Minimum-Effort Waypoint-Following Differential Geometric Guidance Law Design for Endo-Atmospheric Flight Vehicles
Round 1
Reviewer 1 Report
The article presents an interesting and novel proposal for the state of the art, however, there are some points that need to be clarified.
1.- The equation (2) must be expressed clearly.
2.- In equation (4) the term $v$ is not defined. Or if it is $V$ it must be corrected.
3.- A clearer development should be shown when obtaining equations (10) and (12).
4.- It would be interesting to obtain some error indicators such as MSQ in the simulation part.
5.- The quality of the images should be increased.
Author Response
Response to Reviewer 1 Comments
Point 1: The equation (2) must be expressed clearly.
Response 1: We thank the reviewer for this comment. The equation (2) is the relative motion equation between the UAV and the waypoint in the arc-length domain, which is derived in detail in Ref.[30]. Therefore, this paper which used the equation (2) as a basic formula in the arc-length domain is not suitable to put great emphasis on the repeated derivation.
Point 2: In equation (4) the term $v$ is not defined. Or if it is $V$ it must be corrected.
Response 2: We thank the reviewer for this comment. In equation (4), the term $v$ has be corrected as $V$.
Point 3: A clearer development should be shown when obtaining equations (10) and (12).
Response 3: We thank the reviewer for this comment. According to the equation (1), we know that , Substituting it into equation (9), we will get the equation (10) as . In the same way, the equation (12) can be easily obtained.
Point 2: It would be interesting to obtain some error indicators such as MSQ in the simulation part.
Response 2: We thank the reviewer for this comment. It is worthy of noticing that the performance index analysed in this manuscript is sufficient to verify the effectiveness of the proposed guidance law to a certain extent. We believe that there is no need to add other error indicators such as MSQ to verify the robustness of the proposed method in the stage of guidance law design and theoretical analysis. Hence, if the guidance law proposed in this manuscript is further studied in engineering application, we could pay close attention to some error indicators.
Point 2: The quality of the images should be increased.
Response 2: We thank the reviewer for this comment. The simulation results with higher quality have been replaced in the revised manuscript, such as the figure 2, figure 4(a) and 4(c), figure 6, figure 7(c) and so on.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The problem is not well stated and motivated. I suggest authors to clarify better what is the problem they are try to solve.
The manuscript has a clear and comphrensive literature review.
The paper has a sufficient amount of recent references.
There are some missing definitions of variables and constants. Some passages in math expressions are difficult to understand. For example:
Page 3, line 113: LOS is not defined.
Page 3, Eq. (01): "a" and "V" are not defined. In the same Eq. (01), authors should explain if "r_i" may be zero or not.
Eqs (1), (2), and (3) needs more comphrensive math steps.
Page 3, line 113: Instead of "For simplicity's sake ... ", I suggest authors to use "For the sake of simplicity...".
Page 3, Eq. (1): "a" is not defined.
Page 4, line 133: LOS is not defined.
Page 5: As a general overview, Eq. (7) seems to be a sum of energy functions computed in each path length. I suggest authors to clarify this explanation.
Page 5: Eq. (9) is correct if "z_i" and "r_i" are defined in modulus. If they are vectors, then the "z_i" orientation must be highlighted in Figure 2.
Page 11, line 336: "... It can be seen from this figure clearly that the UAV can both follow the desired ...". I suggest dropping the word "both".
Page 15: Please, add the units of the variables in Figure 6.
Page 17, line 454: Please, verify the error message!
The Conclusion is poor. I believe that given the results obtained and the comparisons made with other methods, the work deserves a more comprehensive and detailed conclusion.
Author Response
Response to Reviewer 2 Comments
Point 1: The problem is not well stated and motivated. I suggest authors to clarify better what is the problem they are try to solve.
Response 1: We thank the reviewer for this comment. We have emphatically revised the introduction of the manuscript and restated the problem we want to focus on. In the revised manuscript, the difficulties of the guidance law design for small low-cost UAVs are emphasized in the introduction, especially when the time-varying speed is considered in the guidance law design. As we said in the revised manuscript, “For one thing, the computation burden of small low-cost UAVs is not powerful enough to carry out the complex numerical calculations to find the optimal path. For another, it is difficult to design guidance laws for the time-varying speed UAVs, because the remaining flying time cannot be estimated accurately especially when the time-varying speed of UAV is considered. Motivated by the aforementioned observations, a novel minimum-effort waypoint-following differential geometric guidance law (MEWFDGGL) is proposed in this paper to improve the autonomous flight capability of small low-cost UAVs.”
Point 2: There are some missing definitions of variables and constants. Some passages in math expressions are difficult to understand. For example: Page 3, line 113: LOS is not defined.
Response 2: We thank the reviewer for this comment. We have defined LOS as Line-of-Sight in the revised manuscript.
Point 3: Page 3, Eq. (01): "a" and "V" are not defined. In the same Eq. (01), authors should explain if "r_i" may be zero or not.
Response 3: We thank the reviewer for this comment. We have defined "a" and "V" as the speed and acceleration of the UAV respectively in the revised manuscript and explained that the "r_i" cannot be zero.
Point 4: Eqs (1), (2), and (3) needs more comphrensive math steps.
Response 4: We thank the reviewer for this comment. The equations (1), (2), and (3) are the relative motion equations between the UAV and the waypoint in the arc-length domain, which are derived in detail in Ref.[30]. Therefore, this paper which used the equations (1), (2), and (3) as basic formulae in the arc-length domain is not suitable to put great emphasis on the repeated derivation.
Point 5: Page 3, line 113: Instead of "For simplicity's sake ... ", I suggest authors to use "For the sake of simplicity...".
Response 5: We thank the reviewer for this comment. We have revised the manuscript according to the comment.
Point 6: Page 3, Eq. (1): "a" is not defined.
Response 6: We thank the reviewer for this comment. We have defined "a" as the acceleration of the UAV in the revised manuscript.
Point 7: Page 4, line 133: LOS is not defined.
Response 7: We thank the reviewer for this comment. We have defined LOS as Line-of-Sight in the revised manuscript.
Point 8: Page 5: As a general overview, Eq. (7) seems to be a sum of energy functions computed in each path length. I suggest authors to clarify this explanation.
Response 8: We thank the reviewer for this comment. We have clarified this explanation in the revised manuscript. As we said in the revised manuscript, “It is worthy of noticing that the performance index is a sum of energy functions computed in each path length.”
Point 9: Page 5: Eq. (9) is correct if "z_i" and "r_i" are defined in modulus. If they are vectors, then the "z_i" orientation must be highlighted in Figure 2.
Response 9: We thank the reviewer for this comment. The zi has be highlighted in Figure 2.
Point 10: Page 11, line 336: "... It can be seen from this figure clearly that the UAV can both follow the desired ...". I suggest dropping the word "both".
Response 10: We thank the reviewer for this comment. We have revised the manuscript according to the comment.
Point 11: Page 15: Please, add the units of the variables in Figure 6.
Response 11: We thank the reviewer for this comment. We have revised the manuscript according to the comment.
Point 12: Page 17, line 454: Please, verify the error message!.
Response 12: We thank the reviewer for this comment. We have revised the manuscript according to the comment.
Point 13: The Conclusion is poor. I believe that given the results obtained and the comparisons made with other methods, the work deserves a more comprehensive and detailed conclusion.
Response 13: We thank the reviewer for this comment. We have rewritten the conclusion to reflect the potential application, findings and limitations of our work, especially highlighted the innovation and advantages compared with other methods. As we said in the revised manuscript, “The minimum-effort waypoint-following differential geometric guidance law (MEWFDGGL) and its suboptimal form for varying-speed UAVs is proposed in this paper. The optimal guidance problem is transformed into an optimal space curve design problem using the differential geometric guidance model. The speed change of UAV is theoretically decoupled from the guidance problem rather than directly adopting the constant speed hypothesis. It has been theoretically proven that the proposed MEWFDGGL is a globally energy-optimal guidance law, and the suboptimal MEWFDGGL was proposed in order to solve the problem of complexity and high computation burden, which is advantageous to improve the autonomous flight capability of small low-cost UAVs. Finally, by comparing with the original MEWFG law, the nonlinear numerical simulations show that MEWFDDGGL is more efficient for eliminating the adverse influences on the guidance performance caused by the speed change of UAV. It is worth noting that the MEWFDDGGL proposed in this paper does not break through some limitations of the MEWFG. For example, the optimality of MEWFDDGGL is affected by the estimation accuracy of the remaining path length. However, for theoretical research, the proposed guidance law could also be extended to maneuvering target interception scenarios and also salvo attack scenarios”.
Author Response File: Author Response.docx
Reviewer 3 Report
The article titled “Minimum-Effort Waypoint-Following Differential Geometric Guidance Law Design for UAVs”, relates to my area of interest that’s why I recommend some points which may help in order to improve the readability as well as overall structure of this manuscript. The following are my suggestions, recommendations and questions for this article which may help to improve the quality of this manuscript are as follows.
Minimum-Effort Waypoint-Following Differential Geometric Guidance Law Design for UAVs
1. Title is not proper related to the proposed study so it must be rewritten in a concise and better way.
2. Abstract
· Too many details of your proposed problem solution. It must be summarized
· Must use the contrast word i.e., on one hand and on the other hand.
· By using the given theoretical analysis, how can you relate that the proposed technique is efficient.
3. Introduction
· Background of the area needs to be elaborated more
· Put some pictures of UAVs relevant to your work
· Related work of the proposed area must be elaborated further
· Problem statement with solution is missing
· Motivation part of your work is missing
· Major revision is required in this section
4. Methodology
· Describe your proposed method "sufficiently" detailed such that others can redo your experiment(s).
· Proposed simulations results needs to be compared with other related studies to further discuss the effectiveness of your work.
5. Results
· Validation part in this section is missing so it needs to be added
6. Discussions
· Must include this part before conclusion
· Define the potential application of the findings and limitations in this part
7. General Comments
· Title isn’t proper so it needs to be revised
· Too many details of your proposed problem solution. It must be summarized
· Major observations in the introduction part as I reflect it on point number 3 so kindly incorporate all those points accordingly.
· In methodology part, your proposed method must be sufficient enough so that others can redo your simulations. Also reflect effectiveness of your proposed experimental study with proper details.
· Validation part is missing in the result section so it must be included in your study
· Discussion part may also be added to reflect potential application, findings and limitation of your work.
· Check all the references carefully.
Author Response
Response to Reviewer 3 Comments
Point 1: Title is not proper related to the proposed study so it must be rewritten in a concise and better way.
Response 1: We thank the reviewer for this comment. After discussions, we changed the title of the manuscript as “Minimum-Effort Waypoint-Following Differential Geometric Guidance Law”. Because we think the guidance law proposed in this manuscript is not only suitable for UAVs, but also for loitering munitions and other aircrafts.
Point 2: Too many details of your proposed problem solution. It must be summarized.
Response 2: We thank the reviewer for this comment. We have rewritten the abstract, especially summarized and highlighted the innovation and advantages of the proposed guidance law.
Point 3: Major observations in the introduction part as I reflect it on point number 3 so kindly incorporate all those points accordingly.
Response 3: We thank the reviewer for this comment. We have supplemented the introduction part emphatically. And the background and motivation of the manuscript are further elaborated. It is worth noticing that this manuscript focuses on the waypoint-following guidance method for small low-cost aircrafts, including but not limited to the UAVs, so we don't think it is very necessary to add UAVs’ pictures in the introduction part.
Point 4: In methodology part, your proposed method must be sufficient enough so that others can redo your simulations. Also reflect effectiveness of your proposed experimental study with proper details.
Response 4: We thank the reviewer for this comment. We checked the formulae and variables again, and defined some missing variables to make the description of the manuscript more complete. If necessary, I can share my program with others so that others can reproduce my simulations.
Point 5: Validation part is missing in the result section so it must be included in your study.
Response 5: We thank the reviewer for this comment. It is worthy of noticing that the numerical simulation done in this manuscript is sufficient to verify the effectiveness of the proposed guidance law to a certain extent. We believe that there is no need to add other validation parts to verify the robustness of the proposed method in the stage of guidance law design and theoretical analysis. In the future study, we will rely on a small quadrotor UAV as a flight platform to conduct flight verification of the guidance law proposed in this manuscript.
Point 6: Discussion part may also be added to reflect potential application, findings and limitation of your work.
Response 6: We thank the reviewer for this comment. The discussion part is contained in the conclusion section. We have rewritten the conclusion to reflect the potential application, findings and limitations of our work .As we said in the conclusion section, “The minimum-effort waypoint-following differential geometric guidance law (MEWFDGGL) and its suboptimal form for varying-speed UAVs is proposed in this paper. The optimal guidance problem is transformed into an optimal space curve design problem using the differential geometric guidance model. The speed change of UAV is theoretically decoupled from the guidance problem rather than directly adopting the constant speed hypothesis. It has been theoretically proven that the proposed MEWFDGGL is a globally energy-optimal guidance law, and the suboptimal MEWFDGGL was proposed in order to solve the problem of complexity and high computation burden, which is advantageous to improve the autonomous flight capability of small low-cost UAVs. Finally, by comparing with the original MEWFG law, the nonlinear numerical simulations show that MEWFDDGGL is more efficient for eliminating the adverse influences on the guidance performance caused by the speed change of UAV. It is worth noting that the MEWFDDGGL proposed in this paper does not break through some limitations of the MEWFG. For example, the optimality of MEWFDDGGL is affected by the estimation accuracy of the remaining path length. However, for theoretical research, the proposed guidance law could also be extended to maneuvering target interception scenarios and also salvo attack scenarios”.
Point 7: Check all the references carefully.
Response 7: We thank the reviewer for this comment. We have carefully checked the manuscript and all the references carefully.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Considering the corrected version of the article by the authors. It is shown that the pertinent corrections have been made, which has enriched the proposed article. Considering the above, I consider that the article should be accepted. However, the authors are invited to improve the quality of the figures proposed in the results section.
Author Response
Point 1: Considering the corrected version of the article by the authors. It is shown that the pertinent corrections have been made, which has enriched the proposed article. Considering the above, I consider that the article should be accepted. However, the authors are invited to improve the quality of the figures proposed in the results section.
Response 1: We thank the reviewer for this comment. In order to improve the quality of the figures, we exported and replaced the figures in TIFF format so that the figures have higher quality and more details. If necessary, we can also provide the original electronic artwork as an attachment.
Once again, we sincerely thank all your comments. They really helped us to further clarify our contributions and improve the quality of the paper.
Author Response File: Author Response.docx
Reviewer 3 Report
Submit the revised version by highlighting the text along with the proper justification and comparison analysis is required
Author Response
Point 1: Submit the revised version by highlighting the text along with the proper justification and comparison analysis is required.
Response 1: We thank the reviewer for this comment. We have submitted the revised version and response letter according to the 1st round review report in 26 April 2023. In the revised version, we forget highlighting the text in the manuscript, but we provided a detailed point-to-point response to all comments in the response letter. In the 2nd round of modifications, we have marked up all the revisions to the manuscript by using the “Track Changes” function such that changes can be easily viewed by the editors and reviewers. Once again, we sincerely thank all your comments. They really helped us to further clarify our contributions and improve the quality of the paper.
Author Response File: Author Response.docx
Round 3
Reviewer 3 Report
I recommend to publish in its present form
Author Response
Dear Editor , The authors would like to thank you for the considerate assessment on the quality of this paper and constructive comments. All the comments have been really helpful for improving the quality of the paper. We have sincerely and carefully revised the paper according to all the comments. Based on the comments from the academic editors, we added a flight experiment section to the manuscript, and the full text was checked and revised again. Once again, we sincerely thank all your comments. They really helped us to further clarify our contributions and improve the quality of the paper. Best regards, Xue-Sheng Qin 2023-05-12