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REVIEW 3 major objections 5 minor 42 references

Learning-based Trajectory Tracking for Bird-inspired Flapping-Wing Robots

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A model-free RL policy tracks aerobatic 3D trajectories for a flapping-wing robot in simulation.

desk verdict Solid RL flapping-wing tracking demo in MuJoCo with an overreaching asymptotic-stability claim that should be softened. read the letter →

arxiv 2411.15130 v1 pith:2TXGZAHP submitted 2024-11-22 cs.RO cs.SYeess.SY

Jiaze Cai , Vishnu Sangli , Mintae Kim , Koushil Sreenath This is my paper · ORCID
classification cs.ROcs.SYeess.SY
keywords flapping-wingrobotornithopterreinforcementlearningtrajectorytrackingPPOMuJoCodomainrandomizationsystemidentification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a model-free reinforcement-learning controller can make a high-degree-of-freedom, bird-inspired flapping-wing robot track complex 3D trajectories, including loops, sharp turns, and roll-off-the-bottom maneuvers, while also maintaining stable, periodic flapping and spontaneously switching between flight modes. This is shown entirely in a MuJoCo simulation where the aerodynamic model is a stateless ellipsoid/inertia approximation with hand-tuned coefficients. The authors add that the closed-loop system is locally input-output stable, based on linear transfer functions identified from simulated input-output data, and that the controller remains robust under wind disturbances and randomized aerodynamic conditions. If correct, this would mean that model-free RL can substitute for hand-derived aerodynamic models in ornithopter control, a task that classical model-based methods have only partially solved.

What carries the argument

The load-bearing machinery is the MuJoCo simulation environment with its stateless aerodynamic model, in which lifting bodies are modeled as ellipsoids with five aerodynamic force contributions (added mass, viscous drag, Magnus lift, Kutta lift, and viscous resistance) and the main body is modeled with a simplified inertia model. The fluid coefficients are manually tuned to match a designed lift-to-drag ratio at gliding, and this model is embedded in a curriculum-based PPO training pipeline with three progressively harder stages (constant forward flight, climbing/diving, then turning and arbitrary maneuvers) followed by domain randomization of masses, inertias, aerodynamic coefficients, added mass and inertia, and wind velocity. The identified third-order LTI transfer functions in x, y, and z are the device used to argue closed-loop stability.

What would settle it

Measure the actual lift and drag of the physical flapping-wing platform across the flapping frequencies used by the policy (4–6 Hz) and compare them against MuJoCo predictions under identical kinematics; a substantial discrepancy would indicate that the policy is exploiting a simulation artifact. A more direct test is to deploy the trained policy on the physical robot and check whether it can sustain stable straight-line flight and track a simple trajectory at all.

Watch

Extended reading notes

Core claim

The central claim is that a single model-free RL policy, trained with PPO through a curriculum and heavy domain randomization, can act as a trajectory-tracking controller for a simulated 11-DoF flapping-wing robot (5 actuated wing and tail joints plus a 6-DoF floating base). The policy outputs target joint positions at 50 Hz that are low-pass filtered and passed to a low-level PD controller running at 250 Hz, and this closed loop tracks straight, climbing, diving, gliding, turning, loop, and roll-off-the-bottom maneuvers. The stability argument is built from system identification: a third-order linear time-invariant transfer function is fit in each spatial axis to the closed-loop input-output behavior, and the identified poles all lie in the left-half plane, indicating bounded-input bounded-output (BIBO) stability and minimum-phase behavior. Phase portraits of the wing flap and pitch joints show closed periodic orbits during forward flight, climbing, and turning, which the paper interprets as stable and periodic joint action patterns.

Load-bearing premise

The entire demonstration rests on the assumption that MuJoCo's stateless aerodynamic model, with fluid coefficients manually tuned to match a gliding lift-to-drag ratio, faithfully represents real flapping-wing flight; if that model is not representative of unsteady flapping-wing aerodynamics, the learned tracking and stability conclusions do not transfer to a physical robot.

Editorial extensions

If this is right

  • If the RL control claim holds, model-free reinforcement learning could become a practical alternative to model-based control for bird-sized flapping-wing platforms, removing the need for analytic aerodynamic models.
  • The curriculum-plus-domain-randomization training scheme may transfer to other high-degree-of-freedom, underactuated flying robots, including morphing-wing and bat-like platforms.
  • The stability-analysis method, fitting low-dimensional linear models to the closed-loop input-output behavior of a learned policy, could be reused to certify other learned flight controllers beyond this platform.
  • The single policy's spontaneous switching between flight modes suggests that a single learned controller can replace hand-tuned supervisory logic that selects among separate controllers for cruise, climb, dive, and turn.
  • The policy's robustness to randomized aerodynamic coefficients and wind in simulation points toward a potentially deployable controller once the sim-to-real gap for flapping-wing aerodynamics is closed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An untested but implicit implication is that the same approach would transfer to physical hardware, but the stateless aerodynamics model omits unsteady effects such as leading-edge vortices and wing flexibility, so the demonstrated tracking and stability may not survive on a real robot.
  • A testable extension would be to compare the learned flapping frequency (4–6 Hz) and wing kinematics against measured data from biological birds or existing ornithopters of similar size; a mismatch would indicate that the policy exploits simulation artifacts rather than physical aerodynamics.
  • The BIBO stability of the identified local linear model does not by itself establish asymptotic stability of the full nonlinear stochastic closed-loop system; a stronger certificate, such as a Lyapunov function on the original dynamics, would be needed to make the stability conclusion robust.
  • The paper's sensitivity analysis shows the policy is most affected by the Kutta lift coefficient, which suggests that improving the fidelity of that single aerodynamic term could be the highest-impact step toward real-world transfer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents a model-free reinforcement learning framework for trajectory tracking of a five-joint, bird-inspired flapping-wing robot simulated in MuJoCo. The policy receives joint, attitude, and velocity observations, a history window, and a look-ahead trajectory buffer, and outputs target joint positions passed through a low-pass filter and a joint PD controller. Training uses a three-stage curriculum plus dynamics and aerodynamic randomization. The authors evaluate tracking on longitudinal, lateral, and aerobatic trajectories, test robustness to winds and aerodynamic coefficient randomization, perform linear system identification on the closed-loop input-output behavior, and present phase portraits of wing joints. The central claims are that the RL policy achieves multimodal agile tracking and that the closed-loop system is asymptotically stable.

Significance. The empirical simulation results are a useful contribution: they show that a single model-free policy can track diverse trajectories, including loops and turns, on a high-degree-of-freedom flapping-wing model, and the ablation-style randomization study gives a first-order sensitivity ranking of aerodynamic coefficients. The paper is transparent about the simulation-only status (the hardware platform is still in design) and about the simplifying stateless aerodynamics. Those strengths make the tracking claim credible within the simulator. The theoretical stability claim, however, is not supported by the presented evidence, and the manuscript should be revised to align the formal statements with what the analysis actually shows.

major comments (3)
  1. [IV.A.2, V] The conclusion that the closed-loop system is 'asymptotically stable' is not supported. The only evidence is the left-half-plane pole locations of three identified LTI transfer functions (Eqs. 9–11), which establish bounded-input/bounded-output stability for those fitted linear models. BIBO stability of an approximate input-output model does not imply asymptotic stability of the underlying nonlinear closed-loop system, nor does it address orbital stability of the periodic flapping orbit that the phase portraits in Sec. IV.B exhibit. The text in Sec. IV.A.2 correctly limits the claim to 'locally input-output stable'; the Conclusion should state the same limitation, or the authors should provide a nonlinear stability analysis (e.g., Lyapunov or contraction arguments) or a Poincaré-section analysis.
  2. [IV.A.1] The system-identification section does not report validation on held-out data, excitation conditions, or uncertainty bounds. The MSE of 5.629e-5 appears to be a fit error on the identification input-output pairs; as such it does not certify that the low-dimensional LTI model captures the closed-loop dynamics outside the particular fitted trajectory. Please report train/test MSE, the identification signal (e.g., chirp or random step sequence), and the operating region over which the fit is valid, and restrict stability conclusions to that region.
  3. [II.C, III.E] All flight and robustness results are obtained with MuJoCo's stateless ellipsoid fluid model, whose coefficients are hand-tuned to match only the designed glide lift-to-drag ratio. The paper itself acknowledges this as a simplifying approximation and states that the physical robot is still in design. The tracking and robustness claims therefore should be framed as properties of the simulation model, not of a physical flapping-wing aircraft; an unqualified statement such as 'achieve stable flight' in the abstract may be read as a real-world claim. A concrete step would be to evaluate the trained policy in a higher-fidelity unsteady aerodynamic solver (e.g., UVLM or CFD) or on the physical platform once available, and to report differences in tracking error and success rate.
minor comments (5)
  1. [Fig. 4 caption] The caption lists 'wing pitch angle' twice; the first occurrence should probably be 'wing flap angle'.
  2. [III.E] 'MoJoCo' should be 'MuJoCo' in the paragraph on aerodynamic randomization.
  3. [I.B] The phrase 'Unsteady V ortex Latex Method' appears to be a typo for 'Unsteady Vortex Lattice Method'.
  4. [IV.A.1] The sentence 'the input of the closed system is determined by the input of the policy network' is unclear; please clarify how the desired trajectory is converted into the LTI input u and what exactly is measured as the output.
  5. [General] No code or simulation environment is provided; a public release would improve reproducibility and allow independent assessment of the system-identification and stability analysis.

Circularity Check

1 steps flagged · score 2.0 of 10

No equation-level circularity: tracking results are direct policy evaluations, and the only self-citation supplies methodology, not the result. A minor in-sample fit metric is presented as a 'predicted result.'

  1. fitted input called prediction [Sec. IV.A.1, Eqs. (9)-(11) and the following note]
    "The LTI system is obtained by fitting the input-output pairs, where the input of the closed system is determined by the input of the policy network. The fitted input-output dynamics of position is given by: ... Note that the derived linear model predicted result with a Mean Squared Error (MSE) as low as 5.629 × 10−5."

    The transfer functions in Eqs. (9)-(11) are explicitly 'obtained by fitting the input-output pairs,' so the subsequent 'predicted result' and its MSE are properties of the same fitted model on the fitting data. No held-out or cross-validation split is reported, so as presented the MSE is the in-sample fitting residual renamed as a prediction. This is a minor instance of a fitted quantity being presented as a predictive validation, though it is not load-bearing for the main trajectory-tracking demonstrations.

full rationale

The core results are self-contained in simulation: the RL policy is trained on a reward that penalizes position error, energy, and orientation, and the tracking demonstrations in Sec. IV.C are direct rollouts of that policy against procedurally generated trajectories. These are not derived from the fitted linear model. The stability analysis, by contrast, fits third-order LTI transfer functions to closed-loop input-output data and then reads BIBO stability from pole locations. That inference is an overreach—BIBO stability of an identified LTI model does not establish the asymptotic stability of the nonlinear closed-loop system claimed in Sec. V—but it is an inferential gap rather than a circular reduction: the pole locations are not encoded as an input to the fit, and the fit itself is not claimed to be derived from the stability conclusion. The method is attributed to a self-citation, Ref. [42], whose authors overlap with the present paper, but the paper actually performs the identification and pole check itself, so the citation is methodological rather than load-bearing. The paper also candidly states the limits of its aerodynamic model in Sec. II.C and III.E ('state-less fluid force models ... a simplifying approximation that is made'; 'hand-tuned fluid coefficients may not match the physical model'), and Sec. V notes that hardware validation is future work. These statements weaken the generality of the claims but are not circularity. Overall, the central claims have independent content; the only circular-adjacent feature is the in-sample MSE presented as a 'predicted result,' which warrants a low score rather than a finding of substantive circularity.

Assumptions & free parameters 7 free parameters · 4 assumptions · 1 invented entities

The central claims rest on a custom simulation whose aerodynamic coefficients are hand-tuned, a rigid-body assumption that ignores aeroelasticity, and a stability argument based on a fitted linear model. Removing these parameters leaves a standard PPO training pipeline applied to a bespoke simulator. The paper does not ship code or data, so the ledger entries are assumptions, not independently verified inputs.

free parameters (7)
  • CD_blunt = 0.2
    Manually tuned with the other fluid coefficients to match the designed lift-to-drag ratio at gliding (Section II.C).
  • CD_slender = 0.12
    Manually tuned to match glide lift-to-drag ratio (Section II.C).
  • CD_angular = 1.5
    Manually tuned angular drag coefficient (Section II.C, Table II).
  • CK = 3.14
    Kutta lift coefficient manually tuned to glide performance; paper reports high sensitivity to this coefficient (Section IV.C).
  • CM = 1
    Magnus force coefficient manually tuned (Section II.C, Table II).
  • Reward weights = 0.5, 0.1, 0.2, 0.05
    Hand-selected weights for position, angular rate, orientation, and energy terms in Eq. 8; no ablation is reported.
  • Low-pass filter cutoff = 7 Hz
    Chosen to cap flapping frequency to 4-6 Hz, which shapes the 5.3 Hz behavior presented as a result (Section III.A).
assumptions (4)
  • domain assumption All bodies of the robot are rigid; wing flexibility is handled only through domain randomization.
    Stated in Section II.B despite the introduction identifying aeroelastic coupling as a key challenge.
  • domain assumption MuJoCo's stateless inertia and ellipsoid fluid models approximate the aerodynamic wrench on each body.
    Section II.C says the fluid force depends only on robot state, with no unsteady flow state, so leading-edge vortex and related unsteady effects are absent.
  • ad hoc to paper Hand-tuned aerodynamic coefficients match the real robot's lift-to-drag ratio at gliding and remain valid during dynamic maneuvers.
    Section II.C tunes five coefficients to one steady glide condition; no validation is given for aggressive maneuvers.
  • ad hoc to paper A low-dimensional LTI system fitted to input-output data represents the closed-loop nonlinear dynamics for stability analysis.
    Section IV.A fits transfer functions to simulation data and uses pole locations to conclude closed-loop stability, which is an extrapolation from the fitted model.
invented entities (1)
  • The five-joint bird-inspired flapping-wing robot platform
    purpose: Simulation testbed for the RL controller, with 0.995 m wingspan, 0.31 kg mass, and five actuated joints.
    The hardware is stated in the conclusion to be in the process of design, so the platform exists only in simulation. No physical prototype or measured data support the model.

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Cite this review

Pith. "Pith review of Learning-based Trajectory Tracking for Bird-inspired Flapping-Wing Robots." pith.science (2026). https://pith.science/paper/2TXGZAHP

@misc{pith2026241115130,
  author       = {Pith},
  title        = {Pith review of: Learning-based Trajectory Tracking for Bird-inspired Flapping-Wing Robots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2TXGZAHP}},
  note         = {Machine review of arXiv:2411.15130}
}
read the original abstract

Bird-sized flapping-wing robots offer significant potential for agile flight in complex environments, but achieving agile and robust trajectory tracking remains a challenge due to the complex aerodynamics and highly nonlinear dynamics inherent in flapping-wing flight. In this work, a learning-based control approach is introduced to unlock the versatility and adaptiveness of flapping-wing flight. We propose a model-free reinforcement learning (RL)-based framework for a high degree-of-freedom (DoF) bird-inspired flapping-wing robot that allows for multimodal flight and agile trajectory tracking. Stability analysis was performed on the closed-loop system comprising of the flapping-wing system and the RL policy. Additionally, simulation results demonstrate that the RL-based controller can successfully learn complex wing trajectory patterns, achieve stable flight, switch between flight modes spontaneously, and track different trajectories under various aerodynamic conditions.

Figures

Figures reproduced from arXiv: 2411.15130 by the authors.

Figure 1
Figure 1. The layout of the flapping-wing robot. In simulation, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. The flapping-wing robot follows a loop trajectory generated from simulation. The robot performs an Immelmann [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. The control diagram for flapping-wing robot trajec [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Forward flapping behavior of the controller shown [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: System identification is performed on the closed-loop [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Pole-zero plots for the closed-loop system’s linear [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 9
Figure 9. Figure 9: Example longitudinal and lateral trajectories to dis [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: Effect of randomization on trajectory following. [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 13
Figure 13. Figure 13: An aerobatic trajectory consisting of a loop (orange), [PITH_FULL_IMAGE:figures/full_fig_p008_13.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.