## 6 Conclusions  
This study proposes a multi-plane method for describing the wing deformation of a hovering flapping-wing aircraft, then combines it with a quasi-steady aerodynamic model for aerodynamic analysis. The relationship between the slack angle, wing vein position, and wing deformation are analyzed, along with their effects on aerodynamic forces and efficiency. The wing deformation is investigated, and the relationship between simulated and measured aerodynamic forces is validated through experiments. The results indicate that the wing deformation can be accurately described by adjusting the parameters of the multi-plane method, and that the aerodynamic analysis using this method provides valuable insights into the average lift. Additionally, the multi-plane method establishes a connection between the morphological characteristics and aerodynamic forces of the wing, which has significant implications for aerodynamic analysis.  
The findings of this study are important for the design and optimization of hovering flapping-wing aircraft. The multi-plane method allows for the mapping of two-dimensional wing designs onto the three-dimensional installation space, demonstrating the influence of various design parameters on the three-dimensional wing deformation while ensuring a certain degree of wing deformation freedom. Furthermore, by considering the actual flapping trajectory of the wing and defining the deformation, transient wing deformations can be described.  
By considering the wing deformation characteristics and combining them with a quasi-steady aerodynamic model, the aerodynamic forces and efficiency of the wing can be predicted. Experimental validation of the relationship between simulated and measured aerodynamic forces further confirms the reliability of the multi-plane method. Due to the aerodynamic simulation method being based on quasi-steady blade element theory, special aerodynamic effects such as delayed stall, clap, fling, and wake capture are not taken into account. Subsequent research needs to consider and incorporate these aspects.  
Figure 25: **(a)** Flapping angle of the carbon fiber rod obtained from actual recordings within a normalized time and **(b)** flapping angle of the carbon fiber rod obtained from simulation within a normalized time.  
Moreover, this study reveals the influence of the slack angle and wing deformation on aerodynamic forces and efficiency. Analyzing the relationship between the slack angle and wing deformation provides a better understanding of the aerodynamic characteristics during flapping, which cannot be achieved by single-plane blade element methods that treat the wing as a flat surface. This finding is of great significance for designing more efficient hovering flapping-wing aircraft.  
In summary, this study investigates the aerodynamic characteristics of hovering flapping-wing aircraft by introducing the multi-plane method and considering wing deformation. The research results provide an important theoretical foundation for further optimizing the performance of hovering flapping-wing aircraft.  
Conceptualization, R.Y.; formal analysis, R.Y.; investigation, R.Y. and Z.L.; resources, J.C. and Y.W.; writing--original draft preparation, R.Y.; writing--review and editing, R.Y., J.C. and C.W.; supervision, R.Y., J.C. and Y.W.; project administration, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript  
This research was funded by the Ministry of Education of the People's Republic of China.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to ongoing patent applications and intellectual property concerns with the aim of safeguarding the potential commercial value of the research findings.  
The funders had no role in the design of the study, in the collection, analysis, or interpretation of data, in the writing of the manuscript, or in the decision to publish the results.