Statistical Turbulence and High-Fidelity Disturbance Fields for Quadrotor Flight Control

Xun Huang

physics.flu-dynSep 9, 2026

Statistical Turbulence and High-Fidelity Disturbance Fields for Quadrotor Flight Control

Authors: Xun Huang

Organizations: State Key Laboratory of Turbulence and Complex Systems, Department of Aeronautics and Astronautics, Peking University, Beijing, 100871, People’s Republic of China

Abstract

Reinforcement-learning quadrotor controllers are usually trained under simplified wind models, yet the impact of wind-field fidelity, as opposed to magnitude, on policy robustness remains unquantified. This paper compares five disturbance-fidelity levels, from wind-free flight and discrete 1-cosine gusts through statistical turbulence and synthetic coherent structures to large-eddy-simulation fields of the atmospheric boundary layer, in a full cross-fidelity train test evaluation of proximal policy optimization (PPO) agents, with cascaded PID and geometric SE(3) controllers as training-free references, over a 0-12 m/s wind sweep. Before any controller comparison is made, all disturbance data are validated: every synthetic generator is checked quantitatively against its analytical or certification-standard reference, and the large-eddy-simulation fields against the imposed log law. On a racing-class quadrotor in hover, the train test matrix is remarkably flat, and the cheapest structured training wind, which is discrete-gust domain randomization, ranks first in every test column, a ranking replicated on a wind-sensitive 27 g platform; a once-tuned geometric controller brackets the learned PPO policies at zero crash rate. Mechanism diagnostics show that control authority, not wind realism, bounds robustness, so wind-fidelity investment should scale with platform wind sensitivity.

Figures & tables

Figure 1: 1-cosine gust time histories

w_{g}/w_{m}

of Eq. ( 1 ) at

U_{\infty}=8

m/s for gust lengths

d_{m}=5

20

, and

50

m, spanning the randomized range of the L1 sampler.

Figure 2: Von Kármán and Dryden one-dimensional spectra at

U_{\infty}=8

m/s,

\mathrm{TI}=10\,\%

L_{u}=10

m; the thin line marks the

-5/3

inertial-range rolloff.

Figure 3: Oseen vortex embedded in an L2 statistical-background realization at the hover plane (streamwise velocity perturbation).

Figure 4: Instantaneous streamwise fluctuation $u^{\prime}$ on the lowest exported slice ( $z\approx 1.2$ m, $200\times 200$ m) of the production LES boundary layer.

Panel of Figure 5 — Figure 5: LES validation against the target neutral log law: (a) time-averaged streamwise velocity and (b) fluctuation standard-deviation profiles.

$z$ [m]	$\langle u\rangle$ [m/s]	log law [m/s]	dev. [%]	$\sigma_{u}$ / $\sigma_{w}$ [m/s]
1.17	5.09	4.92	$+3.3$	1.15 / 0.24
2.73	6.19	6.06	$+2.1$	1.03 / 0.40
5.08	6.97	6.89	$+1.2$	0.91 / 0.48
9.77	7.78	7.77	$+0.1$	0.79 / 0.48
19.92	8.79	8.73	$+0.7$	0.64 / 0.40

Table 1: Validation of the production LES against the imposed neutral log law (

u_{*}=0.551

m/s,

z_{0}=0.03

m): time-averaged mean velocity, relative deviation, and fluctuation standard deviations.

Figure 6: Racing-class quadrotor platform: X-configuration, 5-inch propellers, diagonally opposite rotors co-rotating; dimensions in mm,

m=0.83

kg.

Panel of Figure 7 — Figure 7: Cross-fidelity matrix: (a) position-tracking RMSE and (b) crash rate for PPO policies trained at level $L_{i}$ and tested at level $L_{j}$ , each column at its representative test condition.

Training level	test L0	test L1	test L2	test L3	test L4
PPO (L0)	$0.31\pm 0.06$ (6%)	$0.31\pm 0.04$ (6%)	$0.31\pm 0.05$ (8%)	$0.31\pm 0.04$ (6%)	$0.31\pm 0.05$ (10%)
PPO (L1)	$0.29\pm 0.02$ (0%)	$0.29\pm 0.03$ (2%)	$0.29\pm 0.03$ (2%)	$0.29\pm 0.03$ (2%)	$0.30\pm 0.03$ (2%)
PPO (L2)	$0.31\pm 0.06$ (8%)	$0.31\pm 0.05$ (6%)	$0.32\pm 0.06$ (8%)	$0.32\pm 0.06$ (8%)	$0.33\pm 0.08$ (14%)
PPO (L3)	$0.30\pm 0.04$ (2%)	$0.30\pm 0.07$ (2%)	$0.29\pm 0.01$ (0%)	$0.29\pm 0.01$ (0%)	$0.29\pm 0.01$ (0%)
PPO (L4)	$0.30\pm 0.06$ (4%)	$0.29\pm 0.02$ (4%)	$0.29\pm 0.02$ (2%)	$0.29\pm 0.01$ (0%)	$0.29\pm 0.01$ (0%)
PID (baseline)	$0.31\pm 0.00$ (0%)	$0.87\pm 0.01$ (100%)	$0.87\pm 0.01$ (100%)	$0.87\pm 0.01$ (100%)	$0.87\pm 0.01$ (100%)

Table 2: Aggregate cross-fidelity summary at each level’s representative condition: position RMSE (mean

\pm

std), crash rate in parentheses (five seeds

\times

ten episodes per cell).

Figure 8: Crash rate versus mean wind speed under L2 statistical turbulence at

\mathrm{TI}=10\,\%

for policies trained at each fidelity level and the classical baselines.

Panel of Figure 9 — Figure 9: Power spectral density of the collective-thrust command for policies trained at each fidelity level, tested under (a) L2 turbulence and (b) the L4 LES field at $U_{\infty}=8$ m/s (crashed episodes excluded).

Panel of Figure 10 — Figure 10: The three controller families in one shared L3 wind realization: (a) plan view at hover altitude; (b) horizontal deviation from the target versus time.

Figure 11: Plan view of the L4 LES disturbance field at hover altitude: streamwise fluctuation

u^{\prime}

with the L1-trained PPO flight path (dashed) and containment boundary (dotted).

Figure 12: Side (

x

–

z

) view of the trajectories of Fig. 10 at hover altitude:

-.-

, PID;

\dots

, geometric SE(3);

--

, PPO;

\times

, PID termination.

Figure 13: Mean horizontal deviation from the hover target versus time in the L3 wind of Fig. 10 (shaded:

\pm 1

std):

-.-

, PID;

\dots

, geometric SE(3);

--

, L1-trained PPO;

\times

, PID termination at

t\approx 0.2

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Wind-Aware Reinforcement Learning Control of a Small Quadrotor Using Learned Onboard Wind Estimation in Simulated Atmospheric Turbulence

Abdullah Al Tasim, Wei Sun

University of Oklahoma, Norman, Oklahoma, 73019

RL Control Aerial Robotics Quadrotor Control

Open paper AskPDF arXiv

Small multirotor aircraft are increasingly tasked with operations in the atmospheric boundary layer, where turbulent winds comparable to the vehicle's airspeed degrade trajectory tracking and can defeat conventional feedback control. This work illustrates a two-stage learning pipeline that first estimates the local wind from onboard kinematics and dynamics and then exploits that estimate inside a reinforcement learning (RL) flight controller. The wind estimator, an attention-augmented gated recurrent network trained on thousands of simulated flights through von Karman turbulence with power-law shear and veer, recovers the horizontal wind vector with a per-flight root-mean-square error of 0.40 m/s and a direction error of 3.2 degrees on unseen wind regimes, an accuracy near the floor imposed by unresolved turbulence, and generalizes to vertical ascent profiles with a skill score of 0.861 over a constant-wind reference. A proximal policy optimization controller receiving the frozen estimator's output reduces horizontal trajectory tracking error by 48% relative to a wind-blind proportional-derivative baseline across mean winds of 4 m/s to 12 m/s, winning on 100% of evaluation episodes. A three-way ablation decomposes this improvement into a kinematic component, available without wind information, and a wind-perception component; the perception share rises with wind speed, from small in light winds toward roughly half the total benefit in strong winds, consistent with the quadratic scaling of aerodynamic drag. The controller degrades gracefully on out-of-distribution winds of 13 m/s to 15 m/s, where the baseline fails catastrophically.

Abstract

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