On the number of embeddings of minimally rigid graphs

Abstract

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most 2n−4 n−2 ≈ 4 n . We also exhibit several families which realize lower bounds of the order of 2 n , 2.21 n and 2.88 n .For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM 2,n (C) ⊂ P ( n 2 )−1 (C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM 2,n ) zero-dimensional components, and one finds this degree to be D 2,n = 1 2 2n−4 n−2 . The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences.The same approach works in higher dimensions. In particular we show that it leads to an upper bound of 2D 3,n = 2 n−3 n−2 n−6 n−3 for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions.

Undirected rigid formations are problematic

“…whereẽ is the reduced errorẽ =P e(Hx) and F is given by (14). Moreover, if x(t) is a solution to the overall system (7) for which x(t) ∈ A on some time interval [t 0 , t 1 ), then on the same time interval, the reduced error vectorẽ =P e(Hx(t)) satisfies the self-contained differential…”

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

“…Given a rigid graph G, let h(G) = max{h(G, p)}, where the maximum is taken over all generic frameworks (G, p). The graph of Figure 1 shows that h(G, p) need not be the same for all generic realizations (G, p) of a rigid graph G. [3] investigated the number of realizations of minimally rigid frameworks (G, p) with generic edge lengths. (Note that, by Lemmas 3.4 and 3.5, the edge lengths of (G, p) are generic if and only if there is a generic realization (G, q) with the same edge lengths as (G, p).)…”

Multi-loop Position Analysis via Iterated Linear Programming

“…A while loop is then executed until P gets empty (lines [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], by iterating the following steps. Line 4 SOLVE-LINKAGE(B, L, P, H, σ, ρ) 1: S ← ∅ 2: P ← {B} 3: while P = ∅ do 4: B c ← EXTRACT(P ) 5: repeat 6: V p ← VOLUME(B c ) 7: SHRINK-BOX(B c , L, P, H)…”

“…The problem arises, for instance, when solving the inverse/forward displacement analysis of serial/parallel manipulators [1], [2], when planning the coordinated manipulation of an object or the motion of a reconfigurable robot [3], or, as recently shown, in simultaneous localization and map-building [4]. The problem also appears in other domains, such as in the simulation and control of complex deployable structures [5], the theoretical study of rigidity [6], or the conformational analysis of biomolecules [7].…”