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NeuroBEM: Hybrid Aerodynamic Quadrotor Model

Leonard Bauersfeld, Elia Kaufmann, Philipp Foehn, Sihao Sun, Davide Scaramuzza

arXiv:2106.08015v1cs.RO

TL;DR

Aggressive quadrotor flight exposes limitations in conventional first-principles models, while purely learned models can struggle to generalize from sparse real-world data. The paper combines BEM-based rotor modeling with a neural-network residual model, achieving lower prediction error and improved generalization, while retaining computational costs that limit advantages at slow speeds.

  • Problem

    Conventional models omit important aerodynamic effects at high speeds, while purely learned models can struggle to generalize across arbitrary flight conditions.

  • Method

    The paper combines blade-element-momentum rotor modeling with a deep-neural-network residual model for forces and torques.

  • Results

    The hybrid model reduces aerodynamic force and torque prediction error by 50% relative to state-of-the-art modeling and generalizes beyond its training data.

  • Takeaways & Limitations

    The model provides accurate force and torque predictions for aggressive trajectories and improves simulation positional RMSE from approximately 0.8 m to below 0.3 m.

  • Takeaways & Limitations

    Advantages are limited below 5 m s−1, where traditional parametric models perform strongly at lower computational cost.

Abstract

from arXiv · show

Quadrotors are extremely agile, so much in fact, that classic first-principle-models come to their limits. Aerodynamic effects, while insignificant at low speeds, become the dominant model defect during high speeds or agile maneuvers. Accurate modeling is needed to design robust high-performance control systems and enable flying close to the platform's physical limits. We propose a hybrid approach fusing first principles and learning to model quadrotors and their aerodynamic effects with unprecedented accuracy. First principles fail to capture such aerodynamic effects, rendering traditional approaches inaccurate when used for simulation or controller tuning. Data-driven approaches try to capture aerodynamic effects with blackbox modeling, such as neural networks; however, they struggle to robustly generalize to arbitrary flight conditions. Our hybrid approach unifies and outperforms both first-principles blade-element theory and learned residual dynamics. It is evaluated in one of the world's largest motion-capture systems, using autonomous-quadrotor-flight data at speeds up to 65km/h. The resulting model captures the aerodynamic thrust, torques, and parasitic effects with astonishing accuracy, outperforming existing models with 50% reduced prediction errors, and shows strong generalization capabilities beyond the training set.

I. INTRODUCTION

Aggressive quadrotor flight requires models that capture aerodynamic effects beyond low-speed approximations. NeuroBEM combines first-principles rotor modeling with learned residuals and evaluates it on broad real-world flight data and unseen maneuvers.

  • Motivation: High-speed and aggressive maneuvers require quadrotor models that predict platform behavior near physical limits.Aerodynamic forces depend on hidden variables such as airflow and rotor–frame interactions.
  • Limitations of existing models: Traditional quadratic and linear-drag models are efficient at low speeds but develop increasing bias as velocity rises.They neglect how inflow velocity affects generated thrust.
  • Limitations of existing models: BEM improves single-rotor high-wind modeling but omits aerodynamic interactions between rotors and the frame.Parametric gray-box models depend on expert-designed basis functions, while computational-fluid-dynamics models are computationally expensive.
  • Proposed approach: The proposed method combines BEM-based first-principles modeling with a neural network that learns residual forces and torques.This combines first-principles generalization with data-driven function approximation.
  • Evaluation: The model is evaluated using a real-quadrotor dataset spanning 96 flights, 1.8 million data points, and observed speeds up to 65 km h−1.The comparison covers predicted aerodynamic forces and torques and closed-loop simulator integration on unseen test maneuvers.

III. QUADROTOR MODEL

The proposed quadrotor model has a hybrid structure: a rotor model supplies first-principles dynamics, while a learned correction accounts for residual effects.

  • Model structure: The hybrid model consists of a rotor model and a learned correction for quadrotor dynamics.The model development proceeds from rigid-body notation through increasingly complex single-rotor models to the learned residual model.
  • Notation: World and body frames use orthonormal bases, and unit quaternions represent platform orientation.The body frame is located at the quadrotor’s center of mass.
  • Notation: A body-frame point is transformed into world coordinates by translating the body position and rotating the point with the body-to-world quaternion.The quaternion-vector product represents the corresponding vector rotation.

B. Quadrotor Dynamics

The dynamics model represents the quadrotor as a rigid body whose propeller-generated forces and torques are supplemented by learned residual terms. The simplest rotor model uses quadratic speed relationships but omits important aerodynamic effects.

  • Force and torque decomposition: Propeller forces and torques are separated into modeled components and residual terms not explained by the selected rotor model.The residual force and torque terms represent effects such as fuselage, body, and rotor interactions.
  • Quadratic rotor model: The quadratic single-propeller model assumes thrust and torque are proportional to the square of propeller speed Ω.Its coefficients are typically identified using a static propeller test stand.
  • Quadratic rotor model: The quadratic model approximates near-hover flight but ignores motion-dependent lift and induced drag.Induced drag depends on propeller speed and body-relative air velocity and is described as the dominant source of quadrotor drag.

D. Rotor Model: BEM

The BEM rotor model combines momentum theory with blade-element calculations to estimate induced velocity, blade forces, thrust, horizontal force, and drag torque. It also models elastic blade coning and flapping, with a separate approximation for vortex-ring conditions.

  • Model overview: BEM models each rotor separately and accounts for varying relative airspeed when predicting rotor thrust.Unlike quadratic models, it assumes inter-rotor interaction effects are negligible.
  • Induced velocity: Momentum theory relates induced velocity and thrust through momentum balance but does not determine induced velocity by itself.The induced velocity is obtained by jointly satisfying momentum-theory and blade-element thrust calculations.
  • Blade-element forces: Blade elements use radius, azimuth, relative velocities, chord, angle of attack, and lift and drag coefficients to compute differential aerodynamic forces.Overall thrust, horizontal force, and drag torque are obtained by integrating these differential contributions.
  • Blade elasticity: Elastic blades are represented with a hinged, torsionally sprung model that captures coning and azimuth-dependent flapping caused by lift imbalance.The blade moment balance includes weight, gyroscopic, inertial, centrifugal, aerodynamic, and spring contributions.
  • Complete BEM algorithm: The algorithm iteratively estimates induced velocity, calculates coning and flapping angles, and reevaluates integrated forces and torques.Because flapping and coning are typically below 1°, they are neglected initially when calculating induced velocity.
  • Vortex-ring conditions: In vortex-ring state, a quartic empirical approximation replaces momentum-theory induced velocity and uses the maximum of the approximation and horizontal-flight value.Momentum-theory results are valid only when the vehicle does not fly in its own downwash.

E. Learned Residual Dynamics

The learned residual-dynamics component uses a neural network to represent aerodynamic effects omitted by the rotor models. Its architecture is selected by minimizing force and torque prediction error on held-out data.

  • Residual model: A deep neural network predicts residual forces and torques caused by the body, fuselage, and rotor interactions omitted by the rotor models.The network implicitly estimates airflow from a history of observable platform states.
  • Architecture: Candidate models use temporal-convolutional or fully connected encoders followed by separate fully connected force and torque prediction heads.All architectures use leaky-ReLU activations and a linear output layer.
  • Evaluation: Architecture selection minimizes supervised RMSE for force and torque predictions on an unseen test set.Training uses the Adam optimizer and measured labels.

A. Data Collection

The dataset combines motion-capture pose measurements with onboard inertial and motor-speed data from autonomous quadrotor flights. It contains 1.8 million samples from 96 flights and uses trajectory-wise train, validation, and test splits.

  • Measurement setup: The motion-capture arena provides a usable volume of 25 m × 25 m × 8 m, with pose measurements at 400 Hz.Onboard IMU measurements and motor speeds are recorded at 1 kHz.
  • Dataset: The resulting dataset contains 1.8 million data points from 96 flights spanning 1h:15min of flight time.The data is divided into 70% training, 20% validation, and 10% test sets.
  • Data coverage: Each split contains trajectories covering the full range of speeds and accelerations observed in the complete dataset.

B. Quadrotor Platform

The experiments use a custom 6-inch quadrotor platform designed for high-agility flight. Its motors and propellers provide a 4.5 static thrust-to-weight ratio at a total weight of 752 g.

  • Platform specifications: The platform weighs 752 g and produces approximately 33 N of maximum static thrust, yielding a static thrust-to-weight ratio of 4.5.
  • Hardware: It uses an Armattan Chameleon 6-inch frame, Hobbywing XRotor 2306 motors, and 5-inch three-bladed propellers.Its weight and power are comparable to platforms used by professional drone-racing pilots.

C. Control System

The control system combines high-level nonlinear MPC with low-level body-rate tracking, while simulation incorporates integration and motor dynamics for realistic closed-loop evaluation.

  • Control architecture: A nonlinear quadratic MPC controller generates body-rate and collective-thrust commands at 100 Hz, while Betaflight tracks body-rate setpoints at 1 kHz.Human-piloting feed-forward features are disabled, and the same PID parameters are used in simulation and real-world flight.
  • Simulation components: Closed-loop simulation uses rigid-body dynamics, modeled aerodynamic forces and torques, integration, and motor dynamics.This setup evaluates simulation accuracy against the corresponding real-world trajectory.
  • Simulation components: The simulator integrates the dynamics with a symplectic Euler scheme using a 1 ms timestep.The stated advantage is energy conservation compared with standard Euler and Runge–Kutta integration schemes.
  • Motor dynamics: Motors are modeled as first-order systems because the aerodynamic model depends on propeller angular speed.The identified motor time constant for the experimental platform is τΩ = 33 ms.

V. EXPERIMENTS AND RESULTS

The experiments compare aerodynamic models on unseen trajectories spanning near-hover to highly aggressive flight, assessing force and torque prediction and generalization from reduced training data.

  • Evaluation design: The evaluation asks when quadratic thrust and torque models fail, how BEM compares with quadratic models, and what learned residual dynamics contribute.Models are assessed using both predicted forces and torques on unseen flight data and closed-loop simulation mismatch.
  • Evaluation design: Test trajectories range from speeds below 5 m s−1 near hover to more than 18 m s−1 with accelerations up to 46.8 m s−2.Compared models include Fit, BEM, None, and learned augmentations of these approaches.
  • Predictive performance: The proposed BEM-plus-learned-residual model consistently outperforms other models for predicted forces on the unseen test set.The experiments summarize performance with RMSE, including models trained on a reduced dataset for generalization testing.
  • Interpretation: Yaw-torque results are difficult to interpret because the trajectories minimize yaw rate and contain only small yaw torques, with a maximum of 0.072 N m.This limited yaw-torque range explains why naive baselines can perform well on that metric.
  • Generalization: On the reduced dataset, the proposed approach degrades gracefully and still outperforms baselines for predicted forces.The purely learning-based baseline performs best for predicted torques but fails to generalize force predictions, while PolyFit breaks down.

C. Closed-Loop Comparison

Closed-loop simulation compares simulated and real trajectories across unseen flight paths and speeds, showing that the hybrid model remains accurate as maneuvers become faster and training coverage decreases.

  • Evaluation setup: Closed-loop evaluation reports accumulated positional error between simulated and real trajectories for unseen trajectories flown at varying speeds.The comparison uses the same trajectory set in simulation and on the real platform.
  • Closed-loop results: At low speeds, all models perform similarly, but BEM remains competitive through the fastest trajectories while Fit develops increasing bias above 7 m s−1.PolyFit performs well at low speeds but eventually crashes on the fastest trajectory; RotorS performs worse across trajectories.
  • Closed-loop results: BEM+NN achieves competitive performance on slow trajectories and outperforms all baselines on faster trajectories.It also performs consistently better than Fit+NN during fast maneuvers.
  • Generalization: With reduced training data, the proposed approach incurs only about a 20% performance reduction and still outperforms baselines on faster trajectories.The result is presented as evidence of generalization beyond the training data, including test regions outside the reduced training set.

VI. DISCUSSION

The hybrid model combines BEM-based first principles with learned residual dynamics, reducing prediction errors and improving simulation accuracy. Its benefits are strongest for aggressive flight, while slow trajectories and control-pipeline fidelity remain practical boundaries.

  • Model accuracy: 50% decreased aerodynamic force and torque prediction error was achieved versus state-of-the-art modeling.The result was observed in controlled experiments on a large real-world dataset.
  • Model accuracy: Up to 30% better learned residual prediction was achieved when combined with a BEM-based rotor model.The residual dynamics benefit when first principles capture a broader envelope of effects.
  • Practical boundaries: Advantages are limited below 5 m s−1, where traditional parametric models achieve strong performance at lower computational cost.Quadratic and polynomial fits require about 1 µs, whereas BEM and the neural-network pass each require about 100 µs in the reported hardware settings.
  • Practical boundaries: Closed-loop simulation could improve by replacing the human-pilot-oriented BetaFlight controller and modeling motion-capture and communication latency.These changes would reduce differences between simulation and actual experiments.
  • Simulation impact: Positional RMSE decreased from approximately 0.8 m for state-of-the-art approaches to below 0.3 m in simulation.The authors connect this accuracy to faster development and testing of advanced quadrotor control and navigation strategies.
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