Learning-Based Modeling of Soft Actuators Using Euler Spiral-Inspired Curvature

TL;DR

This work addresses the challenge of modeling soft continuum actuators under gravity and payload by introducing a data-driven framework built on an Euler spiral-inspired curvature representation. It combines a polynomial curvature profile $κ(s)$ with a ${G^1}$ Hermite interpolation to extract curvilinear parameters and trains separate MLPs to map actuation and loads to shapes (forward) and to infer payload from observed shapes (inverse). Experimental results demonstrate high accuracy, with forward task-space errors as low as $3.38\%$, $2.19\%$, and $1.93\%$ at the 1/3, 2/3, and tip, and payload estimation errors down to $0.72\%$, validating the method's predictive capability and data efficiency. The approach offers a scalable alternative to traditional continuum-robot models, enabling improved sensing, control, and payload estimation, and sets the stage for extending to multi-segment arms, 3D deformations, and dynamics-driven control.

Abstract

Soft robots, distinguished by their inherent compliance and continuum structures, present unique modeling challenges, especially when subjected to significant external loads such as gravity and payloads. In this study, we introduce an innovative data-driven modeling framework leveraging an Euler spiral-inspired shape representations to accurately describe the complex shapes of soft continuum actuators. Based on this representation, we develop neural network-based forward and inverse models to effectively capture the nonlinear behavior of a fiber-reinforced pneumatic bending actuator. Our forward model accurately predicts the actuator's deformation given inputs of pressure and payload, while the inverse model reliably estimates payloads from observed actuator shapes and known pressure inputs. Comprehensive experimental validation demonstrates the effectiveness and accuracy of our proposed approach. Notably, the augmented Euler spiral-based forward model achieves low average positional prediction errors of 3.38%, 2.19%, and 1.93% of the actuator length at the one-third, two-thirds, and tip positions, respectively. Furthermore, the inverse model demonstrates precision of estimating payloads with an average error as low as 0.72% across the tested range. These results underscore the potential of our method to significantly enhance the accuracy and predictive capabilities of modeling frameworks for soft robotic systems.

Figures (8)

• Figure 1: Schematic illustration of a soft fiber-reinforced bending actuator in different states. The bending actuator is straight when there is no air pressure initially, and bends with a constant curvature when the air inflates the inner chamber in the absence of loading. Under a payload applied on the tip, the actuator deforms into a shape with a continuously varying curvature, which could be approximated by an Euler spiral-inspired representation.
• Figure 2: Illustration of the Euler spiral.
• Figure 3: ${G^1}$ Hermite interpolation schema and notation.
• Figure 4: Illustration of soft actuator under tip loads.
• Figure 5: Regression results using linear and quadratic functions for different loading scenarios. (a–b) Quadratic and linear fitting under payload conditions. (c–d) Quadratic and linear fitting under contact conditions.