Centroidal State Estimation based on the Koopman Embedding for Dynamic Legged Locomotion

TL;DR

Centroidal state estimation for dynamic legged locomotion is challenging due to underactuation and intermittent contact forces that amplify sensor noise. The authors propose a Koopman embedding to linearize centroidal dynamics, implemented via Dynamic Mode Decomposition with Control (DMDC) and a Deep Learning Koopman (DLK) approach, and integrate the resulting linear model into a Moving Horizon Estimator that yields a convex quadratic program for real-time estimation of $\boldsymbol{x}=[\boldsymbol{c}_G,\boldsymbol{l}_G,\boldsymbol{k}_G]$. They evaluate both embeddings on simulated trotting, jumping, and bounding trajectories for a quadruped and show that the linear model from the DMDC family provides robust prediction suitable for MHE, outperforming a nonlinear EKF. The results demonstrate improved accuracy and robustness against end-effector force/torque noise, enabling reliable centroidal state estimation across fast gaits and suggesting practical applicability to real hardware and broader legged platforms.

Abstract

In this paper, we introduce a novel approach to centroidal state estimation, which plays a crucial role in predictive model-based control strategies for dynamic legged locomotion. Our approach uses the Koopman operator theory to transform the robot's complex nonlinear dynamics into a linear system, by employing dynamic mode decomposition and deep learning for model construction. We evaluate both models on their linearization accuracy and capability to capture both fast and slow dynamic system responses. We then select the most suitable model for estimation purposes, and integrate it within a moving horizon estimator. This estimator is formulated as a convex quadratic program to facilitate robust, real-time centroidal state estimation. Through extensive simulation experiments on a quadruped robot executing various dynamic gaits, our data-driven framework outperforms conventional Extended Kalman Filtering technique based on nonlinear dynamics. Our estimator addresses challenges posed by force/torque measurement noise in highly dynamic motions and accurately recovers the centroidal states, demonstrating the adaptability and effectiveness of the Koopman-based linear representation for complex locomotive behaviors. Importantly, our model based on dynamic mode decomposition, trained with two locomotion patterns (trot and jump), successfully estimates the centroidal states for a different motion (bound) without retraining.

Figures (7)

• Figure 1: Moving horizon estimator utilizes the recent history of state measurements and system inputs, to predict the optimal states.
• Figure 2: The neural network architecture to learn the Koopman embedding approximation of centroidal dynamics.
• Figure 3: RMSE for open-loop simulation with learned linearized models on validation dataset. The deep learning Koopman model outperforms the dynamic mode decomposition model in terms of prediction accuracy nearly for all states across various motions.
• Figure 4: Eigenvalues of the state-transition matrices for both learned linear models.
• Figure 5: Motion #1: Trot. Estimation of the CoM position $(m)$ (top row), linear momentum $(\frac{kgm}{s})$ (middle row) and angular momentum $(\frac{kgm^2}{s})$ (bottom row), commanded base linear velocities during motion, $v_x =0.7 (\frac{m}{s})$ and $v_y =0.3 (\frac{m}{s})$. As evident, the intermittent rapid contact switching degrades the performance of EKF especially in estimating the linear momentum, while our estimator effectively maintains accurate state estimation during the motion.