## 1 Introduction
employed an incremental extreme learning machine (IELM) to predict the positioning error of an industrial robot and improve its positioning accuracy [19]. To improve the tracking accuracy of joint angles, [PERSON] et al. proposed a robot-tracking error prediction and compensation method based on time-convolutional networks (TCN) [20]. In [21], [PERSON] et al. used a genetic particle swarm optimization algorithm to optimize a neural network for predicting the positioning error of an industrial robot, thus improving the positioning accuracy of a KUKA robot. Among all the mathematical models mentioned above, the MDH model is the most widely used in the field of industrial robot calibration due to its features of parameter completeness, minimization, continuity, and clear physical significance. [PERSON] et al. established a geometric parameter error model for SCARA robots based on the MDH model and identified the corresponding structural parameters [22]. [PERSON] et al. derived geometric and deformation errors based on the MDH model, combined them into a complete model, and proposed a step-by-step compensation method [23]. In [24], a kinematic error model for FANUC robots was established based on the MDH model, and the Levenberg-Marquardt algorithm (LM) and differential evolution algorithm (DE) were used to identify and optimize the related parameters. [PERSON] et al. analyzed the joint motion of KUKA robots according to the MDH model and proposed a kinematic calibration method based on an improved beetle swarm optimization algorithm [25]. [PERSON] et al. presented a 3 DPP measurement device and kinematic parameter identification method based on the MDH model for the calibration of 6 degrees-of-freedom (DOF) industrial manipulators [26]. However, the majority of the aforementioned industrial robot calibration methods primarily focus on positional errors, neglecting the importance of orientation accuracy. Moreover, when the calibration of the tool coordinate frame and measurement coordinateframe is not sufficiently accurate (i.e., when there is a significant transformation error), the identification results based on the MDH error model often differ greatly from the actual structural parameters of the industrial robot [27]. Additionally, industrial robot parameter identification is susceptible to falling into local optima due to the effects of transformation errors in the tool coordinate frame and measurement coordinate frame and other random noise. To address these issues, this paper presents a more accurate error model and parameter calibration method for industrial robots based on 6-DOF position and orientation to identify the actual structural parameters. The main contributions are as follows: (1) based on the modified Denavit-Hartenberg parameters, the transformation errors of tool coordinate system and measurement coordinate frame were introduced to establish a geometric parameter error model, which uses positional and orientation accuracy as optimization objectives; (2) to overcome the problem of easily falling into local optima when identifying geometric parameters simultaneously, a geometric parameter cross-identification method based on the Levenberg-Marquardt algorithm is proposed; (3) the linear relationships between the parameters of the proposed geometric error model are investigated, and a solution for not calibrating some geometric parameters under specific conditions is provided.