Simultaneous State Estimation and Online Model Learning in a Soft Robotic System

TL;DR

This work tackles simultaneous state estimation and online model learning for a nonlinear soft robotic system using a gray-box formulation. It couples a nominal constant-curvature model with a state-dependent bending stiffness learned online via a reduced-rank Gaussian Process, all integrated through a marginalized particle filter that samples states while marginalizing GP parameters. The method includes online GP hyperparameter learning and yields a posterior over states, stiffness, and GP coefficients, enabling improved multi-step forward predictions. Experiments on real soft-robot data show state estimates comparable to a UKF baseline and substantial gains in predictive accuracy when incorporating the online-learned bending-stiffness model, highlighting the approach’s potential for adaptive, predictive control in soft robotics.

Abstract

Operating complex real-world systems, such as soft robots, can benefit from precise predictive control schemes that require accurate state and model knowledge. This knowledge is typically not available in practical settings and must be inferred from noisy measurements. In particular, it is challenging to simultaneously estimate unknown states and learn a model online from sequentially arriving measurements. In this paper, we show how a recently proposed gray-box system identification tool enables the estimation of a soft robot's current pose while at the same time learning a bending stiffness model. For estimation and learning, we rely solely on a nominal constant-curvature robot model and measurements of the robot's base reactions (e.g., base forces). The estimation scheme -- relying on a marginalized particle filter -- allows us to conveniently interface nominal constant-curvature equations with a Gaussian Process (GP) bending stiffness model to be learned. This, in contrast to estimation via a random walk over stiffness values, enables prediction of bending stiffness and improves overall model quality. We demonstrate, using real-world soft-robot data, that the method learns a bending stiffness model online while accurately estimating the robot's pose. Notably, reduced multi-step forward-prediction errors indicate that the learned bending-stiffness GP improves overall model quality.

Figures (3)

• Figure 1: Nonlinear soft robotic system with the unknown states $\boldsymbol{x} \in \mathbb{R}^{6}$, defining its current pose and velocity, control input pressures $\boldsymbol{u} \in \mathbb{R}^{3}$, and force/torque measurements of the base reactions $\boldsymbol{y} \in \mathbb{R}^{3}$. Simultaneously to estimating the current hidden states $\boldsymbol{x}$, a state-dependent bending stiffness model $k_{\mathrm{bend}} (\boldsymbol{x})$ is learned using a particle filter, marginalized over model parameters. Figure adapted from Mehl.2024 and reprinted with permission from 2024 IEEE Int. Conf. on Robotics and Automation (ICRA). © 2024 IEEE.
• Figure 2: True and estimated hidden states of the soft robot, which is driven by the pneumatic control inputs. Simultaneously with state estimation, the marginalized pf learns a nonlinear bending stiffness model $k_{\mathrm{bend}} (\boldsymbol{q})$.
• Figure 3: Mean and variance of the learned nonlinear bending stiffness gp model (top, evaluated for $\delta L = 0.008\,\mathrm{m}$) and comparison of posterior gp predictions with the particle-based estimates of the bending stiffness (bottom). The particle-based estimates and the gp predictions are consistent with each other.