An Improved Laplace Satellite Tracking Method Based on the Kalman Filter
Round 1
Reviewer 1 Report (Previous Reviewer 1)
Comments and Suggestions for AuthorsComments for author File: 
Comments.docx
Author Response
comment 1: The authors didn’t add in the text any explanation on the orbital motion used for the prediction by the Laplace algorithm. Also there is still no any description in the text how actually the prediction or extrapolation of the satellite track is carried out. So, this previous critical comment is not addressed by the authors.
response 1: In section 2.1, formula (1), (2), (3), (4), (5) are added, first starts from formula (1), by simplifying, formula (2) and formula (4) are obtained, then, deduce formula (5), meanwhile, describes the definition of A, B, C, finally, formula (6) and formula (7) are combined to derive formula (8), which is the motion equation of the satellite.
Formula (11) is added, introduces how to derive formula 12 from formula 10. Explains the issue that “how to obtain (7) from (6), why "R the with top line" (which is not introduced) can be eliminated?”.
The paragraph below formula (12) introduces the entire implementation process of the algorithm. An overdetermined system of equations about the initial orbit can be obtained by replacing the observed data at several times into formula (12), and the initial orbit can be obtained by iteratively solving the overdetermined system. Then, the initial value of the target motion equation is substituted into formula (8) with the initial orbit as the input time t, and the predicted state value of each moment can be obtained by numerical solution. The predicted Angle value of the corresponding longitude axis and latitude axis can be obtained by converting it to the horizontal coordinate system.
comment 2: It seems that the authors ignored this critical comment at all. The comparison of the Section 2.1 in the previous and current paper versions showed no change. No any explanation of the parameters in the formula and the explanation of the calculation process are added (as it is stated by the authors above).
Author response 2: Formula (5) is added, and describes what are A, B, C. Formula (11) is added, introduces how to derive formula 12 from formula 10. Explains the issue that “how to obtain (7) from (6), why "R the with top line" (which is not introduced) can be eliminated?”
New reviewer comment 3: The coordinates in the figures 4 and 5 are indeed now explained in the paper. Though it is still not clear why “the miss distance is extracted as theoretical 0". It seems that the miss is not actually zero, it is less than discreet value of 0.66 arcseconds – accuracy of the servo motor.
Author response 3: Theory 0 is only an assumption, not an actual tracking error, in order to facilitate analysis of the problem.
New reviewer comment 4: The authors responded to the questions about Q and R matrix. However, there is still the question about model of the angles change used by the Kalman Filter (matrix A is considered identity). Some justification on this is strongly required.
Author response 4: The step (1) in section 2.2.2，the improved Laplacian orbit prediction algorithm formula 8 is used to replace the Kalman filter formula 14 to calculate X(k│k-1), that is, to replace matrix A.
New reviewer comment 5: No changes in the Section 3 is detected in the text in new version of the paper. At least, the authors comment above should be added. There are still no details on the experimental data used. What are the satellite orbital parameters of satellites (a), (b) and (c) in Fig. 7 and Fig. 8.
Author response 5: The use of measurement data has been described in detail in the paper and illustrated to explain. The satellites in (a), (b) and (c) in Fig. 7 and Fig. 8 are low-orbit solar synchronous orbit satellites.
Fig. 12 and a paragraph below it are added, introduces the experimental data using process, and the mean of point 0 in the coordinates and the data curve are described. The titles of the figures are modified to make it more meaningful.
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe present study proposes an improved satellite tracking method based on the Laplace algorithm and the Kalman filter to address the frequent loss of the target caused by weak signals, clouds, and unfavorable solar angles. The initial algorithm adapts the equation of motion with limited data, filtering and evaluating the validity of the current data to obtain precise orbit estimates.
The introduction can be improved. I would emphasize more clearly why optical observations are used even for low-orbit targets instead of radio observations. The importance of computational cost of operations in a closed loop might be important to underline if the system is to be responsive.
The title of Chapter 2 is misleading. Furthermore, formula (1) is incomplete. Please verify the correctness of formula (5) as well. The Excel charts presented in Figures 1 and 2 are unclear; using a different scale for the x-axis is recommended. Clarify the considerations made about the two figures (paragraph "As can be seen...") on page 6/20. Figure 3 is unclear and in low resolution. The arrows cover the corners and are not clear. If the diagram is taken from the literature, it must be cited. Additionally, the novelty of using the Kalman filter for the optimal estimation of the measurement data should be emphasized.
The images presented in Figures 6 and 7 could be improved in quality. The workflow formalism is correct, although it is missing the start and end.
Details of the setup of the instrumentation used for the mount (in terms of RA-rate and DEC-rate) and details of the target's orbit are missing.
The discussion on the comparison of results needs to be improved to highlight the significance of your proposed method.
The comparison between the solution proposed in your study using the Kalman filter and the one without it needs to be improved to better highlight the advantages.
At a general level:
* The innovation of using the Kalman filter should be emphasized, particularly about its known applications in similar solutions within the literature.
* International references should also be added.
Comments on the Quality of English LanguageIt is recommended to check the overall quality of the English, particularly focusing on the fluency of the sentences.
Author Response
Comment 1: The present study proposes an improved satellite tracking method based on the Laplace algorithm and the Kalman filter to address the frequent loss of the target caused by weak signals, clouds, and unfavorable solar angles. The initial algorithm adapts the equation of motion with limited data, filtering and evaluating the validity of the current data to obtain precise orbit estimates.
The introduction can be improved. I would emphasize more clearly why optical observations are used even for low-orbit targets instead of radio observations. The importance of computational cost of operations in a closed loop might be important to underline if the system is to be responsive.
Author response 1: The use of optical observations for low-orbit targets mentioned in this article is only a special means, and the method proposed in this article is aimed at this means, it’s not a substitute for radio observations.
The traditional tracking method is closed-loop mode. In this article, we use data leading mode with a running cycle of 20ms.for the computation time of our algorithm, it can fully meet the requirement of use.
Comment 2: The title of Chapter 2 is misleading. Furthermore, formula (1) is incomplete. Please verify the correctness of formula (5) as well. The Excel charts presented in Figures 1 and 2 are unclear; using a different scale for the x-axis is recommended. Clarify the considerations made about the two figures (paragraph "As can be seen...") on page 6/20. Figure 3 is unclear and in low resolution. The arrows cover the corners and are not clear. If the diagram is taken from the literature, it must be cited. Additionally, the novelty of using the Kalman filter for the optimal estimation of the measurement data should be emphasized.
Author response 2:
Because of the formatting and layout issues, formula (1) is cut, now it is already modified.
The correctness of formula (5) has been rechecked and there are no issues.
Use time(20ms) scale for the x-axis of Figures 1 and 2。
In paragraph "As can be seen..."，the reason for the occurrence of Figure 1 and Figure 2 is explained conjunction with Figure 3.
Figure 3 has been modified to high resolution, this figure is not from literature.
Comment 3: The images presented in Figures 6 and 7 could be improved in quality. The workflow formalism is correct, although it is missing the start and end.
Author response 3: Figures 6 and 7 are modified.
Comment 4: Details of the setup of the instrumentation used for the mount (in terms of RA-rate and DEC-rate) and details of the target's orbit are missing.
Author response 4:
Before Figure1, added a paragraph in the methods section, described the experimental setup and summarized the measurements acquired.
In the beginning of section 3.1, the specific parameters of measurement setup are introduced detailed.
In the beginning of section 3.2, “The satellite measured in this experiment is a low-orbit sun-synchronous orbit satellite” is described.
Comment 5: The discussion on the comparison of results needs to be improved to highlight the significance of your proposed method.
The comparison between the solution proposed in your study using the Kalman filter and the one without it needs to be improved to better highlight the advantages.
Author response 5: the above two points are the same. In the last paragraph of section 3.2.3, added a comparison between using and not using Kalman filtering, and described the benefits of using Kalman filtering.
At a general level:
Comment 6: * The innovation of using the Kalman filter should be emphasized, particularly about its known applications in similar solutions within the literature.
Author response 6: Below formula (18), a paragraph is added, which describes the advantages of Kalman filtering in data estimation.
Comment 7: * International references should also be added
Author response 7: International literatures [19-22] are already referenced in this article in the introduction section.
Reviewer 3 Report (New Reviewer)
Comments and Suggestions for Authors
Review of “An Improved Laplace Satellite Tracking Method Based on the Kalman Filter” [aerospace-3054777]
This work explores improvements in satellite tracking techniques, specifically focusing on enhancing the accuracy and stability of tracking low-orbit satellites using photoelectric measurement equipment. The authors aim at presenting improved techniques that can address the challenges associated with tracking satellites with photoelectric instruments when the target is faint or obscured by factors like cloud or low sun angles. This problem is addressed by proposing an improved Laplacian algorithm combined with a Kalman filter to enhance tracking accuracy and stability. Overall, I think that the problem and the goals of the present study are rather clearly presented. The authors build a dynamical model for a spacecraft in low-earth orbit which focuses on short-term dynamics, i.e., only considers the gravity accelerations given by the spherical and degree 2 non-spherical contributions. They use this model to predict the motion of a satellite and to compare these results to observations that they gather through a horizontal photoelectric device. Finally, they show that, by using their methodology which combines an improved Laplace algorithm with Kalman filtering, the tracking accuracy and the stability are improved, validating their methodology. However, I think that the current format and state of this manuscript makes it not suitable for publication in a peer-reviewed journal. Although the goals are clearly presented and the results seem to be achieved, I believe that the paper is confusing in several aspects (methods, equations, figures, see below) and many details have not been taken care of before submission. The result is that it is hard to judge the novelty, and the adequacy of the tests carried out. I believe that this paper needs to be restructured and significantly revised before it can be recommended for publication.
I provide below some more detailed comments, referring to specific sections of the text, which is not very easy given the absence of line numbers.
First of all, the methodology and the workflow are unclear. Starting from the abstract, the authors never mention that they used an actual device and measurements to validate their models, rather than, e.g., using simulated data. I was surprised when I read through Section 3, and I found out that they were actually providing results based on real data. This should be clearly explicated in the abstract and introduction. Furthermore, the experimental setup is not described, which is a bit confusing. I am not an expert in photoelectric tracking of satellites, but in my view, it would be challenging for an expert to judge this paper and evaluate the results. Similarly, at the end of Page 5 the authors state “The improved Laplacian algorithm was implemented and verified on all the measurement data of a satellite that was actually tracked.”. I believe that this is a rather vague statement, and more details should be provided, otherwise it’s very hard to understand what Figures 1 and 2 describe. As a general suggestion to address this comment, I suggest including some sentences about the measurements collected in the abstract and introduction and adding a paragraph in the methods section describing the experimental setup and summarizing the measurements acquired.
One more concern is related to the figures of this manuscript. I believe that they are not adequate for a peer-review manuscript to be published in Aerospace. The x-axis does not really help in understanding the figure, and it should be replaced with units of time or frequency, since the authors know the scale between the current unit and time/frequency. The x-axis label is absent, and the y-axis label is not descriptive, I think it should include more details on which errors are described. Finally, the image captions must provide more details about the figures themselves and a general takeaway message, in their current state they are useless.
Minor comments
Eq 1: this equation is cut. Also, please provide a description of what the various terms refer to. In the current state, not even one symbol is described, and this can be quite frustrating for readers that are not familiar with the spherical harmonics expansion of the gravity potential. The description of the equation should also state if the coefficients are normalized, and which normalization is used.
Between Eq. 1 and Eq. 2: 10-4 m/S2 ïƒ 10-4 m/s2
Eq.2: Please state why only J2 is retained. I agree it’s the largest term by far because of the falloff of the acceleration with the distance and the harmonics degree, but this should be stated.
Eq. 7: what is the solid coordinate system? Do you mean the non-inertial reference frame ECEF?
Section 2.2: “Large- amplitude Gaussian white noise in the measurement data was eliminated to improve the accuracy of the measurement data” - this is unclear, and part of my largest concern that the authors have been approximative in the preparation of the document. Please clarify.
“Because the Kalman filter algorithm has been described in detail in many studies, these five formulas are not explained here.” I am okay with not describing the Kalman filter in its details because it’s a well-established tool, but at the very least the details of the implementation should be described. Kalman filtering usually required some fine tuning of some of the parameters, such as the process and measurement noise covariances, and the initial state covariance matrix. Since the Kalman filter is an important part of this manuscript, I would expect to see these details at the very least. Furthermore, if you do not want to provide the theoretical background of the Kalman filter, then all the Equations 14-15-2-3-4 (numbering is wrong) should be removed because they are not described.
Author Response
Comments 1:First of all, the methodology and the workflow are unclear. Starting from the abstract, the authors never mention that they used an actual device and measurements to validate their models, rather than, e.g., using simulated data. I was surprised when I read through Section 3, and I found out that they were actually providing results based on real data. This should be clearly explicated in the abstract and introduction. Furthermore, the experimental setup is not described, which is a bit confusing. I am not an expert in photoelectric tracking of satellites, but in my view, it would be challenging for an expert to judge this paper and evaluate the results. Similarly, at the end of Page 5 the authors state “The improved Laplacian algorithm was implemented and verified on all the measurement data of a satellite that was actually tracked.”. I believe that this is a rather vague statement, and more details should be provided, otherwise it’s very hard to understand what Figures 1 and 2 describe. As a general suggestion to address this comment, I suggest including some sentences about the measurements collected in the abstract and introduction and adding a paragraph in the methods section describing the experimental setup and summarizing the measurements acquired.
Author response 1:
In the abstract the sentence “This method has been experimentally validated on an actual optical measurement device” was added.
At the beginning of the last paragraph in the introduction, the sentence “Implement this method on actual optical measurement device and conduct experimental verification.” was added.
Before Figure1, added a paragraph in the methods section, described the experimental setup and summarized the measurements acquired.
In the paragraph Below Figure2, the describe of X-axis and Y-axis is modified. The label of X-axis is changed to time(20ms).
Comments 2:One more concern is related to the figures of this manuscript. I believe that they are not adequate for a peer-review manuscript to be published in Aerospace. The x-axis does not really help in understanding the figure, and it should be replaced with units of time or frequency, since the authors know the scale between the current unit and time/frequency. The x-axis label is absent, and the y-axis label is not descriptive, I think it should include more details on which errors are described. Finally, the image captions must provide more details about the figures themselves and a general takeaway message, in their current state they are useless.
Author response 2:
In the paragraph Below Figure2, the describe of X-axis and Y-axis is modified. The label of X-axis is changed to time(20ms). The image captions are modified too.
Comments 3: Eq 1: this equation is cut. Also, please provide a description of what the various terms refer to. In the current state, not even one symbol is described, and this can be quite frustrating for readers that are not familiar with the spherical harmonics expansion of the gravity potential. The description of the equation should also state if the coefficients are normalized, and which normalization is used.
Author response 3:
Because of the formatting and layout issues, formula(1) is cut, now it is already modified.
In the paragraph Below formula(2), some symbols are described, and the value of the constant has been given.
In formula(9)，Vector is a normalized vector, ,,；
Comments 4: Between Eq. 1 and Eq. 2: 10-4 m/S2 à 10-4 m/s2
Author response 4: It’s clerical error. now, changed 10-4 m/S2 to 10-4 m/s2.
Comments 5: Eq.2: Please state why only J2 is retained. I agree it’s the largest term by far because of the falloff of the acceleration with the distance and the harmonics degree, but this should be stated.
Author response 5: Add explanation that “for low orbit short arc segment prediction, it can meet the requirements”.
Comments 6: Eq. 7: what is the solid coordinate system? Do you mean the non-inertial reference frame ECEF?
Author response 6: Yes, solid coordinate system Also known as ECEF
Comments 7: Section 2.2: “Large- amplitude Gaussian white noise in the measurement data was eliminated to improve the accuracy of the measurement data” - this is unclear, and part of my largest concern that the authors have been approximative in the preparation of the document. Please clarify.
Author response 7: In order to understand easily, change the “Large- amplitude Gaussian white noise” into “Large noise”.
Comments 8: “Because the Kalman filter algorithm has been described in detail in many studies, these five formulas are not explained here.” I am okay with not describing the Kalman filter in its details because it’s a well-established tool, but at the very least the details of the implementation should be described. Kalman filtering usually required some fine tuning of some of the parameters, such as the process and measurement noise covariances, and the initial state covariance matrix. Since the Kalman filter is an important part of this manuscript, I would expect to see these details at the very least. Furthermore, if you do not want to provide the theoretical background of the Kalman filter, then all the Equations 14-15-2-3-4 (numbering is wrong) should be removed because they are not described.
Author response 8:
An explanation has been added to the five steps of Kalman filtering, and a field description has been added at the end of the formula to describe the process of Kalman filtering.
The original text introduced the initial settings for measuring noise R and system noise Q in section 3.1.
Round 2
Reviewer 1 Report (Previous Reviewer 1)
Comments and Suggestions for AuthorsThe authors now addressed the previous comments and questiones. In newly added text the following minor correction should be made.
The meaning of the variables Plm, Clm, Slm, ae in eq. (1) are not described. After eq. (2) there is a wrong description “φ is the Earth's radius”.
Author Response
Reviewer's comments: The meaning of the variables Plm, Clm, Slm, ae in eq. (1) are not described. After eq. (2) there is a wrong description “φ is the Earth's radius”.
Author response:
A paragraph has been added under Equation 1 to describe the variables.
After Equation 2, the description "φ is the Earth's radius” is an error. It has been corrected.
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe current version of the paper addresses the reviewer's requests.
Comments on the Quality of English Language.
Author Response
Thank you very much for your review.
Reviewer 3 Report (New Reviewer)
Comments and Suggestions for AuthorsI thank the authors for their response to my comments. I think that the paper is in good shape for publication.
Author Response
Thank you very much for your review
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper “Improved Laplace Satellite Tracking Method Based on Kalman Filter” is devoted to Kalman filter application to the satellite tracking measurements. The paper is not clear from several points of view, the theoretical part of the paper is not correctly described, the presented test results are not clear. So, the paper cannot be published in its present form.
The main critical comment is that the authors claim that the proposed improved method is able to predict the orbit in case of temporary loss of the satellite by the optical tracking system. However, there is no any description in the text how actually the prediction or extrapolation of the satellite track is carried out. The Laplace algorithm is used for initial conditions estimation only, and the Kalman filter just smooth the measurements noise for the Laplace algorithm. The accuracy of the orbital parameters prediction depends on the models of the orbital motion used and on the prediction methods, which are not described in the text.
Too many designations used in the paper are not introduced in the text, which makes it almost impossible to follow the mathematical derivations. All the used coordinate systems should be described in the text (origin and axis directions should be explained). The expression (1) of the force acting on the satellite is questionable, (it seems that it is not gravitational force, why it is linear, what are “A, B, C” and so on). It is completely unclear how to obtain (7) from (6), why "R the with top line" (which is not introduced) can be eliminated?
The figures in the text are not clear. The axis labels should be correctly named. What does ‘frame (20ms)”, “miss (0.6’’)”, “error of Beam” mean? It is not clear why “the miss distance is extracted as theoretical 0"?
According to the (14)-(17) the authors assume that the measured angles are constant (i.e. matrix A is identity). It is quite a strange assumption since the satellite is moving along the observed arc. At least a constant angular velocity model should be used. It seems that the authors not correctly understand the meaning of the Kalman filter parameters R and Q. Q is the covariance matrix of the model errors, it is not correlated with Laplacian algorithm errors. R is the covariance matrix of the measurement errors, not of the servo control accuracy. So, these parameters in Section 3 in line 361 are not correctly set.
In Section 3 it is not clearly described the details of the experimental data. What are the satellite orbital parameters of satellites (a), (b) and (c) in Fig. 7 and Fig. 8. What is the accuracy of the initial measurements? At what moment of time the measurements are not temporary available and the orbit is predicted?
Comments on the Quality of English LanguageThe English in the text should be significantly improved, now it is difficult to understand. For example, “the horizontal type of horizontal transformation of the horizontal type” in line 310, “the horizontal and horizontal transformation of the horizontal form” in line 316, “the ability to improve the time” in line 355, and so on.
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper deals with the use of Kalman filter to improve the tracking of a satellite. The paper is actually unreadable. For sure, one issue is the very poor English language that prevents the major part of the paper reading. Many statements are not understandable also after four/ five readings. There are many repetitions of the same concepts (and same statements) in different paragraphs. Moreover, the paper misses of scientific accuracy, a lot of important data are not present or roughly presented. Nothing has been done by the authors to support the reader oin the understanding the paper contents: e.g. a figure with the adopted reference frame, flow charts describing the method, unreadable figures. Finally, also the technical part is questionable in some part but it results complicated to express a true judgment on this work due to the difficulty to understand the major part of it.
Just some major remarks:
Section1:
The level of English of the first statement is explicative of the rest of the paper.
Problem description and gap of the actual state of art are not clear. It would be clear the role of the Kalman filter but what is the real improvement? What is the final requirement about the tracking of a satellite in the applications considered by the authors?
Statement in lines 46-48, another example of not understandable statement. I could interpret the sense but it is clearly confused.
Short-time and long-time target loss: what does it mean “short” and “long”? ms, seconds, minutes?
Statement in lines 48-50 very confusing description for long term conditions
Lines 64. “as a guarantee” what does it mean?
Statements in lines 65-70 are fragmented and not supported by literature references that show the problem and generate the gap.
What is the logic consequence/accuracy from statement in line 71-72 and the next one?
Statements in line 74-82 present new elements neither described nor introduce before or referenced such as image processing and servo automatic closed loop tracking mode. They are not described and contextualised in this paper. Which is the image process condition? Is servo automatic system intended as the actuator used to move the detection system? Which parameters of this system are important? For example, which is the minimum tracking command and, in consequence, movement? Which is the influence of the jitter? This is an example of incomplete and poor description of the context that gets hard to focus the problem.
Section 2:
The proposed one is a strange way to describe a satellite orbit.
A description of reference frames misses.
Lines 149-168 This part is known but the measurement error due to faster movement of the satellite for high elevation angles depends also on the performance of the tracking mechanism. How is it considered by the authors?
Lines 169-173. Continuous repetitions of the same concepts without demonstrating them.
Lines 178 -183. So a traditional KF is used. Do the authors conform that they are working on a linearised problem?
Line 185-187. Not only by the images processing but also by the image acquisition time, the capability of the processor, the performance of the tracking system. Part of that is confirmed by the following statements. This is a bad way to write a scientific paper when logic consequences and justification of the statements largely miss.
Scale on the axes of fig 3 and 4 is unclear.
Which are the warp and weft angle??? Need an explanation...
Section 2.2.3 a figure with the workflow would help. Now the encoder data appear… bad description again. A description of the entire system on the top the paragraph/paper would help.
Largely confusing description of the method (Section 2.3. a figure with the workflow would help). “Data guidance” what is this?
Section 3:
A figure that supports the test description and execution would help.
Results could be promising (maybe), their description is very limited and sometimes unclear. And, sincerely, I stopped to read the paper in this section because it was a very strong effort and, maybe, unuseful. I definitely suggest to reject this paper
Comments on the Quality of English Language
Low English Language level.
