Let E^d be the d-dimensional Euclidean space and V^d the associated vector space. A graph G=(N,E) is given by a set N of n nodes together with a set E of e edges connecting pairs of‘ nodes. The edge connecting nodes i,j∈N is denoted by ij∈E. Graphs are finite and undirected, without loops or multiple edges. A configuration in E^d for the graph G is an assignment of a position vector pi∈Vd to each node i∈N, so that a corresponding point Pi∈Ed is determined by its position with respect to a chosen origin O ∈ E^d. We denote by p ∈ V^nd the vector grouping all nodal position vectors. A framework is given by a graph together with a configuration, that is, F=(G,p) is a framework with graph G and configuration p.

Associated to a framework is the set L(G,p) of the half-squared edge-lengths,

A configuration q is admissible for (G,p) if L(G,q)=L(G,p). Two configurations p and q are congruent, and we write p ≡ q, if |p_i − p_j| = |q_i − q_j| for every choice of i and j in N. Equivalently, two configurations are congruent if they differ by an isometry of E^d, i.e., a composition of translations, rotations and reflections. A framework (G,p) is rigid if there is an ε > 0 such that any other admissible configuration q for which |p−q| < ε is congruent to p.

The jacobian of L(G,p), which is an e-by-dn matrix, is the rigidity matrix, R. A framework is infinitesimally rigid if the rank of R is equal to nd − d(d + 1)/2, or equivalently, if the only solutions to the system of equations Rṗ = 0 are rigid velocities, i.e., nodal velocities in a rigid motion (Figure 4). For example, the equation corresponding to the edge ij in this system is given by

which is obtained by setting equal to zero the first derivative of λ_ij. The solutions of Rṗ = 0 which are not rigid velocities are called mechanisms.

A framework (G,p) is globally rigid if any admissible configuration q is congruent to p (Figure 5). A framework is universally rigid if it is globally rigid in all dimensions (Figure 6). Universal rigidity implies global rigidity, which implies infinitesimal rigidity, which implies rigidity.

A configuration is generic if the coordinates in p are algebraically independent over the integers, i.e., if the nodal coordinates do not satisfy any nontrivial polynomial equation with integer coefficients. Intuitively, if the configuration is nongeneric, then it is special in some way. For example the framework in Figure 7A is globally rigid, while the one in Figure 7B, where three nodes are aligned on a diagonal, is not. Another example is given in Figure 4, with configurations Figures 4A,B being respectively generic and nongeneric.

A framework (G,p) is generically rigid if it is rigid and p is generic. Rigidity is a generic property, i.e., it is a property of the graph, not the configuration: if a framework is rigid at a generic configuration then it is rigid at every other generic configuration. Moreover, at generic configurations, rigidity and infinitesimal rigidity are equivalent.

The minimum number of edges necessary for generic rigidity are 2n − 3 in 2D and 3n − 6 in 3D. Intuitively, in 2D, we can start with an edge connecting two nodes, then iteratively adding one node connected to the other nodes by two noncollinear edges. In 3D, we can start with a nondegenerate triangle (three vertices and three edges), then iteratively adding one tripod, i.e., a node connected to the other nodes by three noncoplanar edges. These constructions constitute particular Henneberg sequences (Eren et al., 2004b): sequences of operations which preserve minimal generic rigidity.

The characterization of global rigidity has been given in the literature in terms of stress. A stress ω is an assignment of a real number ω_ij to each edge ij of the framework. A selfstress for (G,p) is a stress satisfying at every node i the nodal equilibrium equation

where the summation is extended to every node j connected to node i by an edge. The equilibrium equations can be written in matrix form as

with A the dn-by-e equilibrium matrix. Selfstresses belong to the nullspace of the equilibrium matrix. One classic result is that A = R^T, so that the number of independent selfstresses s and mechanisms m are related to n and e by the following rule

where d(d + 1)/2 is the number of independent rigid motions in E^d. This rule follows from the orthogonality of the fundamental subspaces (nullspace and image of the transposed) of R and A.

A fundamental object is the stress matrix, Ω, a n-by-n matrix whose entries are defined as follows:

where ω is a selfstress. The stress matrix is equal to the weighted Laplacian of the graph, with weights given by the selfstress values on the edges. Notice that the weights can be either positive or negative, so that classic results on positively-weighted Laplacians do not apply.

A useful characterization has been given as follows (Connelly, 1982, 2013). A framework in E^d with the affine span of p₁, …, p_n being all E^d and a nonzero selfstress is superstable if the following conditions hold:

Theorem 1. Connelly (1982), see also Connelly (2013) A superstable framework is universally rigid.

Condition (1) implies that if there is another admissible configuration, then it has the same selfstress; condition (2) then implies that this other configuration is an affine image of the original one, and condition (3) implies that the affine image is actually congruent to the original configuration (cf. Figure 8). A particular class of superstable frameworks is that of cablenets, i.e., externally anchored frameworks where each edge has positive stress (Figure 9A).

Now we turn to generic configurations. A simplex in E^d is a framework on the complete graph on d + 1 nodes, e.g., triangles in E² or tetrahedra in E³. Simplices (and all frameworks on complete graphs) are universally rigid by definition, since admissible configurations must be congruent to each other. Every generic globally rigid framework in E^d which is not a simplex (i.e., it has at least d + 2 nodes) admits at least one independent selfstress. This follows from the next theorem. A framework is redundantly rigid if it is rigid after the removal of an edge. A graph is c-connected if at least c nodes have to be removed from the graph to disconnect it.

Theorem 2. Hendrickson (1992). If a framework with n ≥ d + 2 is generically globally rigid in E^d then it is redundantly rigid and (d + 1)-connected.

For d = 2 the theorem holds with an “if and only if” condition (Berg and Jordan, 2003, cf Connelly, 2013). Since generic redundant rigidity implies that there exist at least a selfstress, s ≥ 0 and that there are no mechanisms, m = 0, it follows from (1) that in a generically globally rigid framework the number of edges is equal to or higher than

e.g., 2n − 2 in 2D or 3n − 5 in 3D.

A complete characterization of generic global rigidity has been given in the following theorem.

Theorem 3. A framework with n ≥ d + 2 is generically globally rigid in E^d if and only if there is a nonzero selfstress whose stress matrix has rank n − d − 1.

The “if” part is due to Connelly (1982), the “only if” part to Gortler et al. (2010).

The next theorem provide the converse of Theorem 1 in the generic case.

Theorem 4. Gortler and Thurston (2014). A universally rigid framework (G,p) with p generic and n ≥ d + 2 is superstable.

It is worth noticing that while global rigidity is a generic property, universal rigidity is not: if framework is universally rigid in a certain generic configuration, it can lose this property in a different generic configuration (compare cases (c) and (d) in Figure 1).

A less strict condition consists in requiring a configuration to be general. A configuration in E^d is general if no d + 1 nodes are affinely dependent, e.g., there are no three collinear nodes in d = 2, or there are no three collinear nodes and no four coplanar nodes in d = 3. In this case we have the following result.

Theorem 5. Alfakih and Ye (2013). A framework (G,p) with p general and n ≥ d + 2 is universally rigid if there is a nonzero selfstress whose stress matrix is positive semi-definite with rank n − d − 1.

It has been shown in Alfakih et al. (2013) that the converse of this theorem holds for (d + 1)-lateration graphs, i.e., graphs obtained from a simplex by applying a sequence of (d + 1)-valent node additions, i.e., the addition of a node connected by d + 1 edges to the other nodes. An analogous result regarding global rigidity has been obtained previously in Anderson et al. (2006). The number of edges of frameworks obtained in this way is

that is e = 3n − 6 for d = 2 and e = 4n − 10 for d = 3.

For n large, these values of e are 50% and 33% higher than the minimum value given by (2), respectively for d = 2 and d = 3.

By considering frameworks with a stress, the notion of tensegrity framework naturally comes into play. Indeed, many results have been first obtained for tensegrity frameworks, and then applied to the particular case of bar-frameworks.

A tensegrity framework is a framework where each edge can be labeled as a bar, a cable, or a strut: bars cannot change length, cables cannot increase in length, and struts cannot decrease in length. It turns out that a tensegrity framework is globally/universally rigid if the corresponding bar-framework is and the stress is proper, that is, cables have positive stress, and struts negative (Connelly, 2013). In other words, there is no difference between a bar-framework with a stress satisfying the theorems above and a tensegrity framework, with same graph and configuration, whose edges are labeled accordingly: cables if the stress is positive, struts if the stress is negative. Bars can be placed anywhere.

We conclude this section by reporting three results about known classes of frameworks. The first one is about convex polygons.

Theorem 6. (Tensegrity polygons, Connelly, 1982). A tensegrity framework with the shape of a convex polygon, with cables on the outside, struts inside, and a proper a selfstress, is universally rigid (Figure 10).

In the next section we will focus on the polygons like those in Figures 2A, 10A, first described by Grünbaum and Shephard (1978).

The second result is about three-dimensional frameworks.

Theorem 7. (Central symmetric tensegrity polyhedra, Lovász, 2001; Bezdek and Connelly, 2006). Every tensegrity framework with the shape of a centrally symmetric polyhedron, with cables outside, bars connecting diametrically opposite pairs of vertices, and a proper selfstress, is universally rigid (Figure 11).

The third result is about combining different frameworks together.

Theorem 8. (Attachments, Ratmanski, 2010). Given two universally rigid frameworks in general position, it is possible to combine them into a universally rigid assembly if they have d + 1 nodes in common.

Analogous results for globally rigid frameworks are presented in Eren et al. (2004a) and Connelly (2011).

Grünbaum polygons are frameworks obtained by placing nodes and edges at the vertices and the sides of a convex polygon, then by choosing one node, the center node (in black in Figure 2A), and by connecting all the other nodes to it with an edge, except the two neighboring nodes (in gray in Figure 2A). The construction is completed by adding one edge connecting the two neighboring nodes.

We provide here a similar construction to assign (2n − 2) edges to a given a set of nodes in E² in order to obtain a universally rigid framework. We will call the resulting framework a nonconvex Grünbaum polygon (Figure 12).

First, the convex hull of the nodes is constructed and three consecutive vertices on its boundary coinciding with three nodes are chosen (Figure 12A), the middle one becomes the center to which all the other nodes are connected (Figure 12B). Then, additional edges are added to form a contiguous sequence of triangles sharing the center as a vertex (Figure 12C), plus the last edge connecting the two neighboring nodes (Figure 12F).

Theorem 9. Every nonconvex Grünbaum polygon is universally rigid.

Proof. Up to the addition of the last edge, the framework can be viewed as forming a kind of fan shape which “unfold” from the center node (Figure 12C). This incomplete framework admits a number of configurations equal to 2^f, where f is the number of internal edges or folds of the fan (Figure 12D). The distance between the two neighboring nodes will reach a maximum only when the fan is completely unfolded. It follows that by adding the last edge between the two neighboring nodes, the unfolded configuration is unique.

By embedding this framework in a higher dimensional Euclidean space, the situation does not change. Since each triangle of a fan is universally rigid by itself and it can only rotate about a fold, relative to its neighboring triangles, the triangle inequality ensure that the distance between the two neighboring nodes has a global maximum when the fan is flat, therefore the Grümbaum polygon is universally rigid. _□

Notice that this proof is valid for both convex and nonconvex Grünbaum polygons. Notice also that the construction works even if the center is aligned with its neighbors, or if two or more fold are collinear. The result holds even if the configuration is nongeneric, the main requirement being that the center and its neighbors are on the boundary of the convex hull.

In three dimensions we can obtain a perfectly analogous result for assigning (3n − 5) edges red to a given set of nodes in E³. We construct the convex hull of this set. There will be at least four vertices of the hull forming two adjacent triangles, sharing one edge of the convex hull. The shared edge is the central edge of the framework, the two nodes on this edge are the central nodes, while the other two are the neighboring nodes. Now, we can add edges connecting each of the neighboring nodes to the central nodes. We do the same with the remaining nodes, by connecting them to the central nodes. In this way, we obtain a set of triangles in space, all sharing one edge (Figure 13A). Then, for each couple of neighboring triangles, we add an edge between the nodes so as to form a tetrahedron. Finally, the last edge of this construction is added between the two neighboring nodes (Figure 13B).

An easy way of visualizing this framework is to project it along the direction of the central edge onto a plane, resulting in a fan-like framework, a nonconvex Grünbaum polygon. Similarly to what we have done before, we can consider the incomplete framework obtained by removing the last edge and argue that this admits a number of configuration equal to 2^f, with f defined for the projected framework as in the two-dimensional case. Among all these configurations, the one which is completely “unfolded” gives the maximum distance between the neighboring nodes, still using this term in analogy with the two-dimensional case. Once we add the last edge in this configuration, we obtain a globally rigid structure, which, by the triangle inequality is also universally rigid. We call frameworks obtained in this way 3D Grünbaum framework and state the following theorem.

Theorem 10. Every 3D Grünbaum framework is universally rigid.

Notice that we can view these kind of frameworks, both in 2D and in 3D, as obtained by anchoring the nodes to a simplex, in the same way as we can anchor a cable-net to a (universally) rigid structure (Figure 9).

Notice also that we can find other generalized Grünbaum frameworks. For example, the one shown in Figure 14 (top right) has two centers, corresponding to two fans with one side in common. It is easy to see that, in order for multiple-fans frameworks to be universally rigid the centers should be on opposite sides of the edge connecting the neighbors. Analogous constructions exist also in three dimensions (Figure 14, bottom right).

In Figure 15 we present two examples of application of Theorems 9, 10 in combination with Theorem 8. These examples shows how to avoid the occurrence of bars of excessive length by considering modular frameworks. In Figure 15 (top), a universally rigid framework in E² is obtained by repetition of a universally rigid module, with adjacent modules having three nodes in common. In Figure 15 (bottom), a universally rigid framework on randomly generated nodal positions in E³ is composed by three universally rigid subframeworks, each sharing four nodes with the adjacent one. Space-filling universally rigid assemblies can be obtained in analogous fashion.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.