⚡ electrical engineering

Critical slowing down for predicting controller induced loss of control in quadrotors

This paper presents a model-free forecasting scheme based on critical slowing down that successfully predicts controller-induced loss of control in quadrotors up to 0.9 seconds in advance, outperforming existing neural network methods and generalizing across different aircraft, controllers, and flight scenarios without requiring re-parameterization.

Original authors: Jasper J. van Beers, Prashant Solanki, Erik-Jan van Kampen, Coen C. de Visser

Published 2026-07-29

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CC BY 4.0

Original authors: Jasper J. van Beers, Prashant Solanki, Erik-Jan van Kampen, Coen C. de Visser

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a tightrope walker. As long as they are steady, a small wobble is easy to fix; they just shift their weight and bounce back to the center. But as they get closer to the edge of the rope, something strange happens. Their recovery gets slower. A tiny nudge that used to be corrected instantly now takes a long time to settle, and their balance feels "sticky." In the world of complex systems—like ecosystems, the human brain, or even a drone—this phenomenon is called Critical Slowing Down. It's a universal warning sign that a system is about to tip over into a completely different, often chaotic state.

Now, picture a drone, a flying robot with four spinning propellers. These machines are amazing, but they can sometimes lose control and crash, often because the computer brain telling them how to fly gets confused by delays or bad signals. Scientists have been trying to predict these crashes, but many methods require the drone to have already crashed thousands of times in a computer simulation to "learn" what a crash looks like. This is a problem because, in real life, we don't want to crash our drones just to teach them how to fly. The big question is: Can we spot the "sticky wobble" before the drone actually falls, without needing a history of past crashes to teach us?

This paper introduces a clever new way to answer that question. The researchers, working at Delft University of Technology, developed a forecasting system called C-BeFore that acts like a "crash detector" for drones. Instead of needing a massive database of past crashes to learn from, this system uses the universal principle of Critical Slowing Down. Think of it like a doctor listening to a patient's heart. The doctor doesn't need to have seen a heart attack before to know that a heart is struggling; they just listen for the specific, slowing rhythm that signals trouble is coming. Similarly, the C-BeFore system listens to the drone's motors. When the drone is flying normally, the motors respond quickly to commands. But as the drone approaches a loss of control, the motors start to "stutter" and recover more slowly from tiny changes.

The team tested this idea on real flight data from four different types of drones, including some that were flying indoors and others outdoors in the wind. They looked at 91 actual crash events where the drones lost control due to unstable computer behavior. The results were impressive: the system could predict a crash up to 0.9 seconds before it happened. For a drone moving fast, that's a lifetime of warning time—enough for a safety system to step in and save the day.

What makes this approach truly special is that it didn't need to be "trained" on crash data. The researchers showed that they could predict crashes on a drone they had never seen before, using a setup that was originally tuned for a different type of drone. It's like having a smoke detector that works perfectly in a kitchen, a garage, or a forest fire, without ever needing to see a fire in that specific room first. The system successfully spotted dangerous "flyaway" events (where a drone shoots up uncontrollably) and yaw-induced crashes, even when the drones had different motors, different controllers, and were flying in different environments.

The paper also compares their method to the current "gold standard" in the field: Recurrent Neural Networks (RNNs), which are a type of artificial intelligence that usually needs to study thousands of examples of crashes to work well. The C-BeFore system beat these AI models in accuracy and, crucially, didn't produce as many false alarms (screaming "crash!" when the drone was just doing a cool trick). While the AI models were good at spotting crashes, they often missed the early warning signs or got confused by normal, aggressive flying. The C-BeFore system, by focusing on the fundamental physics of how systems slow down before they break, managed to distinguish between a brave pilot and a doomed one with much greater precision.

In short, this research suggests that we don't need to wait for a disaster to happen to prevent it. By paying attention to the subtle, slowing rhythms of a system as it approaches a tipping point, we can build safety nets that are generic, reliable, and ready to work on any drone, anywhere, without needing a library of past accidents. It's a step toward making autonomous robots safer by giving them the ability to feel their own instability before it's too late.

Technical Summary: Critical Slowing Down for Predicting Controller Induced Loss of Control in Quadrotors

Problem Statement
As autonomous robots, particularly unmanned aerial vehicles (UAVs) like quadrotors, become more prevalent, ensuring their safe operation is paramount. While significant progress has been made in fault-tolerant control for rotor failures, quadrotors can still lose control (LOC) due to unstable controller characteristics, sensor saturation, or input-output delays, leading to catastrophic events such as "flyaways." Existing safety approaches, such as Control Barrier Functions (CBFs), often constrain navigational states but fail to restrict inner-loop control variables (e.g., motor rates), leaving the system vulnerable to disturbances that push it into an unrecoverable state. Furthermore, model-based Safe Flight Envelope (SFE) methods are computationally expensive and sensitive to model mismatches, while data-driven neural network approaches often suffer from poor generalization and a lack of physical interpretability. The core challenge addressed is the prediction of controller-induced LOC without relying on extensive datasets of failure events or specific system models.

Methodology
The authors propose a system model-free forecasting framework called C-BeFore (Cascaded Bayesian Forecaster) based on the phenomenon of Critical Slowing Down (CSD). CSD is a generic indicator observed in complex systems approaching a critical transition, characterized by a slowing recovery rate from perturbations.

Early Warning Signals (EWS): The method utilizes onboard rotor speed measurements ( $\omega_1, \dots, \omega_4$ ) to compute rolling, pitching, and yawing rotor speed differences ( $u_p, u_q, u_r$ ). These signals are detrended using a moving average filter. The primary indicator used is the lag-1 autocorrelation (AC1) calculated over a sliding window. As the system approaches LOC, the AC1 is expected to approach 1, indicating a loss of resilience.
Bayesian Detection: A Bayesian inference scheme is employed to estimate the probability of LOC, $P(L|E_{k,D})$ $P (L ∣ E_{k, D})$ . This combines a prior probability derived from the AC1 value with likelihoods calculated using the Wasserstein distance. The distance measures the similarity between the current distribution of the EWS and two reference distributions: a nominal distribution ( $N_{E_k}$ $N_{E_{k}}$ ) and a LOC distribution ( $L_{E_k}$ $L_{E_{k}}$ ).
- Data-Driven Mode: If LOC data is available, $L_{E_k}$ is constructed from actual failure events.
- Assumed Mode: If LOC data is unavailable, $L_{E_k}$ is assumed to be a distribution near AC1 = 1, based on CSD theory. This allows the system to function without training on failure data.
Forecasting (Time-to-LOC): The framework utilizes a cascade of Bayesian detectors operating on different window sizes ( $D$ ). A large window detector ( $D_d$ ) serves as the "ground truth" for the moment of LOC. Smaller window detectors ( $D < D_d$ ) are used to anticipate when the larger detector would trigger. By analyzing the temporal difference between detections across these windows and correcting for inter-window errors, the system estimates the time remaining until LOC ( $\Delta t_{LOC}$ ).

Key Contributions

Model-Free Forecasting: The approach does not require a system model or a training phase using LOC data. It leverages the generic principles of CSD to detect instability.
Generic Applicability: The method is validated on real flight data from four different quadrotors (CineGo, DataCan75, SuperKnight5, DamselFly) with varying masses, motor configurations, and flight controllers (Betaflight and Indiflight).
Performance over Neural Networks: The framework is compared against state-of-the-art Recurrent Neural Networks (RNNs) (LSTM, BiLSTM, CNNLSTM, GRU). C-BeFore demonstrates superior classification accuracy and significantly fewer false positives, particularly in distinguishing aggressive nominal maneuvers from actual LOC.
Zero-Shot Generalization: The paper demonstrates that a forecaster parameterized on one quadrotor (CineGo) and one LOC scenario (yaw-induced instability) can successfully detect a different LOC scenario (flyaways due to structural resonance) on different quadrotors without re-parameterization.

Results

Detection Accuracy: On the DataCan75 dataset, the data-driven C-BeFore achieved a 100% true positive rate with zero early false positives, outperforming the best-performing RNNs which exhibited higher false positive rates.
Forecast Lead Time: The system achieves a time-to-LOC forecast of up to 0.9 seconds before the LOC event occurs. In the DataCan75 experiments, the average forecast was approximately 0.44 seconds, which is sufficient for corrective action given the 4kHz control loop.
Assumed Distribution Performance: When using an assumed LOC distribution (without actual LOC data), the performance remained nearly identical to the data-driven version, confirming the method's ability to function without failure datasets.
Generalization: The CineGo-parameterized forecaster successfully detected all flyaway events on the SuperKnight5 and DamselFly, despite differences in controller architecture (PID vs. INDI) and flight conditions (indoor vs. outdoor).

Significance and Claims
The authors claim that this work provides a crucial "add-on" safety monitor that operates independently of the primary controller. By predicting LOC based on generic physical principles (CSD) rather than specific system dynamics or extensive failure data, the approach offers a robust, explainable, and computationally efficient solution ( $O(D \log D)$ complexity) for preventing controller-induced instabilities. The paper emphasizes that while the method does not provide the corrective control action itself, it offers the necessary early warning to enable such actions, bridging the gap between theoretical safety envelopes and real-time operational safety. The ability to generalize across different platforms and failure modes without retraining is highlighted as a major advancement over current data-driven methods.

Technical Summary: Critical Slowing Down for Predicting Controller Induced Loss of Control in Quadrotors

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