## 1 Introduction  
Industrial robots are multi-mechanical-arm or multi-degree-of-freedom robots designed for industrial applications that can replace humans in performing simple and repetitive tasks in harsh environments [1, 2, 3]. Over time, the repeatability of industrial robots can be affected by changes in the precision of the joint gear system and the resolution of the encoder. The accuracy of robots can also be influenced by geometric factors such as deviations in link parameters and coordinate frame definitions, as well as non-geometric factors like temperature and deformations. These factors can lead to deviations in the pose of the end effector of industrial robots, which can impact the quality of associated products. On one hand, industrial robots generally have high repeatability, often reaching the level of 0.1 mm, but their absolute positioning accuracy is usually in the millimeter range. On the other hand, the accuracy of industrial robots gradually deteriorates with increasing operating time, and the rate of deterioration determines their lifecycle. As industrial robots find broader applications, meeting the repeatability requirement alone is no longer sufficient, and the low absolute positioning accuracy becomes a bottleneck for further development. Therefore, improving absolute positioning accuracy has become an urgent problem for industrial robot manufacturers. The calibration of Industrial robots [4; 5] typically involves four steps: kinematic modeling, pose measurement, parameter identification, and error compensation. Kinematic modeling serves as the mathematical foundation in the field of industrial robot calibration. In the 1950s, [PERSON] and [PERSON] proposed the Denavit-Hartenberg model (DH) [6], which expresses the spatial coordinate transformation between adjacent joints by defining four parameters: link length \(a_{i}\), link offset \(d_{i}\), link twist \(\alpha_{i}\), and joint angle \(\theta_{i}\). The DH model has clear physical significance, is convenient to use, and satisfies the principle of minimization [7]. However, a wealth of research studies have demonstrated that the DH model fails to meet the continuity requirement when the axes of adjacent joints are parallel or nearly parallel. To address this singularity issue while maintaining the model's parameter minimization principle, a modified Denavit-Hartenberg model (MDH) [8; 9] was proposed by adding a parameter \(\beta_{i}\) for rotation around the y-axis in parallel or nearly parallel singular joint coordinate frames. In [10], [PERSON] et al. redefined the rules for establishing the link coordinate frame and proposed the S model. In the S model, each link is described by six parameters, including three translation parameters and three rotation parameters, but it does not satisfy the principle of minimization and contains redundant parameters [11]. [PERSON] et al. proposed the complete and parametrically continuous model (CPC) based on the idea that a kinematic model for robot calibration needs to satisfy both \"completeness\" and \"parameter continuity\" [12]. In [13; 14], the kinematic model of industrial robots was described as a product of a series of exponential matrices, known as the product of the exponentials (POE) model, which does not have parameters that mutate in terms of phenomena. To address the issue of model discontinuity commonly encountered in the traditional DH model, [PERSON] et al. proposed a local product of the exponential model [15]. To reduce the difficulty of kinematic modeling, many researchers have considered using neural networks to depict the relationship between the end effector error and the joint angles of industrial robots [16; 17]. [PERSON] et al. proposed a two-step calibration method for a master-slave collaborative robot system based on artificial neural networks (ANNs), using joint angles and output pose errors as the training data [18]. [PERSON] et al.