First Experimental Demonstration of Natural Hovering Extremum Seeking: A New Paradigm in Flapping Flight Physics

TL;DR

This paper presents the first experimental validation of Natural Hovering Extremum Seeking (NH-ES), a model-free, sensory-based control method enabling stable hovering in flapping flight using natural oscillations and local light measurements.

Contribution

It demonstrates experimentally that NH-ES can achieve stable, autonomous hovering in a flapping-wing system without relying on detailed aerodynamic or morphological models.

Findings

NH-ES enables a flapping body to gain altitude and stabilize autonomously.

The method is robust against delays and noise in sensory feedback.

The system successfully hovers around a light source using only local light intensity feedback.

Abstract

In this letter, we report the first experimental demonstration of the recently emerged new paradigm in hovering and flapping flight physics called (Natural Hovering Extremum Seeking (NH-ES)) [doi.org/10.1103/4dm4-kc4g], which theorized that stable hovering flight physics observed in nature by flapping insects and hummingbirds can be generated via a model-free, real-time, computationally-basic, sensory-based feedback mechanism that only needs the built-in natural oscillations of the flapping wing as both the control and the propulsive input. We run experiments of moth-like, light source-seeking, on a flapping-wing body in a total model-free setting that is agnostic to morphological parameters and body/aerodynamic models. We show that the flapping body using NH-ES gains altitude and stabilizes autonomously the servos responsible for flapping, including with pitching dynamics (believed in literature to be a main reason of instability in open-loop hovering). The flapping body effectively/stably hovers about the light source, needing only feedback of local measurements of light intensity. Our results were also achieved under delay/noise effects, supporting earlier observations that NH-ES is robust against potential processing delays and noisy-sensations.

Equations

$$\dot{x}=f(x,\mathbf{u}),$$

$$\mathbf{u}=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right],$$

$$\dot{x}=f(x,u_1,u_2).$$

$${\color[rgb]{0,0,0}u_{1}=c_{1}\,\hat{u}+a_{1}\omega\cos(\omega t),}$$

$${\color[rgb]{0,0,0}u_{2}=c_{2}\,\hat{u}+a_{2}\omega\cos(\omega t),}$$

$$\dot{\hat{u}}=kJ\omega \cos (\omega t),$$

$$J(z)=(z-z_d{)}^2,$$

Full text

††preprint: APS/123-QED

I Introduction

Flapping flight is a natural physical phenomenon that has captivated research interest across many scientific and engineering fields, including physics, biology, computational mathematics, and aerospace, control, and robotic engineering [33, 42, 34, 26, 16]. Physical modeling of flapping flight has long been recognized as a highly complex problem, involving nonlinear multibody, multiple-time-scale, and non-autonomous dynamics [18, 34, 3, 26, 23]. Among flapping maneuvers, hovering is particularly intriguing because it demands lift coefficients far beyond those predicted by conventional aerodynamic principles developed for fixed-wing aircraft [40, 13]. Over time, scientists and engineers have identified unconventional lift mechanisms enabling hovering flight, such as the leading-edge vortex [12, 5], which accounts for the necessary lift in insect flight. While this body of literature has succeeded in answering fundamental questions about the physics of hovering, it remains to explain the mechanism used in nature by flapping insects and hummingbirds to achieve such complex physics in a stable and real-time manner.

The long-held consensus in literature is that hovering flapping flight is inherently unstable in the open-loop sense for most insects, and that an active control mechanism is therefore required for stabilization [34, 33, 3, 4, 37, 38, 28, 39, 22, 35]. At the same time, biological observations and experiments consistently indicate that flapping insects rely heavily on sensory feedback for stable flight and stabilized hovering [15, 29, 37, 28, 36]. These observations suggest that the active control mechanism employed in nature is sensation-based and operates under, arguably, limited computational capabilities [43, 35, 4]. Consequently, the case was made in [10] that a sensation-based control mechanism for hovering is likely to be model-free and independent of explicit aerodynamic or body modeling. The reader can refer to [10, Section I] for more on control methods used in hovering/flapping flight literature.

Very recently, a new paradigm in hovering and flapping flight physics, termed Natural Hovering Extremum Seeking (NH-ES), has emerged [10]. The concept of NH-ES is grounded in a recently new mathematical control framework [9] called Extremum Seeking for Vibrational Stabilization (ES-VS) that takes advantage of a high-amplitude high-frequency input signal to steer a physical system towards stabilization using simple sensory feedback, which operates as a continuously adjusted vibrational stabilization mechanism. Similar to previous extremum seeking control methods (see [30, 1, 27, 7] for example), ES-VS idea is to inject/actuate perturbations into a dynamical system, then via access to corresponding variations in the objective function measurements, a simple feedback estimate loop is used to adjust the input to steer the dynamical system to the vicinity of the extremum point (or optimal state) of the objective function. In essence, extremum seeking methods are an autonomous, systematic and stable “trial and error” or “perturb and observe” control mechanism as they are operable without access to the dynamic system model or the objective function formula; that is, one only needs to actuate the perturbations and measure the objective function. However ES-VS [9, 25] is distinguished from previous extremum seeking methods [30, 1, 27] in that it needs only one high-amplitude, high frequency perturbation signal to vibrationally stabilize and control a second-order mechanical system that can admit forces quadratic in velocities, enabling the inclusion of aerodynamic forces. As a result, ES-VS made it possible to characterize hovering flight physics as NH-ES [10] with the only one high amplitude, high frequency perturbation signal being the flapping motion itself, naturally built in the system for propulsion. The reader is directed to the supplementary file in [11] “Tutorial.pdf” for a step-by-step tutorial on how ES-VS concept works using simple 1D inverted pendulum and mass-spring examples for demonstration.

Since it was theorized in [10], NH-ES attracted significant scientific attention (see for example [41, 6]) after being successfully validated via: (i) positive stability analysis using sophisticated tools from differential geometric control theory [10, Section IV]; (ii) effective and robust simulations across multiple insect species and a hummingbird data set [10, Section III.A] which closely matched reported biological data [10, Table IV], all of which by using the same feedback mechanism and learning-rate parameter [10, Table III]; (iii) its ability to tolerate processing delays and sensation noise [10, Section III.D]; and (iv) demonstrating superior performance, with lesser computational complexity, when compared to the simplest existing control method for hovering: the popular Proportional-Integral-Derivative (PID) controller – see [10, Section III.C] and particularly [10, Table IX]. In summary, the results of [10] revealed a universal and remarkably simple feedback mechanism, in which the natural oscillations of the flapping wings—already inherent to the system—serve as the input required for stabilization and propulsion. Furthermore, NH-ES was shown to generate stable hovering flight in real-time, thereby substantially reducing the complexity of hovering flight physics. Given that the NH-ES feedback mechanism is sensation-based, model-free, closely matched with biological data, and computationally minimal, it has been hypothesized as biologically plausible. Nevertheless, the new paradigm, theory and concept of NH-ES [10] was not tested by simulations that include pitch dynamics (a main reason for instability in open-loop hovering [35, 18]) and was not verified experimentally.

In this letter, we present the first experimental demonstration of NH-ES, aiming to verify, and open the path for, this paradigm-shifting concept in hovering and flapping flight physics. Our aim is to: (1) validate the initial, purely vertical-motion flapping, findings/observations in [10]; and (2) demonstrate the ability of the NH-ES to go beyond vertical-motion flapping and include pitching dynamics. Our experiments are agnostic to (i) morphological parameters, (ii) body model, (iii) aerodynamic model/computations, and/or (iv) objective function formulation. Hence, they confirm the model-free operability of NH-ES. Our results are:

• •

able to confirm that the high-amplitude, high-frequency flapping wing motion results in variations in sensory measurements on the body. This confirms the main NH-ES conceptual premise.

• •

conducted successfully in real-time under inherent processing delays and measurement noises. Hence, they confirm the autonomous operability of NH-ES and its robustness against noise and delays.

• •

successful in demonstrating stable source-seeking to, then hovering about, a desired altitude mimicking a “moth-like” light source-seeking and hovering. The NH-ES design shows stable actuation of the two servos responsible for flapping, including with pitch dynamics.

II Natural Hovering Extremum Seeking Design and Experimental Setup

The proposed Natural Hovering Extremum Seeking (NH-ES) framework is implemented using a generic, model-free feedback structure, illustrated schematically in Fig. 1. Consider a general nonlinear system

[TABLE]

where $x$ denotes the system state and $\bm{u}$ denotes the control input vector. In the experimental flapping-wing system considered here, the control input is composed of two inputs,

[TABLE]

where $u_{1}$ represents the control input applied to the flapping motor, and $u_{2}$ represents the control input applied to the pitch servo. Therefore, the system dynamics can be written, as shown in Fig. 1, as

[TABLE]

The function $f(\cdot)$ is assumed to be completely unknown and represents the mechanical system dynamics of the physical or biological flapping-wing system under consideration. The objective function $J(x)$ is also assumed to be unknown and may be defined implicitly through sensor measurements, such as light intensity, without requiring an explicit analytical expression. Each control input is composed of the same slowly varying estimate $\hat{u}$ and a periodic excitation induced by the natural flapping motion, but with different scaling and excitation-amplitude parameters. Specifically,

[TABLE]

[TABLE]

where $a_{1}$ and $c_{1}$ are the excitation and scaling constants associated with the flapping motor input, $a_{2}$ and $c_{2}$ are the corresponding constants associated with the pitch servo input, and $\omega$ denotes the frequency of the oscillatory input. The adaptive component $\hat{u}$ is updated via induced perturbations resulting from the input signal (flapping motion) on the unknown objective function $J$ measurements:

[TABLE]

where $k$ is the learning rate. Importantly, no model of $f(\cdot)$ or analytical form of $J(\cdot)$ is required, reflecting that insects/hummingbirds do not need to internalize/comprehend their body/aerodynamic models or their morphological parameters; all that they needed is the oscillatory input naturally generated by the flapping motion itself. In the light-source experiments reported here, a high-pass filter (HPF) is used in conjunction with the photoresistive sensor to mitigate measurement biases and low-frequency offsets, consistent with common practice in the implementation of such sensors in literature [2, 8, 24].

The experimental validation of NH-ES is conducted using a laboratory-scale flapping-wing body designed to emulate vertical hovering motion. The body is relatively, but not strictly constrained to one-dimensional vertical translation using parallel guide rails with relaxed tension, allowing isolation of the altitude dynamics while preserving the natural flapping-induced oscillations. Wing flapping is generated by a single actuator operating at a prescribed frequency, and sensory feedback is obtained either from motion-capture measurements or from a body-mounted photoresistive sensor, depending on the experiment. Importantly, no aerodynamic or body model is used in the feedback loop, and no artificial excitation signals are injected. A schematic of the experimental setup and setup configuration is shown in Fig. 2 while more details and specifics are provided in “Setup.pdf”.

In order for us to have the ability of both running experiments with fixed pitch (as in Section III) and with adaptive, controlled pitch dynamics (as in Section IV), we made sure that we have two configurations as shown in Fig.2(a) and Fig.2(b)-(c). Therefore, we have the ability to actuate $u_{2}$, i.e., the pitch servo command independently from $u_{1}$ (the flapping motor command). The reader can refer the Supplemental Material (Video c1 in[17] ) for experimental test we performed to achieve isolated pitch action which will be used to control pitch dynamics throughout the experiments.

II.1 Experimental Verification of NH-ES Input-Output Fundamentals: Natural Flapping Motion Leads to Sensory Perturbation

A central premise of NH-ES is that the high amplitude, high frequency flapping-wing motion (i.e., input perturbation/oscillation) naturally induces perturbations/variations in sensory measurements (i.e, output) taken on the body, thereby providing the excitation required for extremum search and adaptation. Furthermore, if the aforementioned premise is verified and holding, this confirms that NH-ES is in alignment with the fundamentals, findings and predictions of ES-VS. The input-output relation is verified experimentally by fixing the flapping the motor command while maintaining a constant relative position with respect to a light source, and observing the light sensor measurements taken on the body. In this test, the motor input is held constant at a Pulse Width Modulation (PWM) command of $38{,}000$. As shown in Fig. 3, the measured light-intensity signal (output) exhibits clear variations induced solely by the flapping-wing motion (input). These experimentally observed oscillations confirm that the flapping-wing motion itself act not only as a natural probing mechanism, but also results in perturbed sensation that is needed for the feedback as predicted by the NH-ES theory. A corresponding experimental recording is provided in the Supplemental Material (Video S1 in [14]).

III Experimental demonstration of NH-ES with fixed pitch

In this section, we run experiments focused on the vertical motion with minimal pitch dynamics (i.e., Fig.2(a) configuration). That is, in the NH-ES design schematic in Fig.1, only the flapping motor command (i.e., $u_{1}$ control input) is adaptive and in effect while the pitch servo command (i.e., $u_{2}$ control input) is fixed.

III.1 First experiment: set altitude-objective

We first validate NH-ES using a set-altitude objective function defined by the vertical position of the body. In this case, the objective function is explicitly defined as

[TABLE]

where $z$ denotes the vertical position of the flapping body and $z_{d}=700~\mathrm{mm}$ is the desired hovering altitude corresponding to the objective minimum. Motion-capture feedback is used only for performance evaluation, while the control mechanism remains fully model-free. Starting from rest, the flapping-wing body autonomously ascends and converges to the objective minimum, then stable hovering is achieved as shown in Fig. 4. The results demonstrate that hovering stabilization emerges directly from the interaction between intrinsic flapping-induced oscillations and scalar feedback of altitude measurement. A video of this experiment is provided in the Supplemental Material (Video S2 in [31]).

III.2 Second experiment: moth-like, fixed light source seeking.

We next consider moth-like, light source seeking in completely unknown environment by introducing a spatially varying light field and replacing position feedback with a body-mounted photoresistive sensor. In this experiment, the objective function $J$ is not known analytically and is instead implicitly defined by the measured light intensity provided by the sensor on the body, which serves directly as the scalar objective to be maximized. The spatial distribution of the light field and the location of its extremum (i.e., the source/bulb) are completely unknown to the NH-ES control mechanism. As shown in Fig. 4, the flapping-wing body ascends autonomously toward increasing light intensity and converges beneath the light source, where it naturally transitions into a stable hovering state. This behavior is repeatable across trials and relies exclusively on local scalar measurements. The corresponding evolution of the light-sensor objective $J$ is shown in Fig. 5. A corresponding experimental video is provided in the Supplemental Material (Video S3 in [19]).

III.3 Third experiment: moth-like, moving light source seeking

To evaluate adaptability under time-varying conditions, the light source is manually repositioned during flight while keeping the NH-ES control mechanism unchanged and without any re-tuning. Despite the moving extremum, the flapping-wing body adapts in real time and re-converges to the new source location, effectively transiting between ascend and hovering as needed, stably. This behavior is illustrated in Fig. 6, where the body continuously tracks the changing light source and maintains stable hovering. These results confirm the robustness of NH-ES under dynamic environmental conditions. A video of the moving-source experiment is provided in the Supplemental Material (Video S4 in [32]).

IV Experimental demonstration of NH-ES with controlled pitch dynamics

Pitching dynamics (and longitudinal motion) are in the heart of the stability-instability debate in literature, especially for smaller insects [35, 18]. Therefore, in this section, we run experiments that includes pitch dynamics (i.e., Fig.2(b)-(c) configuration). That is, in the NH-ES design schematic in Fig.1 wo have both the flapping motor command (i.e., $u_{1}$ control input) and the pitch servo command (i.e., $u_{2}$ control input) adaptive and in effect. It is worth the reminder that NH-ES has not been tested with pitch dynamics [10]. Hence, the results of Section substantially confirm NH-ES premise as a new paradigm in hovering and flapping flight physics.

IV.1 First experiment: moth-like, fixed light source seeking with pitch dynamics

We run an experiment with the same objective as the moth-like, light source seeking in Section III.B. However, we actuate both $u_{1}$ and $u_{2}$; that is, we control both the flapping motor and pitching servo via NH-ES. The objective of the experiment then is to show that NH-ES is able to adaptively control the longitudinal motion (vertical and pitching) to have the flapping body ascending towards the light source, and then transitions successfully into a stabilized hovering state, particularly, with stabilized pitching – see Fig. 7 (blue). To emphasize more the significance of this result, we averaged out the stabilized pitch servo command (see top-orange in Fig. 7), which is about 28000 PMW. Then, we repeated the same experiment successfully using a fixed pitch servo command of about 28000 PMW – see Fig. 7 (red). The comprehensive view of the results (the whole of Fig. 7) confirms that NH-ES was able to control the pitch adaptively until figuring out the value of about 28000 PMW. Said value, if known beforehand (practically very challenging in unknown environment), could serve as a set value for the pitching command, which NH-ES is able to find autonomously. It is worth to remind the reader that NH-ES was able to achieve model-free, real-time control of both flapping motor and pitch servo using only local light measurements with no global or GPS information. A corresponding experimental video is provided in the Supplemental Material (Video S5) in [20].

IV.2 Second experiment: moth-like, changing light source seeking with pitch dynamics

In this experiment, our objective is to test and verify the ability of NH-ES to respond adaptively and effectively to changing light source position with more focus on testing the pitching adaptability. Hence, we conducted a moth-like, source seeking experiment with both $u_{1}$ and $u_{2}$ (vertical and pitching) being controlled by NH-ES (similar to Section IV.A). Then, we made changes of the light position between “up” and “down” few times and recorded the results – see Fig. 8. The results show the NH-ES ability to respond adaptively to changes, stably transitioning between the changed positions of light in real-time. A corresponding experimental video is provided in the Supplemental Material (Video S6) in [21].

V Comments on Delays and Noise

Our experimental implementation (Sections III-IV) naturally includes several non-ideal effects that are unavoidable in physical robotic systems. These effects include sensor noise, wireless communication delay, computation delay, sampling effects, and actuator response delay. In all light-source-seeking experiments, the measured objective-function signal is obtained from the analog light sensor mounted on the flapper. This signal is transmitted from the Arduino Nano to the Python receiver, where the NH-ES control law is implemented, and the resulting motor and pitch-servo commands are then transmitted to the Flapper Nimble+ board. A simplified illustration of this discrete-time communication and control architecture is provided in Fig. 5 of the supplementary file “Setup.pdf” in [11].

In addition to the natural delay sources associated with sensing, communication, computation, and actuation, two explicit delays were included in the experimental implementation. First, a $50~\mathrm{ms}$ delay was included in the Arduino light-sensor loop after each sensor measurement and transmission step. Second, a $0.02~\mathrm{s}$, i.e., $20~\mathrm{ms}$, delay was included in the Python implementation before transmitting the updated control commands to the flapper board. Therefore, the experiment includes quantified delay effects at both the sensor-measurement side (i.e., the output of NH-ES) and the control-command side (i.e., the input of NH-ES); this is represented by the two rate-transition stages shown in Fig. 5 of the supplementary file “Setup.pdf” in[11].

The light sensor measurement also contains noise due to the photoresistive sensing element, ambient-light variations, analog-to-digital conversion, and wireless data transmission. To illustrate this effect, Fig. 9 shows a $100$-second sample of the light sensor measurement for a fixed source/bulb when the flapper is stationary on the ground. Since the robot and the light source are not moving during this test, the variations in the measured signal represent the natural sensor noise and measurement fluctuations present in the experimental setup in [11].

Despite these practical challenges, the NH-ES controller is implemented directly using the measured objective-function signal. Therefore, the experimental results demonstrate the ability of the proposed model-free feedback law to operate under realistic sensing, communication, actuation, noise, and delay conditions. This is inline with theoretical predictions about NH-ES [10, Section III.D] and its main framework stemming from ES-VS, which in general is able to tolerate significant noise and delays [25, Section 5].

VI Conclusive Remark, Implications, and Future Directions

This letter reported the first experimental demonstration of its kind of natural hovering extremum seeking (NH-ES) as a new paradigm and concept in hovering and flapping flight physics. In our moth-like, light source seeking experiments, we were able to demonstrate and verify NH-ES substantial advantages that were predicted in [10] and hypothesized as biologically plausible, such as being simple, model-free, real-time, operable by access to only local sensation/measurements of the objective function (e.g., local light intensity) without need to any global information (e.g., GPS) or morphological/body/aerodynamic/light-distribution models. That is, the implementer of NH-ES needs only to access local sensory information and perform natural flapping motion that is adaptively updated by said local sensory measurement in a “trial-and-error” or “perturb-and-observe” fashion. Additionally, NH-ES is shown able to operate under considerable levels of delays and noise successfully. In addition to [10], this letter also paved the way for NH-ES potential to address the stability-instability question posed in the literature for open-loop hovering when pitching (longitudinal) motion is in consideration. Experiments in Section IV confirms that NH-ES is able to simultaneously control both vertical and pitching dynamics motion. Hence, we believe NH-ES will have significant implications on the bio-physics and bio-mechanics aspects of hovering and flapping flight. Additionally, the computational simplicity, stability, and easy application of extremum seeking methods, position NH-ES as a new concept with minimal on-board needs that may change preservatives in flapping robotics and Micro-Air-Vehicles (MAVs).

In the future, we aim at developing/expanding NH-ES theoretically and experimentally to include both longitudinal and lateral motions, reaching 3D source seeking and hovering.