Genetic Programming to Optimize 3D Trajectories

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article explores the application of genetic programming to 3D trajectory optimization. The paper is well-considered and addresses the challenging problem of path planning with evolutionary algorithms, such as GA, namely, objective minimization and algorithm execution speed. Therefore, it is recommended to accept it after minor revisions.

Why is the function expression in Figure 6 in this format? What is the significance of choosing this format? Is it feasible to change to another expression? In addition, is this expression generally applicable to different scenarios, or does it need to be modified specifically for more complex scenarios? It is recommended to perform an in-depth analysis.

Comments on the Quality of English Language

It is suggested to modify the entire framework and content of the article to enhance readability.

Author Response

Comment 1: Why is the function expression in Figure 6 in this format? What is the significance of choosing this format? Is it feasible to change to another expression? In addition, is this expression generally applicable to different scenarios, or does it need to be modified specifically for more complex scenarios? It is recommended to perform an in-depth analysis.

Response 1: Thanks for your comment. We are not sure how to interpret the reviewer question. Figure 6, which is now Figure A.3 in Appendix E, depicts visually the sequence of steps as part of the “Transformation” box in Figure 4. In the end, to validate the path, chromosomes represented as function trees are decoded and compiled into a callable function. This function is then plotted in 3D space to produce a finite path between two points. To validate this path, it is mapped directly to the problem interval in geographic space by transforming it geometrically, the curve is reprojected in three dimensions to connect the start and end points. So, Figure A.3 (previously Figure 6) only depicts the sequence of steps mentioned above.

Reviewer 2 Report

Comments and Suggestions for Authors

1- The manuscript is written in a boring style, as there are two sections to fulfill the same purpose of the introduction and review of previous literature.

2- The style of writing a manuscript is completely different from the style of writing a thesis or dissertation in terms of the scientific method.

3- When presenting the main idea for the manuscript, it is preferable to limit the mention of previous references unless necessary.

4- Follow up on commenting on the results as they appear within the manuscript in the Results section.

Author Response

Comment 1: The manuscript is written in a boring style, as there are two sections to fulfill the same purpose of the introduction and review of previous literature.

Comment 2: The style of writing a manuscript is completely different from the style of writing a thesis or dissertation in terms of the scientific method.

Comment 3: When presenting the main idea for the manuscript, it is preferable to limit the mention of previous references unless necessary.

Comment 4: Follow up on commenting on the results as they appear within the manuscript in the Results section.

Response 1 to 4: Thank you for your comments. We have grouped our response to the four comments from reviewer #2 together because they all relate to writing style and better structuring of the manuscript. Overall, the revised version of the manuscript has undergone a drastic restructuring to better fit our work into a journal article style. The introduction and methods have been significantly reduced and most of the content has been moved to the appendices. The results and conclusions are now driven by the research questions posed in the introductions. Figures have been enlarged to improve their readability. Some figures and tables are now in appendices. In summary, the manuscript is now shorter to fit the journal's length restriction and the main ideas and contributions are clearly expressed, eliminating unnecessary subsections in the previous version of the article.

Reviewer 3 Report

Comments and Suggestions for Authors

This study proposed a novel approach for 3D trajectory optimization using genetic programming. Generally, the manuscript is well-structured and written in clear English. However, there are several major issues that need addressing:

(1) The paper includes lengthy and detailed explanations that could be more improved.

l  Improve the whole language style to be concise and suitable for a published paper rather than a thesis.

l  Simplify the introduction to clearly explain the transition from 2D to 3D pathfinding, highlighting the unique challenges and advantages of 3D path planning.

l  Simplify the explanation of GP technology, avoiding overly technical jargon in the Introduction. Focus on key concepts and their application.

l  The literature review would benefit from improved organization. It is suggested to include additional headings and subheadings to separate discussions on pathfinding algorithms and trajectory optimization. This will enhance the clarity and logical flow of the content.

l  The results section could include summaries at the end of each subsection to highlight key findings and their significance. Emphasizing the implications of these results for the optimization process would make the findings more impactful.

(2) The introduction part should provide a more detailed analysis of the limitations of existing GA and GP methods. Emphasize how the proposed GP approach addresses these limitations and specifically discuss its application in 3D trajectory optimization.

(3) The methodology section lacks detailed justification for parameter choices. The paper should explain the rationale behind default parameter values and how these parameters affect the optimization process. If possible, providing experimental results or case studies showing the impact of varying parameters would strengthen this section.

(4) The results section lacks a quantified comparative analysis with existing methods (e.g., Hildemann and Verstegen). There is only a case study of trajectory visualization of different methods (Figure 9). Include a detailed comparison of performance metrics to clearly demonstrate the advantages of the proposed method over existing approaches.

Comments on the Quality of English Language

Extensive editing of English language required.

Author Response

Comments 1: The paper includes lengthy and detailed explanations that could be more improved.

• Improve the whole language style to be concise and suitable for a published paper rather than a thesis.
• Simplify the introduction to clearly explain the transition from 2D to 3D pathfinding, highlighting the unique challenges and advantages of 3D path planning.
• Simplify the explanation of GP technology, avoiding overly technical jargon in the Introduction. Focus on key concepts and their application.
• The literature review would benefit from improved organization. It is suggested to include additional headings and subheadings to separate discussions on pathfinding algorithms and trajectory optimization. This will enhance the clarity and logical flow of the content.
• The results section could include summaries at the end of each subsection to highlight key findings and their significance. Emphasizing the implications of these results for the optimization process would make the findings more impactful.

Response 1: Thank you for your comments. We have grouped our response to the suggestion under Comments 1 together because they all relate to writing style and better structuring of the manuscript. Overall, the revised version of the manuscript has undergone a drastic restructuring to better fit our work into a journal article style. The introduction and methods have been significantly reduced and most of the content has been moved to the appendices. The results and conclusions are now driven by the research questions posed in the introductions. Figures have been enlarged to improve their readability. Some figures and tables are now in appendices. In summary, the manuscript is now shorter to fit the journal's length restriction and the main ideas and contributions are clearly expressed, eliminating unnecessary subsections in the previous version of the article.

Comments 2: The introduction part should provide a more detailed analysis of the limitations of existing GA and GP methods. Emphasize how the proposed GP approach addresses these limitations and specifically discuss its application in 3D trajectory optimization.

Response 2: The Introduction part is now better structured in four subsections in order to emphasize the reviewer suggestions in terms of problem definition, gap addressed, and research questions:

• Background and Problem Definition
• Literature Review, which in turn is divided into “Trajectory Optimisation Algorithms” and “3D Trajectory Optimisation”
• Research Gap
• Research Questions

Comments 3: The methodology section lacks detailed justification for parameter choices. The paper should explain the rationale behind default parameter values and how these parameters affect the optimization process. If possible, providing experimental results or case studies showing the impact of varying parameters would strengthen this section.

Response 3: Thank you for your feedback. The methods and materials section has now been split into two main sections: “Study Area and Data Preparation” and “Research Methods”. The latter section has been simplified and split into two main parts: “Evolutionary Computation Framework”, which explains the DEAP framework we used, and “GP Optimization Workflow”, which refers to our implementation of the DEAP-based trajectory optimization algorithm. The reviewer can find in the “Evolutionary Computation Framework” section a paragraph mentioning customization of DEAP parameters which are, in turn, described in more detail in Appendix D.

Comments 4: The results section lacks a quantified comparative analysis with existing methods (e.g., Hildemann and Verstegen). There is only a case study of trajectory visualization of different methods (Figure 9). Include a detailed comparison of performance metrics to clearly demonstrate the advantages of the proposed method over existing approaches.

Response 4: Thank you for your comment. Apart from the quantitative comparison to  Hildemann and Verstegen, there is a section called "Comparison with similar methods" which compares qualitatively to other methods (studies). Nevertheless, we  would say that, unfortunately, there is no benchmark problem available for 3D trajectories with 3D barriers. The only reproducible benchmark problem available (that we at least know to the best of our knowledge) is the problem we had used. On that problem, we had three objective functions, and the objective function used here deviated. Therefore, it can not be compared based on the objective values (for example, Hildemann and Verstegen used flight time of a specific UAV), just on the calculation time. To satisfy the reviewer comment, we could state that due to the lack of benchmark problems our defined problem can be used for future works as a benchmark.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have completed the required revisions to present the manuscript better than the first version.

Author Response

Comments 1: The authors have completed the required revisions to present the manuscript better than the first version.

Response 1: Thanks for your comment

Reviewer 3 Report

Comments and Suggestions for Authors

The author has addressed most of my concerns, and the paper can be published after adjusting images:

l  Figures 6 and 9 should be resized and their sub-figures placed side by side to prevent too little information from occupying an entire page.

l  For Figure 10, categorize the three trajectory graphs and three curve graphs into two rows of three columns to optimize space.

l  All lines in the figures could have their corresponding labels, and all sub-figures in the images should be numbered and labeled, with appropriate references and summaries included in the figure captions.

Comments on the Quality of English Language

Need to be improved.

Author Response

Comments 1: Figures 6 and 9 should be resized and their sub-figures placed side by side to prevent too little information from occupying an entire page.

Response 1: Thanks. We have arranged subfigures in Fig 9 horizontally. However, we believe that the subfigures in Fig. 9 are more readable vertically.

 

Comments 2: For Figure 10, categorize the three trajectory graphs and three curve graphs into two rows of three columns to optimize space.

Response 2: Thanks. Again, the tradeoff between space and readable is important in this figure. We prefer to keep the subfigures in fig 10 vertically 

 

Comments 3: All lines in the figures could have their corresponding labels, and all sub-figures in the images should be numbered and labeled, with appropriate references and summaries included in the figure captions.

Response 3: Thanks. Figures 11, 12 and 13 have been updated accordingly