Cartesian Stiffness Shaping of Compliant Robots—Incremental Learning and Optimization Based on Sequential Quadratic Programming

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article presents a new method called Locally Weighted Projection Regression (LWPR), which excels in online learning. This technique is specifically designed to enhance the accuracy of modeling-compliant actuators. It significantly improves model-based control in robotics, particularly for systems that include non-rigid links and elastic actuators. The idea is exciting and opens diverse discussions and new branches of research. 

1)  From a reader's perspective, the introduction in the article on compliant robots, while adequately covering their advantages and actuators, could be significantly improved. It would benefit from an expanded historical context, a more apparent comparative analysis with rigid robots, and the inclusion of real-world applications. Additionally, a discussion on current challenges and prospects in the field, accompanied by illustrative diagrams, would enhance the reader's comprehension and set a solid foundation for the article.

2)  In the paper, the adoption of Locally Weighted Projection Regression (LWPR) and Sequential Least Squares Programming (SLSQP) for controlling a 4 DoF planar robot raises some concerns. LWPR, though adept at online learning for modeling compliant actuators, might need help in scenarios with sparse data and struggle with computational efficiency in real-time applications. SLSQP, while effective for minimizing nonlinear functions, is computationally demanding, potentially hindering real-time performance due to the complexity of converting nonlinear functions into quadratic approximations in each iteration. These methods, innovative in theory, require a delicate balance between accuracy and computational practicality, particularly in diverse and rapid-response robotic applications. The authors would benefit from additional clarification on certain aspects of their methodology.

3)  Section 2 of the article adeptly introduces the Locally Weighted Projection Regression (LWPR) approach for modeling VSAs. However, it needs to discuss the practical challenges of LWPR, especially its computational intensity and adaptability in varied real-world scenarios. The section could be strengthened by addressing these limitations and clarifying the generalizability of the findings from the specific actuator model studied to a broader range of VSAs and robotic systems.

4)  Sections 3 and 4 of the article address significant methodologies in robotic control but need more clarity in delineation and exposition. Section 3 introduces an optimization algorithm for shaping end-effector Cartesian stiffness in compliant robots using the Sequential Least Squares Programming (SLSQP) method. While technically sound, this section could benefit from a more lucid presentation of the algorithm's practical implications and limitations in real-world applications. Section 4, detailing experimental validation, overlaps with Section 3 in content, lacking a distinct separation in methodology and results. A clearer demarcation between theoretical development and practical application would enhance these sections' overall coherence and readability.

5)  The article lacks a comprehensive systematization or "big-picture" view that outlines all the steps required to reproduce its results. This absence makes it challenging for readers, especially those in the research community, to fully grasp the scope of the study and apply its methodologies in their work. A more structured approach, with clear, step-by-step guidance and flowcharts or schematic diagrams, would significantly enhance the replicability and practical applicability of the research findings. It is essential for scholarly articles, especially those in technical fields, to provide such detailed frameworks to facilitate further research and experimentation.

6) The difficulty in understanding the beginning and end of the authors' proposal and delineating results and evaluation scenarios indicates a need for clearer structure in the article. This ambiguity can hinder the reader's ability to comprehend the research objectives, methodologies, and conclusions fully. A more explicit and sequential presentation of the research proposal, its implementation, and the evaluation methods used would greatly aid in providing clarity and coherence to the study, enhancing its academic value and utility for the reader.

7) Some open issues after concluding the revision of the paper:

a) In what ways do compliant robotic systems contribute to enhanced safety protocols and increased energy conservation?

b) What complexities are encountered in constructing accurate models and executing control over compliant robotic systems?

c) How is the Locally Weighted Projection Regression (LWPR) methodology effective in overcoming the complexities above?

d) Can the integration of Variable Stiffness Actuators (VSA) within a 4 DoF planar robot be justified in terms of performance benefits?

e) In what manner does the Sequential Least Squares Programming (SLSQP) optimization method facilitate the real-time application of these robotic systems?

Author Response

Dear Reviewer, thank you for kind words on our work presented in this manuscript. Also, thank you for your detailed comments that have helped us improve the paper.

The article presents a new method called Locally Weighted Projection Regression (LWPR), which excels in online learning. This technique is specifically designed to enhance the accuracy of modeling-compliant actuators. It significantly improves model-based control in robotics, particularly for systems that include non-rigid links and elastic actuators. The idea is exciting and opens diverse discussions and new branches of research. 

1)  From a reader's perspective, the introduction in the article on compliant robots, while adequately covering their advantages and actuators, could be significantly improved. It would benefit from an expanded historical context, a more apparent comparative analysis with rigid robots, and the inclusion of real-world applications. Additionally, a discussion on current challenges and prospects in the field, accompanied by illustrative diagrams, would enhance the reader's comprehension and set a solid foundation for the article.

Thank you for your insightful feedback on the introduction to the article. Your suggestions highlight key areas for improvement, and we appreciate the opportunity to enhance the reader's experience. Here's a breakdown of your concerns and proposed improvements:

• To provide a wider picture of the topic, we added a part about the applications that use compliant robot technologies, showcasing applications where compliant robots can impact task performance (lines 112 – 126).

“The above-mentioned collaborative approaches enhance efficiency and flexibility in production processes, reducing the risk of injury by absorbing external forces, which showcase real-world applications where compliant robots excel in industrial settings. Robots with compliant features are employed on assembly lines where they can work safely alongside human workers in the automotive industry [1,2]. Some authors propose the use of impedance control for collaborative human-robot chamfering and polishing applications [3], as well as a null-space search for torque-effective drilling [4]. Collaborative assembly through robot behavior shaping with active and passive compliance was introduced in [5]. Furthermore, a notable real-world application where compliant robots outperform rigid robots is in surgery, specifically in minimally invasive procedures. Compliant robotic systems, such as the da Vinci Surgical System [6], demonstrate superiority over rigid counterparts due to their ability to navigate and manipulate soft and delicate tissues with greater precision and dexterity [7]. The compliance of the robotic arms allows for more natural and adaptive movements, reducing the risk of tissue damage and improving surgeon\textquotesingle s control [8].

[1] Roveda, L. Adaptive interaction controller for compliant robot base applications. IEEE Access 2018, 7, 6553–6561.

[2] Masinga, P.; Campbell, H.; Trimble, J.A. A framework for human collaborative robots, operations in South African automotive industry. In Proceedings of the 2015 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM). IEEE, 2015, pp. 1494–1497.

[3] Kana, S.; Lakshminarayanan, S.; Mohan, D.M.; Campolo, D. Impedance controlled human–robot collaborative tooling for edge chamfering and polishing applications. Robotics and Computer-Integrated Manufacturing 2021, 72, 102199.

[4] Zanchettin, A.M.; Rocco, P.; Robertsson, A.; Johansson, R. Exploiting task redundancy in industrial manipulators during drilling operations, 2011. 2011 IEEE Int. Conf. on Robotics and Automation.

[5] Cherubini, A.; Passama, R.; Crosnier, A.; Lasnier, A.; Fraisse, P. Collaborative manufacturing with physical human–robot interaction. Robotics and Computer-Integrated Manufacturing 2016, 40, 1–13.

[6] Surgical, I. da Vinci. Surgical System. http://www. intusurg. com/html/davinci. html 2013.

[7] Freschi, C.; Ferrari, V.; Melfi, F.; Ferrari, M.; Mosca, F.; Cuschieri, A. Technical review of the da Vinci surgical telemanipulator. The International Journal of Medical Robotics and Computer Assisted Surgery 2013, 9, 396–406.

[8] Burgner-Kahrs, J.; Rucker, D.C.; Choset, H. Continuum robots for medical applications: A survey. IEEE Transactions on Robotics 2015, 31, 1261–1280.

• We introduce a step-by-step pipeline to present the overview of the proposed methodology (lines 143 – 147).

“To exploit the full potential of the proposed methodology, the following pipeline was defined through several steps: 1) learning VSA model parameters using LWPR; 2) continuous parameters relearning through incremental learning; 3) utilization of the learned robotic system model (in our case 4 DoF planar robot with VSAs); and 4) SLSQP algorithm to effectively control behavior by determining the optimal configuration and stiffness on the joint level.”

• We added a part at the end of conclusion to address all limitations and challenges of the proposed methodology (lines 398 – 407).

“The proposed approach has several limitations. Finding of the proper learning parameters for the LWPR algorithm can be time-consuming on occasion, although parameter finding needs to be performed only once during the initial learning process. EE Cartesian stiffness is limited since it is achieved by exploiting the passive stiffness and kinematics of the manipulator. Cartesian stiffness is shaped using an optimization method that cannot guarantee a global minimum. Although time-consuming, this can be overcome by calculating optimal solutions from different initial points. In future work, studies will be conducted on learning techniques that can capture motor dynamics, where a time-series data set will be used. Also, the focus will be on algorithms that combine active and passive stiffness control at the joint level to enhance the algorithm’s performance.”

2)  In the paper, the adoption of Locally Weighted Projection Regression (LWPR) and Sequential Least Squares Programming (SLSQP) for controlling a 4 DoF planar robot raises some concerns. LWPR, though adept at online learning for modeling compliant actuators, might need help in scenarios with sparse data and struggle with computational efficiency in real-time applications. SLSQP, while effective for minimizing nonlinear functions, is computationally demanding, potentially hindering real-time performance due to the complexity of converting nonlinear functions into quadratic approximations in each iteration. These methods, innovative in theory, require a delicate balance between accuracy and computational practicality, particularly in diverse and rapid-response robotic applications. The authors would benefit from additional clarification on certain aspects of their methodology.

Thank you for your thorough examination of the paper discussing the adoption of Locally Weighted Projection Regression (LWPR) and Sequential Least Squares Programming (SLSQP) for controlling a 4 DoF planar robot. We addressed your concerns regarding the potential challenges with LWPR in scenarios of sparse data and computational efficiency, as well as the computational demands of SLSQP:

• The clarification on LWPR's effectiveness in handling sparse data and time efficiency are discussed in lines 151 – 167, and lines 200 -203.

“LWPR is used as a learning technique in order to deal with the uncertainties of the actuator model parameters, as well as the nonlinearity of the actuator and its susceptibility to change due to wear. The LWPR method is designed to overcome the issue of sparse data because it is effective in learning when a small amount of data is available or when the data is noisy. In [1], the authors describe in detail the features of LWPR compared to other state-of-the-art algorithms, like the Gaussian Process and Support Vector Machine. Furthermore, the complexity of the LWPR algorithm increases linearly with problem dimensionality. Regarding computational efficiency, a 70 Hz learning rate has been achieved for a high dimensional learning problem (90 inputs and 30 outputs).

In the proposed application case, a 4 DoF planar manipulator with VSAs, the input dimension is 2x4 (shaft position and joint stiffness of each actuator), and the output dimension is also 2x4 (primal movers position of each actuator).“

[1] Vijayakumar, S., D'souza, A., & Schaal, S. (2005). Incremental online learning in high dimensions. Neural computation, 17(12), 2602-2634.

• The SLSQP computational time needed for calculating robot configuration and joint stiffnesses are added in lines 296 – 297. Since we use the output of the SLSQP algorithm to generate references for robot movement, this can be seen as real-time.

“The time needed to calculate the optimal robot configuration and joint stiffnesses was 0.004 ± 0.001s.”

3)  Section 2 of the article adeptly introduces the Locally Weighted Projection Regression (LWPR) approach for modeling VSAs. However, it needs to discuss the practical challenges of LWPR, especially its computational intensity and adaptability in varied real-world scenarios. The section could be strengthened by addressing these limitations and clarifying the generalizability of the findings from the specific actuator model studied to a broader range of VSAs and robotic systems.

Thank you for your feedback on Section 2 of the article, focusing on Locally Weighted Projection Regression (LWPR) for modeling Variable Stiffness Actuators (VSAs). We appreciate your insights, and we made the necessary improvements to enhance the manuscript:

• Practical Challenges of LWPR is addressed in lines 182 – 188 and in future work part in Conclusion section in lines 398 – 407.

“To successfully incorporate the LWPR approach into a learning problem, the learning rate parameter \alpha and the initial values for distance matrix D_k need to be set properly. The typical approach, which can be applied to various VSAs, involves configuring the parameter D}_k = rI with a small number for the variable r (e.g., r=0.05). Then, the model is retrained by gradually increasing r until the model achieves satisfactory performance. Also, \alpha can be tuned to improve algorithm performance. This methodology might be demanding and time-consuming until satisfactory performance is achieved.”

“Finding of the proper learning parameters for the LWPR algorithm can be time-consuming on occasion, although parameter finding needs to be performed only once during the initial learning process.”

• The LWPR algorithm is developed as a general-purpose learning algorithm so it can be used in a variety of applications. The generalizability to broader VSAs and Robotic Systems to utilize this methodology is presented in the Introduction section by step-by-step pipeline in lines 143 – 147.

“To exploit the full potential of the proposed methodology, the following pipeline was defined through several steps: 1) learning VSA model parameters using LWPR; 2) continuous parameters relearning through incremental learning; 3) utilization of the learned robotic system model (in our case 4 DoF planar robot with VSAs); and 4) SLSQP algorithm to effectively control behavior by determining the optimal configuration and stiffness on the joint level.”

4)  Sections 3 and 4 of the article address significant methodologies in robotic control but need more clarity in delineation and exposition. Section 3 introduces an optimization algorithm for shaping end-effector Cartesian stiffness in compliant robots using the Sequential Least Squares Programming (SLSQP) method. While technically sound, this section could benefit from a more lucid presentation of the algorithm's practical implications and limitations in real-world applications. Section 4, detailing experimental validation, overlaps with Section 3 in content, lacking a distinct separation in methodology and results. A clearer demarcation between theoretical development and practical application would enhance these sections' overall coherence and readability.

Thank you for your feedback. We made the following improvements to enhance the clarity and coherence:

• We provided a clearer exposition of the optimization algorithm for shaping EE Cartesian stiffness in compliant robots using the SLSQP method by detailed presentation of the optimization algorithm in lines 242 -254. This will involve a more concise presentation of the algorithm's practical implications and limitations in real-world applications, ensuring that the technical content is easily understood.
• Some limitation of this method is mentioned in the limitations and future work part of the Conclusion section in lines 398 – 407.
• The sections are organized in a way to present the methodology and then to present and discuss the results. First of all Sections 2 and 3 are reorganized to introduce theoretical concepts followed by use-cases used for the demonstration. Section 2 introduces a general method of LWPR utilization for learning the model of compliant joints and then presents its use case on the QB actuator - the bidirectional antagonistic drive. Section 3 describes a general SLSQP optimization method for Cartesian stiffness shaping, as well as a use case on computing positions and stiffnesses for each of the 4 DoF compliant robot joints for the desired Cartesian stiffness. Finally, Section 4 validates the theory and use cases from Section 2 and Section 3 by introducing external disturbances by a compliant robot equipped with an F/T sensor that exposes the motion to the 4 DoF complaint robot and measures the deviation from the desired position.

To make the concept clear to the readers, we additionally explained the organization of the paper and presented methodology at the end of the Introductory section (lines 148-157).

5)  The article lacks a comprehensive systematization or "big-picture" view that outlines all the steps required to reproduce its results. This absence makes it challenging for readers, especially those in the research community, to fully grasp the scope of the study and apply its methodologies in their work. A more structured approach, with clear, step-by-step guidance and flowcharts or schematic diagrams, would significantly enhance the replicability and practical applicability of the research findings. It is essential for scholarly articles, especially those in technical fields, to provide such detailed frameworks to facilitate further research and experimentation.

Thank you for your insightful feedback regarding the reproducibility and practical applicability of the article's results. We acknowledge the importance of providing a comprehensive systematization view for readers, particularly in technical fields. We address this concern by incorporating the improvement in the Introduction section by providing a step-by-step pipeline and flowchart presented in Fig. 1.

6) The difficulty in understanding the beginning and end of the authors' proposal and delineating results and evaluation scenarios indicates a need for clearer structure in the article. This ambiguity can hinder the reader's ability to comprehend the research objectives, methodologies, and conclusions fully. A more explicit and sequential presentation of the research proposal, its implementation, and the evaluation methods used would greatly aid in providing clarity and coherence to the study, enhancing its academic value and utility for the reader.

Thank you for your valuable feedback regarding the structure of the article. We appreciate your insights. We made a significant improvement by addressing all the above-mentioned comments making the article clear and coherent. We believe that readability, more systematic presentation is achieved especially by addressing your comments 5) and 6).

7) Some open issues after concluding the revision of the paper:

1. a) In what ways do compliant robotic systems contribute to enhanced safety protocols and increased energy conservation?

Compliant robotic systems excel in applications where both safety protocols and energy efficiency are crucial considerations. Compliant robotic systems enhance safety by incorporating inherent compliance, force-limiting algorithms, and advanced safety monitoring systems. They contribute to increased energy conservation through the use of energy-efficient actuators, adaptive control strategies, and regenerative braking mechanisms. These robots optimize energy consumption by adapting to different scenarios and employing efficient motion planning algorithms. In the Introduction section, several papers justify the statements provided above [1-6,8,9].

1. b) What complexities are encountered in constructing accurate models and executing control over compliant robotic systems?

Constructing accurate models and executing control over compliant robotic systems encounter complexities due to unmodeled friction, asymmetry in springs and motors, and spring nonlinearity. The drawbacks of improper modeling can be seen in Fig. 5 (red), where only the static relation between primal movers position and shaft position and joint stiffness is used (friction, asymmetry, and dynamics were neglected). Lines 85-87 added.

1. c) How is the Locally Weighted Projection Regression (LWPR) methodology effective in overcoming the complexities above?

By proper dataset defining, the LWPR algorithm is capable of catching all unmodeled phenomena. In this paper, we presented a static dataset from which the LWPR algorithm can learn static friction and asymmetry between springs and motors. Lines 128-123 are added. Nevertheless, by introducing a time-series dataset in the learning process, the dynamics of motors could also be learned.

1. d) Can the integration of Variable Stiffness Actuators (VSA) within a 4 DoF planar robot be justified in terms of performance benefits?

Compliant robots with VSAs have an elastic element between the actuator and the link, which enables diverse variants of compliant actuators to be systematically designed and engineered by varying actuator configurations and associated elastic elements. They are capable of absorbing sudden impacts and adapting to them. Furthermore, robots with flexible joints can outperform rigid robots in repetitive tasks, or when the high energy impact is needed to perform tasks such as throwing or nailing, which were not possible with rigid robots.

We chose a 4 DoF planar robot because it's redundant. For redundant robots, there is an infinite number of robot configurations for one EE position. EE Cartesian stiffness in compliant redundant robots can be alternated by changing robot configuration, which will not be possible with nonredundant robots, and joint stiffness.

1. e) In what manner does the Sequential Least Squares Programming (SLSQP) optimization method facilitate the real-time application of these robotic systems?

“The time needed to calculate an optimal configuration and joint stiffnesses was 0.004 ± 0.001s.” This sentence is added in lines 296-298 to express the usability of the method in real-time control.

Reviewer 2 Report

Comments and Suggestions for Authors

A Methodology is proposed ( Locally Weighted Projection Regression - LWPR) and its online learning capabilities with the aim  to have the model of complaint actuators accurate and  the  control robust.

The application is a compliant  4 DoF planar robot  driven by Variable Stiffness Actuators.

Cartesian positions  and stiffness control  is studied. In order to estimate   robot behavior, a disturbance is  introduced to the system.

The proposed method is very interesting and results are encouraging.

The authors stat that "Although the proposed methodology is experimentally validated on the 4 DoF planar robot driven by VSAs, the  presented methodology is general and can be exploited by other complaint robots without any additional sensors.

Have demonstrate that? I may opinion I suggest to write "could" insted of "can".

Then, since Cartesian stiffness is influenced by the robot joint  stiffness matrix, are the stiffness joint characterized experimentally or by Finite element methods (FEM) analysis?

I suggest the introduction of the following

Doi:10.21741/9781644902431-33

where a mechanical charcterization of stifness  joint is performed.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

 Dear Reviewer, we are grateful for your review and comments. We completely agree with you that in statement “Although the proposed methodology was experimentally validated on a 4 DoF planar robot driven by VSAs, the methodology is general and could be exploited by other compliant robots without any additional sensors.” should write “could” instead of “can” and we correct that in revised manuscript. Also, the paper you recommended to us is interesting and we incorporate it in Introduction section in lines 32 - 34.

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript addresses an interesting subject. It is well organized and clearly written. It presents simulated and also experimental results, which is valuable. However, some issues still have to be addressed:

1) English has to be revised because are numerous typos: see the yellow marks in the attached document.

2) Lines 168-170. It is unclear to me why you have to put more effort into conducting the mathematical model. It is easy to find q_1 and q_2 from x and S inverting (1) and (2).

3) Why only static values fittings?. Do not you need to include motors dynamics?, do not they influence the responses?. 

4) Your experiments to carry out the iterations take long time. Since you are fitting local zones, why not to use a standard least squares method that would much more fast in providing an accurate local linear model?.

5) Lines 185-186: the fitting obtained in Fig.5 after 4 iterations does not seem to be very good, particularly in joint stiffness. Then the assertion of "succesful learning" is questionable. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English has to be revised because are numerous typos: see the yellow marks in the attached document.

Author Response

Dear Reviewer, thank you for reviewing our manuscript and we are pleased that you found that our work is interesting and valuable to you.

This manuscript addresses an interesting subject. It is well organized and clearly written. It presents simulated and also experimental results, which is valuable. However, some issues still have to be addressed:

1) English has to be revised because are numerous typos: see the yellow marks in the attached document.

Thank you for suggesting and pointing out the typos, we carefully went through the manuscript and revised all of it. We have also enlisted the services of a professional English proofreader to review and refine the language in the manuscript.

2) Lines 168-170. It is unclear to me why you have to put more effort into conducting the mathematical model. It is easy to find q_1 and q_2 from x and S inverting (1) and (2).

We refined the statement concerning your comment (lines 219 -222). You are right that the model generated from equations (1) and (2) is not hard to get, but in this way, all imperfections about spring and motor asymmetries, and friction will not be captured. This can be seen also in Fig. 5 (red), where we used that static model generated exactly from equations (1) and (2). Also, the research community put a lot of effort into modeling friction, so we try to overcome that issue by using a learning algorithm.

3) Why only static values fittings? Do not you need to include motors dynamics?, do not they influence the responses?. 

We completely agree that the motor dynamics will influence the whole actuator response, but with VSAs (QB move pro), we do not have the possibility to control the motor torque, and we wanted to exploit low-level controllers provided by the manufacturer. Still, the research can be improved by incorporating a time-series dataset for learning time/dynamics-dependent relations. We added this in the limitation and future work part in the conclusion section (lines 398 -407).

4) Your experiments to carry out the iterations take long time. Since you are fitting local zones, why not to use a standard least squares method that would much more fast in providing an accurate local linear model?

The advantage of the LWPR algorithm lies in local linear models and its weights that the LWPR algorithm fits incrementally. Also, the total output of the LWPR model is generated as a prediction of each local linear model normalized by its weight (3), which is not the case when using the standard least squares method. We added a more detailed explanation of how the LWPR algorithm works from lines 178 – 188.

5) Lines 185-186: the fitting obtained in Fig.5 after 4 iterations does not seem to be very good, particularly in joint stiffness. Then the assertion of "successful learning" is questionable.

From the perspective of accurately following commanded stiffness reference, we agree that the relearning process can appear to be questionable. However, this iterative learning process should give users the intuition that actuator parameters can be relearned. Also, in our simulation, the spring parameters were changed instantly from one value to another, which will not be the case in real scenarios where the degradation of parameters will occur gradually. For clarification we added: Due to model uncertainties introduced by drastic parameter changes (not likely to happen in real-life scenarios, where parameter degradation occurs gradually), new measurements are introduced in the learning set. (lines 235 – 238)

Reviewer 4 Report

Comments and Suggestions for Authors

The paper "Cartesian Stiffness Shaping of Complaint Robots– Incremental Learning and Optimization Based on Sequential Quadratic Programming" by Kneževi´ et al introduces a new simulation strategy, which referred as "Locally Weighted Projection Regression (LWPR)", that can capture the characteristics of variable stiffness actuators of compliant robot, improving the efficiency and accuracy of the simulation on indicating robotic behaviors including positioning and stiffness control. This new methodology is validated by a 4 DoF planar robot experimentally. In general, this article is in good shape by focusing on an important aspect for soft robot applications, providing solutions, and validating with experimental evidence. However, the authors are still recommended to address the following points.

1. In the Introduction section. The author gave thorough backgrounds on the current limitations on simulation compliant robots with different methodologies that have drawbacks on different scenarios. However, the author should also provide a "big" picture of "why having accurate simulation on compliant robot is necessary". What consequences will people experience when they do not have very accurate modeling? What specific applications would be made from impossible to possible? This information is very necessary for audience that without the robotic backgrounds.

2. The authors need to check their English again, both improving the grammar as well as wording. There are mistakes such as "complaint robot" that go through the entire manuscript. Please check and fix.

Comments on the Quality of English Language

Moderate English edits are necessary

Author Response

Dear Reviewer, we are thankful for your comments and positive feedback on our research presented in the submitted manuscript.

The paper "Cartesian Stiffness Shaping of Complaint Robots– Incremental Learning and Optimization Based on Sequential Quadratic Programming" by Kneževi´ et al introduces a new simulation strategy, which referred as "Locally Weighted Projection Regression (LWPR)", that can capture the characteristics of variable stiffness actuators of compliant robot, improving the efficiency and accuracy of the simulation on indicating robotic behaviors including positioning and stiffness control. This new methodology is validated by a 4 DoF planar robot experimentally. In general, this article is in good shape by focusing on an important aspect for soft robot applications, providing solutions, and validating with experimental evidence. However, the authors are still recommended to address the following points.

1. In the Introduction section. The author gave thorough backgrounds on the current limitations on simulation compliant robots with different methodologies that have drawbacks on different scenarios. However, the author should also provide a "big" picture of "why having accurate simulation on compliant robot is necessary". What consequences will people experience when they do not have very accurate modeling? What specific applications would be made from impossible to possible? This information is very necessary for audience that without the robotic backgrounds.

Thank you for pointing out that some aspects of the Introduction section were not covered adequately. To provide a wider picture of the topic, we added a part about the applications that use compliant robot technologies, showcasing applications where compliant robots can impact task performance (lines 112 – 126). Furthermore, we introduce a step-by-step pipeline to present the overview of the proposed methodology (lines 143 – 148). Additionally, to highlight the importance of accurate modeling, which is the crucial component of every control task, in Section 2 and Section 3, we further explained the proposed algorithm (lines 178 – 188, and lines 242 – 254).

2. The authors need to check their English again, both improving the grammar as well as wording. There are mistakes such as "complaint robot" that go through the entire manuscript. Please check and fix.

Thank you for suggesting and pointing out the typos, we carefully went through the manuscript and revised all of it. We have also enlisted the services of a professional English proofreader to review and refine the language in the manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The paper undertook a major revision. It is possible to navigate the overall text and see several improvements were made. The proposal is now clarified and contributions are well defined. The reviewer's recommendation is to approve the paper in its present form. 

Reviewer 3 Report

Comments and Suggestions for Authors

My questions have been adequately addressed. Then the manuscript can be published.