Neural-Fly Enables Rapid Learning for Agile Flight in Strong Winds

1 INTRODUCTION

• DAIML learns a wind-independent aerodynamic DNN basis from 12 minutes across six wind conditions, then composite adaptation updates wind-specific linear coefficients using tracking and force-prediction errors.Spectral normalization supports generalization and stability guarantees, while a discriminative network places wind-dependent information in the adapted coefficients.Source: §1 · p. 3
• Neural-Fly reports average improvements of 66% over nonlinear tracking, 42% over L1 adaptive, and 35% over INDI controllers.It also achieves centimeter-level position-error tracking and demonstrates agile flight through low-clearance gates in gusty winds on standard quadrotor hardware.Source: Figure 1 · p. 2, §1 · pp. 3, 5

Related Work for Precise Quadrotor Control

• Unlike specialized approaches requiring optical motor sensing or custom stacks, Neural-Fly integrates with PX4; prior GP and drag models lack adaptation to changing winds or fast-flight scope.The cited alternatives include systems requiring high-frequency updates and direct motor-speed feedback, and models limited to no-wind or horizontal-plane flight.Source: Related Work for Precise Quadrotor… · p. 5

Relation between Neural-Fly and Conventional Adaptive Control

• Neural-Fly combines learned compact basis functions with regularized composite adaptation to achieve fast adaptation and closed-loop exponential stability and robustness without persistent excitation.This addresses basis redundancy and stability challenges; direct residual-force estimators can lag on agile trajectories and in strong winds.Source: Relation between Neural-Fly and… · p. 6

Related Work in Multi-environment Deep Learning for Robot Control

• Rather than fine-tuning full DNNs, Neural-Fly adapts low-dimensional weights on a shared representation for control-rate response, achieving real-drone performance with stability and robustness guarantees.This approach avoids the onboard speed, explainability, and stability concerns of adapting whole networks; unlike domain randomization, it targets precise adaptation to specific environments.Source: Related Work in Multi-environment Deep… · p. 7

Experimental Platform

• The wind tunnel generates constant or time-varying wind, with speeds up to 43.6 km/h, while experiments use a custom quadrotor and onboard control at 50 Hz.The tunnel has 1,296 independently controlled fans; the controller runs on the onboard Linux computer and sends commands to the PX4 flight controller.Source: Experimental Platform · p. 9

Offline Learning and Online Adaptive Control Development

• DAIML learns a wind-shared aerodynamic representation whose linear coefficients encode wind-specific information and cluster by similar wind conditions.Training used 36,000 samples from six static wind conditions, collected over two minutes per condition; experiments tested winds up to 43.6 km/h, beyond the training maximum of 6.1 m/s.Source: Offline Learning and Online Adaptive… · p. 10, Figure 4 · p. 10
• Neural-Fly uses a learned representation, Neural-Fly-Transfer uses one trained on another drone, and Neural-Fly-Constant uses a constant mapping; comparisons include nonlinear, INDI, and L1 baselines.Neural-Fly-Constant uses the online adaptation algorithm with a constant mapping; the variants test the contribution of a learned representation and cross-drone transfer.Source: Offline Learning and Online Adaptive… · p. 11

Trajectory Tracking Performance

• Neural-Fly and Neural-Fly-Transfer outperform all other methods across tests, including winds beyond training; learning improves adaptive control, which outperforms the nonlinear baseline.Neural-Fly slightly outperforms Neural-Fly-Transfer, whose representation was trained on a different drone. The tests include constant winds up to 43.6 km/h and time-varying wind.Source: Trajectory Tracking Performance · pp. 11–12, Figure 6 · p. 13

Agile Flight Through Narrow Gates

• The agile-flight evaluation uses two 3 m figure-8 tasks and a 3 m by 2.5 m ellipse, each with a 5 s lap time and gates.One figure-8 task uses a time-varying wind of 3.1 + 1.8 sin(2π 5 t) m/s; the other tasks use 3.1 m/s wind.Source: Agile Flight Through Narrow Gates · pp. 12–13
• Neural-Fly precisely tracks all three agile trajectories and safely passes through gates only slightly wider than the drone.The first task includes about 10 cm minimum clearance; the tasks use 3.1 m/s wind, with the second task adding time-varying wind.Source: Agile Flight Through Narrow Gates · pp. 12–13

Outdoor Experiments

• Outdoor testing with GPS-based state estimates in winds measured up to 17 km/h achieved 7.5 cm mean tracking error on the figure-8 trajectory.The passage notes that GPS provides lower-precision state estimation and less precise aerodynamic residual-force estimation.Source: Outdoor Experiments · p. 14

State-of-the-art Tracking Performance

• Neural-Fly outperforms nonlinear control, INDI, and L1 across tested wind conditions up to 43.6 km/h, with 2.9 cm mean tracking error in 0 km/h wind.Experiments tracked an agile figure-8 using the standard PX4 attitude controller, with Neural-Fly running onboard on a Raspberry Pi 4.Source: State-of-the-art Tracking Performance · p. 14
• The localization system’s roughly 1 cm precision is a practical lower bound; code execution and communication delays measured at least 15 ms and often 30 ms, while attitude control remains a possible extension.Reducing execution and communication delays, and using a nonlinear attitude controller, could help approach localization-level errors; attitude-control implementation still requires study of interactions with cascaded control.Source: State-of-the-art Tracking Performance · pp. 14–15
• With wind around 17 km/h, Neural-Fly maintained precise outdoor tracking with 7.5 cm error using less precise state estimation and no dependence on test-facility features.The outdoor setup used GPS for state estimation; the maximum wind speed during the test was around 17 km/h (4.9 m/s).Source: State-of-the-art Tracking Performance · p. 15, Figure 7 · p. 15

Challenges Caused by Unknown and Time-varying Wind Conditions

• Although trained on static, approximately uniform winds, Neural-Fly quickly identifies unknown changing wind speeds and maintains precise tracking, with average tracking error below 8.7 cm in time-varying wind.The method separates the wind-invariant state representation from wind-specific linear weights, enabling adaptation while following a dynamic trajectory.Source: Challenges Caused by Unknown and… · p. 15

Computational Efficiency of Our Method

• DAIML learns the representation from 12 minutes of flight data, while online adaptation takes 10 ms on a Raspberry Pi 4.Offline training takes 5 minutes on a desktop; onboard feedforward inference takes about 3.5 ms and the adaptation update about 3.0 ms.Source: Computational Efficiency of Our Method · p. 16

Generalization to New Trajectories and New Aircraft

• Neural-Fly supports different trajectories and quadrotors, including a horizontal ellipse demonstration, and its learned representation remains robust to changes in vehicle design.Further work is needed on coupling planning with learning-based control near actuation limits; the authors propose studying a combined MPC planning-and-control structure.Source: Generalization to New Trajectories and… · p. 16

Overview

• The dynamics model treats wind as a potentially time-varying hidden state that changes the UAV’s unmodeled aerodynamic disturbances; Neural-Fly represents these with shared basis functions and condition-specific coefficients adapted online.The shared representation is learned offline with DAIML, while a composite adaptive-control law updates the coefficients and is designed for fast adaptation with global exponential stability and robustness.Source: Overview · pp. 16–17

Data Collection

• Training data came from randomized spline trajectories flown for 2 minutes in each of six static wind conditions spanning 0–22.0 km/h (0–6.1 m/s).The drone tracked splines through three waypoints; fifth-order numerical differentiation estimated acceleration, which was combined with the dynamics equation to obtain noisy measurements of unmodeled dynamics. Data collection was also performed on both the custom-built drone and Intel Aero RTF drone.Source: Data Collection · p. 17

The Domain Adversarially Invariant Meta-Learning (DAIML) Algorithm

• DAIML learns a DNN representation shared across wind conditions, with condition-specific linear coefficients that approximate the unmodeled dynamics.The method seeks a representation ϕ(x) such that ϕ(x)a(w) approximates f(x,w); the paper states that this structure can approximate any analytic f to arbitrary precision with enough DNN neurons.Source: The Domain Adversarially Invariant… · p. 18
• Because wind shifts the state distribution, the representation could otherwise encode wind-specific state patterns and overfit instead of learning wind-invariant dynamics.DAIML adds a discriminator that predicts the wind-condition index from the representation, while training the representation to make that prediction difficult as it approximates the disturbance labels.Source: The Domain Adversarially Invariant… · p. 18

• The method randomly samples a wind domain D_wk, then draws disjoint adaptation and training batches from it.Source: §1 · p. 19
• DNN ϕ is trained using SGD and spectral normalization with a loss.Source: §1 · p. 19
• DNN h is trained using SGD with loss P.Source: §1 · p. 19

10 until convergence

• DAIML alternates least-squares adaptation, representation training, and discriminator regularization, backpropagating through the adapted coefficients and constraining their norm.It uses a closed-form least-squares solution, spectral normalization, and updates the discriminator with probability η ≤ 1 per iteration.Source: §10 · p. 19
• As training progresses, adapted coefficients cluster by wind condition, supporting a shared representation with wind-specific linear weights; random features and α = 0 yield less disentangled coefficients.The t-SNE visualization associates the plot’s upper-right region with higher wind speed.Source: §10 · pp. 19–20

Robust Adaptive Controller Design

• Neural-Fly adapts online with a composite law combining tracking error, force-prediction error, and regularization, with covariance-based gain tuning enabling adaptation without persistent excitation.The Kalman-filter-inspired estimator adapts quickly when a new learned-model mode is excited.Source: Robust Adaptive Controller Design · pp. 20, 22
• Neural-Fly and Neural-Fly-Transfer outperform Neural-Fly-Constant, whose fixed identity basis omits learning, by predicting aerodynamic disturbances online with a compact learned representation.Neural-Fly trains its representation on the same drone; Neural-Fly-Transfer uses data collected with an Intel Aero RTF drone.Source: Robust Adaptive Controller Design · p. 21
• With bounded desired-trajectory derivatives, position tracking error exponentially converges to a ball whose size depends on representation error, measurement noise, and the rate of parameter change.The controller and adaptation law are designed for robustness to imperfect learning and time-varying winds.Source: Robust Adaptive Controller Design · pp. 20–22

Implementation Details

• The quadrotor implementation uses an 11-dimensional input—velocity, quaternion attitude, and rotor PWM—with a four-hidden-layer network shared across the three wind-force components.The architecture is 11 → 50 → 60 → 50 → 4; the previous time step’s PWM is used to compute current desired force, with spectral normalization supporting convergence to a fixed point.Source: Implementation Details · p. 23

Supplementary materials

• Sections S1–S3 cover drone configuration, learning-architecture expressiveness, and DAIML hyperparameters and interpretation.Source: SUPPLEMENTARY MATERIALS · p. 23
• Sections S4–S6 cover the discrete controller, stability and robustness proofs, and gain tuning.Source: SUPPLEMENTARY MATERIALS · p. 23
• Sections S7–S8 cover force-prediction performance and localization-error analysis.Source: SUPPLEMENTARY MATERIALS · p. 23

SUPPLEMENTARY MATERIALS

• The hardware discussion attributes many agile-flight advantages to specialized design, including lower vehicle radius, higher thrust-to-weight ratio, and faster motor response.The authors describe their custom drone as more representative of typical consumer hardware.Source: SUPPLEMENTARY MATERIALS · p. 29
• Neural-Fly achieves state-of-the-art tracking performance on more standard hardware with all control running onboard, while most methods require highly maneuverable quadrotors and onboard or offboard computation.Direct performance comparisons are difficult because configurations, controller tuning, and flight arenas vary.Source: Table S2 · p. 30

The Expressiveness of the Learning Architecture

• Any analytic function of state and wind can be approximated to arbitrary precision as a state-only representation mixed with a wind-dependent coefficient vector.The theorem constructs this factorization using Chebyshev polynomials.Source: The Expressiveness of the Learning… · pp. 30–31
• The wind-dependent coefficient dimension scales as O((log(1/ϵ))^m), where m is the wind-input dimension.This gives a polylogarithmic dependence on inverse approximation error.Source: The Expressiveness of the Learning… · pp. 30–31
• Because deep neural networks approximate polynomials, ϕ(x)a(w) can approximate the underlying dynamics f(x,w) to arbitrary precision.The paper reports that a four-layer network efficiently learns an effective representation in experiments.Source: The Expressiveness of the Learning… · p. 31
• The reported ϕ network architecture is 11 → 50 → 60 → 50 → 4, while the h network is 4 → 128 → 6; both use ReLU activations.Training settings include batch sizes 128 and 256, learning rates 0.0005 and 0.001, discriminator frequency 0.5, normalization constant 10, and regularization degree 0.1.Source: The Expressiveness of the Learning… · p. 32

Section S3

• DAIML was implemented in PyTorch, tuned by trial and error, and reported to be insensitive to most parameters in Table S3.The passage says the hyperparameters are reported in Table S3.Source: Hyperparameters for DAIML and the… · p. 32
• DAIML’s final training loss is compared across training and validation sets, with different trajectories used to examine when the learned model begins to overfit.The validation set uses figure-8 tracking with PID and nonlinear baselines, while training uses random-waypoint tracking.Source: Hyperparameters for DAIML and the… · p. 32
• The discriminator update frequency η controls how often h is updated; η = 0.5 is used for training stability.η = 1 updates both ϕ and h in every iteration.Source: Hyperparameters for DAIML and the… · p. 32
• Setting α = 0 removes adversarial regularization; a proper α can reduce overfitting, while an excessively high value may degrade prediction, so α = 0.1 is recommended.With α > 0, the learned model is described as more explainable, and its linear coefficients a* are disentangled between wind conditions.Source: Hyperparameters for DAIML and the… · p. 33
• Without domain invariance, an expressive network may memorize separate modes for nonoverlapping wind-condition distributions, triggering the wrong mode in a new condition.For example, flying through data from condition 1 while in condition 2 can trigger mode 1, making the representation non-robust and nongeneralizable.Source: Hyperparameters for DAIML and the… · p. 33, Figure S2 · p. 33
• The discriminator encourages latent-space diversity, and DAIML produces a much more disentangled latent space.The passage identifies Figure 4 as showing this result.Source: Hyperparameters for DAIML and the… · p. 33

Discrete Version of the Proposed Controller

• A naive one-step Euler integration can make P non-positive-definite, so the discrete controller uses separate propagation and update steps.This two-step design mirrors a discrete Kalman filter and exactly integrates the continuous-time filter when prediction error, tracking error, and basis functions remain constant between steps.Source: Discrete Version of the Proposed… · p. 34

Section S5

• Neural-Fly’s combined velocity-tracking and adaptation errors exponentially converge to a bounded ball, yielding exponential convergence of velocity error and supporting the paper’s stability bound.The proof combines Theorem 4’s bound with Corollary 5’s bound for position error.Source: Stability and Robustness Formal… · pp. 34, 35, 37
• The stability proof does not extend to arbitrary online estimators: bounded parameter-estimation error is not guaranteed, so Neural-Fly’s adaptation algorithm must establish stability and robustness.The proof relies on P^-1 being uniformly positive definite and uniformly bounded.Source: Stability and Robustness Formal… · p. 35
• Corollary 5 assumes ∥s(t)∥ ≤ A exp(−αt) + B/α for constants A, B, and α, with s = q̃̇ + Λq̃.Source: Stability and Robustness Formal… · p. 37
• Even with ϕ ≡ 0, the analysis gives a strictly positive convergence rate, ensuring robustness without persistent excitation or effective learning.In this case, the P equation is stable and P exponentially converges to Q/(2λ).Source: Stability and Robustness Formal… · p. 37
• For Neural-Fly-Constant, ϕ = I and λ = 0 remove the need for regularization; under the stated conditions, the bound simplifies and constant f permits perfect tracking.The passage notes that f is expected to be constant for quadrotors in hover or steady level flight with constant wind velocity.Source: Stability and Robustness Formal… · p. 38

Gain Tuning

• Position-controller P/D gains were tuned on a 1 m x-direction step using the baseline nonlinear controller, then held constant across methods.The attitude controller was tuned following.Source: Gain Tuning · p. 38
• Integral and adaptation gains were increased separately for each method to reduce response time until overshoot, jitter, or oscillations became large or noticeable.Source: Gain Tuning · p. 38
• For L1 and INDI, tuning produced first-order filters with a 5 Hz cutoff frequency.Source: Gain Tuning · p. 38
• Neural-Fly methods used R = rI and Q = qI, with R gains similar to residual-force measurement noise, Q on the order of 0.1, and λ = 0.01.Source: Gain Tuning · p. 38
• INDI, L1, and Neural-Fly-Constant estimated residual force with sub-second lag; Neural-Fly and Neural-Fly-Transfer reduced lag but showed model mismatch, especially at higher wind speeds.The figure also notes that decreasing lag through gain adjustment increases noise amplification.Source: Figure S3 · p. 39
• The nonlinear baseline’s integral term and adaptive methods’ ˆf term compensated for the measured residual force.Source: Figure S3 · p. 39

Force Prediction Performance

• The nonlinear baseline’s integral term compensates for average wind effects, keeping trajectories roughly centered, but cannot react quickly enough to changing relative velocity.Increasing integral gain may improve responsiveness, but risks overshoot and instability given the combined P, D, and I gains.Source: Force Prediction Performance · p. 40
• INDI, L1, and Neural-Fly-Constant filter measured residual force, so their adaptation lags slightly; increasing adaptation gain reduces lag but amplifies measurement noise.These reactive approaches are consequently limited by system properties such as measurement noise.Source: Force Prediction Performance · p. 40
• After its linear parameters adapt to current wind, Neural-Fly and Neural-Fly-Transfer use a learned model to predict future aerodynamic effects with minimal coefficient changes.This learned-model approach differs from controllers that react to measured residual force.Source: Force Prediction Performance · p. 40
• Under extrapolation to higher wind speeds and time-varying conditions, model mismatch causes predicted and measured forces to diverge, and tracking error scales linearly with representation error.The limitation matches the paper’s theoretical analysis.Source: Force Prediction Performance · p. 40

Localization Error Analysis

• The OptiTrack motion-capture system measures reflective-marker centroids, while the EKF estimates the vehicle’s center of mass; their fixed mean offset reflects this difference.The offline analysis corrects for OptiTrack delay and treats the standard deviation of the position difference as the RMS error between estimates.Source: Localization Error Analysis · p. 40, Figure S4 · p. 41
• 1.0 cm is the estimated standard deviation of localization error, averaged over all experiments in the paper.The estimate comes from comparing delay-corrected OptiTrack positions with EKF position outputs; the mean difference is a fixed offset between the tracked points.Source: Localization Error Analysis · p. 40, Figure S4 · p. 41