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Global Rigidity

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					Global Rigidity

  R. Connelly
Cornell University


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     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.




                                                           2
        Equivalent frameworks
Two frameworks with the same graph G are
 equivalent if the lengths of their edges are the
 same.


                 is eq uivalent to




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     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.




                                                    4
   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)

                              ?


                                                           5
       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.
                                              6
    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)
                                                    7
          Generic global rigidity
A graph G is generically globally rigid in 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:




                                                      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.




                                                        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
             Equilibrium stresses
For a framework G(p), an equilibrium stress w 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 Sj wij(pj-pi)=0 holds.
The following is a square with its equilibrium stresses indicated:
                                    1

                               -1
                          1             1
                              -1

                                1



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                The stress matrix
Given an equilibrium stress w for a framework G(p) with n
  vertices, the stress matrix W 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.


                                                              12
        An example of a stress matrix

                                  1   2   3    4
              1                   1 -1
    2                  3                   1 -1    1


                                 -1 1     -1 1     2
1
         -1       -1   1   W    1 -1     1 -1    3

    1         1        4         -1   1   -1   1   4




                                                   13
   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.



                                                   14
          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.
                                                     15
    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).
                                                   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 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.



                                             18
                       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.




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            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.
                                                       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”.




                                                               21
                  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.




                                                                22
               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.




                                                                 23
        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.
                                                        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 Ed
  if and only if it is generically redundantly rigid.




                                                    26
                    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.



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         QuickTime™ and a
TIFF (Uncompressed) decompressor
   are neede d to see this picture.




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