Design of a Low-Energy Earth-Moon Flight Trajectory Using a Planar Auxiliary Problem

Round 1

Reviewer 1 Report

The English level of this article is not qualified for the publication. The paper is informal and filled with colloquialism. I strongly commend the authors to find some English native editors to revise the article throughout.

The method mentioned in this paper is interesting. By simply the complex four-body transfer problem into several determined sub-problems, the solve process become stable and efficient. Some details should be supplied and adjusted before published.

 

General Comments:

 

1.       The problem was represented by Eq (23). When there’s no intersection point cross the zero distance or the estimation of the brake impulse is too large, how to determine the value of these parameters? The way of correction of coefficients of the form (22) should be mentioned in the paper.

2.       All the complete results of the last stage and the first two-stage should be listed and analyzed in the last section. The problem given in the last section has been changed into the non-coplanar transfer. The author should analyze the adaptability of the initial guess value from the result of the two-stage coplanar design for the non-coplanar problem. The first two-stage process of this method for the last problem should be proposed in this section.

3.       The author mentioned for the purpose of some regularization of the optimization problem, the so-called apogee velocity impulse  is included. What’s the essence of the impulse? If the problem still remains in the same plane, do we need to add another impulse here? Should the aim of the impulse be to change the Earth-Sun flow pattern into Moon-Earth flow pattern?

 

 

Detail comments:

 

1.       There are different coordinates systems used in this paper. The author should introduce all of them once in the specific part, and quote the abbreviations.

2.       The flow chart of these three-stage algorithm should be presented in the proper section.

3.       The physical meanings and the units of all the axes should be given out.

 

4.       The value of the normalized dimension should be given out at the proper section.

Comments for author File: Comments.pdf

Author Response

Thank you very much for your review report and for your time!

It is a shame that our language level is so poor, but we have tried to correct it in a new version of a paper.

Also, we have tried to get an answer for all your questions and suggestions in a new version of our paper and provide all necessary (maybe not all for now) corrections in text, hope you will check it one more time.

General Comments:

1. The problem was represented by Eq (23). When there’s no intersection point cross the zero distance or the estimation of the brake impulse is too large, how to determine the value of these parameters? The way of correction of coefficients of the form (22) should be mentioned in the paper.

Authors’ response: in that case we have started to vary the coefficients (22), especially the first one. The small and slightly variations of it have provided for us an ability to sufficiently easy satisfy the altitude constraint, moreover to reach that it was enough to vary c_L2 within 0.02 (for a different values of the given altitude) from its default (starting) value of 0.96. The variation of first coefficient from (22) provides a direct influence to a value of an axial projection of the SC relative velocity vector in RC4BP Moon-Earth-SC (the corresonding direction of an axial projection coincides with the current Moon to Earth direction – Cx_M, see 20). Also we can say, that the influence from varying of the second coefficient from (22) on satisfying altitude contraint is almost negligible.  

2. All the complete results of the last stage and the first two-stage should be listed and analyzed in the last section. The problem given in the last section has been changed into the non-coplanar transfer. The author should analyze the adaptability of the initial guess value from the result of the two-stage coplanar design for the non-coplanar problem. The first two-stage process of this method for the last problem should be proposed in this section.

Authors’ response: in a current version of manuscript (after revision) this part was added. For solving problem (29) mostly we need to know the optimal “nodal” moments of time (start and arrival), the value of the first impulse, apogee radius (at the moment of start) of the SC transfer orbit, and the estimation of the relative position between a Laplace vector and direction to Sun (at the moment of start). All this info we can collect from the derived solution of the flat auxiliary problem, and define the required time moments from analyzing/ the optimal configuration of the corresponding celestial bodies (by using ephemerids)

 

3. The author mentioned for the purpose of some regularization of the optimization problem, the so-called apogee velocity impulse  is included. What’s the essence of the impulse? If the problem still remains in the same plane, do we need to add another impulse here? Should the aim of the impulse be to change the Earth-Sun flow pattern into Moon-Earth flow pattern?

Authors’ response: we have thought that the adding of the third (presumably small) impulse will help SC to reach the required vicinity of L2 point (for a problem (29) with a “full” or “complete model of motion”) with some admissible state. We assumed that this impulse should take place somewhere near the apogee of the SC flight trajectory: also, we thought that by using it we would truncate (if needed) possible trajectory declinations right before the SC reaches the close vicinity of the L2 point. The usage of the additional impulse here somehow resemblance with the usage of the parameters (22) in auxiliary problem – it helps us to derive a required solution in more robust way. Despite the fact that the number of unknowns increased, the numerical stability also got better.  

Detail comments:

1. There are different coordinates systems used in this paper. The author should introduce all of them once in the specific part, and quote the abbreviations.

Authors’ response: in a current version of manuscript (after revision) this part was added – see Appendix B

 

2. The flow chart of these three-stage algorithm should be presented in the proper section.

Authors’ response: in a current version of manuscript (after revision) we have added a simplified “schematic” flowchart which contains the definition of the main steps of the proposed approach.

 

3. The physical meanings and the units of all the axes should be given out.

Authors’ response: in a current version of manuscript (after revision) this part was added – see Appendix A

 

4. The value of the normalized dimension should be given out at the proper section.

Authors’ response: in a current version of manuscript (after revision) this part was added – see Appendix A

Reviewer 2 Report

This paper propsoed a simple but efficient approach for low energy orbit transfer from Earth to the Moon, the paper is overall well written and organized, and the proposed RC4BP method is legit, and indeed offers simplified excution/implementation, reviewer always welcomes a nontraditional solution to a well studied problem, and the solution of this caliber should not be questioned. Hence,  I have no comments, and recommend an acceptance for its present form.

Author Response

Thank you very much for your report

Reviewer 3 Report

1. Contribution needs to be summarized. What are Novels of this paper

2. How is the initial value of the optimization problem provided?

3. Most of the literature is relatively old and needs to quote relatively new ones

4 Lack of comparison with existing methods

5 What is the accuracy? What are the conditions for trajectory convergence

6 The format and writing are a little messy, and it is uncomfortable to read

Author Response

Thank you very much for your report and your time!

We have tried to get an answer for all your questions and suggestions in a new version of our paper and provide all necessary corrections in text, hope you will check it one more time

 

1. Contribution needs to be summarized. What are Novels of this paper

Authors’ response: the main “novals” of a present work are: the proposed approach for solving problem (29) based on a flat auxiliary problem which is sufficiently simple to solve and has a great numerical stability in comparison with other methods – due to simplified model of motion and usage of the osculating elements. Also, usage a new criterion in a corresponding optimization problem, such as maximizing r_p, - provides an admissible solution of the low-energy trajectory designing.

2. How is the initial value of the optimization problem provided?

Authors’ response: for first stage auxiliary problem it can be used almost arbitrary initial guess for a vector y (except flight time – it should be taken from the diapozone 90…120 days), for other subproblems as initial guess we used the solutions that has derived on a previous step (sequentially).

3. Most of the literature is relatively old and needs to quote relatively new ones

Authors’ response: we have added a few new one positions

4 Lack of comparison with existing methods

Authors’ response: we did add it in a current version of manuscript

5 What is the accuracy? What are the conditions for trajectory convergence

Authors’ response: the accuracy (numerical point of view): auxiliary problem – the relative tolerance of numerical integration was 10^-11, with approximately the same tolerance (abs) for a solution of the corresponding NLP problems; for problem (29) (main) the rtol for numerical integrator was 10^-15, the tolerance (abs) of satisfying the NLP problem constraints typically was not greater then 10^-12. The convergence of the proposed approach for solving problem (29) mostly depends of appropriate choice of the parameters (22) from the auxiliary problem, and usage of the third (corrective) additional velocity impulse, which improves the stability of the algorithm (cause now we have a 3-dimensional motion “in a full model”) on a last step

6 The format and writing are a little messy, and it is uncomfortable to read

Authors’ response: we have tried to fix it

 

Round 2

Reviewer 1 Report

This paper has already been revised according to my comments. I agree to publish it on the journal.

Author Response

Thank you very much for your comments, suggestions and your time.

We are glad that ideas proposed in this study were interesting for you

Reviewer 3 Report

I think that most of the problems have been solved by author. I still have a small problem:

Why you provide initial values by that way?  I think you need to explain the reason for it.

Author Response

Thank you very much for your comments, questions and suggestions!

Please see the attachment 

Author Response File: Author Response.docx