Global Rigidity

1. Global Rigidity R. Connelly Cornell University

2. Pin jointed bar frameworks A (bar) framework is a finite graph G together with a finite set of vectors in d-space, denoted by p = (p1, p2, …, pn), where each pi corresponds to a node of G, and the edges of G correspond to fixed length bars connecting the nodes.

3. Equivalent frameworks Two frameworks with the same graph G are equivalent if the lengths of their edges are the same.

4. Globally rigid frameworks A framework G(p) is globally rigid in Ed if every corresponding equivalent framework G(q) has p congruent to q. For example the following frameworks are globally rigid in the plane.

5. How do you tell when a given framework is globally rigid? Answer: It’s hard!! More precisely, if you could find a polynomial time algorithm for this problem, you could solve a huge list of unsolved equivalent problems, and most likely earn a million $$. This problem is non-deterministically polynomially (NP) complete. For example, even for global rigidity in the line, global rigidity is the uniqueness part of the knapsack problem. (Saxe, 1979)

6. Generic configurations A configuration p is called generic if the coordinates of all of its points are algebraically independent over the rationals. In other words, any non-zero polynomial with rational coefficients will not vanish when the variables are replaced by the coordinates of p. p, e, g, … are algebraically independent.

7. When is a graph G generically globally rigid in Ed? Necessary conditions: • G must be vertex (d+1)-connected. (This means d+1 or more vertices are needed to disconnect the vertices of G.) • G must be generically redundantly rigid. (This means that, for p generic, G(p) must be rigid, even when any edge of G is removed.) B. Hendrickson (1991). Conjecture (Hendrickson): For d=2, these conditions are also sufficient. For d=3, these conditions are not sufficient. (Me 1991)

8. Generic global rigidity A graph G is generically globally rigidin Ed if for some (every?) generic configuration p, the framework G(p) is globally rigid in Ed. The following graphs are not generically globally rigid in the plane:

9. Vertex connectivity If G is not (d+1) vertex connected, then d+1 vertices separate G, and reflection of one of the components of G about the (hyper)-line through those d+1 vertices violates global rigidity.

10. Vertex connectivity If G is not (d+1) vertex connected, then d+1 vertices separate G, and reflection of one of the components of G about the (hyper)-line through those d+1 vertices violates global rigidity.

11. Equilibrium stresses For a framework G(p), an equilibrium stressw is an assignment of a scalar wij= wji to each pair of distinct vertices {i,j} of G, such that wij = 0 when {i,j} is not an edge of G, and for each i, the equilibrium equation Sjwij(pj-pi)=0 holds. The following is a square with its equilibrium stresses indicated:

12. The stress matrix Given an equilibrium stress w for a framework G(p) with n vertices, the stress matrixW is the n-by-n symmetric matrix where the (i,j) entry is -wij and the diagonal entries are such that all the row and column sums of W are 0. Properties of W: • If the affine span of the points of p is d-dimensional, then the rank of W is at most n-d-1. • If the rank of W is n-d-1, and some other configuration q is such that w is an equilibrium stress for G(q), then the points of q are an affine image of the points of p.

13. An example of a stress matrix

14. Tools to show global rigidity Theorem: Suppose that G(p) is a tensegrity in Ed with an equilibrium stress such that • The stress matrix W is positive semi-definite of rank n-d-1. • The only affine motions of the configuration p that preserve the member constraints are congruences. Then G(p) is globally rigid in any EN containing Ed.

15. Infinitesimal rigidity A bar framework G(p) is infinitesimally rigid (= statically rigid) in Ed, if the dimension of the space of equilibrium stresses is m - nd + d(d+1)/2, where n ≥ d -1 is the number of vertices of G, and m is the number of bars (= edges = members) of G. A bar framework G(p) is (locally) rigid in Ed if every configuration q in Ed, sufficiently close to p and equivalent to p, is congruent to p. Theorem: If G(p) is infinitesimally rigid in Ed, then it is locally rigid in Ed.

16. Tool to show generic global rigidity Theorem (Me): If a configuration p is generic in Ed, and the bar framework G(p) has an equilibrium stress w and an associated stress matrix W with maximal rank n-d-1, then G(p) is globally rigid in Ed. Corollary: If the bar framework G(p) is infinitesimally rigid in Ed, has an equilibrium stress w whose associated stress matrix W has (maximal) rank n-d-1, then the graph G is generically globally rigid in Ed (but not necessarily at p).

17. Global Rigidity in the Plane Theorem: (Jackson-Jordan) If a bar graph G is generically redundantly rigid in the plane and vertex 3-connected, then it is generically globally rigid in the plane. (A conjecture of B. Hendrickson)

18. Necessity of stress rank Theorem (Gortler, Healy, Thurston): If a bar framework G(p) is globally rigid for a generic configuration in Ed, G not a bar simplex, then there is an equilibrium stress w with associated stress matrix W that has maximal rank n-d-1.

19. Coning Theorem (R. C. and W. Whiteley): A graph G is generically globally rigid in Ed, if and only if the cone on G is generically globally rigid in Ed+1.

20. Good news - Bad news Good news: In any dimension d for any graph G it is possible to determine in polynomial time whether G is generically globally rigid in Ed. In the plane, the news is even better. Bad news: It is probably not possible to know what it means to be “generic” for global rigidity. For local rigidity “generic” can mean that a certain matrix (the rigidity matrix R(p)) has maximal rank. For global rigidity, there is an open, dense set of configurations that are globally rigid, if G is generically globally rigid, but it is determined by Tarski-Seidenberg type elimination theory.

21. Are there some “good” realizations? Spider webs are naturally globally rigid. They are graphs with some pinned vertices, together with edges that have a positive stress that is in equilibrium at all the non-pinned vertices. Spider webs can be constructed with arbitrary positive stresses on the member (cables). This is essentially Tutte’s constructions as described in his paper “How to draw a graph”.

22. True realizations I call a class of realizations of a graph G true, if each member is infinitesimally rigid and globally rigid. Theorem: Spider webs attached to a triangle in an infinitesimally rigid configuration, where m = 2n-3+1, where G is generically redundantly rigid in the plane and vertex 3-connected, are a true class. This is a corollary of the inductive constructions of Berg-Jordán.

23. Untrue certificates Why aren’t infinitesimally rigid, generically globally rigid frameworks, necessarily globally rigid? If G(p) is infinitesimally rigid in Ed, and has a maximal rank stress matrix, it is only a ‘certificate’ that G is generically globally rigid in Ed. G(p) itself may not be globally rigid. For example, the following framework is not globally rigid in the plane, although it is infinitesimally rigid, and its stress matrix is of rank 2 = 5 - 2 -1 = n - d -1, the maximum.

24. Higher dimensional true realizations? Can the spider web example be extended to higher dimensions? Example (Jiayang Jiang and Sam Frank): There is a graph G containing K6 as a subgraph that is generically redundantly rigid, vertex 6-connected, and yet it is not generically globally rigid in E5. This is a counterexample to Hendrickson’s conjecture, and it contains a simplex. So if the vertices of the simplex are pinned, and if a spider web is hung from them, the graph will not be infinitesimally rigid.

25. Body and bar frameworks So generic infinitesimally rigid information does not seem to be enough for even generic global rigidity. Here is a friendly class of frameworks due to Tay and Whiteley. Consider a bar framework where the vertices are partitioned into sets called bodies. Each body is a globally rigid object, and between some pairs of bodies, there are some number bars connecting them in such a way that any pair of bars are disjoint, even at their end vertices.

26. Bar and bodies Theorem (RC, T. Jordán, W. Whiteley): A bar and body framework is generically globally rigid in Ed if and only if it is generically redundantly rigid.

27. Results • There is a catalog of symmetric globally rigid tensegrity frameworks at http://www.math.cornell.edu/~tens/ This is with R. Terrell and is an update of a previous catalog with A. Back. • There is book “Frameworks: Tensegrities and Symmetry” in progress with S. Guest.