Grey-Box Quadrotor State Estimation

• Grey-box quadrotor state estimation is a hybrid approach that combines known rigid-body dynamics with learned corrections to compensate for unmodeled aerodynamic effects and sensor biases.
• Drift-free EKFs and drag-augmented observers are employed to achieve low RMS velocity errors (≈0.12 m/s) even during aggressive maneuvers, ensuring stability and accuracy.
• Recent advances integrate neural processes and Gaussian process corrections with uncertainty quantification to deliver computationally efficient, real-time state estimation for agile UAVs.

Grey-box quadrotor state estimation refers to methods that interleave physics-based (white-box) modeling and data-driven (black-box) corrections in the design of real-time estimators for multirotor UAVs. The approach exploits partially-known system structure (e.g., rigid-body dynamics, drag laws) augmented by learned models or enhanced correction terms to compensate for unmodeled or highly nonlinear effects. This hybrid modeling paradigm enables improved estimation accuracy, bounded drift, computational efficiency, and, in advanced methods, theoretically-calibrated uncertainty quantification. The recent literature presents multiple algorithmic strategies—ranging from drift-free EKFs exploiting rotor drag, Lyapunov-stable nonlinear observers, and GP-augmented moving horizon estimators, to neural process-based grey-box estimators—validated in both simulation and onboard experiments.

1. Grey-box Modelling Philosophy in Quadrotor State Estimation

Grey-box estimation interposes between pure model-based and fully data-driven approaches by explicitly decomposing the system dynamics into known physics and unknown residual processes. The general framework for the state-discrete system is

$x_{k+1} = g(x_k,u_k,Δk) + r(x_k,u_k,Δk)$

where $g(\cdot)$ is a parameterized physics model (e.g., 6-DOF rigid body with rotor drag), while $r(\cdot)$ is a data-driven or empirically-tuned correction. In practical quadrotor applications, grey-box structures offer critical advantages: they preserve the interpretability and constraint structure of classical dynamics, allow embedding measurement-exploitable relationships (e.g., between accelerations and velocities due to blade flapping), and enable the estimator to remain robust to deviations induced by unmodeled aerodynamics, hardware drifts, or environmental perturbations (Abeywardena et al., 2015, Martin et al., 2015, Hunter et al., 15 Sep 2025, Choo et al., 2023).

2. Drift-free EKF and Drag-augmented Observers

A core thread in grey-box quadrotor literature leverages process models incorporating blade-induced rotor drag to achieve drift-free velocity estimation and robust attitude estimation. In (Abeywardena et al., 2015), a minimal six-state EKF encapsulates roll/pitch, body-frame velocities, and gyro biases. The dynamical model includes

• Coupled attitude kinematics from Euler rates,
• Gyro bias stochastic decay,
• Lateral/longitudinal velocities with blade-flapping drag,

$$\dot{v}_{bx} = -g\sin\theta - \frac{k_1}{m}v_{bx} + w_{\alpha x}$$

• Accelerometer measurements proportional directly to in-plane velocities via drag:

$$a_x = -\frac{k_1}{m} v_{bx} + w_{ax},\quad a_y = -\frac{k_1}{m} v_{by} + w_{ay}$$

The process/measurement structure circumvents the unbounded drift typically associated with naive integration of accelerations; the EKF’s update “directly sees” the velocities through the IMU, bounding estimation error without requiring visual odometry or GPS. Experimental results confirm RMS velocity error ≈ 0.12 m/s over 15 s Vicon-validated flights, with total velocity error envelope within ±0.2 m/s despite aggressive maneuvers (generic inertial-only estimators drift by ≈0.8 m/s over 10 s) (Abeywardena et al., 2015).

A related nonlinear observer (Martin et al., 2015) employs a drag-augmented state space and constant-gain structure, achieving semi-global Lyapunov-stable convergence of the combined attitude/velocity estimate. With proper gain selection, this observer maintains estimation errors within guaranteed bounds for aggressive 6-DOF maneuvers even under sensor biases and significant modeling simplifications (e.g., bounded drag, neglected Coriolis effects).

3. Grey-box Machine Learning and Data-augmented Filtering

Augmenting the classical physics-based filters with modern machine learning, recent work (e.g., (Hunter et al., 15 Sep 2025, Choo et al., 2023)) applies neural network–based or Gaussian process–based corrections to address highly nonlinear, temporally-varying, or otherwise hard-to-model aerodynamic phenomena.

In (Choo et al., 2023), a sparse Gaussian Process (GP) is trained on discrepancies between observed accelerations (IMU) and nominal model predictions, then deployed online to correct the rigid-body model inside a Moving Horizon Estimation (MHE) framework:

$$x_{k+1} = f_\mathrm{dyn}(x_k, u_k) + B_d\hat\delta a_k + w_k\,,$$

where $\hat\delta a_k$ is the single-point GP posterior for unmodeled aerodynamics. This hybrid MHE structure achieves 15–30% reduction in velocity and orientation RMSE versus purely model-based MHE under high-noise and high-speed (v > 10 m/s) conditions, while remaining computationally tractable (5–7 ms/step for N=50 horizon on a commodity CPU).

4. Attention-based Neural Processes with Physics Priors and Conformal Calibration

Physics-informed attentive neural processes (PI-AttNP) (Hunter et al., 15 Sep 2025) instantiate a novel data-driven yet physics-aware grey-box estimator. The key mechanism is explicit decomposition of the one-step state map:

$$f(x_k,u_k,Δk) = g(x_k,u_k,Δk) + \mathrm{NN}(x_k,u_k,Δk\mid Γ)$$

where the neural component learns unmodeled stochasticity and environmental artifacts. The physics model output is injected as a “prior” input to the decoder, biasing its mean and stabilizing learning. Further, uncertainty quantification is calibrated using marginal split conformal prediction, guaranteeing (under exchangeability) that coverage of estimated intervals aligns with nominal rates (e.g., empirical 95% at $\alpha=5\%$). On six-DOF quadrotor state estimation with multimodal measurement noise and strong disturbances, PI-AttNP with conformal calibration delivers RMSE and NLL metrics comparable to the state-of-the-art DKF estimator (but with 15× fewer parameters), and its conformal intervals envelop ground truth trajectories through challenging aerodynamic and actuator spikes (Hunter et al., 15 Sep 2025).

5. Visual-Inertial Grey-box Estimation and Keyframe Strategies

Grey-box state estimators incorporating both inertial and visual measurements, such as the model-aided visual-inertial odometry EKF of (Abeywardena et al., 2016), further generalize the paradigm. Here, the estimator fuses:

• An explicit drag-augmented dynamic model (for in-plane drag, thrust, and gravity),
• Gyroscope/accelerometer measurements (with bias estimation),
• Visual epipolar constraints (tracking pairs between image frames), without requiring explicit feature location estimation.

A key technical challenge—pose/velocity unobservability during low-dynamics hover—is addressed via a keyframe strategy: visual updates are applied only when sufficient disparity from the last keyframe is detected, ensuring persistent observability. The resulting estimator achieves sub-decimeter/sec velocity and sub-degree orientation accuracy in both simulation and real indoor flights on 50 g embedded computers.

6. Algorithmic and Practical Insights

Multiple design principles emerge across the literature:

• Parameter identification: Drag or aerodynamic terms (e.g., $k_1$) should be system-identifed, e.g., via least-squares on Vicon-tracked flights (Abeywardena et al., 2015).
• Observer tuning: In drag-model observers, separation of gain parameters for velocity vs. attitude convergence enables targeted transient response (Martin et al., 2015).
• Computational efficiency: Algebraic (non-EKF) observers and sparse GP approximations ensure real-time onboard feasibility even with high-dimensional models (Choo et al., 2023).
• Uncertainty quantification: Marginal conformal prediction enables rigorous empirical calibration of model error intervals in neural estimators (Hunter et al., 15 Sep 2025).
• Residual learning: For high-speed agile flight, online data-driven correction of aerodynamic forces via GPs or neural networks provides crucial robustness not attainable by white-box models alone (Choo et al., 2023, Hunter et al., 15 Sep 2025).

7. Limitations, Extensions, and Future Research Directions

Limitations of current grey-box estimators include:

• Partial state observability (e.g., lack of out-of-plane or yaw estimation in some drag-based approaches (Martin et al., 2015)).
• Potential breakdown of uncertainty guarantees (e.g., violation of exchangeability assumptions in split-conformal calibration under novel regimes (Hunter et al., 15 Sep 2025)).
• Sensitivity of model-based observers to neglected Coriolis or strong off-axis dynamics.

Proposed extensions encompass:

• Online adaptation or learning of residual models for nonstationary noise or environmental conditions,
• Integration of additional sensors (magnetometer, barometer) for full 6-DOF state reconstruction,
• Joint online identification of physical parameters (e.g., payload mass (Choo et al., 2023)),
• Transfer of conformal calibration wider into multi-agent and model-predictive control pipelines (Hunter et al., 15 Sep 2025).

Experimental validation on physical quadrotors, formal adaptation procedures, and direct-learned measurement models comprise active avenues for further empirical and theoretical development.

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Key References:

• (Abeywardena et al., 2015): Improved State Estimation in Quadrotor MAVs: A Novel Drift-Free Velocity Estimator.
• (Martin et al., 2015): A semi-global model-based state estimator for the quadrotor using only inertial measurements.
• (Abeywardena et al., 2016): Fast, On-board, Model-aided Visual-Inertial Odometry System for Quadrotor Micro Aerial Vehicles.
• (Choo et al., 2023): Data-Based MHE for Agile Quadrotor Flight.
• (Hunter et al., 15 Sep 2025): Hybrid State Estimation of Uncertain Nonlinear Dynamics Using Neural Processes.