How to Model Your Crazyflie Brushless

This paper presents an accurate dynamics model for the newly introduced Crazyflie Brushless quadcopter, validated through simulations and hardware experiments, and demonstrates its effectiveness in training reinforcement learning controllers for complex acrobatic maneuvers that successfully transfer from simulation to reality.

The Gist

Imagine you have a tiny, high-tech drone, about the size of a smartphone, called the Crazyflie. For years, researchers have used this little robot to teach computers how to fly, how to swarm together, and how to do tricks. But recently, in early 2025, the makers released a "Pro" version called the Crazyflie Brushless.

Think of the old version as a car with a standard engine, and the new Brushless version as a car with a turbocharger. It's lighter, stronger, and can fly much faster and do cooler acrobatics.

However, there's a problem: Nobody knew exactly how to write the "instruction manual" for how this new turbo-charged drone moves. Without that manual, it's hard to teach a computer how to control it perfectly.

This paper is the team's solution. They built a digital twin—a perfect virtual copy of the new drone—and used it to teach a computer how to fly the real thing.

Here is the breakdown of what they did, using some everyday analogies:

1. Building the "Digital Twin" (The Model)

To control a drone, you need a mathematical model that predicts: "If I tell the motors to spin this fast, the drone will move up that much."

• The Old Way: Previous models were like a rough sketch. They worked okay for slow, gentle flying but got messy when the drone tried to do something wild.
• The New Way: The authors created a highly detailed "Digital Twin." They measured the new drone's motors, its weight, and how the air pushes against it. They turned all these measurements into a set of equations that act like a physics simulator.
• The Analogy: Imagine trying to learn to ride a bike. The old model was like a drawing of a bike. The new model is like a video game that feels exactly like riding a real bike, including the wind resistance and the wobble of the handlebars.

2. Teaching the Drone to Fly (Reinforcement Learning)

Once they had this perfect video game (the simulator), they didn't just program the drone with rules. Instead, they used Reinforcement Learning.

• How it works: Think of this like training a dog. You don't tell the dog, "Sit, then stay, then jump." You just say, "Do whatever it takes to get a treat."
• The Experiment: They put the "dog" (a neural network computer brain) inside the simulator.
  • Task 1: "Fly to that spot and hover." (Like teaching the dog to sit).
  • Task 2: "Do a backflip!" (Like teaching the dog to roll over).
• The Result: The computer tried millions of times in the simulator. It crashed a lot, but every time it got closer to the goal, it got a "digital treat." Eventually, it learned the perfect way to fly.

3. The "Sim-to-Real" Magic

The biggest challenge in robotics is the Sim-to-Real Gap. This is the difference between the video game and the real world.

• In the game: The wind is perfect, the battery is always full, and the motors are exactly as strong as the math says.
• In reality: There is a breeze, the battery is slightly weaker, and the motors might vibrate.

Usually, a robot trained in a game crashes immediately when you put it in the real world.

The Team's Secret Sauce: Domain Randomization
To fix this, they didn't just train the AI in one perfect world. They trained it in 1,000 slightly broken worlds.

• Sometimes they made the drone 10% heavier.
• Sometimes they made the motors 20% weaker.
• Sometimes they added a fake wind gust.

The Analogy: Imagine you are learning to drive a car. If you only practice on a perfectly smooth, empty track, you'll crash on a rainy, bumpy road. But if you practice on wet roads, icy roads, and bumpy roads, you become a master driver who can handle anything.

By training the AI in these "broken" versions of the simulator, the AI learned to be robust. When they finally uploaded the brain to the real Crazyflie Brushless, it didn't crash. It flew perfectly.

4. The Grand Finale: The Double Backflip

The ultimate test was to see if this new model could handle extreme stunts.

• They trained the AI to do a double backflip (two full rotations in the air).
• The drone did this in a tiny space, only moving up about 6 feet (1.8 meters).
• It spun incredibly fast, stopped, and landed perfectly.

This proved that their "Digital Twin" was accurate enough to teach a robot to do gymnastics that would be impossible with the old, rougher models.

Why Does This Matter?

• For Researchers: They now have a free, open-source "video game" (available on GitHub) that is so accurate they can test new ideas on their computers before ever risking a real, expensive drone.
• For the Future: This helps us build swarms of tiny drones that can fly through forests, rescue people in disasters, or race each other at high speeds, because we finally have a map of how these super-fast little robots actually move.

In short: The authors built a perfect virtual copy of a new, super-fast drone, trained an AI to fly it in a chaotic video game, and then successfully transferred that AI to the real world to perform amazing acrobatics. They bridged the gap between "computer simulation" and "real-life flight."

Technical Summary

Here is a detailed technical summary of the paper "How to Model Your Crazyflie Brushless":

1. Problem Statement

The Crazyflie 2.1 Brushless (CFB) was released in early 2025 as a significant upgrade to the popular Crazyflie 2.1 nano-quadcopter. The CFB features brushless motors and electronic speed controllers (ESCs), providing approximately 50% more thrust and a thrust-to-weight ratio of 3:1 (compared to 2:1 for the brushed predecessor). This increased agility opens new research avenues for high-speed control and acrobatic maneuvers.

However, a critical gap existed: no accurate, open-source dynamics model was available for the CFB. Existing models were designed for the older brushed-motor Crazyflie 2.x, which behaves differently due to distinct motor dynamics, thrust characteristics, and control loops. Without a precise model, researchers could not effectively utilize Simulation-to-Real (Sim2Real) transfer techniques, particularly for Reinforcement Learning (RL), which is highly sensitive to model inaccuracies.

2. Methodology

The authors developed a comprehensive first-principles dynamics model and a high-fidelity simulator to bridge this gap.

• System Modeling:

  • The model treats the CFB as a rigid body with a state-space representation ($\dot{x} = f(x, u)$) involving 17 states (position, velocity, quaternion, angular velocity, and motor speeds).
  • Dynamics: Includes translational and rotational dynamics driven by motor thrust and reactive torques.
  • Motor Dynamics: Unlike the brushed models, the CFB uses brushless motors controlled by ESCs. The authors modeled the motor-ESC system as a first-order linear system characterized by a time constant ($T$) and amplification factor ($K$).
  • Thrust & Torque: Nonlinear relationships between motor rotation speed ($\Omega$) and thrust/torque were modeled using 3rd-order polynomials.
  • Simulation: The model was implemented in MuJoCo XLA (MJX) using the JAX library, enabling highly parallelized GPU simulations.

• System Identification:

  • Motor Parameters: Identified by recording motor commands and RPM (measured via an IR sensor deck) during flight to fit $T$ and $K$.
  • Thrust/Torque Curves: Measured using a custom force/torque testbed with a rotating arm, fitting polynomial curves to steady-state data.
  • Inertia & Mass: Determined by fitting simulated trajectories to measured flight data. The paper provides guidelines for users to manually re-identify moments of inertia when payloads (e.g., propeller guards, decks) are added.

• Reinforcement Learning (RL) Pipeline:

  • Algorithm: Proximal Policy Optimization (PPO) implemented via BRAX on GPUs.
  • Tasks:
    1. Position Control: Learning to fly between setpoints.
    2. Acrobatics: Learning to execute single and double backflips.
  • Domain Randomization: To bridge the Sim2Real gap, key parameters (mass, inertia, motor dynamics, thrust scaling, center of gravity, and attitude measurement offsets) were randomized during training.

3. Key Contributions

1. First Open CFB Dynamics Model: The paper presents the first publicly available, accurate dynamics model specifically for the Crazyflie Brushless, including motor dynamics and parameter identification guidelines.
2. High-Performance Simulator: A GPU-accelerated simulator (MJX-based) that allows for massive parallelization, facilitating rapid RL training.
3. Sim2Real Validation via RL: The authors demonstrated that controllers trained entirely in simulation using their model could be deployed on real hardware without fine-tuning.
   • Position Control: The learned NN controller achieved accuracy comparable to the firmware's PID and Mellinger controllers.
   • Acrobatics: The model enabled the training of an end-to-end neural network capable of executing two consecutive backflips within a vertical space of only 1.8 meters.
4. Domain Randomization Analysis: The study quantified the impact of domain randomization, finding that a 10–20% randomization of mass, inertia, and motor parameters is optimal for bridging the sim-to-real gap for this platform.

4. Results

• Model Accuracy:
  • The model accurately predicts behavior over a 200 ms horizon.
  • Deviations over longer horizons were attributed to unmodeled aerodynamics, sensor noise (Kalman filter estimates vs. ground truth), and center-of-gravity uncertainties.
  • The CFB model showed significantly higher accuracy than the existing PyBullet-drones model for the older CF 2.1.
• Control Performance:
  • Hovering: The NN controller with domain randomization (magnitude 1.0) achieved a mean distance error of 43.16 mm, outperforming the Mellinger controller (54.68 mm) and approaching the PID controller (27.81 mm).
  • Maneuvers: The system successfully executed complex maneuvers, including a double backflip, demonstrating the model's ability to capture high-dynamic behaviors.
• Robustness: Experiments showed that while low randomization leads to reduced vertical accuracy, excessive randomization degrades overall performance. A moderate range (10–20%) provided the best trade-off for robustness.

5. Significance

This work is a foundational step for the nano-quadcopter research community. By providing an accurate, open-source model and simulator for the Crazyflie Brushless, the authors:

• Democratize High-Speed Research: Enable researchers to develop and test agile control algorithms (like RL) without needing extensive hardware-in-the-loop tuning.
• Advance Sim2Real Techniques: Provide empirical evidence on how much domain randomization is necessary for successful transfer on high-agility platforms.
• Enable New Applications: The ability to learn acrobatic maneuvers (like double backflips) in simulation and deploy them on real hardware opens the door to advanced swarm robotics, racing, and complex aerial manipulation tasks using nano-quadcopters.

The entire codebase, including the model, parameters, and simulator, has been open-sourced to facilitate immediate adoption by the community.