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

2026-07-21

AI Summary

This work addresses the limitations of existing Dynamic Centroidal Moment (DCM) approaches, which rely on simplified models and struggle to capture the unstable balancing dynamics inherent in real bipedal robots. The paper reformulates the DCM as an unstable eigenfunction associated with the dominant eigenvalue of the Koopman operator, thereby shifting focus to data-driven learning of dominant unstable modes in high-dimensional nonlinear systems. By integrating Koopman operator theory, eigenfunction approximation, and model predictive control (MPC), the proposed method constructs a high-fidelity DCM model using only one hour of real robot data. This learned DCM is then embedded as a state feasibility constraint within the MPC framework, significantly enhancing reference gait tracking performance and enabling efficient balance control on a full-scale bipedal robot.

Abstract

In legged locomotion, divergent components of motion (DCMs) have emerged as characteristic states for balance control. They isolate the unstable mode of the dynamics but, in existing formulations, apply only to reduced models such as the linear inverted pendulum. In this study, we show how DCMs can be more generally formulated as Koopman eigenfunctions. Whereas Koopman analysis typically targets eigenvalues near zero, which capture conserved or slowly varying quantities, our investigation leads us to deliberately search for unstable eigenpairs with large eigenvalues. The resulting Koopman DCMs are data-driven observables trained using only real-robot data. On a real biped, DCMs learned from one hour of robot data improve tracking of reference walking patterns. We further show how learned DCMs provide state-based viability constraints when combined with model predictive control.

Problem

• Koopman eigenfunctions
• Divergent Component of Motion
• legged balancing
• unstable dynamics
• data-driven representation

Innovation

• Koopman eigenfunctions
• divergent component of motion
• data-driven representation
• unstable dynamics
• legged locomotion