Constraining motion to a flat surface is a fundamental requirement for equipment across science and engineering. Modern precision robotic motion systems, such as gantries, rely on the flatness of components, including guide rails and granite surface plates. However, translating this static flatness into motion requires precise internal alignment and tight-tolerance components that create long, error-sensitive reference chains. Here, we show that by using the geometric inversion of a sphere into a plane, we can produce robotic motion systems that derive planarity entirely from link lengths and connectivity. This allows planar motion to emerge from self-referencing geometric constraints, and without external metrology. We demonstrate these Flat-Plane Mechanisms (FPMs) from micron to meter scales and show that fabrication errors can be attenuated by an order of magnitude in the resulting flatness. Finally, we present a robotic FPM-based 3-axis positioning system that can be used for metrology surface scans ($\pm 12$-mm) and 3D printing inside narrow containers. This work establishes an alternative geometric foundation for planar motion that can be realized across size scales and opens new possibilities in metrology, fabrication, and micro-positioning.

翻译：将运动约束在平坦表面上是科学和工程领域中各类设备的一项基本要求。现代精密机器人运动系统，例如龙门架，依赖于导轨和花岗岩平台等组件的平面度。然而，将这种静态平面度转化为运动，需要精确的内部对准和公差严格的组件，这形成了长且对误差敏感的参考链。在此，我们证明，通过利用球体到平面的几何反演，我们可以制造出完全由连杆长度和连接性产生平面性的机器人运动系统。这使得平面运动能够从自参考的几何约束中产生，而无需外部计量。我们从微米到米尺度展示了这些平面机构，并证明制造误差在最终平面度上可被衰减一个数量级。最后，我们展示了一种基于FPM的机器人三轴定位系统，可用于狭窄容器内的计量表面扫描（$\pm 12$ mm）和3D打印。这项工作为平面运动建立了一种可在不同尺寸尺度上实现的替代几何基础，并为计量、制造和微定位开辟了新的可能性。