Collision Avoidance Second Order Sliding Mode Control of Satellite Formation with Air-Floated Platform Semi-Physical Simulation

Round 1

Reviewer 1 Report

In this paper, the authors present a control algorithm for preventing collisions with obstacles, considering spacecraft applications. Numerical simulations and experimental results are presented to demonstrate the efficiency of the proposed control.

The paper presents interesting and relevant results and may be considered for publication after corrections.

 

1- A wide review must be carried out to suppress typing errors;

2- The authors must include comparative results of the simulated results and the experiment;

3- A robustness analysis for parametric variations must be included in the paper. These results demonstrate that the proposed control is not sensitive to variations in the system parameters;

4- The conclusion is very poor, the authors should improve the discussion in the conclusions, highlighting the contribution of the results and real applications, what is the advantage of the proposed method in relation to other methodologies.

Author Response

1.Author's Reply to the Review Report (Reviewer 1) 

1) A wide review must be carried out to suppress typing errors;

Reply: The full text has been spell checked again.

2) The authors must include comparative results of the simulated results and the experiment;

Reply:Comparative results of numerical simulation and experimental results has been added in section 5.2.

3) A robustness analysis for parametric variations must be included in the paper. These results demonstrate that the proposed control is not sensitive to variations in the system parameters;

Reply:Chapter 4 The simulation parameters add the setting of composite interference, and at the same time, a description of the robustness of this method is added to the analysis of the experimental results in Section 5.2.

4) The conclusion is very poor, the authors should improve the discussion in the conclusions, highlighting the contribution of the results and real applications, what is the advantage of the proposed method in relation to other methodologies.

Reply:The content of the conclusions has been further enriched and the content of this article has been summarized. At the same time, based on the opinions of the reviewers, the issues that need to be studied in depth in the future have been summarized.

Reviewer 2 Report

The presented paper addresses the flight safety problem in spacecraft formation control caused by increased satellite numbers. It presents a second-order terminal sliding mode spacecraft formation obstacle avoidance controller based on artificial potential function (APF).

 

 There is a cross-reference error at line 90. "?? is the conclusion, which makes a summary of the whole paper. "

The mathematical model presented in chapters 2 and 3 is well described.

Results section is too short and incomplete. Please add a conclusion chapter

The English is pretty decent even if there are several errors such as missing punctuation or run-on sentence, expecially in introduction.

Author Response

2.Author's Reply to the Review Report (Reviewer 2) 

1) There is a cross-reference error at line 90. "?? is the conclusion, which makes a summary of the whole paper. "

Reply:The error here has been modified: section 6 is the conclusion, which makes a summary of the whole paper.

2) The mathematical model presented in chapters 2 and 3 is well described.

Reply:Thank you for your approval.

3) Results section is too short and incomplete. Please add a conclusion chapter.

Reply:The content of the conclusions has been further enriched and the content of this article has been summarized. At the same time, based on the opinions of the reviewers, the issues that need to be studied in depth in the future have been summarized.

4) Comments on the Quality of English Language.

The English is pretty decent even if there are several errors such as missing punctuation or run-on sentence, expecially in introduction.

Reply:Has been checked again and modified the error.

Reviewer 3 Report

1. The paper addresses the important issue of motion control of objects in 3D space in the presence of obstacles. In principle, the topic addressed is not new and has been of interest in the field of robotics for more than 30 years. Unfortunately, the authors of this work omit many key solutions in the area of navigation previously proposed in mobile robotics. Particularly relevant is the concept of potential function introduced to rootics by Oussam Khatib and the theoretical work of Rimon and Koditschek. A series of works by Dusan Stipanovic also contributed important results in the control of flying systems. For this reason, I believe that Section 1 should be rewritten.

2. In a theoretical sense, the authors simplify the control design since only fully actuated systems are considered. Therefore, it can be said that the control task considered can be reduced to the task of controlling a mass with three degrees of freedom. Even in the presence of structural uncertainty, this does not seem to be a theoretical challenge.

3. Assumption (7) is very strong. Since the \dot{D} depends on \doq, \dot q and q, it is not justified a priori and should be cocluded from a formal stability analysis.

4. Modifying the potential (14) and expressing it in the form (15) requires an in-depth explanation. On what basis do the authors claim that formula (15) avoids the problem of local minima? Formal analysis and mathematical proof are needed.

5. The experimental studies did not show the control inputs. This is very important to assess the control process.

6. For which purpose z-axis moment used (see lines 283-286)? The experiment is carried out in the plane, thus the forces generated in the x and y directions seem to be sufficient.

Other comments:

- Line 48-49: 'first order sliding mode has problems of slow convergence' - this statement is false. After all, the designer is free to adjust the rate of error convergence. Combining the convergence problem with the chatering effect in my opinion is not justified here.
- Equation (4) can be savely removed, since Eq. (5) is sufficient.
- Line 177: If M is a matrix, the given formula is incorrect.
- Equation (25) and further - the formula x^{p/q} in general is not correct when x < 0. The notation adopted should be explained.
- Line 246-247 - sentence not understood. What is a "shadow sphere".

Some linguistic errors:
- Line 95.
- Line 221
- Line 280 ("through the solution").

Style and presentation:
- The text lacks spaces between the word and brackets [] when citing literature.
- In some places there is no space after the comma.
- A full stop or comma is missing at the end of many mathematical formulas. In scientific texts, mathematical formulas should be part of sentences.
- „Fig.” should be used instead of „fig.” in references to figures.

 

Author Response

3.Author's Reply to the Review Report (Reviewer 3) 

1) The paper addresses the important issue of motion control of objects in 3D space in the presence of obstacles. In principle, the topic addressed is not new and has been of interest in the field of robotics for more than 30 years. Unfortunately, the authors of this work omit many key solutions in the area of navigation previously proposed in mobile robotics. Particularly relevant is the concept of potential function introduced to rootics by Oussam Khatib and the theoretical work of Rimon and Koditschek. A series of works by DusanStipanovic also contributed important results in the control of flying systems. For this reason, I believe that Section 1 should be rewritten.
Reply: The corresponding positions in Section 1 supplement some of the previous key solutions in the field of robot navigation mentioned by the reviewers, including the relevant research of predecessors such as Oussama Khatib and Dusan Stipanovic.It also shows that this article focuses on the application and verification of advanced control algorithms combined with APF methods in engineering practice.

 

2) In a theoretical sense, the authors simplify the control design since only fully actuated systems are considered. Therefore, it can be said that the control task considered can be reduced to the task of controlling a mass with three degrees of freedom. Even in the presence of structural uncertainty, this does not seem to be a theoretical challenge.
Reply:In view of the use of satellite formation flight control, this article combines the second-order sliding mode and APF to carry out high-precision obstacle avoidance research. As you mentioned, the theoretical research of the above two has achieved more research results under the efforts of many predecessors.However, the intention of this article is to combine theory with practice, explore how to better apply theoretical methods to engineering applications, and then explore the practical problems that theory needs to solve through engineering applications.

3) Assumption (7) is very strong. Since the \dot{D} depends on \doq, \dot q and q, it is not justified a priori and should be cocluded from a formal stability analysis.

Reply:Referring to the relevant kinetic modeling literature, relevant assumptionsalways be made. As you mentioned, it is indeed necessary to further analyze the hypothesis. Due to the limited space, and the focus of this article is not on this, in the follow-up work, we will learn from your suggestions and do in-depth research.

4) Modifying the potential (14) and expressing it in the form (15) requires an in-depth explanation. On what basis do the authors claim that formula (15) avoids the problem of local minima? Formal analysis and mathematical proof are needed.
Reply:According to the definitions of the gravitational potential function and the repulsion potential function, a constant positive factor about the desired position is added to the repulsion potential function. By deriving the potential function model of the entire system, it can be proved that there is only a unique minimum value at the desired position.

5) The experimental studies did not show the control inputs. This is very important to assess the control process.
Reply:Reply:Chapter 5 adds a description of the implementation of the control scheme in the experiment.At the end of Chapter 4, the comparison of the effects of this method with the traditional sliding mode scheme is added to highlight the advantages of this method.

6) For which purpose z-axis moment used (see lines 283-286)? The experiment is carried out in the plane, thus the forces generated in the x and y directions seem to be sufficient.
Reply:Section 5.2 adds a description of the z-axis torque：This experiment is mainly used for obstacle avoidance verification of the two-position plane. The rotational movement around the z-axis only needs to maintain the angular orientation.

7) Other comments:
- Line 48-49: 'first order sliding mode has problems of slow convergence' - this statement is false. After all, the designer is free to adjust the rate of error convergence. Combining the convergence problem with the chatering effect in my opinion is not justified here.

Reply:Thestatement of 'first order sliding mode has problems of slow convergence' has been revised.

- Equation (4) can be savely removed, since Eq. (5) is sufficient.
- Line 177: If M is a matrix, the given formula is incorrect.
- Equation (25) and further - the formula x^{p/q} in general is not correct when x < 0. The notation adopted should be explained.
- Line 246-247 - sentence not understood. What is a "shadow sphere".

Reply:Formula (4) and formula (5) have been merged.The M0 of formula (24) is the reversible nominal mass matrix defined above; the definition of formula (25) s^(p/q) is sgn(s)*(abs(s))^(p/q)；shadow sphere indicates the spatial influence range of the obstacle.

Comments on the Quality of English Language

Some linguistic errors:
- Line 95.
- Line 221
- Line 280 ("through the solution").
Style and presentation:
- The text lacks spaces between the word and brackets [] when citing literature.
- In some places there is no space after the comma.
- A full stop or comma is missing at the end of many mathematical formulas. In scientific texts, mathematical formulas should be part of sentences.
- „Fig.” should be used instead of „fig.” in references to figures.

Reply:The above linguistic and format errors have been modified

Reviewer 4 Report

The authors develop a second-order sliding-mode controlling scheme for an ensemble of spacecrafts. Theoretical verification of the scheme was carried out by means of the Lyapunov function. Efficiency of the scheme was confirmed by numerical simulation and the experiment. Taking into account the growing interest to the problem under consideration, I recommend publication after addressing the issues listed below:

1. Implementation of the control scheme in the experiment is poorly described. It is reasonable to compare efficiency of the developed scheme with other known methods of control. It can help to underline value of the result.

2. The authors should study the robustness of the scheme. For example, they can introduce noises, parameter mismatches, or other sources of errors into model for numerical simulation.

English is poor. The phrases like "terminal sliding mode spacecraft formation obstacle avoidance controller" has to be avoided. The authors often use the wrong vocabulary.

Author Response

4.Author's Reply to the Review Report (Reviewer 4) 

The authors develop a second-order sliding-mode controlling scheme for an ensemble of spacecrafts. Theoretical verification of the scheme was carried out by means of the Lyapunov function. Efficiency of the scheme was confirmed by numerical simulation and the experiment. Taking into account the growing interest to the problem under consideration, I recommend publication after addressing the issues listed below:

1) Implementation of the control scheme in the experiment is poorly described. It is reasonable to compare efficiency of the developed scheme with other known methods of control. It can help to underline value of the result.

Reply:Chapter 5 adds a description of the implementation of the control scheme in the experiment.At the end of Chapter 4, the comparison of the effects of this method with the traditional sliding mode scheme is added to highlight the advantages of this method.

2) The authors should study the robustness of the scheme. For example, they can introduce noises, parameter mismatches, or other sources of errors into model for numerical simulation.

Reply:Chapter 4 The simulation parameters add the setting of composite interference, and at the same time, a description of the robustness of this method is added to the analysisof the experimental results in Section 5.2.

3) Comments on the Quality of English Language

English is poor. The phrases like "terminal sliding mode spacecraft formation obstacle avoidance controller" has to be avoided. The authors often use the wrong vocabulary.

Reply:The vocabulary and phrases in the article have been improved.

Round 2

Reviewer 1 Report

The authors made all the requested corrections and clarifications, submitting a version that can be accepted for publication.

Author Response

1) The authors made all the requested corrections and clarifications, submitting a version that can be accepted for publication.

Reply: Thank you very much for your valuable comments.

Reviewer 3 Report

The next iteration of the paper is better, but two issues need further clarification.

1. the problem of the lack of local minima is unfortunately not properly considered in the work. The authors in their response to the reviews state: "By deriving the potential function model of the entire system, it can be proved that there is only a unique minimum value at the desired position."

Unfortunately, this was not demonstrated in the reviewed paper. I think it is necessary to either show a formal proof or indicate the literature where such a proof for the proposed form of the potential function is analyzed. My doubt stems from the fact that with the quadratic form of the attractor function, even for symmetric objects with the assumption of unlimited repulsive potential interaction, minimum points are possible. These issues were considered by Rimon and Koditschek, among others. On the other hand, if harmonic potentials were used, the conclusion about the absence of local minima would be derived immediately. I would like to point out that the issue is not trivial and should be discussed more thoroughly.

The comment about modifying equation (13) and expressing it in the form (14) is not convincing. Why was the quadratic form added to take into account the M matrix?

2 The description of the experiment is unclear. It is really difficult to tell if this experiment was performed - among other things, the waveforms of the input signals and other variables relevant to the method used are not shown. Was the real system shown in Figure 15 used in the experiment whose results show the error waveforms in Figures 14 and 16? Or are only the simulation results shown in Figures 14 and 16? If so, for what purpose is the test stand discussed?
Lines 378-381: "due to the limited engineering technologies such as manufacturing, processing and assembly, the thrust output of the simulator has large installation errors and output errors, and the quality and moment of inertia of the simulator are subject to uncertain changes with the consumption of working fluid" unfortunately do not clarify this confusion.

Punctuation marks are missing from the equations. This is a mistake because equations should be parts of a sentence to which the rules of punctuation apply.

Author Response

1)  the problem of the lack of local minima is unfortunately not properly considered in the work. The authors in their response to the reviews state: "By deriving the potential function model of the entire system, it can be proved that there is only a unique minimum value at the desired position."
Unfortunately, this was not demonstrated in the reviewed paper. I think it is necessary to either show a formal proof or indicate the literature where such a proof for the proposed form of the potential function is analyzed. My doubt stems from the fact that with the quadratic form of the attractor function, even for symmetric objects with the assumption of unlimited repulsive potential interaction, minimum points are possible. These issues were considered by Rimon and Koditschek, among others. On the other hand, if harmonic potentials were used, the conclusion about the absence of local minima would be derived immediately. I would like to point out that the issue is not trivial and should be discussed more thoroughly.
Reply: The reviewer's statement is correct. By deriving the potential function of the entire system, it is still impossible to effectively prove that the modified artificial potential function does not have a minimum point.In this regard, the description of avoiding the minimum point has been modified in the article, and it is only expressed as the value of the modified potential function at the desired position is zero, and the distribution of the potential function is adjusted by setting appropriate parameters to improve the obstacle avoidance efficiency in the case of close range and limited obstacles.It is an honor that under the reminder of the reviewers, we began to pay attention to the research results of Rimon, Koditschek and others, hoping to get a deeper discussion in the future research process and conduct in-depth verification in the experiment.

2) The comment about modifying equation (13) and expressing it in the form (14) is not convincing. Why was the quadratic form added to take into account the M matrix?
Reply: Add the M matrix to facilitate the merger of similar items in the subsequent derivation and proof process.

3) The description of the experiment is unclear. It is really difficult to tell if this experiment was performed - among other things, the waveforms of the input signals and other variables relevant to the method used are not shown. Was the real system shown in Figure 15 used in the experiment whose results show the error waveforms in Figures 14 and 16? Or are only the simulation results shown in Figures 14 and 16? If so, for what purpose is the test stand discussed?

Reply:The test determined the coordinates of the position measurement system and set the position of the test bench and the obstacle.After figure16, the error comparison figure of the the simulation results and actual obstacle avoidance trajectory completed by the test stand during the test is supplemented. The difference between the two has been analyzed below.

 

• Lines 378-381: "due to the limited engineering technologies such as manufacturing, processing and assembly, the thrust output of the simulator has large installation errors and output errors, and the quality and moment of inertia of the simulator are subject to uncertain changes with the consumption of working fluid" unfortunately do not clarify this confusion.

Reply: The relevant description is the analysis of the error between the actual trajectory of the test and the simulation trajectory.

 

• Punctuation marks are missing from the equations. This is a mistake because equations should be parts of a sentence to which the rules of punctuation apply.

Reply: Punctuation marks has been added after the equation.

Reviewer 4 Report

The authors made an attempt to address the issues raised in the preceding report, but English is still poor. Horrible expressions like "... terminal sliding mode spacecraft formation obstacle avoidance controller ... " are stil present, the verb perturb is used as a noun, and so on.

I found no signs of improvement.

Author Response

The authors made an attempt to address the issues raised in the preceding report, but English is still poor. Horrible expressions like "... terminal sliding mode spacecraft formation obstacle avoidance controller ... " are stil present, the verb perturb is used as a noun, and so on.

Reply: Thank you very much to the reviewers for their continued attention to the English expression of this article. By seeking the review of colleagues and the assistance of professional translation software, the English expression has been further revised in this round.Due to the author's limited English proficiency, if there are other expression problems, please feel free to advise.

Round 3

Reviewer 3 Report

1. I believe that the authors correctly decided to remove the information regarding the absence of local minima of the considered potential function. This statement was not supported by the proof, so it should not be given. Nevertheless, it is worth investigating this problem more formally in the future.

2. The description of the methodology used in the experiment should be clear, and unfortunately this is still not the case. As an example, please analyse the lines 378-380: "The satellite simulator analyses the control signal through the industrial computer to control it to reach the desired position at the next time, so as to complete the control task." What exactly does "to control it" mean?

I think the way the experiment is implemented should be shown in a block diagram - the diagram in Figure 11 is not sufficient. If I understand the experiment description correctly, the control process using the main algorithm discussed in this paper is simulated and the reference position trajectory for the real objects is generated from this. At the same time, a local feedback loop is used which is responsible for controlling the movement of the real objects to follow the reference trajectory in real time. I suppose that such a cascaded structure was used because of the implementation difficulties of the main method. Such an operation is acceptable, but the concept of the experiment and the structure of the control system should be described clearly.

3. I think we have not fully understood each other about the rules of punctuation. For example, after formula (1) it should be:

M\ddot{\rho}+ C\dot{\rho} + G = u + d,

where \rho ....

So you should not put a full stop at the end of a formula if there is an explanation of the terms further on. In that case, a comma is used. The same remark applies to (8), (9), (11), (15), (17), (24), (30), etc.

3. Line 235 - Defined -> Let us define
4. Line 338 - no space between consecutive sentences: "long time.Compared"

Author Response

1.I believe that the authors correctly decided to remove the information regarding the absence of local minima of the considered potential function. This statement was not supported by the proof, so it should not be given. Nevertheless, it is worth investigating this problem more formally in the future.

Reply:Thanks again to the reviewer for reminding us of the local minima of the potential function, which makes us fully aware of the importance of this problem. In the future, we will refer to the reviewer's opinions and suggestions to carry out in-depth discussion on this direction.

2. The description of the methodology used in the experiment should be clear, and unfortunately this is still not the case. As an example, please analyse the lines 378-380: "The satellite simulator analyses the control signal through the industrial computer to control it to reach the desired position at the next time, so as to complete the control task." What exactly does "to control it" mean?
   
   I think the way the experiment is implemented should be shown in a block diagram - the diagram in Figure 11 is not sufficient. If I understand the experiment description correctly, the control process using the main algorithm discussed in this paper is simulated and the reference position trajectory for the real objects is generated from this. At the same time, a local feedback loop is used which is responsible for controlling the movement of the real objects to follow the reference trajectory in real time. I suppose that such a cascaded structure was used because of the implementation difficulties of the main method. Such an operation is acceptable, but the concept of the experiment and the structure of the control system should be described clearly.

Reply:In fact, the reviewer's understanding of the experimental description is quite correct, since the lower computer of the satellite simulator is not strong in computing ability, it is necessary to use the reference trajectory for assistance. We describe the experimental procedure in more detail and supplement Figure 11. Meanwhile, Figure 13 is added to describe the control flow.

3. I think we have not fully understood each other about the rules of punctuation. For example, after formula (1) it should be:

M\ddot{\rho}+ C\dot{\rho} + G = u + d,

where \rho ....
So you should not put a full stop at the end of a formula if there is an explanation of the terms further on. In that case, a comma is used. The same remark applies to (8), (9), (11), (15), (17), (24), (30), etc.

Reply:We have made further changes to the punctuation and thank the reviewers for pointing this out.
4. Line 235 - Defined -> Let us define

Reply:We've changed it.
5. Line 338 - no space between consecutive sentences: "long time.Compared"

Reply:This is a very detailed problem, thank you very much for the careful review of the reviewer, we have made changes, the reviewer's valuable comments are of great help to the improvement of this paper, I would like to pay tribute to you again.

Reviewer 4 Report

The authors have made necessary corrections. I recommended publication.

Author Response

1) The authors made all the requested corrections and clarifications, submitting a version that can be accepted for publication.

Reply: Thank you very much for your recognition and thank you again for your valuable comments.

 

Round 4

Reviewer 3 Report

1. The manuscript has been corrected significantly and it is much better now.

2. Lines 348-353. The description of the architecture is now clearer. However, it would be useful to add what control strategy the internal feedback uses to track the reference trajectory. Is this a PID control law or does it use non-linear methods. 

3. See my comments in the PDF file.

Comments for author File: Comments.pdf

- I suggest you work on the punctuation in your paper. Some sentences are not properly separated by a space, which should be placed after the full stop.

- Small spaces between number and unit are missing. In addition, it should be decided whether the units are written in a normal or italic font.

- The style of the bibliography list should be significantly improved.

Author Response

1. The manuscript has been corrected significantly and it is much better now.

Reply:Thank the reviewers for your affirmation of our revision work

2. Lines 348-353. The description of the architecture is now clearer. However, it would be useful to add what control strategy the internal feedback uses to track the reference trajectory. Is this a PID control law or does it use non-linear methods. 

Reply:The lower computer of the satellite simulator can use the PID method for feedback tracking. But in fact, since the operation time of the integration process is long, and the feedback tracking error of each step is small, we finally adopt the PD method for feedback tracking, which can reduce the operation time and ensure the accuracy of feedback tracking.

3. See my comments in the PDF file.

Reply:It has been modified according to the PDF file of the reviewer. Thank you for your careful correction, which will be of great help to our future article writing. My regards to you once again.

- I suggest you work on the punctuation in your paper. Some sentences are not properly separated by a space, which should be placed after the full stop.

Reply:We've changed it. Punctuation is an important part of an article,Thanks for the reviewer's correction. We will pay more attention to this problem when editing.

- Small spaces between number and unit are missing. In addition, it should be decided whether the units are written in a normal or italic font.

Reply:We've made the changes so that the units end up in a normal font with Spaces added.

- The style of the bibliography list should be significantly improved.

Reply:We modified the format of the references to cite mainly in the Chicago way.