折纸顶点分析的拉格朗日方法：运动学
Lagrangian approach to origami vertex analysis: Kinematics
AI总结：
本文利用拉格朗日方法推导刚性折纸顶点的降阶协调条件，获得6度和8度顶点的精确多自由度运动学解，并通过对称性破缺构造低对称性解，为高阶折纸工程设计提供理论支持。
摘要：
近年来，折纸在工程中的应用显著扩大，涵盖跨尺度的可展开结构、折叠机器人和机械超材料。然而，寻找可折叠路径可能是一项艰巨的任务，因为运动学由非线性方程组决定，通常具有多个自由度。在这项工作中，我们利用拉格朗日方法推导出具有反射和旋转对称性的刚性面折纸顶点的降阶协调条件。然后，利用这些降阶条件，我们推导出具有规定对称性的6度和8度顶点的精确多自由度解。这些精确的运动学解使我们能够高效地研究容许运动学的拓扑结构，包括考虑自接触约束，随后通过监测运动学拓扑的变化，直观地解释几何设计参数在这些容许折叠路径上的作用。接着，我们引入一种程序，通过以保持协调性的系统方式打破高阶运动学解的对称性，来构造较低对称性的运动学解。这里发现的多自由度解应有助于建立对高阶折纸顶点运动学可行性的直观理解，并促进折纸工程设计新算法程序的开发。
英文摘要：
The use of origami in engineering has significantly expanded in recent years, spanning deployable structures across scales, folding robotics, and mechanical metamaterials. However, finding foldable paths can be a formidable task as the kinematics are determined by a nonlinear system of equations, often with several degrees of freedom. In this work, we leverage a Lagrangian approach to derive reduced-order compatibility conditions for rigid-facet origami vertices with reflection and rotational symmetries. Then, using the reduced-order conditions, we derive exact, multi-degree of freedom solutions for degree 6 and degree 8 vertices with prescribed symmetries. The exact kinematic solutions allow us to efficiently investigate the topology of allowable kinematics, including the consideration of a self-contact constraint, and then visually interpret the role of geometric design parameters on these admissible fold paths by monitoring the change in the kinematic topology. We then introduce a procedure to construct lower symmetry kinematic solutions by breaking symmetry of higher order kinematic solutions in a systematic way that preserves compatibility. The multi-degree of freedom solutions discovered here should assist with building intuition of the kinematic feasibility of higher degree origami vertices and also facilitate the development of new algorithmic procedures for origami-engineering design.
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