Collision Probability Prediction and Orbit Maneuvering Probability Determination of Non-Cooperative Space Object Orbit

Round 1

Reviewer 1 Report

Dear Authors,

your paper about making predictions on collisions between own orbit spacecraft (OOS) and "Non-Cooperative Space Objects" (NCSO) is very interesting. I appreciate the effort on accounting for the physical properties and their uncertainties of these spacecrafts.

The proposed technique is convincing and supported by the results.

Minor changes are needed in the text: please, follow my comments below and the ones in the attached file.

The abstract should be shortened. The introduction is well written with minor comments. Sections 2 to 5 should avoid repetitions. Improvements are needed for a more readable text, including misleading sentences. Conclusion is ok.
Units in pictures and tables could be improved writing them in square brackets. The Kalman filter is mentioned, but so far obscure how this enter in the algorithms. It is worth to have a short description of the Kalman filter and how this filter is applied in the algorithms.

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Abstract:

1. it can be shortened (especially lines 11-23, that are well explained in the introduction already)
2. remove acronyms in abstract since these are explained in the introduction. Acronyms has to be spelled ones only, i.e. on the first time that they are mentioned.

For the rest of the article see attached file: review16092020

Kind regards

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

      We sincerely appreciate you for your constructive criticisms which effectively helped us to improve the quality of the manuscript.

      Some detailed modifications are as follows:

• The abstract should be shortened, it can be shortened (especially lines 11-23, that are well explained in the introduction already) . Sections 2 to 5 should avoid repetitions.

       In the abstract, lines 11-23 were shortened as: Probability of collision between Non-Cooperative Space Object and the reference spacecraft that is performing a mission has been increased drastically over the past few decades. The traditional method is difficult to identify the maneuvering of Non-Cooperative Space Object.

      Sections 2 to 5 were revised to avoid repetitions

 

• Units in pictures and tables could be improved writing them in square brackets.

       Units in all pictures and tables were improved in square brackets.

 

• Remove acronyms in abstract since these are explained in the introduction. Acronyms has to be spelled ones only, i.e. on the first time that they are mentioned.

       All acronyms were removed in abstract, and acronyms were spelled ones only on the first time that they are mentioned.

 

• The Kalman filter is mentioned, but so far obscure how this enter in the algorithms. It is worth to have a short description of the Kalman filter and how this filter is applied in the algorithms.

        The equations of the extended Kalman filter were adjusted to be two parts, and a short description with the equations was as follows:

    The orbits of NCSO can be determined employing the observational data through an extended Kalman filter [25, 27].

     1) According to Equations (11) and (13), the state vector and its covariance matric are predicted from the time  to  , which is called the time-updated.

     2) Using the observational data according to Equation (20) , the state vector and its covariance matric are estimated at the time , which is called the measurement-updated.

 

• please define what are "enemy spacecraft".

      "enemy spacecraft" is revised to be “the rival spacecraft”, and “the rival spacecraft” is defined as “the non-cooperative space object” that may surveil or harm the reference spacecraft that is performing a mission.

 

• Line 97: is it really "unfit" the verb you want to use?

       The sentence was revised to be “The known classical orbital mechanics models are not suitable for the conventional orbit determination method of NCSO that is maneuvering”

 

• Line 112: I can not understand the sentence, please reformulate.

      Line 112 was modified as follows:

      Since the measurement models and the motion models of a space object are established based on some coordinate systems, defining an appropriate system could make the concerned problem easier to be solved and the computation amount to be reduced.

 

• "ECEF is fixed in the Earth", what you really mean? please, explain it better.

    "ECEF is fixed in the Earth" was removed, and it is replaced with “ECEF rotates with the rotation of the Earth”

 

• Line 152: do we need this sentence?

     We’d better need this sentence , which makes it more readable.

 

• Line 167-168: improve sentence. The section concerning coordinate transformation could be improved, making it more readable. Repetition could be avoided and more explanation on the usage of these transformation could be provided.

     Lines 167-168 were modified as follows:

       and  holds the position and velocity vectors of RS in the ECEF coordinate system, which imply information of the orbital maneuvering of the RS.  and  are the attitude matrix and its derivative of the spacecraft body, which are determined by the attitude angles, including the rolling angle, , the pitch angle, , and the yaw angle, , which can be determined by the space-borne GNSS. According to the transformation of order of from ECEF coordinate system to the spacecraft body coordinate system [24],

 

• Line 214: space objects + LOS is not defined.

       “space objects” was revised to be “NCSO” , and “LOS” is revised to be ” the Line-of-Sight (LOS)”

 

• Line 265-268: where, X_k hat, X_k bar, P_k bar, P_{k-1} ^ and P_{k} ^, K_k, R_k are the estimated state vector at time t_k , the predicted state vector at time t_k , the predicted covariance matrix of the state vector, the estimated value of the covariance matrix of the state vector at time t_{k-1} and t_{k}, the gain matrix, the measurement noise covariance matrix, respectively.

       Lines 265-268 were modified as follows:

       where, , ,  , ,  and  are the estimated state vector, the predicted state vector, the modified state vector, the estimated covariance matrix of state vector, the predicted covariance matrix of state vector, the measurement noise covariance matrix and the gain matrix at the time  , respectively.  and  are the estimated state vector and the estimated covariance matrix of state vector at the time , respectively.

     The same way of description is used throughout the paper.

 

• Line 300: who are they?

       They are the relative position and relative velocity vectors, It was revised in the paper.

 

• Line 332: would posses? please explain better.

       It was revised as follows:

        If the relative position error covariance matrix is allocated to NCSO, the relative position covariance matrix between NCSO and OOS is the summation of Equations (53) and (54).

 

• Line 389: RTN is not defined

      RTN was further defined in the paper as:

     For the RTN system, the origin resides at center of mass of the spacecraft; the R-axis directs along the position vector (radial direction); the N-axis is pointing along the orbit normal vector (normal direction); the T-axis points towards velocity vector (track direction); and they complete the orthogonal triad [23].

 

• Line 483: "are under expectation" express it better. Otherwise, misleading.

      "are under expectation" is wrongly expressed, which is misleading. It is revised to be “are in expectation”.

 

• Take a look at the reference list, e.g. [20]

     References were modified as follows:

8. Liu, B., Space Object Orbit Prediction Based on Space-borne Radar. Chinese Journal of Space Science 2010, 30(6): 532-539
9. Huang, J.; Hu, W.; Zhang, L. Maneuver Detection of Space Object for Space Surveillance, In 6th European Conference on Space Debris, Darmstadt, Germany, 2013.

 

       All the revisions were highlighted in red in the revised manuscript.

       Thank you again for the quick respond and we hope our work can finally be accepted for publication.

 

Sincerely yours,

Yuanlan Wen

On behalf of the co-authors

 

Author Response File: Author Response.pdf

Reviewer 2 Report

1.In formulating Equations (1) to (51), standard text book should be cited where ever possible as reference, so that it will become easy for the reader to understand the paper.

2. In Table 1, the units for x, y, z are wrong.

Author Response

Dear Reviewer

We sincerely appreciate you for your constructive criticisms which effectively helped us to improve the quality of the manuscript.

The detailed modifications are as follows:

• In formulating Equations (1) to (51), standard text book should be cited where ever possible as reference, so that it will become easy for the reader to understand the paper.

   First, the existing standard text books were cited, they are as follows:

23. Xu, G.; Yan, X., Perturbed Orbit and Its Determination. Springer: Berlin, 2016.
24. Markley, F. L.; Crassidis, J. L., Fundamentals of Spacecraft Attitude Determination and Control. Springer: Berlin, 2014.

    Then, two more standard text books were listed as references and were cited, they are as follows:

25. Liu, G, M., Liao, Y., Wen, Y. L., Passive Tracking Technology of Non-Cooperrative Space Target and Application, National Defense Industry Press in China, 2015
26. Montenbruck O., Gill E., Satellite Orbits: Models, Methods and Applications, Springer, 2001

 

• In Table 1, the units for x, y, z are wrong.

       The units for x, y, z are wrong in Table 1, They had been revised to be meters.

 

 

     All the revisions were highlighted in red in the revised manuscript.

    Thank you again for the quick respond and we hope our work can finally be accepted for publication.

 

Sincerely yours,

Yuanlan Wen

On behalf of the co-authors

Author Response File: Author Response.pdf

Reviewer 3 Report

The paper deals with collision probability prediction and orbit maneuvering probability determination on non-cooperative space object orbit. The manuscript is written quite well and the description is clear. My only big concern is that the described problem is not so relevant for the remote sensing journal. Even though the authors submitted to the section 'Satellite mission', I was expecting something that was related somehow to remote sensing. In this case, the authors describe a problem of satellite mission which has no relevance at all with remote sensing problems. According to me, the most useful suggestion for the author is to submit to another journal, such as MDPI Aerospace. However, since such decision is to be made by the Editor, I will provide my comments related to the technical issues of the paper. 

I don't have major comments, as the paper is very clear and well written. Some minor comments are reported below:

1. The English should be slightly revised as sometimes there are some typos and sentences are not perfectly clear.
2. I do not really like the acronym you have chosen for OOS, Our Own Spacecraft. Usually, in research papers you want to use impersonal sentences, avoiding using 'we' or 'I'. Accordingly, I would substitute 'Our Own Spacecraft' with something like 'Reference Spacecraft' or 'Concerned Spacecraft'. These are just suggestions, but I kindly invite you to improve this point.
3. Eq. (44) is derived under the hypothesis that NCSO-OOS relative position vectors are perpendicular in a minimal distance condition. Why is this true? Especially if OOS is maneuvering, can they approach minimum distance in different conditions, with the two vectors that are not perpendicular? How much this hypothesis can influence the results and following discussion?
4. As shown in the results, the technique to estimate the maneuvering probability fails and do not distinguish between maneuvering and not-maneuvering NCSOs. Do you think it is only due to perturbations? In such a case, is it difficult to perform only one experiment to demonstrate this? I understand it is computationally not-consistent for on-board operations, but you can make a simple simulation to show the relevance of perturbations. My suggestion is to discuss a little bit more this point, and if possible to add an example with perturbation, if you really think this is why the technique is currently failing. 
5. The legends of the figures reporting the maneuvering acceleration of NCSO are currently reporting 'OOS with maneuvering' and 'OOS without maneuvering'. Is this a typo? Shouldn't it be 'NCSO with maneuvering' and 'NCSO without maneuvering'?
6. Finally, it is my understanding that the whole algorithm should run on-board. If my understanding is correct, I would stress this relevant point a little bit more. If possible (and if you think it can be relevant), can you also provide some computational times related to the described operations? 

Author Response

Dear Reviewer

     We sincerely appreciate you for your constructive criticisms which effectively helped us to improve the quality of the manuscript.

      Detailed modifications are as follows:

• The English should be slightly revised as sometimes there are some typos and sentences are not perfectly clear.

      Yes, there were some typos and wrong sentences, and they were revised in red character in the paper, such as “under expectation” should be “in expectation”.

 

• I do not really like the acronym you have chosen for OOS, Our Own Spacecraft. Usually, in research papers you want to use impersonal sentences, avoiding using 'we' or 'I'. Accordingly, I would substitute 'Our Own Spacecraft' with something like 'Reference Spacecraft' or 'Concerned Spacecraft'. These are just suggestions, but I kindly invite you to improve this point.

       Thank you for your good suggestions. “Our Own Spacecraft (OOS)” was substituted by “Reference Spacecraft (RS)” in the paper.

 

• (44) is derived under the hypothesis that NCSO-OOS relative position vectors are perpendicular in a minimal distance condition. Why is this true? Especially if OOS is maneuvering, can they approach minimum distance in different conditions, with the two vectors that are not perpendicular? How much this hypothesis can influence the results and following discussion?

        The sentences didn’t express that well in the paper. It should be ”The relative position vector and the relative velocity vector are perpendicular when NCSO approaches to RS in a minimal distance.”

       The distance between NCSO and RS can be expressed as a function of time, and Let the derivative of distance function to time be equal to zero, and the minimum distance is obtained, then the orthogonal result of the relative position and relative velocity can be derived. The process of derivation is added in Equations (44)~(47) in the paper.

      There may be multiple distance minima when NCSO is maneuvering, but it may not influence the results and following discussion.

 

• As shown in the results, the technique to estimate the maneuvering probability fails and do not distinguish between maneuvering and not-maneuvering NCSOs. Do you think it is only due to perturbations? In such a case, is it difficult to perform only one experiment to demonstrate this? I understand it is computationally not-consistent for on-board operations, but you can make a simple simulation to show the relevance of perturbations. My suggestion is to discuss a little bit more this point, and if possible to add an example with perturbation, if you really think this is why the technique is currently failing. 

        In the section 5. Results and Discussion, there was a sentence: ”All the results presented in Table 2 are under expectation”, which was misleading. It should be ”All the results presented in Table 2 are in expectation”.

      In the four sets of simulation, the orbital maneuvering acceleration and orbital maneuver probability of NCSO were estimated well, and the minimum distance and the collision probability between NCSO and RS are also predicted well, although there may be some estimation errors due to the measurement errors and the model errors, including the orbital perturbations, it is necessary to investigate in the future work.

 

• The legends of the figures reporting the maneuvering acceleration of NCSO are currently reporting 'OOS with maneuvering' and 'OOS without maneuvering'. Is this a typo? Shouldn't it be 'NCSO with maneuvering' and 'NCSO without maneuvering'?

       In the simulations, there were two cases:

     case 1: RS does not have any orbit maneuvering operation, NCSO is maneuvering

     case 2: RS have orbit maneuvering operation, NCSO is maneuvering.

      In both cases, NCSO is designed under maneuvering to approach RS, while in case 1, RS does not have any maneuvering operation that it is approached by NCSO; and in case 2, RS with maneuvers operation avoids collision with the upcoming NCSO.

     For RS, with maneuvering or without maneuvering, it should estimate the NCSO orbit, orbital maneuvering acceleration and orbital maneuver probability, and predict the minimum distance and the collision probability between NCSO and RS.

    Simulations show that estimation results of the NCSO orbit, orbital maneuvering acceleration and orbital maneuver probability agree well in both cases, i. e. RS with maneuvering or without maneuvering, and prediction results of the minimum distance and the collision probability between NCSO and RS is in expectation. So a conclusion is that RS orbit maneuverability will not affect the orbit determination and orbit maneuvering identification of NCSO based on the proposed orbit determination method and orbital maneuvering probability algorithm.

 

• Finally, it is my understanding that the whole algorithm should run on-board. If my understanding is correct, I would stress this relevant point a little bit more. If possible (and if you think it can be relevant), can you also provide some computational times related to the described operations? 

       Yes, the whole algorithms should be run on-board, and the algorithm simplification is an important point of the paper.

      The computational time was tested in the simulation as follows:

      The tracking time was 5.496s between RS and NCSO. The measurement cycle was 0.024s, the measurement steps are,

                                                5.496/0.024=229.

      The whole program of the algorithms was run in a laptop, and the average computational time were 2.123s, so the computational time for every measurement step were,

                                                 2.123/229= 0.00927s.

       That shows that the computational time for every measurement step is much smaller than the measurement cycle, those ensure that the whole algorithm is run within the measurement cycle

           Of course, The laptop have a good performance with i5-8250U CPU. The on-board computer, however, is quite different, so the computational times are not written into the paper.

 

       All the other revsions were highlighted in red in the revised manuscript.

      Thank you again for the quick respond and we hope our work can finally be accepted for publication.

 

Sincerely yours,

Yuanlan Wen

On behalf of the co-authors

 

Author Response File: Author Response.pdf