Organizations: Department of Engineering Mechanics,Northwestern Polytechnical University, Xi’an, 710072, China · Shenzhen Research Institute of Northwestern Polytechnical University, Shenzhen, 518063, China · School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, 610031, China · School of Engineering, University of Liverpool, Liverpool, L69 3GH, U.K. · School of Aeronautics and Astronautics, Sun Yat-sen University, No.66 Gongchang Road, Guangming District, Shenzhen, 518107, Guangdong, China · School of Science, Qingdao University of Technology, Qingdao, 266520, China · Department of Mechanical Engineering, University of Birmingham, Birmingham, B15 2TT, UK · School of Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK
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.
Figures & tables
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 d1 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. [ 41 , 30 ] . (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. [ 39 ] . (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 configuration [ 33 ] 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 .
Figure 6: Multistable behavior of the rigid Kresling origami. (a–c) Load–displacement response and corresponding energy landscape of a single rigid Kresling unit under axial displacement loading. (d–f) Load–displacement response and energy distribution under circumferential rotational displacement loading. (g) Dynamic switching simulation of a four-unit rigid Kresling assembly exhibiting multistable transitions. (h) Time histories associated with the dynamic switching process. (i) Deformed configurations at representative time instants during multistable transition.
Figure 7: Dynamic locomotion of a compliant crawling robot actuated by Kresling-based units. (a–d) Mechanical characteristics of a compliant Kresling actuator, including its load–displacement response and multistable behavior. (e) One complete locomotion cycle of the crawling robot [ 47 ] . (f) Schematic of the external magnetic field and orientation of the net magnetization vector in the actuator. (g) Robot configurations under varying magnetic field strengths. (h) Definition of the friction coefficient governing foot–ground interaction. (i) Time histories of the foot displacement during robot locomotion.
IRiS Lab, Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea. · CHARM Lab, Department of Mechanical Engineering, Stanford University, Stanford, CA, USA.
May 15, 2025·Ziyang Zhou, Yogesh Phalak, Vishrut Deshpande +3Robotics
Department of Mechanical Engineering, Virginia Tech,, Blacksburg & 24060, USA. · Department of Electrical Engineering and Computer Science, University of Wyoming, Laramie & 82071, USA.