## 1 Introduction  
Optical components polished with ultra-precision play a vital role in various fields. In the aerospace field, satellite imaging systems rely on them to obtain clear images during space exploration. In biomedical equipment, improving the resolution of microscopes used in medical research and diagnosis cannot be achieved without them. With the growing demand for nanometer-level surfaces, more precisely controlled polishing processes and related technologies have emerged. Computer-controlled optical surfacing (CCOS) technology [1, 2, 3], as a deterministic polishing processing method, is widely used in the ultra-precision manufacturing of optical components. As a deterministic polishing processing method, it precisely controls the material removal process through tool motion and parameter manipulation, minimizing human error and ensuring reproducibility. Its effectiveness relies on optimizing the Tool Influence Function (TIF) since the TIF governs the material removal rate and distribution, which directly impactsthe surface quality of optical components. An optimized TIF can enhance the convergence rate of surface errors and achieve higher processing accuracy. TIF describes the material removal distribution of the polishing tool on the workpiece surface and is vital for polishing efficiency and surface quality determination. TIF is influenced not only by polishing pressure and speed but also by factors such as tool geometry and material properties. The characteristics of the TIF have a profound influence on the entire mirror processing. Among them, the shape of the TIF is an important factor in determining the mirror processing efficiency and processing effect, and it is essential to carry out research on it. The self-rotating small tool polishing device is a commonly used polishing device in CCOS. This device has the advantages of a simple structure, stable operation, and the ability to carry an asphalt polishing layer. It is an indispensable and important device in the polishing process. However, since its TIF is annular and deviates from the Gaussian shape, its convergence ability for intermediate-frequency and high-frequency surface form errors is relatively weak [4; 5; 6]. Therefore, researchers have proposed a planetary motion device based on the self-rotating small tool polishing device, by adding a revolving motion axis, the polishing disk is made to perform combined revolution and rotation motions. Consequently, the polishing speed distribution is changed, causing the shape of the TIF to vary and approach the Gaussian shape [7; 8; 9]. However, since the planetary motion polishing device requires the use of two motors, it has such defects as unsteady transmission, complex structure, and large moment of inertia. Moreover, during the working process, this device needs to move quickly on the optical components being processed. The instantaneous speed distribution of the planetary motion is different from that of the theoretical model at each point on the mirror surface, which will cause the actual TIF to deviate from the theoretical TIF and affect the polishing effect [10]. Therefore, the surface form accuracy that can be achieved by the small tool polishing technology is not as good as that of the magnetorheological polishing technology [11; 12; 13] and the ion beam polishing technology [14; 15]. The TIFs of these two technologies are closer to the Gaussian shape. Different polishing methods have their own characteristics. Traditional single-rotation tools are simple and inexpensive, but they face challenges in achieving high-precision surfaces. Planetary devices can improve the polishing precision to some extent, yet they have a complex structure and will leave intermediate-frequency errors. Magnetorheological polishing can achieve excellent surface finishes, but it requires special equipment and consumables. Ion beam polishing can provide extremely high precision, while it has a slow processing speed and high cost.