Round 1

Reviewer 1 Report

Dear Authors,

The manuscript should be revised before it is published. My review report is in the attachment.

Kind Regards

Comments for author File: Comments.pdf

Author Response

Reviewer 1

1.1 Authors in this paper present an adaptive control approach with state dependent gains for regulating the position of a robotic manipulator carrying a piezoelectric based gripper and solving the tracking trajectory problem of the robot. English of the paper can be polished. Please pay attention to rules of writing academic paper. My basic comments and corrections are as follows.

Thanks for the deep analysis of our paper. We have made a significant effort to reorganize the document and correct it according to your suggestions in this answer letter. We have also polished the English grammar and have made an additional effort to correct typos and misprints when necessary.

1.2 General question about the comparison between your approach and the traditional PID. Have you used only three PID parameters for the comparison? Are they optimum parameters? There may exists a PID parameter set which produces less mean square error and follows trajectory better.

We used PID controller for comparison, considering that this is the controller which is mainly used for robotic manipulators. We evaluated the application of PID control in general, and we did not consider the independent effect of each gain in its structure. These parameters were adjusted using the self-tunning process inserted in the PID formulation proposed in the Matlab algorithm. This algorithm performs a frequency-response estimation experiment that injects signals into the plant and measures the plant output with the feedback loop closed or open, respectively. The method also uses the resulting estimated frequency response to tune PID gains for the plant. The application of such an algorithm allowed us to create a fair comparison with the controller shown as the main contribution of this study. Even when some other parameters may produce lesser mean square errors, they may increase the power related to the controller's application. The combined evaluation of the mean square error and the control power justifies the proposal presented in our study.

1.3 How about the sinusoidal trajectories which have higher frequencies. One period of your trajectory takes 200-300 seconds. Have you tried your approach with higher frequencies (faster than your experiments)?

The proposed sinusoidal trajectories were selected considering the motion developed by the actuator. High-frequency motions do not characterize the piezoelectric actuator. This reason justifies the selection of such slow reference trajectories for the proposed controller. Nevertheless, we used a simulated version of the piezoelectric actuator with no limitation regarding motion velocity. In such simulated cases, the tracking of the reference trajectories is completed without problems, as in the case of the experiments shown in the original version of the study.

1.4 Lines 18-27: Literatures that you use in the first paragraph are relative older. Please add much more recent literatures about piezo actuators and motors. You should take following literatures in this part.

GAO, Xiangyu, et al. Piezoelectric actuators and motors: materials, designs, and applications. Advanced Materials Technologies, 2020, 5. Jg., Nr. 1, S. 1900716.
Delibas, B.; Koc, B.; Thielager, J.; Stiebel, C. A novel drive and control method for piezoelectric motors in microscopy stages. In Proceedings of the Euspen’s 21st International Conference & Exhibition, Copenhagen, Denmark, 7–10 June 2021.
Touairi, S., Khouya, Y., Bahanni, C., Khaouch, Z., & Mabrouki, M. (2019, April). Mechatronic control and modeling of a piezoelectric actuator. In 2019 International Conference on Wireless Technologies, Embedded and Intelligent Systems (WITS) (pp. 1-6). IEEE.

Thanks for your suggestion. We have reviewed the proposed references, where we have detected some relevant information that served to improve the contents of our reviewed manuscript. We believed that including those references contributed to the contribution given in our manuscript. Therefore, we invite this reviewer to check the following paragraph where we have used the listed references.

1.5 Line between 192 and 193: “with the a state feedback“ should be ”with the state feedback”. Please correct it.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.6 Paragraph after equation 16: What is “[? ]“, please correct it.

This question mark appears because of an incorrect reference. We removed this error in the reviewed manuscript.

1.7 Line 210: Please correct the typing mistake.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.8 Line 218: “dc motors” should be “DC motors”, please correct it.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.9 Lines 219 to 220: Please reconstruct the sentence. Grammatically not correct.

We have reconstructed the sentence. The original sentence was:

The engine selected for the movement was the engine of the Servocity brand, which are motors that include an encoder to have control of the positioning, as important technical characteristics the power supply at 12 volts DC for the motor while the encoder is powered 221 at 5 volts, it has a maximum revolution of 32 RPM.

The new phrase is:

The engines selected for performing the movement were 12 volts DC motors (Servocity), including an encoder (32 RPM). The encoder information is used to close the loop in position control.

1.10 “Table 1. selected“ should be “ Table 1. Selected“. Please correct it.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.11 Line 237: “Figures 4, 6 and 8 shows“ should be ” Figures 4, 6 and 8 show”. Please correct it.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.12 Line 239: “as well the relative“ should be “as well as the relative”. Please correct it.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.13 Do Figures 4, 5 and 6 have same figure captions “First articulation θ1”. Are they right? Please check them.

The captions are not correct. We modified the captions. These captions are now as:

Figure 4. Photograph of the electromechanical configuration for the first articulation

Figure 5. Comparison of the controlled motion for the first articulation () when the PID and adaptive control with state constraints are considered.

Figure 6. Photograph of the electromechanical configuration for the second articulation

Figure 7. Comparison of the controlled motion for the second articulation () when the PID and adaptive control with state constraints are considered.

1.14 Figure 9 shows different results that you have written in the caption. Please check it again.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript.

1.15 Figures 5, 7, 9: You should show the unit of the angular movements in these figures like [°].

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript, and we have included the units ([°]) in figures.

1.16 Figures 10 and 11: Same for these figures as well. Please add units for the vertical axes of the figures.

Thanks for pointing out this error. We have made the suggested modification in the revised manuscript, and we have included the units ([°]) in figures.

1.17 Figure 11: From this figure, it is not clearly seen that your approach consumes less energy than the traditional one. Please show it clearly.

The reviewer is right. The comparison presented in Figure 11 is not clearly demonstrating how the proposed controller is using less power than the traditional state feedback formulation. With the aim of removing this imprecision in our argument, we calculated the averaged integral for the norm of the control actions using the proposed barrier-based control and the traditional state feedback (PID control). The integral of the control action is presented when the number of samples was 3000 that correspond to the end of the experimental evaluation. After doing the averaged integration we estimated that the state feedback formulation is 1.3*10⁹ while the obtained when the barrier-based controllers was 5.8*10⁸. Moreover, we calculated the ratio of the averaged mean square error obtained with both controllers (estimated with the information presented in Figure 10) with respect the corresponding averaged value of the integral for the norm of the control actions. These values are evaluating a measure of efficiency in the tracking performance with respect the control power invested in attaining such tracking. For the PID controller, the proposed ratio was 16.92*10^-9, while the value gotten with the application of the barrier controller is 0.89*10^-8. This comparison explains better the benefit of introducing the proposed controller with respect traditional formulations. This description has been included in the revised manuscript.

1.18 Conclusion can be enhanced further.

According to the suggestion, we have modified the conclusions in our manuscript. The new version of such a section is as follows:

This study presents the design of a feedback controller with state-dependent gains for regulating the position of a robotic manipulator carrying a piezoelectric actuator-based gripper. The control method considers the effect of perturbations, uncertain modeling elements, and the presence of state restrictions in an ellipsoid form. The application of an adaptive control ensures asymptotic convergence of the states to an invariant zone whose size depends on the power of uncertainties and perturbations. In addition, the proposed controller's significant characteristics are evidenced by experimental evaluations for a particular developed robotic manipulator. The comparative analysis of the suggested controller concerning the traditional state feedback justified the benefits of the proposed methodology. This comparison demonstrates a minor mean square tracking error with a smaller power associated with the control action. The experimental results show the effective handling of microscopical objects using the developed robotic device. The controller's application overcame the main difficulties, including the necessity of implementing the state-dependent controller based on the barrier-based design.

Reviewer 2 Report

This paper presents a state feedback controller with state dependent gains for regulating the position of a robotic manipulator carrying a piezoelectric based gripper.

The control method considers the effect of perturbations, uncertain modeling elements and the presence of state restrictions in an ellipsoid form. It's interested and useful.

However, the experimental results need to be reorganized. Real images at the micro scale are required. Recent applications used the piezo actuator should be well concluded in Introduction.

Author Response

Reviewer 2

2.1 This paper presents a state feedback controller with state dependent gains for regulating the position of a robotic manipulator carrying a piezoelectric based gripper.

Thanks for the deep analysis of our paper. We have made a significant effort to reorganize the document and correct it according to your suggestions in this answer letter.

2.2 The control method considers the effect of perturbations, uncertain modeling elements and the presence of state restrictions in an ellipsoid form. It's interested and useful. However, the experimental results need to be reorganized. Real images at the micro scale are required.

Thanks for advising about this opportunity area that can help us to improve the manuscript. We have included additional discussions on the outcomes attained in the experimental section. We included a brief study on the effectiveness of tracking the reference trajectories comparing the integral related to the power of the control action with respect to the integral of the mean square tracking error. Also we included a description of the workability of the proposed controller at the microscale dimensions. A sequence of photographs is showing how the control action is forcing the behaving of the gripper formed with piezoelectric actuators. This sequence of photographs is:

Figure 12. Direct visualization of the gripper section attached to the piezolectric actuators

Figure 13. Manipulation of a 160 μm polystyrene sphere using the gripping device made with the piezoelectric actuators.

2.3 Recent applications used the piezo actuator should be well concluded in Introduction.

Thanks for this suggestion. We have added an additional paragraph at the introduction. The mentioned paragraph is:

The piezoelectric actuator is applied in a variety of applications, including industrial, automotive, aviation, aerospace, medical, and consumer electronics. The quiet drive settings make piezo actuators an auto-focusing mechanism in microphone-equipped video cameras and mobile phones. In addition, since piezo actuators require no lubrication, they are used in cryogenic and vacuum environments, among others. Using a stack actuator, extremely fine is possible with controlled voltages corresponding to slow expansion motions. As a result, a piezo actuator can operate billions of times without wear or deterioration. The motion speed is exceptional, and it is limited only by the object's inertia, being moved, and the output capability of the electronic driver. When operating in an energized condition, a piezoelectric actuator consumes small electrical power with limited produced heat.

Author Response File: Author Response.docx

Reviewer 3 Report

This manuscript proposes a robust control approach for a specific robotic manipulator. The article is well written and organized. Some suggestion for improving the readability of the paper:

1) The numerical results of the study should be reported in the abstract section.

2) While only gains of a PD control are given, in the context is written PID. Why?

3) The challenges of the work should be described in the conclusion section.

Author Response

Reviewer 3.

3.1 This manuscript proposes a robust control approach for a specific robotic manipulator. The article is well written and organized. Some suggestion for improving the readability of the paper:

Thanks for the deep analysis of our paper. We have made a significant effort to reorganize the document and correct it according to your suggestions in this answer letter.

3.2 The results of the study should be reported in the abstract section.

Thanks for pointing out this opportunity area. We have added some additional comments in the abstract where we have tried to highlight the more important findings obtained in our study. The new version of the abstract is:

The current study presents an adaptive control approach to solve the tracking trajectory problem for a robotic manipulator that uses a gripper based on bimorph piezoelectric actuators. The development of an adaptive gain state feedback form, that considers the state restrictions, is proposed using a novel class of barrier Lyapunov function that drives the effective control of joints and piezoelectric actuators. The proposed method allows to include complex combinations of state restrictions in the Lyapunov function yielding the construction of differential forms for the gains in the controller that can handle the evolution of trajectories of the robotic arm inside the restricted region. The proposed control design success tracking reference trajectories for both the joints of the robotic arm, as well as the motion of the piezoelectric device, during several operative scenarios. A comprehensive experimental study evaluates the effect of introducing the state-dependent gain considering the state restrictions of ellipsoidal type. The comparison of the mean square error confirms the contributions of the developed control action, showing better tracking quality, and a smaller control power along with the same evaluation, which is a desirable characteristic in the controlled motion of micromanipulators. Mainly, the proposed controller solves the tracking trajectory problem for the micromanipulation system while the motion restrictions are also satisfied, and better tracking performance is also enforced. The obtained trajectories comparison seems the validate the contribution of the proposed controller concerning a feedback form with fixed gains.

3.3 While only gains of a PD control are given, in the context is written PID. Why?

Thanks to the reviewer. This is a misprint in our original manuscript. We have modified Table 1 and now it contains the missed gains of the integral section in the PID.

The new table is

Joint/Gain
First	90.92	5.21	3.4
Second	80.92	15.21	4.6
Third	72.34	24.12	7.6

The modified table is now appearing in the modified manuscript. Once more, thanks for pointing out this mistake.

3.4 The challenges of the work should be described in the conclusion section.

Thanks for the comment. We have modified the conclusions sections. Now the text for the conclusions is

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Dear Authors,

You have modified the manuscript according to my objection and comments. The manuscript can be published after minor revision (methodological errors, text editing etc.). Congratulations.

King Regards

Author Response

1.1 You have modified the manuscript according to my objection and comments. The manuscript can be published after minor revision (methodological errors, text editing etc.). Congratulations.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors added new results. It looks good. However, I do think the authors included too many figures in the manuscript. PLease reorganize these figures and the contents. Keep them less than 9 figures. Combine the figures supporting the same conclusion.

Author Response

2.1 The authors added new results. It looks good. However, I do think the authors included too many figures in the manuscript. Please reorganize these figures and the contents. Keep them less than 9 figures. Combine the figures supporting the same conclusion.

Thanks for the deep analysis of our paper. We have made a significant effort to reorganize the document and correct it according to your suggestions in this answer letter. We also have reduced the number of figures and now we have only nine of them.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

The authors try their best to modify their manuscript.

Review Reports

Reviewer 1

Reviewer 2

Reviewer 3.