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.

翻译：受折纸启发的结构为实现轻量化、可重构和可编程的机器人系统提供了一条有效途径。然而，目前仍缺乏一个统一的力学框架，能够无缝衔接顺应性折纸中的刚性折叠、弹性变形及稳定性驱动的状态转变。本文提出了一种基于离散微分几何的几何一致建模框架，将面板弹性与折痕旋转统一于单一变分公式中。通过将折痕-面板耦合直接嵌入中边几何离散化，该框架自然地在一个力学一致的结构中捕捉了刚性折叠极限、分布弯曲、多稳态以及非线性动态突跳行为。这种统一描述实现了跨越刚性与顺应性区域的稳定性和变形的可编程控制，使折纸结构能够从静态折叠机构转变为主动式机器人模块。进一步地，一个包含重力、接触、摩擦和磁致动在内的隐式动力学公式支持强耦合的多物理场仿真。通过涵盖单折痕分岔、可展开Miura膜、双稳态Waterbomb模块以及基于Kresling结构的爬行机器人等一系列代表性示例，我们展示了几何驱动的力学如何直接影响机器人功能。本工作确立了离散微分几何作为智能折纸机器人的基础设计语言，在一个统一的计算平台内实现了预测性建模、稳定性编程以及力学引导的机器人驱动。