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A Hybrid PEM-GP Framework for Uncertainty-Aware System Identification of Quadcopters
Abdallah Ghoul, Ismail Khalil Bousserhane, Kadri Boufeldja
TL;DR
Reliable quadcopter control needs dynamic models that capture nonlinear behavior without sacrificing interpretability or uncertainty calibration. The paper combines PEM-based physics identification with GP residual learning and finds a favorable accuracy–interpretability–uncertainty trade-off in Duckiedrone experiments. The resulting uncertainty estimates can support confidence-aware deployment decisions.
Problem
Classical identification is interpretable but misses important nonlinear effects, while data-driven models reduce interpretability and lack well-calibrated uncertainty estimates.
Method
The framework identifies a PEM-based parametric model for dominant quadrotor dynamics and trains a GP on residual nonlinear behavior, using a hover-centered state-space structure with decoupled attitude models by default.
Results
Experiments on a physical Duckiedrone show that the hybrid model maintains a more favorable trade-off among prediction accuracy, interpretability, and uncertainty quantification than standalone approaches, with accuracy comparable to LSTM.
Takeaways & Limitations
Hybrid modeling keeps a structurally sound PEM baseline while letting the GP focus on residual dynamics, producing uncertainty estimates that can inform conservative control or model switching.
Takeaways & Limitations
Standalone and full-dynamics GP modeling can be computationally demanding because covariance-matrix inversion scales as O(N^3), limiting training-data size.
Abstract
from arXiv · showhide
Accurate dynamic models play a central role in achieving reliable control of quadcopters. Classical system identification methods remain widely used, mainly because of their interpretability. However, they often fail to capture important nonlinear effects, especially in small-scale aerial platforms where such effects become more pronounced. Data-driven approaches offer a different perspective. They can represent complex nonlinear dynamics more effectively, but this comes at the cost of reduced interpretability and the absence of well-calibrated uncertainty estimates. In this work, we propose a framework that combines physics-based modeling with data-driven learning, while explicitly accounting for uncertainty. A physics-based model is first identified using the Prediction Error Method (PEM), which captures the main structure of the system. The remaining dynamics are then modeled using a Gaussian Process (GP), allowing the residual behavior to be learned directly from data. This separation makes it possible to distinguish between known physical effects and unmodeled dynamics. The proposed framework is validated on a Duckiedrone-like experimental setup. The results show that the PEM-GP model achieves prediction accuracy comparable to that of a Long Short-Term Memory (LSTM) network, while additionally providing calibrated uncertainty estimates. This combination improves model reliability and supports uncertainty-aware decision-making.
1 INTRODUCTION
Quadcopter system identification must balance interpretable physics-based models with the ability to represent nonlinear dynamics and quantify uncertainty. The proposed PEM-GP framework combines these strengths and is evaluated against established baselines on Duckiedrone experiments.
- Motivation: Accurate dynamic models support reliable quadcopter control, simulation, state estimation, and prediction under inputs and disturbances.Small platforms are especially challenging because nonlinear effects and messy sensor data complicate identification.
- Motivation: Classical PEM models are interpretable but become limited during aggressive, coupled maneuvers, while LSTMs capture nonlinear dependencies without explaining outputs or reporting uncertainty.GPs provide probabilistic predictions, but full-dynamics GP modeling can be data-hungry and may neglect known physics.
- Proposed framework: The framework identifies a simplified PEM-based physics model first, then trains a GP to learn residual dynamics that the physical model misses.This separation distributes modeling effort between structured physical effects and unmodeled nonlinear behavior.
- Contributions: The contribution is a structured hybrid system-identification methodology that combines a classical PEM model with a GP residual learner for small-scale quadcopters.The approach is designed to retain physical interpretability while extending the model beyond the classical baseline.
- Contributions: The hybrid model achieves predictive accuracy comparable to a black-box LSTM while providing explicit quantitative uncertainty estimates absent from PEM and LSTM models.The framework is evaluated on a physical Duckiedrone model against standalone PEM, GP, and LSTM baselines across excitation trajectories.
2 Related Work
Prior work spans classical parametric, data-driven, and grey-box identification, each offering different trade-offs in interpretability, nonlinear modeling, uncertainty, and scalability. This work positions PEM-GP residual learning as a structured approach for quadrotor dynamics.
- Classical System Identification Methods: Classical parametric methods provide rigorous, interpretable, and computationally efficient models, but their validity is narrow near hover and degrades during coupled aggressive motion.Linearity assumptions and axis-by-axis identification omit important aerodynamic effects and cross-couplings.
- Data-Driven Methods: Neural networks and recurrent architectures learn nonlinear dynamics directly from data and can outperform linear baselines on complex UAV trajectories.Their principal trade-off is limited physical interpretability and transparency.
- Probabilistic Methods: Gaussian Processes provide probabilistic predictions and calibrated uncertainty that support confidence-aware robotics control.However, full-dynamics GP models can become data-inefficient and computationally difficult as datasets grow.
- Hybrid Methods: Grey-box methods combine a physics model for dominant structure with a learning module for residual dynamics that remain unmodeled.This preserves a physically meaningful core while directing learning toward nonlinearities and cross-axis couplings.
- Research Gap: Systematic PEM-based baselines paired with GP residual learning remain relatively scarce for full quadcopter identification.The proposed pipeline identifies the PEM model first and trains the GP only on unexplained residual nonlinear dynamics.
3 Methodology
The methodology formulates quadcopter identification as supervised sequence modeling and combines an interpretable PEM baseline with GP residual learning. LSTM modeling provides a black-box comparison, while the hybrid model adds predictive uncertainty to residual-corrected predictions.
- Problem formulation: The study casts quadcopter identification as learning a predictive mapping from motor inputs to measured outputs by minimizing prediction error.
- Quadcopter dynamics: The quadcopter dynamics are described using body-frame Newton–Euler equations involving mass, velocity, angular velocity, inertia, forces, and torques.
- PEM baseline: PEM estimates a discrete-time linear state-space baseline around hover, preserving interpretable parameters but omitting actuator and aerodynamic nonlinearities.
- LSTM comparison: The LSTM uses finite histories of outputs and inputs to produce multi-output predictions, with two recurrent layers, a linear head, dropout, and weight decay.
- Hybrid PEM-GP framework: The hybrid framework computes PEM residuals, trains a GP on operational states, and sums the GP residual prediction with the PEM baseline.
- Uncertainty quantification: The hybrid model inherits predictive variance from the GP, retaining a physically interpretable core while representing residual nonlinearities and supporting confidence-aware control.
4 Data Collection Procedure
Data collection uses persistently exciting flight inputs spanning frequency sweeps, transients, coupled manoeuvres, and hover. The 30-second dataset is logged synchronously and split chronologically into training, validation, and held-out test intervals.
- Data requirements: The protocol requires persistently exciting input–output flight data for unbiased identification.
- Input sequence: Excitation combines 0.1–20 Hz multi-sine chirps, steps and doublets, coupled roll–pitch commands, and hover segments.
- Identification procedure: Algorithm 1 estimates PEM parameters, constructs the operational-state regression vector, trains the GP residual model, and returns residual predictions with predictive variance.
- Dataset: The synchronized sequence lasts 30s and contains N = 3000 samples.
- Dataset: Chronological splitting assigns 0–20s to training, 20–25s to validation, and 25–30s to held-out testing, with preprocessing fitted only on training data.
5 Results and Analysis
The evaluation compares PEM, LSTM, GP, and hybrid PEM-GP identification using prediction metrics, prediction profiles, and uncertainty behavior. The hybrid model combines high accuracy with uncertainty estimation and partially preserved interpretability.
- Comparative analysis: The comparison evaluates MSE and R2 alongside interpretability, uncertainty estimation, and nonlinear modeling ability, exposing trade-offs among the identification methods.The hybrid retains partial interpretability through its PEM structure while benefiting from data-driven learning.
- PEM results: The PEM model captured dominant near-hover dynamics but degraded during aggressive manoeuvres, consistent with its linear, decoupled structure.Its total MSE was 3.87 × 10^-3 and its R2 was 0.995.
- LSTM results: The LSTM matched ground truth more closely during high-speed transients and aggressive manoeuvres, achieving an MSE of 2.25 × 10^-3 and R2 of 0.997.It modeled nonlinear temporal behavior and couplings between axes, but remained a black-box without inherent uncertainty quantification.
- GP results: The GP achieved an MSE of 1.76 × 10^-3 and R2 of 0.998 while providing ±2σ confidence intervals that increased when data departed from training conditions or dynamics became highly maneuvering.These probabilistic predictions provide an explicit model-confidence estimate.
- Hybrid PEM-GP results: The hybrid PEM-GP model achieved the strongest reported performance, with an MSE of 2.70 × 10^-9 and R2 of 0.999.It combines the PEM baseline for structured dynamics with GP residual correction.
- Comparative analysis: Hybrid-model uncertainty was low during steady hover-like flight and increased during agile manoeuvres, where the PEM model performed most poorly.This behavior supports using uncertainty to identify less-confident operating regimes.
6 Discussion
The discussion argues that PEM-GP combines complementary strengths: PEM supplies structure and interpretability, while GP models residual nonlinearities and quantifies uncertainty. This balance improves applicability to aggressive quadcopter flight while retaining feasible deployment costs.
- Hybrid modeling can work where physics-based and data-driven paradigms alone do not, according to the experimental results.
- PEM remains interpretable and computationally inexpensive, but its linear structure performs poorly on nonlinear quadcopter dynamics.Its limitations include missing peak responses, phase lag, aerodynamic effects, and dynamic cross-couplings during aggressive flight.
- LSTM provides strong nonlinear prediction accuracy but sacrifices interpretability and computational efficiency, while lacking native uncertainty estimates.
- The GP models residual dynamics using PEM as an informative prior, improving accuracy while reducing training time relative to learning the full dynamics.This decomposition lets the GP focus on platform-specific residual behavior rather than relearning known structure.
- Hybrid uncertainty rises during aggressive maneuvers and falls during steady hover-like flight, identifying regimes where model confidence is lower.This relationship can support conservative control actions or real-time model-switching thresholds.
- Residual-only GP learning alleviates full-system GP complexity, while offline training and inference-only deployment keep operational computation moderate.The reported hybrid prediction cost fits within the sampling period used in the considered quadrotor control setting.
7 Conclusion and Future Work
The paper concludes that PEM-GP provides a more balanced alternative to standalone standard and data-driven models. Future work should use its uncertainty information in control and test generality across broader conditions and environments.
- The proposed framework combines a standard PEM model with a GP residual learner for small-scale quadcopters.
- Real-world Duckiedrone experiments show a more favorable trade-off among accuracy, interpretability, and uncertainty quantification than standalone approaches.
- PEM supplies an interpretable structural baseline, while GP captures complex residuals, producing a model described as superior to PEM or GP alone and more trustworthy than LSTM.
- Future work includes adaptive or robust controllers that explicitly use hybrid-model uncertainty for real-time performance improvement and safety assurance.
- Broader validation across varied flight conditions and environments is required to assess the framework's generality.
- The paper presents the framework as a feasible foundation for high-fidelity, uncertainty-aware system identification and robust UAV control.
DECLARATIONS
The declarations report no competing interests, no external funding, and no requirement for ethical approval. The authors also state that code, minimal reproducibility data, and figure-source CSVs are released with an archived DOI snapshot.
- The authors declare no known competing financial interests or personal relationships that could influence the reported work.
- The study was conducted without external funding.
- The work involved no human participants, animals, or sensitive data, so ethical approval was not required.
- The authors release code, minimal data for reproducing tables and figures, and figure-source CSVs.
- An archived snapshot of the work is available through a DOI.