Explicit-Time Trajectory Tracking for a State-Constraint Continuum Free-Floating Space Robot with Smooth Joint-Path and Low Input

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear authors, here are some suggestions for the paper improvement:

1. Clarity in Explanation:
• Some equations (e.g., Equation 46 and 53) lack intuitive explanations for readers unfamiliar with advanced control theory. Consider including more detailed derivations or supplementary explanations.
• The role of damping coefficients in singularity avoidance (Equation 62) needs further clarification for practical implementation.
2. Experimental Validation:
• The study relies solely on simulations. Adding experimental validation with a physical prototype or hardware-in-the-loop simulation would enhance the credibility and applicability of the results.
• The parameters for the simulations (Tc,m,and ec) are specific to this study. Justify their selection in terms of real-world constraints.
3. Complexity of Control Laws:
• While the explicit-time sliding mode controller is effective, its practical implementation may face challenges due to computational complexity. Consider discussing computational feasibility for real-time applications.
4. Comparative Analysis:
• Expand the discussion on how the proposed method performs compared to other explicit-time or predefined-time control methods, especially in scenarios involving high disturbances or large initial errors.
• Add more visual comparisons (e.g., plots for control signal smoothness across methods).
5. Figures and Visualization:
• Figure 10: Enhance clarity by including gridlines or labels for better visual interpretation of trajectory convergence.
• Figures 13 and 15: Annotate key points or intervals to emphasize the advantages of the proposed method.
6. Future Recommendations:
• Include more specific suggestions, such as exploring reinforcement learning for adaptive control or incorporating real-world disturbances into the model.

Specific Suggestions

1. Introduction:
• Expand the motivation for choosing explicit-time theory over other finite-time or predefined-time methods.
• Clearly articulate the limitations of existing trajectory planning and tracking control methods.
2. Methodology:
• Provide practical guidance on how the proposed control parameters (Tp,m,ec​) can be tuned for different scenarios.
• Explain the trade-offs between planning accuracy and computational efficiency in the proposed method.
3. Simulation Results:
• Quantify the improvement achieved by the proposed method compared to baseline methods in terms of metrics such as convergence time, control input magnitude, and tracking error.
• Discuss the limitations of the simulation setup, such as idealized environmental conditions.
4. Applications and Practical Implications:
• Highlight specific use cases (e.g., servicing a space station or manipulating debris) to connect the research with real-world applications.
• Discuss potential challenges in extending the method to multi-segment or collaborative continuum robots.

The manuscript offers a significant contribution to space robotics, introducing innovative methods for high-precision trajectory planning and tracking. However, addressing the noted weaknesses, particularly by including experimental validation and refining the clarity of explanations, would substantially enhance the paper’s impact and applicability.

Author Response

Comments 1 : Clarity in Explanation:

Some equations (e.g., Equation 46 and 53) lack intuitive explanations for readers unfamiliar with advanced control theory. Consider including more detailed derivations or supplementary explanations.

 

The role of damping coefficients in singularity avoidance (Equation 62) needs further clarification for practical implementation.

 

Response 1：

Thank you forpointing this out,I agree with this comment.

Equation (46) represents the standard form of the Lagrangian modeling method, where the computation process involves deriving the external torque by taking the derivative of kinetic energy minus potential energy. I have added a detailed explanation of this process in the paper and highlighted it in red. Additionally, the physical meanings of the matrices in the equation are explicitly stated in the text.

 

Equation (53) arises from the fact that the free-floating space robot's base is unactuated, requiring its pose to be estimated based on the conservation of kinetic energy. This results in extensive matrix operations in the formulation. To avoid excessive complexity, the equation has been simplified into the form of Equation (53).

 

    In robot inverse kinematics solving, the manipulator may approach singular positions (e.g., an extended state), causing the Jacobian matrix to become non-invertible. By introducing a damping term λI, the matrix can be prevented from becoming singular. In robot control, an empirical value of λ = 0.01 can be chosen.

Comments 2 :

 

The study relies solely on simulations. Adding experimental validation with a physical prototype or hardware-in-the-loop simulation would enhance the credibility and applicability of the results.

 

The parameters for the simulations (Tc,m,and ec) are specific to this study. Justify their selection in terms of real-world constraints.

 

Response 2：

 

Thank you forpointing this out,I agree with this comment.

 

Due to the limitations of the laboratory environment, it is not possible to simulate space conditions. Therefore, a fixed-base single-segment continuum robot experiment was conducted. Using the proposed method for trajectory tracking, the system was able to converge within Tc time, verifying the effectiveness of the algorithm.

 

Tc​ represents the predefined convergence time of the system, i.e., the time required for the sliding surface to reach near zero.Smaller Tc ​: Faster convergence but may cause sharp control signal variations, leading to high-frequency chattering.Larger Tc ​: Slower convergence but results in a smoother control system.

m determines the shape of the sliding mode approaching curve and influences the evolution of the sliding surface.m>1: Fast initial convergence followed by gradual stabilization, suitable for reducing chattering.0<m<1: Slow initial convergence followed by acceleration, useful for avoiding sudden changes in the early stage. In this paper, 0<m<1/2​ is selected. This choice ensures a gradual initial convergence, preventing abrupt changes in the early stage while accelerating in the later stage. It helps maintain system stability and reduces the risk of excessive control effort or chattering.

ec​ characterizes the error term in the sliding surface and determines the steady-state error after Tc​.Larger ec​: Allows for a larger final error, reducing control energy consumption.Smaller ec​: Improves control accuracy but may require higher control gains.

 

Comments 3 :

 

While the explicit-time sliding mode controller is effective, its practical implementation may face challenges due to computational complexity. Consider discussing computational feasibility for real-time applications.

 

Response 3：

Compared to some basic sliding mode algorithms, the explicit-time sliding mode control algorithm does have a slightly higher complexity. However, in the context of sliding mode algorithms, its computational complexity is not particularly high. This can be observed from the design of the sliding surface: explicit-time control involves only a single exponent parameter m, whereas fixed-time control generally includes two parameters, m and α. The simulation results in this paper demonstrate that the convergence speed of the explicit-time method is significantly better than that of the fixed-time method.Additionally, compared to fast terminal sliding mode control, the explicit-time approach involves fewer exponent terms.

The primary source of computational complexity in this paper stems from the spatial continuum robot, which has a high number of joints and dimensions, making the modeling process more intricate. Furthermore, the free-floating base requires consideration of momentum conservation, significantly increasing the complexity of the equations.

 

Response 4：

Expand the discussion on how the proposed method performs compared to other explicit-time or predefined-time control methods, especially in scenarios involving high disturbances or large initial errors.

 

Add more visual comparisons (e.g., plots for control signal smoothness across methods).

 

Response 4：

       Agree.The revised version of this paper includes additional visualized simulation experiments for trajectory planning in the presence of initial pose errors. Moreover, a large disturbance term D was introduced during the trajectory tracking simulation experiments. Finally, physical validation was conducted, further verifying the effectiveness of the explicit-time algorithm.

 

Response 5：

Figure 10: Enhance clarity by including gridlines or labels for better visual interpretation of trajectory convergence.

Figures 13 and 15: Annotate key points or intervals to emphasize the advantages of the proposed method

 

Response 5：

Thank you forpointing this out。I agree with this comment。

After the revision, a grid was added to the 3D plots, and unclear diagrams were redrawn. Additionally, localized zoom-in views were added to highlight less obvious details.

 

Comments 6 :

Include more specific suggestions, such as exploring reinforcement learning for adaptive control or incorporating real-world disturbances into the model.

 

Response 6：

Thank you forpointing this out。I agree with this comment。

I have rewritten the future research , which include:1. Optimizing Control Strategies;2. Environmental Factors and Robustness;3. Hybrid Control Strategies;4.Experimental Validation;5. Application Prospects.

Finally, I would like to thank you once again for your patient comments and suggestions. They have been incredibly helpful for my current and future research work.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is well structured and includes a solid theoretical part, validated by numerical simulations in Matlab.

The subject is topical (trajectory planning and tracking control issues of continuum space robots).

However, the simulations are not accompanied by experimental validation. There is also no discussion/argumentation of how the values used in the simulation process were chosen/set (Tables 1 and 2).

There are also some small editing issues (starting section numbering from 0, capitalization of section titles - section 1, etc)

Figures 11-18 to small, unreadable.

Author Response

Comments 1 :

The paper is well structured and includes a solid theoretical part, validated by numerical simulations in Matlab.

The subject is topical (trajectory planning and tracking control issues of continuum space robots).

 

Response 1：

 

Thank you very much for your recognition of my work. Your comments and suggestions have been invaluable to my future learning and career development.

 

Comments 2 :

However, the simulations are not accompanied by experimental validation. There is also no discussion/argumentation of how the values used in the simulation process were chosen/set (Tables 1 and 2).

 

Response 2：

   Thank you for pointing this out.I agree with this comment.I have provided explanations for the relevant parameters. By designing a 3D model of the continuum robot in SolidWorks and inputting parameters such as density and material properties, the mass, moment of inertia, area, and other relevant parameters of the base and links can be calculated. Table 2 presents kinematics-related parameters. By setting the initial generalized joint angles, other related parameters can be calculated. The calculation method has been provided in the previous kinematic equations, so only the results are given here, omitting the detailed calculation process.

       Due to experimental limitations, it is not possible to simulate a vacuum environment for space robot experiments. Therefore, this paper conducts experimental validation using an existing single-segment continuum robot with a fixed base. The proposed method is applied for trajectory tracking, and the experimental results show convergence within Tc​ time, verifying the effectiveness of the algorithm.

Comments 3 :

There are also some small editing issues (starting section numbering from 0, capitalization of section titles - section 1, etc)

Response 3：

Agree, I have made the modifications according to your suggestions.

 

Comments 4 :

Figures 11-18 to small, unreadable.

 

Response 4：

Agree, I have redrawn the figures after Figure 11 and enlarged them for better clarity.

Finally, I sincerely appreciate your recognition of my work. I am also grateful for your valuable suggestions. Wishing you all the best in the future!

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents a control problem for a continuum free floating space robot with smooth joint-path, low input and the state-constraints. The topic is of interest for robotics and its areas of applications.

In my opinion, the following aspects need to be clarified:

Line 271: Please reformulate: „it can be called” is ambiguous; it is not clear if under the below conditions the function G(y) is or it is not  explicitly time stable;

Line 276: what means „theory 1”, probably it is „Lemma 1”, instead; please check;

Line 362: please check the sign of the first term in the expression fof A; it seems that it must be positive;

Line 369: What means „Theory”?; What means „can be stable”? Please reformulate: is it or it is not?

Line 403: Probably „Theory 2” must be replaced by „Theorem 2”; please check;

Expression of u in Eq. (72) includes a „sign” term which may induce chattering; this may be also seen from the simulations. I recommend to add after Eq. (72) a new expression of the control in which the „sign” term is replaced in order to reduce the chattering effect.

Line 414: I suggest instead of „If considering the uncertainty of the system”,  to write „ If considering the bounded uncertainty D of the system...”

Comments on the Quality of English Language

There are not major problems with the quality of English Language. However, I recommend a careful check of the text; some ambiguous formulations and terms are emphasized in the section "Comments and Suggestions for Authors". 

Author Response

Comments 1 :

The paper presents a control problem for a continuum free floating space robot with smooth joint-path, low input and the state-constraints. The topic is of interest for robotics and its areas of applications.

 

Response 1：

Thank you very much for your recognition of my work. Your comments and suggestions have been invaluable to my future learning and career development.

 

Comments 2 :

Line 271: Please reformulate: „it can be called” is ambiguous; it is not clear if under the below conditions the function G(y) is or it is not  explicitly time stable

 

Response 2：

Thank you for pointing this out.I agree with this comment.

It has already been modified to ："If the function G(y) satisfies the following conditions, it is considered explicitly time-stable. "

 

Comments 3 :

 

Line 276: what means „theory 1”, probably it is „Lemma 1”, instead; please check;

 

Response 3：

Agree，It has already been modified to Lemma 1.

 

Comments 4:

Line 362: please check the sign of the first term in the expression fof A; it seems that it must be positive;

 

Response 4：

Thank you very much for your comments. Based on the issue you raised, it is clear that you have a deep understanding of the field of robot modeling and control. Since the system is a free-floating space robot with no drive on the robot base, the general Lagrangian modeling method typically includes the generalized joint vector of the base. Therefore, we use the conservation of momentum to solve for the position and velocity of the base. Afterward, we substitute the solved values to eliminate the parts related to the base. Based on the results, matrix A remains positive definite.

 

Comments 5:

Line 369: What means „Theory”?; What means „can be stable”? Please reformulate: is it or it is not?

 

Response 5：

Thank you for your suggestion. It has been revised to "Proposition 1," and "can be stable" has been changed to "can achieve stable."

 

Comments 6:

 Probably „Theory 2” must be replaced by „Theorem 2”; please check;

 

Response 6：

Thank you for your suggestion. It has been revised to "Proposition 1," and "can be stable" has been changed to "can achieve stable."

 

Comments7:

Expression of u in Eq. (72) includes a „sign” term which may induce chattering; this may be also seen from the simulations. I recommend to add after Eq. (72) a new expression of the control in which the „sign” term is replaced in order to reduce the chattering effect.

 

Response 7：

Thank you very much for your suggestion. It is clear that you have a deep understanding of sliding mode control. The sign function has been replaced with a saturation function, and the value of phi has been determined.

 

Comments8:

Line 414: I suggest instead of „If considering the uncertainty of the system”,  to write „ If considering the bounded uncertainty D of the system...”

 

Response8:

Thank you for pointing this out.I agree with this comment.

The noise has been set as a random value, and the simulation has been rerun to account for the impact of noise.

 

Finally, I would like to sincerely thank you for your valuable suggestions. Your expertise has been incredibly beneficial to me.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors considered all the reviewer's suggestions and provided valid solutions to all of them:
- they introduced an experimental validation (even if it does not exactly reproduce the environment in which the simulated robot works, due to the difficulty of creating a vacuum), but which largely confirms the theoretical research
- provided the exact values of the parameters used in the simulation and explained how these values were arrived at
- enlarged some figures that were previously difficult to read
- fixed some minor editing errors
I consider that in its current form the paper can be published in the journal.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors made the suggested improvements and I may now recommend the paper publication.