## 1 Introduction
Research on robotic flexible polishing mainly involves controlling the polishing force, modeling and predicting the depth of material removal, and optimizing polishing trajectory space. The polishing force between the polishing tool and the surface to be machined is affected by a number of factors during polishing, including the curvature of the surface, the polishing force applied to the surface, and changes in the posture of the polishing tool [18]. The key to high-quality polishing is to ensure a consistent depth of material removal across the polished surface, which requires the precise control of the polishing force and polishing trajectory space to improve surface roughness [19; 20; 21; 22], but this is challenging for complex curved surfaces [23; 24; 25]. To improve cost-effectiveness and improve process quality, researchers have conducted extensive research into the robotic flexible polishing of complex curved surfaces, focusing on flexible polishing. For example, [PERSON] constructed a contact model between a flexible polishing disk and a surface using mechanical simulation software and established a material removal model by analyzing the contact states of different parameters [26; 27]. [PERSON] enables precise control of the polishing force by designing a pneumatic floating device based on the passive floating flexible control principle [28]. In addition, [PERSON] and co-workers conducted extensive research on the robotic polishing system, identifying trajectory planning methods for the polishing process and exploring the relationship between process parameters and workpiece surface quality. Their work provides a critical parameter basis for subsequent robotic flexible polishing efforts [29]. [PERSON], [PERSON] and co-workers developed a deviation model by adjusting joint stiffness values and fine-tuning the model. The results demonstrate that the created model effectively replicates the behavior of the robotic arm, thereby enhancing machining precision [30]. Despite these advances, there is still a lack of reports on the automatic flexible polishing of marble with complex curved surfaces. To fill this gap, this paper presents the use of sol-gel (SG) flexible tools, with sol-gel technology, for the robotic polishing of marble with complex curved surfaces. This study presents a preparation method for sol-gel (SG) flexible tools, which are used in the robotic polishing process of marble with complex curved surfaces. The robotic polishing method is based on the Preston material removal theory. Constant-pressure polishing of surfaces of different curvatures is achieved by simulating the contact force distribution between the SG flexible tools and the marble and constant-pressure material removal profiles. By optimizing the polishing trajectory space, the uniformity of the material removal depth is increased, and the consistency of the surface quality is improved. The material removal profile of surfaces of different curvatures is predicted by building a model of the relationship between the optimum polishing trajectory space and the curvature. This approach aims to use robots to polish marble with complex curved surfaces and to achieve a consistent high-quality surface.