A Tube Linear Model Predictive Control Approach for Autonomous Vehicles Subjected to Disturbances

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

What this paper sets out to achieve is worthy and important. I have some suggestions that would make this paper more robust

 

1.      Some Sentences are overloaded with information, please rephrase them  to improve clarity, Example

a.      Introduction: Line 55- 56 “To decrease the computational burden, the paper proposes a tube linear MPC framework of autonomous vehicles for obstacle avoidance conditions under various disturbances". Sentence is overloaded with information, please rephrase it to improve clarity,

b.      Introduction: Line 57 -58  “In the proposed framework, the vehicle dynamic models are linearization and dimension reduction to a bicycle model.” rephrase this sentence to make more sense.

c.      Section 6: Line  239-241 “ In this section, the tube MPC system is applied to the autonomous system to avoid one obstacle on the road under state disturbance, uncertain road friction coefficients, and wind disturbance.” rephrase this sentence to make more sense.

2.      Section 2.1 - line 72-80 details of the architecture are discussed step by step while refereeing to the Figure 2. To enhance the readability, it would be beneficial to number the corresponding steps in the figure, providing readers with clear and easy reference points.

Figure 2 is very similar to Figure 1 in the following reference, could be supported with suitable references here

“Yu, J., Guo, X., Pei, X., Chen, Z., Zhou, W., Zhu, M., & Wu, C. (2020). Path tracking control based on tube MPC and time delay motion prediction. IET Intelligent Transport Systems, 14(1), 1-12.”

3.      References to the definitions on page 3, Could have been added.

4.      Experimental set up were discussed but could be expanded.

5.      The Limitations of the investigation- could be discussed.

 

Comments on the Quality of English Language

 Minor editing of English language required

Author Response

Point-by-Point Responses

We truly appreciate the comments raised by reviewers and Editor in Chief and have carefully taken their suggestion to revise our paper. For convenience of further review, all the revision has been highlighted in red in the text. A point-by-point response is provided as follows.

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Comments from the editors and reviewers:

Reviewer #1:

1. Some Sentences are overloaded with information, please rephrase themto improve clarity, Example

a.Introduction: Line 55- 56 “To decrease the computational burden, the paper proposes a tube linear MPC framework of autonomous vehicles for obstacle avoidance conditions under various disturbances". Sentence is overloaded with information, please rephrase it to improve clarity,

b.Introduction: Line 57 -58“In the proposed framework, the vehicle dynamicmodels are linearizationand dimension reduction to a bicycle model.” rephrase this sentence to make more sense.

c.Section 6: Line239-241 “ In this section, the tube MPC system is applied to the autonomous system to avoid one obstacle on the road under state disturbance, uncertain road friction coefficients, and wind disturbance.” rephrase this sentence to make more sense.

Thanks for your suggestions. We have rewritten the above sentences in the revised manuscript with the highlighted part.

a. To improve autonomous vehicles' tracking performance and computational efficiency under disturbances, the paper proposes a tube linear model predictive controller (MPC) framework.

b. The proposed framework simplifies the complex vehicle dynamic models to a bicycle model with 3 degrees of freedom.

c. The proposed tube MPC system is applied to obstacle avoidance under common disturbances such as state disturbance, uncertain road friction coefficients, and wind disturbance.

2. Section 2.1 - line 72-80 details of the architecture are discussed step by step while refereeing to the Figure 2. To enhance the readability, it would be beneficial to number the corresponding steps in the figure, providing readers with clear and easy reference points. Figure 2 is very similar to Figure 1 in the following reference, could be supported with suitable references here. “Yu, J., Guo, X., Pei, X., Chen, Z., Zhou, W., Zhu, M., & Wu, C. (2020). Path tracking control based on tube MPC and time delay motion prediction. IET Intelligent Transport Systems, 14(1), 1-12.”

Thanks for your suggestions. We have numbered the corresponding steps in Figure 2 and cited the suitable reference in the revised manuscript.

3. References to the definitions on page 3, Could have been added.

Thanks for your suggestions. We have cited the references about the definitions on page 3.

4. Experimental set up were discussed but could be expanded.

Thanks for your suggestions. The paper mainly compares the simulation result between the proposed tube liner MPC approach and the traditional MPC approach. Due to the limitation of experiment resources, the paper did not experiment. we will design and conduct a whole vehicle experiment to validate the proposed approach in the future.

5. The Limitations of the investigation- could be discussed.

Thanks for your suggestions. We have discussed the limitations of the investigation on page 8 highlighted part. When the simplified linear bicycle model cannot precisely describe the vehicle motion, the proposed linear MPC system is not available. For example, the bicycle model cannot describe the vehicle’s motion on the icy or slippery road precisely. The proposed linear MPC system is not available for the above two conditions.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors, thank you for your research. Please specify:

- "A Tube Linear MPC Approach ..." why was this Approach chosen for research? Evidence must be provided in the manuscript as confirmation.

- "...Improve the Track Performances..." what Performances are we talking about? It is necessary to give numerical values and perform a comparison.

- "the paper proposes a tube linear model predictive controller (MPC) framework for obstacle avoidance conditions under various disturbances":

1) Where is this structure in the manuscript?

2) "various disturbances" what do you mean?

- There is no "Methodology" section in the manuscript. Why?

- What Fig. 5 and 10? What useful information can readers see?

Kind regards,

Author Response

Point-by-Point Responses

We truly appreciate the comments raised by reviewers and Editor in Chief and have carefully taken their suggestion to revise our paper. For convenience of further review, all the revision has been highlighted in red in the text. A point-by-point response is provided as follows.

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Comments from the editors and reviewers:

Reviewer #2:

1."A Tube Linear MPC Approach ..." why was this Approach chosen for research? Evidence must be provided in the manuscript as confirmation.

Thanks for your questions. Compared with the traditional tube non-linear MPC approach. The proposed Tube Linear MPC Approach can significantly reduce the computational burden and is more suitable for dealing with the disturbances in obstacle avoidance conditions in engineering applications. The traditional tube non-linear MPC is solved with MPT and Tube Linear MPC is transferred to a linear programming (LP) problem instead of MPT to solve. Section 6.1 compared the computational burden of LP and MPT. The results show that the computational burden of  LP is much less than that of the MPT. The methodology of tube linear MPC is described in section 2.

2. "...Improve the Track Performances..." what Performances are we talking about? It is necessary to give numerical values and perform a comparison.

Thanks for your questions. It is a good question. Only the track performance is discussed in the paper. The paper compares the track performance between the traditional MPC and tube linear MPC. The track performance in the paper means the possibility of trajectory to avoid the obstacle successfully. If the trajectory fails to avoid the obstacle, the track performance of the trajectory is 0 scores. If the trajectory avoids the obstacle successfully, the track performance of the trajectory is 100 scores. Therefore, the traditional MPC and tube linear MPC are compared for the number of trajectories avoiding the obstacle successfully.

3. "the paper proposes a tube linear model predictive controller (MPC) framework for obstacle avoidance conditions under various disturbances":

1）Where is this structure in the manuscript?

Thanks for your questions. The structure of the proposed tube linear model predictive controller is in section 2.1 on page 2~3. the detailed solving process of the tube MPC optimization problem is on page 8 lines 241-252.

2） "various disturbances" what do you mean?

Thanks for your questions. "various disturbances" means common disturbances such as state disturbance, uncertain road friction coefficients, and wind disturbance. The reader may misunderstand the various disturbances. I have replaced the "various disturbances" with common disturbances.

4. There is no "Methodology" section in the manuscript. Why?

Thanks for your questions. The methodology of Tube Linear MPC has been described in section 2.

5. What Fig. 5 and 10? What useful information can readers see?

Thanks for your questions. Fig. 5 and 10 show the random state disturbances with a uniform distribution over the bound [0.05; 0.2; 0.005; 0.02].

 

Reviewer 3 Report

Comments and Suggestions for Authors

1) The introduction section is poor and needs to be improved. The problem should be motivated in a good manner. Also, consider providing a list of contributions and give sufficient evidences to support your claim. 

 

2) What is the computing time of the proposed MPC? Is it appropriate for real-time applications?

 

3) Following my previous email, one of the main challenges with implementing MPC schemes is their high computational cost. This challenge can be addressed by using methods described in [R1] and [R2]. The authors need to address this comment as a remark; consider using the suggested articles to conduct the explanations in the remark. 

[R1]. "Robust to Early Termination Model Predictive Control", 2023. [http://doi.org/10.1109/TAC.2023.3308817]

[R2]. "Sparsity-Exploiting Anytime Algorithms for Model Predictive Control: A Relaxed Barrier Approach", 2020. [http://doi.org/10.1109/TCST.2018.2880142]

 

4) Provide the source of equations if they are from specific source; otherwise, the authors need to provide more details on how to derive the equations. For instance, equations (18) and (19). 

 

5) Is the optimization problem (35) convex? Provide its specifics, like number of decision variables, dimension, etc. 

 

6) This paper lacks a proper comparison study. Consider comparing your method with a state-of-the-art method; this would help the readers to understand pros and cons of the proposed method. 

Comments on the Quality of English Language

Acceptable. 

Author Response

Point-by-Point Responses

We truly appreciate the comments raised by reviewers and Editor in Chief and have carefully taken their suggestion to revise our paper. For convenience of further review, all the revision has been highlighted in red in the text. A point-by-point response is provided as follows.

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Comments from the editors and reviewers:

Reviewer #3:

1) The introduction section is poor and needs to be improved. The problem should be motivated in a good manner. Also, consider providing a list of contributions and give sufficient evidences to support your claim.

Thanks for your questions. We have rewritten the introduction and given sufficient evidence to support my claim in the highlighted part on pages 1 and 2.

2) What is the computing time of the proposed MPC? Is it appropriate for real-time applications?

Thanks for your questions. It is a good question. The computational burden of the proposed tube linear MPC is 0.03s with my laptop. The proposed tube linear MPC is appropriate for real-time applications. The reason can be explained as follows.1) The computing power in engineering is stronger than my laptop (Intel i7 9750 with 8 cores and 16G RAM). 2) Recently, the MPC has been employed for real-time applications. For example, Hosseinzadeh et. al proposes a robust to early termination MPC, and simulations are carried out on a F-16 aircraft to demonstrate the effectiveness of the proposed scheme. Feller et. al presented and analyzed a novel class of stabilizing and numerically efficient model predictive control (MPC) algorithms for discrete-time linear systems and demonstrated the effectiveness of the proposed approach. Therefore, it is appropriate for real-time applications. We have added the above comments in the highlighted part on page 9.

3) Following my previous email, one of the main challenges with implementing MPC schemes is their high computational cost. This challenge can be addressed by using methods described in [R1] and [R2]. The authors need to address this comment as a remark; consider using the suggested articles to conduct the explanations in the remark.

[R1]. "Robust to Early Termination Model Predictive Control", 2023. http://doi.org/10.1109/TAC.2023.3308817].

[R2]. "Sparsity-Exploiting Anytime Algorithms for Model Predictive Control: A Relaxed Barrier Approach", 2020. [http://doi.org/10.1109/TCST.2018.2880142].

Thanks for your suggestions. We have addresses the comment as a remark in the revised manuscript on page 9 highlighted part.

4) Provide the source of equations if they are from specific source; otherwise, the authors need to provide more details on how to derive the equations. For instance, equations (18) and (19).

Thanks for your suggestions. I have cited the reference to provide the source of equations in the revised paper on page.

5) Is the optimization problem (35) convex? Provide its specifics, like number of decision variables, dimension, etc.

Thanks for your questions. The optimization problem is convex. We have provided the specifics of the optimization problem in the revised manuscript on page 8 highlighted part. From the above description, it can be noted that the optimization problem's decision variable is the front tires' steering input sequence. Though the steering input sequence is the 4*1 column vector, it has only one effective dimension.

6）This paper lacks a proper comparison study. Consider comparing your method with a state-of-the-art method; this would help the readers to understand pros and cons of the proposed method.

Thanks for your suggestions. Compared with other controllers such as LQR (linear Quadratic Regulator), fuzzy logic control, and sliding mode control, traditional model predictive control (MPC) has better real-time performance, stability, and robustness[13] and it is widely used in autonomous vehicles. Therefore, the paper compared the tracking performances of the proposed tube linear MPC with that of traditional MPC instead of other methods. Compared with traditional MPC, the proposed method can improve effectively the path tracking performance of autonomous vehicles in obstacle avoidance conditions under common disturbances. Though the computational burden is a little bigger than that of traditional MPC, it can meet the requirements of real-time applications. We have added the above comments in the revised manuscript on page 8 highlighted part.

 

Reviewer 4 Report

Comments and Suggestions for Authors

This study proposed a framework with good results to improve the track performance of autonomous vehicles under obstacle avoidance conditions in disturbance situations.

I think this is an interesting research topic for obstacle avoidance in autonomous vehicles.

In order to differentiate themselves from existing research, the authors proposed a tube linear model predictive controller (MPC) framework for obstacle avoidance conditions under various disturbances.

 

There are several improvements to the paper:

1. Rather than grouping the references in the introduction, we recommend that you separate them and explain their characteristics. It may feel like the existing research background is somewhat lacking. It would be nice if you provided a more detailed explanation of the references.

 

2. There are some missing explanations of variables in the formulas in Chapter 4. I would like to clarify the definitions of the variables mentioned in this paper.

 

3. There are mistakes in the titles of 6.1 and 6.2. Please check.

 

4. In Figure 4, it will be helpful for readers to understand if you faithfully explain the basis and reason why you think 20 steps are sufficient.

Reducing the computational burden is an important content of this paper, and I hope that you will consider sufficient supplementary explanations for this part in the main text.

 

5. I can understand by referring to the text and reading the characteristics from Figure 5 to Figure 9. However, although it is important to describe related results in the text, readability appears to be improved if they are compared in separate tables.

Author Response

Point-by-Point Responses

We truly appreciate the comments raised by reviewers and Editor in Chief and have carefully taken their suggestion to revise our paper. For convenience of further review, all the revision has been highlighted in red in the text. A point-by-point response is provided as follows.

***************************************************************************

Comments from the editors and reviewers:

Reviewer #4:

1. Rather than grouping the references in the introduction, we recommend that you separate them and explain their characteristics. It may feel like the existing research background is somewhat lacking. It would be nice if you provided a more detailed explanation of the references.

Thanks for your questions.  We have separated the references in the introduction and provided a more detailed explanation of the references in the highlighted part on pages 1 and 2.

2. There are some missing explanations of variables in the formulas in Chapter 4. I would like to clarify the definitions of the variables mentioned in this paper.

Thanks for pointing out the issue. We have added explanations of variables in the formulas in Chapter 4 on page 7 and highlighted this part.

3. There are mistakes in the titles of 6.1 and 6.2. Please check.

Thanks for pointing out the issue. We have corrected the issue in the revised manuscript.

4. In Figure 4, it will be helpful for readers to understand if you faithfully explain the basis and reason why you think 20 steps are sufficient.

Thanks for your questions. It is a good question. The predicted horizon has a significant effect on the efficiency and accuracy of model predictive control. When the predicted horizon adopts 20 steps, it can trade off the accuracy and efficiency of model predictive control well [34]. Therefore, the predicted horizon is set to 20 steps in the paper.

[34] Chen J, Tian G, Fu Y. A novel multi-objective tuning strategy for model predictive control in trajectory tracking[J]. Journal of Mechanical Science and Technology, 2023, 37(12): 6657-6667.

Reducing the computational burden is an important content of this paper, and I hope that you will consider sufficient supplementary explanations for this part in the main text.

Thanks for your suggestion. We have explained the reason why the proposed tube MPC can reduce computational burden in the revised manuscript on page 9 highlighted part. The reason can be explained as follows. According to the algorithm of minimal robust positively invariant set algorithm 1 [32] and parameters in Table 1, the minimal robust positively invariant set will be obtained by recursively adding Minkowski sum 80 times (s in (17) is 80). However, the 29030 Minkowski sums are employed in the MPT in [31]. the numbers of Minkowski sums by the LP are much less than that by the MPT.

5. I can understand by referring to the text and reading the characteristics from Figure 5 to Figure 9. However, although it is important to describe related results in the text, readability appears to be improved if they are compared in separate tables.

Thanks for your suggestion. We have added the separate tables in the revised manuscript on pages 11 and 12 highlighted part.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Fig. 5 and 10 will be understandable to a wide range of readers?

Reviewer 4 Report

Comments and Suggestions for Authors

I confirmed that many parts of this paper have been improved through additional explanation.