Harmonic Decomposition Models of Flapping-Wing Flight for Stability Analysis and Control Design

This paper demonstrates the extension of the harmonic decomposition methodology, originally developed for rotorcraft applications, to the study of the nonlinear time-periodic dynamics of flapping-wing flight. A harmonic balance algorithm based on harmonic decomposition is applied to find the periodic equilibrium and approximate linear time-invariant dynamics about that equilibrium of the vertical and longitudinal dynamics of a hawk moth. These approximate linearized models are validated through simulations against the original nonlinear time-periodic dynamics. Dynamic stability using the linear models is assessed and compared to that predicted using the averaged dynamics. In addition, modal participation factors are computed to quantify the influence of the higher harmonics on the flight dynamic modes of motion. The study shows that higher harmonics play a key role in the dynamics of flapping-wing flight, inducing a vibrational stabilization mechanism that increases the pitch damping and stiffness while reducing the speed stability. This results in stabilization of the pitch oscillation mode and thus of the longitudinal hovering cubic. In addition, the proposed methodology is used to derive open-and closed-loop control laws for attenuating arbitrary state harmonics and to enhance the dynamic response characteristics. © 2022 by Umberto Saetti. Published by the American Institute of Aeronautics and Astronautics, Inc.