig. 3(b). Repeat above growing process of bypass branched, until $U^{\prime}$ is an empty set, that is, when all control points are connected, it will jump out of the loop. Finally, the random branch curve has been generated entirely. Figure 3(c) and 3(d) show the generated 4-direction and 6-direction random branch curves.
Step 3: RTSP points extraction.
For any non-closed curve, a closed disjoint curve can always be obtained by using the method of one-sided stroke. Based on the non-closed branch curve that generated in Step 2, a closed, disjoint, and uniformly distributed random polishing path can be generated according to the requirements of the process by using the method of one-sided stroke. Taking the right stroke as an example, the following describes the specific steps of extracting RTSP points.
- (a) Arbitrarily select a point on branch curve as a starting position and specify the forward direction $Dir$ on the guide curve. $Dir$ is positive or negative, and assuming the direction from the root to the top of the branch growth is positive, set the starting point as the reference point ${P_o}$, and generate the first path point ${P_{t1}}$ on the branch curve between reference point ${P_o}$ and its adjacent point $P_o^\mathrm{^{\prime}}$ on forward direction $Dir$, as shown in Fig. 4(a).
- (b) The path points can be obtained in turn by RTSP generation module, the following will take the tree branches (including the trunk and branches) as an example to describe the three main situation of this processing module.
Situation 1: stroking along another branch that grows from point ${P_o}$, and take a point at the distance p/2 on the right side of the vertical line between ${P_o}$ and $P_o^\mathrm{^{\prime}}$ as ${P_{t2}}$, as shown in Fig. 4(b).
Situation 2: reverse stroking along the current branch when ${P_o}$ is the terminal point. As shown in Fig. 4(c).
Situation 3: stroking along the current branch, as shown in Fig. 4(d).
Fig. 4. Process of path extraction. (a) First path point determination; (b, c, d) processing module: along one side of the curve; (e) return to the starting point and end of path extraction; (f) Initial obtained path points.
Step 4: RTSP fitting with NURBS curve.
NURBS curve can help modern manufacturing more efficient and accurate to complete the processing target. It is more and more widely used in CNC machine tools and six-degree-of-freedom robot path planning. The definition of NURBS curve is as Eq. (1) [24,25],
According to the fitting results in Fig. 5, it can be seen that NURBS degree has a significant impact on the path smoothness and uniformity. Specifically, when the degree p is 1, the path achieves unobvious smoothing effect compared with original curve. As the degree increases, the curve becomes more and more smooth, however, the equidistant advantage of path spacing is getting worse, which deteriorates the uniform coverage of paths. In order to balance the smoothness of the actual robot motion process and the uniformity of polishing removal, the quadratic NURBS curve is preferred to smooth the proposed RTSP.
Fig. 5. Path smoothing based on NURBS technique. (a) Fitting results and (b) locally smoothed corners by different NURBS degrees.
2.3 Path generation performance
On the basis of the generation principle of the random tree path algorithm above, it can be found that the path is generated in accordance with the layout of the control points. Therefore, evenly laying out points on the workpiece surface with different boundary shapes can generate a RTSP that is uniformly traversed in effective region for polishing. In order to verify the performance of proposed RTSP generation algorithm, a series of path generation cases for various boundary conditions were carried out, and the results are shown in Fig. 6. It can be observed that uniform traversal and random tree paths can be generated within the boundaries of different shapes, and the path trajectories within each boundary are different. These results show that the path not only has strong boundary adaptability, but also has high randomness. In addition, the generation process of proposed random path does not need repeated time-consuming search and iteration, but a process of random arrangement and selection of data, which can generate random paths more efficiently within a few seconds.
Fig. 6. Generate random branch curve in (a) hexagon, (b) circle, (c) concentric circle and (d) Bing Dwen Dwen and corresponding RTSPs in (e-h).
3. Experimental setup, design and metrology
3.1 Experimental setup for bonnet polishing
The experimental setup used in this paper is given in Fig. 7(a) and 7(b). A 6-Degrees-Of-Freedom (6-DOF) robot (KUKA KR1420) is used to control the positioning and travel speed of a bonnet tool. As shown in Fig. 7(a), an end effector equipped with a constant force controller is installed on the robot flange. The force controller can detect contact force variation between the bonnet and the workpiece along the tool axis and realize specified command force with an error of ± 1 N by closed-loop feedback and real-time adjustment of an inner cylinder displacement. In this way, the decoupling control of polishing force and position can be achieved, the uncontrollable contact force caused by robot positioning error in the polishing process can be avoided, which is conducive to improving the controllability of polishing quality. The bonnet tool spindle is driven by an offset servo motor through the synchronous belt, and the radial rotation error of the spindle is about 2 µm and the available spindle speed ranges from 100 to 3000 RPM. During polishing, slurry is continuously supplied by peristaltic pump through universal bamboo joint nozzle and protected by a slurry cover made of acrylic material to prevent slurry sputtering. In addition, the used slurry can flow back to the storage tank from the outlet at the bottom of the cover to make the polishing process sustainable.
Fig. 7. Experimental setup. (a) Robotic bonnet polishing configuration; (b) Bonnet pad dressing process and bonnet tool before and after dressing; (c) Polishing principle and (d) typical footprints in bonnet polishing.
The radius of the bonnet tool used in experiment is 10 mm and covered with LP27 polyurethane pad. The rotational symmetry of polishing pad that pasted on the bonnet will significantly affect the stability of the etching footprint the stability of etched footprint. In this experiment, a cup-shaped diamond grinding wheel with a diameter of 8 mm is used to dress the polishing pad to ensure its good rotational symmetry. During dressing, the axis of the grinding spindle and the tool axis are in a plane and intersect near the center of the bonnet at a certain angle. In this way, on the one hand, it can ensure that the envelope formed by the relative rotation of the grinding wheel and the bonnet head can effectively dress the whole surface of the polishing pad, on the other hand, it can ensure the improvement of rotational symmetry under the condition of removing as little material as possible so as not to destroy the gap state on the surface of the polishing pad. The dressing process and the bonnet tool before and after dressing are shown in Fig. 7(b), respectively.
Figure 7(c) shows a schematic diagram of the principle of bonnet polishing (BP), which realizes efficient material removal from the workpiece surface by controlling the rotation of a flexible rubber bonnet filled with pressurized air and covered with specified polishing pad. The process parameters that affect the removal performance mainly include feed rate, path spacing, bonnet diameter, tool speed, procession angle, bonnet inner pressure and compression offset. Compression offset is usually used to indirectly control the contact force. It is worth mentioning that in this experiment, the contact force is directly set through the force controller. The controllable balance between the contact flexibility and stiffness of the rubber bonnet makes it well adapt to the changes of the workpiece surface. Therefore, bonnet polishing has the advantages of high removal efficiency and high processing precision, and is especially suitable for polishing of aspheric surfaces and freeform surfaces.
To verify the stability of material removal in BP, a series of spot polishing tests were carried out on planar K9 glass with an initial surface roughness of 1.5 nm, a diameter of 50 mm. Table 1 lists the detailed experimental parameters. The polishing slurry adopts a CeO2 solution with a particle size of 0.8 µm and a concentration of 25 g/L mixed with water. The dwell time of each spot was controlled to 8 s and the corresponding tool influence function (TIF, also called footprint) is normalized to unit time, as shown in Fig. 7(d). Results show that the footprints for 4 spot tests have high similarity, and the average material removal rate is 9.976×10-4 mm3/s, with a removal rate deviation of less than 5%, which confirms the stability of material removal in BP.
Table 1. Experimental parameters in BP.
3.2 Simulation and experimental design
The actual footprint (Fig. 7(d)) in BP experiment not only includes the profile information of material removal, but also includes the scribed topography introduced by abrasive particles along angular velocity direction of the bonnet, corresponding to low-spatial frequency and MSF, respectively. Through filtering, the included removal form and scribed waviness topography can be obtained, as shown in Fig. 8, respectively. This scribed topography is mainly caused by the comprehensive influence of nonlinear deformation of bonnet pad at the contact interface and scratch of abrasive particles between pad and workpiece, which will inevitably affect the waviness topography of the polished surface. Therefore, in order to obtain the simulation results that are close to the actual polishing results, footprint that including the low and mid-spatial frequencies should be used in simulation. This can not only simulate the macro removal distribution introduced by the bonnet tool along the polishing path, but also can simulate the influence of the waviness resulted by bonnet contact state itself on the final waviness topography after polishing.
To better understand the generation of MSF waviness in polishing and validate the suppression effectiveness of MSF waviness by proposed RTSP. Three simulations were conducted in Section 4, the simulation adopts the same experimental parameters given in Table 1, and the TIF adopts the complete influence function obtained by spot polishing in experiment, as shown in Fig. 7(d). Section 4.1 studies the influence of path spacing on MSF resulted by RTSP. Section 4.2 compares the MSF waviness topographies introduced by different path patterns. In Section 4.3, the evolution of the MSF waviness topography as a function of polishing runs, in which the path is randomly generated in each round is studied.
Further, a set of comparative experiments corresponding to Section 4.2 were carried out in Section 5 to verify the effectiveness of the proposed algorithm. In Section 6, a planar metal mirror is polished by proposed RTSP with multiple runs to demonstrate its practicability in automatic polishing. All experimental conditions are consistent with the parameter settings in Table 1.
3.3 Metrology
The spatial wavelength corresponding to the MSF waviness in optical applications is generally less than 1/10 of the aperture size of the optical element and greater than λ/10 (λ = 628 nm) [27]. In order to better identify the MSF waviness introduced by polishing, high-pass filtering is used to remove the surface form error and extract the interesting waviness topography characteristics. It worth mentioning that the spatial cutoff wavelength is selected as 1.0 λ by trial and error, which can better present the waviness topography caused by different paths under the experimental conditions. In addition, for the rationality of comparison, the topography distribution of the MSF waviness is normalized to the unit removal depth.
The measuring equipment used for surface characterization in the experiment includes optical microscope and white light interferometer. An optical microscope with a 20x optical objective is used to visually observe the scratches, pits and other defects on surfaces before and after polishing. The surface topography was measured with 10x magnification objective lens by a white light interferometer (Superview W1 series). The measuring field of view corresponding to the 10x magnification objective lens is limited to 0.5 mm × 0.5 mm. In order to measure a footprint (3.5 mm × 3.5 mm) or an effective polishing removal area (5 mm × 5 mm), stitching measurements were used to characterize the entire polished area. In addition, in order to obtain more accurate surface roughness, the measurement is switched to 50x magnification objective lens, and several points are randomly sampled from the polishing area for characterization.
4. Simulation results
4.1 Effect of path spacing on MSF waviness
In polishing process, path spacing (also called path interval) is an important factor affecting the generation of MSF waviness, which was investigated by numerical simulation for 4-direction and 6-direction random paths in this section. The experimental bonnet TIF corresponding to experimental parameter in Table 1 is used for process simulation. The path spacing values are set to 0.3 mm, 0.4 mm, 0.5 mm and 0.6 mm, respectively. For each case, a region with size of 10 mm × 10 mm is polished by simulation and a central with a size of 5 mm × 5 mm of the simulated topography was cropped for analysis to avoid the interference of edge effect.
Figure 9 shows the finally processed MSF waviness corresponding to the 4-direction and 6-direction random paths under different path spacing, respectively. Figure 10 shows the corresponding RMS and PV curves as function of the path spacing. It can be seen that with the increase of path spacing, the MSF waviness introduced by the polishing path shows a basic trend of increasing. When the path spacing is 0.3 mm, the MSF waviness introduced by the 4-direction or 6-direction paths is the minimum. But for 4-direction paths, the periodic features introduced are most significant when the path spacing is 0.5 mm, this is caused by the complex convolution of irregular TIF shape along adjacent paths in sub-aperture polishing.
Fig. 9. Simulated MSF waviness topography evolution as function of path spacing for (a) 4-direction and (b) 6-direction random paths.
Interestingly, it can be intuitively seen from Fig. 9(a) and 9(b) that the MSF waviness exhibits a more significant strip-like features in the horizontal direction than that in vertical direction. This phenomenon is mainly caused by the asymmetric distribution of footprint and the MSF waviness caused by the abrasive grains scribing at the contact area. The bonnet rotational linear velocity in path planning is fixed along Y direction, thereby superimposing and enhancing the path-induced MSF in the X direction. For the 6-direction random path, the MSF waviness difference between X and Y directions is not significant compared with that of 4-direction path, since the 6-direction path has more random directions.
4.2 Simulation results by different paths
The MSF waviness topographies induced by polishing along 4-direction and 6-direction paths are compared with traditional raster path and Hilbert path under the same path spacing of 0.5mm, respectively, as shown in Fig. 11(a)–11(d). It can be seen that with the improvement of the randomness of path direction, the regular MSF waviness on polished surface are gradually suppressed. In addition, due to the regularity of the raster and Hilbert path, obvious periodic waviness was introduced on the workpiece surface. The MSF waviness of raster path are generated perpendicular to the path because of the discontinuity of path along X direction. For the Hilbert path, the MSF waviness shows clearer ripple along X direction than that of Y direction due to the fixed attitude of bonnet tool in path planning.
Fig. 11. Simulated waviness topography and corresponding PSD curves comparison. MSF waviness polished by (a) raster path, (b) Hilbert path, (c) 4-direction path and (d) 6-direction path. PSD curves (e) along X direction and (f) along Y direction.
The PSD analysis are calculated for MSF waviness along X and Y direction, respectively, as shown in Fig. 11(e) and 11(f). It can be seen that the main peak occurs at the spatial frequency of 2 mm−1, corresponding to the selected path spacing of 0.5 mm. In addition, the PSD curves corresponding to raster and Hilbert paths have higher peak intensity than that of 4-direction and 6-direction random paths. And 6-direction path can generate weakest peak intensity, which verifies the effectiveness of random path in suppressing MSF waviness and also proves that the stronger the randomness of the path direction, the better the effect of suppressing MSF waviness.
4.3 Polishing simulation by multiple runs
For the actual processing application, polishing operation of the workpiece surface is usually not completed by one round. In repeated polishing, the traditional paths not easily to suppress or even aggravate the waviness of the polished surface, it often needs to change the polishing direction or the attitude of polishing tool to control the MSF waviness. These methods not only reduce the processing efficiency, but also put forward strict requirements on the repetition precision of polishing system. The algorithm proposed in this paper has superior randomness and can ensure that the paths generated each time are random and different, so as to achieve the effective suppression of MSF waviness without affecting the main polishing configuration. MSF waviness topographies by multiple polishing with random paths are simulated in this section.
The simulated MSF waviness topography evolution as function of run time along 4-direction and 6-direction random paths are shown in Fig. 12. Figure 13 gives the corresponding RMS and PV trend of the MSF waviness. These results clearly show that with the increase of polishing runs, RMS and PV values of the two path gradually decrease and tend to be stable, which confirms the effectiveness of the proposed random path to suppress MSF waviness through multiple polishing runs.
Fig. 12. Simulated MSF waviness topography evolution as function of run time for (a) 4-direction and (b) 6-direction random paths.
5. Experimental verification
Polishing experiments were carried out to verify the correctness of the predicted MSF error and the effectiveness of RTSP in suppressing MSF waviness. Four square regions are scheduled on a planar K9 glass (with an initial surface roughness of 1.5 nm and a diameter of 50 mm), each with a size of 10 mm×10 mm, and polished by raster, Hilbert, 4-direction and 6-direction random paths, respectively. The generated raster paths, Hilbert paths and random paths are compiled into the KRL programming language recognized by KUKA robot controller and executed. It worth mentioning that the central region with size of 5 mm × 5 mm is cropped and processed by filtering for comparison. In addition, the path spacing adopted is 0.5 mm and CeO2 solution with a particle size of 0.8 µm and a concentration of 25 g/L mixed with water is used.
The processed MSF waviness topographies polished by different patterns are shown in Fig. 14(a)–14(d), respectively. It can be seen that the MSF waviness topographies are consistent with that predicted by numerical method given in Fig. 11. It worth noting that the experimental MSF waviness by raster path is not continuous along path direction, which may be due to the vibration in the actual polishing. Specifically, during the polishing experiment, the response limit of force controller makes it difficult to accurately compensate the fluctuation of contact force along the path caused by equipment vibration or the sudden change of the initial surface error, thus resulting the fluctuation of the removal depth along the polishing path.
Fig. 14. Experimental waviness topographies and corresponding PSD curves comparison. MSF waviness polished by (a) raster path, (b) Hilbert path, (c) 4-direction path and (d) 6-direction path; PSD curves (e) along X direction and (f) along Y direction.
Similarly, the PSD analysis results of four path patterns along X and Y directions were shown in Fig. 15(e) and 15(f), respectively, which further demonstrates the correctness of the predicted MSF waviness and the effectiveness of proposed RTSP in suppressing MSF waviness. And it can be seen that no matter experiment or simulation, the effect of the four paths on the suppression of MSF error in the Y direction is not significant. This is mainly because the MSF error along Y direction is seriously disturbed by the internal waviness of the footprint caused by abrasive scraping, while the vibration of the actual polishing equipment, especially polishing along the random path with sharp corners, further weakens the differences caused by path patterns. Therefore, it is worth noting that the 6-direction path has better directional randomness than 4- direction or one pattern path, which can better suppress the MSF waviness in polishing, however, the sharp corner existing in the path poses more challenges for the dynamic performance of the polishing machine.
6. Application
In this section, the proposed RTSP is applied to the polishing of a planar metal mirror to verify its practicability in automatic polishing. The sample to be polished is made of Ti-6Al-4V titanium alloy and the initial surface is prepared by grinding with #2000 grit sandpaper, reaching an initial surface roughness of Sa 87.447 nm. The polishing slurries used is self-prepared and consists of 8 wt% colloidal silica with a particle size of 80 nm, 100 mM ammonium persulfate, 80 mM NaF, 20 mM citric acid and deionized water, as listed in Table 2. The pH of the solution is 4, which is adjusted by citric acid. In addition, ammonium persulfate is an oxidant, which can oxidize titanium alloy and form an oxide film on the surface. Sodium fluoride is a complexing agent, in which fluorine ions can destroy the oxide film formed on the surface of passivated titanium alloy. During the experiment, three rounds of polishing were performed on the same central region of the sample, and the 4-direction polishing paths for each round were randomly generated. It worth mentioning that the actual polishing time of one round is 40 minutes while the predicted runtime is 39.93 minutes. This is mainly caused by the limitation of the dynamic performance of the polishing machine, and the commanded feed rate cannot be accurately followed at the sharp corner of the path because of unexpected acceleration and deceleration.
Table 2. Polishing slurry components.
Figure 15 shows the surface roughness topographies and corresponding RMS and PV values as function of polishing runs. It can be seen that there are obvious grinding marks on the initial titanium alloy surface. As polishing round increases, the grinding marks are gradually removed and replaced by fine scratches introduced in polishing. In addition, the RMS value of the polished surface roughness decreases rapidly from the initial Sa 83.077 nm to Sa 5.57 nm after one round of polishing. After two and three rounds of polishing, the surface roughness remains stable, reaching Sa 5.23 nm and Sa 5.44 nm, respectively. The PV value of the surface roughness decreases from the initial 874.47 nm to 47.20 nm, 37.57 nm and 42.22 nm, respectively, after three rounds of polishing, and also tends to be stable.
Figure 16(a) and 16(b) show the surface morphology pictures taken by optical microscope before and after polishing. Results shows that there are obvious scratches, pits and other defects on the unpolished surface and these surface defects are then effectively removed after polishing. The planar samples of titanium alloy before and after polishing are shown in Fig. 16(c) and 16(d), respectively and the surface after polishing achieves a mirror effect. Figure 17 gives the MSF waviness topographies before and after polishing and corresponding PSD curves along X and Y directions. It can be seen that the PSD curve after polishing shifts downward significantly which indicates that MSF waviness caused by previous grinding has been significantly removed and no obvious MSF waviness was introduced by polishing. These results demonstrate the effectiveness and high efficiency of the proposed RTSP to suppress MSF waviness and improve polishing quality.
Fig. 16. Experimental results in polishing TC4 alloy. (a) Optical micrographs before polishing and (b) after polishing; (c) Unpolished surface and (d) polished surface.
Fig. 17. MSF waviness comparison. MSF waviness topographies (a) before and (b) after polishing; PSD curves (c) along X direction and (d) along Y direction.
7. Conclusions
In this paper, a universal random tree-shaped path (RTSP) generation method is proposed, which efficiently realizes the randomness and uniformity of paths. Based on numerical simulation, the influence of process parameters and path pattern on polished MSF waviness is investigated. The consistency of the simulation and experimental results confirms the correctness of the predicted MSF waviness and the effectiveness of the proposed RTSP to suppress MSF waviness. The conclusions and contributions of this paper are as follows:
- 1) A RTSP generation method with strong versatility is proposed by imitating the branch growth rules, which can efficiently generate random polishing paths with 4 or 6 directions or even any direction, and can meet the requirements of automatic high-quality polishing.
- 2) The proposed RTSP is compared with traditional paths through simulation and experiments, and the results consistently show that the RTSP, especially 6-direction, can effectively suppress the MSF waviness.
- 3) Multiple runs polishing can effectively suppress the MSF waviness by randomly generated RTSP in each run which can reduce the complexity of the process in practice.
- 4) A planar mirror made of Ti-6Al-4V titanium alloy with an initial roughness of Sa 87.1 nm is polished by one round that took about 40 minutes and the surface roughness is rapidly reduced to Sa 5.5 nm, achieving a mirror effect. After three rounds of polishing along different 6-direction random paths, results show that no significant MSF waviness is introduced on the polished surface.
Funding
National Natural Science Foundation of China (51991371); Natural Science Foundation of Sichuan Province (2022NSFSC1927); China Postdoctoral Science Foundation (2022M712634); Fundamental Research Funds for the Central Universities (2682021CX026).
Acknowledgments
The authors acknowledge support from Mr. Menghuan Yu for English improvement of this paper.
Disclosures
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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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