## 1 Introduction
In some studies, the error weight coefficients calculated using relevant mathematical models were used as sensitivity indicators. These methods are vital for compensating errors in specific machine tools. However, the weighting coefficients vary depending on the selected model. Currently, few studies have assessed the effect of high weights on the accuracy of measuring machines through geometric error-weighting analysis to guide machine design and manufacturing. In ultra-precision measuring machines, controlling Abbe errors is crucial for achieving high-precision measurement. In 2015, [PERSON] detailed the characteristics of the Abbe error of a parallel coordinate measuring machine 3-PUU (Three-Prismatic-Universal-Universal) model during motion and established an Abbe error transfer model using the full differential theory [16]. In 2018, [PERSON] et al. [17] analyzed the causes of Abbe offset, reduced the length of the Abbe arm to zero by adjusting the measurement equipment and the measured object, and proposed a method for compensating for these errors when the measurement system is limited by the conditions of the object. In 2020, [PERSON] et al. [18] developed a spatial error model for three-axis machine tools focusing on correcting the Abbe error during the measurement of single geometric errors based on [PERSON]'s and [PERSON]'s principles to improve the accuracy of the transfer matrix error model. In 2022, [PERSON] et al. [19] established a machine tool volume error model including Abbe error, based on the traditional machine tool volume error model, by analyzing its mechanism to form the tool volume, based on 21 items of geometric error measurement data and the Abbe arm and angle function relationship. In 2024, [PERSON] et al. [20] developed a high-precision X-Y positioning platform based on [PERSON]'s principle by using a self-developed real-time embedded six-degree-of-freedom feedback system. The positioning error in the X- and Y-directions is \(\pm 10\) nm. In recent years, many high-precision coordinate measuring machines (CMMs) have been designed and manufactured based on [PERSON] principles, such as the Zeiss F25 [21],Nanometer Accuracy Non-contact Measurement of Free-form Optical Surface of Dutch United Instruments [22], Panasonic's Ultrahigh Accurate 3-D Profilometer [23], and the NPL (National Metrology Institute) compact CMM [24]. [PERSON] [25], [PERSON] [26], and [PERSON] [27] have also studied CMMs. These high-precision measuring machines that satisfy the Abbe principle only need to detect and compensate for positioning errors and straightness motion errors to achieve high-precision measurements. However, these machines require high-precision measuring frames, high-precision laser interferometers (for positional feedback), and high-precision displacement sensors (probes), among other requirements, thereby leading to high cost, low space utilization, and small measurement stroke caused by high-precision measurement mechanism. Currently, most research works mainly focus on the compensation of the error of coordinate measuring machines and Computerized Numerical Control (CNC) machining machines. However, high-precision large-diameter measuring machines based on cylindrical coordinate systems remain scarce. This highlights the need for weighting analysis of the volumetric error of the measuring machine error to reduce compensation costs and guide the design of these machines. This study focuses on volumetric error modeling and error weight analysis of a column CMM with zero Abbe error. The findings were applied to the accurate design and manufacturing of a column CMM with zero Abbe error.