Topology of randon linkages

TL;DR

This paper investigates the average topological complexity of random mechanical linkages as the number of links grows, revealing measure-independent asymptotic behavior and providing insights relevant to robotics, shape analysis, and molecular structures.

Contribution

It introduces a probabilistic framework for analyzing Betti numbers of linkage configuration spaces and proves measure-independent asymptotic results for large linkages in both planar and 3D cases.

Findings

Asymptotic average Betti numbers are measure-independent for large n.

Results apply to both planar and spatial linkages.

Higher moments of Betti numbers are also characterized.

Abstract

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -- the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R^3. We also prove results about higher moments of Betti numbers.