Transcript Global Rigidity - Cornell University
Global Rigidity
R. Connelly Cornell University 1
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
= (
p
1 ,
p
2 , …,
p
n ), where each
p
i corresponds to a node of G, and the edges of G correspond to fixed length bars connecting the nodes.
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Equivalent frameworks
Two frameworks with the same graph G are
equivalent
if the lengths of their edges are the same.
is equivalent to 3
Globally rigid frameworks
A framework G
(p
) is
globally rigid
in E d if every corresponding equivalent framework G(
q
) has
p
congruent to
q
. For example the following frameworks are globally rigid in the plane.
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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) ?
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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.
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When is a graph G generically globally rigid in E
d
?
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) 7
Generic global rigidity
A graph G is
generically globally rigid in E d
if for some (every?) generic configuration
p
, the framework G(
p
) is globally rigid in E d .
The following graphs are
not
in the plane: generically globally rigid 8
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.
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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.
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Equilibrium stresses
For a framework G(
p)
, an
equilibrium stress
of a scalar w ij = w ji w is an assignment to each pair of distinct vertices {i,j} of G, such that w ij = 0 when {i,j} is not an edge of G, and for each i, the equilibrium equation S j w ij (
p
j -
p
i )=0 holds. The following is a square with its equilibrium stresses indicated: 1 -1 1 -1 1 1 11
The stress matrix
Given an equilibrium stress w vertices, the
stress matrix
where the (i,j) entry is w ij W for a framework G(
p
) with n is the n-by-n symmetric matrix 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
rank of W is at most n-d-1.
is d-dimensional, then the • If the rank of W such that w is n-d-1, and some other configuration is an equilibrium stress for G(
q q
is ), then the points of
q
are an affine image of the points of
p
.
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An example of a stress matrix
2 1 3 1 -1 1 1 -1 1 4 W 1 -1 1 -1 1 2 -1 1 -1 1 3 1 -1 1 -1 4 -1 1 -1 1 1 2 3 4 13
Tools to show global rigidity
Theorem
: Suppose that G(
p
) is a tensegrity in E d with an equilibrium stress such that • The stress matrix W rank n-d-1.
is positive semi-definite of • The only affine motions of the configuration
p
that preserve the member constraints are congruences.
Then G(
p
) is globally rigid in any E N containing E d .
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Infinitesimal rigidity
A bar framework G(
p
) is
infinitesimally rigid
(=
statically rigid
G.
) in E d , 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 A bar framework G( configuration
q p
) is
(locally) rigid
in E d , sufficiently close to
p
equivalent to
p
, is congruent to
p
.
in E d if every and
Theorem
: If G(p) is infinitesimally rigid in E d , then it is locally rigid in E d . 15
Tool to show generic rigidity global
Theorem
(Me): If a configuration
p
is generic in E d , 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 E d .
Corollary
: If the bar framework G(
p
) is infinitesimally rigid in E d , 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 E d (but not necessarily at
p
). 16
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) 17
Necessity of stress rank
Theorem
(Gortler, Healy, Thurston): If a bar framework G(
p
) is globally rigid for a generic configuration in E d , G not a bar simplex, then there is an equilibrium stress w with associated stress matrix W that has maximal rank n-d-1. 18
Coning
Theorem
(R. C. and W. Whiteley): A graph G is generically globally rigid in E d , if and only if the cone on G is generically globally rigid in E d+1 . 19
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 E d . 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. 20
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”.
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True realizations
I call a class of realizations of a graph G
true
, if each member is infinitesimally rigid and globally rigid.
Theorem
class.
: 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 This is a corollary of the inductive constructions of Berg-Jordán. 22
Untrue certificates
Why aren’t infinitesimally rigid, generically globally rigid frameworks, necessarily globally rigid?
If G(
p
) is infinitesimally rigid in E globally rigid in E d . G(
p
d , and has a maximal rank stress matrix, it is only a ‘certificate’ that G is generically ) 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.
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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 K 6 as a subgraph that is generically redundantly rigid, vertex 6-connected, and yet it is not generically globally rigid in E 5 .
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. 24
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. 25
Bar and bodies
Theorem (RC, T. Jordán, W. Whiteley): A bar and body framework is generically globally rigid in E d if and only if it is generically redundantly rigid.
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Results
• There is a catalog of symmetric globally rigid tensegrity frameworks at http://www.math.cornell.edu/~tens/ A. Back.
This is with R. Terrell and is an update of a previous catalog with • There is book “Frameworks: Tensegrities and Symmetry” in progress with S. Guest.
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