Real-Time Extremum Seeking for Natural Hovering and Source Seeking in Bio-Inspired Flapping Flight: Further Simulations

The problem of achieving hovering and source seeking in flapping insects and hummingbirds, as well as its potential replication in micro aerial vehicles (MAVs), has been extensively investigated over the past decades by researchers in both the scientific and engineering communities. Traditionally, hovering in flapping systems has been analyzed through open-loop or closed-loop formulations to examine stability and design control strategies. A fundamentally novel insight has recently emerged in the literature in our work [ 1 ]: the hovering behavior of flapping insects and hummingbirds can be effectively characterized and emulated as a stable, real-time, model-free natural extremum seeking control (ESC) feedback system. Inspired by this insight and building upon our recently proposed ESC framework, this paper extends the applicability of ESC to capture two critical aspects of flapping-wing flight: stable natural hovering and one-dimensional source (height) seeking. The proposed approach exploits the inherent high-frequency oscillatory dynamics generated by wing flapping, using them as a built-in perturbation signal in an ESC loop. The controller requires only output measurements of a directional signal (e.g., altitude) and introduces a projected thrust component that enables convergence to the maximum of an unknown objective function (e.g., light source). To validate the effectiveness of the proposed ESC framework, simulation studies are performed on two flapping insects with distinct wingbeat, the hawkmoth (low-frequency flapper) and the hoverfly (high-frequency flapper), in addition to the hummingbird. Simulation results further confirm the ESC system’s ability to achieve fast source seeking convergence and robust hovering. We also shed the light on our recent experimental demonstration results that confirm both source seeking and hovering in a model-free, real-time fashion by a flapping robot.