💻 computer science

Koopman DCM: Unstable Eigenfunctions as Data-driven Representations for Legged Balancing

This paper proposes a data-driven approach to formulate Divergent Components of Motion (DCMs) as unstable Koopman eigenfunctions using real-robot data, demonstrating improved walking pattern tracking and viability constraints for bipedal balancing.

Original authors: Stéphane Caron

Published 2026-07-22

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Stéphane Caron

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

=== DRAFT ===

The Balancing Act: Teaching Robots to Stay Upright

Imagine trying to balance a broomstick on the palm of your hand. It's a classic party trick, but for a robot, it's a life-or-death calculation. This is the world of legged locomotion, where scientists teach machines with legs—like bipeds or quadrupeds—to walk without toppling over. The core challenge is that these robots are naturally unstable; gravity wants to pull them down, and they have to constantly fight to stay upright.

To solve this, engineers often use simplified mental models, like imagining the robot as a single point mass on a stick (a "linear inverted pendulum"). In this simplified world, there's a special "magic number" called the Divergent Component of Motion (DCM). Think of the DCM as a "tipping point" indicator. If you know where the DCM is, you know exactly how much the robot is leaning toward a fall. If you can control the DCM, you can keep the robot balanced. However, these old-school models are like training wheels: they work well on paper but often fail to capture the messy, complex reality of a real robot moving on real ground. They rely on perfect assumptions that rarely exist in the wild.

The Paper's Big Idea: Letting the Data Do the Talking

This paper, titled "Koopman DCM: Unstable Eigenfunctions as Data-driven Representations for Legged Balancing," asks a bold question: What if we didn't force the robot to fit our simplified model, but instead let the robot teach us its own version of the "tipping point"?

The authors, led by Stéphane Caron, propose a new way to find the DCM. Instead of calculating it based on a textbook formula, they use a mathematical tool called the Koopman operator to hunt for "unstable eigenfunctions." In plain English, they are looking for a specific pattern in the robot's movement data that grows rapidly when the robot starts to fall. They call this pattern a "Koopman DCM."

Here is the twist: Most scientists using the Koopman operator look for patterns that stay the same or change slowly (like a spinning top that keeps spinning). But Caron's team deliberately looked for the opposite—they searched for the patterns that grow the fastest, the ones that scream "I'm falling!" This allowed them to train a neural network using one hour of real data from a wheeled biped robot named "Cookie." They didn't need a perfect physics model; they just needed the robot to walk, wobble, and correct itself while recording what happened.

What They Found: Smarter Balancing

The results were impressive. When they tested this new, data-driven DCM on the real robot, it outperformed the traditional, model-based method.

Better Tracking: When asked to follow a wavy walking path, the robot using the learned DCM made fewer mistakes. In one test, it tracked the path 2.4 times better than the old method.
Fewer Falls: They also tested this on a simulation of a humanoid robot (the Unitree G1). When they used the learned DCM inside a "Model Predictive Control" system (a brain that plans steps ahead of time), the robot never fell in 90 trials. In contrast, the robot using the old method fell in 33% of the trials in that specific simulation.

The paper shows that by learning directly from the robot's own experience, we can create a "tipping point" detector that is more accurate and robust than the one we designed by hand.

The Catch: It's Not Magic (Yet)

While the results are strong, the authors are careful not to overhype. They admit their method makes some specific choices to keep things simple. For instance, they assumed the robot's control inputs (like how hard it pushes its wheels) affect the balance in a very specific, linear way. In the real, messy world, this relationship might be more complicated. They also note that while their method worked great on a wheeled robot in the real world, they haven't yet tested it on a full, complex walking robot in the real world without wheels.

Furthermore, the "unstable" patterns they found are specific to the data they collected. If the robot's legs were different or the floor was slippery, they would likely need to collect a new hour of data to retrain the system.

Why It Matters

This work is a step toward robots that can learn to balance on their own, rather than relying on engineers to guess the perfect physics equations. It suggests that if we give robots the right tools to listen to their own movements, they can figure out how to stay upright even when the world is messy and unpredictable. It's a shift from "teaching a robot the rules of physics" to "letting the robot discover the rules of its own balance."

Technical Summary: Koopman DCM: Unstable Eigenfunctions as Data-driven Representations for Legged Balancing

Problem Statement
Legged robot balance control has historically relied on reduced-order models, most notably the Linear Inverted Pendulum (LIP). In the LIP framework, the Divergent Component of Motion (DCM) serves as a critical control coordinate that isolates the system's unstable mode. However, existing DCM formulations face two primary limitations:

Model Dependency: The DCM is strictly defined for reduced models (like the LIP) and does not naturally extend to the full, nonlinear dynamics of a real robot.
Parameter Mismatch: The natural frequency $\omega = \sqrt{g/h}$ used in standard DCM formulations is derived from a theoretical center-of-mass (CoM) height $h$ . This model-based parameter often fails to capture the true dynamics of complex, real-world systems, leading to suboptimal control performance.

The paper addresses these issues by re-formulating the DCM not as a property of a specific reduced model, but as a Koopman eigenfunction of the full system dynamics, learnable directly from real-robot data.

Methodology
The authors propose a data-driven pipeline to learn "Koopman DCMs" as unstable eigenfunctions. The core methodology involves:

Theoretical Reformulation:
- The authors frame the search for a DCM as an eigenfunction search problem for the Koopman operator.
- Unlike typical Koopman applications that target eigenvalues near zero (conserved quantities), this work deliberately searches for unstable eigenpairs with large positive eigenvalues ( $\omega > 0$ ).
- For a control-affine system $\dot{x} = f(x) + B(x)u$ , the learned observable $\xi$ is constrained to satisfy the linear differential equation $\dot{\xi} = \omega(\xi - u)$ , mirroring the structure of the LIP DCM but derived from data.
Learning Pipeline:
- Input: The system uses a delay embedding of measurement-action histories ( $z_k$ ) rather than a predefined state space.
- Architecture: A Multi-Layer Perceptron (MLP) learns the observable $\xi_\theta$ .
- Loss Functions: The training objective combines two components to avoid trivial solutions (where $\omega \to 0$ $ω \to 0$ and $\xi \to 0$ $ξ \to 0$ ):
  - DCM Dynamics Loss: Enforces the linear recursion $\xi_{k+1} = e^{\omega \delta t}\xi_k + (1 - e^{\omega \delta t})u_k$ .
  - CoM Prediction Loss: Enforces the relationship between the learned DCM and the CoM position, ensuring the learned observable anchors the stable CoM dynamics.
- Temporal Extension: To prevent convergence to low eigenvalues on real data, the authors employ Eligibility Traces (TD( $\lambda$ ) loss), evaluating the encoder against multi-step predictions rather than just one-step residuals.
Control Application:
- Real-Robot: The learned DCM is used in a proportional feedback loop to track Zero-Moment Point (ZMP) references on a wheeled biped (Cookie).
- Simulation (MPC): The learned DCM serves as the state variable in a Model Predictive Control (MPC) framework for a humanoid (Unitree G1). The DCM dynamics are used to formulate viability constraints (ensuring the robot remains within the support polygon), which are enforced as hard constraints in the optimization.

Key Contributions

Re-casting DCMs: The paper frames Divergent Components of Motion as unstable Koopman eigenfunctions, generalizing the concept beyond reduced models.
Data-Driven Training: A novel pipeline is proposed to learn these eigenfunctions directly from closed-loop measurement-action data without requiring a reduced-order kinematic model.
Real-Robot Deployment: The method is deployed on a physical biped, demonstrating improved tracking of walking patterns compared to model-based baselines.
Viability Constraints: The authors demonstrate how learned DCMs can be integrated into MPC to enforce state-based viability constraints, improving stability in simulation.

Results

Real-Robot Performance (Cookie Biped):
- The system was trained on 1 hour of real-robot data.
- In ZMP tracking experiments with sinusoidal references, the learned DCM (with a delay horizon $H=10$ ) significantly outperformed the kinematic LIP baseline.
- Specifically, at an amplitude of 0.2, the learned DCM achieved a Root Mean Square Error (RMSE) of 0.05, compared to 0.12 for the kinematic baseline (a 2.4x improvement).
- The learned controller consistently reduced tracking errors across various amplitudes and periods.
Simulation Performance (Unitree G1):
- In a lateral balancing simulation, a controller using the learned DCM within an MPC framework achieved a 0% fall rate over 90 trials.
- In contrast, a proportional DCM controller (without the learned eigenfunction constraints) resulted in a 33% fall rate (30/90 trials) under the same challenging conditions.
- The MPC approach also reduced ZMP tracking error (0.59 vs 0.73).

Significance and Claims
The paper claims that by treating DCMs as data-driven Koopman eigenfunctions, it is possible to bypass the limitations of reduced-order models. The primary significance lies in the ability to:

Learn System-Specific Dynamics: The method captures the true unstable modes of the robot directly from data, rather than relying on theoretical parameters like CoM height.
Enable Robust Control: The learned observables provide a more accurate state representation for feedback control and viability analysis.
Avoid Sim-to-Real Transfer Issues: Because the pipeline trains directly on closed-loop data from the real robot (or a high-fidelity simulator), it sidesteps the domain adaptation challenges typically associated with training in simulation and deploying on hardware.

The authors remain modest regarding the scope, noting that the current formulation assumes a specific linear input coupling ( $\dot{\xi} = \omega(\xi - u)$ ) and that future work could explore more general bilinear dynamics or variable-height models.