Hovering Flight in Flapping Insects and Hummingbirds: A Natural Real-Time and Stable Extremum Seeking Feedback System

TL;DR

This work reframes hovering in flapping insects and hummingbirds as a natural extremum-seeking (ES) control problem, leveraging intrinsic wing flapping as the perturbation and altitude-related sensation as feedback to achieve real-time, model-free stabilization. A reduced 2-DOF vertical dynamics model provides the testing ground, with torque perturbation $\tau = \hat{\tau} + a \Omega \cos(\Omega t)$ and a gradient-like update $\dot{\hat{\tau}} = K J a \Omega \cos(\Omega t)$ guiding the system toward the hovering extremum. Simulations across hawkmoth, cranefly, bumblebee, dragonfly, hoverfly, and hummingbird demonstrate robust hovering for objectives $J = z^2$ and $J = \dot{w}^2$, including robustness to delays and noise, and a stability analysis using VOC averaging supports stronger stability than open-loop or PID controllers. The results suggest ES-based, model-free strategies could bridge gaps between theory and biological hovering mechanisms, with future work extending to pitching dynamics and experimental validation; key relations include $\tau$ and $\dot{\hat{\tau}}$ as described above.\n

Abstract

In this paper, we take an initial and novel step toward characterizing the physics of the hovering phenomenon in flapping insects and hummingbirds as a new 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.

Figures (20)

• Figure 1: Schematic body diagram of the flapping insect system which moves vertically via the vertical velocity $w$ (to gain or lose altitude $z$) and the flapping angle $\phi$.
• Figure 2: Sketch of main wing parameters.
• Figure 3: Natural hovering extremum seeking system for flapping insects and hummingbirds. The perturbation of the torque $\tau$ is consistent of the average/mean torque estimate $\hat{\tau}$ added to it the modulation signal $a \Omega cos(\Omega t)$, i.e., $\tau = \hat{\tau} + a \Omega \cos(\Omega t)$. This perturbation in $\tau$ affects the flapping angle rate $\dot{\phi}$, causing variation in the flapping angle $\phi$. Said variations in $\phi$ is reflected in the dynamic system $\bm{\dot{x}} = f(\bm{x}, \tau)$ (the insect/hummingbird). As a result, the insect/hummingbird sensation (i.e., objective function measurements) is also varied in a domino-effect fashion. The sensation (measurements) of the objective function reflects the altitude (or altitude-related signals) or acceleration (or acceleration-related signals). Measurements of the objective function is the feedback which is processed by demodulation, providing the update for the torque average/mean $\hat{\tau}$ which takes the insect/hummingbird closer to the minimum of $J$, seeking hovering in real-time.
• Figure 4: State variables response of the natural hovering ES system \ref{['eq:Full_model_ESC']} for hawkmoth. The left column represents results under the objective function $J = z^2$, while the right column represents results under the objective function $J = {\dot{w}}^2$. The ES system (in black) stabilizes by oscillating about the hovering equilibrium (red).
• Figure 5: For the hawkmoth case, hovering condition by stabilization about a constant altitude $z$ is observed in the most left figure. Similarly, the hovering condition of balance between lift and weight is observed in the most right figure and the relevant objective function in the middle figure.