Flatness-Preserving Residual Learning for Real-Time Tight Quadrotor Formation Flight

Authors: Pei-An Hsieh, Fengjun Yang, Nikolai Matni, M. Ani Hsieh · IROS 2026

Licences

Open Access
Paper: CC BY 4.0

Everything below draws on the full paper under its CC BY 4.0 licence; its figure is credited to the authors.

TL;DR

A downwash residual that depends only on positions and velocities keeps two-quadrotor dynamics flat, so a cheap flatness controller matches NMPC tracking with about 20x less solve time.

Why it matters

Downwash compensation becomes a few small network evaluations inside a classical flatness-based controller that can be tuned with linear control methods. On hardware it needed under 30 seconds of training data and ran at a 5 ms loop rate, which suits small vehicles and compute-limited flight stacks.

Problem

When quadrotors fly close together, downwash from the upper vehicle pushes the lower one off its path and can cause collisions. Residual models learned from flight data capture these effects, but plugging a general learned model into the controller usually needs nonlinear MPC, which is computationally heavy and scales poorly with model complexity.

Proposed solution

The residual force and torque are restricted to depend only on the team's joint positions and velocities, which the authors show keeps the multi-quadrotor system differentially flat. The residual is a physics-based reduced-order downwash model plus a small neural network on rotation-equivariant features (vertical and radial separation), trained with knowledge-based neural ODEs. A cascaded feedback-linearization controller then uses the residual and its Jacobian for feedforward thrust and attitude compensation.

Key insights

Restricting the learned residual to the team's positions and velocities preserves differential flatness, so thrust, attitude and angular velocity can still be recovered from flat outputs with small corrections
A physics-based reduced-order downwash model plus a small network generalises better than learning the residual alone, especially at close separation where coupling is strongest
Rotation-equivariant features (vertical and radial separation in a local flow frame) keep the network tiny: one hidden layer of nine neurons in simulation, four neurons on hardware
The same learned residual also improves a standard SE(3) geometric controller without extra data collection or retraining

Main results

Hardware: the full method (Learned-FBL) cut RMSE by a further 29% and maximum vertical error by 43% relative to the physics-only ROM-FBL controller, which itself beat the nominal controller by 24% in RMSE and 30% in maximum vertical error
Hardware: Learned-FBL reached an average RMSE of 5 cm and maximum vertical error of 10 cm over 60 runs (five repeats per case), matching the reported performance of the KNODE-DW MPC baseline
Simulation vs NMPC with the same residual: 0.99 ms vs 27.6 ms average solve time at 0.3 m separation and 1.3 ms vs 26.2 ms at 0.5 m, with similar 3D RMSE (0.016 m vs 0.020 m and 0.018 m)
Learned-SE(3) and ROM-SE(3) reduced hardware RMSE by 43.7% and 34% respectively against the nominal SE(3) controller
Residual trained on 28 s (6,720 points) of real flight data; two Crazyflies merged from a 1.1 m stack to 0.1 m vertical separation to pass a 0.4 m high window

Limitations

The authors list relaxing the structural assumptions of the flatness-preserving residual (dependence on positions and velocities only) as future work
Experiments cover two-quadrotor teams only; scaling up the experiments is left to future work
On hardware the controller outputs thrust and angular-velocity commands, so only residual forces are compensated and residual torques are left to the onboard PID loop
At 0.5 m separation the learned model's angular-acceleration error was slightly worse than the physics model alone, which the authors attribute to mild overfitting to close-proximity data; NMPC also kept a lower maximum vertical error in simulation
Hardware tests used Vicon motion capture and an off-board laptop computing commands, not onboard sensing or compute

Field impact

Finalist for the IROS 2026 Best Paper / Best Student Paper Award. The paper reports the first hardware application of flatness-preserving residual learning to tight quadrotor formation flight. No code or project page is listed in the paper.

Abstract

Quadrotors flying in tight formations are severely affected by turbulent aerodynamic interactions, such as downwash, that can cause catastrophic collisions if left unmodeled. To compensate for these effects, we propose a physics-informed residual dynamics learning framework that captures complex aerodynamic interactions while ensuring the joint multi-quadrotor system remains differentially flat. We leverage this preserved flatness to design a computationally efficient feedback linearization controller that is easily tunable with linear control techniques and cancels aerodynamic disturbances via feedforward compensation. Hardware experiments demonstrate our framework reduces average tracking errors by 31% compared to nominal baselines. Crucially, our lightweight approach matches the tracking performance of state-of-the-art nonlinear model predictive control (NMPC) while requiring an order of magnitude less computation. We are the first to show that stable, tight formation flight can be achieved with under 30 seconds of training data and a 5ms loop rate, unlocking high-fidelity aerodynamic compensation for compute-constrained flight stacks.

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