## 4 Experimental Validation
### Polishing Experiment Verification  
The polishing force (F) and optimum trajectory space in different regions were set to plan the polishing trajectory and process. The field processing is shown in Figure 16, where different regions were polished individually. The polished Han white marble workpiece A, as shown in Figure 17, has various curvature distributions on its surface, with some regions showing significant changes in curvature. However, after grinding and polishing, the overall appearance is smooth, delicate, and uniform, with a remarkable luster effect. Figure 14: Boundaries of regions by curvature partition. Figure 13: 3D model of curvature partition. Figure 15: Curvature distribution of different regions by curvature partition. In this section, the luster effect of workpiece A, polished by the robotic system, is compared with workpiece B, polished by hand, using the text and image reflection method. As shown in Figure 18, the surface of the Han white marble workpiece A, polished by the robotic flexible polishing, effectively reflects the text in the image, maintaining clear text outlines and showing a good mirror effect and consistency. Conversely, Figure 19 shows that the surface gloss of workpiece B, manually polished Han white marble, is significantly inferior to the results of robotic flexible polishing. Although the surface has some gloss, the uniformity and consistency are poor, and the surface is unable to clearly reflect the text and image. Figure 16: Flexible polishing of complex-shaped marble. Figure 17: Workpiece A, polished Hanbaiyu marble. In this study, roughness (Ra) and glossiness were assessed to evaluate the surface quality of the marble tiles. Roughness was measured with a TIME 3220 portable surface roughness tester. In addition, a WG60G was used to measure the glossiness with a 60\({}^{\circ}\) angle for standard measurement. This device has a measurement range from 0 to 2000 gloss units and a resolution of 1 gloss unit. Different regions of the Hanbaiyu marble workpieces polished by the two processing methods were measured for surface roughness  
Figure 19: Surface gloss of workpiece B, hand-polished Hanbaiyu marble. Figure 18: Surface gloss of workpiece A, robot olished Hanbaiyu marble. and glossiness, as shown in Figure 20. Workpiece B, which was manually polished and waxed, exhibited a surface roughness of 0.2843 \(\upmu\)m and a glossiness of only 63.25. The large standard deviations in both measurements indicate poor surface quality and low consistency, suggesting that this polishing method has limited effectiveness in enhancing the surface quality of Hanbaiyu marble workpieces. In contrast, workpiece A, polished by a robot using SG flexible tools, achieved a surface roughness of 0.0233 \(\upmu\)m with excellent consistency and a glossiness of 102.5, demonstrating a high level of gloss with minimal variation across different regions. Through a comparative analysis of manual polishing and the SG robotic polishing process in Figure 21, it is evident that adopting robotic processing not only effectively reduces the number of polishing steps but also significantly improves surface quality and consistency. Utilizing SG flexible tools for robotic polishing not only enhances the gloss effect of Hanbaiyu marble workpieces but also further increases their added value. In comparison, the Hanbaiyu marble workpiece A, polished with the SG flexible tool in a robotic polishing system, has a surface roughness of 0.0233 \(\upmu\)m and excellent consistency. In addition, it has a high gloss level of 102.5, with minimal differences in gloss between different regions. This indicates that robotic polishing with the SG flexible tool not only achieves a high-quality surface but also ensures good consistency, which effectively improves the gloss of the Hanbaiyu marble workpiece and further increases its added value. The SG flexible tool after robot flexible polishing shown in Figure 22 only has slight wear.