Research on Vertical Parking Path Planning Based on Circular Arcs, Straight Lines, and Multi-Objective Evaluation Function

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper explores an ambitious research area in vertical parking path planning, emphasizing the smoothness and efficiency of various path planning methods. Its innovation lies in introducing novel mathematical approaches by optimizing vertical parking path planning through a combination of circular arc-line tangency and a multi-objective optimization function. Existing methods, including geometric and numerical optimization techniques, are thoroughly analysed, with their respective strengths and limitations clearly outlined.

The cited references are appropriate, well-suggested, and up-to-date within the last five years. However, some of them focus on alternative parallel parking (e.g., [6]), and some references appear to be missing in the text (e.g., [7]). The reviewer also notes the absence of citations for other relevant technical standards, such as ISO 16787 and ISO 23374-1.

Unfortunately, the simulation methods are described vaguely, including a lack of raw data. The authors use simplified assumptions that may confuse readers, and these should be highlighted in the results. For example, the steering angle is not strictly equal to the turning of the vehicle's wheels, and their relationship is not linear. Additionally, the reviewer notes the lack of discussion on other technologies currently used in vehicle systems that significantly improve the parking process, such as wheel mounting, variable camber of the wheels, steered rear axles, or rear-wheel steering. This omission makes it difficult to compare the proposed method with existing market solutions. Furthermore, it is unclear how topological parameters of the parking area, such as slope and curvature, are taken into account.

From a scientific and mathematical standpoint, this work contributes to advancing current knowledge. However, the manuscript could be made more engaging by incorporating insights from a broader review and market perspective, such as the potential adoption of this methodology in the vehicle systems or additional progress for its innovation.

Author Response

Comments 1:
The reviewer notes that the cited references are appropriate, well-suggested, and up-to-date within the last five years. However, some of them focus on alternative parallel parking (e.g., [6]), and some references appear to be missing in the text (e.g., [7]). Additionally, the reviewer highlights the absence of citations for other relevant technical standards, such as ISO 16787 and ISO 23374-1.

Response 1:
Thank you for your valuable feedback. We appreciate the reviewer's insights and have carefully reviewed and updated the reference list to ensure it accurately reflects the focus of the paper, particularly regarding the discussion of parallel parking. In response to the reviewer's comment, we have included more references directly related to vertical parking to provide clearer context for the methodology used.

Furthermore, we have incorporated the missing references, such as [7], and included citations for relevant technical standards, including ISO 16787 and ISO 23374-1, to strengthen the academic foundation of our paper.

These updates can be found on page 4, section 2.2, and in the revised reference list.

We hope these revisions adequately address the reviewer's comments. Thank you once again for your insightful feedback, which has greatly contributed to improving the manuscript.

Comments 2:
The reviewer has raised concerns regarding the vague description of the simulation methods in the manuscript, particularly pointing out the lack of raw data. The reviewer also highlighted that the simplified assumptions, such as the assumption that the steering angle is strictly equal to the turning of the vehicle's wheels (which is not linear in reality), might confuse readers. Furthermore, the reviewer noted that the manuscript does not discuss other relevant vehicle technologies that could significantly improve the parking process, such as wheel mounting, variable camber of the wheels, steered rear axles, and rear-wheel steering. This omission makes it difficult to compare the proposed method with existing market solutions. Additionally, the reviewer pointed out that the topological parameters of the parking area, such as slope and curvature, are not considered in the current study, which could impact the real-world applicability of the method.

Response 2:
Thank you for your valuable feedback. We greatly appreciate the reviewer's in-depth insights regarding the simulation methods and the potential confusion caused by the simplifying assumptions. In response to your comments, we have made the following revisions and clarifications in the revised manuscript:

• Clarification of Simplifying Assumptions:
  We acknowledge that the simplifying assumption regarding the relationship between the steering angle and the turning of the vehicle's wheels may not have been sufficiently clear. In the revised manuscript, we have explicitly highlighted these assumptions in the results section to ensure that readers understand their limitations. We have also clarified that the steering angle assumption is linear, although we recognize that this is not entirely accurate in real-world scenarios. Future research will further explore this nonlinear relationship to improve our model.

• Lack of Raw Data:
  We understand the concern about the lack of raw data. In the revised manuscript, we have added more details regarding the simulation setup and the data collection process, ensuring that readers can better understand the experimental design and the data source.

• Discussion of Other Vehicle Technologies:
  We appreciate the reviewer's suggestion to discuss other vehicle technologies, such as wheel mounting, variable camber of the wheels, and rear-wheel steering. In the revised manuscript, we have expanded this section to include these technologies and explained how they contribute to improving the parking process. Although these technologies are not considered in the current model, we recognize their relevance and propose them as potential directions for future research.

• Topological Parameters of the Parking Area:
  We agree that topological parameters of the parking area, such as slope and curvature, are important aspects of real-world parking scenarios. In the revised manuscript, we explicitly state that these topological parameters are not considered in the current study. However, we emphasize that future research will incorporate slope and curvature variations to enhance the robustness and applicability of our method.

These revisions can be found below Table 2 on page 5, Section 4 on page 11, and Table 3 on page 14.

We hope that these revisions address the reviewer's concerns and provide a clearer understanding of the assumptions and limitations of our study. Once again, we greatly appreciate the constructive feedback, which has played an important role in improving our manuscript.

Comments 3:
The manuscript presents a valuable contribution to advancing scientific and mathematical knowledge, particularly in the field of vertical parking path planning. However, the reviewer suggests that the manuscript could be made more engaging by integrating insights from a broader perspective, such as the potential adoption of this methodology in vehicle systems or by discussing additional progress for its innovation.

Response 3:
Thank you for your valuable feedback. We agree with the reviewer's suggestion and appreciate your insights. To address this, we have incorporated a broader perspective on the potential application of the proposed method in vehicle systems, particularly in autonomous or semi-autonomous parking systems, in the revised manuscript. We have also expanded the discussion on the future innovations of this method, including the integration of new technologies such as rear-wheel steering and advanced obstacle detection systems. These additions can be found in Section 1, and on page 2, we further elaborate on the future development potential of our research and its impact on autonomous parking solutions.

We hope these revisions make the manuscript more engaging and provide a clearer outlook on the future development of the research. Thank you once again for your insightful feedback.

Closing Remarks:
We sincerely appreciate your constructive feedback and insightful comments. We have fully addressed all of your concerns in the revised manuscript, and we believe these changes have enhanced the clarity and depth of our research. We are confident that these revisions have significantly improved the manuscript.

Should you have any further questions or require additional clarifications, please feel free to contact us. We deeply appreciate the time and effort you dedicated to reviewing our work and look forward to any additional suggestions you may have.

Once again, thank you for your valuable contribution to improving this manuscript. Wishing you success in your work and good health.

Reviewer 2 Report

Comments and Suggestions for Authors

I have no comments. Everything is fine. I believe that the problem is up to date, especially for partially or fully automated vehicles.

The main question in my opinion is if there is a possible solution to a car's vertical parking problem with omitting and avoiding the possible obstacles and finding the correct position in order to fit in the parking spot at the same time.

I find it interesting and original, because the paper presents some simple solutions which could be used in real, especially in autonomous vehicles or in case of the parking assistance systems installed in real cars. It adresses a gap of semi- or fully autonomous parking with little papers dealing with it previously.

The topic is surely relevant in the field of vehicles, especially autonomous vehicles, part of which can be electric. Whether it is electric or non-electric vehicle, it must in the future be supported with a system enabling smooth parking, especially in case of the plug-in parking places. This paper could be at some point a reference one for the forecoming research.

It adds an alternative view of the problem of space calculating based on both space analysis and vehicle kinematics. Of course, more factors, such as weather and road conditions could be included in the future.

I think that in some future papers the authors could add the non-movable obstacles such as the plug-in spot and some random things such as a tree which would enable a 3d parking model.

The conclusions are consistent and, moreover contain the diagram which could be expanded and serve as a kind of know-how for the future parking algorithms.

 

 

Author Response

Response:

We sincerely appreciate the reviewer’s positive comments and constructive suggestions regarding our paper. We are delighted to hear that the reviewer finds our research topic and methodology both relevant and innovative, particularly in the context of autonomous vehicles and parking assistance systems. We are also grateful for the reviewer’s recognition of our work, noting that our research helps fill a gap in the field of semi- or fully automated parking.

Regarding the reviewer’s suggestion to incorporate non-movable obstacles, such as charging stations or random objects like trees, in future studies, we fully agree with this approach. We believe it will significantly enhance the robustness and realism of the model. Considering these obstacles will indeed provide an exciting direction for future research, contributing to the development of a more comprehensive 3D parking model. We plan to explore how to integrate these factors into our path planning system in future work.

Additionally, we acknowledge the reviewer’s suggestion to include environmental factors, such as weather and road conditions. Although this study primarily focuses on vehicle kinematics and parking space analysis, we understand that these external factors are crucial in real-world applications. We intend to investigate how these elements can be incorporated into the model in future versions of the research.

Once again, thank you for your valuable feedback. We are pleased that our paper has contributed to the ongoing discussion of autonomous parking, and we are committed to refining and expanding our research based on these insightful suggestions.

Wishing you continued success in your work and good health.

Reviewer 3 Report

Comments and Suggestions for Authors

1. These ‘circular arc-line’，‘upper and lower boundaries’ and ‘multi-objective optimization function’  are  all common approaches in parking path planning, what's the contibution of this work? Please highlight that.
2. Relevant references are not fully refered.
3. How to determine the weight coefficients of the evaluation function?
4. How does this method perform in real - world physical parking scenario considering factors like sensor accuracy, vehicle control precision, and external disturbances ?

Author Response

Comments 1:

These ‘circular arc-line’，‘upper and lower boundaries’ and ‘multi-objective optimization function’  are  all common approaches in parking path planning, what's the contibution of this work? Please highlight that.

Response 1:

Thank you for your valuable feedback. We understand your concern regarding the methods such as "circular arc-line," "upper and lower boundaries," and "multi-objective optimization function," which are indeed common approaches in parking path planning. We have made clear responses in the manuscript and highlighted our research contributions, especially in the context of these common methods, by proposing an innovative optimization framework and integration strategy.

Firstly, although the circular arc-line method and multi-objective optimization function have been applied in path planning, we have introduced a new optimization framework that combines both approaches, effectively addressing the issues of path continuity and vehicle kinematic constraints in vertical parking. Furthermore, based on the analysis of the upper and lower boundaries of the parking area, we are able to define the feasible parking path region more accurately, thus improving the accuracy and efficiency of the path planning process.

Secondly, our multi-objective optimization function integrates multiple factors such as path curvature, adjustable margin, and path length. Compared to traditional methods, our approach optimizes several aspects, making it more valuable for real-world applications.

We have clarified these innovative contributions in the revised manuscript, particularly in Sections 3 and 4, where we discuss in detail how these conventional methods are improved through our unique framework and optimization strategy.

Once again, thank you for your feedback. We believe these revisions and additions will better highlight the uniqueness and contributions of our research.

Comments 2:

Relevant references are not fully refered.

Response 2:

Thank you for your valuable feedback. We greatly appreciate your comment regarding the incomplete references. In response, we have carefully reviewed the manuscript, adjusted the placement of citations, and added relevant references to ensure all pertinent sources are properly cited. The specific revisions can be found in the reference section.

We hope these revisions adequately address your concerns regarding the references. Once again, thank you for your insightful feedback, which has played a significant role in improving the quality of the manuscript.

Comments 3:

How to determine the weight coefficients of the evaluation function?

Response 3:

Thank you for your valuable comment. In this study, the weight coefficients in the evaluation function were carefully selected based on both theoretical considerations and practical requirements of vehicle dynamics and path planning.

The selection of weight coefficients takes into account the relative importance of different factors in parking path planning, such as smoothness, path length, and adjustable margin. The primary goal was to balance the need for a smooth and controllable parking path with the goal of minimizing the path length and providing sufficient flexibility for adjustments during parking. Here is a detailed explanation of how we determined the weight coefficients:

Path Curvature: We assigned a higher weight to path curvature to ensure smooth steering and vehicle controllability during parking. Given that sharp turns can negatively affect vehicle maneuverability, this higher weight helps prevent abrupt curvature, which could compromise both comfort and safety during the parking process.

Path Length: A moderate weight was assigned to path length to minimize the distance the vehicle travels while parking. This is critical for improving parking efficiency, reducing parking time, and optimizing space usage in often limited parking areas.

Adjustable Margin: We assigned a moderate weight to the adjustable margin term to ensure that the vehicle can adjust its trajectory when necessary, avoiding collisions with obstacles or parking boundaries. This margin allows the vehicle to adapt to variations in the environment, improving the robustness of the system.For details, please refer to 3.24.

Comments 4:

How does this method perform in real - world physical parking scenario considering factors like sensor accuracy, vehicle control precision, and external disturbances ?

Response 4:

Thank you for your insightful question regarding the performance of our method in real-world physical parking scenarios, considering factors such as sensor accuracy, vehicle control precision, and external disturbances.

We fully recognize that real-world parking environments introduce several challenges, including sensor inaccuracies, vehicle control limitations, and disturbances from the surrounding environment, all of which can significantly affect the performance of path planning methods. In the current study, we have primarily focused on the theoretical modeling of the parking path, assuming ideal conditions such as perfect sensor accuracy, precise vehicle control, and an obstacle-free environment. This simplification was made to clearly demonstrate the core concepts and functionality of our proposed method.

However, we agree that real-world applications require a more comprehensive consideration of these factors. To address the challenges posed by sensor inaccuracies, for example, future research could explore the integration of sensor fusion techniques to improve the reliability of environmental perception. Additionally, we plan to consider vehicle control precision in future work by incorporating more realistic models of vehicle dynamics and feedback control mechanisms that account for steering errors and motion disturbances.

Moreover, external disturbances such as road surface variations, weather conditions, and other dynamic factors should be considered in future versions of the model. This could involve introducing uncertainty into the path planning process and developing robust optimization techniques that can adapt to these real-world factors.

We hope to explore these aspects in future research to further validate the proposed method in real-world scenarios, ensuring its practical applicability and robustness.

Thank you again for your valuable feedback. We look forward to addressing these challenges and enhancing the real-world applicability of our approach in future studies.

Closing Remarks:
We sincerely appreciate your constructive feedback and insightful comments. We have fully addressed all of your concerns in the revised manuscript, and we believe these changes have enhanced the clarity and depth of our research. We are confident that these revisions have significantly improved the manuscript.

Should you have any further questions or require additional clarifications, please feel free to contact us. We deeply appreciate the time and effort you dedicated to reviewing our work and look forward to any additional suggestions you may have.

Once again, thank you for your valuable contribution to improving this manuscript. Wishing you success in your work and good health.

Reviewer 4 Report

Comments and Suggestions for Authors

In this manuscript, it is presented a novel method for vertical parking path planning that addresses curvature discontinuity and turning-in-place issues. The proposed approach uses circular arcs, straight lines, and a multi-objective evaluation function to generate an optimal path. Simulation results demonstrate that the method ensures path curvature continuity and meets vehicle kinematic constraints. The study presents an innovative path planning method for vertical parking. However, there are some minor points that need to be revised as follows.

 

1. The manuscript does not sufficiently address computational efficiency in real-time applications. While the proposed method claims to balance geometric and numerical optimization techniques, the manuscript lacks quantitative benchmarks comparing the computational time with state-of-the-art methods.
2. The simulations conducted were restricted to a single parking scenario using predefined parameters. The manuscript would benefit from testing various parking conditions, including different parking space sizes, vehicle dimensions, and environmental constraints.
3. It must be conducting a sensitivity analysis to assess how different weight combinations affect the results. Additionally, the authors must provide references or prior studies to support the chosen weights.
4. It must be Include more diagrams or step-by-step illustrations that link the mathematical formulas in the parking environment.

Author Response

Comments 1:
The manuscript does not sufficiently address computational efficiency in real-time applications. While the proposed method claims to balance geometric and numerical optimization techniques, the manuscript lacks quantitative benchmarks comparing the computational time with state-of-the-art methods.

Response 1:
Thank you for your valuable comment. In response to this concern, we have included a comprehensive comparative analysis of computational efficiency in the revised manuscript. Specifically, we have provided a detailed comparison of path length, computation time, and path smoothness between the proposed method, traditional geometric methods, and numerical optimization techniques. These comparison results are presented in Figures 7-9 (page 12) and Table 3 (page 13), which highlight the efficiency advantages of our method. We hope this revision will better demonstrate the computational advantages of our approach in real-time applications. Thank you again for your constructive feedback, which has significantly contributed to improving the manuscript.

Comments 2:
The simulations conducted were restricted to a single parking scenario using predefined parameters. The manuscript would benefit from testing various parking conditions, including different parking space sizes, vehicle dimensions, and environmental constraints.

Response 2:
Thank you for your valuable feedback. We appreciate your suggestion to test different parking conditions, such as varying parking space sizes, vehicle dimensions, and environmental constraints. While we understand the importance of testing our method under a wider range of conditions, we chose to focus on a single vertical parking scenario in this study to clearly demonstrate the core concepts and functionality of the proposed method. However, we acknowledge that future research could extend the current work by incorporating more diverse parking scenarios, such as parallel parking and angled parking, to further assess the robustness and adaptability of our method in more complex and varied environments. Once again, thank you for your valuable suggestion. Your feedback will certainly guide our future improvements.

Comments 3:
It must be conducting a sensitivity analysis to assess how different weight combinations affect the results. Additionally, the authors must provide references or prior studies to support the chosen weights.

Response 3:
Thank you for your valuable feedback. We completely agree with your suggestion and recognize the importance of conducting a sensitivity analysis to assess how different weight combinations influence the results. In the revised manuscript, we have included a sensitivity analysis and selected two additional sets of starting point coordinates to assess the path planning performance from different parking starting points. The analysis results show how different weight combinations impact the path planning outcomes, ensuring the robustness of the proposed method across different scenarios. These changes can be found in the revised manuscript (Figure 13, page 15).

We believe these modifications effectively address the sensitivity of the method to different weight combinations, further validating the reasonableness of our parameter selection. Once again, thank you for your valuable suggestions, which have played a crucial role in improving our manuscript.

Comments 4:
The reviewer suggests including more diagrams and step-by-step illustrations that link the mathematical formulas to practical applications.

Response 4:
Thank you for your valuable feedback. We appreciate the reviewer’s suggestion to include more diagrams and step-by-step illustrations that link the mathematical formulas to practical applications. In response to this suggestion, we have made the following modification in the revised manuscript:

We have added a diagram immediately following Equation (1) that visually illustrates the vehicle's kinematic model, including key variables such as the coordinates of the front and rear axle centers (x_f,y_f  and x_r,y_r​), as well as the heading angle θ. This addition is intended to better demonstrate how the mathematical model is applied in the parking scenario.

This update can be found in Section 2.1, following Equation (1).

We hope this revision effectively addresses the reviewer’s comment and provides a clearer understanding of the vehicle’s motion in relation to the equations. Thank you once again for your insightful feedback.

Closing Remarks:
We sincerely appreciate your constructive feedback and insightful comments. We have fully addressed all of your concerns in the revised manuscript, and we believe these changes have enhanced the clarity and depth of our research. We are confident that these revisions have significantly improved the manuscript.

Should you have any further questions or require additional clarifications, please feel free to contact us. We deeply appreciate the time and effort you dedicated to reviewing our work and look forward to any additional suggestions you may have.

Once again, thank you for your valuable contribution to improving this manuscript. Wishing you success in your work and good health.