Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
Proc Natl Acad Sci U S A. 2019 Jan 2;116(1):90-95. doi: 10.1073/pnas.1809796115. Epub 2018 Dec 19.
Linkages are assemblies of rigid bodies connected through joints. They serve as the basis for force- and movement-managing devices ranging from ordinary pliers to high-precision robotic arms. Aside from planar mechanisms, like the well-known four-bar linkage, only a few linkages with a single internal degree of freedom-meaning that they can change shape in only one way and may thus be easily controlled-have been known to date. Here, we present "Möbius kaleidocycles," a previously undiscovered class of single-internal degree of freedom ring linkages containing nontrivial examples of spatially underconstrained mechanisms. A Möbius kaleidocycle is made from seven or more identical links joined by revolute hinges. These links dictate a specific twist angle between neighboring hinges, and the hinge orientations induce a nonorientable topology equivalent to the topology of a [Formula: see text]-twist Möbius band. Apart from having many technological applications, including perhaps the design of organic ring molecules with peculiar electronic properties, Möbius kaleidocycles raise fundamental questions about geometry, topology, and the limitations of mobility for closed loop linkages.
链接是通过关节连接的刚体组件。它们是从普通钳子到高精度机器臂等力和运动管理装置的基础。除了平面机构，如著名的四杆连杆，到目前为止，只有少数具有单个内部自由度的连杆是已知的——这意味着它们只能以一种方式改变形状，因此很容易控制。在这里，我们提出了“Möbius 万花筒环”，这是一类以前未被发现的具有单个内部自由度的环连杆，其中包含空间欠约束机构的非平凡例子。Möbius 万花筒环由七个或更多相同的通过转动铰链连接的连杆组成。这些连杆决定了相邻铰链之间的特定扭转角度，而铰链的方向诱导出不可定向的拓扑，相当于[公式：见正文] -扭转 Möbius 带的拓扑。除了具有许多技术应用，包括可能设计具有奇特电子性质的有机环分子之外，Möbius 万花筒环还提出了关于几何、拓扑和闭环连杆的可动性限制的基本问题。
