Rigidity of linearly-constrained frameworks

TL;DR

This paper extends the characterization of the generic rigidity of linearly-constrained bar-joint frameworks from 2D to higher dimensions, considering various affine subspace constraints, and relates it to body-bar frameworks with linear constraints.

Contribution

It generalizes the rigidity characterization to higher dimensions and different affine subspace constraints, building on prior 2D results and body-bar framework theories.

Findings

Extended rigidity characterization to d≥3 with affine subspace constraints

Connected results to body-bar frameworks with linear constraints

Provided conditions for rigidity in various constrained scenarios

Abstract

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Theran in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We then point out that results on body-bar frameworks obtained by N. Katoh and S. Tanigawa in 2013 can be used to characterise when a graph has a rigid realisation as a $d$-dimensional body-bar framework with a given set of linear constraints.