Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

As the papers gives the insight into a well-designed study on the fuzzy identification and control of a four-bar parallel robot for upper-limb rehabilitation, it connects solid foundation with high-level modeling, simulation, and experimental validation. 

Single actuators are considered only in the paper, thus the physical experiments should include full robot system tests with the fuzzy controller operating the entire parallel robot in a rehabilitation scenario, to fit it to the reality. Eventually, human subject tests would also be valuable. Secondly, every patient is an individual, thus the fuzzy logic controller should be shown to adapt to variability among patients, and also selected smooth trajectories and motion ranges should be connected to typical rehabilitation protocols for elbow and forearm. What are clinical benchmarks or criteria in this context? Is it possible, furthermore, any benchmarking across classical PID (well-tuned), sliding mode controllers or standard Lyapunov-based adaptive controlers in terms of tracking accuracy, robustness, and computational complexity? Finally, the limits of the current work should be more stressed - modeling asumptions made, absence of patient trials, lack of integration of a haptic feedback approach, more complex trajectories, etc. 

Last but not least, some additonal performance metrices could also be introduces, such as control-based settling times, overshoots, RMSEs, energy-related indices, as well the discussion how do external disturbances act on the control scheme? such as unexpected forces or vibrations on the manipulator.

Try to highlight the relative advantages, drawbacks, and suitability of the proposed controller in diverse scenarios, in comparison to a family of other controllers, and for different regimes. 

The paper uses formal academic English. Please remove repetition in lines 176-181, reduce the redundancy in giving description of ANFIS methodology, and break longer sentences into two or more. A well known fuzzy control concept is used in the paper, along with TS models, and GWO approach. This is a standard part. The previous papers of the authors are cited, but there is no risk of plagiarism, as the new modelling approach, and fuzzy identification using anfis is presented. The paper incluydes new exerimental esults on phusical actuators, with no risk to use the prior works extensively. 

The paper fits well into recent trends to use fuzzy and hybrid fuzzy control for in reduction of uncertainties and variability in rehabilitation systems. Its physical implementation experiments improve the contribution. References to recent related research and methods are properly cited

The main contributions is th e development of fuzzy dynamic models of actuators using ANFIS for nonlinear identification, and design of PDC fuzzy controllers tuned via GWO metaheuristic for precise trajectory tracking. A demonstration of  robustness, accuracy, and feasibility, particularly through experimental validation is given. Simulation present show good tracking of smooth elliptical trajectories with acceptable error margins and controlled oscillations. The physical implementation on a microcontroller with real actuators provides clear evidence of control robustness and practical implementation feasibility. Quantitative evaluation is included. 

Consider introduction of deeper discussion onf possible limitations and generalization of the results. 

Author Response

Reviewer 1
Dear Reviewer,

We sincerely appreciate your thoughtful comments and the time you dedicated to evaluating our manuscript. Below, we respond point by point to your observations.

We have revised the Abstract section to precisely delimit the scope of the work presented and to outline future phases. In particular, our intention in subsequent stages is to evaluate the actuators coupled to the rehabilitation robotic system in a controlled clinical setting, conducting performance studies under different external loads and patient anthropometric variations, and incorporating clinical criteria and metrics.

In this contribution, we restrict ourselves to the fuzzy identification of the actuators decoupled from the robotic rehabilitation system and to their closed-loop evaluation using a PDC controller, which operates jointly with the identified model and across the entire nonlinear universe of discourse.

Finally, we acknowledge that a comparison with other control techniques could be performed to assess the efficiency of the proposed controller; however, such a comparison falls outside the scope of the present study and is part of our planned future work.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript focuses on an upper-limb rehabilitation robot with a parallelogram structure, proposing a trajectory tracking scheme that integrates fuzzy identification, Parallel Distributed Compensation (PDC) fuzzy control, and the Grey Wolf Optimizer (GWO) algorithm. It establishes dynamic models for the actuators and the mechanism, completes identification using the Adaptive Neuro-Fuzzy Inference System (ANFIS), and optimizes pole placement via GWO. The feasibility of the proposed method is verified through simulations and single-actuator experiments. However, some confusing issues requires further clarification and extended discussion to enhance the quality of the paper.

Major Comments：

1. In Figure 3, the authors directly use symbols such as points A, B, C, D, and link lengths L1–L5. While some symbols are labeled in the figure, their geometric or physical meanings lack systematic explanation in the main text. The authors shouldsupplement the definition of the geometric model, providing clear descriptions of the physical meanings of relevant points and links.
2. In Section 3.5, although the time-varying nature of friction, resistance, and inertia is mentioned at the end of the section, the key simplifying assumptions adopted in the modeling process are not explicitly stated. It is suggested that the authors clearly elaborate on these assumptions in the modeling section and analyze their potential impacts on control performance in the results discussion.
3. In Section 3.8, in Lines 336–339, the authors mention that different poles are configured for the two actuators due to differences in joint motion ranges and amplitudes. However, no quantitative data or model analysis is provided to support this design choice. The authors shouldsupplement relevant comparative data or theoretical analysis to enhance the adequacy and rigor of the argument.
4. In the comparative experiments of Table 6, eightfuzzy rules are fixed regardless of whether the input dimension is 2 or 3. The manuscript fails to explain the relationship between the number of rules and input dimensions, nor does it clarify why other input dimensions are not considered. It is suggested that the authors supplement the basis for determining the number of rules and analyze the potential impacts of different input dimensions on model complexity and performance in the discussion section, thereby improving the completeness and persuasiveness of the results.
5. In Section 5, the manuscript claims to have achieved trajectory tracking for the four-bar rehabilitation robot and designed therapeutic movements simulating elbow joint motion. However, based on the content in Section 4, the verification is mainly based on simulations, and physical experiments are limited to the single-actuator level, lacking verification at the end-effector or full-machine level. So, the authors are advisedto express the research scope more cautiously in the conclusion and supplement full-machine experiments in future work to avoid inconsistencies between the research scope and verification results.

Minor Comments：

1. Section 2 reviews numerous applications of fuzzy control on different platforms but directly shifts to the parallel robot theoretical framework from References [15–17] at the end, taking it as the research foundation. It is suggested that the authors further clarify the reasons for selecting References 15–17 as the research basis and explain the connection between this framework and the subsequent fuzzy identification and control methods to enhance logical coherence.
2. There is an inconsistency between the caption of Figure 2 and the main text. The main text describes ‘Figure 2b presents the distances along the y-axis,’ while the figure title is labeled ‘Diagram of distance along the Z-axis.’. It is suggested that the authors verify and uniformly correct this inconsistency.
3. Figure 2(a) only shows the directions of the x and y axes, with no indication of the z-axis direction. Since the D-H method requires the z-axis to be strictly aligned with the joint axis, it is suggested to supplement the z-axis label in the figure to ensure consistency between the diagram and the derivation.
4. In Table 4, the authors only use color highlighting for the optimal values of RMS1_SIMU and RMS2_SIMU, without applying the same treatment to the corresponding RMS1_GWO and RMS2_GWO. It is recommended that the authors maintain consistency in the highlighting method or clearly explain the reason for only emphasizing the SIMU results in the table note to avoid ambiguity.
5. In the simulation experiment of the dual-actuator system, the authors only conducted 5 independent runs. Given that GWO is a stochastic optimization algorithm, this sample size is insufficient to demonstrate the algorithm’s stability and statistical significance. So, the authors shouldincrease the number of experiments, report the mean, standard deviation, or confidence interval, and provide convergence curves to improve the reliability of the results.

Author Response

Reviewer 2

We sincerely appreciate your thoughtful comments and the time you dedicated to evaluating our manuscript. Below, we respond point by point to your observations

1.- In Figure 3, the authors directly use symbols such as points A, B, C, D and link lengths L1–L5. Although some symbols are labeled in the figure, their geometric or physical meanings lack a systematic explanation in the main text. It is suggested to supplement the definition of the geometric model, providing clear descriptions of the physical meanings of the relevant points and links.
= Thank you very much—your observation is spot on; the symbols and link lengths were ambiguous and not sufficiently clear beyond the graphic. In the revised manuscript (extended file), we have added text explaining each symbol and geometric relationship, as well as a clarification of the parallelogram mechanism. Please see Lines 152–157.

2.- In Section 3.5, although at the end the time-varying nature of friction, resistance, and inertia is mentioned, the key simplifying assumptions adopted in the modeling process are not explicitly stated. It is recommended to clearly present these assumptions in the modeling section and analyze their possible impacts on control performance in the discussion of results.
= Thank you for your comment; those aspects of the fuzzy model were indeed quite ambiguous. The assumptions underlying the fuzzy modeling and their implications for control performance are now presented explicitly. Please see Lines 234–245.

3.- In Section 3.8, in Lines 336–339, the authors indicate that different poles are configured for the two actuators due to differences in joint motion ranges and amplitudes. However, no quantitative data or model analysis is provided to support this design decision. It is suggested to supplement with pertinent comparative data or theoretical analysis to improve the sufficiency and rigor of the argument.

=Many thanks for your comment—you are absolutely right. The way it was presented lent itself to misunderstanding. To address this, we have moved the GWO-derived results to the Results section, immediately after the ANFIS fuzzy modeling results. In that section, we explicitly justify the change in amplitude ranges by referencing the elliptical trajectory used in the simulation tests of the actuators coupled to the rehabilitation system via inverse kinematics, thereby clarifying why distinct pole assignments were configured for the two actuators.

4.- In the comparative experiments of Table 6, eight fuzzy rules are fixed regardless of whether the input dimension is 2 or 3. The manuscript does not explain the relationship between the number of rules and the input dimensions, nor does it clarify why other input dimensions were not considered. It is recommended to present the basis for determining the number of rules and to analyze, in the discussion section, the possible impacts of different input dimensions on model complexity and performance, thereby improving the completeness and persuasiveness of the results.
= Thank you very much for your comment—you are absolutely right. Nowhere did we explain how the number of rules is determined from the inputs or the criteria used to select the identified model.

We have clarified this in the revised manuscript: the formula for computing the number of fuzzy rules is now provided in Equation 36 (Lines 360–362), and the rationale for not considering models with an excessive number of rules is discussed in Lines 364–369.

5.- In Section 5, the manuscript claims to have achieved trajectory tracking for the four-bar rehabilitation robot and designed therapeutic movements that simulate elbow motion. However, based on the content of Section 4, verification is mainly based on simulations, and physical experiments are limited to the single-actuator level, lacking verification at the end-effector or full-machine level. Therefore, the authors are advised to express the research scope more cautiously in the conclusion and to supplement full-machine experiments in future work to avoid inconsistencies between the stated scope and the verification results.
= Thank you for the comment. In the Conclusions section, we have provided a detailed explanation of the results obtained, clearly delimiting the scope of the project. We also acknowledge the current limitations and outline the planned next steps to extend the validation accordingly.

Minor Comments:
1.- Section 2 reviews numerous applications of fuzzy control on different platforms, but at the end it directly moves to the theoretical framework of the parallel robot from References [15–17], taking it as a research basis. It is suggested to better clarify the reasons for selecting References 15–17 as the foundation and to explain the connection between this framework and the subsequent fuzzy identification and control methods, in order to improve logical coherence.
= Thank you for your comment. Section 2 discusses various control techniques applied to rehabilitation systems; this serves to show their contributions. However, in the following section, where References [15–17] are cited, these are used for the kinematic modeling of the parallelogram type, which is a very different issue.

2.- There is an inconsistency between the caption of Figure 2 and the main text. The text describes “Figure 2b presents the distances along the y-axis,” while the figure title indicates “Diagram of the distance along the Z-axis.” It is suggested to verify and correct this inconsistency uniformly.
= We appreciate for your observation. The figure shows distances along the z-axis, so the inconsistency was in the text; this inconsistency, it was corrected.

3.- Figure 2(a) only shows the directions of the x and y axes, without indicating the direction of the z-axis. Since the D–H method requires the z-axis to be strictly aligned with the joint axis, it is suggested to add the z-axis label in the figure to ensure consistency between the diagram and the derivation.
= Thank you for pointing this out. By the right-hand rule in robotics, the z-axis in the five coordinate systems shown is directed out of the computer screen. The z-axis label was added to the figure to indicate that it is considered.

4.- In Table 4, the authors only use color highlighting for the optimal values of RMS1_SIMU and RMS2_SIMU, without applying the same treatment to the corresponding RMS1_GWO and RMS2_GWO. It is recommended to maintain consistency in the highlighting method or to clearly explain in the table note the reason for emphasizing only the SIMU results, to avoid ambiguities.
= Thank you for your comment. Highlighting was applied to both sections of the best selected poles.

5.- In the simulation experiment of the dual-actuator system, the authors only performed 5 independent runs. Given that GWO is a stochastic optimization algorithm, this sample size is insufficient to demonstrate the algorithm’s stability and statistical significance. Therefore, it is suggested to increase the number of experiments, report the mean, standard deviation or confidence interval, and provide convergence curves to improve the reliability of the results.
= We appreciate for your suggestion. We understand your point; however, for future work, we intend to ensure stability with the gains obtained by GWO using Lyapunov theory.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The article deals with the issue of control of a robotic system based on the parallelogram principle. This system is designed for upper-limb rehabilitation. The fundamental problem is the design of actuators for this robotic system and the design of the control system. The development of such rehabilitation devices is extremely important and useful, and therefore this article is beneficial and topical.

In the introduction, the authors describe the issue of upper limb diseases and justify the need for a rehabilitation device. Similar works are listed with a relevant reference to the publication. It is useful to create an overview of currently used fuzzy control strategies.
The authors created a mathematical model of this robotic system.
The problem of inverse kinematics is further solved.
Next, the fuzzy control of the robot is solved so as to ensure smooth tracking of the trajectory of the end point movement.
The simulation results are then presented. The simulation results showed satisfactory results.

Comments:
Figure 1 is very nice but not illustrative. In Fig. 1 it is necessary to indicate the main parts of the device and provide a legend with the individual names of the parts of the device. This image should also indicate the method of application of the device at least schematically so that it is clear what the requirements for the device are. You can indicate a figure of how the rehabilitation is taking place. Alternatively, indicate the individual end positions of the device during movement. The render image is nice, but in addition to it it would be appropriate to show a kinematic diagram that corresponds to the render image so that the arrangement of the device is clear.

The kinematic diagram in Fig. 2 is not anchored. It should indicate where the joints are located, which connect the device to the frame and where the end point is. In such images, the coordinate system is usually also indicated.

The table on line 130 should have a name and number so that it can be referenced in the text.

Line 315: Figure 10 provides a flow chart and not pseudocode.

Table 2: The data are the result of an experiment that is burdened by measurement errors and so it is unnecessary to state so many decimal places in the results. It should be rounded according to the measurement error or measurement uncertainty.
The same is true in Table 4.

Figure 11 lacks units (Angular position).

Equation 38 is incorrectly stated.

Page 22 is unnecessarily blank

The conclusion is extremely brief. In conclusion, the authors should critically comment on the achieved results and rely on numerical results from simulations. And it is also necessary to expand it, for example, by discussing future plans for further research in this area.

Reference 36 is not completely defined.

Author Response

Reviewer 3

We sincerely appreciate your thoughtful comments and the time you dedicated to evaluating our manuscript. Below, we respond point by point to your observations

1.- Figure 1 is very nice but not illustrative. In Fig. 1 it is necessary to indicate the main parts of the device and provide a legend with the individual names of the parts of the device. This image should also indicate the method of application of the device at least schematically so that it is clear what the requirements for the device are. You can indicate a figure of how the rehabilitation is taking place. Alternatively, indicate the individual end positions of the device during movement. The render image is nice, but in addition to it it would be appropriate to show a kinematic diagram that corresponds to the render image so that the arrangement of the device is clear.

= Thank you very much for your comment; I agree. The original figure did not clearly indicate the device’s main components or where the patient’s limb is positioned on the robotic rehabilitation end-effector. We have revised it to include labels for each joint and link and to depict the patient’s arm at the device’s end-effector.

2.- The kinematic diagram in Fig. 2 is not anchored. It should indicate where the joints are located, which connect the device to the frame and where the end point is. In such images, the coordinate system is usually also indicated.

= Thank you for the observation—I agree. Figure 2 does not clearly indicate where each joint or motor is located. However, in Lines 134–139 we now reference Figure 1 (the render of the robotic device), which has been revised to explicitly label these joints and show their placement on the frame and at the end-effector.

3.- The table on line 130 should have a name and number so that it can be referenced in the text.

= Thank you for the observation. In the LaTeX structure we changed from tabular to table, added a caption and numbering, and included the corresponding in-text reference.

4.- Line 315: Figure 10 provides a flow chart and not pseudocode.

=Excellent point—thank you. We corrected both the text and the figure label to “flowchart,” which is the proper term.

5- Table 2: The data are the result of an experiment that is burdened by measurement errors and so it is unnecessary to state so many decimal places in the results. It should be rounded according to the measurement error or measurement uncertainty.

The same is true in Table 4.

= Very good observation, thanks. We edited both tables and rounded the values according to the measurement uncertainty.

6- Figure 11 lacks units (Angular position).

= Thank you for the comment. The figure was updated to include the appropriate unit.

7.- Equation 38 is incorrectly stated.

= Many thanks for pointing this out; you are correct. The RMSE equation has been corrected.

8.- Page 22 is unnecessarily blank

= Thank you for the observation. We adjusted the layout to remove the blank page.

9.- The conclusion is extremely brief. In conclusion, the authors should critically comment on the achieved results and rely on numerical results from simulations. And it is also necessary to expand it, for example, by discussing future plans for further research in this area.

= We appreciate the comment. The conclusions section was expanded, incorporating numerical performance results and a more detailed discussion, as well as future research plans.

10.- Reference 36 is not completely defined.

= Thank you for noting this. The .bib entry is complete; however, the MDPI template’s reference style controls which fields appear in the compiled PDF. We verified consistency and adjusted fields so the output is as complete as possible under that format.

Author Response File: Author Response.pdf