Adaptive SINDy: Residual Force System Identification Based UAV Disturbance Rejection

This paper proposes an Adaptive SINDy framework that integrates data-driven Sparse Identification of Non-Linear Dynamics with Recursive Least Squares adaptive control to effectively identify residual forces and reject wind disturbances, demonstrating superior trajectory tracking performance compared to baseline PID and INDI controllers on a lightweight Crazyflie drone in turbulent environments.

The Gist

Imagine you are trying to ride a bicycle through a chaotic, windy park. You want to follow a specific path drawn on the ground (like a circle or a figure-eight), but strong gusts of wind keep pushing you off course.

The Problem:
Traditional bicycle riders (standard controllers) try to fix this by just pedaling harder or steering back when they feel a push. But if the wind is unpredictable and changes direction constantly, a simple "push back" strategy often fails. The rider might overcorrect, wobble, and eventually crash.

The Old "Smart" Solutions:
Scientists have tried two main ways to fix this:

1. The "Guesswork" Model: They try to write a perfect math textbook describing exactly how wind blows. But wind is messy and non-linear; the textbook is never quite right, and the rider still gets blown off course.
2. The "Black Box" AI: They use a super-smart computer brain (Neural Networks) that learns from experience. While this works well, it's like a magic trick. The computer knows what to do, but it can't explain why. If the computer makes a mistake, no one knows why, and it's hard to trust it with a fragile drone.

The New Solution: "Adaptive SINDy"
This paper introduces a new method called Adaptive SINDy. Think of it as giving the drone a detective's notebook combined with a smart autopilot.

Here is how it works, broken down into simple steps:

1. The Detective (SINDy)

Instead of trying to predict the wind from the sky, the drone looks at its own body.

• The Observation: When the wind hits the drone from the left, the drone's internal computer notices: "Hey, I'm tilting to the right to fight the wind!"
• The Notebook: The drone uses a special mathematical tool called SINDy (Sparse Identification of Non-Linear Dynamics). Imagine SINDy as a detective who has a giant list of possible clues (like "tilt angle," "speed," "thrust").
• The Shortcut: Most of these clues are irrelevant. SINDy is smart enough to throw away 99% of the list and keep only the few essential clues that actually explain why the drone is wobbling. It creates a simple, easy-to-understand rule: "When the wind pushes left, I tilt right by X amount."
• Why it's cool: Unlike the "Black Box" AI, this rule is written in plain language (math you can read). It's interpretable, meaning we know exactly what the drone is thinking.

2. The Smart Autopilot (Adaptive Control)

Once the detective (SINDy) figures out the rule for the wind, the autopilot (Adaptive Control) uses it immediately.

• Real-Time Adjustment: As the wind changes, the detective updates the notebook instantly. The autopilot then adjusts the drone's motors to cancel out the wind before it pushes the drone too far off course.
• The "Leaky" Memory: The system uses a technique called "Recursive Least Squares" (RLS). Think of this as a student who learns from every test but doesn't get stuck on old mistakes. If the wind pattern changes, the student quickly forgets the old rule and learns the new one.

3. The Test Drive

The researchers tested this on a tiny, lightweight drone called a Crazyflie (which is very fragile, like a paper airplane).

• The Setup: They put the drone in a room with four giant fans blowing air at it from all sides, creating a chaotic, turbulent storm.
• The Competition: They pitted their new "Detective Autopilot" against a standard "Old School" controller (PID) and a "Black Box" AI.
• The Result:
  • The Old School controller panicked and crashed immediately. It couldn't handle the chaos.
  • The Black Box AI did okay, but it was a bit less precise.
  • The Adaptive SINDy drone flew perfectly. It followed complex paths (circles, figure-eights, and spirals) without crashing, even in the storm. It kept its errors incredibly small (within the width of a human hand).

The Big Picture

This paper proves that you don't need a super-complex, unexplainable AI to fly a drone in a storm. Instead, you can use a smart, lightweight detective that watches the drone's own movements, figures out the simple rules of the wind, and adjusts the flight instantly.

It's like teaching a child to ride a bike in the wind not by giving them a physics lecture on aerodynamics, but by teaching them: "When you feel the wind push you left, lean right just enough to stay balanced." Simple, effective, and safe.

Technical Summary

Here is a detailed technical summary of the paper "Adaptive SINDy: Residual Force System Identification Based UAV Disturbance Rejection."

1. Problem Statement

Unmanned Aerial Vehicles (UAVs) face significant challenges maintaining stability and trajectory tracking in turbulent wind environments. The core difficulties include:

• Non-linearity and Uncertainty: Wind dynamics are highly non-linear and difficult to model analytically (e.g., via Dryden or von Kármán models) for control design purposes.
• Limitations of Existing Methods:
  • Classical Control (PID, LQR): Often fail to reject strong disturbances or require extensive tuning.
  • Disturbance Observers: Can be limited to non-complex environments or specific system configurations.
  • Learning-Based Approaches (Neural Networks, RL): While effective, they often lack interpretability (acting as "black boxes") and require large datasets for training, hindering generalization to new UAV models or conditions.
• The Gap: There is a need for a control framework that combines the interpretability of physical modeling with the adaptability of data-driven methods to handle unknown residual forces (wind) in real-time.

2. Methodology

The authors propose a novel hybrid framework integrating Sparse Identification of Non-Linear Dynamics (SINDy) with Recursive Least Squares (RLS) adaptive control.

A. System Modeling

The UAV dynamics are decomposed into known and unknown components:
$$\dot{X} = f(x, u)_{\text{known}} + g(x, u, w)_{\text{unknown}}$$
Where $x$ represents the 12 states (position, velocity, orientation, etc.), $u$ is the control input, and $w$ is the unknown wind disturbance.
The goal is to identify the residual force ($\dot{Y}$), defined as the difference between actual dynamics and the known model:
$$\dot{Y} = \dot{X} - f(x, u)_{\text{known}}$$

B. Sparse Identification (SINDy)

Instead of using a neural network, the authors use SINDy to find a sparse set of coefficients $\Xi$ that best approximate the residual dynamics using a library of candidate functions $\Theta(x, u)$:
$$\dot{Y} = \Theta(x, u) \Xi$$

• Physics-Informed Library: Based on the observation that wind induces a tilt (roll $\phi$ and pitch $\theta$) in the drone to counteract the force, the library includes:
  $$\Theta = [1, \theta, \phi, T, T\sin(\theta), T\cos(\theta), T\sin(\phi), T\cos(\phi)]$$
  where $T$ is thrust. This ensures the model remains interpretable and physically relevant.
• Optimization: The authors use Sparse Relaxed Regularized Regression (SR3) to solve the optimization problem, allowing for constraints and improved numerical stability over standard STLSQ.

C. Adaptive Control (RLS)

Once the SINDy structure is identified, an RLS adaptive controller updates the coefficients ($A$) online to track changing wind conditions.

• Residual Estimation: The controller estimates the disturbance force $\hat{f}_{dist} = \phi A$.
• Control Law: The desired force $F_d$ is computed by compensating for the estimated disturbance:
  $$F_d = m a_{cmd} + m g_{vec} - \hat{f}_{dist}$$
• Update Law: The adaptation matrix $A$ is updated using a leaky RLS law driven by the error between the estimated and actual residual forces, ensuring the system adapts to new wind patterns in real-time.

3. Key Contributions

1. Novel Integration: First application of SINDy-based system identification coupled with RLS adaptive control specifically for UAV wind disturbance rejection.
2. Interpretability: Unlike deep learning approaches, the resulting model uses a sparse, physics-informed library (tilt angles and thrust), making the disturbance dynamics understandable.
3. Data Efficiency: The method requires relatively small amounts of data for training compared to Reinforcement Learning or Deep Neural Networks.
4. Robust Architecture: Successfully demonstrated on both simulation (ArduPilot SITL and Crazyflie) and real-world flights on a lightweight Crazyflie drone in a highly turbulent environment (up to 2 m/s wind from four directions).

4. Experimental Results

The framework was tested on three trajectories (Circle, Lemniscate/Infinity, Spiral) under wind disturbances.

A. Simulation (ArduPilot SITL & Crazyflie)

• Comparison: Adaptive SINDy was compared against a baseline PID, an Incremental Non-Linear Dynamic Inversion (INDI) controller, and a pre-trained DAIML (Deep Learning) adaptive controller.
• Performance:
  • Adaptive SINDy consistently outperformed the baseline PID, reducing RMSE by significant margins (e.g., from ~0.5m to ~0.3m on circular paths).
  • On the Lemniscate trajectory, Adaptive SINDy slightly outperformed the DAIML model (RMSE: 0.350m vs 0.373m) and the INDI controller, demonstrating superior handling of complex, non-convex paths.
  • It maintained tight error distributions (low P95 values), indicating robustness against worst-case gusts.

B. Real-World Flight (Crazyflie)

• Setup: A Crazyflie 2.1 drone flown in a Gazebo-simulated wind tunnel environment using four ducted fans (1–2 m/s wind).
• Outcome:
  • PID Baseline: Failed completely, crashing on every attempt due to inability to reject disturbances.
  • Adaptive SINDy: Successfully completed all 15 flight runs (6 circles, 4 lemniscates, 5 spirals) without crashing.
• Metrics (Real Flight):
  • Lemniscate: RMSE = 12.2 cm, MAE = 10.5 cm.
  • Circle: RMSE = 17.6 cm, MAE = 13.7 cm.
  • Spiral: RMSE = 32.7 cm (highest due to expanding radius and increasing disturbance demand).

5. Significance and Conclusion

This paper presents a significant step forward in UAV control by bridging the gap between classical control theory and modern data-driven methods.

• Safety: The ability to keep a lightweight drone stable in turbulent winds where PID fails is critical for applications like search and rescue or delivery in urban canyons.
• Generalizability: The approach does not rely on massive datasets or black-box models, making it easier to deploy on different UAV platforms with minimal retraining.
• Future Work: The authors suggest extending this to larger UAVs, more turbulent environments, and implementing on-board system identification using state history to adapt to changing dynamics in real-time without external data collection.

In summary, Adaptive SINDy offers a robust, interpretable, and data-efficient solution for UAV disturbance rejection, proving effective in both simulation and challenging real-world flight conditions.