用几何反演让机器人自动生成平面运动，无需高精度对齐。
Robots That Generate Planarity Through Geometry
- 通过球面到平面的几何反演，仅靠连杆长度和连接关系实现平面运动
- 制造误差导致的平面度偏差降低一个数量级（约10倍），跨微米到米级有效
- 适合高精度定位、微制造与狭小空间内3D打印等场景
将运动限制在平面上是科学与工程设备的基本需求。现代精密机器人系统（如龙门架）依赖导轨和花岗岩平板等部件的静态平面性。然而，将这种静态平面性转化为精确运动，需依赖严格的内部对齐和高精度组件，形成冗长且易出错的参考链。本文展示，通过球面到平面的几何反演，可构建完全由连杆长度与连接关系决定平面性的机器人系统。该方法使平面运动自发产生于自参照几何约束，无需外部测量。我们验证了从微米到米级的平面机构（FPMs），并发现制造误差在最终平面度中被削弱约10倍。最后，我们设计了一种基于FPM的三轴定位系统，可实现±12毫米范围内的表面扫描与狭窄容器内的3D打印。这项工作为平面运动提供了替代性的几何基础，适用于多种尺度，拓展了计量学、制造与微定位的新可能。
原文摘要 · Abstract (English)
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.
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