Tropical geometry reveals detailed mechanism motion possibilities
C-space Analysis using Tropical Geometry
Robotics
Summary
Mechanisms like robot arms move in certain ways described by their configuration space, which can be complex and have tricky spots where movement options branch or fold. The authors studied these tricky spots using a math tool called tropical geometry to better understand how the mechanism can move around these points. Their method detects complicated intersections and cusps in movements that previous methods find hard to analyze. They showed this works on some common folding linkages.
What this means in practice
- •For mechanical engineers: Analyze complex joint behavior in foldable linkages to improve design robustness around singular motion points.
- •For robotics developers: Predict nuanced robot arm configurations by detecting singularities where multiple motion branches meet or end.
Authors
Abhilash Nayak
Abstract
Configuration space~(C-space) of a mechanism is a real variety describing the set of feasible configurations that it can attain. To understand the behavior of a mechanism, it is crucial to identify and scrutinize especially the singular points of its C-space. They usually appear when the variety intersects itself, leading to different branches of motion. There exist many approaches to detect those intersections if they are transversal. However, the problem remains challenging if there are tangential, cuspidal, inter-dimensional or a combination of these intersections. This paper exploits an approach acquired from tropical geometry to analyze the neighborhood of any point on C-spaces of 1-degree-of-freedom~(\emph{dof}) mechanisms. This is done by finding the approximate rational parametrization of the curve(s) passing through the given point using Puiseux series. The proposed approach is shown to succesfully detect the transversal branchings in two foldable four bar mechanisms and a cusp in the configuration curve of the double Watt mechanism.