Hovering flight in flapping insects and hummingbirds: A natural real-time and stable extremum-seeking feedback system

In this paper, we take a step toward characterizing the physics of the hovering phenomenon in flapping insects and hummingbirds as a class of extremum seeking (ES) feedback systems. By characterizing hovering flight in insects and hummingbirds as a natural hovering ES system, we achieve (1) very simple, (2) stable, (3) model-free, and (4) real-time hovering. More importantly, our hovering ES characterization only needs the natural oscillations of the wing as the ES input. That is, unlike other control techniques in the literature, the natural hovering ES system only needs the natural flapping action built in the system and feedback of local sensations (measurements) related to the altitude where the insect seeks to stabilize itself. Said ES characterization can become an important initial step in starting a new line of research that may succeed in resolving the long-standing gap between model-based control theory and the biologically observed mechanisms that stabilize hovering flight. We provide simulation trials, including comparisons with some approaches from literature, to demonstrate the effectiveness and robustness of our results. We used literature data for hawkmoth, cranefly, bumblebee, dragonfly, hoverfly, and a hummingbird.