## 1 Introduction  
The support system of the primary mirror is a key system in ground-based large-aperture telescopes, as it determines the surface shape and pose accuracy of the primary mirror and plays a decisive role in the imaging quality of the telescope. The primary mirror support system must resist the influence of gravity, thermal load, and wind load, which are often dynamic. Therefore, active optics has been proposed and applied. Since its first application to the New Technology Telescope (NTT) at the European Southern Observatory, active optics technology has become a key technology for large-aperture telescopes. Active optics refers to the active correction of the surface shape of the primary mirror, which can significantly reduce the surface shape error caused by various loads and improve the imaging quality and observation performance of the telescope [1; 2; 3; 4]. It can achieve the closed-loop control of the surface shape of the primary mirror, providing feedback through the monitoring of the primary mirror shape and adjusting and correcting the primary mirror shape through the use of actuators [5; 6]. The active support system based on active optics technology has two subsystems, the surface control system and the positioning system. The surface control system usesproper force actuators to output axial force to the primary mirror to correct the surface shape, and the positioning system is used to constrain the rigid displacement of the primary mirror relative to the mirror cell. Active optics technology is currently applied to more than ten telescopes, which can be divided into three categories according to the positioning system scheme. The first type uses fixed positioning points to achieve primary mirror positioning and includes the NTT [7], VLT Survey Telescope (VST) [8], Subaru Telescope [9], and Southern Astrophysical Research Telescope (SOAR) [10]. The second type uses virtual positioning points and includes the Advanced Electro-Optical System 3.6 m telescope (AEOS) [11], Very Large Telescope (VLT) [12], and Gemini Telescope [13]. The third type uses a six-hardpoint positioning mechanism for primary mirror positioning, including the MMT Telescope [14], Large Binocular Telescope (LBT) [15,16], and the unfinished Large Synoptic Survey Telescope (LSST) [17,18]. Although a rough classification is possible, the structure of the active support system and primary mirror are not standardized, and each support system is customized for the specific telescope and primary mirror [19]. Therefore, in order to ensure the surface accuracy of a 4m SiC lightweight primary mirror, this article designs an active support system for the primary mirror. Considering that primary mirror positioning based on hardpoints can maintain and adjust the pose of the primary mirror, the designed system uses hardpoints for primary mirror positioning. The optical design of the telescope requires the active support system for the 4m SiC primary mirror to ensure a root mean square (rms) of the primary mirror shape error of less than \(\lambda/30\) (\(\lambda=632.8\) nm). The designed system uses pneumatic actuators for conducting surface corrections and to also provide force to support the gravity of the primary mirror. Some studies have shown that there is mutual interference between the surface control system and the positioning system. For example, [PERSON] et al. [20] studied the interference of fixed points in surface shape correction and proposed using compensation planes to solve it and achieved a good surface shape. However, there is currently no research to solve the problem of reduced surface accuracy caused by force on hardpoints. Therefore, a method is proposed to compensate for the force on the hardpoints, which solves the problem of protrusions and dents on the primary mirror surface caused by the force on the hardpoints. In order to achieve closed-loop control of the primary mirror surface shape, the required correction force must be calculated based on the primary mirror shape error.