Robust Self-Calibration of Subreflector Actuators Under Noise and Limited Workspace Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authos

Based on our review, we found that the novelty and contribution of the proposed method are not clear. Most of the content is already well-known and exists in the literature. The authors should highlight the main advantages of the proposed calibration method compared with the existing methods. Details about the review comments, please refer to the attachment.

Best Regards

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

Thank you very much for your time and thoughtful review of our manuscript. Your valuable comments have greatly contributed to improving the quality and clarity of our work. We have carefully addressed each of your suggestions and incorporated the necessary revisions accordingly.

Please find our detailed point-by-point responses and the corresponding revisions in the attached document.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the integration of Huber loss and L2 regularization into the LM algorithm is a well-motivated approach to address outliers and overfitting in kinematic calibration. The focus on subreflector actuators in large radio telescopes addresses a clear gap in high-precision systems where environmental disturbances and limited workspace constraints are critical yet underexplored challenges. The combination of simulated and experimental evaluations under varying noise levels and workspace constraints strengthens the credibility of the method. The reported 90% error reduction over traditional LM is compelling. However, some improvements should still be made.

1. The choice of 42 kinematic error parameters requires explicit justification. Are these parameters derived from the telescope’s specific degrees of freedom, or is this number system-agnostic?
2. Please provide pseudocode or a flowchart for the enhanced LM algorithm.
3. Please specify whether the simulated noise levels (e.g., Gaussian vs. non-Gaussian) and workspace constraints reflect real-world telescope operating conditions. Justify thresholds for "reduced excitation" scenarios.
4. Potential biases in sensor data (e.g., drift) and their mitigation should be addressed for ground truth measurements.
5. Due to computational complexity relative to traditional LM, please discuss potential scenarios where the method might fail (e.g., extreme non-linearities or correlated sensor faults).
6. It might be better to emphasize the method’s applicability beyond radio telescopes (e.g., robotic manipulators, aerospace systems) to widen reader interest.

Author Response

Dear Reviewer,

    We sincerely appreciate your careful evaluation of our manuscript and the constructive feedback you provided. Your insights have been extremely helpful in refining the quality of this work. We have thoroughly considered each of your comments and made corresponding revisions to the manuscript.

Comment 1: The choice of 42 kinematic error parameters requires explicit justification. Are these parameters derived from the telescope’s specific degrees of freedom, or is this number system-agnostic?
Response 1: Thank you for the insightful question. We have clarified in the revised manuscript that the 42 error parameters originate from the specific 6-DOF Stewart platform configuration used in the subreflector actuator. Each of the six legs contributes seven independent parameters: three for base joint errors, three for platform joint errors, and one actuator length offset, totaling 6 x 7 = 42 parameters. A new explanation has been added after Equation (13).

Comment 2: Please provide pseudocode or a flowchart for the enhanced LM algorithm.
Response 2: We appreciate this suggestion. Algorithm 1 has been added in Section 3 to describe the proposed method clearly. The pseudocode outlines each step of the robust LM procedure including initialization, residual computation, Huber weighting, LM update, and convergence criteria. This improves readability and reproducibility.

Comment 3: Please specify whether the simulated noise levels (e.g., Gaussian vs. non-Gaussian) and workspace constraints reflect real-world telescope operating conditions. Justify thresholds for "reduced excitation" scenarios.
Response 3: We have clarified that Gaussian white noise was assumed in simulations, based on typical sensor behavior. Workspace constraints mimic the physical limits of the subreflector actuators. "Reduced excitation" refers to calibration with 50% of the full actuator range to simulate real-world measurement constraints. This has been detailed in Section 4.1 and Section 5.

Comment 4: Potential biases in sensor data (e.g., drift) and their mitigation should be addressed for ground truth measurements.
Response 4: Thank you for this important comment. We have expanded the discussion in the revised manuscript to address potential sensor biases and drift during the acquisition of ground-truth pose data.

In the experimental setup, a structured-light 6-DOF measurement system was used to obtain reference poses. To minimize systematic errors and long-term drift, the system was carefully calibrated before data collection, and all measurements were conducted under stable indoor conditions within a short time window. Nonetheless, we acknowledge that real-world sensors may still be affected by non-ideal factors such as low-frequency noise, surface reflection errors, or slight mounting shifts. These effects are difficult to model precisely and may introduce small biases into the data. While the proposed robust LM algorithm effectively suppresses transient outliers, it does not fully eliminate consistent bias.

In addition, we have clarified in Section 5 that the higher errors observed in experimental results (compared to simulation) are partly attributed to these practical measurement imperfections, which are not fully captured by the idealized noise models used in simulation.

Comment 5: Due to computational complexity relative to traditional LM, please discuss potential scenarios where the method might fail (e.g., extreme non-linearities or correlated sensor faults).
Response 5: Thank you for highlighting this important aspect. We have clarified in the revised manuscript (Section 5, Discussion) the potential limitations and failure scenarios of the proposed robust LM method.

Specifically, while the robust LM approach significantly improves robustness and convergence stability compared to the conventional LM method, it may still encounter challenges under certain extreme conditions. For instance, when facing highly nonlinear error landscapes or significantly inaccurate initial guesses, the algorithm could experience slower convergence or converge to suboptimal local minima. Moreover, if sensor measurements contain strongly correlated or systematic errors (such as uniform sensor drift affecting all measurements similarly), the robust estimation using Huber weighting might fail to identify these errors as outliers, potentially resulting in biased parameter estimations.

Comment 6: It might be better to emphasize the method’s applicability beyond radio telescopes (e.g., robotic manipulators, aerospace systems) to widen reader interest.
Response 6: We appreciate this suggestion. We have modified the introduction and conclusion to emphasize that the proposed robust calibration approach is general and can be applied to other systems beyond the specific radio telescope use-case. In particular, we now mention that any multi-DOF platform or robotic manipulator requiring precise calibration (such as parallel robots in manufacturing or alignment systems in aerospace) could benefit from this method.

Thank you again for your valuable contribution to the review process. 

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents a robust self-calibration method for the kinematic calibration of subreflector actuators in radio telescopes, based on an enhanced Levenberg-Marquardt (LM) algorithm. The proposed approach integrates Huber loss to suppress outlier influence and L2 regularization to enhance numerical stability, addressing calibration accuracy under noise and limited workspace conditions. Simulation and experimental results demonstrate that the method achieves over 90% reduction in both position and orientation errors compared to traditional LM algorithms, while maintaining stable convergence under low excitation and high-noise environments. The study highlights the method’s practical potential for complex parallel mechanisms, such as subreflector actuators. The method’s robustness to measurement noise, outliers, and workspace constraints aligns with challenges faced by radio telescopes in harsh environments, making it highly applicable to real-world scenarios.

The manuscript should be improved to meet the publication standard. 

1. While emphasizing accuracy and robustness, the computational overhead of the enhanced LM algorithm (e.g., iterative weight updates, regularization tuning) is not quantified, which could impact real-time applications.
2.The selection of Huber threshold (δ) and regularization coefficient (λ) is briefly mentioned but lacks a systematic analysis of their sensitivity or optimization strategies.
3.While traditional LM is used as a baseline, comparisons with other hybrid calibration frameworks are absent, limiting insights into relative advantages.

Author Response

Dear Reviewer,

    We sincerely thank you for your thorough review and constructive comments on our manuscript. Your feedback has been instrumental in improving the clarity and rigor of our work. Below, we provide detailed point-by-point responses to each of your concerns. Corresponding revisions have been made in the manuscript to address them appropriately.

Comment 1:  While emphasizing accuracy and robustness, the computational overhead of the enhanced LM algorithm (e.g., iterative weight updates, regularization tuning) is not quantified, which could impact real-time applications.

Response 1: We thank the reviewer for this insightful comment. Indeed, our proposed robust LM algorithm introduces additional computational steps, such as iterative Huber weight updates and regularization terms. However, these operations are lightweight compared to the primary matrix operations of the LM solver. In practice, we observed that the robust LM converges in a similar number of iterations as the traditional LM across all simulation and experimental scenarios tested. Consequently, the overall computational time remains largely unchanged, and we did not encounter any real-time performance issues.

To address this point more explicitly, we have added a clarifying statement in Section 4, stating:
"The additional weight updates and regularization terms introduce negligible computational overhead, and the robust algorithm’s convergence speed was similar to that of conventional LM in our tests, preserving near real-time performance."
We sincerely thank the reviewer for prompting this clarification.

Comment 2: The selection of Huber threshold (δ) and regularization coefficient (λ) is briefly mentioned but lacks a systematic analysis of their sensitivity or optimization strategies.

Response 2: We appreciate the reviewer’s comment on the importance of parameter selection. In the revised manuscript (Section 3), we have expanded the explanation regarding the choice of the Huber loss threshold δ and the L₂ regularization coefficient . Specifically, δ was set to approximately 1.5 times the standard deviation of the expected measurement noise, aligning with conventional robust estimation practice. Meanwhile, λ was fixed at a small value (10⁻⁵) to enhance numerical stability without over-penalizing parameter updates.

We also suggest practical tuning strategies in broader applications: for instance, δ  can be estimated through cross-validation or preliminary noise characterization, and λ may be dynamically adjusted following LM’s built-in damping scheme. Although fine-tuning may further optimize performance, our results show that the method is robust to moderate variations in these parameters, and no significant degradation was observed within the tested ranges. We thank the reviewer for encouraging us to elaborate on this point.

Comment 3: While traditional LM is used as a baseline, comparisons with other hybrid calibration frameworks are absent, limiting insights into relative advantages.

Response 3: We appreciate the reviewer’s suggestion to compare our approach with other hybrid calibration frameworks. In this work, we chose to focus on the traditional LM algorithm as a baseline because it remains a widely adopted and well-understood method in practical calibration tasks, including the engineering context of our current project.The presented robust LM variant builds directly on this baseline to achieve improved stability and accuracy under real-world noise and excitation constraints. Future work will extend the comparison to other robust and hybrid methods, as also noted in the revised Discussion section.

    We are grateful for your thoughtful and constructive feedback, which has helped us significantly improve the quality and presentation of our work. We hope the revised manuscript meets your expectations and addresses your concerns.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors

Thanks for your significant effort in improving the quality of the presented paper.

After the detailed review, we think that the current form now meets the criterion of the acceptance.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have improved the manuscript and answered all the questions. The reviewer has no further comments of this manuscript.