Robots That Generate Planarity Through Geometry

TL;DR

The work addresses how to achieve precise planar motion without relying on external reference flats by embedding planarity in the geometry of a mechanism. It introduces the Flat-Plane Mechanism (FPM), which realizes planar motion as a self-referencing geometric inversion of a sphere to a plane, implemented with thirteen links and six joints. The paper demonstrates scalable FPMs from $L_c=100\mu$m to $2.12$ m, shows fabrication-errors attenuated by about an order of magnitude, and introduces an iterative self-fabrication approach that avoids external metrology, plus a robotic FPM for metrology scans and constrained-space 3D printing with micron-scale repeatability and good agreement with a lab-grade CMM. This geometric framework enables high-precision metrology and fabrication across size scales while reducing reliance on precise external references, potentially democratizing access to accurate planar motion.

Abstract

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.

Figures (14)

• Figure 1: Fig. 1. The Flat-Plane Mechanism (FPM).(A) Stereographic projection defines the mapping between the Riemann sphere and the complex plane. (B) The FPM embodies this mapping in a physical structure, converting spherical input motion into planar output motion. Points $O$ and $F$ are grounded; point $B$ is constrained to a sphere, constraining the endpoint $D$ to a plane. (C) The characteristic length, $L_c$, is defined as the tip-to-base distance ($\overline{OD}$) when the mechanism is at the origin ($\overline{OF}$ is collinear with $\overline{FB}$). (D) A compliant micro-FPM printed via two-photon lithography. (E) A centimeter-scale FPM constructed from wooden dowels and magnetic joints. (F) A meter-scale FPM fabricated from carbon fiber rods. (G to I) Workspace and flatness measurements for the micro-, centimeter-, and meter-scale FPMs, respectively.
• Figure 1: Fig. S1. FPM proof. The geometry of the FPM with labeled points and reference surfaces used in the proof (A-B).
• Figure 2: Fig. 2. Design space and sensitivity analysis.(A) We define the FPM design space using its characteristic length and bi-pyramid dimensions when the mechanism is at the origin. (B) The kinematic sensitivity landscape reveals a family of designs that attenuate fabrication errors. (C) Mechanisms in this family achieve RMS flatness an order of magnitude lower than the RMSE within their link lengths. (D) We experimentally validate these results for FPMs of varying sizes. (E) We also map out the trade-off between workspace size and flatness. Error bars for the simulated data show the 95% confidence interval from $n=50$ randomized instances.
• Figure 2: Fig. S2. FPM naming conventions. The FPM naming convention for (A) links and (B) nodes/joints.
• Figure 3: Fig. 3. A robotic FPM for metrology and fabrication.(A) The actuated FPM, overlaid with the fixed-distance constraints. This FPM's workspace spans 200 in the X (B) and Y (C) directions. (D) Surfaces scanned using the FPM: extruded aluminum, milled aluminum, and cast acrylic. (E) Comparison of the FPM's scans with a lab-grade CMM. (F) The robot scanning the bottom of a narrow container and then 3D printing a model of Yom Tov Lipkin, one of the inventors of the first exact straight-line mechanism.