Extending hybrid quaternion attitude control of an underactuated flapping-wing aerial vehicle to practical autonomy: aggressive regimes, onboard sensing, a priori tuning, energy benchmarking, and tailless configurations

A 2019 IEEE/ASME Transactions on Mechatronics paper by Qian, Fang, and Li demonstrated, on a 19.1 g self-made flapping-wing aerial vehicle (FWAV) without ailerons, a quaternion-based hybrid attitude controller that globally asymptotically stabilizes hovering and forward flight in indoor motion-capture experiments. The paper closes by identifying five open directions that together define the gap between this laboratory result and a practical, design-transferable flight-control capability: extension to aggressive flight regimes with explicit handling of velocity-induced aerodynamic torques; replacement of the external motion-capture system with onboard sensing (IMU/vision) robust to high-frequency flapping vibration; a comprehensive energy-consumption comparison against other control methods; a priori quantification of the controller parameters (hysteresis thresholds and the singular-perturbation time-scale bound) for a given vehicle design; and extension of the hybrid framework to other underactuated configurations such as tailless two-wing vehicles.

Background

A flapping-wing micro aerial vehicle (FWMAV) is a bird- or insect-inspired aircraft of roughly tens of grams whose lift and thrust come from oscillating wings rather than a propeller or rotor. The vehicle studied in the source paper (Qian, Fang, and Li, IEEE/ASME Transactions on Mechatronics, 2019) weighs 19.1 g, flaps at 13–16 Hz, and, to save payload and avoid downwash interference, has no ailerons: only the tilt of the body (the reduced attitude) can be directly controlled through the tail and flapping inputs, while rotation about the body's vertical axis (yaw) is uncontrollable. Such a system is called underactuated — the number of independent control inputs is smaller than the number of degrees of freedom to be stabilized.

Attitude (orientation) of a rigid body is conveniently represented by a unit quaternion $\boldsymbol{q}$, a four-parameter object on the unit sphere $\mathcal{S}^3$ that avoids the singularities of Euler-angle representations; its double-covering property ($\boldsymbol{q}$ and $-\boldsymbol{q}$ denote the same physical attitude) prevents any continuous feedback from globally stabilizing a desired attitude without undesired ``unwrapping'' rotations, which is why hybrid (switching) controllers are used. For the yaw-uncontrollable FWAV, the relevant feedback signal is the reduced attitude $\boldsymbol{\Gamma} = R(\boldsymbol{q})\boldsymbol{e}_3$, i.e. the body's vertical axis expressed in the world frame (with $R(\boldsymbol{q})$ the rotation matrix and $\boldsymbol{e}_3 = [0,0,1]^T$); the paper constructs a reduced quaternion error $\boldsymbol{q}_e$ (with the third component of its vector part constrained to zero) that discards the uncontrollable yaw component while preserving quaternion kinematics.

Two further phenomena shape the control problem. First, translational velocity induces aerodynamic torques on the flapping body — forward speed produces a pitch torque like a pendulum restoring itself, and lateral speed produces a yaw torque like a weather vane — so attitude and translation dynamics are coupled. Second, when the vehicle is nearly horizontal ($\Gamma_z$ close to 0), the yaw angular velocity cannot be decoupled from the tilt dynamics by any continuous control law; the proposed remedy is a hybrid controller with two hysteretic switching functions $h$ and $h_\omega$ (hysteresis means the control mode changes only when the switching variable crosses a threshold by a margin $\delta$, preventing rapid chattering) that switches between a pull back to the stable flight region'' mode and a precise regulate to the desired attitude'' mode. The stability proof for forward flight uses singular perturbation analysis — a two-time-scale decomposition that writes the inertia matrix as $J = \varepsilon J_0$ with a small parameter $\varepsilon$, treating angular velocity as a fast boundary layer and attitude as the slow system, and yielding sufficient stability conditions on the small parameter $\varepsilon$ — together with a composite Lyapunov function (an energy-like scalar function that decreases along closed-loop trajectories and thereby certifies stability), establishing global asymptotic stability of the equilibrium set.

All experiments in the source paper were conducted in an indoor motion-capture laboratory: an external camera system measured the vehicle's position and attitude, and control commands were computed on the ground and transmitted to the vehicle. Onboard sensing with micro-electromechanical inertial sensors (an IMU measures angular rates and specific force; a MARG sensor adds a magnetometer) is difficult for flapping vehicles because the wing beats inject severe periodic oscillations at the flapping frequency into every measurement. The paper's experimental comparison covered attitude tracking accuracy against two representative controllers from the literature, but not energy consumption, and the hysteresis thresholds and the bound on $\varepsilon$ are derived only as abstract stability conditions, never quantified for the actual vehicle's measured parameters. Subsequent work by the same group and others has advanced neighboring capabilities — neural-network 3D position control, simulation-to-real transfer, minimum-snap trajectory generation with differential flatness (the property that all states and inputs of a vehicle can be rewritten as algebraic functions of a suitable output and its derivatives, so trajectories can be planned directly in that output space), and attitude estimation that learns flapping-induced oscillations — but the field's 2025–2026 reviews still list full autonomy, energy efficiency, and robust onboard estimation for small flappers as open challenges.

Problem Statement

The problem, as stated by the source paper and still open, is to carry the demonstrated quaternion-based hybrid attitude-control framework for the underactuated (yaw-uncontrollable, aileron-less) flapping-wing aerial vehicle from its laboratory proof of concept to a practical, verifiable, and design-transferable flight-control capability, along the five linked open directions that the paper identifies:

1. Aggressive flight regimes: extend the controller beyond hovering and slow forward flight to higher speeds and sharp turns, where velocity-induced aerodynamic torques (pendulum-like pitch and wind-vane-like yaw couplings) are present, within the limited control authority of the tail-based actuation.
2. Onboard autonomy: replace the external motion-capture system with onboard sensing (IMU/MARG, optionally vision) and onboard computation, which requires state estimation robust to the severe periodic oscillations at the 13–16 Hz flapping frequency (and to flapping-frequency drift), enabling autonomous flight outside the laboratory.
3. A priori parameter quantification: replace the purely abstract stability-derived conditions with constructive tuning guidelines that map the measurable parameters of a given vehicle design (inertia, aerodynamic coefficients, actuator limits) to the hysteresis thresholds $\delta$ and the singular-perturbation time-scale bound $\varepsilon$ (with $J=\varepsilon J_0$), including, optionally, adaptive alternatives, so that the controller parameters are quantified before flight rather than tuned empirically.
4. Energy accounting: deliver a comprehensive energy-consumption analysis of the controller compared with at least two named baseline control methods under identical experimental conditions on the source vehicle's parameters, covering at least hovering and forward flight.
5. Framework generality: extend the hybrid reduced-quaternion control framework to other underactuated flapping-wing configurations, in particular tailless vehicles with only two wings, where removal of the tail eliminates control surfaces and aggravates the yaw–tilt coupling.

The common objective uniting these directions is practical, onboard-autonomous, aggressively-capable, and design-transferable flight control for underactuated flapping-wing vehicles under the hybrid quaternion framework. The five directions are independent open targets identified in the paper's future work under this common objective; they are not a conjunctive requirement, and a solution may treat any one of them in depth, the remaining directions staying open targets rather than prerequisites. Throughout, practical'' and design-transferable'' are understood operationally: a method or tuning rule counts as such when it is specified in terms of a vehicle's measurable design parameters (inertia, aerodynamic coefficients, actuator limits) rather than a single hand-tuned vehicle, and is supported by simulation and/or hardware validation. A chosen direction resolves the task when the solver delivers, for a named vehicle design, a documented method, result, or analysis that addresses that direction and is validated by simulation and/or hardware; treating a direction in depth means producing such a deliverable, not a preliminary sketch or an unbounded survey.

References

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