## 5 Conclusions  
This study analyzed the robot error distribution in configuration space, constructed a robot error prediction model, and achieved a method for joint angle compensation. After comprehensive experimental validations, the following conclusions were drawn:  
(1) A more stable and significant spatial similarity of robot errors is shown in configuration space compared to Cartesian space, especially under heavy loads and various postures.  
(2) The spatial similarity of robot errors in configuration space shows marked anisotropy across different joint dimensions, and the distribution characteristics of the error components along the x-, y-, and z-axis also differ.  
(3) By weighting the sampling data in configuration space using a set of joint weights, a spatial-interpolation-based unbiased estimation method with joint weights optimization can effectively address the interference caused by the anisotropy of robot error spatial similarity and accurately predict the robot errors. The optimal joint weights are selected using the particle filter method for the prediction of x, y, and z error components individually.  
(4) A robot error measurement experimental system is established using a laser tracker and a six-joint industrial robot with a 120 kg end-effector. The SMR is mounted at the TCP of the end-effector for position measurement. In the error measurement experiment, 80 sampling points with diverse poses are selected within the task workspace to measure the real robot errors, thereby evaluating the accuracy of the robot error prediction model. The proposed error prediction method achieves a maximum error estimation accuracy of 0.172 mm across all sampling points, representing a 52.1% and 81.7% improvement in prediction accuracy compared to the MLP model replicated in Cartesian space and the spatial grid IDW method, respectively.  
(5) In the robot position compensation experiment, the proposed method is used to adjust the joint angles of the sampling points, and the real robot errors after compensation are measured. Across all 80 sampling points, the robot error is reduced from 4.96 mm to 0.28 mm, meeting the stringent requirement for positioning accuracy in the assembly hole machining process of aerospace components. By considering the common characteristics of robot error distribution, the proposed method is applicable for enhancing the positioning accuracy of six-axis industrial robots, especially in challenging scenarios involving heavy loads and varying orientations of the end-effector.  
Figure 14: Experimental results of robot positioning accuracy compensation.  
However, the proposed method does not consider the distribution patterns or evaluation methods for the orientation errors of the robot end-effector, which will become an important direction for our future research.  
Conceptualization, F.M. and J.W.; data curation, Q.F.; formal analysis, F.M. and Q.F.; funding acquisition, J.Y.; investigation, Q.F.; methodology, F.M. and Q.F.; project administration, Z.D.; resources, R.K. and D.G.; software, F.M.; supervision, R.K.; validation, Z.D., R.K. and J.Y.; visualization, J.W.; writing--original draft, F.M.; writing--review and editing, D.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Fund of China, grant number 52325506. This research was funded by the National Science and Technology Major Project, grant number 2024 ZD01 NL00105.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.