A Robot Error Prediction and Compensation Method Using Joint Weights Optimization Within Configuration Space

Response to Reviewer 1 Comments

1. Summary

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files. 

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Is the research design appropriate?

Must be improved

We have added detailed descriptions and data analysis in both the model construction and experimental validation sections to ensure a balance between theoretical discussion and empirical validation.

Are the methods adequately described?

Yes

Are the results clearly presented?

Must be improved

We have reorganized the comparative experiment results, clarified the data presentation, added more details, and introduced a new Table 3 to better summarize the findings.

Are the conclusions supported by the results?

Must be improved

We have reorganized the conclusions section to better align with the experimental results, highlighted the contributions of this study, clearly outlined its limitations, and provided a forward-looking discussion on future work.

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1: There is only one aspect that could be improved, and that is to carry out more experimental tests. There seems to be an imbalance here. For this reason, I propose adding more experimental data, so the text should be reformulated with more experimental data.

Response 1: Thank you for your valuable comments, which allow us to improve our manuscript. The authors should place greater emphasis on addressing the balance between theoretical discussion and experimental validation.

The authors would like to clarify that the manuscript includes extensive experimental analysis and data presentation throughout Chapters 2, 3, and 4, supporting the foundational analysis, model construction, and experimental validation phases of this study.

In response to your feedback, the authors have placed greater emphasis on ensuring a balanced representation of theory and experiment. Specifically:

(1)    In Section 2.3, we conducted comprehensive robot error measurement experiments to analyze the distribution characteristics of robot errors in both Cartesian space and configuration space. The preparation process and conditions of these experiments are detailed in lines 203-221 and Figure 3 of the revised manuscript, with the experimental design described in lines 225-240. The resulting data and analysis are presented in Figures 4-7 and Tables 1-2.

(2)    In Section 3.1, a sampling experiment of robot errors is necessary to establish the robot error prediction model. However, the original manuscript inadequately described this experimental process. In the revised manuscript, we have included an introduction to this sampling experiment in lines 390-395:

“To achieve this objective, it is essential to first define the range of the task workspace based on the operational requirements and actual environmental constraints of the robot. According to this defined range, uniform sampling of joint angles is performed in the configuration space, following the method shown in Figure 9, and the actual robot errors at these sampled points are measured.”

Additionally, Section 4.1 provides a comprehensive description of the robot error sampling experiment and the parameters obtained for the error prediction model.

(3)    In Section 4.2, we detail the experimental design and result analysis for the error prediction model comparison experiments and the robot precision compensation experiments. We apologize that the original manuscript inadequately described the estimation accuracy comparison experiments for the robot error prediction model. To address this, the revised manuscript now includes additional comparative experiment subjects and their results, which are summarized in the newly added Table 3.

We believe that these revisions will better support the conclusions drawn from our work and strengthen the overall quality of the manuscript. Thank you again for your valuable feedback.

4. Response to Comments on the Quality of English Language

None

5. Additional clarifications

None

Response to Reviewer 2 Comments

1. Summary

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Yes

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Yes

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1: The abstract does not mention the specific problem being solved, the general problem is mentioned, but the particular problem is missing.

Response 1: Thank you for raising this point. We agree with your comment and have revised the abstract to include a more detailed description of the specific problem being addressed. The revised section (lines 12-15 of the revised manuscript) now reads as follows:

“To enhance robot positioning accuracy, one crucial approach is to analyze the distribution patterns of robot errors and leverage spatial similarity for error prediction and compensation. However, existing methods in Cartesian space struggle to achieve accurate error estimation when the robot is loaded or the end-effector orientations are varied.”

Comments 2: In point 4.2, the results of the experimentation are shown, but there is no mention of a comparison with other methods to estimate the error. I consider it useful to compare with other methods, in order to highlight the contribution of this work.

Response 2: We apologize for the confusion caused to the reviewer. In response to the comment, we would like to clarify that in Section 4.2, a comparative experiment was indeed conducted to evaluate the proposed error prediction model against other existing methods. The estimation accuracy of the proposed method (Method 1) was assessed at 80 sampling points with diverse poses within the robot task workspace and compared with the MLP error prediction model implemented in Cartesian space (Method 2) and the spatial grid IDW method (Method 3). The results of this comparison are illustrated in Figure 13.

To address this comment more explicitly and provide a clearer presentation of the comparison experimental results, we have added a detailed description in lines 585-586 of the revised manuscript, along with the newly added Table 3. Table 3 provides the maximum and average prediction errors for the three error prediction methods, offering a more intuitive comparison that supports the contribution of this study.

Comments 3: It is not clear whether the algorithm can be applied to different robots, or under what conditions it can be used on robots of different models.

Response 3: Thank you for pointing this out. We agree with this comment. Therefore, we have reorganized the conclusion section to better address this concern.

The principle behind our proposed method is grounded in both theoretical derivation and experimental validation, demonstrating that the distribution of industrial robot errors in configuration space exhibits significant and stable characteristics, regardless of the payload conditions at the end-effector or the range of pose variations. Consequently, the error prediction and compensation approach presented in this paper is designed with considerations for the common characteristics of robot error distribution, making it applicable for improving the positioning accuracy of six-axis industrial robots, particularly in scenarios involving heavy loads and varying orientations of the end-effector.

As a response to this comment, the following statement has been added to the revised manuscript at lines 631-634:

” 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.”

Comments 4: Conclusions should be made by referring to the graphs obtained in the results, highlighting the contribution and its importance.

Response 4: Thank you for your valuable suggestion. We have reorganized the conclusions section to better align with the experimental results, to highlight the contributions and importance of this study. The specific revisions are reflected in the manuscript from lines 618-634 as follows:

” (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 0.172mm 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.”

4. Response to Comments on the Quality of English Language

None

5. Additional clarifications

None

Response to Reviewer 3 Comments

1. Summary

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Can be improved

Is the research design appropriate?

Can be improved

We have added detailed descriptions and data analysis in both the model construction and experimental validation sections to ensure a balance between theoretical discussion and empirical validation.

Are the methods adequately described?

Can be improved

Are the results clearly presented?

Can be improved

We have reorganized the comparative experiment results, clarified the data presentation, added more details, and introduced a new Table 3 to better summarize the findings

Are the conclusions supported by the results?

Can be improved

We have reorganized the conclusions section to better align with the experimental results, highlighted the contributions of this study, clearly outlined its limitations, and provided a forward-looking discussion on future work.

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1: In abstract, include a brief overview of the experimental setup or data collection process.

Response 1: Thank you for raising this point. We agree with your comment and have revised the abstract to include a more detailed description of the experimental setup and data collection process. The revised section (lines 23-26 of the revised manuscript) now reads as follows:

“An experimental system is established using an industrial robot equipped with a 120 kg end-effector and a laser tracker. 80 sampling points with diverse poses are randomly selected within the task workspace to measure the robot errors before and after compensation.”

Comments 2: In abstract, add a sentence on the broader implications or potential applications of this research in the aerospace industry.

Response 2: Thank you for your suggestion. Following this comment, we have added a sentence in the abstract to introduce the effectiveness of this study and provided potential application scenarios, as shown in lines 26-28 of the revised manuscript:

“The proposed method achieves an error prediction accuracy of 0.172 mm, reducing the robot error from the original 4.96 mm to 0.28 mm, thus meeting the stringent accuracy requirements for hole machining in robotic aerospace assembly processes.”

Comments 3: In abstract, explicitly state the gap in existing research or methods that this study addresses.

Response 3: Thank you for raising this point. We agree with your comment and have revised the abstract to include a more detailed description of the specific problem being addressed. The revised section (lines 12-15 of the revised manuscript) now reads as follows:

“To enhance robot positioning accuracy, one crucial approach is to analyze the distribution patterns of robot errors and leverage spatial similarity for error prediction and compensation. However, existing methods in Cartesian space struggle to achieve accurate error estimation when the robot is loaded or the end-effector orientations are varied.”

Comments 4: Precision vs. Repeatability:

- Clarify whether the focus is on precision or repeatability. The statement "With a positional accuracy requirement of ±0.5 mm for assembly holes in aerospace component, which is significantly higher than the original accuracy of robot, error prediction and accuracy compensation method for robots are essential" may need revision. Note that articulated robots (6-axis) often have repeatability in the range of ±0.03 mm to ±0.05 mm, while SCARA robots can achieve repeatability values as fine as ±0.01 mm to ±0.02 mm. Explain why precision is necessary if repeatability of common industrial robot is sufficient for the task.

Response 4: Thank you for your valuable comments. Your suggestion to distinguish between precision and repeatability is well-taken and helps to clarify the focus of our research. In response, we would like to further elucidate that this paper emphasizes the robot's absolute positioning accuracy rather than its repeatability.

In discussing the application of industrial robots in aerospace manufacturing, it is crucial to distinguish between precision and repeatability of the robot. Precision refers to the accuracy with which a robot can reach a specified target position, quantified by the distance between the robot actual position and the target position. However, repeatability refers to the consistency of a robot in reaching the same target position, measured by the maximum deviation between the robot actual positions achieved for the same target across multiple trials. Typically, a robot's repeatability accuracy is higher than its positioning accuracy. While repeatability can reach ±0.05 mm to ±0.08 mm for industrial robots and ±0.01 mm to ±0.02 mm for SCARA robots. In contrast, positioning accuracy, which is influenced by both geometric and non-geometric factors, is generally greater than ±2 mm.

In aerospace manufacturing, particularly for hole-making tasks on aircraft, robots usually perform one-time, non-repetitive operations at multiple unique positions. Each target hole location is distinct, and the robot only visits each position once. At this point, the robot's positioning accuracy directly influences the hole positioning errors for the aircraft assembly process.

Clearly, a robot without error calibration and precision compensation has a position error that falls far short of meeting the stringent accuracy requirement of ±0.5 mm. Therefore, despite the high repeatability of the industrial robot, it remains essential to enhance the robot’s positioning accuracy for aerospace hole-making processes to meet the manufacturing requirements.

Comments 5: Terminology:

- Clarify the phrase "anisotropic distribution characteristics" and explain its implications in the context of the research.

Response 5: Thank you for your suggestion. We apologize for the unclear description of this phenomenon, which may have led to confusion for the reviewer. The anisotropic distribution of robot errors in configuration space, as observed from the analysis of experimental data (see Figure 10), reveals distinct differences in sensitivity and nugget effects across different joint dimensions. Specifically, in some joint dimensions, robot positioning errors exhibit more pronounced variations, whereas in others, the repeatability deviations due to nugget effects are relatively smaller. The anisotropy of robot error distribution can significantly impact the effectiveness of error prediction models, as it underscores the need to consider the unique properties of each joint when constructing these models. In response to this comment, we have revised the manuscript (lines 437-445) to provide a more detailed definition:

“According to the analysis of experimental results, although robot positioning errors exhibit significant and consistent spatial similarity in configuration space, this characteristic also shows markedly different nugget effects and sensitivities across various joint angle dimensions. Notably, the distribution of robot errors exhibits anisotropy across different joint dimensions in configuration space, which can directly impact the effectiveness of error prediction. Therefore, when constructing the error prediction model based on robot error spatial similarity, emphasis should be placed on enhancing the influence of joint dimensions with more significant spatial similarity while suppressing those with a pronounced nugget effect.”

Comments 6: Limitations and Future Work:

- Acknowledge the limitations of the proposed approach.

- Include a paragraph on future work in the concluding section.

Response 6: Thank you for pointing this out. We agree with this comment. The limitations of the proposed approach and a discussion on future work are indeed essential elements that should be clearly addressed in the concluding section. In response to this feedback, we have added a paragraph in the revised manuscript (lines 635-637) that recognizes the limitation of this research regarding the omission of evaluation methods for the orientation errors of the robot end-effector. Acknowledging this gap, we highlight it as an important direction for our future research. The specific modification reads as follows:

“However, the proposed method has not considered 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.”

Comments 7: Minor Corrections:

- Add a space between numbers and units throughout the paper (e.g., "120 kg" instead of "120kg").

Response 7: Thank you for pointing this out. The format has been revised according to the reviewer’s advice. “120kg” has been changed to “120 kg” on line 24. The texts on line 206, 282, 294, 313 and 623 have also been modified according to your suggestions.

Comments 8: Minor Corrections:

- Remove the definitive article "the" at the beginning of captions (e.g., line 252).

Response 8: Thank you for pointing this out. The format has been revised according to the reviewer’s advice. I've deleted the definite article “the” at the beginning of the title on line 252. The text in line 269, 281 and 293 has also been modified according to your suggestions.

4. Response to Comments on the Quality of English Language

None

5. Additional clarifications

None