Fourth-order kinematic analysis: Advanced decomposition methods for particle motion in modified orthogonal frame

Fatimah Alghamdi; Ayman Elsharkawy

doi:10.1371/journal.pone.0337900

Peer Review History

Original SubmissionJune 1, 2025
30 Sep 2025 Decision Letter - Alessandra Aldieri, Editor PONE-D-25-29661Fourth-Order Kinematic Analysis: Advanced Decomposition Methods for Particle Motion in Modified Orthonormal FramePLOS ONE Dear Dr. Elsharkawy, 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. Although there are no huge technical issues, all the Reviewers have highlighted issued in how the study is presented, which taken singularly would be minor, but become major considered all together. Please submit your revised manuscript by Nov 14 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Alessandra Aldieri Academic Editor PLOS ONE Journal requirements: When submitting your revision, we need you to address these additional requirements. 1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf 2. Please update your submission to use the PLOS LaTeX template. The template and more information on our requirements for LaTeX submissions can be found at http://journals.plos.org/plosone/s/latex. 3. Thank you for stating the following financial disclosure: [The authors extend their appreciation to the Deputyship for Research \& Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project number MoE-IF-UJ-R2-22-3029-1.]. Please state what role the funders took in the study. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript." If this statement is not correct you must amend it as needed. Please include this amended Role of Funder statement in your cover letter; we will change the online submission form on your behalf. 4. Please provide a complete Data Availability Statement in the submission form, ensuring you include all necessary access information or a reason for why you are unable to make your data freely accessible. If your research concerns only data provided within your submission, please write "All data are in the manuscript and/or supporting information files" as your Data Availability Statement. 5. When completing the data availability statement of the submission form, you indicated that you will make your data available on acceptance. We strongly recommend all authors decide on a data sharing plan before acceptance, as the process can be lengthy and hold up publication timelines. Please note that, though access restrictions are acceptable now, your entire data will need to be made freely accessible if your manuscript is accepted for publication. This policy applies to all data except where public deposition would breach compliance with the protocol approved by your research ethics board. If you are unable to adhere to our open data policy, please kindly revise your statement to explain your reasoning and we will seek the editor's input on an exemption. Please be assured that, once you have provided your new statement, the assessment of your exemption will not hold up the peer review process. 6. If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Additional Editor Comments (if provided): [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #2: N/A Reviewer #3: N/A ****** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ****** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: This work introduces new methods to decompose the snap vector of a particle on a 3D curve, with formulations based on orthonormal bases and geometric planes. The approach is validated through computational examples. The topic is both novel and interesting. However, the mathematical content appears disconnected from a clear explanation of the context in which it is applied, making its overall usefulness somewhat unclear. The abstract jumps directly into the aim and content of the paper. I recommend adding a sentence at the beginning to introduce the general topic and its potential applications. The introduction is well-written, and the references cited are appropriate. Nonetheless, I suggest briefly explaining the concept of the Frenet frame to support readers who may be less familiar with it. Additionally, the authors should more clearly emphasize the novelty of their contribution in light of existing gaps in the literature. Sections 2 and 3 would benefit from introductory sentences, similar to the one provided at the beginning of Section 4. Furthermore, I recommend adding a final comment that discusses the relevance and implications of the application presented in Section 4. The conclusions should be expanded to better highlight the strengths of the proposed method, particularly in comparison with existing approaches. Reviewer #2: The title of the study is appropriately chosen and accurately reflects the essence of the research. In the Abstract, the authors summarize the essence of the study, which explores novel decomposition techniques for the fourth-order derivative of position (the snap vector) of a particle moving along spatial curves in three-dimensional Euclidean geometry. In the Introduction, the authors review a significant portion of the most recent and relevant literature on the topic. Thirty-four sources are cited, seventeen of which have been published within the last ten years. Section 2 – Preliminaries presents fundamental concepts related to modified frames and their relationship to Frenet frames. The authors introduce the model of motion for a point with constant mass along a three-dimensional curve in Euclidean space. Section 3 – Main Results contains the derivation of snap vector decompositions using modified frames and alternative tangential-radial decompositions. Two theorems and one corollary concerning the fourth-order derivative of position are introduced and their solutions are presented. In Section 4 – Application, the authors provide an illustrative example based on a particle moving along a helical curve in Euclidean 3-space, demonstrating the practical use of the results obtained in the previous section. In the Conclusion, the authors summarize the contributions of the study and validate the proposed method through the results obtained. Comments and Suggestions for Improvement: 1.Technical Error in Citation: In the Introduction, there is a technical error in the citation of references – [16, 17, 18, 19, 16]. The reference [16] is repeated and should be corrected. 2. Missing Formula: Formula (3.4) is missing in the manuscript and should be included or corrected. 3. Overstated Contribution: The claim in the Introduction about the study’s contribution to higher-order kinematic analysis appears to be overstated: “Our research contributes novel analytical techniques and alternative decomposition approaches that extend current understanding of higher-order kinematic analysis.” Similar analytical and decomposition techniques for jerk and snap vectors in tangential and radial directions have been previously presented by researchers such as Guner [13], Hamouda [15], Elsharkawy [17], and Tawfiq [18]. A more accurate claim would be: “This paper contributes to the presentation and further development of analytical techniques and alternative decomposition approaches in higher-order kinematic analysis.” 4. Lack of Proper Acknowledgment of Prior Work: Sections 2 (Preliminaries) and 3 (Main Results) closely follow the methodology, analytical techniques, and decomposition approaches outlined in Hamouda [15], with only minor differences in parameter designations, leading to slight variations in the results. These sections should begin with a clear statement that the work builds upon the methods developed in Hamouda [15]. Additionally, the paper should explicitly outline the differences and advantages of the current approach compared to the referenced work. 5. Unacknowledged Use of Prior Example: In Section 4 – Application, the example of a particle moving along a helical curve (Fig.3) is used to illustrate the results. However, this example was already presented by Guner [13], and this should be clearly acknowledged. The paper should also highlight any differences in the results obtained in this example. The article demonstrates academic value and contributes to the ongoing development of higher-order kinematic analysis. However, minor revisions are needed to correct the remarcs mentioned before. In conclusion, I recommend that this work be included for publication after minor revision. Reviewer #3: Please, see the attached review report. Primary Concerns: 1. In the last paragraphs of the Introduction, the handiness of modified frame applications was explained. However, it is not clear how this frame solves the problem that occurs when curves exhibit vanishing curvature. This must be clarified in the Preliminaries section, exactly when the formal definition of the modified orthogonal frame is given. For instance, after Equations (2.5) (2.6), an expression is needed to ensure that curvature is not zero; otherwise, undefined cases occur for the derivative formula. But we said the modified orthogonal frame solves the problem that occurs when curves exhibit vanishing curvature. I propose to check and refer to the paper doi.org/10.3390/axioms13110800. This issue was explained in detail in that paper. 2. In the given Example, a modified frame is not needed because the Frenet frame works for such a helix. We must understand how the new frame solves the deficiency in the example. Minor Concerns: 1. On page 2, line 13: “Th” should be corrected as “The”. In fact, the inner and cross products are well-known in Euclidean spaces, a more suitable entry may be selected to start. 2. The citations are absent on page 2. All information that is not your own must be cited. 3. On page 2, line 6: “The modified orthogonal frame along γ (u) defined as” should be corrected as “The modified orthogonal frame along γ (u) is defined as” 4. On page 2, line 10: “For the relationship the modified frame and Frenet frame, The relation matrix along γ (u) given by” should be corrected as “For the relationship between the modified frame and Frenet frame, the relation matrix along γ (u) is given by” 5. In Theorem 1 and in proof: “synchronize” should be “synchronizes” 6. Theorem 2 and Corollary 1: it is unusual to give an equation number to an assumption. 7. On page 8, Proof: What does it mean “Assume that P in Figure 1 moves along γ (u)”. Do you mean “Assume that P moves along γ (u) as can be seen in Figure 1 ” 8. On page 8, equation (3.9): Put a full stop at the end of the equation. 9. It is recommended that Figures 2 and 3 be slightly enlarged to enhance readability. ****** 6. PLOS authors have the option to publish the peer review history of their article (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: Yes: Dr. Krasimir Ganchev Reviewer #3: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. Attachments Attachment Submitted filename: 13.09.2025.pdf https://doi.org/10.1371/journal.pone.0337900.r001
Revision 1
2 Oct 2025 Author Response Response to Reviewers Comments We thank the Reviewers for their thoughtful feedback and valuable suggestions for improving our manuscript. We have carefully considered all comments and have made substantial revisions to address them. Below, we provide a point-by-point response to each comment. Reviewer 1: Comment 1: "The abstract jumps directly into the aim and content of the paper. I recommend adding a sentence at the beginning to introduce the general topic and its potential applications." Response: We have added an introductory sentence at the beginning of the abstract to provide better context: "The analysis of higher-order derivatives of position, particularly the snap (fourth derivative), provides critical insights into complex motion dynamics in fields such as robotics, aerospace, and biomechanics." Comment 2: "I suggest briefly explaining the concept of the Frenet frame to support readers who may be less familiar with it." Response: We have added an explanatory paragraph in the Introduction (Section 1) that briefly describes the Frenet frame: "The Frenet frame consists of three orthonormal vectors: the tangent vector (indicating instantaneous direction of motion), the normal vector (pointing toward the center of curvature), and the binormal vector (orthogonal to both)." Comment 3: "The authors should more clearly emphasize the novelty of their contribution in light of existing gaps in the literature." Response: We have added a dedicated paragraph in the Introduction highlighting the specific gaps our research addresses: "Our research addresses several gaps in the existing literature by: (1) providing the first comprehensive decomposition of the fourth-order snap vector using modified orthonormal frames, (2) introducing an innovative tangential-radial decomposition approach that offers new geometric insights, and (3) developing formulations that remain valid even when traditional Frenet frames become undefined due to vanishing curvature." Comment 4: "Sections 2 and 3 would benefit from introductory sentences, similar to the one provided at the beginning of Section 4." Response: We have added introductory paragraphs at the beginning of both Sections 2 and 3: • Section 2: "This section establishes the fundamental mathematical framework and notation necessary for our subsequent analysis. We review key concepts from differential geometry, including curve parameterization, Frenet frames, and modified orthonormal frames, which form the foundation for our decomposition methods." • Section 3: "In this section, we present our principal theoretical contributions: two novel decomposition theorems for the snap vector. The first theorem provides a decomposition using the modified orthonormal frame, while the second theorem offers an alternative formulation using tangential and radial components." Comment 5: "I recommend adding a final comment that discusses the relevance and implications of the application presented in Section 4." Response: We have added a concluding paragraph in Section 4 that discusses the practical implications: "This application highlights how our approach can be implemented computationally and provides insights into motion analysis for engineering applications such as robotic path planning, vehicle dynamics, and animation systems where higher-order smoothness is crucial." Comment 6: "The conclusions should be expanded to better highlight the strengths of the proposed method, particularly in comparison with existing approaches." Response: We have completely rewritten the Conclusion section (Section 5) to: • Explicitly compare our method with existing approaches • Highlight the mathematical stability advantages over traditional Frenet frames • Emphasize the dual decomposition approach as a key strength • Discuss potential applications and future research directions Reviewer 2: Comment 1: "Technical Error in Citation: In the Introduction, there is a technical error in the citation of references – [16, 17, 18, 19, 16]. The reference [16] is repeated and should be corrected." Response: We have corrected this citation error. The duplicate reference [16] has been removed from the citation list in the Introduction. Comment 2: "Missing Formula: Formula (3.4) is missing in the manuscript and should be included or corrected." Response: We have identified and corrected this oversight. Comment 3: "Overstated Contribution: The claim in the Introduction about the study's contribution to higher-order kinematic analysis appears to be overstated... A more accurate claim would be: 'This paper contributes to the presentation and further development of analytical techniques and alternative decomposition approaches in higher-order kinematic analysis.'" Response: We have revised the statement as suggested to provide a more accurate description of our contribution: "This paper contributes to the presentation and further development of analytical techniques and alternative decomposition approaches in higher-order kinematic analysis." Comment 4: "Lack of Proper Acknowledgment of Prior Work: Sections 2 (Preliminaries) and 3 (Main Results) closely follow the methodology, analytical techniques, and decomposition approaches outlined in Hamouda [15], with only minor differences in parameter designations... These sections should begin with a clear statement that the work builds upon the methods developed in Hamouda [15]." Response: We have added explicit acknowledgments at the beginning of both Sections 2 and 3 that: • Clearly state our work builds upon the methodology developed by Hamouda et al. [15] • Outline the key differences in our approach (use of modified orthonormal frames vs. quasi-frames) • Highlight the advantages of our modified frame approach, particularly regarding mathematical stability Comment 5: "Unacknowledged Use of Prior Example: In Section 4 – Application, the example of a particle moving along a helical curve (Fig.3) is used to illustrate the results. However, this example was already presented by Guner [13], and this should be clearly acknowledged." Response: We have added a clear acknowledgment at the beginning of Section 4 that: • Explicitly references Guner [13] for the helical curve example • Explains the differences in our application (snap vector vs. jerk vector analysis, modified frames vs. Frenet frames) • Highlights how our results provide complementary insights Reviewer 3: Major Comment 1: "In the last paragraphs of the Introduction, the handiness of modified frame applications was explained. However, it is not clear how this frame solves the problem that occurs when curves exhibit vanishing curvature. This must be clarified in the Preliminaries section, exactly when the formal definition of the modified orthogonal frame is given." Response: We have added a detailed clarification in the Preliminaries section (after Equation 2.6) that explicitly explains how the modified frame addresses the vanishing curvature problem. The added text states: "The modified orthogonal frame addresses a critical limitation of the Frenet frame: it remains well-defined even when curves exhibit vanishing curvature. While the Frenet frame becomes undefined when κ=0 (as N=T′/‖T′‖ becomes indeterminate), the modified frame vectors Nₘ=T′ and Bₘ=T×T′ remain mathematically valid regardless of curvature values." Major Comment 2: "In the given Example, a modified frame is not needed because the Frenet frame works for such a helix. We must understand how the new frame solves the deficiency in the example." Response: We have added an explanation in the Application section clarifying our rationale: "While the Frenet frame is well-defined for this helix (since κ≠0), we employ the modified orthogonal frame to demonstrate the consistency of our decomposition methodology and to provide a foundation for future applications where Frenet frames may become problematic. This example serves as a validation case where both frames are applicable, allowing readers to compare results and build intuition before applying these methods to more complex curves with potential curvature singularities." Minor Concerns - All Addressed: 1. "Th" correction: Changed "Th inner" to "the inner product and vector cross product" in the Preliminaries section. 2. Missing citations: Added appropriate citations for the definitions of regular curves and inner/cross products. 3. Grammar correction: Changed "defined as" to "is defined as" for the modified orthogonal frame definition. 4. Grammar correction: Changed "For the relationship the modified frame" to "For the relationship between the modified frame" and corrected capitalization. 5. Verb agreement: Changed "synchronize" to "synchronizes" in Theorem 1 and its proof. 6. Equation numbering: Removed the equation number from the assumption in Theorem 2 and Corollary 1. 7. Figure reference clarity: Changed "Assume that P in Figure 1" to "Assume that P moves along γ(u) as shown in Figure 1". 8. Punctuation: Added a period at the end of Equation (3.9). 9. Figure sizing: Increased the size of Figures 2 and 3 from 0.4/0.2 textwidth to 0.5/0.3 textwidth respectively to enhance readability. ________________________________________ We believe these revisions have significantly improved the clarity and rigor of our manuscript. The added explanations provide better context for the modified frame's advantages, and the grammatical corrections enhance the overall readability. We thank the reviewers for their detailed feedback that has helped strengthen our paper. Attachments Attachment Submitted filename: Response to reviewers.pdf https://doi.org/10.1371/journal.pone.0337900.r002
27 Oct 2025 Decision Letter - Alessandra Aldieri, Editor PONE-D-25-29661R1Fourth-Order Kinematic Analysis: Advanced Decomposition Methods for Particle Motion in Modified Orthonormal FramePLOS ONE Dear Dr. Elsharkawy, 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. There are still a couple of issues which have not been sufficiently addressed accoriding to one of the Reviewers. Please submit your revised manuscript by Dec 11 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Alessandra Aldieri Academic Editor PLOS ONE Journal Requirements: If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #1: All comments have been addressed Reviewer #3: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #3: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #3: N/A ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #3: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #3: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: Authors have addressed all comments in a proper way, improving the content and the clarity of their work. Reviewer #3: The minor revisions were done in a satisfactory manner. Nevertheless, previously directed major issues still exist: 1) The authors responded that "We have added a detailed clarification in the Preliminaries section (after Equation 2.6) that explicitly explains how the modified frame addresses the vanishing curvature problem." But the problem is that if $\kappa=0$, how could Eq. (2.5) or (2.6) be possible? The reviewer still proposes that authors focus on the previous suggestion. 2) The reviewer still believes that the example must be chosen for a curve where the Frenet frame does not work. Then, referring to the Modified Orthogonal Frame may be meaningful. Otherwise, the readers cannot sense the benefit of MOF in the authors' theoretical results. 3) In title and elsewhere, "modified orthonormal frame" or "modified orthogonal frame"? ****** 7. PLOS authors have the option to publish the peer review history of their article (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 #3: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. https://doi.org/10.1371/journal.pone.0337900.r003
Revision 2
30 Oct 2025 Author Response Response to Reviewer Comments We thank the reviewer for their insightful comments and suggestions. We have carefully addressed each point as follows: Comment 1: Clarification on Vanishing Curvature “The authors responded that ’We have added a detailed clarification in the Preliminaries section (after Equation 2.6) that explicitly explains how the modified frame addresses the vanishing curvature problem.’ But the problem is that if κ = 0, how could Eq. (2.5) or (2.6) be possible? The reviewer still proposes that authors focus on the previous suggestion.” Response: We appreciate this important clarification. We have now added a detailed explanation in the Preliminaries section (after Equation 2.6) that addresses this concern directly. The key points are: 1. The modified orthogonal frame {T, Nm, Bm} is defined directly from the curve derivatives: Nm = T′ and Bm = T × T′, independent of curvature. 2. Equations (2.5) and (2.6) are derived under the assumption κ ̸= 0 specifically to establish the relationship between modified and Frenet frames. However, the modified frame itself remains mathematically valid when κ = 0. 3. When κ = 0, the derivative formulas would need to be derived separately without relying on the Frenet relations, but the frame vectors themselves remain well-defined. This clarification emphasizes that the modified frame’s primary advantage is its definitional robustness, while the specific equations (2.5)-(2.6) represent special cases for non-vanishing curvature. Comment 2: Example Selection “The reviewer still believes that the example must be chosen for a curve where the Frenet frame does not work. Then, referring to the Modified Orthogonal Frame may be meaningful. Otherwise, the readers cannot sense the benefit of MOF in the authors’ theoretical results.” Response: We completely agree with this suggestion. We have now added another example of a curve where the Frenet frame does not work, but the Modified Orthogonal frame is defined. Comment 3: Terminology Consistency “In title and elsewhere, ’modified orthonormal frame’ or ’modified orthogonal frame’?” Response: Thank you for catching this inconsistency. We have standardized the terminology throughout the paper to use “modified orthogonal frame” because: • The vectors T, Nm, and Bm are orthogonal but not necessarily orthonormal, since Nm = T′ is not normalized. • We have updated the title and all instances in the text to maintain consistency. These revisions significantly strengthen our paper by clearly demonstrating the modified orthogonal frame’s advantages and providing concrete examples where it outperforms traditional Frenet analysis. Attachments Attachment Submitted filename: reply round 2.pdf https://doi.org/10.1371/journal.pone.0337900.r004
16 Nov 2025 Decision Letter - Alessandra Aldieri, Editor Fourth-Order Kinematic Analysis: Advanced Decomposition Methods for Particle Motion in Modified Orthogonal Frame PONE-D-25-29661R2 Dear Dr. Elsharkawy, 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. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. For questions related to billing, please contact billing support. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Alessandra Aldieri Academic Editor PLOS ONE Additional Editor Comments (optional): Reviewers' comments: https://doi.org/10.1371/journal.pone.0337900.r005
Formally Accepted
Acceptance Letter - Alessandra Aldieri, Editor PONE-D-25-29661R2 PLOS One Dear Dr. Elsharkawy, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS One. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset You will receive further instructions from the production team, including instructions on how to review your proof when it is ready. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few days to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. You will receive an invoice from PLOS for your publication fee after your manuscript has reached the completed accept phase. If you receive an email requesting payment before acceptance or for any other service, this may be a phishing scheme. Learn how to identify phishing emails and protect your accounts at https://explore.plos.org/phishing. If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Alessandra Aldieri Academic Editor PLOS One https://doi.org/10.1371/journal.pone.0337900.r006

Open letter on the publication of peer review reports

PLOS recognizes the benefits of transparency in the peer review process. Therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. Reviewers remain anonymous, unless they choose to reveal their names.

We encourage other journals to join us in this initiative. We hope that our action inspires the community, including researchers, research funders, and research institutions, to recognize the benefits of published peer review reports for all parts of the research system.

Learn more at ASAPbio .