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  • Jackson T. Hurley1,*,
  • Kshitij Mall1 and
  • Zhenbo Wang2

Reviewer 1: Anonymous Reviewer 2: Anonymous Reviewer 3: Anonymous

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

For the low-thrust orbit transfer trajectory optimization problem, the paper combines the UTM framework with continuation strategies, aiming to solve the high sensitivity of traditional indirect methods to initial guesses and the resulting boundary value instability issues, and provides a promising method for long-duration orbit transfer problems.

Comments and suggestions:

1.Compared with traditional indirect methods, How much has the UTM method expanded the tolerance for initial guess deviations When initial guesses (such as costate variables and initial control values) have different degrees of deviation, what is the convergence success rate of UTM? What about the performance of traditional indirect methods?

  1. Compared with other control input limiting techniques, what are the advantages of UTM?

3.Does the proposed method have advantages in computational efficiency compared with GPOPS-II? Please provide comparative results under different scenarios.

4.Does the algorithm rely on the assumption of unperturbed planar orbits? How adaptable is the algorithm under high-fidelity models?

5.Based on the UTM method, could it lead to frequent engine on-off cycling? How to ensure the smoothness of control inputs?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper uses the Uniform Trigonometrization method to solve the LEO to GEO low-thrust transfer problem, a notoriously tough problem to achieve convergence in. The paper is generally well written, but 12/29 citations are self-citations. While I understand that the papers cited may be relevant to the work presented, the number should be reduced, and instead, papers from peers should be cited. Also, some of the citations appear twice in the references section, and the authors should pay attention to this. I also have other important comments regarding the technical aspects of the manuscript below: 

  1. Introduction 
    I would suggest mentioning the general challenge of solving end-to-end low-thrust transfer problems in various regimes. Suggest citing the following papers that have solved the indirect low thrust TPBVP in other domains, which remains challenging for several reasons, and are pertinent to the discussion.
    [1] In the CR3BP, transitioning through the lunar gravitational SOI: Alvarado, K. I., & Singh, S. K. (2025). Exploring the Design Space of Low-Thrust Transfers with Ballistic Terminal Coast Segments in Cis-Lunar Space. Aerospace, 12(3), 217.
    [2] From LEO to LLO end-to-end: Singh, S. K., Anderson, B. D., Taheri, E., & Junkins, J. L. (2021). Exploiting manifolds of L1 halo orbits for end-to-end Earth–Moon low-thrust trajectory design. Acta Astronautica, 183, 255-272.
    [3] Pontani, M., & Conway, B. (2014). Optimal low-thrust orbital maneuvers via indirect swarming method. Journal of Optimization Theory and Applications, 162(1), 272-292.

    Page 2, Line 66-69: It is important to mention the analytical guarantees that are embedded in the traditional indirect treatment of the problem, which may help the discussion on comparing direct and indirect approaches and their respective converged solutions.

    Page 2, Line 81: You cannot claim that this is the first time without explicitly mentioning your thrust acceleration levels.  I am sure there are papers where a strategic continuation approach, coupled with an indirect formulation, has achieved convergence for similar acceleration levels. Please rewrite this so you do not make such a strong claim.

    Page 5, Table 1: Are the Isp values realistic when coupled with the Tmax values? Did you base this on a known propulsion system?

    Page 5: The accurate term is "Minimum-Time" or "Time-Optimal". Correct this in the section heading.

    Page 5, Line 136: Please clarify that it's a planar transfer and LEO is equatorial.

    Page 8, Line 189: Table reference missing.

    Page 9: Figures 1,2: It is impossible to make out the markers for the solutions you are comparing. Different colors may help. Also, marker size should definitely be larger.

    Page 9, Line 200: Again, Minimum-Time for section heading

    Page 11, Section 5: I do not follow the logic behind investigating case 3. I think that is a much easier problem to solve than case 2. I think a better use case to show would be an out-of-plane or orbit rotation type transfer, where you relax the planar and circular assumption for the LEO. Does the UTM-based approach achieve convergence reliably in those cases?

    Figure 10 should not be included here because other details regarding the fuel-optimal transfer you are solving is not explicitly stated. 

    Page 14,15: Author contributions etc. appear twice in the paper. Correct this. 


     


     

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper applies the Uniform Trigonometrization Method (UTM) to multiple difficult low-thrust LEO-to-GEO orbit transfer problems. Using the UTM framework along with continuation strategies, the authors achieve smoother control of variables and significantly improve the convergence properties of the standard boundary value problem solver. The paper demonstrates the potential of the UTM method in low-thrust orbit optimization, but needs improvement in terms of method rigor, experimental depth, and expression standardization.

 

There are several areas in which the paper could be improved:

  1. The term 'p roblems' in the abstract should be revised to 'problems', and it is recommended to polish the language throughout the entire text.
  2. The paper is relatively weak in terms of innovation, as it seems to only apply UTM to the low-thrust LEO-GEO orbit transfer problem at actual thrust levels, without making breakthroughs in the algorithm itself.
  3. In the “General Optimal Control Problem Formulation”, Equation (1) does not specify the dimensions and physical meanings of the state variable x and the control variable u. Suggest supplementing the specific composition and physical meanings of variables to avoid readers' misunderstanding due to vague definitions.
  4. The Equation numbering in the paper is duplicated and confused, and needs to be uniformly numbered and checked for citations.
  5. Equation (6a) selects a sine function for controlling variable transformation, but lacks a physical intuitive explanation. It is recommended to supplement the physical or mathematical basis for function selection.
  6. In Table 7 of Case 3, when Isp increased from 6000N to 10000N, the flight time only increased from 3.9716 days to 4.0297 days, with a small change but without analyzing the physical mechanism. It is recommended to supplement the mechanism analysis.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed my comments to my satisfaction. Manuscript quality has definitely improved from the original version.

Reviewer 3 Report

Comments and Suggestions for Authors

The author has addressed all review comments in a comprehensive manner. The revised manuscript has a well-structured format, the case design is logical, and the benefits of using UTM for complex LEO-GEO transfer have been successfully demonstrated. The data presented is ample and the charts are easily understandable. The scientific rigor and innovation of the study meet the standards of the journal. Therefore, I suggest accepting this paper.