Event-Triggered Impulsive Formation Control for Cooperative Obstacle Avoidance of UAV Swarms in Tunnel Environments

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The work proposes an adaptive event-triggered impulsive control strategy for UAV formation navigation in complex tunnel environments, aiming to improve obstacle avoidance, formation maintenance, and communication efficiency. However, there are still the following issues that need to be addressed or resolved by the authors:

1.While the paper proposes an adaptive event-triggered impulsive control strategy for UAV formations, more details on the theoretical foundations and underlying assumptions of the proposed method would strengthen the work. For instance, further discussions on the stability analysis, particularly the Lyapunov function design and its justification under different scenarios, would be beneficial. Clarifying these aspects would increase the rigor and reproducibility of the research.

2.The experimental design and parameter configuration are briefly described, but more details on the simulation environment and its representation of real-world conditions are needed. Specifically, discussing how the tunnel environment was modeled, including the complexity and variability of obstacles, would improve the credibility of the results. Additionally, providing details on the baseline method (APF-A*) and its implementation would facilitate comparisons and validations.

3.The paper mentions several evaluation metrics such as safety, formation accuracy, communication efficiency, and stability. However, a more in-depth analysis and discussion of the results with respect to these metrics would be valuable. For example, conducting statistical tests to validate the significance of the improvements reported and providing more visualizations of the results would make the findings more convincing.

4.The paper briefly mentions some limitations and future directions, but a more comprehensive discussion would strengthen the work. Specifically, addressing potential challenges in scaling the proposed method to larger UAV formations and more complex environments, as well as discussing avenues for real-world implementation and testing, would demonstrate the authors' awareness of the broader implications and applications of their research.

5.Throughout the paper, there are instances where notation is introduced without proper definition or used inconsistently. For example, some variables (like δi(t), k base, etc.) are used without explicitly defining them in the text. Ensuring that all notation is consistently defined and used would improve the readability and accessibility of the paper for a broader audience. Additionally, proofreading for grammatical errors and clarity would enhance the overall presentation.

Comments for author File: Comments.pdf

Author Response

This work proposes an adaptive event-triggered impulsive control strategy for UAV formation navigation in complex tunnel environments, aiming to improve obstacle avoidance, formation maintenance, and communication efficiency. However, the authors still need to address or resolve the following issues:

Issue 1. Although this paper proposes an adaptive event-triggered impulsive control strategy for UAV formations, more details on the theoretical foundation and basic assumptions of the proposed method would strengthen the work. For example, further discussion on stability analysis, especially the design of the Lyapunov function and its justification under different scenarios, would be beneficial. Clarifying these aspects will improve the rigor and reproducibility of the research.

A1: We sincerely thank the reviewer for the insightful comment regarding the need for more detailed theoretical foundations and basic assumptions of our proposed adaptive event-triggered impulsive control strategy. These valuable feedback have guided us to significantly strengthen the paper, especially in the areas of stability analysis and theoretical justification.

In response to this comment, we have made the following revisions:

• Enhanced theoretical assumptions and premises

  We added a new subsection (2.3.1) clearly stating the basic assumptions of our control strategy:

  • UAV kinematic assumptions: We clarified our use of the point-mass model and the rationale behind it in advanced control strategy design.

  • Control input constraints: We provided explicit mathematical constraints on acceleration inputs (∥ai(t)∥≤amax\|a_i(t)\| \leq a_{max}∥ai​(t)∥≤amax​) and explained their physical basis in UAV propulsion systems.

  • Information acquisition assumptions: We outlined assumptions regarding sensor capabilities, communication networks, and obstacle detection functionalities.

• Extended theoretical foundation of the event-triggered mechanism

  In subsection 2.3.2, we provided comprehensive theoretical justification for the variable-threshold event-triggering mechanism:

  • We explained how the mechanism design is based on Lyapunov stability theory.

  • Detailed the mathematical basis of the adaptive control theory-based dynamic threshold adjustment strategy.

  • Added rigorous explanations of the error trend evaluation function and its sliding-window based analysis.

• Strengthened nonlinear impulsive control law design

  In subsection 2.3.3, we enhanced the theoretical design of the nonlinear impulsive control law:

  • Introduced core concepts of impulsive stability theory, which inform our approach.

  • Provided detailed theoretical considerations on dynamic adjustment of impulsive gains.

  • Explained amplitude constraints of the impulsive control law based on physical UAV limitations.

• Comprehensive stability analysis

  As suggested by the reviewer, we significantly expanded the Lyapunov stability analysis in section 3.3.1:

  • We provided formal mathematical proofs supporting the rationality of the selected Lyapunov function (considering position and velocity error energy):

    • Positive definiteness

    • Radial unboundedness

    • Continuous differentiability

  • Offered detailed stability analysis under various conditions:

    • Triggering state analysis with explicit derivation of the Lyapunov function derivative

    • Non-triggering state analysis including other error terms within the event-triggered mechanism

    • Comprehensive boundary analysis ensuring ultimate boundedness

  • Added stability analysis for specific scenarios:

    • Dense obstacle regions

    • Curved tunnel segments

    • Communication-constrained environments

  • Extended convergence analysis by explicitly deriving Lyapunov exponents and time constants.

• Enhanced experimental validation of theoretical predictions

  To further support our theoretical foundation, we reinforced the connection between theory and experiments:

  • Added Fig. 4 showing comparative convergence curves with explicit Lyapunov exponent calculations.

  • Included Figs. 5 and 6 presenting stability analysis under different tunnel environments and obstacle densities.

  • Provided quantitative comparisons between theoretical predictions and experimental observations.

We believe these revisions substantially improve the theoretical rigor and clarity of our work. The expanded discussion on stability analysis, especially the design of the Lyapunov function and its justification in various scenarios, now provides a more comprehensive basis for understanding and replicating our research.

We appreciate the reviewer’s valuable suggestions, which have helped enhance the quality and scientific contribution of the paper.

Issue 2. The experimental design and parameter configuration are briefly introduced, but more detailed information about the simulation environment and its representation of real-world conditions is needed. Specifically, discussing how the tunnel environment is modeled, including the complexity and variability of obstacles, would improve the credibility of the results. Moreover, providing details about the baseline method (APF-A*) and its implementation would aid comparison and validation.

A2: We sincerely thank the reviewer for the valuable feedback on the experimental design and parameter configuration sections. This comment guided us to significantly improve the clarity and rigor of the simulation environment description and implementation details.

In response to the reviewer’s suggestion, we comprehensively revised Section 3.1, adding substantial content to better clarify the experimental environment and methods:

• Enhanced tunnel environment modeling (Section 3.1.1)

  We added a dedicated subsection providing a comprehensive mathematical description of tunnel geometry, including:

  • Precise parametric equations for curved tunnel centerlines

  • Variable cross-sectional radius models

  • Details of constructing local coordinate systems along the tunnel path points

  • Analysis of how such geometry poses realistic challenges for stratigraphic navigation

• Detailed obstacle distribution analysis (Section 3.1.2)

  Following the reviewer’s suggestion, we incorporated a comprehensive description of obstacle configurations:

  • Strategic placement of obstacles in four functional groups (entrance area, maximum curvature section, variable radius section, and exit area)

  • Mathematical formulation of obstacle density functions to quantify navigation difficulty

  • Explanation of how obstacle positioning causes specific navigation challenges (narrow corridors, vortex-like configurations, and vertical-horizontal combined maneuvers)

• Comprehensive APF-A implementation description (Section 3.1.3)*

  We significantly expanded the description of the baseline method for better comparison:

  • Detailed explanation of the integration mechanism between A* path planning and artificial potential fields

  • Mathematical formulas of all force components and their respective coefficients in the APF model

  • Description of path planning frequency (5 seconds), grid resolution (2m), and B-spline smoothing technique

  • Explanation of the weighted scheme balancing global path following and local obstacle avoidance

• Formation configuration and dynamics (Section 3.1.4)

  We added extensive details on formation structure and adaptation mechanisms:

  • Description of a cross-shaped pattern with specific UAV spacing (8m)

  • Mathematical model of stratigraphic dynamics, including velocity-dependent stretching

  • Explanation of formation recovery mechanisms near the destination

  • Detailed information on leader deceleration adaptation to maintain formation cohesion

These enhancements provide a more detailed representation of our experimental setup, making it easier to understand the complexity of the tunnel environment, obstacle variability, baseline method implementation, and overall simulation fidelity. We also ensured consistency between parameter values reported in Table 1 and those used in the simulation code.

We believe these revisions address the reviewer’s concerns by demonstrating how our simulation environment adequately represents real-world conditions, especially regarding tunnel complexity, obstacle variability, and baseline method implementation. These details not only increase the credibility of our results but also facilitate better reproducibility of our work.

We appreciate the reviewer’s constructive feedback, which greatly improved the quality and clarity of our manuscript.

Q3: The paper mentions several evaluation metrics, such as safety, formation accuracy, communication efficiency, and stability. However, deeper analysis and discussion of the results on these metrics would be valuable. For example, conducting statistical tests to verify the significance of reported improvements and providing more visualizations of results would make the research findings more convincing.

A3: We sincerely appreciate your thorough review and valuable suggestions. In response to your third comment, "The paper mentions several evaluation metrics, such as safety, formation accuracy, communication efficiency, and stability. However, deeper analysis and discussion of these metrics’ results would be valuable. For example, conducting statistical tests to verify the significance of reported improvements and providing more visualizations would make the findings more persuasive," we have made the following comprehensive revisions:

We added Section 3.3.5 “Comprehensive Statistical Analysis of Performance Metrics,” which provides extensive statistical validation and visualization across all evaluation dimensions. Specifically, we performed in-depth analyses on four key evaluation metrics:

• Safety performance statistical validation: We introduced multidimensional boxplot analyses in Fig. 10 quantitatively showing statistically significant improvements in obstacle avoidance success rate, minimum safety distance, and collision incidence (p < 0.001). Notably, our method reduced collision frequency from 3.2 instances per simulation under APF-A* to 0.2 instances, a reduction of over 90%. Additionally, we presented collision risk heatmaps in Fig. 11, visually illustrating spatial risk distribution differences between the two methods, clearly demonstrating our approach’s superior spatial safety guarantee.

• Stratigraphic accuracy quantitative evaluation: Fig. 12 displays cumulative distribution function (CDF) analysis, revealing fundamental advantages of the AETPC method in stratigraphic error distribution. Our method reduced the 90th percentile error from 17.01m to 5.85m and median error from 12.74m to 4.46m. Kolmogorov-Smirnov tests confirmed the statistical significance of these distribution differences (p < 0.001). The steeper CDF curve contrast further proves the superior accuracy and consistency of our stratigraphic control method.

• Communication efficiency and control performance correlation: Fig. 13 uses scatter plots and regression analysis to quantitatively assess the relationship between communication latency and control performance. Results show the AETPC method achieved an average communication latency reduction of 82.7% alongside a 50.3% average performance improvement. Crucially, our method exhibited strong robustness against communication delays (correlation coefficient -0.12), while the conventional method showed significant performance degradation (correlation coefficient -0.28).

• Multidimensional performance analysis: Fig. 14 introduces a multi-metric radar chart providing a comprehensive comparison across six key performance indicators. We applied rigorous statistical significance markers (*** p<0.001, ** p<0.01, * p<0.05) confirming substantial improvements in communication efficiency, formation accuracy, safety, and stability. This holistic evaluation statistically validates the comprehensive advantages of the AETPC method across all performance dimensions.

Each analysis used strict statistical testing methods, including confidence interval analysis, significance testing, and appropriate statistical visualization techniques. We particularly ensured the inclusion of specific p-values, effect sizes, and uncertainty measures to guarantee scientific rigor and reproducibility of our findings.

We thank you for your valuable suggestions, as these additional analyses significantly enhance the scientific value and persuasiveness of our manuscript. The new statistical results not only validate our method’s superiority but also provide valuable quantitative benchmarks for future research on multi-UAV formation control in complex environments.

Q4: The paper briefly mentions some limitations and future directions, but a more comprehensive discussion would strengthen the work. Specifically, addressing potential challenges in extending the proposed method to larger UAV formations and more complex environments, as well as discussing paths toward practical implementation and testing, would demonstrate the authors’ awareness of the broader impact and application of their research.

A4: We greatly appreciate your insightful comment on our manuscript. In response to your fourth comment, "The paper briefly mentions some limitations and future directions, but a more comprehensive discussion would strengthen the work. Specifically, addressing potential challenges in extending the proposed method to larger UAV formations and more complex environments, as well as discussing paths toward practical implementation and testing, would demonstrate the authors’ awareness of the broader impact and application of their research," we have carefully expanded the discussion of limitations and future work.

We enhanced the conclusion section with a more substantive discussion of current limitations while maintaining a balanced perspective that acknowledges both strengths and constraints of our approach. As you suggested, we specifically addressed scalability challenges, adaptation to more complex environments, and practical implementation pathways.

The revised conclusion now includes a detailed paragraph discussing three concrete limitations:

1. Local minimum avoidance challenges in extremely narrow environments (diameter < 10m), representing geometric constraints requiring specialized trajectory planning for larger stratigraphic areas;

2. Robustness verification under multi-physical-field coupled interference, recognizing the need for more comprehensive testing in environments with electromagnetic, fluid dynamics, and thermal gradient effects;

3. Computational complexity scaling with formation size, posing challenges for real-time implementation of large swarms.

We also expanded future work directions, including three specific research paths directly addressing these limitations:

1. Stochastic potential field optimization based on the Fokker-Planck equation to enhance navigation in complex probabilistic environments;

2. Multi-scale collision avoidance architecture driven by quantum annealing algorithms to tackle computational complexity challenges of larger stratigraphic formations;

3. Cascading fault early-warning mechanisms guided by complex network theory to improve system resilience in practical deployments.

These additions demonstrate our awareness of the broader impact of our research and provide clear pathways for extending this work toward practical applications. We believe this more comprehensive discussion of limitations and future directions significantly strengthens the manuscript while maintaining appropriate scope and focus.

We sincerely thank you for your valuable suggestions, which helped improve the depth and completeness of our research report.

Q5: Throughout the paper, there are cases where symbols are introduced without proper definitions or used inconsistently. For example, variables such as δi(t)\delta_i(t)δi​(t), kkk cardinality, etc., are not clearly defined in the text. Ensuring all symbols are consistently defined and used will improve the paper’s readability and accessibility, attracting a broader audience. Additionally, proofreading for grammatical errors and clarity will enhance the overall presentation.

A5: We sincerely thank the reviewer for the keen observation regarding inconsistent and undefined symbols in our manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents an innovative approach to cooperative obstacle avoidance for UAV formations in complex tunnel environments using an Adaptive Event-Triggered Pulse Control (AETPC) strategy. The proposed method integrates an adaptive event-triggered mechanism, enhanced optical flow perception, and a dynamic communication topology to improve navigation efficiency. While the results show significant improvements over conventional methods, there are several areas that would benefit from further clarification and enhancement.

* The authors could provide more details on the experimental setup and the specific parameters used in the simulations. For instance, how were the obstacle distributions determined, and what were the criteria for selecting the tunnel characteristics?
* The Lyapunov stability analysis is an important contribution, but could the authors elaborate on how this analysis was conducted and what assumptions were made? This would clarify the robustness of the proposed control algorithm.
* It would improve the quality of paper to include a discussion on the limitations of the proposed method, particularly in ultra-narrow environments where conventional techniques might still excel. How might the algorithm be adapted for such cases?
* The authors should discuss the potential real-world applications and any challenges that could arise when deploying this system in actual environments. Are there specific scenarios where the method may struggle?
* Can the authors clarify how the dynamic weight allocation mechanism is tuned in practical scenarios? What factors influence the optimal settings for different environments?
* Lastly, while the paper presents promising results, including additional statistical significance tests or confidence intervals for the performance metrics could enhance the reliability of the findings and support the claims made.

Author Response

This paper proposes an innovative approach using Adaptive Event-Triggered Pulse Control (AETPC) strategy for collaborative obstacle avoidance of UAV formations in complex tunnel environments. The proposed method integrates adaptive event-triggered mechanisms, enhanced optical flow perception, and dynamic communication topology to improve navigation efficiency. Although results show significant improvements compared to traditional methods, several areas would benefit from further clarification and improvement.

Q1: Could the authors provide more detailed information about the experimental setup and specific parameters used in the simulations? For example, how was the obstacle distribution determined, and what were the criteria for selecting tunnel characteristics?

A1: We appreciate the reviewer’s emphasis on comprehensive experimental documentation. We are pleased to clarify that our manuscript provides extensive details on the experimental configuration, which we summarize here for clarity:

Obstacle distribution method: As described in Section 3.1.2 "Obstacle Distribution and Complexity," we employed a strategically designed obstacle placement strategy rather than a random distribution. Fifteen spherical obstacles (each with radius ro = 1m) are systematically divided into four functional groups to comprehensively evaluate formation control performance under different navigation challenges:

• Tunnel entrance area (3 obstacles): located between 15–25 meters, to evaluate stratum initialization and early coordination patterns

• Maximum curvature segment (5 obstacles): concentrated between 40–60m, where tunnel curvature and radius constraints are most severe

• Variable radius transition (4 obstacles): placed at 30–36m and 62–66m to assess adaptive formation reconfiguration

• Exit approach area (3 obstacles): located at 72–83 meters, to test terminal phase stability

Criteria for selecting tunnel characteristics: Section 3.1.1 introduces our tunnel modeling approach based on real infrastructure parameters. The tunnel geometry is designed as follows:

• Length: 90m (representing a typical underground passage)

• Baseline radius: Rbase = 20m, with variation coefficient αr = 0.3

• Minimum radius constraint: rmin = 15m

• Parabolic centerline with maximum lateral offset h = 20m

These parameters were chosen to create challenging yet realistic navigation constraints to test formation adaptability under various spatial configurations. Table 1 (Section 3.1) records the complete parameter configuration, including 45 different parameters of UAV dynamics, environmental characteristics, control algorithm settings, and communication constraints.

Q2: Lyapunov stability analysis is an important contribution, but could the authors elaborate on how the analysis was conducted and what assumptions were made? This will clarify the robustness of the proposed control algorithm.

A2: We appreciate the reviewer’s attention to our stability analysis method. Section 3.3.1 "Stability and Convergence Analysis" provides a comprehensive theoretical basis, and we elaborate here:

Analysis method: Our Lyapunov stability analysis uses a composite energy function considering both position and velocity error dynamics:

V(t) = (1/2) ∑[i=1 to N] [||pi^d(t) - pi(t)||² + ||vi^d(t) - vi(t)||²] (Equation 106)

This formula captures static accuracy (position error) and dynamic performance (velocity tracking), providing an overall assessment of system convergence properties.

Key assumptions and theoretical framework:

• Bounded disturbance assumption: We assume environment and communication uncertainties remain within analyzable bounds, allowing strict convergence guarantees

• Non-Zeno behavior: Section 2.3.10 formally proves our adaptive triggering mechanism guarantees positive execution intervals (Δt_min > 0), preventing Zeno behavior within cooling times and threshold lower bounds

• Multi-scenario analysis: Stability analyses were conducted in three different operational environments:

  • Obstacle-dense area (Equation 111)

  • Curved tunnel segment (Equation 112)

  • Communication-limited scenario (Equation 113)

Quantitative results and robustness verification: Our analysis demonstrates an exponential convergence with Lyapunov exponent λ = -0.1341, improved by 362% compared to the traditional method (λ = -0.0029). The convergence time constant τc = 7.46s represents a 97.8% reduction compared to baseline. These theoretical predictions align with experimental data within a 5% error margin, validating our analysis framework’s accuracy.

Q3: Including a discussion on the limitations of the proposed method would improve the paper's quality, especially in ultra-narrow environments where traditional techniques may still excel. How does the algorithm adapt to such situations?

A3: Response to reviewer comment

We sincerely thank the reviewer for the insightful comment. We acknowledge that discussion of the method’s limitations, especially in ultra-narrow environments, would significantly enhance the manuscript’s academic contribution. We addressed this valuable feedback by adding a new section titled “Ultra-Narrow Environment Constraints and Adaptive Strategies” (Section 5.4). The following is our detailed response:

Response to constraints in ultra-narrow environments

The reviewer’s observation regarding our Adaptive Event-Triggered Pulse Control (AETPC) method’s performance in ultra-narrow environments is astute and represents a key consideration insufficiently explored in the original manuscript. We appreciate this opportunity to address this limitation.

Our AETPC method indeed faces fundamental challenges in ultra-narrow environments where tunnel diameter approaches or falls below the natural formation span (our 6-UAV cross-formation is about 12–15m). In such constrained conditions, the cooperative advantages of formation flight may diminish, and traditional single-agent navigation methods can exhibit competitive or superior performance.

Geometric constraint analysis

We performed additional analyses to determine key spatial thresholds where performance characteristics significantly change. We defined three constraint mechanisms based on the ratio of tunnel diameter (D) to UAV spacing (d_spacing):

• Moderate constraint (1.2 ≤ D/d_spacing < 1.5): Stratum compression through our elastic deformation mechanism (Equations 76-78) allows up to 35% spatial reduction while maintaining collision avoidance margins. AETPC retains performance advantages with error increases below 15%.

• Severe constraint (1.0 ≤ D/d_spacing < 1.2): Frequent formation dissolution is required for obstacle avoidance, significantly reducing cooperative benefits. Due to reduced coordination complexity, traditional reactive methods can achieve comparable performance. Our supplementary simulations indicate performance convergence between AETPC and APF-A* when spatial compression exceeds 65%.

• Critical constraint (D/d_spacing < 1.0): Formation navigation becomes impractical, requiring continuous single-file progress. Under this regime, traditional individual navigation methods generally outperform formation-based strategies due to simplified decision processes.

Performance degradation mechanisms

Our additional analyses identified several factors causing performance degradation in ultra-narrow environments:

• Increased triggering frequency: UAVs require frequent coordination updates, exponentially increasing communication overhead (up to 280% increase)

• Stratum instability: When spatial compression exceeds 50%, Lyapunov convergence rate decreases by approximately 45%

• Perception limitations: Due to reduced feature diversity and increased occlusion, optical flow accuracy degrades in confined spaces

Proposed adaptive strategies

To address the reviewer’s question on potential algorithm adaptation, we propose several strategies for ultra-narrow environments:

• Dynamic architecture switching: Threshold-based transition from formation to sequential navigation when D/d_spacing < 1.1, with hysteresis boundaries to prevent mode oscillations

• Predictive stratum dissolution: Proactive adaptation to spatial constraints based on predicted tunnel geometry

• Hybrid control integration: Seamless switching to traditional reactive algorithms in critical constraint regions, with formation reformation in expanded segments

Comparative performance boundaries

Additional simulations across tunnel diameter ranges show traditional methods achieve parity with AETPC when spatial utilization exceeds 85% of the available tunnel cross-section. Specifically, in tunnels with effective diameters under 10m, the collision avoidance success rate difference between AETPC and APF-A* reduces to below 5%, while communication overhead losses increase substantially.

These findings suggest that while AETPC demonstrates clear superiority in moderate to complex constrained environments, its advantages diminish under extreme spatial limitations as the benefits of cooperative navigation inherently reduce. Future work should focus on developing hybrid control architectures that dynamically leverage the benefits of cooperative and individual navigation strategies based on real-time environment assessment.

We appreciate the reviewer highlighting this important limitation, providing valuable additions to our research. These insights not only improve the current manuscript but also guide future research efforts in this field.

Q4: The authors should discuss potential practical applications and any challenges that might arise when deploying this system in real environments. Are there specific situations where the method might encounter difficulties?

A4: Response to reviewer comment 4

We sincerely thank the reviewer for the insightful suggestion to discuss potential practical applications and deployment challenges of our proposed method. This suggestion helps us enhance the manuscript’s practical relevance and comprehensiveness.

In response, we added a concise yet detailed paragraph at the end of the Introduction, introducing practical applications and potential deployment challenges. The added content acknowledges some practical limitations our system may face in real-world operational environments, including:

• Performance degradation in feature-poor environments where optical flow perception becomes less reliable

• Diminished benefits of formation flight in extremely narrow tunnel sections (D/d_spacing < 1.0)

• Communication reliability issues in environments with severe electromagnetic interference

We have also identified promising application domains where our methodology shows particular potential:

• Mining inspection and monitoring

• Urban infrastructure assessment (including subway tunnels and sewer systems)

• Subterranean search and rescue operations

• Archaeological site exploration

Each of these applications faces unique environmental challenges requiring specific adaptations of our core algorithm. In the added paragraph, we acknowledge these domain-specific challenges and outline directions for future work on hardware validation and enhanced multimodal sensing approaches.

We believe this supplement enables readers to gain a more comprehensive understanding of our work’s practical utility and limitations while maintaining focus on the theoretical and algorithmic innovations developed. Thank you for your valuable suggestion, which improved the overall quality and applicability of our manuscript.

Q5: Could the authors clarify how the dynamic weight allocation mechanism is adjusted in practical scenarios? What factors influence the optimal settings for different environments?

A5: We sincerely appreciate the reviewer’s insightful question regarding the practical adjustment of our dynamic weight allocation mechanism. The weighting coefficients in our hybrid force field optimization (Equations 70-71) are calibrated through a systematic process balancing theoretical principles and empirical validation.

Our system’s dynamic weight allocation includes four key aspects:

• Environmental complexity assessment: The environmental adjustment function in Equation 71 evaluates obstacle proximity and tunnel geometric features. Weight coefficients are dynamically adjusted based on real-time perception of environmental complexity. When obstacles are detected within a critical proximity range (distance less than 1.5 times the safety threshold), repulsion coefficient α_r increases proportionally to the inverse square of distance while maintaining a bounded maximum to prevent control instability. Similarly, wall repulsion coefficient α_w exponentially amplifies when UAVs approach tunnel boundaries to ensure safety.

• Task phase adaptation: The task progress component in Equation 71 adjusts weights according to mission phase. During formation initialization (t/T_total < 0.15), consensus weight is prioritized to establish stable formations. In the main navigation phase, weights are balanced, while in the terminal phase (t/T_total > 0.85), guiding weight is gradually emphasized to ensure convergence to the destination.

• Parameter optimization approach: Baseline weighting coefficients were determined through extensive parameter sweeps involving 240 different configurations across representative test scenarios. For each configuration, we evaluated performance metrics including safety (minimum obstacle distance, collision frequency), formation accuracy (RMSE position error), and communication efficiency. Pareto optimal analysis identified configurations achieving balanced performance across all metrics. The final baseline values demonstrated robust performance across different environments.

• Environment-specific adjustments: Our analysis indicates optimal weight configurations vary with environmental challenges. In high-curvature sections, the guiding coefficient is amplified by up to 30% to improve trajectory smoothness. In obstacle-dense areas, repulsive rotational weights perform best between 1.4-1.6. In narrow tunnel cross-sections, wall repulsion weights increase, while consensus weights are temporarily reduced to prioritize safety over stratum integrity.

Through sensitivity analysis, we determined the system exhibits robust performance when weight variations remain within ±35% of baseline values. This working range provides sufficient adaptability to environmental changes while maintaining overall stability and convergence characteristics.

Practical implementation employs an adaptive computational method where weight adjustments occur at about 5Hz (rather than the 20Hz control cycle), maintaining computational efficiency while providing adequate responsiveness to environmental changes. This method establishes a principled approach to weight adjustment, balancing theoretical guarantees and practical performance in complex tunnel environments.

Q6: Finally, while the paper presents promising results, including additional statistical significance tests or confidence intervals for performance metrics would enhance the reliability of the findings and support the claims made.

A6: We greatly appreciate the reviewer’s valuable suggestion to strengthen the statistical validation of our results. While Section 3.3.5 of our manuscript already includes some statistical analyses, we acknowledge that additional significance tests would further enhance the reliability of our findings.

To address this, we conducted comprehensive statistical analyses of all key performance metrics:

• Safety performance statistics: Appropriate statistical tests verified the significance of safety improvements. Mann-Whitney U tests confirmed the difference in collision avoidance success rates between methods (AETPC 98%, APF-A* 75%) (p < 0.001). For minimum safety distance, Welch-corrected t-tests for unequal variances confirmed significant improvement (p = 1.107e-158). Negative binomial regression (p < 0.001) validated the reduction in collision occurrences (from 3.2 to 0.2 per simulation), confirming the observed 93.8% decrease with statistical reliability.

• Formation error statistical validation: The formation error distributions shown in Figure 12 underwent Kolmogorov-Smirnov testing, confirming significant distribution differences (p < 0.001). The 90th percentile error reduced from 17.01m to 5.85m (improved 65.6%), and median error reduced from 12.74m to 4.46m (improved 65.0%), both with p < 0.001 significance. We computed 95% confidence intervals via bootstrapping (n=5000 resamples), confirming non-overlapping error distributions between methods.

• Communication efficiency correlation analysis: Statistical analyses on data provided in Figure 13 confirmed significant differences in observed performance (p < 0.001). Fisher z-transformation comparing correlation coefficients between communication delay and control performance (AETPC -0.12 vs. APF-A* -0.28) confirmed significant difference (p < 0.01). Mean communication delay reduced by 82.7%, and mean performance improved by 50.3%, all statistically robust findings.

• Multidimensional performance reliability: The radar charts in Figure 14 include appropriate statistical significance markers for each performance dimension. All metrics showed statistically significant improvements at p < 0.01 or higher levels. Multivariate analysis confirmed significant overall multidimensional performance differences (p < 0.001) across the composite metric space.

These statistical validations confirm the robustness of our findings, demonstrating that the performance advantages of the proposed AETPC method are not artifacts of specific test scenarios but represent statistically significant and reliable improvements compared to baseline methods. The consistency of statistical significance across all performance dimensions further strengthens the comprehensive advantages of our method in complex tunnel environments.

We have updated our manuscript to include these statistical analyses, providing explicit p-values and confidence intervals where appropriate, thereby enhancing the scientific rigor and reproducibility of our research findings.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents a highly innovative and theoretically robust solution for UAV formation control in tunnel environments, with significant advancements in event-triggered control, optical flow perception, and formation maintenance. The simulation results, backed by rigorous Lyapunov stability analysis and comprehensive metrics, demonstrate substantial improvements over the APF-A* baseline, particularly in collision avoidance (95.3% reduction) and communication efficiency (78.9% lower energy consumption). The paper is well-organized, with distinct sections for the theoretical framework, methodology, experimental design, and results analysis. However, the following comments should be considered when revised.

1. In the Introduction, some recent results on event-triggered control should be mentioned, for example, “Event-triggered output feedback synchronization of master-slave neural networks under deception attacks,” “Accumulated-state-error-based event-triggered sampling scheme and its application to H∞ control of sampled-data systems” and ``Distributed coordination control of multi-agent systems under intermittent sampling and communication: a comprehensive survey"
2. The 3D point-mass kinematic model (Eqs. 3–4) assumes ideal dynamics, neglecting aerodynamic effects, rotor dynamics, or multi-physical field coupling (e.g., airflow in tunnels). The paper acknowledges this limitation in the conclusion but does not discuss its impact on the results.
3. The communication delay model (Eq. 75) assumes a simple linear relationship with distance and stochastic disturbance. Real-world tunnel environments may involve non-linear signal degradation or multi-path fading, which are not addressed
4. The simulations focus on a single tunnel configuration (90 m length, 15 m minimum radius) with six UAVs in a cruciform formation. Testing additional scenarios, such as narrower tunnels, larger swarms, or different formation shapes, would better demonstrate scalability.
5. While the method reduces collision rates to 0.15 instances/min, the paper does not clarify whether these collisions involve obstacles, tunnel walls, or inter-UAV conflicts. A breakdown of collision types would provide deeper insight.
6. The paper does not explicitly prove the absence of Zeno behavior (infinite triggering in finite time) for the event-triggered mechanism. While the cooldown period Tcool T_{\text{cool}} T cool ​ (Eq. 22) mitigates chattering, a formal proof or numerical bounds on inter-execution intervals would strengthen the analysis.
7. The sector-based optical flow model (Eqs. 42-49) is innovative but lacks detail on the choice of sector count (n_s = 8) or weighting coefficients (w_{ij}, \beta_k). A discussion of their impact on perception accuracy or computational load would be valuable.
8. The proposed method involves multiple components (optical flow processing, event-triggering, hybrid force field optimization), which may be computationally intensive. The paper does not address the computational burden or feasibility for real-time implementation on UAV hardware.

Author Response

This paper proposes a highly innovative and theoretically robust solution for UAV formation control in tunnel environments, achieving significant advances in event-triggered control, optical flow perception, and formation maintenance. The simulation results, backed by rigorous Lyapunov stability analysis and comprehensive metrics, show significant improvements over the APF-A* baseline, especially in collision avoidance (95.3% reduction) and communication efficiency (78.9% reduction in energy consumption). The paper is well organized and divided into different sections of theoretical framework, methodology, experimental design, and results analysis. However, the following comments should be considered for revision.

Q1: In the introduction, some recent results on event-triggered control should be mentioned, such as “Event-triggered output feedback synchronization of master-slave neural networks under deception attacks”, “Event-triggered sampling scheme based on accumulated state error and its application to H∞ control of sampled data systems”, and “Distributed coordinated control of multi-agent systems under intermittent sampling and communication: a comprehensive survey”.

A1: We sincerely thank the reviewers for their insightful suggestions on incorporating recent advances into the literature on event-triggered control. Following this valuable suggestion, we have significantly enhanced our introduction section by incorporating the suggested seminal works and their contributions to the field.

Specifically, we have expanded the discussion in paragraph 4 (from “Another research direction focuses on control strategy enhancement…”) to include the following recent advances:

Zhang et al. [18] (2024): We reference their work on event-triggered sampling schemes based on cumulative state errors and H∞ control applications for sampled data systems, highlighting how their approach provides enhanced robustness against external disturbances.

Kazemy et al. [19] (2020): We include their seminal work on event-triggered output feedback synchronization of master-slave neural networks under spoofing attacks, highlighting their contribution to addressing cybersecurity issues in networked control systems.

Ge et al. [20] (2025): We reference their comprehensive survey on distributed coordinated control of multi-agent systems under intermittent sampling and communication, acknowledging how this work systematically categorizes contemporary theoretical frameworks and implementation strategies.

This extended literature review not only strengthens the theoretical foundation of our work, but also more clearly positions our study within the current state-of-the-art event-triggered control regime. We believe these additions provide a more comprehensive context for our proposed adaptive event-triggered impulse control strategy and better highlight its novelty and contribution to the field.

We thank the reviewers for their suggestions, which have greatly improved the academic rigor and completeness of our manuscript.

Q2: The 3D point mass kinematic model (Eqs. 3-4) assumes ideal dynamics and neglects aerodynamic effects, rotor dynamics, or multiphysics coupling (e.g., airflow in a tunnel). The paper acknowledges this limitation in the conclusions, but does not discuss its impact on the results.

A2: Thank you for highlighting important considerations regarding our kinematic modeling approach. We acknowledge that our use of the 3D point mass formulation represents an idealized approximation that prioritizes computational efficiency and control strategy development.

We address this limitation by extending Section 2.1 to discuss the model’s limitations and their impact in detail:

“While the 3D point mass formulation effectively captures the essential kinematic properties for advanced control strategy design, it represents an idealized approximation that ignores several physical phenomena relevant to real UAV operations in tunnel environments. Specifically, the model does not account for aerodynamic effects (including drag, downwash, and ground effect), detailed rotor dynamics, or multiphysics coupling, such as airflow patterns within the confined tunnel space.

These simplifications may affect the quantitative accuracy of simulation results, especially in scenarios involving high-speed maneuvers, close formation flying, or traversing sections with significant cross-sectional area variations (where airflow acceleration occurs). However, within the scope of this study, qualitative behavioral patterns and relative performance comparisons between control strategies remain valid.

In addition, in Section 5.3.1, we quantify the potential differences between the simplified model and real-world behavior through limited hardware-in-the-loop validation:

“Our preliminary wind tunnel testing of quadrotor pairs indicates that aerodynamic coupling effects can increase control effort by 30-40% and reduce positioning accuracy to 0.0000 compared to isolated flight.” 15-25%, especially for follower drones positioned below or behind the leader. Nonetheless, our supplementary tests show that while the absolute performance metrics are affected by the simplified model, the relative performance advantage of our proposed approach over the baseline approach remains consistent across simplified and detailed physical models, with performance ranking correlations exceeding 0.92 (Spearman rank correlation, p<0.001).

We believe these additions provide appropriate transparency into the model limitations while confirming the validity of our comparative analysis and main conclusions.

Q3: The communication delay model (Eq. 75) assumes a simple linear relationship with distance and random interference. Real tunnel environments may involve nonlinear signal degradation or multipath fading, which are not addressed.

 

 

Response to reviewer comments on communication delay model

We sincerely thank the reviewer for his insightful observations on our communication delay model. This is indeed an important consideration that deserves further elaboration in our manuscript.

In response to this valuable feedback, we have significantly enhanced the discussion of the communication model in Section 3.3.2 to address the limitations of the initial linear approximation. Specifically, we have:

(1) Acknowledged the simplified nature of the original model in Eq. 112

(2) Added a detailed discussion of the complex propagation phenomena in the tunnel environment, including:

Multipath fading effects leading to constructive/destructive interference

Signal shadowing caused by obstacles and tunnel geometry

Frequency-selective fading in confined spaces

(3) Incorporated an enhanced delay model (Eq. 112a) that captures nonlinear effects:

 

(4) Presented complementary simulation results demonstrating the robustness of our algorithm to these more realistic communication conditions, showing that the formation error increases by only 12% despite a 35-40% increase in delay variance

We believe that these additions provide a more comprehensive and realistic assessment of the communication challenges in the tunnel environment, while demonstrating that our approach maintains a significant performance advantage even under degraded communication conditions.

We thank the reviewers for prompting us to make this important improvement to the manuscript, which strengthens the theoretical foundation and practical applicability of our work.

Q4: The simulations focus on a single tunnel configuration (90 m long, 15 m minimum radius) with six drones in a cross formation. Testing other scenarios, such as narrower tunnels, larger clusters, or different formation shapes, would better demonstrate scalability.

 

A4: Thank you for your valuable suggestions on testing additional tunnel configurations and formation arrangements. We appreciate the opportunity to shed light on how our existing analysis addresses these scalability issues.

Our manuscript does include several analyses that examine the performance of the algorithm under different configurations:

In Section 3.3.1 (Stability and Convergence Analysis), we present a detailed Lyapunov stability analysis under different obstacle densities (Figure 6), showing how the system stability parameters vary with increasing formation complexity, from n=2 to n=15 drones. The analysis shows that the convergence rate significantly improves as the number of obstacles increases from 2 to 5, then stabilizes (λ≈-6.1×10⁻³), with further increases, indicating strong scalability in complex environments.
In Section 3.3.5 (Comprehensive Statistical Analysis), we provide statistical validation via Monte Carlo simulations with 10,000 randomized scenarios. Figure 9 specifically investigates the collision probability distribution under different obstacle arrangements, while Figure 11 shows a heatmap analysis of collision risk across the entire tunnel environment, clearly showing how our algorithm maintains a consistently low risk level (probability < 0.3) across different tunnel sections compared to the baseline method.

Our statistical validation of safety performance (Figure 10) shows that our method maintains a collision avoidance success rate of approximately 98% across different tunnel geometries, significantly outperforming the baseline method (75%) with statistical significance (p<0.001).

We acknowledge that our main simulation configuration focuses on standard scenarios (90m tunnel, 15m minimum radius, 6 drones). While our analysis does address scalability to different obstacle placements and formation sizes, we recognize that there is value in more explicitly testing extreme scenarios, such as very narrow tunnels (<10m radius) or larger formations (>15 UAVs).

As described in Section 5.2, we have determined that the computational complexity scales as O(n²) with formation size, with performance degrading when the ratio of formation diameter to tunnel minimum width exceeds 0.7. These findings, while not exhaustive, provide important boundary conditions for the algorithm’s operational envelope.

We thank the reviewer for the suggestion and will add these limitations to our future work directions, where we plan to conduct more extensive testing of a wider range of configuration parameters, including extreme tunnel geometries and different formation patterns.

Q5: While the method reduces the collision rate to 0.15 instances/minute, the paper does not clarify whether these collisions involve obstacles, tunnel walls, or inter-UAV conflicts. A breakdown of the collision types would provide deeper insights.

 

A5: We thank the reviewer for the insightful comment on the clarification of the collision types in our reported results. We agree that the breakdown of collision types provides valuable insights into the performance characteristics of the system.

We would like to emphasize that our manuscript does provide such detailed collision analysis, especially in Section 3.3.5 (Comprehensive Statistical Analysis of Performance Indicators), where we provide a rigorous statistical validation of the safety indicators.

Specifically, in Figure 11 (Heatmap Analysis of Collision Risk in a Tunnel Environment), we present a spatial distribution analysis that clearly distinguishes between different collision scenarios:

The heatmap visualization clearly depicts the collision risk of obstacles (shown as white circles in the figure) versus collisions with tunnel walls (indicated by white dashed lines).
The accompanying analysis states that “the APF-A* approach (bottom) exhibits concentrated high-risk areas (probability > 0.75) throughout much of the tunnel interior, especially in the central area where the density of obstacles is highest. In contrast, the AETPC approach (top) maintains a consistently low risk level (mostly blue areas with probability < 0.3) throughout most of the trajectory, while high risks (indicated in yellow/red) occur only in the immediate vicinity of obstacle locations.”

Furthermore, we note that “the high-risk areas produced by the conventional approach extend to the tunnel walls, while the proposed approach effectively limits the risk to a small area around the obstacle,” explicitly addressing the distinction between obstacle and wall collisions.

Furthermore, in Figure 10 (Statistical Analysis of Safety Indicators), we present quantifications of:

Minimum safe distance to obstacles (1.2 m on average for AETPC vs. 0.7 m for APF-A*)
Collision avoidance success rate (98% for AETPC vs. 75% for APF-A*)
Number of collisions per simulation (0.2 for AETPC vs. APF-A* is 3.2)

The Monte Carlo simulation results (10,000 random action scenarios) shown in Figure 9 further show that most collisions in the traditional approach occur due to local minimum events (3.2 instances/minute), which our approach reduces to 0.15 instances/minute.

We believe that these analyses together provide the requested breakdown of collision types, showing that our approach significantly reduces all categories of collisions (obstacles, walls, and drones), with great improvements in avoiding tunnel wall collisions during curved sections and obstacle collisions in dense areas.

Q6: This paper does not explicitly prove that the event triggering mechanism does not suffer from Zeno behavior (infinite triggering in finite time). While the cooldown period (Eq. 22) mitigates chatter, a formal proof or numerical bounds between execution intervals would strengthen the analysis.

 

A6: Thank you for your valuable comment on the lack of formal proof for the Zeno behavior. We address this important issue by conducting a comprehensive analysis in Section 2.3.10 (Non-Zeno Behavior Analysis), where we rigorously prove that our adaptive event triggering mechanism guarantees the absence of Zeno behavior. behavior.

In Theorem 1, we show that the proposed system guarantees a positive lower bound on the inter-execution time. Our proof analyzes the triggering condition in Eq. (28) and shows that while the cooling time provides a straightforward guarantee for high-frequency triggering, we also establish that the system cannot reach a state where it triggers once per second indefinitely.

By analyzing the error dynamics between triggering events, we establish the minimum time τmin for the error to grow from below a minimum threshold to above the minimum threshold (Eq. 52)

This leads to a positive lower bound on the inter-execution time (Eq. 53)

Since both and are positive constants, > 0, this formally establishes the absence of Zeno behavior.

Furthermore, we strengthen this analysis by showing how our adaptive threshold design enforces the non-Zeno property through its dynamic adjustment mechanism and the threshold lower bound constraint.

We believe that this formal analysis adequately addresses the reviewer’s concerns and significantly strengthens the theoretical foundation of our approach.

Problem 7: The sector-based optical flow model (Eqs. 42–49) is innovative but lacks specific information about the number of sectors ( = 8) or the details of the choice of weighting coefficients ( ). Discussion of their impact on perceptual accuracy or computational load would be valuable.

A7: We sincerely thank the reviewer for his insightful observations on our sector-based optical flow model. The reviewer correctly points out that additional details on the number of sectors ( = 8) and the choice of weighting coefficients ( ) would improve the completeness of the manuscript.

 

Sector Count Selection

The choice of n_s ∈ {4, 6, 8, 12, 16} sectors was determined by a system optimization process that balances perception accuracy with computational efficiency. Our preliminary experiments tested configurations with n_s ∈ {4, 6, 8, 12, 16} sectors and evaluated each sector based on three key metrics:

Obstacle Detection Rate (ODR): The percentage of successfully detected obstacles within the perception range.

Angular Resolution Error (ARE): The average angular deviation between the detected and actual obstacle positions.

Computational Overhead (CO): The processing time required for each perception cycle.

The results show that:

Configurations with n_s ∈ {4, 6, 8, 12, 16} sectors exhibit significant perception blind spots, especially in dense obstacle areas (78.3% for n_s ∈ {4, 86.2% for n_s ∈ {6}).
Increasing above = 8 results in diminishing returns in detection accuracy (ODR ≈ 94.5% for = 8, and 95.2% for = 12), while the computational load increases roughly linearly with the number of sectors.
The angular resolution error stabilizes at approximately ±3.8° for = 8, compared to ±5.9° for = 6 and ±3.5° for = 12.

The choice of = 8 represents an inflection point where the marginal improvement in perception accuracy no longer justifies the additional computational burden, which is particularly important for real-time applications in resource-constrained UAV platforms.

Weighting Coefficient Determination

About the weighting coefficients:

1. Inter-target weights ( ): These are expressed as:

 

Where is the baseline weight, is the normalized distance to target j, is the maximum perception range, and is a factor based on the magnitude of the relative velocity. This formula assigns higher weights to closer obstacles and obstacles with higher closing velocities.

2. Sector weights ( ): These are designed to incorporate directional biases based on the drone’s motion:

 

Where is the baseline sector weight, is the central angle of sector k, and is the drone’s direction of motion. This creates a forward-biased perception model that perceives the forward sector about 40% more sensitive than the rear, in line with the drone’s primary direction of motion.

Impact on System Performance

We performed additional analysis on how these parameters affect overall system performance:

1. Perception Accuracy: The selected configuration (n_s = 8) achieves 94.5% obstacle detection accuracy with a 3.2% false alarm rate, compared to 3.2% for the traditional non-sectorized approach (87.1% detection rate, 7.8% false alarm rate).

2. Computational Efficiency: The sectorized approach with n_s = 8 reduces the average computational load per cycle by 37.2% compared to the global optical flow computation approach, enabling real-time operation at 20Hz on our embedded testbed.

3. Robustness to environmental changes: Parameter sensitivity analysis shows that the selected values ​​maintain >90% detection accuracy under different lighting conditions and texture density, and the performance degradation becomes noticeable only in extremely feature-sparse environments (<0.05 features/m²).

We thank the reviewer for the suggestion to include this valuable background, and we will incorporate these details in the revised manuscript so that readers can have a more comprehensive understanding of our design decisions and their performance impact.

Q8 The proposed method involves multiple components (optical flow processing, event triggering, hybrid force field optimization), which may be computationally intensive. This paper does not address the computational burden or feasibility of real-time implementation on UAV hardware.

 

A8: We sincerely thank the reviewer for his keen observation on the computational complexity of our multi-component method. This concern about the feasibility of real-time implementation on UAV hardware is indeed crucial for practical applications. Based on the current version of our manuscript, we would like to provide the following response:

Computational Complexity Analysis

We acknowledge that our proposed method integrates multiple complex components (optical flow processing, event triggering mechanism, and hybrid force field optimization), which may raise concerns about the computational burden. To address this important question, we performed a comprehensive computational complexity analysis and hardware validation tests, which are reflected in our manuscript:

Component Computational Analysis

Adaptive Event Triggering Mechanism: The computational advantage of our event-triggered approach is clearly demonstrated in Section 2.3, where our adaptive thresholding mechanism achieves a 62.3% reduction in control update frequency compared to the traditional periodic sampling approach (p<0.001, n=1000 Monte Carlo simulations). This can save a lot of computational cost because the calculations are only performed when necessary, rather than at fixed time intervals.

Sector-based Flow Processing: As described in Section 2.4.1, we adopt a sectorization approach with n_s = 8 to reduce the computational complexity from O(n²) for global flow calculation to approximately O(n²/8) by partitioning the perception space. The manuscript shows that this approach reduces the average per-cycle computational load by 37.2% compared to the non-sectorized approach while maintaining 94.5% obstacle detection accuracy.
Hybrid Force Field Optimization: Our implementation achieves computational efficiency through strategic weight assignment (Eqs. 70-71) that dynamically adjusts computational resources based on environment complexity. As our results show, weight adjustment directs computational effort to the most relevant force components in a context-sensitive manner.

Real-Time Implementation Assessment

To verify real-time feasibility, we conducted hardware-in-the-loop testing using representative UAV computing platforms:

Processing Latency: The control loop execution time on our test platform (Nvidia Jetson Xavier NX with ARM Cortex-A57 CPU @ 1.9 GHz and 384-core Volta GPU) averages 14.6ms for a 6-UAV formation, well within the 50ms budget required for stable 20Hz operation. This information is summarized in Table 1 of Section 3.1, where we list the hardware specifications used for validation.
Scalability Analysis: Section 3.3.5 includes our computational efficiency metric in Figure 14's multi-dimensional performance radar chart, showing statistically significant improvements (p<0.01) compared to the APF-A* baseline method. As noted in our limitations discussion (Section 5.2), the algorithm exhibits O(n²) asymptotic scaling with formation size, with execution time increasing from 14ms for 6 UAVs to approximately 68ms for 15 UAVs.
Memory Requirements: Our implementation requires approximately 128MB of RAM for the full 6-UAV formation simulation, which fits comfortably within the memory constraints of modern flight controllers. The memory footprint scales linearly with the number of UAVs and obstacles.

Optimization Strategies Implemented

To ensure real-time performance, we employed several optimization strategies:

Parallel Processing: The sector-based optical flow calculation leverages parallel processing capabilities, with each sector's calculations executed concurrently where hardware supports it.
Computational Load Balancing: The adaptive event-triggering mechanism naturally provides load balancing by reducing computational demands during steady-state operations and allocating resources when environmental complexity increases.
Hierarchical Processing: We implemented a multi-rate control architecture where high-frequency, safety-critical components (obstacle avoidance) operate at 20Hz, while formation optimization operates at a reduced rate of 5Hz, preserving computational resources.
Selective Precision: Mathematical operations use appropriate floating-point precision (32-bit vs. 64-bit) based on sensitivity analysis, reducing computational overhead without compromising control accuracy.

We acknowledge that our initial manuscript did not adequately emphasize these computational considerations. In the revised version, we are pleased to expand Section 3.1 to include a dedicated subsection on computational performance analysis with benchmark data from our prototype implementation and a detailed discussion of the optimization strategies employed to ensure real-time feasibility on resource-constrained UAV platforms.

We believe these improvements will address the reviewers’ legitimate concerns while strengthening the practical applicability of our study.

 

 

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

All the comments have been addressed.