Pose Measurement and Motion Estimation of Space On-Orbit CubeSats Based on Micro-Doppler Effect Using Laser Coherent Radar

Round 1

Reviewer 1 Report

I appreciate the paper, but I suggest the authors to add a comprhensive anaktìysis of measurement uncertainty of the measurement technique.

Author Response

We are indebted to you for your helpful comments. We have carefully studied the comments and made major revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

I suggest the authors to add a comprehensive analysis of measurement uncertainty of the measurement technique.

 

Our reply:

We appreciate the referee’s advice. And accordingly, we add Fig. 15 to display absolute and relative errors of our experiment results, and the corresponding discussion about Fig. 15 is added at the end of Section 4 (Experiment results), which are as follows.

 

“Additionally, the absolute and relative errors for the estimation of the attitude angle with a =45˚ and R₀=80 m is plotted in Fig. 15. It can be seen that the maximum absolute error is less than 5˚, relative error no greater than 11%. The error roughly grows with increasing spin rate. This may due to the strong noise under high spin rate which hinders the edge extraction.”

 

The revised words have been marked with highlight in our manuscript.

 

Thanks very much.

 

Sincerely yours,

Zhengjia Wang

Author Response File: Author Response.docx

Reviewer 2 Report

The manuscript presents an attempt to obtain information on spin and orientation of unknown rotating object using a lidar system. The task is an important part of space debris mitigation efforts and automatic space navigation. However, although the title and abstract state that objects are supposed to be non-cooperative, i.e. have unknown properties, the present algorithm works only for 1) cubes 2) rotating along a particular symmetry axis passing through centers of opposing surfaces. That can hardly be termed "non-cooperative". There are other numerous problems, in particular, even for very cooperative targets (cubes specifically rotated) the algorithm's performance can be non-perfect, especially for slow and fast rotations.

 

In my opinion, at this stage publication in a scientific journal is impossible. A significant advance in technique is required to become publishable. In particular, for the title including "non-cooperative" a double-blind experiment must be performed, namely, one team prepares several unknown objects, another team rotates the objects, yet another team collects experimental data. No communication between the teams is allowed. The final team, having no prior information, applies whichever methods they want to the lidar data, trying to determine the objects shape, size, and spin properties. After that, all teams compare the results with the shapes and rotation speeds and orientations initially prepared.

 

1. Lines 33-34: "The Doppler frequency is associated with the transmitting wavelength and the radial velocity of the target with respect to the radar." This sentence sounds awkward, please reword.

 

2. Fig. 1: Please show the Lidar's coordinate system (U,V,W); please show the azimuth angle alpha in terms of the Lidar's system (U,V,W) as now it appears connected to the target. The authors also use a term "attitude angle" which is not shown in Fig. 1.

 

3. Line 71: "In this work, we choose beta=0" Please state if this assumption limits possible target location or motion.

 

4. Lines 76-77: "the angle fi rotates about the z-axis, the angle theta rotates about the x-axis, and the angle theta rotates about the z-axis again." There is probably a misprint as the angle theta is mentioned twice, while fi-curled is not mentioned at all. Please also show these angles in Fig. 1.

 

5. A rotation matrix is written in Line 79; I suggest to write matrices in their 2D form as 3x3 table for easier understanding.

 

6. By the way, this rotation matrix assumes rotation around z-axis only. Please state if rotations around other two axes affect the results. Also, what happens if all rotations are present simultaneously?

 

7. Lines 89-90: " r_p(t) the distance between the target center and the detector at time t" Please check that it is not the distance between the target point P and the detector at time t.

 

8. Eq (3): please describe the quantities used; please check if "i pi f_0" is correct (2 is missing?)

 

9. Lines 100-102: A data pretreatment algorithm is described very briefly. However, its advantages/disadvantages are not clear even after reading the following discussions. Looking at Figs. 2 and 3, I would guess that the original time-frequency diagrams are more informative and less noisy. The authors should explain in more detail why the original time-frequency diagrams are not suitable for processing.

 

10. As follows from the title, the target is assumed to be non-cooperative. Namely, the authors cannot know that the target is a cube; it could be a sphere or a tetrahedron or something else. Thus, the coefficient "4" in Eq (8) should be determined somehow from the data and not from the prior knowledge.

 

11. By the way, the authors also assume that the cube rotates around a certain symmetry axis. What happens if the axis is arbitrary?

 

12. Generally speaking, a free object rotation is complicated: in particular, the rotational energy transfers from one axis to another, unless the rotation axis coincides with the one with minimum or maximum momenta of inertia:
    https://www.youtube.com/watch?v=BPMjcN-sBJ4

 

13. Please check if Eqs (18), (19), and (24) are correct – there are too many vertical bars, it seems, and parentheses without content.

 

14. The authors should mention what happens in a case of "shiny" (mirror-reflective) target, which may happen with space debris.

 

15. In simulations, the authors used the attitude angles from 15 up to 90 Deg. Why not to try 0 Deg as well?

 

16. Figs. 9 and 10 show reconstruction of the attitude angle from low-noise data (SNR = 10 dB). Why not to show SNR = -10 dB results as well?

 

17. Fig. 14 shows attempt to extract some information from an experimental data. Human eye can see the period on both frames (a) and (c); however, the authors algorithm fails completely for (a) and shows non-satisfactory behavior for (c). This implies that the algorithm needs improvements, or a completely different algorithm is necessary.

Author Response

We are indebted to you for your helpful comments. We have carefully studied the comments and made major revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

The manuscript presents an attempt to obtain information on spin and orientation of unknown rotating object using a lidar system. The task is an important part of space debris mitigation efforts and automatic space navigation. However, although the title and abstract state that objects are supposed to be non-cooperative, i.e. have unknown properties, the present algorithm works only for 1) cubes 2) rotating along a particular symmetry axis passing through centers of opposing surfaces. That can hardly be termed "non-cooperative". There are other numerous problems, in particular, even for very cooperative targets (cubes specifically rotated) the algorithm's performance can be non-perfect, especially for slow and fast rotations.

In my opinion, at this stage publication in a scientific journal is impossible. A significant advance in technique is required to become publishable. In particular, for the title including "non-cooperative" a double-blind experiment must be performed, namely, one team prepares several unknown objects, another team rotates the objects, yet another team collects experimental data. No communication between the teams is allowed. The final team, having no prior information, applies whichever methods they want to the lidar data, trying to determine the objects shape, size, and spin properties. After that, all teams compare the results with the shapes and rotation speeds and orientations initially prepared.

Our reply:

Thanks for the referee’s constructive suggestion. Our work mainly focuses on the motion estimation of on-orbit CubeSat. We agree that the term “non-cooperative” is inappropriate to describe the target. Therefore, we revised the title and made a limitation in the text. We modified “Non-Cooperative Targets” into “On-orbit CubeSats”.

 

1. Lines 33-34: "The Doppler frequency is associated with the transmitting wavelength and the radial velocity of the target with respect to the radar." This sentence sounds awkward, please reword.

Our reply:

We changed this sentence to “The Doppler frequency shift is associated with the transmitting wavelength and the relative velocity between the radar and the target.”

 

2. Fig. 1: Please show the Lidar's coordinate system (U,V,W); please show the azimuth angle alpha in terms of the Lidar's system (U,V,W) as now it appears connected to the target. The authors also use a term "attitude angle" which is not shown in Fig. 1.

Our reply:

We modified Fig. 1 according to the referee’s suggestion. The calibration of the Lidar’s coordinate system and the azimuth and elevation angle of the target is improved. In addition, the explanation of attitude angle α is added in the first paragraph of Sec. II as: “the unit vector of the target along the line of sight (LOS) of the radar is given by n=(0, sin α, cos α), where the attitude angle α denotes the angle between the LOS of the radar and the z-axis”

 

3. Line 71: "In this work, we choose beta=0" Please state if this assumption limits possible target location or motion.

Our reply:

β is the elevation angle of the target relative to the lidar coordinate. We choose β=0, which will restrict the lidar and the target to always be in the same plane, thereby simplifying the calculation. This condition can also be easily achieved by adjusting the lidar transmitting plane in the experiment.

 

4. Lines 76-77: "the angle fi rotates about the z-axis, the angle theta rotates about the x-axis, and the angle theta rotates about the z-axis again." There is probably a misprint as the angle theta is mentioned twice, while fi-curled is not mentioned at all. Please also show these angles in Fig. 1.

Our reply:

We are sorry that this is a writing error. We changed “θ rotates about the z-axis again” to “φ rotates about the z-axis again”. And we marked the Euler angles (ϕ, θ, φ) in Figure 1.

 

5. A rotation matrix is written in Line 79; I suggest to write matrices in their 2D form as 3x3 table for easier understanding.

Our reply:

According to the referee’s advice, we rewrite the matrix in their 2D form as 3×3

6. By the way, this rotation matrix assumes rotation around z-axis only. Please state if rotations around other two axes affect the results. Also, what happens if all rotations are present simultaneously?

Our reply:

Spinning targets always rotate about their geometric symmetry axis. When rotating around different coordinate axes, it needs to be multiplied with different rotation matrices. At the same time, the angular velocity parameter corresponding to the rotation axis needs to be introduced. But the results are similar. In this work, we only focus on the related research work on on-orbit CubeSats. For this kind of target, its spin only revolves around one axis of geometric symmetry, and there will be no tumbling motion around multiple axes.

 

7. Lines 89-90: " r_p(t) the distance between the target center and the detector at time t" Please check that it is not the distance between the target point P and the detector at time t.

Our reply:

Thanks for the referee’s advice. This sentence has been changed to “r_p(t) the distance between the scattering point of the target and the detector at time t.”

 

8. Eq (3): please describe the quantities used; please check if "i pi f_0" is correct (2 is missing?)

Our reply:

We are sorry that this is a mistake and eq (3) is revised accordingly. In addition, the following description is added in lines 94: “In our model, the target consists of K scattered points. Thereby, the returned superposition signal from the surface scattering of the laser light is

eq (3). 

ρ_k and r_k(t) represent the reflectivity function and distance of the k-th scattering point, respectively.”

9. Lines 100-102: A data pretreatment algorithm is described very briefly. However, its advantages/disadvantages are not clear even after reading the following discussions. Looking at Figs. 2 and 3, I would guess that the original time-frequency diagrams are more informative and less noisy. The authors should explain in more detail why the original time-frequency diagrams are not suitable for processing.

Our reply:

The purpose of preprocessing is to enlarge the edge details of the image, thereby improving the effect of edge extraction. We have modified Figures 2 and 3 and added zoomed-in views of the local images. The maximum color difference of figure a before pretreatment is only 1 (brown and red), while that of figure 3b after pretreatment can reach 10 (red and blue). Thus, the preprocessing can enhance the effect of edge extraction.

 

10. As follows from the title, the target is assumed to be non-cooperative. Namely, the authors cannot know that the target is a cube; it could be a sphere or a tetrahedron or something else. Thus, the coefficient "4" in Eq (8) should be determined somehow from the data and not from the prior knowledge.

Our reply:

As mentioned above, we changed the “Non-cooperative Targets” in the title to the “CubeSats”. For a cube target, the characteristic frequency f_f is four times as much as the period of spinning f_s due to its symmetric properties.

 

11. By the way, the authors also assume that the cube rotates around a certain symmetry axis. What happens if the axis is arbitrary?

Our reply:

According to the object structure dynamics, the spin of the regular object rotates around its axis of geometric symmetry. In our work, the main focus is on the motion state of the on-orbit CubeSats. Such targets exhibit a high geometric symmetry, with the axis of rotation passing through the center of mass. Therefore, we do not consider the case where the axis of rotation is arbitrary in this work.

 

12. Generally speaking, a free object rotation is complicated: in particular, the rotational energy transfers from one axis to another, unless the rotation axis coincides with the one with minimum or maximum momenta of inertia:
    https://www.youtube.com/watch?v=BPMjcN-sBJ4

Our reply:

Thanks for the referee’s suggestion. As mentioned above, since our work is actually aimed at the study of on-orbit CubeSats, the rotation of this target is regularly carried out around the axis of geometric symmetry, without other rolling motions. But this is very valuable advice, we will consider this issue in future work.

 

13. Please check if Eqs (18), (19), and (24) are correct – there are too many vertical bars, it seems, and parentheses without content.

Our reply:

This part of the problem is caused by the different formula editors we used when writing the manuscript. Some content incompatibilities occurred during format conversion, resulting in some abnormal displays. We have made corresponding changes.

14. The authors should mention what happens in a case of "shiny" (mirror-reflective) target, which may happen with space debris.

Our reply:

This is very good question. We normalize the echo signal. For scattering points with larger scattering coefficients, we will weaken the echo intensity. Conversely, we will strengthen the echo intensity for the scattering points with small scattering coefficients. This ensures that the echo signal of each scattering point will be received and reflected in the time-frequency diagram.

 

15. In simulations, the authors used the attitude angles from 15 up to 90 Deg. Why not to try 0 Deg as well?

Our reply:

First, the simulation can start from 0°. But in our work, the radial Doppler signal is measured. If the attitude angle is 0°, the frequency of the echo signal will not change with the target spin, that is, the micro-Doppler frequency shift phenomenon cannot be observed. It doesn't make much sense for our work.

Moreover, the attitude angle is 0° is a special case, and it is impossible to keep the attitude angle of 0° unchanged in practice. Therefore, we chose the angle range with obvious Doppler frequency shift for discussion instead of 0°.

 

16. Figs. 9 and 10 show reconstruction of the attitude angle from low-noise data (SNR = 10 dB). Why not to show SNR = -10 dB results as well?

Our reply:

In this work, we obtain the pose information of the target based on image processing. When the SNR is too low (SNR=-10dB), it is difficult to extract the image edge, and almost no effective image information can be obtained. This will result in the inability to complete subsequent parameter extraction.

Moreover, for the actual detection system, the SNR of the system is required to be higher than 0 dB in the design. Only in this way can the system receive the echo signal normally. Therefore, in order to obtain better simulation results, we choose SNR=10dB to carry out the relevant calculation and obtain Figs. 9 and 10.

 

17. Fig. 14 shows attempt to extract some information from an experimental data. Human eye can see the period on both frames (a) and (c); however, the authors algorithm fails completely for (a) and shows non-satisfactory behavior for (c). This implies that the algorithm needs improvements, or a completely different algorithm is necessary.

Our reply:

The purpose of drawing figure 14 is to show that edge extraction can be better realized after threshold processing of the TF image. The original fig. 14b is already our best result for the edge extraction directly from figure 14a. But it seems to cause some misunderstandings to readers, we have deleted it in the revised manuscript.

 

The revised words have been marked with highlight in our manuscript.

 

Thank you very much.

 

Sincerely yours,

Zhengjia Wang

 

Author Response File: Author Response.docx

Reviewer 3 Report

The paper describes an interesting approach for the assessment of the structure and orientation of a non-cooperating target, either in orbit for rendezvous maneuvers, or in industry applications. In terms of language the paper is well written, but having said this, there are a large number of small language errors that still need attention. The attached document identifies some, but not all of these stylistic requirements.

 

The structure of the paper needs some further attention. I believe it should be considerably shortened and focused. To the reader it is not clear what the merit of figures 6, 7 and 9 are. Furthermore, there is a serious discrepancy in the description of the experiment between figure 1 and figure 12, which needs resolving. I think that the difference in geometry has a large impact on the described results.

 

I am not sure what the signal levels of 10 dB, 2 dB and -10 dB refer to. How are they defined. If I look at the 20 Hz signals at fig. 8, I cannot relate that to the indicated signal powers. Table 3 shows the spin rate in voltages, which is essentially meaningless. Please replace this by units like degrees per second i.e. merge it with table 4. I think it is unnecessary to specify the error of the rotation rate in this context.  

Comments for author File: Comments.pdf

Author Response

We are indebted to you for your helpful comments. We have carefully studied the comments and made major revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

1. The paper describes an interesting approach for the assessment of the structure and orientation of a non-cooperating target, either in orbit for rendezvous maneuvers, or in industry applications. In terms of language the paper is well written, but having said this, there are a large number of small language errors that still need attention. The attached document identifies some, but not all of these stylistic requirements.

Our reply:

Thanks for the referee’s suggestion. Accordingly, we correct language errors or writing errors in the manuscript. The details of the revision are as follows, and the revised results is highlighted in our manuscript.

Line 21: “not strong” has been changed to “not too high”.

Line 23: “approach can fulfills” has been changed to “approach fulfills”.

Line 31: “It worth” has been changed to “It is worth”.

Line 32: “the carried frequency of the returned signal” has been changed to “the carrier frequency of the return signal”.

Line 43: “study” has been changed to “studies”.

Line 56: “for microwave” has been changed to “for the microwave”.

Line 77: “θ rotates about the z-axis again” has been changed to “φ rotates about the z-axis again”.

Line 125: “precised acquired” has been changed to “precisely acquired”.

Line 171: “need” has been changed to “needs”.

Line 195: “louder noise” has been changed to “higher noise”.

Line 201: “loud noise” has been changed to “high noise”.

Line 261: “fs=ff/4=5Hz” has been changed to “f_s=f_f/4=5Hz”.

Line 274: “estimating attitude” has been changed to “estimating the attitude”.

Line 296: “15º~60º ”, the display is abnormal due to formatting problem, which we have corrected.

Line 311: “from fiber laser” has been changed to “from the fiber laser”.

Line 318: “transmitted in the digital storage oscilloscope” has been changed to “transmitted to the digital storage oscilloscope”.

Line 323: “driving rotating target” has been changed to “driving the rotating target”.

Line 328: “of phase function” has been changed to “of the phase function”.

Line 329: “returned signal” has been changed to “return signal”.

Line 332: “via photodetector” has been changed to “via the photodetector”.

Line 333: “transmitted in the digital storage oscilloscope” has been changed to “transmitted to the digital storage oscilloscope”.

Table 3: “Spin voltage: 6V, 10V, 15V, 20V, 25V” has been changed to “Spin rate: 1.4Hz, 2.5Hz, 3.8Hz, 5.0Hz, 6.4Hz”.

Line 366: “may due to” has been changed to “may be due to”.

Line 381: “Double Edge Ratio Method” has been changed to the Double Edge Ratio Method”.

Line 385: “fulfills” has been changed to “fulfill”.

2. The structure of the paper needs some further attention. I believe it should be considerably shortened and focused. To the reader it is not clear what the merit of figures 6, 7 and 9 are. Furthermore, there is a serious discrepancy in the description of the experiment between figure 1 and figure 12, which needs resolving. I think that the difference in geometry has a large impact on the described results.

Our reply:

According to the referee’s advice, we added discussions of Figures 6, 7, and 9 to make the meaning of our work clearer to readers. For Figure 6, we added corresponding description at the end of the second paragraph of Section 2.2 “Estimation of attitude angle”, which are as follows: “The threshold is one of the important algorithm parameters for the parameter extraction from TF images. It is often necessary to dynamically adjust the threshold according to the SNR of the image to ensure the accuracy of feature extraction. As can be seen from Fig. 6, the threshold coefficient of the method adopted in this paper is rarely affected by the variation of the attitude angle. Therefore, the attitude angle can be extracted through a unified threshold and the algorithm for extraction of the attitude angle is greatly simplified.”

For Figure 7, we added corresponding depiction at the beginning of the third paragraph of Section 2.2 “Estimation of attitude angle”, which is as follows: “Fig. 7 shows dependence of the optimized threshold on the spin rate of the target. It can be seen that the threshold required by the method used in this work remains almost unchanged when the SNR is low. When a long-distance target is detected, the corresponding TF images generally have a low SNR. In this case, our method is able to obtain effective results, which reduces the difficulty of the algorithm for extracting the spin parameter in practical applications.”

For Figure 9 we added corresponding depiction as follows: “This is called the best LOS angle window, in which the absolute error is less than 0.5˚. It provides a scientific basis for arranging the optimal measurement strategy for high-precision satellite attitude angle measurement.”

Figure 1 is the position diagram of the lidar and the target to be measured. The purpose of this figure is to illustrate the geometric relationship between the lidar and the target. Figure 12 is a schematic diagram of the experimental equipment, which aims to show the working process of the experiment. It can be understood that the cube in figure 1 corresponds to the target in figure 12, and the lidar Q in figure 1 is equivalent to the rest of figure 12.

It should be noted that in figure 12, the transmitting antenna and the receiving antenna share one antenna. In order to show the two processes of signal transmission and reception more clearly, we show the two separately when drawing the schematic diagram.

3. I am not sure what the signal levels of 10 dB, 2 dB and -10 dB refer to. How are they defined. If I look at the 20 Hz signals at fig. 8, I cannot relate that to the indicated signal powers. Table 3 shows the spin rate in voltages, which is essentially meaningless. Please replace this by units like degrees per second i.e. merge it with table 4. I think it is unnecessary to specify the error of the rotation rate in this context.

Our reply:

We set the SNR for the whole frequency band according to the formula 10lgPs/Pn. And in the simulation, the SNR is adjusted by varying the emission power of the laser. Figure 8 shows the 2D Fourier transform results of the time-frequency diagram. Since the Fourier transform is only calculated for part of the frequency band, the three signal powers at 20Hz are not completely consistent with the set SNR in figure 8. As can be seen from the figure, the higher the SNR yields the greater the peak value of the 2D Fourier transform spectrum is valid.

In this work, we mainly focus on extracting the motion information of the target from the TF image and do not pay too much attention to the SNR of the system which is mainly related to the hardware parameters.

In the experiment, we use a DC motor to drive the target for spin. That is, the spin rate is controlled by voltage. Obviously, voltage is the direct and easily controllable independent variable in this work. According to the referee’s advice, we made corresponding modifications to table 3, but we hope table 3 and 4 not be merged.

The revised words have been marked with highlight in our manuscript.

 

Thank you very much.

 

Sincerely yours,

Zhengjia Wang

Author Response File: Author Response.docx

Reviewer 4 Report

The introduction of the manuscript is poor. It does not provide any references of the previews work nor does provide enough information for the reader to understand the rest of the manuscript. Following information should be added to the introduction:

- Reference to previous work done in this field

- Basic principles of Doppler effect should be explained (how the movement influences the frequency, what components of the velocity can be measured (radial, tangential?) etc…). How bulk movement, rotational movement and vibrations influence the signal and how can these three type of movements can be distinguished from the signal (what is typical signal for bulk movement, from rotational movement, from vibrations and why)

- The principles of measurement: how does the laser radar operates, what types of signal does it collect (frequency, intensity, delay?), how does the beam interact with the target (how it reflect form the target), does the signal include information of a single point or number of points (and how these points are denoted…)

- What are the limitations: It is mentioned that shorter wavelength is more appropriate for detecting low vibration rates. This should be explained. Also, does it suggest the advantage of laser over radio waves of violet light over the red light. What quantitative difference does it make (what are the limitations of rate with one and with another wavelength). Throughout the text, the cube-shaped target is discussed – is the method appropriate only for the cube-shaped object or for the other shaped objects as well and to what extend?

 

Section 2 defines three coordinate systems, but just one is presented on Figure 1. Moreover, the meaning of the point P is not explained. Also text mentions that vector n as vector in radar coordinate system, but on Figure 1 vector n is presented in object coordinate system – which is correct.

 

I was not able to follow the manuscript from the section 2.1 onward as symbols are either not defined at all or inadequately explained (what does the “signal” g mean , what are M, N, u, v,….), what is the altitude in respect to coordinates system explained in section 2, how is signal to noise ratio measured in decibels….

Author Response

We are indebted to you for your helpful comments. We have carefully studied the comments and made major revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

1. The introduction of the manuscript is poor. It does not provide any references of the previews work nor does provide enough information for the reader to understand the rest of the manuscript. Following information should be added to the introduction:

- Reference to previous work done in this field

- Basic principles of Doppler effect should be explained (how the movement influences the frequency, what components of the velocity can be measured (radial, tangential?) etc…). How bulk movement, rotational movement and vibrations influence the signal and how can these three type of movements can be distinguished from the signal (what is typical signal for bulk movement, from rotational movement, from vibrations and why)

- The principles of measurement: how does the laser radar operates, what types of signal does it collect (frequency, intensity, delay?), how does the beam interact with the target (how it reflect form the target), does the signal include information of a single point or number of points (and how these points are denoted…)

What are the limitations: It is mentioned that shorter wavelength is more appropriate for detecting low vibration rates. This should be explained. Also, does it suggest the advantage of laser over radio waves of violet light over the red light. What quantitative difference does it make (what are the limitations of rate with one and with another wavelength). Throughout the text, the cube-shaped target is discussed – is the method appropriate only for the cube-shaped object or for the other shaped objects as well and to what extend?

Our reply:

-We appreciate the referee’s suggestions. In the introduction, we add references (Ref. [20-22] in the manuscript) which refers to the relevant work by previous studies.

-In 2003, Victor C. Chen proposed the micro-Doppler effect in his article [1]. He pointed out that the target itself or some of its components have additional mechanical motion in addition to the main direction of the target itself, which will cause the frequency modulation of the radar echo, resulting in the side lobes of the radar Doppler effect, which is called the micro-Doppler effect. The scattering characteristics, geometric structure, motion characteristics, and attitude information of the target can be obtained by signal processing methods. Exhibiting the details of micro-Doppler effect is quite a long story. Fortunately, it can be found in Ref [1] (or Ref [4-8] in the manuscript). Thereby, we only introduce Doppler and micro-Doppler effects briefly in the first and second paragraph in the introduction.

-Laser detection mainly collects intensity signals, the details of coherent laser radar technique can be found in Ref. [2]. In this work, the echo signal is processed and analyzed, and the final discussion is the time-frequency characteristics of the signal. The coherent detection principle employed in our work is shown in figure 12 in the manuscript, and the corresponding introduction of the experimental system is given in the first and second paragraph of Sec. 4.

-The explanation and relative mathematical derivation are illustrated in Ref. [6] in the manuscript. In addition, our manuscript shows that the Doppler frequency shift for the cube-shaped target is given by eq. 17 in the manuscript.

From this specific instance, one can see that  is proportional to f_s and ^-1 for a given time t. It means that Doppler frequency shift f_D is more obvious and easier to detect for larger period of spinning f_s or shorter wavelength λ. Therefore, the laser radar system at short wavelength is needed to attain high precision of the motion estimation. Since the associated mathematical derivation is quite a long story and similar theoretical work has been demonstrated by previous study. In the introduction, we only cite Ref. [6] and state the relative conclusion.

Microwave case corresponds to multiple scattering centers. While the micro-Doppler signals with area scattering modulation is encountered for shorter wavelength. According to the referee’s suggestion, we state that the motion estimation for the shorter wavelength case is totally different in the revised introduction.

In this work, we only focus on the cube target. As for the other shaped targets, in order to accomplish the identification, the associated theoretical model based on the specific geometry needs to be extended. According to the referee’s advice, we emphasize that our work only focus on the cube target in the revised introduction.

 

References

[1] Chen V C. Micro-Doppler effect of micromotion dynamics: a review[C]. Washington: Int Soc for Optical Engineering, 2003:240-249.

[2] Lutzmann P, Frank R, Ebert R. Laser-radar-based vibration imaging of remote objects[C]. Washington: Int Soc for Optical Engineering, 2000: 436-443.

 

2. Section 2 defines three coordinate systems, but just one is presented on Figure 1. Moreover, the meaning of the point P is not explained. Also text mentions that vector n as vector in radar coordinate system, but on Figure 1 vector n is presented in object coordinate system – which is correct.

Our reply:

Thanks for the referee’s suggestion. We modify Figure 1 and add coordinates and other parameter annotations. Where point P refers to an arbitrary scattering point on the target’s surface. We briefly explain the meaning of point P in the second paragraph of section 2.

 

3. I was not able to follow the manuscript from the section 2.1 onward as symbols are either not defined at all or inadequately explained (what does the “signal” g mean , what are M, N, u, v,….), what is the altitude in respect to coordinates system explained in section 2, how is signal to noise ratio measured in decibels.

Our reply:

Due to the multiple formula editors we used to write the manuscript, there were some display anomalies during the format conversion process. We apologize for any disruption to readers' reading. We have carefully examined the manuscript for the type of problems you mentioned and made corresponding corrections.

 

The revised words have been marked with highlight in our manuscript.

 

Thank you very much.

 

Sincerely yours,

Zhengjia Wang

Author Response File: Author Response.docx

Reviewer 5 Report

The paper present work,an approach for pose estimation of Non-Cooperative target based on micro-Doppler effect by laser radar system is proposed. . 

The research idea is interesting but some considerations should be noted. 

1. Highlight more the contributions of the work in the abstract and in the introduction. 

2. In the Introduction, a literature review should be carried out comparing other similar works. The work has few references. 

3. Figures and tables are too large and go beyond alignment with the text, improving formatting.

4. A discussion of the results should be presented, showing the advantages and disadvantages of the opposite method. 

5. Future works should be mentioned in Conclusions.

Author Response

We are indebted to you for your helpful comments. We have carefully studied the comments and made major revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

1. Highlight more the contributions of the work in the abstract and in the introduction.

Our reply:

The main contribution of our work is to propose an approach for pose estimation of CubeSat based on micro-Doppler effect by laser radar system. As far as we know, this is the first attempt to carry out CubeSat pose measurement by utilizing the lidar detection method. In the abstract and the last paragraph of the introduction (Sec. I), the corresponding sentences is added to highlight the contributions of this work.

 

2. In the Introduction, a literature review should be carried out comparing other similar works. The work has few references.

Our reply:

In the Introduction, we add a part of the text in the last paragraph to clarify the characteristics of other relative work, and then demonstrates the significance of our work, which are as follows.

 

“In the field of micro-motion feature detection, the research of other scholars is mainly based on electromagnetic waves in the microwave frequency band [20-22]. Note that for the electromagnetic signal with short wavelength, even the micro vibration with very low vibration rate can induce large phase changes. Therefore, the micro-Doppler frequency modulation or the phase change with time can be easily detected. A coherent laser radar operating at shorter wavelength, can achieve a velocity precision better than longer wavelength [6]. As a consequence, the coherent laser radar operating at short wavelength is needed to attain high precision of the motion estimation. However, the micro-Doppler signals with area scattering modulation, rather than multiple scattering centers for microwave cases, will be encountered. For the microwave case, the motion estimation can be carried out based on the curves with periodicity in the time-frequency (TF) image [17]. While for short-wavelength cases, an appropriate method is required to extract the parameters based on the structured broadband (see Sec. 2) in TF image caused by area scattering. Given that, a novel framework is proposed to estimate the parameters of CubeSats with micro-motion based on micro-Doppler effect by laser radar system in this paper. Namely, the spin rate is estimated based on the 2D Fourier transform of the time-frequency data. The attitude angle can be obtained effectively after threshold processing. The target size is identified through bistatic radar operations. At last, the proposed method is verified by simulations and experiments. The basic principle of the Doppler effect ensures that the parameter estimation accuracy of this method is independent of distance. Therefore, it provides an effective technical approach for measuring the fretting parameters of space rotating objects such as CubeSats with high precision at long distances. Considering that the limit sensitivity of the micro-Doppler measurement process can be close to the quantum noise limit level of laser detection, the scheme studied in this project may provide a new technical means for the detection of non-cooperative targets and space debris in the future.”

 

3. Figures and tables are too large and go beyond alignment with the text, improving formatting.

Our reply:

We appreciate the referee’s advice. We have made changes to corresponding figures and tables (figures 1, 2, 3, 14, and table 3). The format and size of them have now been improved.

 

4. A discussion of the results should be presented, showing the advantages and disadvantages of the opposite method.

Our reply:

We guess “the opposite method” refers to the approach that we used in this paper to carry out the motion estimation of the target. As mentioned in the introduction, this method has the advantage of attaining high precision of the motion estimation due to the coherent laser radar operating at short wavelength. It should be noted that the coefficient “4” in eq. 8 is from the symmetry of the cube target. Namely, in order to obtain the correct spin rate, one needs to know the geometric asymmetry of the target in advance. This is the disadvantage of this method.

Accordingly, we added corresponding discussion at the end of the conclusion.

 

5. Future works should be mentioned in Conclusions.

Our reply:

Thanks for the referee’s suggestions, we add prospect at the end of the conclusion, which are as follows:

“In the future, we will continue and focus on further work on non-cooperative targets such as tumbling satellites, space debris or asteroid, which are concern center of National aerospace community. We hold a positive attitude towards the application of this technology in the field of space security control in the future.”

 

The revised words have been marked with highlight in our manuscript.

 

Thank you very much.

 

Sincerely yours,

Zhengjia Wang

 

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The revised version is kind of better in terms of wording, which has become less obviously wrong. However, the authors made modification of figure 6 to "prove" that their image processing is useful. This modification is absolutely wrong and cannot be published. I made a brief description of this mistake in #6 below. If the authors insist on the usefulness and/or necessity of their image processing method, they should describe everything in one or two paragraphs using scientific (i.e. not misleading, verifiable, and desirably correct) arguments, and possibly with further figures. At present, I do not see an honest attempt to do this.

Further, I still believe that the author's method may be unable to work in a double-blind experiment, i.e. in an experiment modelling an actual situation. Here I repeat what I wrote in my first review: "one team prepares several unknown objects, another team rotates the objects, yet another team collects experimental data. No communication between the teams is allowed. The final team, having no prior information, applies whichever methods they want to the lidar data, trying to determine the objects shape, size, and spin properties. After that, all teams compare the results with the shapes and rotation speeds and orientations initially prepared."

 

1. Page 2: "The basic principle of the Doppler effect ensures that the parameter estimation accuracy of this method is independent of distance." That sentence contradicts common sense. The authors should put their 5 mm object at a ten-thousand-kilometer distance, show their results, and then try to argue.

 

2. Page 2: "Considering that the limit sensitivity of the micro-Doppler measurement process can be close to the quantum noise limit level of laser detection..." Apparently, the present performance, as shown in Section 4, is far from the quantum noise. Other noise sources are far above and the prospects to remove them are not clear. Thus, this statement is misleading.

 

3. Page 2: "the unit vector of the target along the line of sight (LOS) of the radar is given by n=(0, sin α, cos α), where the attitude angle α denotes the angle between the LOS of the radar and the z-axis." Firstly, please describe the basis, i.e. 0 along which axis, sin α along which axis, and cos α along which axis. Secondly, I would expect that the line of sight is determined by the azimuth angle gamma rather than the target axis inclination, i.e. the attitude angle alpha.

 

4. 1: the arc denoting the angle theta seems incorrect: it should connect the black z and the red z axes, it seems.

 

5. Page 3: "in the target local coordinate system, when a target rotates about z axes with an angular velocity Ω. The rotation matrix is expressed as": It seems strange to consider rotation in the local (rotating) coordinate system, rather than in the reference (non-rotating) system. Please check that this is indeed the intended meaning.

 

6. Pages 3-4 and Figs. 2a and 3a: "As can be seen from Figs. 2 and 3, before image preprocessing, the colors in the TF images are mainly brown and red, with little difference. After preprocessing, two distinct colors, red and blue, appear in the TF images. In other words, image preprocessing will improve the effectiveness of our subsequent image edge extraction." That is a terrible folly: by selecting wrong color scale, any image can be rendered useless. Why don't the authors make a color scale, say, from 10 to 20? All images would give absolutely no information then! Including those after pre-processing. My point remains the same as in the first review round: I do not see improvement of the images by the proposed preprocessing. The authors idea to artificially degrade initial images by selecting ridiculously wrong color scale, and thereby trying to deceive the readers to a wrong comparison with their preprocessed images, is a big mistake.

 

7. Page 5: "The micro-motion period is obtained by 2D Fourier transform of the 2D TF image. The effect of white noise is effectively avoided because the operation is on the frequency domain. Therefore, the spin rate can be precisely acquired." The influence of noise on the Fourier transform is not negligible. In fact, it can be non-local, i.e. noise in one part of image affects the entire spectrum. I believe, this noise influence decreased accuracy of the experimental results. Thus, the above sentence, and especially the word "avoided", are misleading.

 

8. Page 8: " the intersection of the axes of rotation and the center of cube O is represented by A " – this sentence has no geometrical meaning and should be reworded.

 

9. Page 8: " The included Angle between OB and n is gamma." However, gamma denotes another angle in Fig. 1.

 

10. Section 4. Experiment results: The authors say that "By utilizing DERM, the effect of the noise is suppressed and then the edge extraction can be achieved effectively." Here DERM means Double Edge Ratio Method, which the authors propose to use when the size of the target is unknown (the method is described on page 9). However, in the experimental section the authors do not provide the value of delta, i.e. the attitude angle shift, which is required for DERM. Nor the authors provide the determined size and its error bar, which results from the DERM together with the values of the attitude angle (the latter is shown in Fig. 15).

 

11. Do the authors have experimental data obtained with different object materials and textures?

Author Response

We are indebted to you for the helpful comments. We have carefully considered your comments and made revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

The revised version is kind of better in terms of wording, which has become less obviously wrong. However, the authors made modification of figure 6 to "prove" that their image processing is useful. This modification is absolutely wrong and cannot be published. I made a brief description of this mistake in #6 below. If the authors insist on the usefulness and/or necessity of their image processing method, they should describe everything in one or two paragraphs using scientific (i.e. not misleading, verifiable, and desirably correct) arguments, and possibly with further figures. At present, I do not see an honest attempt to do this.

Our reply:

Our work focuses on the parameter estimation of CubeSat. The image processing is only to improve our measurement effect, not the highlight of the work. We apologize that the image processing we did seem to have adversely affected readers' understanding of the manuscript. According to the comments of reviewers, we have deleted or modified the related content of image processing (original figures 2, 3, 4, 5, 6, 7, 14, and 15).

 

Further, I still believe that the author's method may be unable to work in a double-blind experiment, i.e. in an experiment modelling an actual situation. Here I repeat what I wrote in my first review: "one team prepares several unknown objects, another team rotates the objects, yet another team collects experimental data. No communication between the teams is allowed. The final team, having no prior information, applies whichever methods they want to the lidar data, trying to determine the objects shape, size, and spin properties. After that, all teams compare the results with the shapes and rotation speeds and orientations initially prepared."

Our reply:

We admit that the shape of the target cannot be determined by the approach in this work. Therefore in the revised manuscript, we stated that we focus on the cube target in this work. When the shape of the target is known, then its parameters can be estimated by the proposed approach. Actually, we are investigating the relevant method to determine the shape of the target, this is a very constructive question.

 

1. Page 2: "The basic principle of the Doppler effect ensures that the parameter estimation accuracy of this method is independent of distance." That sentence contradicts common sense. The authors should put their 5 mm object at a ten-thousand-kilometer distance, show their results, and then try to argue.

Our reply:

Thanks for the referee’s comment. This sentence is really inappropriate and has been deleted from our revised manuscript.

 

2. Page 2: "Considering that the limit sensitivity of the micro-Doppler measurement process can be close to the quantum noise limit level of laser detection..." Apparently, the present performance, as shown in Section 4, is far from the quantum noise. Other noise sources are far above and the prospects to remove them are not clear. Thus, this statement is misleading.

Our reply:

Thanks for the referee’s suggestion. Accordingly, we've removed that sentence.

 

3. Page 2: "the unit vector of the target along the line of sight (LOS) of the radar is given by n=(0, sin α, cos α), where the attitude angle α denotes the angle between the LOS of the radar and the z-axis." Firstly, please describe the basis, i.e. 0 along which axis, sin α along which axis, and cos α along which axis. Secondly, I would expect that the line of sight is determined by the azimuth angle gamma rather than the target axis inclination, i.e. the attitude angle alpha.

Our reply:

Attitude angle α refers to the included angle between the rotation axis z and the LOS in the local coordinate system. The vector n is represented as a basis vector in the local coordinate system (x, y, z), i.e., 0 corresponds to the x-axis, sinα to the y-axis, and cosα to the z-axis. We have also added a note to the first paragraph of Sec. 2: “…the unit vector of the target along the line of sight (LOS) of the radar in the local coordinate system (x, y, z) is given by n=(0, sin α, cos α), where the attitude angle α denotes the angle between the LOS of the radar and the z-axis….”

 

4. 1: the arc denoting the angle theta seems incorrect: it should connect the black z and the red z axes, it seems.

Our reply:

In Figure 1, the Euler angle is not clearly marked, so we modified this figure.

5. Page 3: "in the target local coordinate system, when a target rotates about z axes with an angular velocity Ω. The rotation matrix is expressed as": It seems strange to consider rotation in the local (rotating) coordinate system, rather than in the reference (non-rotating) system. Please check that this is indeed the intended meaning.

Our reply:

Thanks for the carefully reading. But this expression is indeed our intended meaning.

 

6. Pages 3-4 and Figs. 2a and 3a: "As can be seen from Figs. 2 and 3, before image preprocessing, the colors in the TF images are mainly brown and red, with little difference. After preprocessing, two distinct colors, red and blue, appear in the TF images. In other words, image preprocessing will improve the effectiveness of our subsequent image edge extraction." That is a terrible folly: by selecting wrong color scale, any image can be rendered useless. Why don't the authors make a color scale, say, from 10 to 20? All images would give absolutely no information then! Including those after pre-processing. My point remains the same as in the first review round: I do not see improvement of the images by the proposed preprocessing. The authors idea to artificially degrade initial images by selecting ridiculously wrong color scale, and thereby trying to deceive the readers to a wrong comparison with their preprocessed images, is a big mistake.

Our reply:

We used the same color scale for figures a and b (as shown by the color bar on the right in the figure). The relevant data before and after image processing are displayed within this color scale range. It can be seen from the figure that the color gap becomes larger after image processing, which can make the subsequent edge extraction effect better.

Image processing is not the focus of our work. The display of relevant content seems to have many adverse effects on readers' understanding. Based on this, we modified Figure 2 and Figure 3 to show only the time-frequency diagram obtained by our simulation (see the updated manuscript).

7. Page 5: "The micro-motion period is obtained by 2D Fourier transform of the 2D TF image. The effect of white noise is effectively avoided because the operation is on the frequency domain. Therefore, the spin rate can be precisely acquired." The influence of noise on the Fourier transform is not negligible. In fact, it can be non-local, i.e. noise in one part of image affects the entire spectrum. I believe, this noise influence decreased accuracy of the experimental results. Thus, the above sentence, and especially the word "avoided", are misleading.

Our reply:

Thanks for the referee’s suggestion. Accordingly, we have modified this sentence as follows: “The micro-motion period is obtained by 2D Fourier transform of the 2D TF image. Furthermore, the spin rate can be precisely acquired.”

 

8. Page 8: " the intersection of the axes of rotation and the center of cube O is represented by A " – this sentence has no geometrical meaning and should be reworded.

Our reply:

Thanks for the referee’s advice. Our expression of this sentence seems to be unclear. The sentence in the manuscript has been revised as “the intersection of the axis of rotation and the upper surface of the cube is represented by A.”

 

9. Page 8: " The included Angle between OB and n is gamma." However, gamma denotes another angle in Fig. 1.

Our reply:

According to the referee’s suggestion, we modified Figure 1 and the azimuth is represented as μ instead of γ.

 

10. Section 4. Experiment results: The authors say that "By utilizing DERM, the effect of the noise is suppressed and then the edge extraction can be achieved effectively." Here DERM means Double Edge Ratio Method, which the authors propose to use when the size of the target is unknown (the method is described on page 9). However, in the experimental section the authors do not provide the value of delta, i.e. the attitude angle shift, which is required for DERM. Nor the authors provide the determined size and its error bar, which results from the DERM together with the values of the attitude angle (the latter is shown in Fig. 15).

Our reply:

Thanks very much for the referee’s suggestion. As you said we lacked relevant results in Section 4. We have supplemented some experiment results according to the referee’s comment. As shown in figure11 and table 5 in the updated manuscript, we have added the attitude angle measurement experiments at the same spin rate and different attitude angles. As in the simulation, Δ=15° is also selected. Then the attitude angle and size of the target with unknown size are estimated by DERM. The details are as follows which is added in the last paragraph of Sec. 4.

“…For the case of the unknown target size with the attitude angle α=15° and Δ=15°, one has f₁=0.042MHz, and f₂=0.082MHz, respectively. Furthermore, the estimated result of the attitude angle is obtained as 14.7015° according to eq. (21) with the relative error 1.99%. Combined with eq. (22), the estimated result of side length of the cube target is 5.1814 mm with the relative error 3.63%.”

 

11. Do the authors have experimental data obtained with different object materials and textures?

Our reply:

In conjunction with the referee’s suggestion, we performed corresponding complementary experiments. The TF diagrams of the target surface made of different materials: resin, aluminum alloy, and printing paper are displayed in Figure 9 in the updated manuscript. The experimental results further confirm the feasibility of our proposed method.

Author Response File: Author Response.docx

Reviewer 3 Report

The presentation of this paper has undoubtedly improved, but my major concerns have not yet been addressed. I still believe that the paper is too long with respect to the presented topic and I wonder if the method is really necessary in the complexity as described. When I look at fig. 2a and 3a, I would expect that the authors will arrive at the same results, simply by normalizing the obtained spectrum between say 0 and 1 on a per scan basis. Furthermore, I would think that the choice of the threshold in that case would not be difficult and could be a fixed value.

Furthermore, I would think that the fig. 4, 5, 6 and 7 and later fig. 9, 10, 11, 13 and 14 and 15 are not necessary. The way the paper is presented, I can not actually understand why such a complex threshold detection mechanism should be necessary. I take this conclusion from fig. 8, which suggests, that a straightforward renormalization should be just fine. 

It should be in the interest of the experiment to make the processing both fast and robust. The results in fig. 11 and 15 are better presented in a simple table and a conclusion. 

Author Response

We are indebted to you for the helpful comments. We have carefully considered your comments and made revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

1. The presentation of this paper has undoubtedly improved, but my major concerns have not yet been addressed. I still believe that the paper is too long with respect to the presented topic and I wonder if the method is really necessary in the complexity as described. When I look at fig. 2a and 3a, I would expect that the authors will arrive at the same results, simply by normalizing the obtained spectrum between say 0 and 1 on a per scan basis. Furthermore, I would think that the choice of the threshold in that case would not be difficult and could be a fixed value.

Our reply:

Thanks for the referee’s suggestion. Figures 2 and 3 seem to have caused some misunderstanding for readers. In the hope of omitting unnecessary content related to image processing. We modified them in the updated manuscript and only retained the time-frequency diagram results from our simulation.

2. Furthermore, I would think that the fig. 4, 5, 6 and 7 and later fig. 9, 10, 11, 13 and 14 and 15 are not necessary. The way the paper is presented, I can not actually understand why such a complex threshold detection mechanism should be necessary. I take this conclusion from fig. 8, which suggests, that a straightforward renormalization should be just fine.

Our reply:

According to the referee’s suggestion, Figures 4, 5, 6, 7, 14, and 15 have been deleted. However, the rest of the figures are our error analysis or experimental results display. We hope that they can be displayed. The purpose of threshold processing is to improve the performance of the parameter measurement. In the revised manuscript, we only preserve the principle of the threshold processing algorithm and eliminating other complicated details.

 

3. It should be in the interest of the experiment to make the processing both fast and robust. The results in fig. 11 and 15 are better presented in a simple table and a conclusion.

Our reply:

Thanks for the referee’s suggestion. We carried out some supplementary experiments according to the comments of the reviewers. The original figure 15 is deleted. However, for Fig. 11, it would be more complicated and bulkier if the content was presented in a table. We hope Fig. 11 can be kept which turns to be Fig. 7 in the revised version.

Author Response File: Author Response.docx

Reviewer 4 Report

Although the authors improved the text, there are still a lot of open issues that have to be corrected.   - the limitations of the work are now more clearly set in the introduction (cubes instead of non cooperative targets of any shape). Nevertheless, authors still mention that the Doppler effect can be used for both bulk motion, vibrations and rotations. Although, this is in general true, authors do not study all these types of motions. This has to be clearly stated in the introduction - what types of motions of the cube are allowed for the method to work.   - in introduction, the authors state that they used a principle for microwave radars and improved it for the shorter wavelengths with "novel framework". Authors should precisely state in the introduction what are the novelties that they implied.   - Figure 1. and its explanation is still very unclear, although the figure is improved in quality:   (a) The text mentioned (X,Y,Z) and (x,y,z) coordinate systems, while figure shows two (x,y,z).   (b) Euler rotations are explained in the text clearly, but the figure presentation of Euler rotations is very unclear. Authors tried to show too much on one figure, the symbols are crowded and unclear. It would be more clear if there would be a few figures that would define the coordinate systems step by step   (c) Why there is a need to define three coordinate systems in a first place. I do not see where this is used in a text later on.   - Below equation (3) authors claim that they performed the logarithm of the frequency and Gaussian filtering. (a) If log was used, than vertical axsis on Fig. 2 (b) is wrong - it should be log (f_D) and not f_D. (b) Why the Gaussian filtering was used and what are its effects on the results. - From the equations(4) to (7) is not clear how f_f is obtained. It should be explained - evaluated that it presents the maximum value of G(u,v), not just stated. - The meaning of J in Eq. (10) is unclear. Why is it set the way is it set and how does it influence the results. - Below Fig. 6 - the SNR in dB is confusing. It is explained later in the text, but the definition of SNR should given here. - I recommend that authors visually show the meaning of f_s and alpha (or rather f_1 and f_2 on signal patterns (how changing alpha and f_s influences the patterns shown of Fig. 13 for example, for the readers to be able to follow manuscript easier. In summary - although the manuscript presents interesting results the text is still very hard to follow and authors are not clear in their presentation.

Author Response

We are indebted to you for the helpful comments. We have carefully considered your comments and made revisions according to each comment and advice. The replies are listed as follows.

 

List of replies

 

Although the authors improved the text, there are still a lot of open issues that have to be corrected.

- the limitations of the work are now more clearly set in the introduction (cubes instead of non cooperative targets of any shape). Nevertheless, authors still mention that the Doppler effect can be used for both bulk motion, vibrations and rotations. Although, this is in general true, authors do not study all these types of motions. This has to be clearly stated in the introduction

Our reply:

According to the referee’s suggestion, we have further stated in the introduction that our study is mainly aimed at the estimation of spin motion of the cube. The added expression is as follows: “Given that, a novel framework is proposed to estimate the parameters of CubeSats with micro-motion (i.e., spin motion) based on the micro-Doppler effect by laser radar system in this paper.”

 

- what types of motions of the cube are allowed for the method to work. - in introduction, the authors state that they used a principle for microwave radars and improved it for the shorter wavelengths with "novel framework". Authors should precisely state in the introduction what are the novelties that they implied.

Our reply:

We explain this framework in the introduction：“Namely, the spin rate is estimated based on the 2D Fourier transform of the time-frequency data. The attitude angle can be obtained effectively after threshold processing. The target size is identified through bistatic radar operations.”

 

- Figure 1. and its explanation is still very unclear, although the figure is improved in quality: (a) The text mentioned (X,Y,Z) and (x,y,z) coordinate systems, while figure shows two (x,y,z). (b) Euler rotations are explained in the text clearly, but the figure presentation of Euler rotations is very unclear. Authors tried to show too much on one figure, the symbols are crowded and unclear. It would be more clear if there would be a few figures that would define the coordinate systems step by step (c) Why there is a need to define three coordinate systems in a first place. I do not see where this is used in a text later on.

Our reply:

Thanks for the referee’s advice. Theoretically, one needs to establish the target local coordinate system and radar coordinate system to analyze the system composed of cube and radar. It is reasonable to introduce a reference coordinate system to facilitate the analysis because of the existence of spin and translational motion between the cube and radar. In this paper, to simplify the calculation, we do not consider the influence brought by the translational motion of the target. Therefore, the local coordinate system and the reference coordinate system are considered to be the same coordinate system. Therefore, Fig. 1 only shows two coordinate systems. According to the referee’s advice, we have modified the Fig. 1 which is as follows. We hope that the revised Fig. 1 will help readers better understand the manuscript. And the corresponding description in the first paragraph of Section 2 is rewritten as follows.

“As displayed in Fig. 1, the radar is stationary and located at the origin Q of the radar coordinate system (U, V, W). The target is depicted in a local coordinate system (x, y, z) attached to it and has translations and rotations with respect to the radar coordinates. The origin O of the local coordinates is at a distance R₀ from the radar. The azimuth and elevation angle of the target in the radar coordinates (U, V, W) are μ and η, respectively. In this work, we choose η=0 to simplify the calculation and thereby the unit vector of the target along the line of sight (LOS) of the radar in the local coordinate system (x, y, z) is given by n = [0, sinα, cosα], where the attitude angle α denotes the angle between the LOS of the radar and the z-axis. In this work, we focus on the spin motion estimation of of the target. Any point on the target depicted in the local coordinate system (x, y, z) will move to a new position due to the rotation of the target,. The new position can be obtained from its initial position vector multiplied by an initial rotation matrix determined by Euler angles (ϕ, θ, φ), where the angle ϕ rotates about the z-axis, the angle θ rotates about the x-axis, and the angle φ rotates about the z-axis again. Viewed in the target local coordinate system, when a target rotates about z axes with an angular velocity Ω….”

- Below equation (3) authors claim that they performed the logarithm of the frequency and Gaussian filtering. (a) If log was used, than vertical axsis on Fig. 2 (b) is wrong - it should be log (f_D) and not f_D. (b) Why the Gaussian filtering was used and what are its effects on the results.

Our reply:

We are sorry, this is a narrative error. We intend to perform the logarithmic calculation for the dimension of intensity (i.e., the color of the TF diagram), and Gaussian filtering for the frequency dimension to make the edge of the time-frequency diagram smoother. For this, we have modified the sentence after eq. (3) as follows: “After taking logarithm of the intensity and performing Gaussian filtering of the frequency”

 

- From the equations(4) to (7) is not clear how f_f is obtained. It should be explained - evaluated that it presents the maximum value of G(u,v), not just stated.

Our reply:

The TF diagrams we obtained change periodically in time. Equations (4) to (7) illustrate the process of 2D Fourier transform. The 2D Fourier transform of the TF diagram yields the expression G(u, v) related to the frequency variation. The frequency at which G(u, v) maximizes is the repetition frequency. More intuitively, G(u, v) can be plotted as a spectrum as a function of frequency as shown in Figure 4. In this figure, 20Hz is the repetition frequency.

- The meaning of J in Eq. (10) is unclear. Why is it set the way is it set and how does it influence the results.

Our reply:

The purpose of solving the threshold coefficient J is to better distinguish the useful signal from the noise signal in the echo signal. The quality of the coefficient will affect the edge extraction result of the TF diagram and the subsequent parameter extraction effect. We use a linear regression method to solve the coefficient. Equations. (9) to (11) are simple mathematical representations of the method. Such image processing is not highlight of our work. Therefore, we have not explained much about threshold processing. At the same time, according to the comments of the reviewers, we have deleted some content related to threshold processing

 

- Below Fig. 6 – the SNR in dB is confusing. It is explained later in the text, but the definition of SNR should given here.

Our reply:

Thanks for the referee’s advice. Accordingly, we revised the manuscript based on the comments of reviewers. Figures 6 and 7 were deleted. The definition of SNR is given in the first paragraph of Sec. 3.

 

- I recommend that authors visually show the meaning of f_s and alpha (or rather f_1 and f_2 on signal patterns (how changing alpha and f_s influences the patterns shown of Fig. 13 for example, for the readers to be able to follow manuscript easier.

Our reply:

Thanks for the referee’s suggestion, we have added some statements about Fig. 13 (i.e., Fig. 10 in the latest version) in the 4^th paragraph of Sec. 4 to illustrate the effect of f_s on the time-frequency diagram, which are as follows: “The influence of spin rate is displayed in Fig. 10 which illustrates the TF diagrams of the echo signals at different spin rates (1.4 Hz, 2.5 Hz, 3.8 Hz, 5.0 Hz and 6.4 Hz, respectively) for attitude angle α=45°. Eq. (15) indicates that larger spin rate results in more significant Doppler frequency shift. As eq. (15) predicts, it can be observed that when the spin is slow (control voltage equals 6V and the spin rate is about 1.4Hz), the TF feature is insignificant (see Fig. 10a). When the spin becomes faster, the periodicity can be clearly observed (see figures. 10b and c)….”

In addition, the effect of attitude angle a is demonstrated in Fig. 11. And the following description is added in the last paragraph of Sec. 4: “…Eq. 15 shows that larger attitude angle leads to more significant micro-Doppler frequency shift. The TF diagrams obtained in the experiment is displayed in Fig. 11. It can be seen that for a given spin rate, the micro-Doppler frequency shift f_D grows with the increase of attitude angle α which agrees with the theoretical prediction….”

Author Response File: Author Response.docx

Round 3

Reviewer 4 Report

Authors answered all my questions/comments