## 1 Introduction  
The surface gloss of marble products with a complex shape determines the value of the product. Polishing is the most critical phase in shaping the surface gloss of marble [1], and the cost of polishing accounts for more than 40% of the total product cost. Optimizing the polishing process will help to improve the final quality of the product, increase economic value, and minimize costs, and it has a significant impact on the value of all other pre-polishing processes [2; 3; 4; 5]. In the production of marble products with complex shapes, the grinding and polishing processes determine the gloss effect of the finished surface. At present, manual and semi-automatic processes are the main methods used to process marble products with complex shapes. Manual polishing not only poses significant health risks to workers but is also associated with the potential for respiratory and musculoskeletal disorders [6]. With the continuous improvement of technology and the rapid development of the stone processing industry, traditional processing methods are facing many challenges and cannot meet future demands. Consequently, automatic polishing solutions for marble products with complex shapes are urgently needed. Polishing is the continuous grinding of the marble's surface with progressively finer abrasives. In this process, the abrasives remove scratches left by the previous steps and introduce finer scratches [3; 7; 8; 9; 10]. In the field of flat marble polishing, a mature research and industrialization system has been successfully established, and automatic grinding and polishing have been achieved. The polishing of marble with regularly complex curved surfaces has been extensively studied by scientists, and automatic grinding and polishing processes have been successfully realized. Both mainly use resin-bonded abrasive tools [11; 12]. However, compared to flat and regularly shaped marbles, marbles with complex curved surfaces, characterized by intricate double-yielding effects and variable surface curvatures, still heavily rely on manual polishing. It is inefficient and will result in inconsistent surface gloss. Until now, there has been no solution for the automatic polishing of marble with complex curved surfaces. Recently, robotic polishing has attracted the interest of researchers and companies [13]. Robotic polishing systems provide an effective and economical solution for polishing complex curved surfaces and can be applied to process differently shaped workpieces [14]. But the robotic polishing of complex curved surfaces requires a more flexible fit between the polishing tool and the marble with a complex curved surface. The central focus of flexible automatic polishing technology is to achieve efficient, high-quality, and stable polishing surfaces by polishing with compliant contact and using automation technology to control the polishing contact force to stabilize the process [4; 15; 16; 17]. This technology includes active flexible control and passive floating flexible control. In the former, the sensor is used for information feedback and control, allowing the system to actively adapt to changes in the workpiece surface. The control system is complex and more difficult to control. In contrast, the latter integrates floating force control technology with flexible polishing tools, allowing the tools to passively adapt to changes in the workpiece. In the latter, polishing position and force are controlled separately, and the overall processing control is simplified. Compared to active flexible control, passive floating flexible control is easier to implement and more cost-effective, so it is widely used in the grinding and polishing of workpieces with complex shapes. Therefore, this paper uses the passive floating flexible control method combined with robots to achieve flexible polishing of marble with complex curved surfaces.