Review Reports - An Improved Red-Billed Blue Magpie Optimization Algorithm for 3D UAV Path Planning in Complex Terrain

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript proposes an improved Red-Billed Blue Magpie Optimization Algorithm (IRBM-OA) for 3D UAV path planning in complex terrain. The topic is relevant and timely, and the use of nature-inspired optimization for UAV path planning is promising. However, several sections require clarification, stronger experimental validation, improved presentation, and detailed comparisons to ensure scientific rigor and reproducibility.

1. The manuscript briefly mentions improvements to the Red-Billed Blue Magpie Optimization Algorithm, but the specific novelty is not clearly articulated. Please clearly define:

What limitations exist in the original RBMO algorithm?
What exact modifications were introduced?
How these modifications improve exploration, exploitation, convergence, or stability.

A dedicated subsection comparing original vs. improved algorithm (mathematically and conceptually) would strengthen the contribution.

2. The mathematical formulation of the improved algorithm is not sufficiently detailed. Please clarify:

The role of each control parameter
How the improved update rules operate in 3D search space
How obstacles, no-fly zones, and terrain elevation data are incorporated into the cost function

3.The problem formulation should explicitly describe:

UAV kinematic constraints (max turning angle, climb/descent rate, velocity limits)
Minimum safe altitude over terrain
Collision avoidance constraints
Environmental assumptions (static terrain, wind ignored, etc.).

4. Some key figures, especially 3D terrain maps, convergence curves, and generated paths lack sufficient clarity, resolution, or labeling. Please improve:

Axis labels and units
Color legend for terrain elevation
Visual distinction between optimal path, initial path, and obstacles

5. The evaluation is limited and needs stronger benchmarking. Please include:

Comparison with more state-of-the-art algorithms (e.g., A*, RRT*, PSO variants, DE variants, WOA, GWO)
Statistical results over multiple runs
More performance indicators (computational time, success rate, smoothness index, energy consumption)

6. The manuscript should be thoroughly checked for grammar errors and typos. In addition, abbreviations with full form repeated several times, full form with abbreviation should be used once and then abbreviations only.

7. The manuscript presents results but lacks deeper analysis. Please enhance by discussing:

Why the proposed algorithm performs better
Specific scenarios where the algorithm struggles
Trade-offs between computation cost and solution quality
Practical implications for real UAV systems

Comments on the Quality of English Language

Moderate editing is required.

Author Response

Dear Editors and Reviewers:

On behalf of the co-authors, I would like to express our sincere gratitude for the opportunity to revise our manuscript. We are very grateful to the editor and reviewers for their positive and constructive comments and suggestions on our manuscript titled "An Improved Red-crowned Crane Optimization Algorithm for 3D UAV Path Planning in Complex Terrain". We apologize for not being able to update the revised manuscript more quickly due to the need to add some necessary experimental data.

In this revised version, we have addressed all the reviewers' concerns. A detailed response to each reviewer’s comments is attached. For clarity, all revised sections are marked in red within the manuscript. We hope these targeted changes successfully address each comment and requirement, making the manuscript acceptable. We sincerely appreciate your consideration and look forward to your positive response.

Response to Reviewer 1 Comments：

The manuscript briefly mentions improvements to the Red-Billed Blue Magpie Optimization Algorithm, but the specific novelty is not clearly articulated. Please clearly define:

Response : We sincerely thank the reviewer for the valuable comments. In response, we have revised the manuscript to provide more detailed explanations of each improvement and clearly outline the specific modifications to highlight the novelty. The corresponding changes are discussed under each specific comment below, and the locations of the modifications have been highlighted in bold in each response to facilitate the reviewer’s verification. We sincerely appreciate the reviewer’s constructive suggestions once again.

1-1. What limitations exist in the original RBMO algorithm?

Response : In the revised manuscript, we have added a clearer explanation of the limitations of the original RBMO algorithm in the abstract, including its insufficient population diversity, limited global exploration ability, and tendency to fall into local optima. This addition also clarifies the motivation for the improvements proposed in our work. Thank you again for your valuable suggestion. ( Page 1, Lines 9-13 )

1-2. What exact modifications were introduced?

Response : To clarify the specific improvements of the proposed CTWRBMO, we have added a dedicated subsection (Section 3.5) providing a detailed comparison between the original RBMO and the improved algorithm. This subsection clearly describes all introduced modifications, including chaotic initialization, adaptive search probability, nonlinear decaying disturbance weight, neighborhood-based differential update, and elite retention mechanisms. ( Page 11, Lines 282-313 )

1-3. How these modifications improve exploration, exploitation, convergence, or stability.

Response : Thank you for pointing this out. In the newly added Section 3.5, we have explained in detail how each modification contributes to performance enhancement. Specifically, we describe how chaotic initialization improves search diversity, how adaptive ε(t) balances exploration and exploitation dynamically, how exponential disturbance decay enhances convergence refinement, and how neighborhood-based differential updates improve stability and reduce the risk of local stagnation. ( Page 11, Lines 282-313 )

1-4. A dedicated subsection comparing original vs. improved algorithm (mathematically and conceptually) would strengthen the contribution.

Response : We appreciate this valuable suggestion. Following the reviewer’s recommendation, we have added a new subsection (Section 3.5) that systematically compares RBMO and CTWRBMO from both mathematical and conceptual perspectives. This includes a comparison table 1and detailed explanations of the structural improvements introduced in the proposed algorithm. ( Page 11, Lines 274-313 )

The mathematical formulation of the improved algorithm is not sufficiently detailed. Please clarify:

Response : Thanks for pointing out the ambiguity. We have supplemented the manuscript with additional explanations and mathematical formulations of the improved algorithm, enabling readers to better understand the working principles of the enhanced RBMO algorithm, the role of each improvement strategy.

2-1. The role of each control parameter

Response : Thank you for your reminder，we have added the functions of each control parameter in the manuscript, and to demonstrate rigor, we have provided the reason and basis for setting the values of each parameter, such as a, k and c. (Page 6, Lines 159-161 ,165-166; Page 7, Lines 194-195)

2-2. How the improved update rules operate in 3D search space

Response : We thank the reviewer for this insightful comment. In the revised manuscript, we have clarified how the improved update rules operate in the three-dimensional search space. Specifically, each individual is represented as a three-dimensional position vector , corresponding to a UAV path node in the 3D environment. The update rules are applied in a vectorized manner, where neighborhood difference vectors determine the movement direction in the 3D search space, and the displacement magnitude along each dimension is jointly regulated by random perturbations and dynamic weighting factors. This mechanism enables a smooth transition from large-scale global exploration to refined local exploitation in three-dimensional space. Moreover, a greedy acceptance criterion is adopted, such that a newly generated 3D position is accepted only if it improves the fitness value, ensuring continuous convergence toward better solutions in the 3D search space. Relevant explanations have been added to the manuscript (Page 8, Lines 252–261).

2-3. How obstacles, no-fly zones, and terrain elevation data are incorporated into the cost function

Response : We have further refined the terrain-related penalty design by explicitly stating that higher altitude penalties are imposed when UAV flight paths approach the minimum safe terrain clearance. This clarification more clearly explains how terrain elevation information is incorporated into the cost function during the path planning process (page 24, lines 652–665). In addition, we have supplemented the description to indicate that UAV paths incur significant penalties when traversing no-fly zones or obstacles, thereby further clarifying how no-fly zones and obstacles are integrated into the cost function (Page 24, lines 672–679).

The problem formulation should explicitly describe:

Response : We appreciate the thoughtful review and constructive feedback provided by the reviews. We acknowledge that the original manuscript did not provide sufficiently detailed descriptions in the problem modeling section, which may have made certain constraints or concepts unclear to readers. In response, we have revised the manuscript to provide a more detailed and explicit description of the problem modeling, clearly defining all constraints, variables, and symbols, to ensure that the UAV path planning formulation is fully understandable.

3-1. UAV kinematic constraints (max turning angle, climb/descent rate, velocity limits)

Response : We sincerely thank the reviewer for the valuable suggestion. In response, the following revisions have been made in the manuscript:

1) Section 5.3.3 has been updated to include a detailed description of the maximum turning angle; (Page 21, lines 541–543)

2)A new Section 5.3.4 has been added to provide a detailed explanation of the maximum climb and descent rate constraints; (Page 21, lines 544–556)

3) A new Section 5.3.5 has been added to describe the flight speed constraints in UAV path planning. (Page 21, lines 557–571)

These modifications have been highlighted in the manuscript to facilitate the reviewer’s verification.

3-2. Minimum safe altitude over terrain

Response : Thank you for your suggestion. We have added a quantitative description of the minimum terrain altitude in the Problem Modeling section of the manuscript, Specifically, the minimum safe altitude over terrain is designed to ensure that the UAV maintains a necessary vertical safety distance from the ground throughout the flight. An additional minimum safety height ℎ is added on top of the equivalent terrain to prevent potential collision risks caused by terrain variations. (Page 20, Lines 502-504)

3-3. Collision avoidance constraints

Response : Thank you for your constructive comment. We have carefully considered your suggestion and provide the following clarifications:

Collision zones: Threat areas such as mountains and no-fly zones are modeled as spherical regions that UAVs are strictly prohibited from entering, ensuring that the generated flight paths explicitly comply with collision avoidance constraints.
Path refinement and node insertion: To prevent UAVs from entering these spherical threat zones, additional nodes are dynamically inserted between consecutive path nodes, enabling more precise collision detection during the path optimization process.
Constraint satisfaction: By combining obstacle modeling with path optimization, the generated flight paths effectively avoid all spherical threat areas, fully satisfying the collision avoidance constraints. The corresponding revisions and explanations have been detailed in Section 5.3.6 of the manuscript. (Page 22, Lines 579-582，588-591)

3-4. Environmental assumptions (static terrain, wind ignored, etc.).

Response : We have supplemented the description of the environmental assumptions in the problem modeling section of the text (Page 19, Lines 496-498) to present the conditions and prerequisites on which the drone path planning is based in a clearer and more systematic manner, thereby facilitating readers' understanding of the rationality of the algorithm design and experimental setup.

Some key figures, especially 3D terrain maps, convergence curves, and generated paths lack sufficient clarity, resolution, or labeling. Please improve:

Response : Thank you very much for your constructive suggestions. We agree that clarity and readability are crucial for accurately conveying the experimental results. In the revised manuscript, we have enhanced the quality of key figures, including the 3D terrain map, convergence curve, and planned flight path, by increasing their resolution and refining visual details. Additionally, we have added clearer labels, legends, and annotations where appropriate to improve readability. These revisions aim to make the figures more informative and easier to interpret.

4-1. Axis labels and units

4-2. Color legend for terrain elevation

4-3. Visual distinction between optimal path, initial path, and obstacles

Response : Thank you for the valuable suggestions from the reviewers. We have improved the clarity and readability of the key graphics in the revised version, including adding coordinate axis labels and units, increasing the color legend for terrain height, and enhancing Visual distinction between optimal path, initial path, and obstacles. (Page 26，Linescapable721-727) At the same time, the resolution of all relevant graphics has been raised to 600 dpi. The relevant modifications can be found in Figures 4 and 9.

To facilitate the reviewers' intuitive review, the partial diagrams of Figure 4 and Figure 9 are also provided below this text.

Fig. 4:

Fig. 9:

The evaluation is limited and needs stronger benchmarking. Please include:

5-1. Comparison with more state-of-the-art algorithms (e.g., A*, RRT*, PSO variants, DE variants, WOA, GWO)

Response : We sincerely thank the reviewer for the valuable suggestion to strengthen the experimental benchmarking.

Accordingly, we have expanded the convergence performance evaluation by including two additional state-of-the-art optimization algorithms, namely L-SHADE and GWO. L-SHADE is an advanced adaptive differential evolution framework, while GWO is a widely adopted bio-inspired swarm optimization algorithm. Their inclusion enables a more rigorous and up-to-date assessment of the convergence behavior and optimization efficiency of the proposed CTWRBMO. The corresponding results and discussions have been added to the revised manuscript (Fig. 4 and Table 4).

In addition, ablation experiments have been conducted to further investigate the individual contributions of the proposed mechanisms integrated into CTWRBMO, thereby strengthening the overall experimental validation. (Section 4.4 for details, Page 19, Lines 426-480)

Regarding the algorithm selection in the 3D UAV path planning experiments, we appreciate the reviewer’s suggestion to include classical planners (e.g., A*, RRT*) and additional metaheuristic variants (e.g., PSO, DE, WOA, GWO). However, in strongly constrained three-dimensional environments involving complex terrain, obstacles, no-fly zones, and UAV kinematic constraints, these methods typically require problem-specific modeling, constraint-handling techniques, or trajectory repair strategies to reliably generate feasible and smooth paths. Introducing heterogeneous constraint-processing mechanisms would make a fair and consistent comparison difficult.

Therefore, we focused on algorithms with comparable optimization frameworks and constraint-handling principles, allowing the evaluation of path quality and feasibility to remain controlled and interpretable. This rationale has now been explicitly clarified in the revised manuscript (Page 26, Lines 711–723).

We fully agree that extending the 3D path planning comparisons to classical planners and additional advanced heuristic algorithms is valuable, and we acknowledge this as a limitation. Such extensions will be considered as important future work to further validate the generality and applicability of the proposed approach.

We sincerely thank the reviewer again for the insightful comments, which have significantly helped improve the rigor and clarity of this work.

5-2. Statistical results over multiple runs

Response : We have revised Figure 4 and Table 4 (page 28, lines 399–411) as follows:
1. Figure 4 now shows the average convergence curves over 30 independent runs, clearly illustrating the convergence behavior and stability of each algorithm.
2. Table 4 has been updated to report the best values, mean values, and standard deviations across the 30 runs, providing a statistically rigorous basis for comparing algorithm performance and robustness.

5-3. More performance indicators (computational time, success rate, smoothness index, energy consumption)

Response : We think this is an excellent suggestion. In response, we have incorporated the following metrics in Table 10 of the manuscript (Page 26, Lines 735–743; Page 29, Lines 770–797) to systematically evaluate each algorithm: average computational time (in seconds) to assess algorithm efficiency; success rate, defined as the proportion of feasible paths that avoid obstacles, terrain violations, and excessive curvature; path smoothness, measured as the mean turning angle between consecutive waypoints (in radians); and energy consumption, estimated using a simplified model that accounts for the Euclidean distance between consecutive waypoints, path curvature (reflecting UAV turning limitations), and vertical motion energy (reflecting ascent and descent). All metrics are averaged over 30 independent runs to ensure statistical reliability. While the energy model is simplified, it provides a consistent and fair basis for comparing different algorithms and demonstrates the practical implications of each approach.

The manuscript should be thoroughly checked for grammar errors and typos. In addition, abbreviations with full form repeated several times, full form with abbreviation should be used once and then abbreviations only.

Response : Thank you for your careful review. We have thoroughly checked the manuscript for grammar and spelling errors, addressed all identified issues, and revised abbreviations that were not clearly explained by providing their full forms upon first occurrence, which have been highlighted in yellow.

The manuscript presents results but lacks deeper analysis. Please enhance by discussing:

7-1. Why the proposed algorithm performs better

Response : Thank you for your valuable suggestions. In the revised manuscript (Page 13, Lines 363 to 387), we explained from a statistical perspective based on the experimental results why the proposed CTWRBMO algorithm can achieve better performance.

Specifically:

(1) In terms of convergence accuracy, the optimal values obtained by CTWRBMO on F1, F4, and F6 are very close to the theoretical optimal values, while on F3, F5, F7, F8, F9, and F10, CTWRBMO consistently outperforms all the compared algorithms in terms of the best solution.

(2) The average results of all algorithms after 30 independent executions show that CTWRBMO achieved the best average performance on F1, F2, F3, F4, F7, F8 and F9. This indicates that its superior performance is statistically stable rather than an accidental result.

(3) On most test functions, the standard deviation of CTWRBMO is significantly lower than that of other algorithms, reflecting lower performance volatility and stronger robustness, and demonstrating stable convergence behavior in independent runs.

Furthermore, to provide more statistical support, we introduce the Wilcoxon rank-sum test in Section 4.3 to verify whether the observed performance improvement is statistically significant. At the same time, in Section 4.4, we conduct ablation experiments to evaluate the contribution of each individual improvement strategy, thereby further understanding how these components jointly enhance the overall optimization performance.

7-2. Specific scenarios where the algorithm struggles

Response : Thank you for this insightful comment. This issue has been addressed in the revised manuscript(Page 14, Lines 388-393). The experimental results indicate that the performance advantage of CTWRBMO is not absolute across all benchmark functions. Specifically, on functions F6 and F10, the final convergence accuracy of CTWRBMO is slightly inferior to that of L-SHADE, and on function F5, it is marginally worse than that of RBMO.

However, it should be noted that the differences in the final optimal values among these algorithms on the above functions are relatively small. Moreover, CTWRBMO still demonstrates a faster overall convergence speed in these cases, indicating that it remains highly competitive when dealing with complex optimization problems, even in scenarios that require intensive local exploitation.

7-3. Trade-offs between computation cost and solution quality

Response : To address the trade-off between computational cost and solution quality, we have included a dedicated time complexity analysis in the revised manuscript (Section 3.6, Page 27, Lines 315 - 329).

Specifically, we theoretically analyzed the computational complexity of the proposed CTWRBMO algorithm and demonstrated that its overall time complexity is O(T(N^2 D + Nf(D))), which is at the same level as that of the original RBMO algorithm. Although CTWRBMO includes additional mechanisms such as chaotic initialization, dynamic weight factors, and exponential convergence control terms, these components only introduce constant-level or O(ND) computational overhead, thus not increasing the asymptotic complexity. It is worth emphasizing that in the CEC2020 benchmark experiments and unmanned aerial vehicle path planning experiments, all the compared algorithms were executed under the same computational budget, including the same population size and maximum number of iterations. Therefore, the improvement in solution quality achieved by CTWRBMO is not due to an increase in computational cost, but rather due to more efficient utilization of the same computing resources.

Overall, CTWRBMO achieves higher convergence accuracy and stronger stability without sacrificing computational efficiency. This indicates that the proposed algorithm achieves a good balance between computational cost and solution quality, making it highly suitable for practical optimization tasks with limited computational budgets.

7-4. Practical implications for real UAV systems

Response : We think this is an excellent suggestion. We have revised the manuscript to clearly explain the practical significance of the proposed CTWRBMO algorithm for actual unmanned aircraft systems in practical applications. In the updated manuscript (Page 31, Lines 838 - 844), we explain that CTWRBMO can generate shorter and smoother paths, thereby reducing energy consumption and extending flight endurance time - this is particularly important for unmanned aircraft platforms with limited battery capacity. Moreover, its high path feasibility and low variance ensure safe navigation in environments with obstacles, threat areas, or complex terrain, thereby improving the reliability of the mission. These results indicate that CTWRBMO can effectively balance the quality of the solution, operational safety, and computational efficiency, highlighting its practical value in actual unmanned aircraft operations.

Comments on the Quality of English LanguageModerate editing is required.

Response : We tried our best to improve the manuscript and made some changes to it. These changes will not influence the content and framework of the paper. We sincerely thank the reviewers for their enthusiastic work and hope that the corrections will be approved.

Thank you again for your review work and look forward to your feedback.

Yours sincerely

Xu Yong,Xue Ning ,Zhang Yi

16 December 2025

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper proposes an improved Cuckoo Bird Random Mountaineering Optimization (CTWRBMO) algorithm for three-dimensional path planning of unmanned aerial vehicles (UAVs) in complex terrain. While the proposed improvement holds certain value, the paper exhibits shortcomings in explaining the algorithm's mechanism and conducting parameter sensitivity analysis. Specific issues are as follows:

1. The introduction provides a rather general description of the limitations of the RBMO algorithm. Please clearly analyze the strengths and weaknesses of the original algorithm and explain the rationale for selecting it for this study.
2. The analysis of the latest swarm intelligence algorithms is insufficient. Recent research from the past three years should be supplemented and analyzed.
3. The origin of the circular mapping parameter in Equation (7) is unclear. Provide the reason and basis for setting this parameter to 0.2.
4. Provide justification for the fixed coefficient of 0.62 in the dynamic ε adjustment formula (Equation 8).
5. Include pseudocode or a flowchart for the elite perturbation mechanism (Equations 10–14) to enhance clarity.
6. Present a time complexity analysis for the improved algorithm.
7. Supplement the experiments with statistical significance tests (e.g., Wilcoxon or Friedman test).
8. The values for weights w1–w4 in the path cost function (20–26) are not explained.

Author Response

Dear Editors and Reviewers:

Thank you for your letter and the comments concerning our manuscript, which we submitted to Biomimetics. Those comments are all valuable and helpful for revising and improving our paper and the essential guiding significance of our research. We have studied the comments carefully and have made corrections, which we hope to meet with approval.

In the revised manuscript, we have used blue text to indicate the modifications made based on your suggestions. The comments made by the reviewing experts and the corresponding responses are summarized below:

Response to Reviewer 2 Comments：

The introduction provides a rather general description of the limitations of the RBMO algorithm. Please clearly analyze the strengths and weaknesses of the original algorithm and explain the rationale for selecting it for this study.

Response : We sincerely appreciate the valuable comments. In the revised manuscript, we have addressed this concern as follows:

Advantages of RBMO (Introduction, Page 2, Lines 76–81): We explicitly highlight the cooperative population-based search mechanism of RBMO, its effective information sharing among individuals, and its demonstrated performance in continuous optimization problems. We also explain why these characteristics are particularly relevant for three-dimensional UAV path planning, where the optimization variables are continuous and the objective landscape is highly nonlinear due to terrain and obstacle distribution.

Limitations of RBMO ( Introduction, Page, Lines 82–88): We provide a detailed analysis of the canonical RBMO’s limitations, including insufficient early-stage population diversity, lack of adaptive control over search dynamics, and susceptibility to premature convergence. We further discuss how these limitations may lead to infeasible or suboptimal UAV trajectories under complex constraints such as terrain variations, obstacles, no-fly zones, and kinematic limitations.
Justification for Selecting RBMO as the Research Foundation(Introduction, Page 2, Lines 89–95): Despite its limitations, RBMO was deliberately chosen due to its flexible structure, clear behavioral mechanisms, and proven effectiveness in continuous optimization. Compared with more rigid frameworks, RBMO provides a suitable baseline for systematic enhancement, allowing targeted improvements to be incorporated while retaining its inherent cooperative search advantages.
2. The analysis of the latest swarm intelligence algorithms is insufficient. Recent research from the past three years should be supplemented and analyzed.

Response : Thank you for your suggestion. Reviewing the latest swarm intelligence algorithms can reflect the advancement and effectiveness of the algorithms we have selected, which is very helpful to us. We have supplemented and analyzed the research literature from the past three years [11-13] in the manuscript. ( Page 2, lines 64-71)
3. The origin of the circular mapping parameter in Equation (7) is unclear. Provide the reason and basis for setting this parameter to 0.2.

Response : We have added explanations for the roles of a and k as per the reviewers' suggestions, as well as the reasons for choosing a = 0.2 and k = 0.5. Additionally, we have included three references [18-20] to provide theoretical basis for the values of a and k. (Page 4, Lines 159–170)
4. Provide justification for the fixed coefficient of 0.62 in the dynamic ε adjustment formula (Equation 8).

Response : We think this is an excellent suggestion. We have conducted a sensitivity analysis experiment on parameter c to explain the basis of the coefficient 0.62 (Figure 2，Page 6, Lines 196–206)
5. Include pseudocode or a flowchart for the elite perturbation mechanism (Equations 10–14) to enhance clarity.

Response : Thank you for your reminder. Flowcharts can visually reflect the process of algorithms and better help readers understand. We have added a flowchart of the elite mechanism to the manuscript. (Figure 3, Page 8, Lines 267–273)
6. Present a time complexity analysis for the improved algorithm.

Response : Time complexity is an important indicator for evaluating the performance of an algorithm and can directly reflect the effect of the algorithm. We have added Section 3.6 to analyze the time complexity for the improved algorithm. (Page 11, Lines 315–330)

Supplement the experiments with statistical significance tests (e.g., Wilcoxon or Friedman test).

Response : Thank you very much for your valuable suggestion. In response, we have incorporated the Wilcoxon rank-sum test into Section 4.3 of the revised manuscript to provide a more rigorous statistical comparison of the algorithmic performance. (Page 16, Lines 412–426)

The values for weights w1–w4 in the path cost function (20–26) are not explained.

Response : Thank you for your constructive comment regarding the setting of the weight coefficients ?1–?4. We have provided a detailed explanation in the revised manuscript. Specifically（Page 23, Lines 603-619）：

Threat cost weight ?4=5: The threat cost is the most critical safety constraint in UAV path planning. Entering a threat area may result in mission failure. Therefore, we assign the highest weight to ensure that the algorithm always prioritizes path safety.
Distance weight ?1=1: Path length directly affects UAV energy consumption and flight time, making it the second most important factor after safety. A medium-to-high weight encourages the search for shorter paths without compromising safety.
Height weight ?3=0.5 and angle weight ?2=0.05: The height and angle costs mainly influence path smoothness and UAV maneuverability. Height is assigned a medium weight to maintain safe altitude while not overly penalizing path efficiency, whereas angle is given a smaller weight to suppress unnecessary large turns and avoid excessive curvature penalties. This also accounts for the fact that height costs typically have larger magnitudes; a smaller weight ensures balanced optimization without dominating distance and threat considerations.

These explanations have been added in detail in the manuscript to clarify the rationale behind the chosen weight settings.

Thank you again for your review work and look forward to your feedback.

Yours sincerely

Xu Yong,Xue Ning ,Zhang Yi

14 December 2025

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents an improved metaheuristic optimization algorithm termed the Circle-Mapping Transition and Weighted Red-Billed Blue Magpie Optimizer (CTWRBMO) to address the challenges of 3D UAV path planning in complex environments. The authors introduce four modifications to the standard Red-Billed Blue Magpie Optimization (RBMO) algorithm: (1) Circle chaotic mapping for population initialization, (2) a dynamic adjustment strategy for the ε parameter to balance exploration/exploitation, (3) a nonlinear dynamic weighting factor wd to control disturbance amplitude, and (4) an elite perturbation and retention strategy to improve local search and avoid premature convergence. The proposed CTWRBMO is benchmarked against several other MAs on the CEC2020 and a simulated 3D UAV path planning problem. The results suggest that CTWRBMO offers superior performance in terms of convergence speed, solution accuracy, and stability.
Here are my concerns.
1. Why authors choose RBMO as the basis of this research? There are numerous MAs in recent decades, the motivation of this paper is unclear.
2. Authors integrate four strategies into RBMO to form CTWRBMO. Authors need to know that more strategies do not mean higher novelty of the proposed method. Besides, the ablation experiments should be added to investigate the performance of the integrated strategies.
3. Authors use PSO, HHO, GTO, AVOA, and RBMO as competitor algorithms, but why authors choose these? State-of-the-art optimizers such as L-SHADE should be involved as the competitor.
4. Why PSO, GTO, and WOA are missed in the 3D UAV path planning problem?
5. The authors introduce new hyperparameters, such as the initial value of ε = 0.62 (Eq 8). This value appears arbitrary. A sensitivity analysis is needed to show how the algorithm's performance is affected by changes to these new parameters.
Overall, this paper has many flaws that should be addressed carefully.

Author Response

Dear Editors and Reviewers:

In the revised manuscript, we have used green text to indicate the modifications made based on your suggestions. The comments made by the reviewing experts and the corresponding responses are summarized below:

Response to Reviewer 3 Comments：

Why authors choose RBMO as the basis of this research? There are numerous MAs in recent decades, the motivation of this paper is unclear.

Response: We thank the reviewer for this insightful comment. Although many optimization algorithms have been proposed, RBMO was selected due to its simple and flexible framework and its effectiveness in continuous optimization, which are well suited to the continuous and constrained nature of three-dimensional UAV path planning. Moreover, RBMO exhibits certain limitations in complex constrained environments, making it an appropriate baseline for targeted improvements and for clearly demonstrating the effectiveness of the proposed strategies. The motivation for choosing RBMO has been clarified in the revised manuscript. (pages 2, Lines 89-95).

As Reviewer 2 provided a similar comment, the corresponding modification in the manuscript has been highlighted in blue to kindly indicate to Reviewer 3 that this suggestion has also been addressed.

Authors integrate four strategies into RBMO to form CTWRBMO. Authors need to know that more strategies do not mean higher novelty of the proposed method. Besides, the ablation experiments should be added to investigate the performance of the integrated strategies.

Response: We fully agree with your point of view that integrating multiple strategies does not necessarily imply greater innovation. At the same time, a clear and systematic analysis of the specific contributions of each strategy is necessary.

In response, we have added a new ablation study in Section 4.4 to investigate the individual effects of the four strategies integrated into CTWRBMO. Specifically, four RBMO variants (IRBMO1–IRBMO4) were constructed, each incorporating only one strategy: circular chaotic initialization, nonlinear dynamic weighting, dynamic ε adjustment, and elite disturbance with elite retention, respectively.

All variants were evaluated on six representative CEC2020 benchmark functions under identical parameter settings, with 30 independent runs for each algorithm. The comparative results demonstrate that each strategy contributes to performance improvement from different aspects, and their integration in CTWRBMO leads to complementary and enhanced optimization performance.

The corresponding experimental results and analysis have been added and discussed in Section 4.4. (Page 17, Lines 428–482)
3. Authors use PSO, HHO, GTO, AVOA, and RBMO as competitor algorithms, but why authors choose these? State-of-the-art optimizers such as L-SHADE should be involved as the competitor.

Response: We sincerely thank the reviewer for this valuable suggestion. In the revised manuscript, we further clarify the rationale for selecting the comparison algorithms and include the state-of-the-art optimizer L-SHADE as an additional competitor. The specific reasons for choosing the above algorithms as benchmarks have been explicitly stated in the manuscript. Specifically, PSO and GWO are selected as representative classical swarm intelligence algorithms, while HHO, WOA, GTO, and AVOA are included as bio-inspired optimization algorithms that have demonstrated strong performance on continuous optimization problems in recent years. RBMO is chosen as the baseline algorithm to highlight the improvements introduced by CTWRBMO; meanwhile, the inclusion of L-SHADE, a state-of-the-art differential evolution algorithm, enables a more comprehensive and convincing performance comparison. These revisions have been incorporated into Section 4.1 (Page 11 Lines 333-340). The parameter settings of all algorithms are summarized in Table 2. The updated convergence curves are shown in Figure 4, and the corresponding mean, variance, and best optimization results are reported in Table 4. (Page 12, Lines 363–411)

Why PSO, GTO, and WOA are missed in the 3D UAV path planning problem?

Response: In the three-dimensional UAV path planning experiments, PSO, GTO, and WOA were not included mainly due to feasibility and fairness considerations. Preliminary experiments indicated that, in strongly constrained 3D environments with complex terrain and kinematic constraints, these general-purpose algorithms have difficulty consistently generating feasible paths without introducing additional constraint-handling or path-repair mechanisms. To ensure a fair comparison, no algorithm in this study was equipped with such customized strategies. Therefore, representative classical and recently proposed algorithms more suitable for constrained continuous optimization were selected for comparison. The rationale for this algorithm selection has been clarified in the revised manuscript. We sincerely thank the reviewer for this valuable suggestion. (See Details on Page 26, Lines 708–720)

The authors introduce new hyperparameters, such as the initial value of ε = 0.62 (Eq 8). This value appears arbitrary. A sensitivity analysis is needed to show how the algorithm's performance is affected by changes to these new parameters.
Response :We sincerely thank the reviewer for this valuable comment. In response, we have conducted a sensitivity analysis on the initial coefficient in the dynamic adjustment, which controls the global exploration intensity of the CTWRBMO algorithm in its early stage. For each candidate value of , the algorithm was independently run 30 times using the CEC2020 benchmark function F2 to evaluate its impact on convergence behavior and overall performance. (Page 6, Lines 195–205)

Thank you again for your review work. Your suggestions have made our manuscript more rigorous and standardized, and we attach great importance to them. We have carefully revised the manuscript in response to your suggestions. Look forward to your feedback.

Yours sincerely

Xu Yong,Xue Ning ,Zhang Yi

16 December 2025

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Authors have addressed all concerns in the revised version, it can be accepted for publication.