Harnessing Discrete Differential Geometry: A Virtual Playground for the Bilayer Soft Robotics

TL;DR

This work addresses the challenge of accurately simulating bilayer soft robots undergoing complex, environment-driven deformations. It introduces a Discrete Differential Geometry (DDG)–based Discrete Elastic Rod (DER) framework with a bilayer-specific energy formulation and inter-layer constraints, enabling robust modeling of stretching, bending, twisting, and coupling, along with frictional contact and fluid drag. The approach is validated against classical 2D and 3D benchmarks, demonstrating correct bending-curvature and bend–twist coupling, and is applied to gripping, crawling, jumping, and swimming demonstrations, showing strong agreement with experiments. The resulting simulator provides a general, efficient platform for design and control of advanced bilayer soft robotic systems in interactive environments, paving the way for model-based optimization and real-time applications.

Abstract

Soft robots have garnered significant attention due to their promising applications across various domains. A hallmark of these systems is their bilayer structure, where strain mismatch caused by differential expansion between layers induces complex deformations. Despite progress in theoretical modeling and numerical simulation, accurately capturing their dynamic behavior, especially during environmental interactions, remains challenging. This study presents a novel simulation environment based on the Discrete Elastic Rod (DER) model to address the challenge. By leveraging discrete differential geometry (DDG), the DER approach offers superior convergence compared to conventional methods like Finite Element Method (FEM), particularly in handling contact interactions -- an essential aspect of soft robot dynamics in real-world scenarios. Our simulation framework incorporates key features of bilayer structures, including stretching, bending, twisting, and inter-layer coupling. This enables the exploration of a wide range of dynamic behaviors for bilayer soft robots, such as gripping, crawling, jumping, and swimming. The insights gained from this work provide a robust foundation for the design and control of advanced bilayer soft robotic systems.

Figures (9)

• Figure 1: Examples of bilayer soft robots and structures. (A) The helix bilayer hydrogel inspired from the chiral pea pod armon2011geometry. (B) Moisture-driven crawling robot consisting of a hygroscopically active layer and a hygroscopically inactive layer shin2018hygrobot. (C) Dual-responsive jumping actuator driven by light and humidity, composed of a photothermal expansion layer, an interfacial adhesion layer, and a moisture-responsive layer li2022dual. (D) Light-driven jellyfish-like swimming robot fabricated from the striped composite hydrogel yin2021visible. (E) Tunable helical ribbon fabricated by bonding an unstrained elastic adhesive sheet to a pre-strained latex sheet ChenZi2010. (F) Inverse design of the target space curve by encoding the microstructure of the morphing bilayer ribbons lijh2025.
• Figure 2: Schematic representation of a bilayer structure. (A) Diagram of a bilayer system that composed of an top layer (red) and bottom layer (blue), with arc length ${s}_1$ and ${s}_2$, and thickness ${h}_1$ and ${h}_2$, respectively. (B) Discretization diagram of the bilayer structure. The subscripts and superscripts $i$ and $j$ denote the discrete element indices of top layer and bottom layer, respectively, e.g., node position $\mathbf{x}$, edge material frame $\{\mathbf{m}^{i}_{1}, \mathbf{m}^{i}_{2}, \mathbf{m}^{i}_{3}\}$ and the relative position $\mathbf{p}^{ij}$ between top layer and bottom layer.
• Figure 3: Model validation of bilayer structure. (A) Schematic diagram illustrating planar bending behavior of bilayer structure. The top and bottom layers are configured with Young's modulus $E_1$ and $E_2$, and thickness $h_1$ and $h_2$, respectively. The ratios $m$ and $n$ represent Young's modulus and thickness of the bottom layer relative to the top layer, respectively. (B) The normalized bending curvature $\kappa h / \eta$ plotted against Young's modulus ratio $m$. (C) The normalized bending curvature $\kappa h / \eta$ plotted against thickness ratio $n$. Comparisons are highlighted among the simulation results (circles), the simplified beam model theory (squares), and Timoshenko theory (solid line) as expressed in Eq. (\ref{['eq:timoshenko']}).
• Figure 4: (A) The state of bilayer structures before deformation, in which the top layer has a natural configuration of a spiral helix and the bottom layer has a natural configuration of a straight line. (B) The state of the bilayer structure after deformation, in which the bilayer structure morphs into a complex state.
• Figure 5: Gripping demonstration using a bilayer gripper to grasp a ping pong ball. A sequence of snapshots illustrate the dynamic grasping process in both the experiment wang2019soft and the simulation. The gripper transitions over time from its initial straight configuration to a fully wrapped state, showcasing its ability to conform to the object's shape.