Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper addressed an interesting issue, but it could be improved with some modifications.

1. Please provide more detailed explanations regarding the dynamic that has been employed. It needs further clarification for the readers.
2.  Please provide additional information related to transferring to LMIs.
3. Please add the simulation results and then present experimental results. Comparing these two results can provide insight to readers.
4.  The quality of the figures is not satisfactory. Please improve them.
5.  The literature review is not comprehensive. Please include more up-to-date references. Here are some suggestions:  (Naderi, M., Typaldos, P. and Papageorgiou, M., 2025. Lane-free signal-free intersection crossing via model predictive control. Control Engineering Practice, 154, p.106115.)   (Yanumula, V.K., Typaldos, P., Troullinos, D., Malekzadeh, M., Papamichail, I. and Papageorgiou, M., 2023. Optimal trajectory planning for connected and automated vehicles in lane-free traffic with vehicle nudging. IEEE Transactions on Intelligent Vehicles, 8(3), pp.2385-2399.)  (Malekzadeh, M., Papamichail, I., Papageorgiou, M. and Bogenberger, K., 2021. Optimal internal boundary control of lane-free automated vehicle traffic. Transportation Research Part C: Emerging Technologies, 126, p.103060.) (Yanumula, V.K., Typaldos, P., Troullinos, D., Malekzadeh, M., Papamichail, I. and Papageorgiou, M., 2021, September. Optimal path planning for connected and automated vehicles in lane-free traffic. In 2021 IEEE International Intelligent Transportation Systems Conference (ITSC) (pp. 3545-3552). IEEE.)

Author Response

Dear Editors and Reviewers,

Thank you for your valuable feedback and constructive suggestions on our manuscript. We have carefully addressed all the comments and revised the manuscript accordingly. Below, we provide a point-by-point response to each reviewer’s concerns.

Changes in the revised manuscript are highlighted in red text.

 

Response to Reviewer #1
[The paper addressed an interesting issue, but it could be improved with some modifications.]

Response:

We sincerely appreciate the reviewer’s positive assessment of our work and their constructive suggestion to further improve the manuscript.

 

 

 

Comment 1:

[Please provide more detailed explanations regarding the dynamic that has been employed. It needs further clarification for the readers.]

Response:

Thank you for highlighting the need for a clearer explanation of the dynamic model.

 

We have significantly expanded the explanation of the model assumptions and simplifications in Section 2.1 (pages 4, lines138-146) of the revised manuscript:

“The vehicle dynamics model is developed for model predictive control, prioritizing computational efficiency while retaining essential planar motion characteristics. Key simplifications include:

(1) planar motion constraints (longitudinal, lateral, yaw) under flat-road assumptions;

(2) rigid-body suspension and chassis;

(3) decoupled pure lateral tire slip properties;

(4) single-track bicycle model eliminating load transfers;

(5) quasi-static longitudinal dynamics;

(6) neglected aerodynamic forces. These assumptions reduce the system to three degrees of freedom with front-wheel-drive actuation.”

 

These simplifications reduce the system to a three-degree-of-freedom model while preserving the essential behavior for controller design. We believe these additional details clarify the modeling framework and improve the manuscript’s clarity.

 

 

 

Comment 2:[Please provide additional information related to transferring to LMIs.]

Response:

Thank you for the suggestion to elaborate on the Linear Matrix Inequality (LMI) formulation in our control design. We have thoroughly revised the manuscript to address these two critical issues. Below are the key additions:

• We sincerely apologize for the notation inconsistency in the original formulation. As rightly observed:The parameter originally labeled as in Optimization Problem 1 (Page 9, Lines 301) has been corrected toto align with the definitions.
• We acknowledge that the original manuscript lacked sufficient explanation when transitioning from Optimization Problem 1 to 2. The following additions have been made (Page 9, Lines 301-306):

“Consider constraints：

Where is a compact set containing the origin.

To ensure that the actual perturbated system satisfies the input constraint , the constraints of the nominal system must be tightened through a constraint-tightening procedure, as formalized by in the aforementioned optimization problem 1.”

 

These revisions provide a further explanation of the LMI transformation process. We hope this addresses the reviewer’s concerns and enhances the reproducibility of our method.

 

 

 

Comment 3:[Please add the simulation results and then present experimental results. Comparing these two results can provide insight to readers.]

Response:

Thank you for your thorough review and constructive feedback. We have revised the manuscript as follows

• Enhanced Simulation Results

Data inconsistencies and missing data points (yaw angle error) during re-examination have been rectified through comprehensive verification protocols. All affected analyses are now systematically updated in Tables 1–2 (Page 11/12, Lines 357/358)with validated results

• Systematic Comparative Analysis

All modifications are highlighted in red text for easy identification. These revisions strengthen the manuscript’s theoretical-practical integration while addressing your insightful critique. We are grateful for your guidance and welcome further suggestions to improve this work. Below are the key additions (Page 12, Lines 365-383):

“Moreover, Reduction in Yaw angle error in S-curve: Tube-RMPC achieves a 1.77% reduction in the peak deviation compared to MPC and a 5.66% improvement over PID; Lower Yaw angle RMSE: Tube-RMPC achieves a 7.00% reduction in peak deviation compared to MPC and an 9.44% improvement over PID.

Moreover, Reduction in Yaw angle error in double lane change: Tube-RMPC achieves a 23.06% reduction in the peak deviation compared to MPC and a 28.10% improvement over PID; Lower Yaw angle RMSE: Tube-RMPC achieves a 0.03% reduction in the peak deviation compared to MPC and a 6.83% improvement over PID.

Tube-RMPC outperforms MPC and PID in both S-curve and double lane change scenarios, achieving up to 71.17% lower RMSE in tracking accuracy and superior yaw-angle stabilization, especially in dynamic maneuvers. Its robustness is evident in the near-identical yaw RMSE (0.03% difference) compared to MPC during double lane change, underscoring its precision and reliability for high-performance autonomous navigation,”

 

 

 

Comment 4:[The quality of the figures is not satisfactory. Please improve them]

Response:

Thank you for your suggestion. We have improved the quality of all simulation figures in the revised manuscript to ensure better clarity and readability.

 

 

 

Comment 5:[The literature review is not comprehensive. Please include more up-to-date references. Here are some suggestions: (Naderi, M., Typaldos, P. and Papageorgiou, M., 2025. Lane-free signal-free intersection crossing via model predictive control. Control Engineering Practice, 154, p.106115.)  (Yanumula, V.K., Typaldos, P., Troullinos, D., Malekzadeh, M., Papamichail, I. and Papageorgiou, M., 2023. Optimal trajectory planning for connected and automated vehicles in lane-free traffic with vehicle nudging. IEEE Transactions on Intelligent Vehicles, 8(3), pp.2385-2399.) (Malekzadeh, M., Papamichail, I., Papageorgiou, M. and Bogenberger, K., 2021. Optimal internal boundary control of lane-free automated vehicle traffic. Transportation Research Part C: Emerging Technologies, 126, p.103060.) (Yanumula, V.K., Typaldos, P., Troullinos, D., Malekzadeh, M., Papamichail, I. and Papageorgiou, M., 2021, September. Optimal path planning for connected and automated vehicles in lane-free traffic. In 2021 IEEE International Intelligent Transportation Systems Conference (ITSC) (pp. 3545-3552). IEEE.).]

Response:

We appreciate the reviewer’s suggestion to enhance the literature review with more recent and relevant works. The recommended references are indeed valuable and closely related to our research topic. Accordingly, we have updated Section 1 to incorporate these studies and provide a broader context for our work. The discussion now better reflects the current developments in lane-free traffic modeling and MPC-based control strategies. Below are the key additions:

“Addressing these challenges necessitates a focused exploration of advanced control strategies that are capable of ensuring stability under varying conditions, making this a crucial direction in autonomous driving technology research[4].(Page 2, Line 4

3)

To address this critical challenge, a variety of advanced control methodologies have been developed, including fuzzy control[5], linear quadratic regulators[6], sliding-mode control[7], and model predictive control (MPC)[8–11].(Page 2, Line 54)

In addition to these efforts, recent research has extended MPC applications to novel traffic paradigms, such as lane-free environments[15,16] proposed an optimal path planning framework for connected and automated vehicles (CAVs) operating in lane-free traffic conditions, demonstrating the versatility of MPC in unstructured scenarios where traditional lane-based assumptions are not applicable.(Page 2, Lines 71-76)”

We once again thank the reviewer for your detailed and insightful comments, which have helped us improve both the theoretical foundation and presentation quality of our work. We hope the revised manuscript addresses your concerns satisfactorily.

Reviewer 2 Report

Comments and Suggestions for Authors

This work proposes a Tube-RMPC strategy for trajectory tracking under uncertainties. It shows simulation results. Methodolgy is good and shows improvement from other methods. However, I have some comments.

Major concerns:

1. A clearer derivation of the Tube-RMPC strategy  would enhance readability of the paper.
2. Assure a fair comparison between MPC, PID and Tube-RMPC considering tuning.
3. In order to show robustness, sensitivity analysis would be a must.

Minor concerns:

4. Please correct typos i.e. "Dnamaic".

5. Include description of each signal in figure captions.

6. Is correct data in tables? Please check.

 

 

 

 

Comments on the Quality of English Language

The English language quality should be improved.

Author Response

Dear Editors and Reviewers,

Thank you for your valuable feedback and constructive suggestions on our manuscript. We have carefully addressed all the comments and revised the manuscript accordingly. Below, we provide a point-by-point response to each reviewer’s concerns.

Changes in the revised manuscript are highlighted in red text.

 

Response to Reviewer #2
[This work proposes a Tube-RMPC strategy for trajectory tracking under uncertainties. It shows simulation results. Methodolgy is good and shows improvement from other methods. However, I have some comments.]

Response:

We sincerely thank the reviewer for the positive assessment of our work, particularly recognizing the novelty of the proposed Tube-RMPC strategy and its performance advantages over baseline methods. We have carefully addressed all comments and revised the manuscript accordingly to improve clarity, completeness, and technical rigor. The following responses detail the specific modifications made in response to your valuable feedback.

 

Comment 1: [A clearer derivation of the Tube-RMPC strategy would enhance readability of the paper.]

Response:
We sincerely thank the reviewer for the constructive comment. We fully agree that the original derivation of the Tube-RMPC strategy contained certain logical gaps, which may hinder readers’ understanding.

In response, we have carefully revised this section. Specifically, we added a clarifying sentence prior to Theorem 1 to better connect the derivation of the error dynamics and the computation of the feedback gain . Below are the key addition (Page 7, Lines 249-250):“The feedback gain matrix can be obtained by applying Theorem 3.1 presented below.”

Furthermore, we have expanded the explanations between Optimization Problem 1 and Optimization Problem 2 to provide a more continuous and transparent derivation flow. These additions aim to improve the clarity and logical coherence of the controller design process. Below are the key additions(Page 9, Lines 301-306):

“Consider constraints：

Where is a compact set containing the origin.

To ensure that the actual perturbated system satisfies the input constraint , the constraints of the nominal system must be tightened through a constraint-tightening procedure, as formalized by in the aforementioned optimization problem 1.”

 

We hope these revisions adequately address the reviewer’s concerns and enhance the readability of the manuscript.

Comment 2: [Assure a fair comparison between MPC, PID and Tube-RMPC considering tuning].

Response:
We sincerely appreciate the reviewer’s insightful comment regarding the importance of fair tuning in controller comparisons.

In response, we would like to clarify that all three control strategies—Tube-RMPC, MPC, and PID—were tuned based on the same performance criterion, specifically the minimization of trajectory tracking error. This approach ensures a consistent and fair basis for comparison.

As stated in the literature review, both PID and MPC are commonly used and well-established methods for trajectory tracking. Therefore, we selected them as representative baseline controllers to benchmark against the proposed Tube-RMPC strategy.Moreover, we conducted multiple sets of experiments for each controller under identical simulation conditions. For each method, we reported the best-performing result in terms of tracking performance and stability. Below are the key additions(Page 10, Lines 342-346):

“To ensure a fair comparison, all three controllers—Tube-RMPC, MPC, and PID—were tuned using the same performance criterion: minimizing trajectory tracking error under identical simulation conditions.As noted in the literature, PID and MPC are widely used in vehicle trajectory tracking and serve as effective baselines for evaluating the proposed Tube-RMPC approach.”

We have added this explanation to Section 4.1 of the revised manuscript to explicitly address this concern. We hope these clarifications adequately address the reviewer’s suggestion and improve the rigor of the comparative analysis.

Comment 3: [In order to show robustness, sensitivity analysis would be a must.]

Response:
We sincerely thank the reviewer for pointing out the importance of sensitivity analysis in demonstrating robustness.

In Section 4.2 of our manuscript, we have included RMSE-based evaluations under different test scenarios to reflect the system's sensitivity and robustness. We believe these results provide a preliminary yet meaningful indication of the controller's performance under varying conditions.

However, if the current analysis does not fully meet the reviewer’s expectations, we would greatly appreciate further clarification or suggestions on specific aspects to improve.We are also willing to include additional sensitivity experiments in future revisions, as per the reviewer’s guidance.

Comment 4: [Please correct typos i.e. "Dnamaic".]

Response:
We apologize for the typographical errors. All identified typos, including “Dnamaic,”(Page , Line 135) have been corrected in the revised manuscript. We have also performed a thorough proofreading to eliminate similar issues.

Comment 5: [Include description of each signal in figure captions.]

Response:
Thank you for the suggestion. All figure captions have been revised to include clear and complete descriptions of each signal presented. This improves the interpretability and self-contained nature of each figure.

Comment 6: [Is correct data in tables? Please check.]

Response:
We sincerely thank the reviewer for carefully checking the data presented in the manuscript.

Upon review, we found that there were indeed inaccuracies in the data reported in Tables 1 and 2 (Page 11/12, Lines 357/358). We sincerely apologize for this oversight. The tables have been corrected in the revised version, and corresponding discussions and analyses have been updated accordingly to reflect the accurate results.

We appreciate the reviewer’s attention to detail, which helped us improve the quality and reliability of the manuscript.

We sincerely thank the reviewer once again for the constructive feedback, which has greatly improved the clarity and completeness of our work. We remain open to further comments and are grateful for the opportunity to revise our manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

The topic of the paper is current and concerns the control of a car vehicle under conditions of an uncertain dynamic model. Unfortunately, the research contribution is unclear. In particular, I find it difficult to agree with the two original contributions indicated in the introduction. The vehicle model under slip conditions is well known in the literature, and the Tube RMPC method is also not new. Therefore, in my opinion, the only real contribution of this work is the software implementation and the performance of simulation studies using an environment with high fidelity in modelling vehicle motion.
The structure of the paper does not meet the standards expected of high-quality scientific articles. This is a significant issue that reduces its readability and makes it difficult to identify original results. There is no clear formulation of the control problem investigated in the paper. Despite the title of Section 2, no such problem is defined. Instead, the reader is presented with a description of technical details that do not elucidate the main idea of the paper.

Specific comments:
1. The Magic Tyre Formula is fundamentally highly nonlinear, whereas equation (3) presents a linear model. It is evident that the authors have applied a Taylor linear approximation, but this should be explicitly stated.
2. Lines 140–141: The model is nonlinear only due to kinematics, while the dynamics have been simplified to a linear form.
3. Formula (5): How should the input \Delta \delta_f​ be interpreted? I believe a comment here is necessary.
4. In subsection 2.2, the authors recall a concept they refer to as linearisation. From the perspective of control theory, it would be more appropriate to consider this as a linear approximation, to distinguish it from state or feedback linearisation. Moreover, the introduction of the approximation is unconvincing, as the original nonlinear dynamics with the state function f are not presented. I do not believe this paragraph is necessary, as linear approximation is a well-known technique. Note that an approximation has already been used in equation (3), although this was not introduced earlier. Also, note that this paragraph misuses notation. What is the relationship between the state \xi_{dyn}​ in (5) and \xi in (6)? Why not simply apply the approximation procedure to (5) without digressing into theoretical discussion?
5. Lines 157–158: The claim that time-continuous models cannot be used in the design of predictive controllers is too strong. While time-discrete models are typically used in practice, this is primarily for implementation reasons, not a limitation of the concept. I suggest rewriting this paragraph to convey a more balanced view.
6. The time-discrete model (8) and the matrices presented afterwards are the result of a specific discretisation approach. In my view, the discretisation method employed by the authors should be explicitly named, as other discretisation schemes could have been considered.
7. Figure 2 is unclear. In particular, the content of the block diagram is unreadable.
8. The matrix D in equation (9) is not explained.
9. Section 3.1 is more of a sketch of the applied method than a proper algorithm presentation. The control law should be clearly highlighted. Does equation (14) describe it? If so, we are dealing with a simple linear controller with a feedforward/compensating term represented by \bar{u}'.
10. Section 3.2 is not well structured and fails to present the algorithm in a satisfactory manner, needlessly including digressions and commentary. The authors do not adopt the narrative structure typically used in academic publications, where formal methods (e.g., Corollary, Remark, etc.) are employed. What is the purpose of proving theorems that are already known in the literature? Furthermore, some theoretical concepts introduced are not directly used. For instance, Lemma 1 introduces a Minkowski sum that is not used later. If that is the case, what is the purpose of including this lemma?
In my view, this section (like 3.1) should be rewritten, adhering to the best practices of scientific writing, with a clear identification of key results and proposals.
11. I am concerned with the title of Section 4, where the authors imply that experimental results are presented. In my opinion, these are simulation results - realistic thanks to the CarSim tool, but still simulations. I believe this distinction should be clearly explained in the description of the simulation scenarios.
12. The algorithm comparison, in my view, does not allow for an objective evaluation. How can we be certain that the PID and MPC methods (by the way—what specific MPC implementation?) cannot achieve better results? How were these algorithms tuned, and what criteria were used? What was held constant (e.g., energy consumption, oscillatory behaviour, stability) in the comparison?

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please see the attached file. In addition, that there are o lot of typos - in many places, there is a lack of spacing before parentheses when referencing equations and literature.

Author Response

Dear Editors and Reviewers,

Thank you for your valuable feedback and constructive suggestions on our manuscript. We have carefully addressed all the comments and revised the manuscript accordingly. Below, we provide a point-by-point response to each reviewer’s concerns.

Changes in the revised manuscript are highlighted in red text.

 

Response to Reviewer #3
[The topic of the paper is current and concerns the control of a car vehicle under conditions of an uncertain dynamic model. Unfortunately, the research contribution is unclear. In particular, I find it difficult to agree with the two original contributions indicated in the introduction. The vehicle model under slip conditions is well known in the literature, and the Tube RMPC method is also not new. Therefore, in my opinion, the only real contribution of this work is the software implementation and the performance of simulation studies using an environment with high fidelity in modelling vehicle motion.
The structure of the paper does not meet the standards expected of high-quality scientific articles. This is a significant issue that reduces its readability and makes it difficult to identify original results. There is no clear formulation of the control problem investigated in the paper. Despite the title of Section 2, no such problem is defined. Instead, the reader is presented with a description of technical details that do not elucidate the main idea of the paper.]

 

Response:

We sincerely thank the reviewer for the detailed and insightful comments. We acknowledge the importance of clearly articulating research contributions and presenting the control problem in a rigorous and structured manner.

In response, we have made several key revisions to the manuscript:

• Clarified Contributions and Motivation:
  Although both the vehicle dynamic model under slip conditions and the Tube-based RMPC strategy have been previously established in the literature, our work focuses on a practical extension of these frameworks by explicitly modeling road–tire adhesion coefficient uncertainties within the Tube-RMPC formulation. This factor, while acknowledged in earlier studies, is often simplified or omitted during controller implementation. As highlighted in the revised Introduction, control robustness under varying surface conditions is a crucial challenge in autonomous driving. Our study builds upon existing RMPC theory and contributes a computationally feasible control framework that integrates these uncertainties into the tube structure and validates its effectiveness using a high-fidelity CarSim–Simulink co-simulation environment, thereby enhancing its practical applicability.The following additions have been made (Page 2, Lines 43-49):

“In this context, the design of robust and adaptive vehicle control strategies plays a pivotal role in ensuring safe navigation and trajectory tracking. Effective control systems must be capable of handling not only modeling uncertainties but also real-world disturbances such as variations in road-tire adhesion. These variations, which are difficult to measure directly in real-time, significantly affect lateral dynamics and must be addressed within the control framework to enhance safety and performance.”

 

• Improved Structure and Terminology:
  The manuscript has been restructured for clarity and consistency. To improve readability and theoretical clarity, we have made several targeted revisions throughout the manuscript. Specifically:

we provide a detailed explanation of the modeling assumptions used in the simplified 3-DOF vehicle model, clarifying the physical basis and control-oriented simplifications.The following additions have been made (Page 4, Lines138-146):

“The vehicle dynamics model is developed for model predictive control, prioritizing computational efficiency while retaining essential planar motion characteristics. Key assumptions include: (1) planar motion constraints (longitudinal, lateral, yaw) under flat-road assumptions; (2) rigid-body suspension and chassis; (3) decoupled pure lateral tire slip properties; (4) single-track bicycle model eliminating load transfers; (5) quasi-static longitudinal dynamics; (6) neglected aerodynamic forces. These assumptions reduce the system to three degrees of freedom with front-wheel-drive actuation.”

 

We acknowledge that the original manuscript lacked sufficient explanation when transitioning from Optimization Problem 1 to 2. The following additions have been made (Page 9, Lines 301-306):

“Consider constraints：

Where is a compact set containing the origin.

To ensure that the actual perturbated system satisfies the input constraint , the constraints of the nominal system must be tightened through a constraint-tightening procedure, as formalized by in the aforementioned optimization problem 1.”

These changes collectively improve the theoretical rigor and flow of the control design section.

 

• Explicit Problem Statement (Section 2):

In the revised manuscript, we have addressed this issue by adding a short but explicit paragraph at the beginning of Section 2, just before Section 2.1. This new paragraph bridges the modeling content with the main research objective and introduces the purpose of the Tube-RMPC design.The added paragraph includes the following statement (Page 3, Lines 129-134):

“To formulate this control problem in a mathematically tractable way, a unified dynamic vehicle–tire model is first constructed and simplified into a linearized form. This model serves as the basis for the development of a Tube-based Robust Model Predictive Control (Tube-RMPC) strategy. The aim is to ensure that the actual system trajectory remains within a predefined robust tube surrounding the nominal prediction, despite the presence of external disturbances and parameter variations.”

This addition provides a clearer problem framing and allows readers to better understand the motivation behind the modeling and control strategy that follows.

We greatly appreciate the reviewer’s comments, which have helped us improve both the clarity and rigor of the manuscript.

 

 

Comment 1: [The Magic Tyre Formula is fundamentally highly nonlinear, whereas equation (3) presents a linear model. It is evident that the authors have applied a Taylor linear approximation, but this should be explicitly stated.]

Response:
We appreciate the reviewer’s insightful observation. We agree that our original description was unclear and placed undue emphasis on the Magic Tire Formula itself, rather than on the specific approximation we intended to use. Our actual intention was to refer to the small-angle assumption commonly employed in the context of tire modeling, which leads to an approximately linear relationship between lateral force and sideslip angle.

The revised content is as follows (Page 4, Lines 161-162): 

“Under the small-angle assumption derived from the Magic Tire model in[32], the lateral force is approximately linearly related to the sideslip angle of the tire, as shown below:”

We believe this revision better reflects our modeling intent and improves the clarity of the mathematical formulation.

 

Comment 2: [Lines 140–141: The model is nonlinear only due to kinematics, while the dynamics have been simplified to a linear form.].

:
We thank the reviewer for this accurate observation. In general, whether a dynamic model is nonlinear or linear depends on the physical properties of the system, the modeling purpose, and the simplifications applied during derivation.  In our case, although the overall vehicle model originates from nonlinear dynamics, we have applied standard assumptions to obtain a control-oriented linear approximation of the dynamic equations. We acknowledge that our original manuscript did not clearly articulate these assumptions. To address this, we have revised the relevant section to clarify the modeling assumptions and their role in simplifying the dynamics as follows (Page 4, Lines 138-146):

“The vehicle dynamics model is developed for model predictive control, prioritizing computational efficiency while retaining essential planar motion characteristics. Based on the prior knowledge of vehicle dynamics, a simplified three-degree-of-freedom vehicle model is developed under specific assumptions[31]. Key assumptions include: (1) planar motion constraints (longitudinal, lateral, yaw) under flat-road assumptions; (2) rigid-body suspension and chassis; (3) decoupled pure lateral tire slip properties; (4) single-track bicycle model eliminating load transfers; (5) quasi-static longitudinal dynamics; (6) neglected aerodynamic forces. These assumptions reduce the system to three degrees of freedom with front-wheel-drive actuation.”

Additionally, we note that the system equations used in our study (Eq. (1)) are derived based on the trigonometric relationships illustrated in Figure 1, which describe the force components on the front and rear tires in both longitudinal and lateral directions.  

We believe these revisions improve the transparency of our modeling methodology and better reflect the structure of the vehicle dynamics used for controller design.

Comment 3: [Formula (5): How should the input \Delta \delta_f​ be interpreted? I believe a comment here is necessary]

Response:
We sincerely thank the reviewer for the careful and detailed examination. This is indeed our mistake. As can be seen from Equation (4) (Page 5, Line 168), the control input has consistently been the front wheel steering angle , and there is no definition or usage of a control increment in the actual formulation. The appearance of in Equation (5) (Page 5, Line 171) was a typographical error.

We have corrected this in the revised manuscript by replacing with in Equation (5), to maintain consistency throughout the control model derivation. This correction ensures that the input variable is clearly and consistently represented as the actual steering angle applied to the system.

 

Comment 4: [In subsection 2.2, the authors recall a concept they refer to as linearisation. From the perspective of control theory, it would be more appropriate to consider this as a linear approximation, to distinguish it from state or feedback linearisation. Moreover, the introduction of the approximation is unconvincing, as the original nonlinear dynamics with the state function f are not presented. I do not believe this paragraph is necessary, as linear approximation is a well-known technique. Note that an approximation has already been used in equation (3), although this was not introduced earlier. Also, note that this paragraph misuses notation. What is the relationship between the state \xi_{dyn}​ in (5) and \xi in (6)? Why not simply apply the approximation procedure to (5) without digressing into theoretical discussion?]

Response:
We thank the reviewer for this detailed and insightful comment. Your suggestions have greatly improved the clarity and rigor of our manuscript.

First, we fully agree that the term “linearisation” was not precise in this context. As the reviewer correctly points out, exact feedback or state linearisation typically requires system-specific conditions and is not generally applicable. In practical applications of model predictive control (MPC), linear approximation via first-order Taylor expansion around an operating point is a standard and widely accepted approach. Accordingly, we have replaced the use of “linearisation” with “linear approximation” (Page 5, Line 177) in the revised manuscript to avoid confusion.

Second, we acknowledge that the original explanation of the approximation was not sufficiently clear. Our intention in introducing the function  was to express the deviation between the current and reference system dynamics in a general form. In particular, our controller is designed based on the error dynamics between the actual system and a reference trajectory. Although linear approximation is a well-known method, we believe it is still necessary to mention it explicitly to show that our linear model is not arbitrarily assumed, but obtained via Taylor expansion, and to connect it with the structure of our robust controller design.

Regarding the reviewer’s suggestion to directly apply the approximation to Equation (5), we respectfully note that doing so would eliminate the explicit role of the Taylor expansion and the transition from general nonlinear dynamics to a linear predictive framework. Therefore, we chose to retain this step.

We believe these revisions improve both the logical flow and the interpretability of the modeling process.

Comment 5: [ines 157–158: The claim that time-continuous models cannot be used in the design of predictive controllers is too strong. While time-discrete models are typically used in practice, this is primarily for implementation reasons, not a limitation of the concept. I suggest rewriting this paragraph to convey a more balanced view.]

Response:
We sincerely thank the reviewer for the helpful correction. We agree that our original wording was too strong and did not reflect the theoretical validity of continuous-time predictive control.

In the revised manuscript, we have modified the relevant statement to present a more balanced and realistic perspective. While continuous-time models are indeed applicable in predictive control from a theoretical standpoint, their use often leads to high computational complexity, especially when dealing with nonlinear dynamics. This makes them less suitable for high-frequency control tasks and real-time implementation.

Accordingly, we now write(Page 5, Lines 185-188):

“The derived equation is continuous and continuous-time models are theoretically applicable to predictive control, but they often result in excessive computational effort in real-time scenarios. Therefore, it is a reasonable and effective approach to discretized the continuous linear system.”

This revision more accurately reflects both the theoretical scope and the practical considerations of model predictive control design. We appreciate the reviewer’s suggestion, which helped us improve the accuracy and tone of our discussion.

 

Comment 6: [The time-discrete model (8) and the matrices presented afterwards are the result of a specific discretisation approach. In my view, the discretisation method employed by the authors should be explicitly named, as other discretisation schemes could have been considered.]

Response:
We thank the reviewer for this important observation. The original manuscript did not explicitly state the discretization method applied to obtain the time-discrete model, which may have caused ambiguity.

In our study, we employed the Forward Euler method to discretize the continuous-time reference system. This approach was chosen for its simplicity and efficiency, particularly in real-time implementations of model predictive control. It allows the continuous-time dynamics to be approximated as a linear time-invariant discrete system that is suitable for subsequent controller design.

To clarify this, we have added the following sentence to Section 2.2, prior to Equation (8) (Page 5, Line 189):

“The continuous linear system is converted into its discrete-time form using the Forward Euler method[35] as follows”

We believe this revision improves the clarity and completeness of the modeling methodology.

 

Comment 7: [Figure 2 is unclear. In particular, the content of the block diagram is unreadable.]

Response:
We appreciate the reviewer’s feedback. We acknowledge that the resolution and visual quality of Figure 2 in the original submission were insufficient, making the content of the block diagram difficult to read.

In the revised manuscript, we have redrawn Figure 2 with improved layout, font size, and resolution, ensuring that all components, labels, and interconnections are clearly legible.

We believe this revision significantly improves the readability and presentation quality of the figure.

Comment 8: [The matrix D in equation (9) is not explained.].

Response:
We thank the reviewer for highlighting this omission. In our formulation, the matrix  is the disturbance distribution matrix, which describes how external uncertainties influence the system dynamics. The term  corresponds to a bounded parameter disturbance, primarily representing variations in road–tire adhesion conditions.

To clarify this, we have added the following explanatory sentence after Equation (9) in the revised manuscript(Page 6, Lines 214-215):

“ is the disturbance distribution matrix, and w(k) corresponds to a bounded parameter disturbance.”

This addition makes the role of  and the disturbance term explicit, improving the clarity of the robust control formulation.

Comment 9: [Section 3.1 is more of a sketch of the applied method than a proper algorithm presentation. The control law should be clearly highlighted. Does equation (14) describe it? If so, we are dealing with a simple linear controller with a feedforward/compensating term represented by \bar{u}.]

Response:
We sincerely thank the reviewer for emphasizing the importance of clearly presenting the control law. We fully agree that the control law is a key component of any controller design, and that it should be prominently identified.

In our manuscript, Equation (14) does represent the final Tube-RMPC control law. While we considered placing this expression at the beginning of Section 3.1 for emphasis, we ultimately chose to retain its current position to maintain a smooth and logical transition from the derivation in Equation (13) to the final form in Equation (15).This arrangement was intended to ensure mathematical continuity and avoid introducing the final result prematurely, without a proper derivation path. Nevertheless, we have revised the text to make it explicitly clear as follow(Page 7, Line 235)：

“The Tube-RMPC control law, which constitutes the core of the proposed controller, is expressed as”, and we hope this presentation is acceptable to the reviewer.

Comment 10: [Section 3.2 is not well structured and fails to present the algorithm in a satisfactory manner, needlessly including digressions and commentary. The authors do not adopt the narrative structure typically used in academic publications, where formal methods (e.g., Corollary, Remark, etc.) are employed. What is the purpose of proving theorems that are already known in the literature? Furthermore, some theoretical concepts introduced are not directly used. For instance, Lemma 1 introduces a Minkowski sum that is not used later. If that is the case, what is the purpose of including this lemma? In my view, this section (like 3.1) should be rewritten, adhering to the best practices of scientific writing, with a clear identification of key results and proposals.]

Response:
We sincerely thank the reviewer for the detailed and critical feedback. We agree that in the initial version of the manuscript, the structure of Section 3.2 lacked clarity and may have caused confusion regarding the relevance and purpose of certain theoretical components.

In the revised version, we have strengthened the logical progression of the derivation and addressed the issues you raised as follows:

• In particular, in response to the reviewer’s concern regarding the purpose of including known theoretical results, we have revised the wording around Line 234to provide a more natural and context-driven introduction to the theorem. Rather than presenting the theorem in isolation, we now explain its necessity in the robust invariance analysis of the error dynamics, thereby directly linking it to the controller’s performance guarantees. We have also revised the transition between Optimization Problem 1 and Optimization Problem 2. The previous version lacked sufficient explanation and appeared abrupt. In the revised text, we now explicitly explain how the constraint tightening and disturbance handling motivate the reformulation, providing a more intuitive and logical flow between the two problems. The following additions have been made (Page 9, Lines 301-306):

“Consider constraints：

Where is a compact set containing the origin.

To ensure that the actual perturbated system satisfies the input constraint , the constraints of the nominal system must be tightened through a constraint-tightening procedure, as formalized by in the aforementioned optimization problem 1.”

• we have clarified the purpose of this lemma in Lines 281–287, where we explain that the Minkowski sum is essential for computing the minimal robust positively invariant set. Specifically, the formula for characterizing this set involves the Minkowski sum between the disturbance set and the system dynamics, which forms the basis for deriving invariant tube bounds. Moreover, the method we use for computing the minimal invariant set follows the approach described in Reference [40].

These revisions were made with the aim of improving readability, removing unnecessary complexity, and highlighting the essential theoretical justifications of the proposed control strategy. We hope the updated structure meets the reviewer’s expectations and reflects best practices in academic writing.

 

Comment 11: [I am concerned with the title of Section 4, where the authors imply that experimental results are presented. In my opinion, these are simulation results - realistic thanks to the CarSim tool, but still simulations. I believe this distinction should be clearly explained in the description of the simulation scenarios.]

Response:
Thank you for your constructive suggestion. We agree that the current title “Experimental Results” may be misleading, as the results presented in Section 4 are obtained from a co-simulation environment rather than physical experiments.

Based on your valuable comment, we have revised the title of Section 4 (Page 9, Line 322)to “Co-Simulation Result and Analysis”, which better represents the simulation-based validation of the proposed Tube-RMPC control strategy. In addition, the subsection titles have been updated as follows: Section 4.1(Page 9, Line 323) → “Co-simulation Process”; Section 4.2(Page 9, Line 347) → “Co-simulation Results”

These changes enhance the clarity and consistency of the manuscript, ensuring that readers are properly informed about the scope and context of the presented evaluation.

 

Comment 12: [The algorithm comparison, in my view, does not allow for an objective evaluation. How can we be certain that the PID and MPC methods (by the way—what specific MPC implementation?) cannot achieve better results? How were these algorithms tuned, and what criteria were used? What was held constant (e.g., energy consumption, oscillatory behaviour, stability) in the comparison?]

Response:
We thank the reviewer for this important question regarding the fairness and transparency of the controller comparison.

First, we sincerely acknowledge that the initial version of the manuscript had an oversight in the presentation of simulation results, where some performance data—particularly related to yaw angle errors—were either incomplete or missing. This may have affected the clarity and completeness of the result analysis.

we would like to clarify that all three control strategies—Tube-RMPC, MPC, and PID—were tuned based on the same performance criterion, specifically the minimization of trajectory tracking error. This approach ensures a consistent and fair basis for comparison.As stated in the literature review, both PID and MPC are commonly used and well-established methods for trajectory tracking. Therefore, we selected them as representative baseline controllers to benchmark against the proposed Tube-RMPC strategy. Moreover, we conducted multiple sets of experiments for each controller under identical simulation conditions. For each method, we reported the best-performing result in terms of tracking performance and stability. Below are the key additions(Page 10, Lines 342-346):

“To ensure a fair comparison, all three controllers—Tube-RMPC, MPC, and PID—were tuned using the same performance criterion: minimizing trajectory tracking error under identical simulation conditions.As noted in the literature, PID and MPC are widely used in vehicle trajectory tracking and serve as effective baselines for evaluating the proposed Tube-RMPC approach.”

In Section 4.2 of our manuscript, we have included RMSE-based evaluations under different test scenarios to reflect the system's sensitivity and robustness. Below are the key additions (Page 12, Lines 365-383):

“Moreover, Reduction in Yaw angle error in S-curve: Tube-RMPC achieves a 1.77% reduction in the peak deviation compared to MPC and a 5.66% improvement over PID; Lower Yaw angle RMSE: Tube-RMPC achieves a 7.00% reduction in peak deviation compared to MPC and an 9.44% improvement over PID.

Moreover, Reduction in Yaw angle error in double lane change: Tube-RMPC achieves a 23.06% reduction in the peak deviation compared to MPC and a 28.10% improvement over PID; Lower Yaw angle RMSE: Tube-RMPC achieves a 0.03% reduction in the peak deviation compared to MPC and a 6.83% improvement over PID.

Tube-RMPC outperforms MPC and PID in both S-curve and double lane change scenarios, achieving up to 71.17% lower RMSE in tracking accuracy and superior yaw-angle stabilization, especially in dynamic maneuvers. Its robustness is evident in the near-identical yaw RMSE (0.03% difference) compared to MPC during double lane change, underscoring its precision and reliability for high-performance autonomous navigation,”

We believe these results provide a preliminary yet meaningful indication of the controller's performance under varying conditions.

We sincerely thank the reviewer once again for the constructive feedback, which has greatly improved the clarity and completeness of our work. We remain open to further comments and are grateful for the opportunity to revise our manuscript.

Reviewer 4 Report

Comments and Suggestions for Authors

An interesting Model Predictive Control, called Tube RMPC, has been proposed in the present work. The paper is well-structured and theoretical explanation are clear, but some points must be revised:

1) The impact of road-tire adhesion variations is explored a lot by the scientific community, especially by researchers which study tires for road vehicle. Therefore, the end of the Introduction should be modified underlining the potential of the proposed control technique, which is another possibility for considering the adhesion variations during control activities on road vehicles.

2) In Figure 2, the name "Experimental Platform" must be changed due to the non-presence of real experimental data, but the Tube RMPC has been tested by using a co-simulation approach. Further works can be adopted for validating the control strategy on a real autonomous road veichle

3) Line 240, there is an END that should be deleted.

4) Section 4 is not related to Experimental Results. The name of the Section 4 and of related subsections must be modified.

5) The comparisons in the Result Section must be increased. There are no results concerning the robustness to noises and uncertainties in measurements adopted for providing the feedback to the proposed control strategy. Furthermore, the proposed strategy can be compared with a nonlinear model predictive control? What is the gain in terms of computational effort of the Tube RMPC compared with the other ones?

 

Author Response

Dear Editors and Reviewers,

Thank you for your valuable feedback and constructive suggestions on our manuscript. We have carefully addressed all the comments and revised the manuscript accordingly. Below, we provide a point-by-point response to each reviewer’s concerns.

Changes in the revised manuscript are highlighted in red text.

 

Response to Reviewer #4
[An interesting Model Predictive Control, called Tube RMPC, has been proposed in the present work. The paper is well-structured and theoretical explanation are clear, but some points must be revised:]

Response:

We sincerely thank the reviewer for the positive evaluation of our work, especially regarding the structure and theoretical explanations. We appreciate your constructive comments and have carefully addressed each of the suggested revisions to further improve the manuscript.

 

Comment 1: [The impact of road-tire adhesion variations is explored a lot by the scientific community, especially by researchers which study tires for road vehicle. Therefore, the end of the Introduction should be modified underlining the potential of the proposed control technique, which is another possibility for considering the adhesion variations during control activities on road vehicles.]

Response:
Thank you for your insightful comment. We fully agree that the impact of road-tire adhesion variations has been extensively studied by the scientific community, particularly in the context of tire modeling and friction estimation. To address your suggestion and better highlight the novelty of our approach, we have revised the end of the Introduction to emphasize the importance of robust vehicle control strategies that explicitly handle adhesion uncertainties. We have clarified that our proposed Tube-RMPC method provides a complementary and control-oriented perspective by incorporating road-tire adhesion variation as a bounded disturbance, thereby enhancing system robustness in real-world driving conditions.

The following sentences have been added at the end of the Introduction section (Page 2, Lines 43-49):“In this context, the design of robust and adaptive vehicle control strategies plays a pivotal role in ensuring safe navigation and trajectory tracking. Effective control systems must be capable of handling not only modeling uncertainties but also real-world disturbances such as variations in road-tire adhesion. These variations, which are difficult to measure directly in real-time, significantly affect lateral dynamics and must be addressed within the control framework to enhance safety and performance.”

We believe this addition clarifies the potential of the proposed control strategy and its relevance to the handling of road adhesion variations.

 

Comment 2: [In Figure 2, the name "Experimental Platform" must be changed due to the non-presence of real experimental data, but the Tube RMPC has been tested by using a co-simulation approach. Further works can be adopted for validating the control strategy on a real autonomous road veichle.]

 

Response:
Thank you for pointing out this issue. We acknowledge that the term “Experimental Platform” may cause confusion, as the presented results are based on a co-simulation environment rather than physical experiments.

In response to your suggestion, we have modified the title of Figure 2 from “Experimental Platform” to “Co-Simulation Framework” to accurately reflect the virtual testing approach employed in this study.

Modifications in the manuscript include:

The title of the middle block in Figure 2 has been changed to “Tube-RMPC Strategy”, which better represents the control method being implemented.

The title of the rightmost block has been modified to “Co-Simulation Framework”, clearly indicating that the evaluation environment is based on MATLAB/Simulink and CarSim.

We appreciate your helpful feedback, which has improved the clarity and precision of our presentation.

Comment 3: [Line 240, there is an END that should be deleted.]

Response:
Thank you for pointing out this detail. We have carefully reviewed the manuscript and confirmed that there was an unintended and redundant “END” at Line 240. This has now been deleted in the revised version to avoid confusion and improve readability.

We appreciate your careful review and attention to detail.

 

Comment 4: [Section 4 is not related to Experimental Results. The name of the Section 4 and of related subsections must be modified.]

Response:
Thank you for your constructive suggestion. We agree that the current title “Experimental Results” may be misleading, as the results presented in Section 4 are obtained from a co-simulation environment rather than physical experiments.

Based on your valuable comment, we have revised the title of Section 4 (Page 9, Line 322) to “Co-Simulation Result and Analysis”, which better represents the simulation-based validation of the proposed Tube-RMPC control strategy. In addition, the subsection titles have been updated as follows: Section 4.1(Page 9, Line 323) → “Co-simulation Process”; Section 4.2(Page 9, Line 347) → “Co-simulation Results”

These changes enhance the clarity and consistency of the manuscript, ensuring that readers are properly informed about the scope and context of the presented evaluation.

Comment 5: [The comparisons in the Result Section must be increased. There are no results concerning the robustness to noises and uncertainties in measurements adopted for providing the feedback to the proposed control strategy. Furthermore, the proposed strategy can be compared with a nonlinear model predictive control? What is the gain in terms of computational effort of the Tube RMPC compared with the other ones?]

Response:
Thank you for your insightful comments. In response to your suggestion that the result section lacks sufficient comparative analysis, robustness evaluation against measurement noise and uncertainties, and a comparison with nonlinear model predictive control (NMPC), we provide the following clarifications:

First, we sincerely acknowledge that the initial version of the manuscript had an oversight in the presentation of simulation results, where some performance data—particularly related to yaw angle errors—were either incomplete or missing. This may have affected the clarity and completeness of the result analysis. In the revised version, we have conducted a thorough re-examination of the simulation data and applied strict consistency checks. As a result, all relevant metrics, including yaw angle RMSE and peak errors under varying road adhesion conditions, have been corrected and updated. The revised Tables 1 and 2 (Pages 11/12, Lines 357/358) now comprehensively reflect the comparative performance of Tube-RMPC, MPC, and PID controllers. At the same time, we have conducted detailed quantitative analysis of these results to illustrate the advantages of the proposed controller in terms of tracking accuracy and yaw stability.
Moreover, the robustness of the proposed Tube-RMPC controller has been evaluated through extensive co-simulation tests under varying road adhesion conditions (μ = 0.4, 0.6, 0.8), which introduce uncertainties in tire-road interaction modeling. These variations serve as realistic representations of parameter uncertainty and external disturbances.

Second, regarding the absence of NMPC as a comparison baseline, we emphasize that the primary focus of this work is to design a computationally efficient and real-time applicable controller suitable for autonomous vehicle embedded systems. Although NMPC provides higher modeling fidelity, its computational complexity and real-time infeasibility make it less practical for onboard implementation. Therefore, we adopt a model linearization strategy and construct a Tube-RMPC framework, which enables the incorporation of uncertainty bounds through constraint tightening and disturbance invariant sets. This approach significantly reduces the online optimization burden while maintaining robust control performance. The linearized formulation facilitates real-time computation and provides a favorable trade-off between control precision and computational efficiency. Modifications in the manuscript include (Page 5 , Lines 174-176):

“Although nonlinear MPC can more accurately represent vehicle dynamics, its high online computational burden limits its applicability in high-frequency trajectory tracking[33].”

These modifications reinforce the practical value and engineering applicability of our control strategy, especially in the context of embedded autonomous vehicle systems.

We sincerely thank the reviewer once again for the constructive feedback, which has greatly improved the clarity and completeness of our work. We remain open to further comments and are grateful for the opportunity to revise our manuscript.

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The current version of the manuscript has been improved, mostly in accordance with my suggestions from the first review.

Nevertheless, I still have two main substantive comments:

1. The authors describe the control objectives in lines 130–135; however, they do so without a rigorous mathematical formulation. I believe this issue could still be addressed later in the manuscript, around line 223. Specifically, there is a lack of a formal definition of the control problem, which could be stated, for example, as: “Find a control law such that…” and supplemented with the required properties expressed using mathematical formulas.

2. In Section 4.2, no explanation is provided regarding the reference trajectories considered. More precisely, it would be important to clarify whether the reference trajectories are feasible - that is, whether they are actual solutions of the system dynamics (equation (4)) - or if they have been selected arbitrarily. If the trajectories are indeed feasible, one might expect that asymptotic tracking is possible, and any tracking error arises due to structural or parametric uncertainties. Were these trajectories chosen as typical for the system under investigation?

The manuscript also contains numerous typesetting and formatting issues that very negatively impact its overall quality:

- Lines 240–243: The order in which the components of the control law (equation (14)) are discussed is reversed. The first term is the feedback component, while the second term, u′, is the nominal control.
- The style of presenting mathematical formulas is incorrect. For instance, punctuation marks are missing at the end of equations. In lines 266–272, each sentence begins with an indentation, which breaks the text’s consistency. The same issue appears in lines 232–236.
- The font used for mathematical formulas is entirely different from that used in the main text. I believe this should be harmonised to improve readability.
- In several places, references to the literature are formatted incorrectly - for example, there are no spaces before the citation brackets (e.g., “word[number]” instead of “word [number]”).
- Lines 244–245: Part of the sentence is split across two lines without a clear reason.
- Lines 304–305: It is unclear why this sentence is split across lines. The word “Where” should not be capitalised, as it is part of the same sentence. Furthermore, it is not clear whether these lines relate to Optimisation Problem 1 or to the paragraph below. This should be clarified.

Author Response

Dear Editors and Reviewers,

Thank you for your valuable feedback and constructive suggestions on our manuscript. We have carefully addressed all the comments and revised the manuscript accordingly. Below, we provide a point-by-point response to each reviewer’s concerns.

Changes in the revised manuscript are highlighted in red text.

 

Response to Reviewer #3
[The current version of the manuscript has been improved, mostly in accordance with my suggestions from the first review.Nevertheless, I still have two main substantive comments:]

Response:

We sincerely appreciate the reviewer’s recognition of the improvements made in the revised manuscript in response to the initial round of comments. We are grateful for your continued constructive feedback. Regarding the two remaining substantive comments, we have carefully addressed each point in the revised manuscript and provide detailed responses below.

 

Comment 1: [The authors describe the control objectives in lines 130–135; however, they do so without a rigorous mathematical formulation. I believe this issue could still be addressed later in the manuscript, around line 223. Specifically, there is a lack of a formal definition of the control problem, which could be stated, for example, as: “Find a control law such that…” and supplemented with the required properties expressed using mathematical formulas.]

Response:
We sincerely thank the reviewer for the constructive suggestion. Indeed, the objective of the proposed Tube-RMPC strategy was previously described in Lines 221–226; however, we acknowledge that its position after the presentation of the control law (Equation 14, Line 237) may have affected the clarity and logical flow of the manuscript.

In response, we have restructured the section by moving the explanation of the control objective to immediately precede the control law(Lines 235-240). This adjustment improves the coherence of the section and ensures that readers are first introduced to the goal before seeing the control formulation. We believe this revision resolves the ambiguity and enhances the manuscript’s readability.

Thank you again for your helpful comment..

 

 

Comment 2: [In Section 4.2, no explanation is provided regarding the reference trajectories considered. More precisely, it would be important to clarify whether the reference trajectories are feasible - that is, whether they are actual solutions of the system dynamics (equation (4)) - or if they have been selected arbitrarily. If the trajectories are indeed feasible, one might expect that asymptotic tracking is possible, and any tracking error arises due to structural or parametric uncertainties. Were these trajectories chosen as typical for the system under investigation?]

 

Response:
We sincerely thank the reviewer for this insightful comment.

In the original version of the manuscript, while we did indicate in Section 4.1 that representative driving scenarios—specifically, a double-lane change and an S-curve—were selected for simulation, we acknowledge that an explicit explanation of the nature and feasibility of the reference trajectories was not clearly provided. We agree that such clarification is necessary for a complete understanding of the control evaluation.

In the revised version, We have added detailed explanations in Sections 4.1 and 4.2, respectively, where we believe such clarifications are most appropriately placed, as it introduces the simulation setup. Specifically, we clarified the following:

The reference trajectories are predefined offline and are not exact solutions of the full nonlinear system dynamics (Equation (4));However, under nominal conditions, they are approximately feasible, and were intentionally designed to represent typical autonomous driving scenarios, such as double-lane changes, S-curves, and straight-line tracking;These reference paths serve as practical benchmarks for evaluating the proposed Tube-RMPC controller's robustness under structural modeling errors and external disturbances;Due to model simplifications and the presence of disturbances, asymptotic tracking cannot be guaranteed, which is consistent with the bounded tracking errors observed and analyzed in the simulation results. The original text is revised as follows(Lines 343-346):

“Moreover, the predefined reference trajectories, though not exact solutions of the nonlinear dynamics, are approximately feasible and serve as practical benchmarks for robustness evaluation under modeling errors and disturbances, and provide detailed simulation and theoretical support for autonomous system design.”

(Lines 390-393):

“However, asymptotic tracking cannot be strictly guaranteed due to model simplifica-tions and external disturbances, which aligns with the bounded tracking errors ob-served and discussed in the simulation analysis.”

 

We thank the reviewer again for pointing out this omission. The revised explanation in Section 4.2 improves both the clarity and rigor of our simulation study.

 

Comment 3: [The manuscript also contains numerous typesetting and formatting issues that very negatively impact its overall quality:

- Lines 240–243: The order in which the components of the control law (equation (14)) are discussed is reversed. The first term is the feedback component, while the second term, u′, is the nominal control.
- The style of presenting mathematical formulas is incorrect. For instance, punctuation marks are missing at the end of equations. In lines 266–272, each sentence begins with an indentation, which breaks the text’s consistency. The same issue appears in lines 232–236.
- The font used for mathematical formulas is entirely different from that used in the main text. I believe this should be harmonised to improve readability.
- In several places, references to the literature are formatted incorrectly - for example, there are no spaces before the citation brackets (e.g., “word[number]” instead of “word [number]”).
- Lines 244–245: Part of the sentence is split across two lines without a clear reason.
- Lines 304–305: It is unclear why this sentence is split across lines. The word “Where” should not be capitalised, as it is part of the same sentence. Furthermore, it is not clear whether these lines relate to Optimisation Problem 1 or to the paragraph below. This should be clarified.]

Response:
We sincerely thank the reviewer for the detailed feedback regarding the formatting and typesetting issues. We have carefully addressed each of the specific problems raised, as detailed below:

Lines 240–243: Reversed explanation of control law components
Thank you for pointing this out. We have revised the description to correctly present the components of the control law in Equation (14). The feedback term is now introduced first, followed by the nominal control input, consistent with the mathematical expression. The following are the modified parts(Lines 244-248):

“The first component serves as an auxiliary feedback controller that ensures the actual system's state(9)closely follows the nominal state.The second component   is a con-trol strategy tailored for the nominal system(10), which guides the nominal state to-ward the equilibrium point.”

Punctuation missing at the end of equations; inconsistent indentation (Lines 266–272, 232–236)
We have carefully reviewed all displayed equations throughout the manuscript. All equations now include appropriate punctuation marks at the end (such as commas or periods).

Additionally, the indentation at the beginning of paragraphs following these equations has been standardized. We removed unnecessary indentations in the specified lines (now in 235-240 and 267-276), thereby ensuring visual consistency and alignment with academic writing standards.

Inconsistent font for mathematical expressions
We appreciate this observation. In the revised manuscript, we have harmonised the font and size of all mathematical expressions to match the main text, using the recommended typesetting style for equations. This enhances the overall readability and presentation quality of the manuscript.

Incorrect citation formatting (e.g., “word[number]” instead of “word [number]”)
We have performed a comprehensive proofreading of all references and citation formats. All citation usages now follow the correct format: a space is included before the bracket, i.e., “word [number]”.

Line breaks in Lines 244–245 and 304–305
We have addressed the unjustified sentence breaks in both locations:

In Lines 244–245, the split sentence has been merged into a single coherent line to maintain sentence flow.

In Lines 304–305, we revised the structure to avoid an unnatural line break. The word “Where”(line 308) is now correctly written in lowercase and treated as a continuation of the preceding sentence.

 

Once again, we thank the reviewer for these precise comments. All formatting and typesetting issues have been carefully resolved in the revised version to enhance the overall quality and readability of the manuscript.

Reviewer 4 Report

Comments and Suggestions for Authors

The paper has been updated in accordance with the suggestions. Therefore, it can be accepted in the present form.

Author Response

[The paper has been updated in accordance with the suggestions. Therefore, it can be accepted in the present form.]

Response:

We sincerely thank you for your positive evaluation and for acknowledging the revisions we have made. We greatly appreciate your constructive feedback during the review process, which helped us significantly improve the quality and clarity of our manuscript.

We are delighted to learn that the revised version meets your expectations and is acceptable in its current form. Your support and guidance throughout this process are deeply appreciated.

Thank you once again for your time and valuable suggestions.

Sincerely,
Yang chen