Rigidity of Linearly Constrained Frameworks

Abstract

We consider the problem of characterising the generic rigidity of bar-joint frameworks in 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 ≥ 3 and each vertex is constrained to lie in an affine subspace of dimension t, when t = 1, 2 and also when t ≥ 3 and d ≥ 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.

Rigidity through a Projective Lens

“…In the case where the linear constraints are generic, a 2-dimensional analogue of Laman's theorem (closely analogous to Theorem 4) was proved by Streinu and Theran [46] and this has been extended to all dimensions, under additional hypotheses on the dimension of the affine subspaces each vertex is restricted to, first in [81] and then in [79]. Moreover if one restricts to body-bar frameworks or to 2-dimensions but allows nongeneric linear constraints, as in Theorem 4, then combinatorial characterisations are also known [82].…”

Rigidity of symmetric frameworks in normed spaces

“…Linearly constrained frameworks. Following [5], we define a linearly constrained framework in R d to be a triple (G, p, q) where G = (V, E, L) is a looped simple graph, p : V → R d is injective and q : L → R d . For v i ∈ V and ℓ j ∈ L we put p(v i ) = p i and q(ℓ j ) = q j .…”

“…It would be an interesting future project to extend our analysis to higher dimensions. While for bar-joint frameworks little is known when d ≥ 3, in the linearly constrained case characterisations are known when suitable assumptions are made on the affine subspaces defined by the linear constraints [5,10].…”

“…Streinu and Theran [24] proved a characterisation of generic rigidity for linearly constrained frameworks in R 2 . The articles [5,10] provide an analogous characterisation for generic rigidity of linearly constrained frameworks in R d as long as the dimensions of the affine subspaces at each vertex are sufficiently small (compared to d), and [8] characterises the stronger notion of global rigidity in this context.…”

On the Rank Function of Generic Linearly Constrained Frameworks in $\mathbb{R}^2$

“…Similar to the bar-and-joint case we can construct a generic linearly constrained rigidity matroid R lc 2 (G) of a looped simple graph G = (V, E, L) from a linearly constrained rigidity matrix R(G, p, q) for which (p, q) is generic. For more information on the linearly constrained frameworks as well as some higher dimensional results we refer the interested readers to [3,6].…”