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From Folding Mechanics to Robotic Function: A Unified Modeling Framework for Compliant Origami

Bohan Zhang, Bo Wang, Huajiang Ouyang, Zhigang Wu, Haohao Bi, Jiawei Xu, Mingchao Liu, Weicheng Huang

Abstract

Origami inspired architectures offer a powerful route toward lightweight, reconfigurable, and programmable robotic systems. Yet, a unified mechanics framework capable of seamlessly bridging rigid folding, elastic deformation, and stability driven transitions in compliant origami remains lacking. Here, we introduce a geometry consistent modeling framework based on discrete differential geometry (DDG) that unifies panel elasticity and crease rotation within a single variational formulation. By embedding crease panel coupling directly into a mid edge geometric discretization, the framework naturally captures rigid folding limits, distributed bending, multistability, and nonlinear dynamic snap through within one mechanically consistent structure. This unified description enables programmable control of stability and deformation across rigid and compliant regimes, allowing origami structures to transition from static folding mechanisms to active robotic modules. An implicit dynamic formulation incorporating gravity, contact, friction, and magnetic actuation further supports strongly coupled multiphysics simulations. Through representative examples spanning single fold bifurcation, deployable Miura membranes, bistable Waterbomb modules, and Kresling based crawling robots, we demonstrate how geometry driven mechanics directly informs robotic functionality. This work establishes discrete differential geometry as a foundational design language for intelligent origami robotics, enabling predictive modeling, stability programming, and mechanics guided robotic actuation within a unified computational platform.

From Folding Mechanics to Robotic Function: A Unified Modeling Framework for Compliant Origami

Abstract

Origami inspired architectures offer a powerful route toward lightweight, reconfigurable, and programmable robotic systems. Yet, a unified mechanics framework capable of seamlessly bridging rigid folding, elastic deformation, and stability driven transitions in compliant origami remains lacking. Here, we introduce a geometry consistent modeling framework based on discrete differential geometry (DDG) that unifies panel elasticity and crease rotation within a single variational formulation. By embedding crease panel coupling directly into a mid edge geometric discretization, the framework naturally captures rigid folding limits, distributed bending, multistability, and nonlinear dynamic snap through within one mechanically consistent structure. This unified description enables programmable control of stability and deformation across rigid and compliant regimes, allowing origami structures to transition from static folding mechanisms to active robotic modules. An implicit dynamic formulation incorporating gravity, contact, friction, and magnetic actuation further supports strongly coupled multiphysics simulations. Through representative examples spanning single fold bifurcation, deployable Miura membranes, bistable Waterbomb modules, and Kresling based crawling robots, we demonstrate how geometry driven mechanics directly informs robotic functionality. This work establishes discrete differential geometry as a foundational design language for intelligent origami robotics, enabling predictive modeling, stability programming, and mechanics guided robotic actuation within a unified computational platform.
Paper Structure (25 sections, 40 equations, 7 figures)

This paper contains 25 sections, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A Geometry-Consistent Discrete Mechanics Framework for Compliant Origami
  3. Unified Geometric and Variational Formulation
  4. Discrete Elastic Energy of Compliant Panels
  5. Geometry-Consistent Modeling of Crease Interfaces
  6. Dynamic Formulation for Active Origami Systems
  7. Gravity.
  8. Normal Contact.
  9. Tangential Contact.
  10. Magnetic Actuation.
  11. Programmable Stability in Compliant Origami Units
  12. Tensile Loading: Bending–Crease Coupling
  13. Shear Loading: Nonlinear Response and Bifurcation
  14. Bridging Rigid and Compliant Origami Regimes
  15. Kinematic Folding in the Rigid Limit
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Schematic illustration of a flexible origami structure and its discrete modeling framework. (a) Minimal structural composition of a flexible origami unit, consisting of a single crease (highlighted in orange) connecting two shell panels (highlighted in blue). (b) Discretization of the shell module: the smooth panel surface is approximated by a triangular mesh, within which the discrete first and second fundamental forms are defined to characterize in-plane stretching and bending. (c) Discretization of the crease module: the crease is modeled as a combination of rotational elements and additional bending elements.
  • Figure 2: Mechanical response of the single-fold structure. (a) Deformation configuration under tensile displacement loading. (b,c) Boundary profile evolution for varying applied displacement $d_1$ and different crease stiffness values. (d) Variation of the folding angle as a function of crease stiffness. (e) Two distinct deformation modes under shear displacement loading: mirror-symmetric and centrally-symmetric configurations. (f,g) Deformed boundary profiles corresponding to the two modes. (h,i) Evolution of nodal displacement and total energy in the two modes.
  • Figure 3: Mechanical behavior of rigid and flexible Miura origami structures. (a) Compressive deformation of the rigid Miura configuration, and (b) corresponding load--displacement response. (c) Evolution of the effective in-plane Poisson's ratio. (d) Four-point bending configuration applied to the flexible Miura structure. The theoretical model is derived from Ref. wei2013geometricschenk2013geometry. (e) Evolution of the folding angle. (f) Comparison between the simulated curvature ratio and the analytical prediction. (g,h) Two deformed configurations under bending, exhibiting saddle-shaped and cylindrical surface morphologies.
  • Figure 4: Deployment simulation of Miura origami membranes. (a) Evolution of the deployed configuration at representative time instants. The deployable membrane configuration is derived from Ref. wang2023simulation. (b,c) Load--response curves under displacement-controlled and force-controlled actuation, respectively. (d) Influence of crease stiffness on the surface flatness of the deployed membrane.
  • Figure 5: Bistable mechanical behavior of the Waterbomb origami. (a) Two stable states of the flexible Waterbomb configuration and the associated flip. (b) Deformation profiles at representative instants during the state-switching process. (c,d) Load--displacement response and corresponding energy evolution. (e) Two stable states of the rigid Waterbomb configurationta2022printable and its snap-through behavior. (f,g) Time--displacement histories under free fall from heights $z=5$ and $z=10$, leading to recovery and flip, respectively. (h,i) Quasi-static load--displacement response and energy distribution .
  • ...and 2 more figures