Dear Dr. Mehdizadeh,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

While the reviewers were generally positive about the study, there are some points where additional discussion are needed. The NN section in particular needs to be strengthened and the reviewers give some suggestions about how to do so.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Partly

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2. Has the statistical analysis been performed appropriately and rigorously? -->?>

Reviewer #1: Yes

Reviewer #2: N/A

Reviewer #3: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available??>

The PLOS Data policy

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English??>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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Reviewer #1: PONE-D-25-37501 Review

Thank you for the opportunity to review this interesting paper.

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Here are some comments based on PLOS ONE criteria.

1. Original Research

The manuscript presents new modeling and control frameworks for gait adaptation, analyzes the limits of standard actuation methods, proposes and tests a novel two-stage hip actuation policy, and evaluates a neural network-based controller. I believe this is original research and I am not aware of papers dealing with the same analysis.

2. Results Not Published Elsewhere

A version of this paper is available as a preprint on bioRxiv, but as far as I know, it is not published in a scientific journal.

3. Technical Quality and Detail

The study includes mathematical modeling, stability and controllability analyses, and simulations. Model details, optimization routines, mathematical formulations, and simulation conditions are presented in most parts (see below) clearly and in detail (e.g., equations, algorithms, parameter values, methods for generating figures and tables).

The code used for the analysis is made publicly available at:

GitHub - SFULocomotionLab/biped_models.

This is very much appreciated.

4. Conclusions Supported by Data

The primary conclusions [namely that standard control is insufficient for robust gait adaptation, and that adding a second hip spring with an independent stiffness to actuate the second half of the swing phase is a way to solve the problem] are backed by computational experiments.

5. Intelligible English

The manuscript is well-written in academic English. The abstract, author summary, introduction, discussion, methods, and references are clearly organized.

6. Ethical and Integrity Standards

This is a computational and modeling study, so human or animal ethics approval is not required, I believe. Competing interests and funding sources are fully disclosed.

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Here are some comments about the content of the manuscript.

Model.

I feel that some analytical work on your model 1a,1b,1c can be done, instead of relying only on numerics. At the end, the angle swept during stance should be relatively easy to compute especially under some parameter regimes and this should be enough to determine, at least, an approximate apex return map for the model. See for example what is done in these papers:

Geyer H, Seyfarth A, Blickhan R. Spring-mass running: simple approximate solution and application to gait stability. J Theor Biol. 2005 Feb 7;232(3):315-28. doi: 10.1016/j.jtbi.2004.08.015. PMID: 15572057.

Selvitella AM, Foster KL. An approximate solution of the SLIP model under the regime of linear angular dynamics during stance and the stability of symmetric periodic running gaits. J Theor Biol. 2024 Dec 7;595:111934. doi: 10.1016/j.jtbi.2024.111934. Epub 2024 Sep 4. PMID: 39241821.

Dhawale N, Mandre S, Venkadesan M. 2019 Dynamics and stability of running on rough terrains. R. Soc. open sci. 6: 181729. http://dx.doi.org/10.1098/rsos.181729

These are simpler or different models, but maybe at least a comment on why finding analytical or approximate solutions is a further challenge wrt what you are addressing in the paper would be useful. It feels to me this might also help your struggles in finding limit cycles numerically, as at least you can guess where parameters for stability are more easily.

Stability.

Your result in Figure 5 is not surprising, but there is something I am confused about. Depending on the controller, you should be able to cover all the parameter space with stable regions, or not? I think a comment on this early on would make your Figure 5 experiment a little more meaningful, otherwise I do not completely understand the use of it.

Feedback Policy.

Why as an evaluation method you used the root mean square error between the tracked path and the target gait only and did not include the RMSE also of the velocity curves?

Neural Network Controller.

I do not understand this section. I think you jumped from your controller to a feed forward neural network with a not too deep, arbitrary architecture, while there are many other options to compare your controller with other NN ones. It just seems unmotivated and there does not seem to me that there are strong reasons for which this was chosen.

The way the data is split into training and validation and testing is quite obscure. Of the 90% what percentage is training, what is validation? How was the split determined? Also the number of epochs seems arbitrary and in situation like this, where the loss function is very non convex, I would not be to confident that if you go from 200 epochs to 2000 epochs, the algorithm that seemed to converge to a specific gait does not end up visiting another gait.

I understand it is tempting to use a NN for this, but there are some cons to consider. First of all, whatever architecture you choose, there is not really a good motivation to support it. I know you mention biologically plausibility, but the network you are using is not more plausible than any others in practice and not even more plausible of more simple models. The fact that you save parameters in your architecture, well it’s because you chose that architecture and that policy, but, then, convincingly explaining that a randomly chosen NN architecture with smaller size saves parameters and matches your policy in terms of efficiency is cumbersome and this fact is quite a negative message about your policy.

The training-validation-test split is quite puzzling no matter what. I know cross-validation might be computationally expensive, but at least you might want to do a second split with the same percentages and similar other two splits with a different percentage in the split (start with one random seed for the first percentage split; then change the seed and repeat). The fact is that CV itself is not theoretically well understood in the iid case and here is so much more complicated that you risk that the true confidence bands of your estimates are enormous.

At the end, although you are doing your best to be rigorous, you might end up adding a section that damages the paper. It might sound extreme, but there are reasons that make me recommend removing this section (anything NN related) in full.

Some of your conclusions.

Isn’t it already known that open-loop stability is insufficient for gait adaptation? I mean it is already something rare without adaptation involved I would say. I think there needs to be a stronger or more precise conclusion on this, otherwise you might probably benefit from removing this paragraph.

When you add further parameters, you surely increase stability. You made a choice that works, but also this does not seem surprising to me. I think a stronger motivation of why your way is a better way to proceed is definitely needed. What I am trying to say in different words is that you are comparing no-control with control and of course this increases the stability region, basically no matter what type of control you use. Am I confused? If not, I think the motivation and the explanations of why your results are a step forward need to be stronger. And on top of this, there is the issue above of a randomly chosen NN with less parameters performing as efficiently as your policy.

Reviewer #2: This study investigates the controllability of a simple walking model with one or two simple controllers. It argues that while open-loop control cannot achieve human-like step length and step frequency, it is possible to design a system by adjusting the control inputs for each step. The study comprehensively investigates control performance using the model, and the theoretical development is carefully presented. Overall, the paper is well organized. However, while reading the paper, I encountered a few points that were unclear.

One of the main claims of this study is that open-loop control cannot control the entire range of human step length and step frequencies. I agree that this claim is assumed to be correct. The issue is whether this claim is fully supported by the results. I understand that the method used to demonstrate this is to numerically calculate the model's behavior to generate a limit cycle and then analyze the Poincare map around that limit cycle. This limit cycle seems to depend on the initial values, where the paper states that the initial values have been explored through trial and error (lines 152-153). My question is whether this trial-and-error exploration has any effect on the results. Personally, I think that it is probably not a problem in terms of the overall trend, but I have doubts about whether it is possible to make a statement as in line 469, "we showed that ... cannot fully control the ...". If necessary, I would like to see some discussion on whether there is any dependence on the initial values.

The second point concerns the evaluation method for the control system. In "Performance of the feedback policy" in the "Method" section, optimization is used to determine the response time. I found it difficult to understand how this was decided, especially the evaluation function and decision coefficient in lines 258-261, which seemed to appear suddenly. I felt that a more detailed explanation was needed for this part.

Minor comments:

* Line 166: It would be helpful to explain how the disturbance magnitude of 1e-7 was determined. If possible, it would be helpful to know how much influence this 1e-7 has on the motion.

* An explanation of the meaning of x in equation (4) would be desirable. I assume that this x is the same as the x in equation (3), but this was not clear when I first read the manuscript.

* Line 629, Reference 17: The year of publication is missing.

Reviewer #3: The authors investigate simple walking controllers which are sufficient for adapting to various step lengths and frequencies. Their methodology involved: generating a grid of limit-cycle gaits, designing feedback policies for these gaits, and testing two interpolation methods; linear and nonlinear. I find this work methodologically novel and interesting as well as rigorous. I do have some suggestions to further improve the rigor and clarity of the findings below:

One notable and surprising finding is the necessity of a third actuation, namely a second torque during the latter half of swing, to achieve control across the full gait pattern space. The paper concludes that a single hip spring and a single push-off are insufficient for such controllability. They show that adding a second hip spring with independent stiffness solves this problem and explains aspects of the swing phase hip muscle activation patterns. Especially because this is an interesting finding, it would be good to test or at least consider alternative explanations or controllers other than a second hip torque: have the authors tested a single spring and damper instead? Can they propose other controllers that may achieve similar stability?

The study also provides an interesting comparison between the two interpolation methods. The neural network based interpolation successfully approximates the control policy but is inferior to the simpler linear interpolation policy in the challenging region of high step lengths. In this region, the neural network either required more steps to complete a path or failed entirely, while the interpolation policy succeeded. This result, while interesting, could be discussed more explicitly and in detail in terms of its biological significance. The authors make an intriguing connection to the findings in Bertram and Ruina, stating “people report difficulty…” at these challenging combinations. Can the authors elaborate what is the nature of this difficulty and how does it relate to the neural network’s difficulties in interpolation?

The authors could discuss more prominently how their assumption of a deadbeat and linear controller might affect their findings and conclusions. What if the controller was instead optimized for some other cost function or was a nonlinear feedback control law?

Overall, the paper's writing could be more direct. I recommend simplifying complex sentences and using active voice to improve the clarity of the writing. The logical flow of the paragraphs can also be improved.

The methodology for evaluating the feedback policy's tracking performance could be more clearly articulated. While the paper mentions that the walker "starts from the initial gait" to track a path, it does not specify how this initial gait is chosen or if it is varied across different trials. The policy's performance could be dependent on the initial conditions, so clarifying this aspect would strengthen the analysis.

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what does this mean? ). If published, this will include your full peer review and any attached files.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Controlling a simple model of bipedal walking to adapt to a wide range of target step lengths and step frequencies

PONE-D-25-37501R1

Dear Dr. Mehdizadeh,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Anne E. Martin

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions??>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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3. Has the statistical analysis been performed appropriately and rigorously? -->?>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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4. Have the authors made all data underlying the findings in their manuscript fully available??>

The PLOS Data policy

Reviewer #1: (No Response)

Reviewer #2: Yes

Reviewer #3: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English??>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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Reviewer #1: The authors have addressed all the main concerns I had. In particular, I appreciate the explanations about the role of the control used and the decision of removing the neural network component of the work. Thank you for the opportunity to review this manuscript.

Reviewer #2: I am grateful to the authors for their sincere response to the items I commented on. I have no further comments to add.

Reviewer #3: The authors have adequately addressed my comments. I still believe this work is interesting and insightful. look forward to seeing this work in print.

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what does this mean? ). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy

Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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