Data-driven Discovery and Formulation Refines the Quasi-steady Model of Flapping-Wing Aerodynamics for Free-Flight

Insects skillfully navigate three-dimensional space by modulating unsteady aerodynamic forces on flapping wings. While understanding these forces is vital for biology, physics, and engineering, existing evaluation methods face trade-offs: high-precision computational fluid dynamics or robotic models are computationally or experimentally expensive and lack explanatory power on underlying mechanisms, whereas theoretical models based on quasi-steady assumption offer insights but suffer from poor accuracy, particularly in free-flight with body motion. To address this, we applied a data-driven model identification approach to enhance the accuracy of the quasi-steady aerodynamic model by discovering and formulating previously overlooked but critical mechanisms. Through automated selection from 5,000 candidate kinematic functions, we identified mathematical expressions for three key additional mechanisms — the effect of advance ratio, effect of spanwise kinematic velocity, and rotational Wagner effect — which were qualitatively recognized in prior studies but lacked formalization or were dismissed intuitively. By integrating these mechanisms into the quasi-steady model, we demonstrate significant improvements in prediction accuracy, using computational fluid dynamics results as the ground truth, both in hawkmoth forward flight (at high Reynolds number) and fruit fly maneuver (at low Reynolds number), making it the most accurate model among those of comparable complexity. This study demonstrates that mathematically capturing such latent mechanisms further advances the understanding of insect flight dynamics. The improved quasi-steady model, being both accurate and computationally efficient, serves as a valuable tool for optimizing wing kinematics in various metrics such as power consumption and dynamic stability, contributing to a deeper understanding of evolutionary adaptations in insect flight and the development of bio-inspired flying robots.