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Peer-Review Record

Gait Stability Under Hip Exoskeleton Assistance: A Phase-Dependent Analysis Using Gait Tube Methodology

Appl. Sci. 2025, 15(13), 7530; https://doi.org/10.3390/app15137530

by Arash Mohammadzadeh Gonabadi^1,2,*

and Farahnaz Fallahtafti²

Reviewer 1: Anonymous

Reviewer 2: Anonymous

Reviewer 3: Anonymous

Appl. Sci. 2025, 15(13), 7530; https://doi.org/10.3390/app15137530

Submission received: 29 May 2025 / Revised: 30 June 2025 / Accepted: 1 July 2025 / Published: 4 July 2025

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The introduction is well-grounded from a theoretical point of view, and the study also presents a novelty factor: the first application of gait tube analysis for evaluating gait stability under robotic exoskeleton assistance.

The authors have outlined the hypothesis in the introductory part; however, the study also requires a precisely formulated aim, which must also be stated in the abstract.
After presenting the purpose of the study, the authors should highlight the novelty of the study more clearly, as mentioned in the first paragraph of the report.
Please specify exactly what were the inclusion and exclusion criteria.
It is recommended that the calculation of sample size, statistical power or both be incorporated in the paper.
The recommendation is to specify that the study has complied with the Declaration of Helsinki (DoH)—Ethical Principles for Medical Research Involving Human Participants (1964) and its latest amendments adopted by the 75th General Assembly of the World Medical Association (WMA) in Finland on October 19, 2024.
Given the journal's prestige, please remove or replace the following outdated bibliographical sources from the study: 20, 25, 29, 34, 37, and 45.

Author Response

Dear Editor and Reviewers,

We appreciate your time and effort in evaluating our manuscript, Gait Stability Under Hip Exoskeleton Assistance: A Phase-Dependent Analysis Using Gait Tube Methodology. In response to your feedback, we have edited key sections of the manuscript using Word’s track changes mode. A summary of our edits (in blue font color) is provided on the following pages, and we believe the content and clarity of the manuscript have been further strengthened.

As always, we are grateful for the opportunity to refine the content of our manuscript.

Sincerely,

Arash Mohammadzadeh Gonabadi

Farahnaz Fallahtafti

Editor

Comments and Suggestions for Authors

EQ1 - Ensure all references are relevant to the content of the manuscript.

Thank you for the helpful suggestion. We carefully reviewed all references cited in the manuscript to ensure their relevance to the content. A few references that were either outdated or loosely related to the core topics were removed. We also updated several citations to support key concepts and recent findings.

EQ2 - Highlight any revisions to the manuscript, so editors and reviewers can see any changes made.

Thank you for the reminder. All revisions to the manuscript have been marked using track changes in the submitted Word document to allow editors and reviewers to identify the modifications easily. This includes additions, deletions, and updated content responding to reviewer and editor comments. Please let us know if any further formatting or highlighting is needed.

EQ3 - Provide a cover letter to respond to the reviewers’ comments and explain, point by point, the details of the manuscript revisions.

We have provided a detailed cover letter that responds point by point to all reviewer comments. This document outlines each revision made to the manuscript and specifies where changes were applied to address the reviewers’ suggestions.

EQ4 - If the reviewer(s) recommended references, critically analyze them to ensure that their inclusion would enhance your manuscript. If you believe these references are unnecessary, you should not include them.

Thank you for this guidance. One of the reviewer-suggested references, “A new method proposed for realizing human gait pattern recognition: Inspirations for the application of sports and clinical gait analysis” (Zhang et al., 2023), was reviewed and found to be highly relevant to our topic. Although its focus is on gait pattern classification rather than stability quantification, it aligns with our methodological innovation and highlights current directions in advanced gait analysis. We therefore included this reference in the Introduction to enrich the context and support the novelty of our proposed gait tube approach. Its inclusion helps bridge traditional and data-driven methodologies in the field.

EQ5 - If you found it impossible to address certain comments in the review reports, include an explanation in your appeal.

We have thoroughly addressed all reviewer comments and suggestions. In addition to our responses, we conducted further analyses based on the reviewers’ recommendations and integrated the results into the manuscript. These additions have helped improve the paper's clarity, depth, and overall quality.

Reviewer 1

Comments and Suggestions for Authors

We sincerely thank the reviewer for this encouraging and positive feedback. We appreciate your recognition of the theoretical foundation of the introduction and the novelty of our proposed approach using gait tube analysis under exoskeleton assistance. This motivation is central to our study, and we are grateful that it was clearly conveyed.

R1Q1 - The authors have outlined the hypothesis in the introductory part; however, the study also requires a precisely formulated aim, which must also be stated in the abstract.

We appreciate the reviewer’s suggestion regarding the importance of clearly articulating the aim of the study. In response, we have revised both the Abstract and the Introduction to explicitly state the study’s aim. This addition helps clarify the study’s objective and distinguishes it from the hypotheses.

Added text to the Abstract:

“This study aimed to evaluate how wearable hip exoskeleton assistance affects phase-dependent gait stability in healthy adults using a novel visualization technique known as gait tube analysis. Hip exoskeletons offer significant potential to enhance locomotion by augmenting joint torque, yet their effects on stability throughout the gait cycle remain underexplored.”

Added text to the Introduction:

“Therefore, the aim of this study was to investigate the phase-dependent effects of hip exoskeleton assistance on center of mass variability during walking, using gait tube analysis to quantify multidimensional stability throughout the gait cycle.”

R1Q2 - After presenting the purpose of the study, the authors should highlight the novelty of the study more clearly, as mentioned in the first paragraph of the report.

Thank you for this insightful recommendation. In response, we have revised the Introduction to explicitly emphasize the novelty of our approach, specifically, the application of gait tube analysis to assess stability under assistive exoskeleton conditions. We have added clarifying text immediately after stating the study’s aim to highlight the methodological innovation and its distinction from existing stability assessment methods.

Added text to the Introduction:

“This approach represents a novel application of gait tube methodology to human-exoskeleton interaction. Unlike traditional gait stability metrics that rely on discrete gait events or stride-averaged statistics, gait tube analysis provides a continuous, multidimensional visualization of center of mass variability. To our knowledge, this is the first study to apply this method to evaluate the destabilizing or stabilizing effects of wearable hip assistance throughout the gait cycle.”

R1Q3 - Please specify exactly what were the inclusion and exclusion criteria.

Thank you for your valuable suggestion. We have now added explicit statements regarding the inclusion and exclusion criteria used in participant recruitment. These details were derived from the original experimental protocol as described in our previously published study [1] and are now included in the revised Materials and Methods section of the manuscript.

Added text to the Materials and Methods section:

“Participants were included if they were between 18 and 40 years of age, had no known neuromuscular or musculoskeletal impairments, and were capable of walking unaided for prolonged periods. Exclusion criteria included any recent lower limb injury, neurological disorder, cardiovascular condition, or prior use of wearable robotic devices that could interfere with gait mechanics. These criteria ensured the safety of participants and the consistency of gait measurements under exoskeleton conditions, as detailed in the original data collection protocol [1].”

R1Q4 - It is recommended that the calculation of sample size, statistical power or both be incorporated in the paper.

We appreciate the reviewer’s great suggestion regarding sample size and statistical power analysis. As noted (in R1Q3), the current study is a secondary analysis based on data collected and published in a prior study focused on the biomechanics and metabolic cost of hip exoskeleton assistance [1]. Since no additional participants were recruited, a prospective power analysis was not feasible. However, we acknowledge this limitation and have added a clarification to the Limitations section of the manuscript.

Added text to the Limitations section:

“Additionally, the study utilized existing data from a previously published protocol, and therefore, no a priori power analysis was conducted. While our sample size (n = 10) is consistent with similar biomechanical studies, it may limit the generalizability and statistical power to detect small effect sizes. Future research should include prospective power analyses to optimize sample size and improve the robustness of statistical comparisons.”

R1Q5 - The recommendation is to specify that the study has complied with the Declaration of Helsinki (DoH)—Ethical Principles for Medical Research Involving Human Participants (1964) and its latest amendments adopted by the 75th General Assembly of the World Medical Association (WMA) in Finland on October 19, 2024.

Thank you for pointing this out. We have revised the manuscript to explicitly state that the study complied with the Declaration of Helsinki and its most recent amendments. This has now been added to the Ethics Statement within the Methods section.

Added text to the Methods section:

“All procedures involving human participants were conducted in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Declaration of Helsinki and its later amendments adopted by the 75th World Medical Association General Assembly (October 19, 2024, Helsinki, Finland).”

R1Q6 - Given the journal's prestige, please remove or replace the following outdated bibliographical sources from the study: 20, 25, 29, 34, 37, and 45.

We thank the reviewer for this constructive and insightful comment. In response, we have replaced Reference 20 with a more comprehensive and up-to-date tutorial on entropy measures:

“Delgado-Bonal, A.; Marshak, A. Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy 2019, 21, 541. https://www.mdpi.com/1099-4300/21/6/541”

Unfortunately, for References 25, 29, 34, 37, and 45, after an extensive literature search, we were unable to identify more recent publications that address the specific concepts or methods applied in our analysis with equivalent clarity or relevance. These sources were therefore retained, as they continue to provide foundational knowledge directly applicable to the context and aims of our study. We have carefully ensured that all references are accurate, technically sound, and appropriately support the scientific content of the manuscript.

We respectfully hope that the retention of these references is acceptable given their relevance and the lack of suitable modern replacements.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,

Thank you for inviting me to review the manuscript entitled "Gait Stability Under Hip Exoskeleton Assistance: A Phase-Dependent Analysis Using Gait Tube Methodology." This study introduces gait tube analysis as a novel method for visualizing center of mass velocity trajectories to quantify phase-dependent gait stability under hip exoskeleton assistance. The main findings revealed that powered conditions significantly increased vertical variability during early-to-mid stance, with strong correlations between ellipsoid volume and total variability validating the method's robustness.

Overall, this manuscript presents innovative methodology and addresses an important topic in wearable robotics and gait analysis. The gait tube analysis approach offers a unique perspective on phase-dependent stability assessment. However, there are several areas that require attention to improve the clarity, methodological rigor, and impact of the manuscript.

General Comments

Introduction: The introduction provides comprehensive background but could benefit from clearer justification for the novel gait tube methodology and more explicit hypotheses regarding expected phase-specific effects.
Methods: The methodology is generally well-described, but some aspects require clarification, particularly regarding the gait tube analysis implementation and statistical approach for phase-dependent comparisons.
Results: The results are well-presented with appropriate visualizations, but the manuscript would benefit from more detailed interpretation of the phase-specific findings and their biomechanical significance.
Discussion: The discussion effectively contextualizes findings but could be strengthened by addressing the clinical implications more thoroughly and acknowledging limitations more comprehensively.

Specific Comments

Introduction

Page 1, lines 37-40: The statement about hip extension accounting for "up to 45% of mechanical power" needs a more recent and specific citation, as the current reference [4,5] may not directly support this specific claim.
Page 2, lines 53-64: The critique of MOS interpretation in the AP direction is well-articulated, but consider providing more balanced discussion of when MOS might still be valuable despite these limitations.
Page 2, lines 90-99: The introduction of gait tube analysis is somewhat abrupt. Consider providing more theoretical foundation for why this approach would be superior to existing methods before introducing the technique.
Page 3, lines 106-115: The hypotheses could be more specific about expected effect sizes and the physiological mechanisms underlying the predicted phase-specific patterns.

Methods

Page 3, lines 118-125: More details about the exclusion criteria and data quality assessment would strengthen the methodology. How were "clean strides" defined beyond the minimum of five strides?
Page 4, lines 153-164: The gait tube analysis description needs more technical detail. How were the 3×3 covariance matrices computed, and what was the rationale for pooling data across participants rather than computing individual matrices?
Page 4, lines 162-164: The reference to supplementary MATLAB code is helpful, but key algorithmic details should be included in the main text for reproducibility.
Page 4, lines 179-201: The statistical analysis section lacks clarity on multiple comparison corrections. With 100 gait cycle points being compared, how was the family-wise error rate controlled?
Page 4, lines 183-186: The use of a five-point sliding window for smoothing should be justified. How was this window size chosen, and how might it affect the temporal resolution of the analysis?

Results

Page 5, lines 202-208: The description of "more dispersed trajectories" in powered conditions could be quantified more precisely. Consider providing specific metrics for this dispersion.
Page 5-6, lines 219-230: The phase-specific ellipsoid volume findings are interesting, but the biomechanical interpretation of why variability peaks occur at 10-50% rather than the expected 50-60% needs more detailed explanation.
Page 6, lines 239-250: The directional analysis results are valuable, but the finding that only VT was consistently elevated warrants more discussion about the biomechanical implications of vertical COM control during exoskeleton assistance.
Page 7, Table 1: The metabolic cost data provides important context, but the relationships between stability metrics and energy expenditure could be explored more quantitatively through correlation analysis.

Discussion

Page 11, lines 329-343: The discussion appropriately addresses the unexpected timing of peak variability, but could benefit from more detailed exploration of the biomechanical mechanisms underlying these phase-specific effects.
Page 11, lines 344-348: The comparison with traditional stability metrics is valuable, but consider discussing the computational and practical advantages/disadvantages of gait tube analysis relative to these established methods.
Page 12, lines 384-402: The discussion of the stability-efficiency trade-off is insightful, but would benefit from more specific recommendations for exoskeleton control strategies based on these findings.
Page 13, lines 403-451: The extensive comparison with other stability metrics is comprehensive but somewhat overwhelming. Consider condensing this section and focusing on the most relevant comparisons.
Page 14, lines 463-486: The limitations section appropriately acknowledges study constraints, but should also discuss the generalizability of gait tube analysis to other populations and conditions.

Conclusion

Page 14, lines 487-500: The conclusion effectively summarizes findings but could be strengthened by providing more specific recommendations for future exoskeleton design and more explicit statements about the clinical significance of the phase-dependent stability findings.

This manuscript presents novel and potentially impactful methodology for assessing gait stability under assistive device conditions. The gait tube analysis approach offers unique insights into phase-dependent stability that could inform exoskeleton design and control strategies. The experimental design is sound, and the findings contribute meaningfully to the field of wearable robotics and gait analysis.

However, the manuscript would benefit from clearer presentation of the methodological details, more focused discussion of the most relevant findings, and stronger emphasis on the clinical and practical implications of the phase-dependent stability patterns observed. I recommend minor revisions for this manuscript.

Best regards,

The Reviewer

Author Response

Dear Editor and Reviewers,

As always, we are grateful for the opportunity to refine the content of our manuscript.

Sincerely,

Arash Mohammadzadeh Gonabadi

Farahnaz Fallahtafti

Reviewer 2

Comments and Suggestions for Authors

We sincerely thank the reviewer for their thoughtful summary and positive assessment of our work. We are pleased that you recognized the novelty of the gait tube analysis and its potential for providing insight into phase-dependent gait stability under exoskeleton assistance. Your recognition of the method's robustness and the significance of our main findings is highly appreciated. We hope our revisions further strengthen the clarity and impact of the manuscript.

R2Q1 - Introduction: The introduction provides comprehensive background but could benefit from clearer justification for the novel gait tube methodology and more explicit hypotheses regarding expected phase-specific effects.

Thank you for this helpful suggestion. We note that your comment aligns closely with the points raised in Reviewer 1’s comments R1Q1 and R1Q2. In response to those comments, we revised the final paragraphs of the Introduction to explicitly articulate the aim of the study, clarify the novelty of applying gait tube analysis in this context, and present our hypotheses regarding expected phase-specific effects, particularly increased gait variability during late stance under powered assistance. We believe these revisions now address your comment as well, and we appreciate your contribution to improving the clarity and rigor of the manuscript.

Added text to the Introduction:

And,

R2Q2 - Methods: The methodology is generally well-described, but some aspects require clarification, particularly regarding the gait tube analysis implementation and statistical approach for phase-dependent comparisons.

Thank you for your thoughtful feedback. In response, we have revised the Methods section to further clarify how the gait tube analysis was implemented and how phase-dependent comparisons were conducted statistically. Specifically, we expanded the explanation of ellipsoid volume computation, added details on the 3×3 covariance matrix, smoothing window, and projection process, and elaborated on how Wilcoxon signed-rank tests were applied at each phase point. We also clarified the correlation analysis between ellipsoid volume and total variability. These additions enhance the transparency and reproducibility of our methodology.

Added text to the Section: 2.2. Gait Tube Analysis:

“Specifically, for each normalized gait phase (0–100%), we computed a 3×3 covariance matrix of COM velocity (Vx, Vy, Vz) across all valid strides. The eigenvalues and eigenvectors of this matrix were used to define ellipsoids representing local variability. These ellipsoids were then projected onto the plane perpendicular to the mean trajectory using the Frenet–Serret frame to preserve the directional context of variability [2]. A five-point moving window was applied to smooth temporal fluctuations in ellipsoid volume, improving signal clarity while maintaining phase resolution. This process yielded a temporally continuous map of variability that forms the gait tube structure.”

Added text to the Section: 2.3. Statistical Analyses:

“Phase-dependent comparisons were performed at each of the 100 normalized points across the gait cycle. For each participant and condition, ellipsoid volumes were smoothed using a five-point moving window, and differences from the NoExo condition were assessed at each time point using paired Wilcoxon tests. The minimum p-value across the cycle was recorded for each condition to summarize statistically significant deviations. To reduce the risk of false positives, we interpreted significant effects only if they persisted across multiple contiguous gait phases.”

R2Q3 - Results: The results are well-presented with appropriate visualizations, but the manuscript would benefit from more detailed interpretation of the phase-specific findings and their biomechanical significance.

We thank the reviewer for this valuable suggestion. In response, we have expanded the Discussion section to provide a more detailed interpretation of the phase-specific findings. In particular, we discuss how increases in variability during early and mid-stance phases may relate to biomechanical demands such as weight acceptance and single-limb support transitions. We also highlight the biomechanical relevance of increased vertical variability and its implications for dynamic balance under exoskeleton assistance. These additions enhance the physiological context and translational value of our findings.

Added text to the Discussion:

“These early-to-mid stance increases in variability may reflect biomechanical challenges during weight acceptance and transition from double to single limb support. During these phases, the body's center of mass rapidly shifts over the stance limb while absorbing impact forces and maintaining forward progression. The presence of assistive torque from the hip exoskeleton may disrupt the natural timing and coordination of joint moments, thereby amplifying vertical excursions and postural adjustments. The observed increase in VT, particularly in conditions with late or high-magnitude torque, suggests a potential destabilizing effect on vertical balance control mechanisms, which are critical for trunk and head stabilization. These findings indicate that early stance is a sensitive period for stability disruptions under powered assistance and should be considered when designing exoskeleton controllers that adapt torque delivery in a phase-specific manner.”

R2Q4 - Discussion: The discussion effectively contextualizes findings but could be strengthened by addressing the clinical implications more thoroughly and acknowledging limitations more comprehensively.

We thank the reviewer for this excellent suggestion. In response, we have revised the Discussion section to elaborate on the clinical implications of our findings. Specifically, we discuss how phase-specific gait variability patterns could inform the development of adaptive exoskeleton controllers for populations at higher fall risk, such as individuals with stroke or Parkinson’s disease. Additionally, we expanded the Limitations paragraph to address factors such as the short adaptation period, treadmill use, and the generalizability of results beyond healthy adults. These changes enhance the translational relevance and transparency of the study.

Added text to the Discussion:

“The phase-specific variability patterns identified in this study have important clinical implications. Individuals with impaired balance, such as stroke survivors or people with Parkinson’s disease, may be more vulnerable to destabilizing effects during early stance when vertical variability is elevated. By integrating gait tube metrics into real-time control strategies, exoskeletons could be designed to modulate torque in a stability-aware manner, enhancing support during phases prone to instability while minimizing unnecessary assistance elsewhere. This approach could improve safety and confidence during walking in clinical populations, potentially reducing fall risk and supporting more personalized rehabilitation protocols.”

And:

“Furthermore, all participants were young, healthy adults, and their responses may not reflect the behavior of clinical populations with impaired neuromotor control or balance. Future studies should validate these findings in populations with gait impairments to better understand how assistive torque affects pathological movement patterns and fall risk.”

R2Q5 - Page 1, lines 37-40: The statement about hip extension accounting for "up to 45% of mechanical power" needs a more recent and specific citation, as the current reference [4,5] may not directly support this specific claim.

We thank the reviewer for catching this important detail. In response, we have added more recent and specific supporting references. These include studies addressing muscle energy expenditure models, metabolic cost estimation, and variability analysis in gait biomechanics. We believe the updated citations now more accurately support the statement and strengthen the context of hip joint contributions during walking.

New References to Add:

“Gonabadi, A.M.; Antonellis, P.; Malcolm, P. Differentiating Fallers from Nonfallers Using Nonlinear Variability Analyses of Data from a Low-Cost Portable Footswitch Device: A Feasibility Study. Acta Bioeng. Biomech. 2021, 23, 139–145. https://www.actabio.pwr.wroc.pl/Vol23No4/23.pdf
Arones, M.M.; Shourijeh, M.S.; Patten, C.; Fregly, B.J. Musculoskeletal Model Personalization Affects Metabolic Cost Estimates for Walking. Bioeng. Biotechnol. 2020, 8, 588925. https://doi.org/10.3389/fbioe.2020.588925
Umberger, B.R.; Gerritsen, K.G.M.; Martin, P.E. A Model of Human Muscle Energy Expenditure. Methods Biomech. Biomed. Engin. 2003, 6, 99–111. https://doi.org/10.1080/1025584031000091678”

R2Q6 - Page 2, lines 53-64: The critique of MOS interpretation in the AP direction is well-articulated, but consider providing more balanced discussion of when MOS might still be valuable despite these limitations.

Thank you for this constructive suggestion. We have revised the paragraph to preserve the original critique while adding a more balanced discussion of the contexts in which MOS remains valuable. Specifically, we acknowledge its biomechanical grounding, interpretability during discrete events, and ongoing relevance in clinical and perturbation-based studies. These additions clarify that MOS and gait tube analysis can serve complementary roles in gait research.

Added text to the Introduction:

“Nonetheless, MOS remains a widely used and biomechanically grounded metric that provides valuable insights into stability at discrete gait events, particularly heel strike [3,4]. It has been applied successfully in perturbation studies, clinical assessments, and investigations of fall risk where event-specific deviations are informative. Rather than replacing MOS, our approach offers a complementary tool that expands stability analysis to a continuous, multidimensional framework across the entire gait cycle [3,4].”

R2Q7 - Page 2, lines 90-99: The introduction of gait tube analysis is somewhat abrupt. Consider providing more theoretical foundation for why this approach would be superior to existing methods before introducing the technique.

We appreciate the reviewer’s observation. In response, we have revised the Introduction to include a stronger theoretical foundation before introducing gait tube analysis. The added text explains the limitations of existing event-based and unidimensional approaches and motivates the need for a continuous, multidimensional method. This addition provides a more logical transition into the gait tube methodology and highlights its relevance for analyzing phase-specific effects of assistive devices.

Added text to the Introduction:

“While traditional metrics such as MoS, Lyapunov exponents, or entropy provide valuable insights, they are typically constrained to single dimensions or isolated events within the gait cycle. These approaches may overlook how stability evolves continuously in three dimensions, particularly in response to dynamic inputs like assistive torque. Moreover, unidimensional metrics may not fully capture the coupled variability between directions (e.g., vertical and mediolateral sway), which can be critical for maintaining balance. A method capable of continuously characterizing gait variability in all three directions throughout the stride would allow for richer interpretation of stability mechanisms and a better understanding of how assistive devices influence balance across different gait phases.”

R2Q8 - Page 3, lines 106-115: The hypotheses could be more specific about expected effect sizes and the physiological mechanisms underlying the predicted phase-specific patterns.

We thank the reviewer for this valuable suggestion. In response, we have revised the Introduction to elaborate on our hypotheses by specifying the expected direction of effects (e.g., increased variability in vertical direction) and linking these predictions to physiological mechanisms such as the influence of assistive torque on limb loading, trunk control, and push-off dynamics. These clarifications help ground our hypotheses in biomechanical theory and improve the interpretability of our aims.

Added text to the Introduction:

“We specifically expected that ellipsoid volumes would increase during the late stance phase (approximately 50–60% of the gait cycle), coinciding with the peak in exoskeleton torque delivery. Based on prior studies of push-off mechanics, we anticipated that this added torque would introduce greater vertical and mediolateral variability due to alterations in limb propulsion and trunk control demands. While we did not perform an a priori power analysis due to the secondary nature of the data, our study was designed to detect medium-to-large effects in variability metrics, particularly during stance phases under powered assistance.”

R2Q9 - Page 3, lines 118-125: More details about the exclusion criteria and data quality assessment would strengthen the methodology. How were "clean strides" defined beyond the minimum of five strides?

We thank the reviewer for this important observation. In response, we have expanded the Participants subsection of the Methods to describe the stride quality assessment process in more detail. Specifically, we clarify that clean strides were defined as those free of signal dropouts or gait cycle anomalies, based on stride time consistency and visual inspection of COM trajectories. These clarifications improve the transparency of our preprocessing criteria.

Added text to the Method Section:

“Clean strides were defined as those free of marker loss, tracking artifacts, or irregular timing patterns that deviated more than ±2 standard deviations from the subject’s average stride time [5]. Strides with inconsistent COM trajectories or discontinuities in velocity profiles were excluded based on visual inspection. This ensured that each included stride reflected typical, uninterrupted walking behavior under the given condition.”

R2Q10 - Page 4, lines 153-164: The gait tube analysis description needs more technical detail. How were the 3×3 covariance matrices computed, and what was the rationale for pooling data across participants rather than computing individual matrices?

We appreciate the reviewer’s request for additional technical clarity. In response, we have expanded the Gait Tube Analysis subsection of the Methods to describe how the 3×3 covariance matrices were calculated from COM velocity components at each gait phase. We also explained our rationale for pooling across participants—to generate a generalized gait tube representing group-level variability patterns. This pooling approach was chosen to highlight how assistive strategies influenced the overall stability envelope, while individual comparisons were preserved in the ellipsoid volume and correlation analyses.

Added text to the Method Section:

“At each normalized time point in the gait cycle (0–100%), all valid strides across participants within a given condition were aligned in time. The COM velocity vectors (Vx, Vy, Vz) at each time point were assembled into a data matrix, and a 3×3 covariance matrix was computed using standard covariance estimation:

(1)

where represents the COM velocity vector for stride , and is the mean vector. This matrix captures the stride-to-stride variability in all three directions.

We chose to pool data across participants to generate a generalized group-level representation of the gait stability envelope for each condition. This approach enhances the visibility of consistent, condition-dependent variability patterns while controlling for individual idiosyncrasies. For participant-level comparisons (e.g., ellipsoid volume statistics and correlations), separate analyses were conducted to retain individual-specific features.

”

In response, we have created a new supplementary document titled “Gait Tube Analysis Mathematical Model”, which provides a detailed explanation of the mathematical formulation, derivation framework, and implementation logic behind the proposed method. This supplement complements the annotated MATLAB code already provided and supports reproducibility and clarity for researchers interested in applying or adapting the technique. Please see the response to R2Q11 and the explained Supplementary Material.

R2Q11 - Page 4, lines 162-164: The reference to supplementary MATLAB code is helpful, but key algorithmic details should be included in the main text for reproducibility.

We thank the reviewer for highlighting the importance of reproducibility. The supplementary MATLAB code provided with the manuscript includes comprehensive comments and in-line explanations to guide users through the implementation. In addition, and in response to Comments R2Q2 and R2Q10, we have already expanded the Methods section to describe the computation of the 3×3 covariance matrices, ellipsoid projection process, data pooling strategy, smoothing, and statistical analysis in detail. These additions provide the key algorithmic elements directly within the main text and, together with the annotated code, ensure that the study's procedures can be fully replicated.

In response, we have created a new supplementary document titled “Gait Tube Analysis Mathematical Model”, which provides a detailed explanation of the mathematical formulation, derivation framework, and implementation logic behind the proposed method. This supplement complements the annotated MATLAB code already provided and supports reproducibility and clarity for researchers interested in applying or adapting the technique.

Added text to the Method section:

“A detailed mathematical formulation of the gait tube analysis method is provided in the Supplementary Material titled “Gait Tube Analysis Mathematical Model”, which accompanies this manuscript.”

The Gait Tube Analysis Mathematical Model:

“

Supplementary Material: Mathematical Formulation of Gait Tube Analysis for Phase-Dependent Gait Stability

Authors: Arash Mohammadzadeh Gonabadi, Farahnaz Fallahtafti

Introduction

This supplementary material responds to the reviewer’s request for a detailed mathematical formulation of the gait tube analysis methodology presented in the study “Gait Stability Under Hip Exoskeleton Assistance: A Phase-Dependent Analysis Using Gait Tube Methodology”. Gait tube analysis is a novel method for visualizing and quantifying phase-dependent gait stability by representing center of mass (COM) velocity trajectories in a three-dimensional (3D) state space and assessing variability through covariance-based ellipsoids. This document outlines the mathematical framework, key formulas, and computational steps implemented in the MATLAB code (GaitTubePlot_Supplementary_MATLAB_Code.m) to construct gait tubes and compute ellipsoid volumes across the gait cycle.

Mathematical Formulation of Gait Tube Analysis

2.1 Overview

Gait tube analysis constructs a tubular representation of COM velocity trajectories in a 3D state space defined by anteroposterior , vertical , and mediolateral velocities. For each phase of the gait cycle (normalized to 100 points, 0–100%), the method:

Computes the mean COM velocity trajectory across subjects to form the central path of the gait tube.
Calculates a 3×3 covariance matrix at each gait phase to capture stride-to-stride variability.
Projects the covariance onto a plane perpendicular to the trajectory’s tangent to generate bivariate ellipsoids.
Visualizes these ellipsoids as cross-sections of the gait tube and computes their volumes to quantify phase-dependent variability.

The resulting gait tube provides a continuous, phase-resolved visualization of gait stability, with ellipsoid volumes serving as a metric for local variability.

2.2 Input Data

The input data consist of COM velocity trajectories for subjects across conditions, time-normalized to gait cycle points:

array of anteroposterior velocities (mm/s).
array of vertical velocities (mm/s).
array of mediolateral velocities (mm/s).

Velocities are converted from m/s to mm/s for enhanced numerical resolution:

2.3 Mean Trajectory Computation

For each condition , the mean COM velocity trajectory is computed across subjects at each gait phase :

where:

Missing or invalid data (NaN) are excluded using MATLAB’s omitnan option. The mean trajectory serves as the central axis of the gait tube.

2.4 Covariance Matrix Calculation

At each gait phase , a data matrix is formed:

Rows with NaN values are removed, and the 3×3 covariance matrix is computed:

Where is the sample covariance, and (with as the 3×3 identity matrix) is a regularization term to ensure positive definiteness, particularly when .

2.5 Tangent Vector Estimation

The tangent vector at phase is estimated using finite differences to define the local direction of the mean trajectory:

where is the Euclidean norm. If is invalid (NaN or infinite), a fallback is used.

2.6 Normal and Binormal Vectors

The plane perpendicular to is spanned by normal and binormal vectors, computed using the Frenet-Serret framework. Starting with an arbitrary vector

If is invalid, a fallback is used. The binormal vector is:

If is invalid, a fallback is used. The projection plane basis is:

2.7 Projected Covariance and Ellipse Generation

The covariance matrix is projected onto the plane perpendicular to :

The eigen decomposition of is:

contains the eigenvalues, and contains the eigenvectors. The ellipse radii are:

If is invalid, a fallback is used. Ellipse points in the 2D plane are generated parametrically:

with sampled at 50 points . The 2D ellipse is projected back to 3D:

where represents the ellipse, points centered at .

2.8 Ellipsoid Volume Calculation

The volume of the 3D ellipsoid at each phase quantifies variability using the full covariance matrix . The eigen decomposition is:

The ellipsoid volume is calculated as:

where are the semi-axes of the ellipsoid. The volume is smoothed using a Gaussian filter:

If insufficient data are available , the volume is set to NaN.

2.9 Visualization

The gait tube is visualized in two subplots:

3D Gait Tubes: Ellipses are plotted every 5th phase alongside the mean trajectory , using condition-specific colors and line styles (e.g., dash-dot for NoExo, dotted for PowerOff).
Phase-Dependent Ellipsoid Volumes: Smoothed volumes are plotted against gait cycle percentage (0–100%):

Axis limits are set with 10% padding based on the extrema of ellipse points and mean trajectories.

“

R2Q12 - Page 4, lines 179-201: The statistical analysis section lacks clarity on multiple comparison corrections. With 100 gait cycle points being compared, how was the family-wise error rate controlled?

We thank the reviewer for pointing out this important statistical consideration. In response, we have revised the Statistical Analyses section to clarify how we addressed the issue of multiple comparisons. Specifically, we note that although we performed point-wise Wilcoxon signed-rank tests across 100 gait phases, we mitigated family-wise error inflation by reporting the minimum p-value per condition and requiring effects to persist across at least five consecutive gait points to be considered meaningful. While this does not represent a formal correction like Bonferroni, this cluster-based thresholding approach helps control false positives while maintaining phase sensitivity. This strategy has been used in prior biomechanics studies for temporal data comparisons.

Added text to the Statistical Analyses:

“To account for multiple comparisons across 100 gait phases, we applied a temporal clustering approach. Specifically, a condition was only considered statistically different from the NoExo reference if significant p-values (p < 0.05) occurred across at least five consecutive gait points. This method balances sensitivity to phase-dependent effects with control of family-wise error rate and has been applied in prior studies analyzing time-normalized biomechanical data.”

R2Q13 - Page 4, lines 183-186: The use of a five-point sliding window for smoothing should be justified. How was this window size chosen, and how might it affect the temporal resolution of the analysis?

Thank you for this insightful comment. We selected a five-point sliding window—equivalent to smoothing over 5% of the gait cycle—to reduce high-frequency, stride-to-stride noise while preserving localized phase-specific trends. This window size aligns with best practices in cyclic human movement analysis, where smaller windows (around 5 % of cycle length) strike a balance between temporal precision and signal clarity [6]. Larger windows were avoided to prevent smoothing away short-duration stability changes that are critical for our phase-dependent analysis.

Added text to the Statistical Analyses:

“This window size (5% of the gait cycle) was chosen based on sliding‑window practices in cyclic human movement analyses, which recommend small windows (≈5% of the cycle) to balance noise reduction and temporal resolution [6]. It preserves meaningful phase‐specific trends while attenuating high‐frequency fluctuations that could obscure short‐duration effects [6]. Larger windows were avoided to prevent over-smoothing of critical transient changes in gait stability [6].”

R2Q14 - Page 5, lines 202-208: The description of "more dispersed trajectories" in powered conditions could be quantified more precisely. Consider providing specific metrics for this dispersion.

We thank the reviewer for this helpful suggestion. In response, we have revised the Results section to more precisely quantify dispersion using ellipsoid volume and component-wise variability metrics. We now explicitly reference increased vertical variability and report the average ellipsoid volume values in powered conditions versus unpowered conditions. These additions improve the interpretability and scientific rigor of our description of trajectory dispersion.

Added text to the Results:

“This increased dispersion was quantified by greater ellipsoid volumes, with powered conditions exhibiting an average volume of 75,232 mm³/s³, compared to 52,097 mm³/s³ in the NoExo condition and 47,285 mm³/s³ in the PowerOff condition. Additionally, vertical variability (standard deviation of vertical COM velocity) was consistently higher in powered conditions, particularly during early-to-mid stance phases (see Table 1). These quantitative differences reflect greater stride-to-stride deviations in COM control during exoskeleton-assisted walking.”

R2Q15 - Page 5-6, lines 219-230: The phase-specific ellipsoid volume findings are interesting, but the biomechanical interpretation of why variability peaks occur at 10-50% rather than the expected 50-60% needs more detailed explanation.

We thank the reviewer for this insightful comment. In response, we have expanded the Discussion section to offer a more detailed biomechanical interpretation of the observed variability peaks occurring between 10–50% of the gait cycle. We suggest that these early-to-mid stance phases—characterized by weight acceptance, single-limb support initiation, and transition to mid-stance—are mechanically demanding and sensitive to external perturbations. The earlier-than-expected peak may also reflect anticipatory or compensatory adjustments triggered by exoskeleton torque. These additions provide greater depth to the interpretation of our findings.

Added text to the Discussion:

“This earlier-than-expected variability peak may reflect biomechanical instability during weight acceptance and early single-limb support, where the body transitions from double to single support while managing vertical loading. These phases are characterized by rapid changes in limb stiffness, postural control demands, and motor coordination. Additionally, participants may have made anticipatory or compensatory adjustments in response to the onset of hip assistance, which could increase variability earlier in the cycle than the torque peak itself. These findings suggest that exoskeleton-induced perturbations can influence gait stability not only at the moment of maximal assistance, but also during preceding preparatory phases when balance demands are already elevated.”

R2Q16 - Page 6, lines 239-250: The directional analysis results are valuable, but the finding that only VT was consistently elevated warrants more discussion about the biomechanical implications of vertical COM control during exoskeleton assistance.

We thank the reviewer for highlighting this important point. In response, we have expanded the Discussion section to interpret the biomechanical significance of increased vertical COM variability under powered conditions. We discuss how vertical control is crucial for balance during stance loading and trunk stabilization, and how added exoskeleton torque during push-off may disrupt natural force coordination, increasing vertical variability. These additions enhance the physiological insight provided by our directional analysis.

Added text to the Discussion:

“From a biomechanical perspective, vertical COM control is closely linked to managing ground reaction forces, trunk posture, and head stabilization during stance. The increased VT variability observed under powered conditions may reflect perturbations in the force generation and absorption mechanisms responsible for smooth vertical motion. Hip torque assistance—especially during push-off—can alter leg stiffness and disrupt the neuromuscular coordination of proximal and distal joints, amplifying fluctuations in vertical displacement. This result suggests that exoskeleton designs should consider how vertical force control is affected during late stance to avoid destabilizing the upper body. Notably, vertical ground reaction force analysis during powered exoskeleton walking has shown distinctive loading patterns in such devices, linking vertical loading control to stability maintenance [7–9]. Moreover, studies modeling the vertical stiffness of the hip and knee during gait demonstrate that vertical stiffness varies significantly across the gait cycle and is critical for attenuating vertical oscillations [7].”

R2Q17 - Page 7, Table 1: The metabolic cost data provides important context, but the relationships between stability metrics and energy expenditure could be explored more quantitatively through correlation analysis.

We appreciate the reviewer’s insightful comment. In our preliminary analysis, we explored the relationship between metabolic cost and several stability-related parameters (including AP, ML, VT variability, and ellipsoid volume). While trends were observable, the relationships appeared nonlinear and inconsistent, without strong monotonic correlations. Given that the central aim of this study was to introduce and validate the gait tube methodology for assessing stability, we chose not to extend the analysis to detailed metabolic-stability modeling. However, we agree that this is a valuable future direction and have added this point to the Limitations section of the manuscript.

Added text to eh Discussion, Limitation:

“While we included metabolic cost values to contextualize exoskeleton effects, we did not observe strong linear correlations between energy expenditure and stability metrics such as ellipsoid volume or directional variability. The relationship appeared nonlinear and variable across participants. As our focus in this study was on developing and validating the gait tube analysis for stability assessment, we did not pursue deeper modeling of metabolic-stability interactions. Future work could incorporate nonlinear modeling techniques to investigate these relationships more rigorously.”

R2Q18 - Page 11, lines 329-343: The discussion appropriately addresses the unexpected timing of peak variability, but could benefit from more detailed exploration of the biomechanical mechanisms underlying these phase-specific effects.

We thank the reviewer for encouraging a deeper interpretation of the phase-specific findings. In response, we expanded the Discussion to further explore the biomechanical underpinnings of increased variability during early-to-mid stance. This includes the role of ground reaction force modulation, limb stiffness transitions, postural demands during single-limb support, and anticipatory responses to exoskeleton torque. These additions strengthen the physiological interpretation of the timing and directional nature of the observed stability changes.

Added text to the discussion:

“This timing corresponds to key biomechanical transitions within the gait cycle. Between 10% and 30%, the body transitions from double to single limb support, requiring coordinated control of vertical and lateral ground reaction forces—critical for maintaining trunk and head stability during weight acceptance and limb stiffness modulation [10–12]. During this phase, lower limb joints adjust stiffness to manage vertical loading and ensure smooth progression. The addition of hip torque assistance may alter typical joint moment patterns, introducing instability prior to the peak torque application later in the cycle [10–12]. From 30% to 50%, the limb continues weight support while preparing for push-off, a phase that is biomechanically sensitive to force and coordination perturbations [10–12]. These findings indicate that phase-specific adjustments in ground reaction force modulation and neuromuscular control underlie the observed early-to-mid stance variability, and they highlight that assistive torque strategies should consider stability demands beginning earlier in the gait cycle than traditionally expected.”

R2Q19 - Page 11, lines 344-348: The comparison with traditional stability metrics is valuable, but consider discussing the computational and practical advantages/disadvantages of gait tube analysis relative to these established methods.

We thank the reviewer for this thoughtful suggestion. In response, we have revised the Discussion to compare gait tube analysis with traditional gait stability metrics in terms of computational and practical considerations. We now elaborate on the advantages of phase-resolved, multidimensional visualization as well as the increased computational burden and data requirements. These clarifications provide a more balanced and realistic perspective on the potential for clinical and research applications of this method.

Added text to the Discussion:

“From a computational standpoint, gait tube analysis requires high-quality, time-normalized kinematic data and the construction of covariance matrices and ellipsoids at each phase, making it more resource-intensive than scalar metrics like MOS or Lyapunov exponents. However, this additional complexity allows for richer phase-specific insights and the visualization of multidimensional COM variability that event-based methods cannot provide. Practically, the gait tube method may not yet be well suited for real-time applications or clinical environments without automation and streamlined preprocessing tools. Nevertheless, its intuitive 3D visual outputs and potential to capture subtle stability changes make it a powerful complement to traditional metrics in research and potentially in high-resolution rehabilitation contexts.”

R2Q20 - Page 12, lines 384-402: The discussion of the stability-efficiency trade-off is insightful, but would benefit from more specific recommendations for exoskeleton control strategies based on these findings.

We thank the reviewer for highlighting this opportunity to strengthen the practical implications of our findings. In response, we have expanded the Discussion to include specific recommendations for exoskeleton control strategies. Based on the observed increases in variability during early-to-mid stance and particularly in the vertical direction, we suggest that controllers should modulate torque magnitude and timing to avoid destabilizing users during these sensitive phases. These insights could inform the development of adaptive or phase-dependent assistance profiles that balance energy efficiency and gait stability more effectively.

Added text to the Discussion:

“Based on these findings, we recommend that exoskeleton controllers incorporate phase-specific modulation strategies to reduce torque during early stance (10%–30%), where vertical variability was elevated, and apply assistance more conservatively during mid-stance transitions. Adaptive controllers that monitor real-time stability indicators—such as vertical COM fluctuations or variability thresholds—could dynamically adjust torque delivery to prioritize safety without fully sacrificing energetic gains. This phase-aware approach may be particularly beneficial for clinical populations who are more sensitive to balance perturbations.”

R2Q21 - Page 13, lines 403-451: The extensive comparison with other stability metrics is comprehensive but somewhat overwhelming. Consider condensing this section and focusing on the most relevant comparisons.

We sincerely appreciate this thoughtful comment. As this manuscript is centered on introducing and validating a new methodological framework—gait tube analysis—we aimed to provide a comprehensive comparison with the most commonly used stability metrics in the literature (e.g., Margin of Stability, Lyapunov exponents, entropy). This in-depth comparison was essential not only to highlight the conceptual differences but also to establish the strengths and limitations of our method relative to well-established approaches. We believe this level of detail provides important context and critical insight for researchers considering the application of gait tube analysis in future work. While we understand the concern about length, we respectfully chose to retain this section in its current form to support methodological transparency and offer a richer set of options to the research community.

R2Q22 - Page 14, lines 463-486: The limitations section appropriately acknowledges study constraints, but should also discuss the generalizability of gait tube analysis to other populations and conditions.

Thank you for this excellent suggestion. In response, we have revised the Limitations section to address the generalizability of the gait tube methodology. We now discuss its potential applicability to other populations (e.g., older adults, individuals with neurological impairments) as well as to walking environments beyond treadmill conditions. These additions improve the clarity of the method’s scope and outline directions for future validation and adaptation.

Added text to the Limitation:

“Furthermore, while the gait tube method demonstrated strong utility in a healthy, treadmill-based walking context, its generalizability to overground walking, uneven terrains, or clinical populations with movement disorders (e.g., stroke, Parkinson’s disease) remains to be tested. These populations often exhibit atypical stride timing, asymmetries, or reduced walking consistency, which may affect the stability of the constructed gait tube and the interpretability of ellipsoid volume trends. Future studies should evaluate whether this framework can be adapted to account for non-periodic or highly variable gait and whether it retains sensitivity to phase-specific instability in diverse populations.”

R2Q23 - Page 14, lines 487-500: The conclusion effectively summarizes findings but could be strengthened by providing more specific recommendations for future exoskeleton design and more explicit statements about the clinical significance of the phase-dependent stability findings.

We appreciate the reviewer’s helpful suggestion. In response, we have revised the Conclusion section to include more actionable recommendations for future exoskeleton design, specifically regarding the timing and modulation of assistive torque. We also added a more explicit statement on the clinical significance of detecting phase-dependent changes in gait stability, particularly for populations at increased risk of falls. These additions enhance the practical relevance and translational value of the study.

Added text to the Conclusion:

“To support safer and more effective exoskeleton use, future device designs should consider implementing phase-specific control strategies that reduce assistance during early-to-mid stance—when users appear most sensitive to stability disruptions—and increase assistance during late stance with caution. Adaptive control systems that monitor individual stability responses in real time could further personalize support to optimize both energy efficiency and balance. Clinically, the ability to detect phase-dependent instability offers significant potential for fall-risk assessment and rehabilitation planning in older adults or individuals with neurological impairments. Incorporating gait tube analysis into clinical gait evaluations could help identify specific instability-prone phases and guide targeted interventions.”

R2Q24 - This manuscript presents novel and potentially impactful methodology for assessing gait stability under assistive device conditions. The gait tube analysis approach offers unique insights into phase-dependent stability that could inform exoskeleton design and control strategies. The experimental design is sound, and the findings contribute meaningfully to the field of wearable robotics and gait analysis.

We sincerely thank the reviewer for the thoughtful evaluation and encouraging feedback. Based on your insightful suggestions, as well as those from Reviewer 1, we have significantly revised and strengthened the manuscript. The updated version now includes clearer methodological descriptions, expanded discussion of the most relevant phase-specific findings, and a more detailed interpretation of the clinical and translational implications. We are grateful for your input, which has helped us produce a clearer, more rigorous, and more impactful contribution to the field.

References

Mohammadzadeh Gonabadi, A.; Antonellis, P.; Myers, S.; Pipinos, I.; Malcolm, P. Designing and Developing a New Semi-Rigid Bilateral Exoskeleton to Assist Hip Joint Motion. In Proceedings of the 44th Annual Meeting of the American Society of Biomechanics (ASB2020); 2020.
Erkan, E.; Yüce, S. Serret-Frenet Frame and Curvatures of Bézier Curves. Mathematics 2018, 6, 321, doi:10.3390/math6120321.
Bruijn, S.M.; Meijer, O.G.; Beek, P.J.; van Dieën, J.H. Assessing the Stability of Human Locomotion: A Review of Current Measures. J. R. Soc. Interface 2013, 10, 20120999, doi:10.1098/rsif.2012.0999.
Fallahtafti, F.; Mohammadzadeh Gonabadi, A.; Samson, K.; Yentes, J.M. Margin of Stability May Be Larger and Less Variable during Treadmill Walking Versus Overground. Biomech. 2021, 1, 118–130, doi:10.3390/biomechanics1010009.
Mohammadzadeh Gonabadi, A.; Antonellis, P.; Dzewaltowski, A.C.; Myers, S.A.; Pipinos, I.I.; Malcolm, P. Design and Evaluation of a Bilateral Semi-Rigid Exoskeleton to Assist Hip Motion. Biomimetics 2024, 9, 211, doi:10.3390/biomimetics9040211.
Banos, O.; Galvez, J.-M.; Damas, M.; Pomares, H.; Rojas, I. Window Size Impact in Human Activity Recognition. Sensors 2014, 14, 6474–6499, doi:10.3390/s140406474.
Zhao, H.; Cao, J.; Liao, W.-H. Simultaneous Estimation of the Vertical Stiffness in the Knee and Hip for Healthy Human Subjects during Walking. Bioengineering 2023, 10, 187, doi:10.3390/bioengineering10020187.
Fineberg, D.B.; Asselin, P.; Harel, N.Y.; Agranova-Breyter, I.; Kornfeld, S.D.; Bauman, W.A.; Spungen, A.M. Vertical Ground Reaction Force-Based Analysis of Powered Exoskeleton-Assisted Walking in Persons with Motor-Complete Paraplegia. J. Spinal Cord Med. 2013, 36, 313–321, doi:10.1179/2045772313Y.0000000126.
Normand, M.A.; Lee, J.; Su, H.; Sulzer, J.S. The Effect of Hip Exoskeleton Weight on Kinematics, Kinetics, and Electromyography during Human Walking. J. Biomech. 2023, 152, 111552, doi:10.1016/j.jbiomech.2023.111552.
Little-Letsinger, S.E.; Cook, R.W.; Wilson, D.; Truitt, K.; Schmitt, D. Gait Compliance Alters Ground Reaction Forces in Human Walking: Implications for the Evolution of Bipedalism. J. Exp. Biol. 2025, 228, doi:10.1242/jeb.250219.
Leal-Junior, A.; Frizera-Neto, A. Gait Analysis: Overview, Trends, and Challenges. In Optical Fiber Sensors for the Next Generation of Rehabilitation Robotics; Elsevier, 2022; pp. 53–64.
Schrade, S.O.; Devittori, G.; Easthope, C.A.; Shirota, C.; Lambercy, O.; Gassert, R. Exoskeleton Knee Compliance Improves Gait Velocity and Stability in a Spinal Cord Injured User: A Case Report. 2019, doi:https://doi.org/10.48550/arXiv.1911.04316.
Xu, D.; Zhou, H.; Quan, W.; Jiang, X.; Liang, M.; Li, S.; Ugbolue, U.C.; Baker, J.S.; Gusztav, F.; Ma, X.; et al. A New Method Proposed for Realizing Human Gait Pattern Recognition: Inspirations for the Application of Sports and Clinical Gait Analysis. Gait Posture 2024, 107, 293–305, doi:10.1016/j.gaitpost.2023.10.019.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This article is centered on the stage-dependent assessment of gait stability with the assistance of a hip exoskeleton and incorporates the newly proposed “gait tube analysis” method. However, there are still many deficiencies in the structural logic, details of the experimental design, interpretation of the methodology, and discussion of the results. Substantial modifications are suggested to improve the scientific quality and clarity of the paper.

The research motivation and hypotheses are not clear enough. It is suggested to further highlight the practical significance proposed by the "gait tube analysis" method and the key differences from the existing methods.
The introduction part is lengthy in structure and some of its contents have little to do with the research topic. It should focus more on the current research status, problems and core innovation points of gait stability analysis. In addition, regarding recent research on gait analysis, to provide more effective evidence, the authors may consider referring to the following updated relevant studies: A new method proposed for realizing human gait pattern recognition: Inspirations for the application of sports and clinical gait analysis (https://doi.org/10.1016/j.gaitpost.2023.10.019).
Sample sizes were small (n=10), and authors were required to provide the basis for sample size calculations, explaining statistical efficacy or the impact on extrapolation.
There is a lack of transparency in the methodology section, e.g., how COM trajectories are specifically handled, the criteria for dividing the gait into each condition, and the specific criteria for rejecting anomalous step lengths should be clearly stated.
The methodology of the “gait tube analysis” is not sufficiently formulated and it is recommended that the main mathematical model or derivation framework be supplemented.
Graphs with low information density, such as Figures 1, 2, and 4, should simplify the color scheme, standardize fonts, and add trend lines or prominence labels to improve readability.
Some statistical results were not labeled with specific p-values or confidence intervals, especially the Wilcoxon test results, and it is recommended that a complete description of statistics and effect sizes be added.
The conclusions section is slightly general in its description and should kick back to the study hypothesis and clarify the specific implications of this study for exoskeleton control strategies or clinical rehabilitation
MATLAB code is mentioned in the Supplementary, but its calling details are not described in the main text, and it is recommended that key code segments or processing flows be written in the Methods section to enhance reproducibility.

Author Response

Dear Editor and Reviewers,

As always, we are grateful for the opportunity to refine the content of our manuscript.

Sincerely,

Arash Mohammadzadeh Gonabadi

Farahnaz Fallahtafti

Reviewer 3

Comments and Suggestions for Authors

We sincerely thank the reviewer for the careful evaluation and constructive feedback.

R3Q1 - The research motivation and hypotheses are not clear enough. It is suggested to further highlight the practical significance proposed by the "gait tube analysis" method and the key differences from the existing methods.

Thank you for this valuable comment. This point aligns closely with the suggestions previously raised by Reviewer 1 (R1Q1) and Reviewer 2 (R2Q1 and R2Q8). In response to these comments, we revised the Introduction to clearly articulate the motivation behind the study, emphasize the practical and clinical relevance of the gait tube analysis, and explicitly state our hypotheses regarding the expected effects of hip exoskeleton assistance on gait variability. These revisions also clarify how our method differs from traditional stability metrics such as Margin of Stability and Lyapunov exponents. We kindly refer you to the detailed responses and revised text under those reviewer items for further information.

Added text to the Abstract:

Added text to the Introduction:

And,

R3Q2 - The introduction part is lengthy in structure and some of its contents have little to do with the research topic. It should focus more on the current research status, problems and core innovation points of gait stability analysis. In addition, regarding recent research on gait analysis, to provide more effective evidence, the authors may consider referring to the following updated relevant studies: A new method proposed for realizing human gait pattern recognition: Inspirations for the application of sports and clinical gait analysis (https://doi.org/10.1016/j.gaitpost.2023.10.019).

Thank you for your sharp observation and insightful feedback. We appreciate your recommendation. Because this study introduces a new methodological framework, we intentionally included a comparison with existing methods for gait stability and variability assessment. We believe that outlining the strengths and limitations of established approaches is necessary to position the novelty and relevance of the proposed gait tube method. This broader context helps readers and researchers understand where the new method fits and why it may be valuable.

That said, we acknowledge your point about focus and have worked to ensure that the content remains directly relevant to the core topic. In addition, we have integrated the suggested study—"A new method proposed for realizing human gait pattern recognition..."—into the Introduction to enhance the review of recent innovations in gait analysis and strengthen the foundation for our methodological contribution.

Added text to the Introduction:

“Recent work has also highlighted the potential of combining biomechanical signals with data-driven models for gait pattern recognition and classification. For example, Zhang et al. [13] introduced a novel method to identify gait patterns based on joint motion signals, offering insights applicable to both sports and clinical rehabilitation settings [13]. While their approach focuses on identifying movement patterns, our work extends this direction by offering a continuous, phase-dependent metric of stability rather than discrete classification, complementing such approaches and supporting applications in exoskeleton control, real-time feedback, and individualized therapy planning.”

R3Q3 - Sample sizes were small (n=10), and authors were required to provide the basis for sample size calculations, explaining statistical efficacy or the impact on extrapolation.

Thank you for the thoughtful comment and careful attention to methodological detail. This point was also raised by Reviewer 1 (R1Q4), and in response, we have addressed the issue in both the manuscript and response letter under that comment. As explained there, this study was a secondary analysis using data from a previously published experimental protocol, and no a priori sample size calculation was feasible. However, we now acknowledge this limitation explicitly in the manuscript and discuss its implications for statistical power and generalizability. We thank you again for your valuable observation and shared attention to rigor.

Response to R1Q4:

We appreciate the reviewer’s great suggestion regarding sample size and statistical power analysis. As noted (in R1Q3), the current study is a secondary analysis based on data collected and published in a prior study focused on the biomechanics and metabolic cost of hip exoskeleton assistance [1]. Since no additional participants were recruited, a prospective power analysis was not feasible. However, we acknowledge this limitation and have added a clarification to the Limitations section of the manuscript.

Added text to the Limitations section:

R3Q4 - There is a lack of transparency in the methodology section, e.g., how COM trajectories are specifically handled, the criteria for dividing the gait into each condition, and the specific criteria for rejecting anomalous step lengths should be clearly stated.

Thank you for this important observation. We agree that more explicit methodological detail improves transparency and reproducibility. In response, we have clarified in the manuscript how the COM trajectories were processed, how each walking condition was defined, and what specific criteria were used to reject anomalous strides. Specifically, we now explain that COM trajectories were segmented and time-normalized to 100 gait cycle points using heel-strike events, and that walking conditions corresponded to distinct exoskeleton control settings (e.g., LowEarly, HighLater). We also specify that strides were excluded if they showed marker loss, discontinuities, or deviated more than ±2 standard deviations from the participant’s average stride time or step length. These details have been added to Sections 2.1 and 2.2 of the revised manuscript to address your comment and improve methodological clarity.

Added text to the Method Section:

And:

“COM trajectories were first segmented based on heel-strike events and time-normalized to 100 points per stride using linear interpolation. Each walking condition corresponded to a specific exoskeleton control profile from the original protocol, defined by variations in assistive torque magnitude (Low vs. High) and timing (e.g., Early, Mid, Later, Latest). The PowerOff and NoExo conditions served as unpowered baselines.”

R3Q5 - The methodology of the “gait tube analysis” is not sufficiently formulated and it is recommended that the main mathematical model or derivation framework be supplemented.

Thank you for this excellent and constructive suggestion. In response, we have created a new supplementary document titled “Gait Tube Analysis Mathematical Model”, which provides a detailed explanation of the mathematical formulation, derivation framework, and implementation logic behind the proposed method. This supplement complements the annotated MATLAB code already provided and supports reproducibility and clarity for researchers interested in applying or adapting the technique.

Added text to the Method section:

The Gait Tube Analysis Mathematical Model:

“

Supplementary Material: Mathematical Formulation of Gait Tube Analysis for Phase-Dependent Gait Stability

Authors: Arash Mohammadzadeh Gonabadi, Farahnaz Fallahtafti

Introduction

Mathematical Formulation of Gait Tube Analysis

2.1 Overview

Computes the mean COM velocity trajectory across subjects to form the central path of the gait tube.
Calculates a 3×3 covariance matrix at each gait phase to capture stride-to-stride variability.
Projects the covariance onto a plane perpendicular to the trajectory’s tangent to generate bivariate ellipsoids.
Visualizes these ellipsoids as cross-sections of the gait tube and computes their volumes to quantify phase-dependent variability.

The resulting gait tube provides a continuous, phase-resolved visualization of gait stability, with ellipsoid volumes serving as a metric for local variability.

2.2 Input Data

The input data consist of COM velocity trajectories for subjects across conditions, time-normalized to gait cycle points:

array of anteroposterior velocities (mm/s).
array of vertical velocities (mm/s).
array of mediolateral velocities (mm/s).

Velocities are converted from m/s to mm/s for enhanced numerical resolution:

2.3 Mean Trajectory Computation

For each condition , the mean COM velocity trajectory is computed across subjects at each gait phase :

where:

Missing or invalid data (NaN) are excluded using MATLAB’s omitnan option. The mean trajectory serves as the central axis of the gait tube.

2.4 Covariance Matrix Calculation

At each gait phase , a data matrix is formed:

Rows with NaN values are removed, and the 3×3 covariance matrix is computed:

Where is the sample covariance, and (with as the 3×3 identity matrix) is a regularization term to ensure positive definiteness, particularly when .

2.5 Tangent Vector Estimation

The tangent vector at phase is estimated using finite differences to define the local direction of the mean trajectory:

where is the Euclidean norm. If is invalid (NaN or infinite), a fallback is used.

2.6 Normal and Binormal Vectors

The plane perpendicular to is spanned by normal and binormal vectors, computed using the Frenet-Serret framework. Starting with an arbitrary vector

If is invalid, a fallback is used. The binormal vector is:

If is invalid, a fallback is used. The projection plane basis is:

2.7 Projected Covariance and Ellipse Generation

The covariance matrix is projected onto the plane perpendicular to :

The eigen decomposition of is:

contains the eigenvalues, and contains the eigenvectors. The ellipse radii are:

If is invalid, a fallback is used. Ellipse points in the 2D plane are generated parametrically:

with sampled at 50 points . The 2D ellipse is projected back to 3D:

where represents the ellipse, points centered at .

2.8 Ellipsoid Volume Calculation

The volume of the 3D ellipsoid at each phase quantifies variability using the full covariance matrix . The eigen decomposition is:

The ellipsoid volume is calculated as:

where are the semi-axes of the ellipsoid. The volume is smoothed using a Gaussian filter:

If insufficient data are available , the volume is set to NaN.

2.9 Visualization

The gait tube is visualized in two subplots:

3D Gait Tubes: Ellipses are plotted every 5th phase alongside the mean trajectory , using condition-specific colors and line styles (e.g., dash-dot for NoExo, dotted for PowerOff).
Phase-Dependent Ellipsoid Volumes: Smoothed volumes are plotted against gait cycle percentage (0–100%):

Axis limits are set with 10% padding based on the extrema of ellipse points and mean trajectories.

“

R3Q6 - Graphs with low information density, such as Figures 1, 2, and 4, should simplify the color scheme, standardize fonts, and add trend lines or prominence labels to improve readability.

Thank you for this valuable and detailed suggestion. We have carefully considered your feedback and implemented several improvements to enhance figure clarity, consistency, and interpretability.

Regarding the color scheme and line styles, we intentionally used a consistent visual design throughout the manuscript. The colors represent the timing of torque application (Early, Earlier, Mid, Later, Latest), and we used the same five colors for both Low and High torque groups to maintain clarity. Line thickness represents the magnitude of applied torque, where thinner lines correspond to Low torque conditions and thicker lines indicate High torque. Dashed or dotted lines are used consistently to represent reference conditions (NoExo and PowerOff), helping to visually differentiate powered from unpowered trials.

We acknowledge your point about font formatting, and in response, we have standardized all figure fonts to match the manuscript’s formatting requirements and made them of higher quality and resolution.

To improve visual clarity, we have also added a new figure (Figure 2) showing the gait tube structure for a single representative condition, which helps readers understand the shape and flow of COM trajectories in isolation before comparing across multiple overlays. In addition, we grouped related plots together within the same figure panels where appropriate, making it easier for readers to directly compare conditions and observe differences in gait variability patterns.

We believe these revisions significantly improve the readability and information density of the figures, and we thank you again for the thoughtful recommendation.

Modified and added Figures:

“

Figure 1. Gait tubes were pooled across subjects under each walking condition. A) Three-dimensional view of the center of mass (COM) velocity trajectories in velocity state space. B) Sagittal view showing COM vertical versus anteroposterior components. C) Transverse view showing COM mediolateral versus anteroposterior components. D) The frontal view shows COM mediolateral versus vertical components. Each color represents a distinct walking condition. The NoExo condition is shown as a bold dashed line, and PowerOff is a dotted line, which is included for reference. Ellipsoids were computed at each gait phase and plotted to represent local variability in COM velocity across strides.

Figure 2. Gait tubes were pooled across subjects under NoExo walking condition. A) Three-dimensional view of the center of mass (COM) velocity trajectories in velocity state space. B) Sagittal view showing COM vertical versus anteroposterior components. C) Transverse view showing COM mediolateral versus anteroposterior components. D) The frontal view shows COM mediolateral versus vertical components. The NoExo condition is shown as a bold dashed line. Ellipsoids were computed at each gait phase and plotted to represent local variability in COM velocity across strides.

Figure 3. Phase-dependent metrics and supporting signals across walking conditions. A) Phase-dependent ellipsoid volumes across the gait cycle for all walking conditions. Based on Wilcoxon signed-rank tests, statistically significant differences from the NoExo condition are indicated by corresponding p-values in the legend. B) Torque assistance profiles for powered conditions, showing the magnitude and timing of applied hip torque. C) Mean normalized ground reaction force (GRF) trajectories averaged across subjects and strides in the anteroposterior, mediolateral, and vertical directions.

Figure 4. Summary of gait tube metrics averaged across the gait cycle. A) Mean ellipsoid volume. B) Mediolateral variability. C) Anteroposterior variability. D) Vertical variability. Bars represent condition-wise means across subjects, and error bars indicate standard deviations. Statistical significance is determined using Wilcoxon signed-rank tests relative to the NoExo condition.

Figure 5. Summary of variability metrics and correlations across walking conditions. A) Raw values of ellipsoid volume and total variability for each condition. B) Z-scores computed relative to the NoExo condition for both metrics. C) Pearson correlation coefficients (r-values) and corresponding p-values comparing ellipsoid volume and total variability across conditions. Dashed bars indicate reference conditions (NoExo and PowerOff).

Figure 6. Z-scores and correlations of gait stability metrics across twelve walking conditions. A) Z-scores of ellipsoid volume, Lyapunov exponent, local dynamic stability, Hurst exponent, sample entropy, maximum Floquet multiplier, and Poincaré variance, all computed relative to the NoExo condition. The legend identifies each metric. B) Pearson correlation coefficients (r) between ellipsoid volume and each of the six alternative stability metrics. C) Corresponding p-values for these correlations across conditions.

Figure 7. Raw values of gait stability metrics compared to ellipsoid volume across twelve walking conditions. Each subplot displays ellipsoid volume (left y-axis) alongside a secondary stability metric (right y-axis), allowing for visual comparison of condition-specific trends. A) Lyapunov exponent. B) Local dynamic stability. C) Hurst exponent. D) Sample entropy. E) Maximum Floquet multiplier. F) Poincaré variance.

”

R3Q7 - Some statistical results were not labeled with specific p-values or confidence intervals, especially the Wilcoxon test results, and it is recommended that a complete description of statistics and effect sizes be added.

We appreciate this helpful comment. Regarding effect sizes, this point was also raised by Reviewer 1 (R1Q4) and Reviewer 3 (R3Q3). As described in our responses to those comments, we clarified that this study was based on secondary analysis of an existing dataset and was not originally powered to detect specific effect sizes. This limitation has been acknowledged explicitly in the revised manuscript.

Regarding p-values, we have already reported all Wilcoxon test results in Table 1, including condition-wise comparisons at each gait phase. To further enhance statistical transparency, we have now added Table 2, which reports Confidence Intervals for selected key outcome variables. These additions improve the statistical clarity and completeness of the manuscript.

Table 2 and the text added to the Results:

“

*Table 2.* 95% confidence intervals (CIs) for gait stability metrics across powered, unpowered, and baseline walking conditions. Confidence intervals are provided for ellipsoid volume, mediolateral variability, anteroposterior variability, and vertical variability. These intervals help interpret the precision of the observed differences and complement the p-values reported in Table 1.
*Condition*	*Ellipsoid* *(mm³/s³)*		*Variability* *(mm/s)*
	*Ellipsoid* *(mm³/s³)*		*Anteroposterior (AP)*		*Mediolateral (ML)*		*Vertical (VT)*
	CI		CI		CI		CI
*Low Early*	72385.14	74342.55	14.30	38.32	21.13	50.41	9.95	40.23
*Low Earlier*	62600.66	62657.68	13.43	34.61	19.81	51.82	8.28	39.24
*Low Mid*	70521.85	70600.06	11.56	35.63	19.85	49.10	8.14	43.66
*Low Later*	78548.18	78576.63	14.06	35.21	21.03	49.81	8.65	45.68
*Low Latest*	69858.14	69902.69	13.32	36.15	19.74	46.41	8.79	41.56
*High Early*	67194.32	67251.67	13.22	31.70	22.03	49.40	8.11	39.49
*High Earlier*	64386.53	64431.67	12.64	33.25	21.18	51.36	8.53	36.37
*High Mid*	72614.53	72673.19	12.17	35.07	21.32	48.54	9.86	43.96
*High Later*	99915.70	99959.18	13.66	35.34	26.15	59.74	8.73	47.25
*High Latest*	71921.06	71982.23	12.20	34.32	22.39	52.82	8.61	37.81
*No Exo*	52053.73	52139.39	9.91	26.88	18.66	57.82	6.55	35.82
*Power Off*	47259.04	47311.88	10.43	30.68	17.41	47.19	7.86	33.82

“

R3Q8 - The conclusions section is slightly general in its description and should kick back to the study hypothesis and clarify the specific implications of this study for exoskeleton control strategies or clinical rehabilitation.

Thank you for this thoughtful suggestion. In response, we have revised the Conclusions section to directly link back to our original hypothesis and to clarify the specific implications of our findings for exoskeleton control strategies and clinical rehabilitation. We now emphasize that increased gait variability during early-to-mid stance under powered conditions highlights the importance of phase-specific torque modulation, and we discuss how this information could inform safer and more personalized exoskeleton use, particularly in clinical populations.

Added text to the Conclusion:

“These findings confirm our hypothesis that the timing of assistance plays a critical role in modulating gait stability and suggest that exoskeleton controllers should implement phase-specific torque strategies to reduce variability during early stance. Clinically, this method could be used to identify instability-prone gait phases and guide individualized rehabilitation protocols that address balance control in users with neurological or age-related gait impairments.”

R3Q9 - MATLAB code is mentioned in the Supplementary, but its calling details are not described in the main text, and it is recommended that key code segments or processing flows be written in the Methods section to enhance reproducibility.

We thank the reviewer for highlighting the importance of reproducibility. The supplementary MATLAB code provided with the manuscript includes detailed comments and in-line explanations to guide users through each step of the implementation. In addition, and in response to Reviewer Comments R2Q2, R2Q10, R2Q11, and R3Q5, we have already expanded the Methods section to explain the full processing pipeline. This includes how the 3×3 covariance matrices are computed, how ellipsoids are projected, how data is pooled and smoothed, and how phase-specific statistical comparisons are performed. These detailed additions ensure that the algorithmic structure is fully transparent in the main text (See R2Q10 for more detail and the added text to the manuscript).

Moreover, we have created a new supplementary document titled “Gait Tube Analysis Mathematical Model,” which provides a structured derivation and explanation of the method's theoretical framework and computational steps. This new supplement complements the annotated MATLAB code and is intended to further support clarity and reproducibility. We kindly refer you to the response to R3Q5 and to the updated Supplementary Material now provided.

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Gait Stability Under Hip Exoskeleton Assistance: A Phase-Dependent Analysis Using Gait Tube Methodology

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