Research on the Stability and Trajectory Tracking Control of a Compound Steering Platform Based on Hierarchical Theory
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper investigates the unique composite steering system to enhance the steering efficiency and dynamic response of distributed-drive unmanned platforms under low driving torque conditions. Specifically, a composite steering dynamics model is established, and a hierarchical stability control strategy along with a model predictive control-based trajectory tracking algorithm is innovatively proposed. The topic and the results could be interesting for readers. However, the reviewer has the following suggestions for the authors to improve the quality of the manuscript.
- The key contributions are missing. The authors should highlight the key contributions.
- "Where:" is wrong. Please check and revise.
- More descriptions about the fuzzy controller can be given in the introduction.
- The authors should be careful to check the whole paper as there are some typos and grammatical errors.
Author Response
Please check the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper studies a composite steering dynamics systems model and a hierarchical stability control strategy along with a MPC based trajectory tracking algorithm. Examples presented show that the method is effective. The following need to be revised:
1. The English needs to be improved in many places such as "he" in line 61,"Where" in line 146, 193,226, and so on.
2. The application of punctuation marks is too casual and not standardized such as ";" in line 233,258,273,315,..many places,please check the whole paper.
3. It is recommended to provide the main contributions of this article in the introduction section.
4. It is recommended to provide symbol explanations for this article in the introduction section.
5. Please give an detailed explanation for equation (41)(42), especially the explanation of subscripts.
6. It is better to propose a Theorem to show the stability of the system, it is formal in this type of paper.
Moreover, I suggest that the structure of the article needs to be organized and refined. In order to clearly demonstrate the contribution of this article. And the main contribution of this article should also be clearly stated in the text.
Comments on the Quality of English Language
The English Can be improved.
Author Response
Comment 1: The English needs to be improved in many places such as "he" in line 61,"Where" in line 146, 193,226, and so on.
Response: We are grateful to the reviewer for the careful examination of our manuscript and for pointing out the improper usage of “Where:” in several instances. Your observation is entirely correct and very helpful. Upon thorough inspection, we found that most occurrences of “where” in our manuscript are used to explain variables following equations or tables. According to academic writing conventions, such explanations should be grammatically connected to the preceding formula or sentence, meaning they must be part of the same sentence. Therefore, “where” should be written in lowercase, should not begin a new line with indentation, and must not be followed by a colon.
In response to your comment, we have carefully revised all such instances to comply with proper formatting. Specifically, the following representative modifications have been made (highlighted in red in the revised manuscript):
- The dynamic equilibrium equation of yaw motion can be expressed as a composite of the lateral forces generated by the front and rear wheels and the differential moment induced by skid steering, as shown below:
where denotes the yaw moment generated by skid steering, as mentioned above.
- Therefore, the system is first approximately linearized using the Taylor series ex-pansion. After linearization, the following expression is obtained:
where , , , and are the discrete-time system matrices, …
- The above optimal control problem can be converted into a quadratic programming (QP) problem for solution. The general form of the quadratic programming problem is as follows:
where .
All similar issues throughout the manuscript have been corrected following this pattern. The revised content has been marked in red font for your convenience.
Comment 2: The application of punctuation marks is too casual and not standardized such as ";" in line 233,258,273,315,..many places, please check the whole paper.
Response: We sincerely thank the reviewer for pointing out the issue regarding improper and inconsistent use of punctuation marks in the manuscript. We fully accept this suggestion and have carefully reviewed the entire text to ensure that all punctuation, including semicolons and commas, is applied in a standardized and grammatically correct manner.
In particular, we have corrected the inappropriate use of semicolons at lines 233, 258, 273, 315, and other relevant locations. A thorough proofreading of the manuscript has also been conducted to improve overall linguistic accuracy and formatting consistency.
- To ensure the stability of the compound steering platform, the output equation of the system is established. Therefore, the system state error is chosen to be zero as the control objective, leading to the following output equation:
Since the above state-space equations are in continuous time, and for the convenience of subsequent calculations, the forward Euler method is used to discretize them. For a time step , the discretized state equation is:
- All modifications have been marked in red in the revised manuscript.
- where is the future reference trajectory, is the output error weighting matrix and is the control increment weighting matrix:
By substituting the predictive model into the above equation for , the objective function is obtained as:
Since excessively small control increments can result in sluggish system responses, while overly large control increments may cause instability in control, it is necessary to impose constraints on the control increment to ensure control accuracy and driving safety:
Comment 3: It is recommended to provide the main contributions of this article in the introduction section.
Response: We would like to express our sincere gratitude to the reviewer for your careful reading and constructive comments. We fully agree with your suggestion that the key contributions of this work should be clearly highlighted to enhance the clarity and accessibility of the manuscript.
In response to your suggestion, we have revised the end of the Introduction section to explicitly emphasize the key contributions of our work and clearly distinguish them from prior studies. In addition, we have added a technology roadmap to enhance the organization of the main contributions.
The modified content has been marked with red font in the text. The specific modifications are as follows:
In summary, existing studies have laid a solid foundation for compound steering systems in both trajectory tracking and lateral stability control. Prior work has demon-strated effective control schemes for skid-steering platforms and highlighted the potential of fuzzy logic in handling nonlinearities and actuator coordination. However, most of these methods treat trajectory tracking and stability control independently, lack inte-grated hierarchical control frameworks, and often rely on oversimplified torque alloca-tion strategies that are insufficient under complex, coupled dynamic conditions.
To address these limitations, this paper proposes a hierarchical control framework for compound steering intelligent driving platforms, enabling coordinated trajectory tracking and lateral stability control across multiple actuators. Compared to existing re-search, the main contributions of this paper are as follows:
(1) A hierarchical control framework is proposed, integrating model predictive tra-jectory tracking and fuzzy-based stability control into a unified structure. This framework enables coordinated control between front-wheel steering and differential torque actua-tion.
(2) A model predictive controller (MPC) is developed for trajectory tracking under curvature constraints, incorporating vehicle dynamics and future reference paths to im-prove path-following accuracy.
(3) A fuzzy logic controller is designed for real-time yaw moment allocation, which adaptively distributes the corrective moment between steering angle and drive torque based on vehicle speed and curvature. This enhances system robustness under nonlinear, uncertain, and highly coupled conditions.
(4) Comprehensive simulation results are provided to verify the effectiveness of the proposed method, demonstrating superior performance in both trajectory tracking and lateral stability compared to conventional decoupled control approaches.
The remainder of this paper is organized as follows: Section II presents the modeling of the compound steering vehicle. Section III details the design of the model predictive trajectory controller and fuzzy-based stability controller. Section IV provides simulation results and performance analysis. Section V concludes the paper.
Comment 4: It is recommended to provide symbol explanations for this article in the introduction section.
Response: We would like to express our sincere gratitude to the reviewer for your careful reading and constructive comments. We fully agree with your suggestion that the key contributions of this work should be clearly highlighted to enhance the clarity and accessibility of the manuscript.
In response to your suggestion, we have revised the end of the Introduction section to explicitly emphasize the key contributions of our work and clearly distinguish them from prior studies. In addition, we have added a technology roadmap to enhance the organization of the main contributions.
Furthermore, as you recommended, we have provided a table of symbol explanations in the Introduction section to help readers better understand the notations used throughout the manuscript. This symbol table summarizes key variables and system parameters related to the vehicle dynamics and control models.
The modified content has been marked in red font in the text. The specific modifications are as follows:
To facilitate understanding of the mathematical modeling and control formulation used in this paper, the key symbols and notations are summarized below, as shown in These include vehicle dynamic variables, control inputs, and system parameters frequently referenced throughout the manuscript.
Table 1. .Symbol Descriptions
|
Symbol |
Descriptions |
|
|
Longitudinal velocity |
|
Lateral velocity |
|
|
Yaw rate (angular velocity around vertical axis) |
|
|
Front wheel steering angle |
|
|
, |
Front and rear cornering stiffness |
|
, |
Drive torques on left and right wheels |
|
Yaw moment of inertia |
|
|
Vehicle mass |
|
|
, |
Distance from center of mass to front/rear axle |
|
B |
Track width |
|
ψ |
Vehicle heading angle |
|
Vehicle sideslip angle |
|
|
Longitudinal and lateral tire forces |
|
|
Control increment at time step |
|
|
, |
Minimum and maximum allowable control increments |
|
κ |
Trajectory curvature |
|
Sampling period (e.g., 0.02 s) |
|
|
, |
rediction horizon, control horizon in MPC |
|
, |
State and control weighting matrices |
Comment 5: Please give an detailed explanation for equation (41)(42), especially the explanation of subscripts.
Response: We thank the reviewer for highlighting the need to clarify Equations (41) and (42). In the revised version of the manuscript, these equations have been renumbered as Equations (51) and (52) due to formatting adjustments. In response to your suggestion, we have added a clear explanation of the subscripts used in these equations. Specifically,
the subscript k denotes the current time step, while represents the previous step. indicates the control increment at time step kkk, and is the control input at that step. The bounds [, ] [, ] define the allowable range of control inputs and their rate of change, respectively.
This explanation has been inserted directly following Equations (51) and (52) in the revised manuscript, and the changes have been marked in red.
Since excessively small control increments can result in sluggish system responses, while overly large control increments may cause instability in control, it is necessary to impose constraints on the control increment to ensure control accuracy and driving safety:
The control quantity constraint conditions after obtaining the cumulative control increment are:
In Equations (51) and (52), the subscript k indicates the current time step, and denotes the previous one. is the control increment between two steps, and is the updated control input. These constraints ensure smooth control actions and keep the input within allowable bounds.
Comment 6: It is better to propose a Theorem to show the stability of the system, it is formal in this type of paper.
Response: We sincerely thank the reviewer for this professional and constructive suggestion. We fully understand the importance of providing formal theoretical analysis, such as a stability theorem, in control-oriented research.
However, in this work, we adopt a hierarchical control architecture tailored for intelligent driving platforms with compound steering systems. The upper-level controller is designed using Model Predictive Control (MPC) to achieve curvature-constrained trajectory tracking, while the lower-level controller employs a Fuzzy Logic Controller (FLC) to allocate the corrective yaw moment between front-wheel steering and differential drive torque. Due to the nonlinear, data-driven, and rule-based nature of the fuzzy controller, it is challenging to derive a unified Lyapunov-based stability proof for the entire closed-loop system.
Instead of formal proof, we demonstrate the stability and effectiveness of our control strategy through extensive simulation-based evaluation, which forms a core part of our contribution. As described in Section IV of the manuscript:
We tested the system under various reference trajectories, including high-curvature path tracking and S-curve lane changes, to verify tracking precision.
Lateral error, yaw rate, and sideslip angle were closely monitored and shown to remain bounded and convergent throughout all scenarios.
We further compared the fuzzy torque allocation strategy against a conventional rule-based allocation method, and results showed that the proposed approach effectively reduced lateral deviation and improved dynamic stability, especially at higher vehicle speeds and under transient steering inputs.
These results confirm that, although the control system lacks a formal stability proof, it exhibits consistent and reliable behavior across complex dynamic conditions relevant to real-world applications.
We truly appreciate the reviewer’s insightful comment and fully agree that theoretical generalization would be valuable in future work. We plan to explore modeling approaches such as hybrid systems or Takagi–Sugeno fuzzy approximations to enable more formal stability analysis in subsequent studies.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThis manuscript establishes a composite steering dynamics model, and a hierarchical stability control strategy along with a model predictive control-based trajectory tracking algorithm is innovatively proposed to enhance the steering efficiency and dynamic response of distributed-drive unmanned platforms under low driving torque conditions. First, a composite steering platform dynamics model is established by combining Ackermann steering and slip yaw moment methods. Then, a trajectory tracking controller is designed using a model predictive control algorithm. Finally, the additional yaw moment is calculated based on lateral velocity error and yaw rate error, with stability control allocation performed using a fuzzy control algorithm. Their experimental results show that the composite steering technology enables unmanned platforms to achieve trajectory tracking tasks with lower torque, faster speed, and higher efficiency.
Minor issues
Line 99, check the Figure 1 citation because it is mentioned twice.
Line 101, it would be better to leave a space after the figure caption.
We know that some variables are known. However, defining their meaning is recommended.
In the equation (12), what is the meaning of 21.15?
Figure 1 is centered, but Figures 2 and 3 are not. Check it.
Line 198 a comma needs to be deleted.
Line 291, the equation number is not in the correct position, and the symbol # is correct?
Line 294 an equation number is missing
Line 361, a space after Figure 5 is missing.
Line 395, several spaces are missing.
Line 453, after Figure 9, a space is missing
The Figure caption in Figure 15 seems a different format, please check it.
I think line 502 is the figure caption of figure15.
In line 503, why is FIGURE 15 in capital letters?
Line 561, I think the Section Patents is not correct, check it.
I think the use of HIL in your works is important, I recommend add in the keywords.
The colon after the subtopic is unnecessary. For example:
4.1 Hairpin bending condition： the suggested: 4.1 Hairpin bending condition
Figure 18 looks different; it means the image quality is less than the others. Check it please
Major issues
What is your sampling rate? And why did you decide on that sampling rate?
You describe the software CarSim; however, could you provide a validation of the use of the software in your work? Maybe a reference from the literature or any other way about how you can trust the results obtained from the software.
In line 457 “Ackermann steering, though all errors remain acceptable” How much is acceptable and why?
Could you explain the great difference in Figure 14?
Could you explain the great difference in Figure 18?
Could you explain the great difference in Figure 20?
Comments for author File: Comments.pdf
The quality of English is not so bad. However, grammatically has to make several improvements.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsPlease see the attachments.
Comments for author File: Comments.pdf
Please see the attachments.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsI have no further comments.
Author Response
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Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsIt is much improved in this version. I recommend this to be published afer some minor revsion for the punctuation after the equations in the text..
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsThe revised manuscript can be accepted directly.
Author Response
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Author Response File: Author Response.pdf
