## 1 Introduction  
A heavy-load longitudinal and transversal swing table (HLTST) is a crucial mechanism for carrying large components and adjusting their position, especially compact and small-sized tables, which are commonly used in aerospace and military applications, automated production, marine operations, intelligent manufacturing, and other important areas [1, 2], such as spacecraft assembly [3, 4, 5], precision CNC machining [6], wave compensation [7], large-scale 3D printing, and other heavy-duty precision control application scenarios. In important military security links such as aircraft bomb loading and missile assembly, the precision and efficiency of the table's attitude adjustment has an important impact on the aircraft re-deployment preparation time [8] and the quality of strategic and tactical equipment production [9]. Consequently, enhancing the positioning accuracy and reducing the adjustment time are key strategies for improving the operational performance of heavy-duty precision positioning equipment. Defects in machining and the assembly of parts, coupled with the presence of transmission clearance--an original positioning deviation--make it challenging for various axes of motion in a table to achieve an ideal geometric state. These cumulative errors are collectively referred to as the system's geometric errors [10; 11]. In the process of multi-axis combined motion of the table, geometric errors will generate coupling relations and crosstalk with each other through the motion chain while gradually accumulating to the end of the system [12; 13; 14], which will have a serious impact on the positioning accuracy and efficiency of the table, so how to accurately model and compensate for the geometric errors is an urgent problem that needs to be solved. In addressing the aforementioned challenges, numerous studies have been undertaken by researchers. [PERSON] et al. [15] took the six-axis motion platform in a laser welding system as the research object. Based on the stream of variation theory, they established an error transfer model. The experimental results show that this method can intuitively reflect the process of error propagation and accumulation. It can effectively improve the accuracy and alignment efficiency of the optical device and help understand the trend of power loss in the laser welding process. [PERSON] et al. [16] established a geometric and kinematic error model for the six-axis motorized fiber alignment stage based on the multi-body system error theory and developed an error compensation strategy according to the error sensitivity analysis, which significantly improved the fiber alignment accuracy and efficiency of the automatic alignment system for optical waveguide devices. [PERSON] et al. [17] employed crew theory and the exponential product formula to create a geometric error model for the rotational axes of a five-axis motion platform. The experimental results showed that this method reduced the platform's pose error by 28.1%. [PERSON] et al. [18] established a coupling error model of the contact probe of a three-dimensional screw-thread-measuring machine based on mechanical and geometrical analysis methods. The experimental results showed that when the measuring device moved by 10 \(\upmu\)m, the probe's displacement deviation increased by 2 \(\upmu\)m, and the calculated deflection angle was consistent with the theoretical value. This model offers a foundational framework for probe error decoupling and enhances the accuracy of the probe-guiding mechanism. [PERSON] et al. [19] addressed the significant issue of geometric error coupling in five-axis machine tools and proposed an Optimal Proportional Compensation Method considering multiple-direction coupling effects, which increased the compensation accuracy by 58.9% and reduced the number of compensation terms by 44.4%.