Round 1

Reviewer 1 Report

Comments for author File: Comments.pdf

Please see the attached report

Author Response

Revision of the manuscript universe-2470206

Trajectory Optimization and Control Applied to Landing Maneuvers on Phobos from Mars-Phobos Distant Retrograde Orbits

 

Vittorio Baraldi and Davide Conte

The reviewed work presents valuable findings in the field of Distant Retrograde Orbit (DRO) implementations and landing trajectory optimization. The results support the feasibility of utilizing DROs for in-orbit operations around Phobos and highlight the potential for establishing a Mars Base Camp for efficient exploration of Mars and its surroundings. The study comprehensively discusses the problem and its results. However, there are a few areas that could be improved to enhance clarity and readability before the paper can be accepted for publication.

Thank you very much for taking the time to read and thoroughly review our paper and recommend it for publication.

Introduction

• The introduction lacks sufficient context or references to related studies in the field. It is important to establish the current state of research, highlight existing gaps, and explain how the proposed study contributes to the existing body of knowledge.

o Thank you for your comment. We implemented a brief explanation of how the gravitational potential and density of Phobos was taken into account

• Although the introduction briefly acknowledges the unusual shape and challenges associated with orbiting and landing on Phobos, it does not delve into the specific implications and difficulties arising from these characteristics. Phobos, with its irregular potato-like shape, presents significant obstacles for spacecraft attempting to orbit or land on its surface. These challenges primarily stem from the complex gravitational potential and density distribution of Phobos.

Various methods have been developed to calculate the gravitational potential around irregular-shaped asteroids, highlighting the efforts to overcome such challenges. One such approach is the mascons method, where the asteroid’s shape is approximated by a set of point masses (Muller and Sjogren, 1968, 1969). Another method, pioneered by Werner (1994); Werner and Scheeres (1996); Werner (1997), precisely evaluates the gravitational potential of a homogeneous polyhedron, employing a combination of planar triangles on its surface. This method is considered particularly effective in describing the gravitational field near or on the surface of a constant density polyhedron (Scheeres et al., 1998, 2000).

Further advancements in modeling the gravitational fields of irregular-shaped asteroids include the application of the mascon gravity framework using a shaped polyhedral source, as demonstrated by Chanut et al. (2015) and Aljbaae et al. (2017). These studies have shown the effectiveness of modeling the gravitational field by calculating the mass of each tetrahedron in the polyhedral shape and assigning it to a point mass at the center of the tetrahedron. This method provides a more accurate estimation of the potential compared to harmonic coefficients and significantly reduces computational processing time (Aljbaae et al., 2021).

Considering that the polyhedral shape of Phobos is available in the Small Bodies Data Ferret (https://sbnapps.psi.edu/ferret/), it is important for the authors to explicitly state how they incorporate the gravitational potential of Phobos in their study. Since the shape information is accessible, the authors should outline the specific methodology or approach they employ to consider the gravitational potential of Phobos in their analysis. Providing clarity on this aspect will enhance the transparency and credibility of their research.

  Thank you for your comment and references. We briefly explained how the gravitational potential of Phobos and polyhedral shapes was studied in the literature to provide a better context for our research. Your references was a precious starting point. You can find the explanation in the Phobos paragraph.

• While the authors mentioned their use of the CR3BP model to simulate Mars and Phobos, there is no explicit indication of an investigation into the effects of solar radiation pressure, the gravitational influence of the Sun, or any potential close approaches of asteroids on the Mars system. These factors have significant implications for spacecraft operations, mission planning, and the overall stability of the system.

Considering the importance of these aspects in comprehending the dynamics of the Mars system, I would like to inquire whether the authors’ research includes an analysis of these effects. Understanding the interplay between these external influences and spacecraft trajectories within the Mars system is crucial for ensuring the accuracy and comprehensiveness of the analysis. Furthermore, it would contribute to a more comprehensive understanding of the challenges and considerations associated with missions to Mars and its moons.

Thank you. You can find a brief explanation of which effects we took into consideration and why we decided not to consider solar radiation pressure, Sun’s gravity, and asteroid approaches into account in the final paragraph of the introduction.

Background Theory

• Distant Retrograde Orbits: The authors’ discussion lacks any mention of limitations or potential drawbacks associated with DROs. While they highlight the advantages of DROs, such as stability and ease of access, they did not mention the possible challenges and trade-offs involved. For instance, although DROs are known for their relative stability and resistance to perturbations, they are not entirely impervious to external influences. Significant perturbations or disturbances, such as variations in gravitational forces from other celestial bodies or non-uniform mass distributions, can impact the stability and longevity of DROs. Consequently, maintaining a DRO over extended periods may necessitate continuous adjustments and corrections. Another limitation stems from the fact that spacecraft in DROs tend to have longer orbital periods and slower velocities compared to closer orbits. This can lead to limited payload capacity due to the higher energy requirements for orbital maneuvers and transfers.

o Thank you. We briefly addresses the DROs drawbacks and outlined how DROs are the best choice for this purpose anyway, despite such drawbacks, along with references.

• Particle Swarm Optimization: The authors focus on describing and implementing the PSO algorithm without discussing any potential limitations or drawbacks associated with its application. It would be beneficial to address factors such as convergence speed, sensitivity to initial conditions, and the possibility of encountering local optima. While they briefly mention that the constants used in the PSO algorithm are derived from the literature and optimized for the specific problem, they do not provide a thorough discussion on the determination of these parameters

or their potential impact on the algorithm’s performance. Additionally, it would be informative if the authors explained their rationale for selecting this particular optimization algorithm and whether they compared it to other commonly used optimization methods in astrodynamics problems. Including such information would generate further interest and enhance the overall understanding of their research.

  Thank you for your comment. Although a brief outline of PSO’s drawbacks and limitations was provided in-text, we implemented a deeper explanation of how such limitations can affect the analysis.

• Mars-Phobos DRO: The authors briefly mention that the constants used in the PSO algorithm are derived from the literature and optimized for the specific problem. However, they do not provide an in-depth discussion on the determination of these parameters or their potential impact on the algorithm’s performance. Additionally, the text mentions that computational issues may arise during implementation, but it lacks clarification on the nature of these issues or potential remedies. Although parallel programming and computing are suggested as potential solutions to improve computation time, no further information or explanations are provided.

o Thank you, We defined the parameters to optimize and their boundary values starting from existing values of the orbital velocity and orbital period for DROs at different amplitudes and were educated guesses based on such known values (see 13,17).

 

Landing Trajectory Optimization

• It is not clear how the authors evaluate the gravitational forces of Phobos. Is the mass distribution of Phobos considered or at least in this work? In fact, the mass distribution of Phobos could potentially have an influence on the trajectory optimization, but it is not explicitly mentioned or discussed in the manuscript. The optimization approach described seems to be based on the assumption of a simplified model or representation of Phobos, focusing more on the trajectory planning and propulsion requirements.

o Thank you. We took into consideration the gravitational coefficients of Phobos up to J2, J3. Added a brief explanation in-text.

• The authors mentioned that the parameters to optimize are chosen to minimize the dimensionality of the trade space, but there is no clear justification or explanation for this choice. Further elaboration on the selection process and the impact of these parameters on the optimization results would enhance the understanding of the methodology.

o Added a brief explanation of how the trade space was simplified and minimized in Section 3.2 and Section 3.

• The manuscript mentions that the landing trajectory optimization is more computationally complex than the previous optimization, but it does not specify the specific challenges or difficulties that arise. Providing more information on the computational complexities and potential limitations would help readers understand the scope of the problem and the potential trade-offs involved.

o Thank you. We better specified how the computational complexity of the landing optimization is influenced by the number of particles initialized and the number of iterations performed in-text.

• There is incomplete information on the penalty scaling factor and the cost function. Please provide a comprehensive explanation of their significance and how they are derived. Additional details on the rationale behind the choice of these factors and their impact on the optimization process would enhance the clarity of the methodology.

o This penalty scaling factor formula comes from existing references (see 50, 51). We now have added a deeper explanation of where this factor comes from and how it is necessary to have a better outcome for the optimization.

 

 

Reviewer 2 Report

The presentation of the paper is very good and the notations are well defined.

Mathematical expressions are well presented.

Therefore i recommend the paper for publication without any change. 

 

 

 

Author Response

Thank you for your comments. We appreciate you taking the time to read and review our paper and suggesting that it be published for its merit

Reviewer 3 Report

In this study, the authors delve into the utilization of trajectory design, optimization, and control techniques for an orbital transfer. Specifically, the focus is on transitioning from Mars-Phobos distant retrograde orbits to the surface of Phobos. The objective is to determine landing trajectories that minimize the total ∆v required for a direct 2-burn maneuver. To achieve this, a particle swarm optimization is employed, utilizing the ∆v and time of flight as optimization parameters. The work is really interesting and overall well-written, I believe it is worth publication. I report here just a few minor suggestion regarding mainly some typos and similar:

 

Nomenclature sections: maybe it could be necessary to include the units of measurement also for the position and velocity vectors, initial and final position and velocity vectors;

 

Line 17 and in the rest of the text: why Mars Base Camp is reported with capital letters?

 

Line 33: the acronym MPDRO has been already defined so it can be used here

 

Line 56: spacecrafts  spacecraft

 

Line 70: Figure1  Figure 1 (a space is missing)

 

Line 71: the acronym DRO can be used being already defined

 

Line 104: “larger”  “largest”

 

Line 119 +2: (Notice that there is a problem with lines numbering in this page) here I would suggest rephrasing “used several times”  “widely used”

 

Equation 1: is it necessary to report all these digits while defining c_c and c_s?

 

Figure 4 caption: Flow Chart  Flowchart

 

Line 137: I would suggest to replace “grater”  “larger”

 

Line 130: I would suggest to replace “higher”  “larger”

 

Figure 5: here the masses are indicated with m1 and m2, while in equation 3 are indicated with a different terminology

 

Line 154, 155: are all these digits necessary?

 

Line 155: isn’t better to report T in hours?

 

Table 1: are all the digits needed?

 

Table 2: on the basis of Table 2, lines 3 and 4, it seems that the lower limit for  T is ~7.6 h, while the upper limit is 2*π seconds … is that correct?

 

Figure 6: which is the reference system of this Figure?

 

Equation 10: are all these digits needed?

 

Figure 7: please increase the font size because the labels cannot be read at the current state; also, the indication of the x-amplitude is covered by the other images so please fix this;

 

Author Response

“Trajectory Optimization and Control Applied to Landing Maneuvers on Phobos from Mars-Phobos Distant Retrograde Orbits” by Vittorio Baraldi and Davide Conte

Response to Reviewer #3 (our replies are in bold)

 

In this study, the authors delve into the utilization of trajectory design, optimization, and control techniques for an orbital transfer. Specifically, the focus is on transitioning from Mars-Phobos distant retrograde orbits to the surface of Phobos. The objective is to determine landing trajectories that minimize the total ∆v required for a direct 2-burn maneuver. To achieve this, a particle swarm optimization is employed, utilizing the ∆v and time of flight as optimization parameters. The work is really interesting and overall well-written, I believe it is worth publication. 

Thank you for your comments. We appreciate you taking the time to read and review our paper and suggesting that it be published for its merit. We have made the changes requested and provided appropriate responses to your comments in the new version of the paper. 

 

I report here just a few minor suggestion regarding mainly some typos and similar:

Nomenclature sections: maybe it could be necessary to include the units of measurement also for the position and velocity vectors, initial and final position and velocity vectors;

 

Line 17 and in the rest of the text: why Mars Base Camp is reported with capital letters?

The Mars Base Camp is a mission concept such as the “NASA Artemis” missions or the “Lunar Gateway”. Thus, Mars Base Camp is reported in capital letters.

 

 

Line 33: the acronym MPDRO has been already defined so it can be used here

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 56: spacecrafts  spacecraft

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 70: Figure1  Figure 1 (a space is missing)

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 71: the acronym DRO can be used being already defined

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 104: “larger”  “largest”

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 119 +2: (Notice that there is a problem with lines numbering in this page) here I would suggest rephrasing “used several times”  “widely used”

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Equation 1: is it necessary to report all these digits while defining c_c and c_s?

For a matter of transparency and ability to replicate these results, we believe that it is necessary to explicitly report all the parameters that we used in our algorithm.

 

Figure 4 caption: Flow Chart  Flowchart

Thank you for this comment. We have implemented this in the new version of the paper. 

Line 137: I would suggest to replace “grater”  “larger”

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Line 130: I would suggest to replace “higher”  “larger”

Thank you for this comment. We have implemented this in the new version of the paper. 

 

Figure 5: here the masses are indicated with m1 and m2, while in equation 3 are indicated with a different terminology 

Thank you for this comment. We have specified these definitions in line 134+3.

 

Line 154, 155: are all these digits necessary?

Thank you for this comment. These values are taken from a reference that is cited at the end of the sentence.

Line 155: isn’t better to report T in hours?

Thank you for this comment. This value is taken from a publication, which is cited at the end of the sentence, and is reported in such reference. 

 

Table 1: are all the digits needed?

Thank you for this comment. For a matter of transparency and ability to replicate these results, we believe that it is necessary to explicitly report all the parameters that we used in our algorithm.

Table 2: on the basis of Table 2, lines 3 and 4, it seems that the lower limit for  T is ~7.6 h, while the upper limit is 2*π seconds … is that correct?

Thank you for this comment. No, we used non-dimensional parameters for this analysis.We have added a brief explanation of the non-dimensionalization and the characteristic length and time used in line 156+4 on.

 

Figure 6: which is the reference system of this Figure?

Thank you for this comment. It is the Mars-Phobos frame of reference.

 

Equation 10: are all these digits needed?

Thank you for this comment. For a matter of transparency and ability to replicate these results, we believe that it is necessary to explicitly report all the parameters that we used in our algorithm.

 

Figure 7: please increase the font size because the labels cannot be read at the current state; also, the indication of the x-amplitude is covered by the other images so please fix this;

Thank you for this comment. The image has been fixed and it should be more readable.

 

Reviewer 4 Report

This paper should be published because it is very unique and contains contents of great importance for the future development of planetary science.  However, many of the readers of Universe are not “rocket scientists”, so I think some explanations should be added. I will point out where this is the case below;

1. the term "retrograde orbits" is often used, but why "retrograde"?  Some explanation is better to be provided..

2. page 3, The text in the blue oval planet is expressed by too strong blue background and difficult to read Phobos.  Phobos is not an aqua planet, so why not make it orange or light brown?  Also, the green trails in the lower right figure are too weak and ugly; they should be drawn by the dark green.

3. 4p, m2 definition should be written clearly.

4. 13p,  write units in kg as well as mt. Please use also MKS unit. 

     It would be not so familiar to use milli-tons.

5．Write the definitions of the words PSO, DRO and CR3BP at the end of the Nomenclature.

6．Finally, it is important to write explicitly  the original equations of motion from which equations (6) and (7) are derived.  This may be given in either Appendix or in Supplementary information.  It should be written in 1-2p of the Supplementary Information.

Author Response

“Trajectory Optimization and Control Applied to Landing Maneuvers on Phobos from Mars-Phobos Distant Retrograde Orbits” by Vittorio Baraldi and Davide Conte

Response to Reviewer #1 (our replies are in bold)

 

This paper should be published because it is very unique and contains contents of great importance for the future development of planetary science. 

Thank you for your comments. We appreciate you taking the time to read and review our paper and suggesting that it be published for its merit. We have made the changes requested and provided appropriate responses to your comments in the new version of the paper. 

 

However, many of the readers of Universe are not “rocket scientists”, so I think some explanations should be added. I will point out where this is the case below;

1. the term "retrograde orbits" is often used, but why "retrograde"?  Some explanation is better to be provided..

Thank you for this comment. We have implemented this at line 76. 

 

2. page 3, The text in the blue oval planet is expressed by too strong blue background and difficult to read Phobos.  Phobos is not an aqua planet, so why not make it orange or light brown?  Also, the green trails in the lower right figure are too weak and ugly; they should be drawn by the dark green.

Thank you for this comment. However, this image was taken from another publication (Wallace, M.; Parker, J.; Streange, N.; Grebow, D. Orbital Operations for Phobos and Deimos Exploration. AIAA/AAS Astrodynamics 333 Specialist Conference 2012 ,5067). Thus, it cannot be changed. On the other hand, this image is not crucial to this paper’s analysis, but it only serves as a graphical representation of some families of Mars-Phobos periodic orbits. 

3. 4p, m2 definition should be written clearly.

Thank you for this comment. We have implemented this in the new version of the paper. 

 

4. 13p,  write units in kg as well as mt. Please use also MKS unit. 

     It would be not so familiar to use milli-tons.

Thank you for this comment. We have changed all mass units to kg for clarity and consistency, as suggested.

 

5．Write the definitions of the words PSO, DRO and CR3BP at the end of the Nomenclature.

Thank you for this comment. We have implemented this in the new version of the paper. 

 

6．Finally, it is important to write explicitly  the original equations of motion from which equations (6) and (7) are derived.  This may be given in either Appendix or in Supplementary information.  It should be written in 1-2p of the Supplementary Information.

Thank you for this comment. These equations are presented in their final form following a derivation in Conte, D.; Spencer, D. Interplanetary Astrodynamics; Routledge, 2023. We have added this citation to the paper in the text directly above Equations (6) and (7).