Servo State-Based Polynomial Interpolation Model Predictive Control for Enhanced Contouring Control

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript discusses an important topic. The overview of existing methods for improving machining accuracy under limited dynamic response of multi-axis machine tools, presented in the Introduction, is mostly correct. The explanation of the proposed method in Section 2 (Contour Error Estimate Based on Polynomial Interpolation), including the problem shown in Figure 1, is clear. Section 3 (State-Space Model Predictive Control Method) also appears to be well prepared. However, the manuscript has several weaknesses. In its current form, it cannot be accepted for publication. After the authors address the following issues, the manuscript may be reconsidered for publication.

In the Introduction (lines 38–40), the authors correctly write:“Theoretically, the contour error will be reduced to zero when tracking error reaches zero ... However, servo response mismatch across axes and other practical constraints prevent complete elimination of tracking errors.” However, in the rest of the paper, the authors focus only on contour error modeling and contour error control, which means they focus only on compensating existing errors. The causes of these errors [Gao2023] and other possible ways to reduce them in CNC systems are not discussed in enough detail. This weakens the justification for the proposed method.

The authors should clearly address the following points:

1. Could a similar contour error reduction be achieved by shortening the control cycle?
   The authors mention in lines 86–88 that a basic limitation of feedback-based compensation is its time delay. Error is calculated based on the previous control cycle, which causes a delay in response. As shown in Figure 1 and in the rest of the paper, the proposed method includes intermediate points between the main reference points P^{r} and P^{r+1}. It uses third-order polynomial trajectories and selects three interpolation points (at s=1/3,2/3,1s = 1/3, 2/3, 1).
   Would reducing the CNC system’s cycle time by a factor of three give a similar (or better) result without the proposed compensation?
   Do the authors assume that cycle time cannot be shortened for technical reasons? If so, this should be explained clearly and supported by references. Without this explanation, the reader may be misled.
2. In Section 4 (Experimental Verification), it is stated that the interpolator runs with a 2 ms cycle. This is quite long for modern CNC systems. As shown in Table 1 of [Osornio-Rios2008] for a cycle time of 2 ms and feedrate of 30 m/min, the distance between reference points is 1.0 mm. This is a large distance and can cause significant machining errors. A shorter cycle time would reduce this distance and may improve machining accuracy. The central controller (Beckhoff industrial PC, Model: C6640-0040) and EtherCAT communication interface should not limit the cycle time. EtherCAT and similar systems can work with much shorter cycles, as shown in many publications. For example, in Table 1 of [Przybyl2018], a system with sixty distributed nodes operates with a 0.25 ms cycle time. This system is more complex than the five-axis system in the manuscript. So, the system described in this paper could likely run with a shorter cycle than 2 ms. As already mentioned, a shorter cycle time (especially for the interpolator) may reduce contour error. The revised version of the manuscript should clearly include this discussion.
3. The manuscript does not provide enough technical details about the experimental setup, which are important for evaluating the work. For example, it is not clearly stated where the algorithm was implemented — fully in the central controller (which seems likely), or partly in the servo drives. There are also no details on the programming language or the way the algorithm was implemented in the controller.
4. The manuscript does not say whether the servo drives have the same 2 ms cycle time as the communication cycle, or a shorter one. This is important, because modern servo drives can operate with very fast control loops — up to several tens of kHz [Osornio-Rios2008], [Konvicny2021], [Przybyl2023]. This requires high-quality drives. As mentioned before, shorter control cycles (of the interpolator, drives, or both) can help reduce contour error. Please include this in the revised version.
5. In lines 286–287, the interpolator is mentioned only briefly:
   “The interpolator proposed in Reference [37] is adopted to improve interpolation efficiency.”
   Even though this may be justified, it would be helpful to include a few general sentences explaining how the interpolator works.
6. The description of Figure 4 is too short. Only Table A1 in Appendix A explains what the figure shows.
7. In lines 295–296, the abbreviation NCK is used without explaining what it means.
8. The data in Table 1 show relatively high contour errors — up to 0.450 mm for TTPCE. This may be caused by the long time step (2 ms), or by some other reason. Please explain why the error is so large. Is it related to a special type of machining?
   

    

Literature that should be referenced:

[Gao2023] Gao, Wei ; Ibaraki, Soichi ; Donmez, M. Alkan et al. / Machine tool calibration : Measurement, modeling, and compensation of machine tool errors. In: International Journal of Machine Tools and Manufacture. 2023; Vol. 187.

[Osornio-Rios2008] Roque Alfredo Osornio-Rios, Rene de Jesus Romero-Troncoso, Gilberto Herrera-Ruiz, Rodrigo Castañeda-Miranda, The application of reconfigurable logic to high speed CNC milling machines controllers, Control Engineering Practice, Volume 16, Issue 6, 2008, Pages 674-684, ISSN 0967-0661, https://doi.org/10.1016/j.conengprac.2007.08.004.

[Przybyl2018] Andrzej Przybył, Hard real-time communication solution for mechatronic systems, Robotics and Computer-Integrated Manufacturing, Volume 49, 2018, Pages 309-316, ISSN 0736-5845, https://doi.org/10.1016/j.rcim.2017.08.001

[Konvicny2021] Daniel Konvicny, Pavol Makys, Marek Furmanik, Effect of increasing the sampling frequency with respect to the bandwidth of the PI controller of current control loop, Transportation Research Procedia, Volume 55, 2021, Pages 935-940, ISSN 2352-1465, https://doi.org/10.1016/j.trpro.2021.07.190

[Przybyl2023] Przybył, A. FPGA-Based Optimization of Industrial Numerical Machine Tool Servo Drives. Electronics 2023, 12, 3585. https://doi.org/10.3390/electronics12173585

 

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

please see the attached file

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

• While the paper raises valid concerns about sampling-based contour evaluation, similar points have been made in prior studies, so its originality may be limited unless a novel metric or control method is clearly proposed and validated.
• While the paper effectively demonstrates improvements in contour error, contour error alone may not sufficiently capture critical dynamic issues such as phase delay, oscillations, and abrupt changes in speed or acceleration profiles. These factors can significantly affect machining quality, especially in high-speed operations. To strengthen the practical relevance and robustness of the proposed method, it is recommended that additional experiments be conducted to evaluate machining time, jerk and acceleration profiles, and—ideally—the physical quality of machined parts (e.g., surface finish, burrs, or residual marks). Such evidence would further validate the method's effectiveness beyond contour error metrics alone.
• It is advisable to include clear explanations of the variables to improve understanding of the equation.
• There appears to be a dimensional inconsistency in Equation (21), as (AX_r + BU _DF) represents the full predicted future state vector per Equation (18), whereas P_ref likely denotes only the reference position vector. Suggesting that it should instead be X_ref​ to maintain consistency between predicted and reference state trajectories.
• The disturbance fff mentioned in the statement appears to be a typographical error and should be F_d. 

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have adequately addressed all of the indicated comments. The manuscript, in its current form, can be accepted for publication.

I have only one minor remark: I kindly ask the authors to verify whether there is a (likely linguistic) inconsistency in the sentence “These persistent errors inevitably lead to contour deviations, especially in high-speed or high-precision machining tasks, thereby necessitating explicit contour error modeling and compensation.” The sentence appears to be logically imprecise, as the phrase “or high-precision” seems to contradict the overall meaning. Would it not be more appropriate to remove this phrase? Alternatively, the sentence could be rephrased as follows:“These persistent errors inevitably lead to contour deviations, which become particularly critical in high-speed and high-precision machining tasks, thereby necessitating explicit contour error modeling and compensation.”

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

see the attached file

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

- While the limitations of the current platform are understandable, I believe the manuscript could be further strengthened by incorporating the following within the present experimental capabilities:

1) Including of quantitative plots of acceleration and jerk over representative test trajectories, comparing the proposed method with the baseline would help demonstrate improvements in motion smoothness and peak load reduction, even without actual cutting.

2) Or adding a clear numerical evaluation of phase lag between commanded and actual axis trajectories, as this is directly linked to contour fidelity in high-speed operation.

Author Response

Dear Reviewer,

Thank you for your thoughtful comments. Please find our detailed responses in the attached reply document.

Best regards！

Author Response File: Author Response.pdf