Rigidity and Tensegrity

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Rigidity and Tensegrity

• Robert Connelly
• Cornell University

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Part I The local theory

• Statics

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What determines rigidity?
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What determines rigidity?

• The physics of the materials.

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What determines rigidity?

• The physics of the materials.
• The external forces on the structure.

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What determines rigidity?

• The physics of the materials.
• The external forces on the structure.
• The combinatorics/topology of the structure.

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What determines rigidity?

• The physics of the materials.
• The external forces on the structure.
• The combinatorics/topology of the structure.
• THE GEOMETRY OF THE STRUCTURE.

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What model?
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What model?

• My favorite is a tensegrity.

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What sort of rigidity/stablility?

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What sort of rigidity/stablility?

• A very basic choice is prestressability.
• This means that there is potential function Eij
  defined for each member ( cable, bar, or strut)
  i,j, and the configuration p (p1, , pn) is
  at a unique local minimum, modulo congruences, of
  the total potential
• E(p)Sij Eij(pi - pj2)
• using the Hessian for the second-derivative test.

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What energy function to choose?
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What energy function to choose?

• It does not matter too much.
• It only matters what the first and second
  derivatives are of Eij at pi-pj2.
• (Eij )(pi-pj2) wij the (pre)stress
  coefficient.
• (Eij )(pi-pj2) cij gt 0, the stiffness
  coefficient.

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The basic stress-stiffness decomposition

• Theorem The Hessian H of the energy potential
  is the sum of the quadratic forms
• H stress energy stiffness energy,
• and for the displaced pi pi for each i,
• stress energy Sij wij pi - pj2,
• stiffness energy Sij cij (pi - pj) (pi -
  pj)2.

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The stiffness matrix

• The matrix of the quadratic form giving the
  stiffness energy is
• R(p)TDR(p),
• where R(p) is the rigidity matrix, D is the
  diagonal matrix with cijs as diagonal entries,
  and ()T is the transpose. This stiffness matrix
  is always positive semi-definite.

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The stress matrix

• Note that the stress energy quadratic form is the
  sum of d identical forms if the structure is in
  d-dimensional space. Then each of these forms
  has its symmetric matrix W, where each
  off-diagonal entry is -wij-wji, and the row and
  column sums are 0.

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An example of a stress matrix
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Equilibrium stresses

• For a tensegrity 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.
• When a configuration is at a critical point (e.g.
  a local minimum), then the corresponding stress
  is in equilibrium.
• A cable corresponds to wij gt 0, i.e. tension.
• A strut corresponds to wij lt 0, i.e. compression.

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Properties of the stress matrix

• If the affine span of the points of the
  configuration p (p1, , pn) 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.

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Part II The global theory

• Stress matrices applied again.

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Dominance

• A tensegrity G(q) dominates G(p) and we write
  G(p) G(q) if
• Each cable ij of G(p) is no longer than the
  corresponding cable ij of G(q).
• Each strut ij of G(p) is no shorter than the
  corresponding strut ij of G(q).
• Each bar ij of G(p) is the same length as the
  corresponding bar ij of G(q).

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Global Rigidity

• A tensegrity G(p) in Ed is globally rigid, if
  G(q) G(p) implies the configuration q in Ed is
  congruent to the configuration p.

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How do you tell when a given framework is
globally rigid?
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How do you tell when a given framework is
globally rigid?

• Answer Its hard, even when there are only
  bars!!
• 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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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.

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• This tool is sufficient for many of the popular
  tensegrities that are built by artists and
  others.
• You can see several examples of symmetric
  tensegrities at my web page
• http//mathlab.cit.cornell.edu/visualization/tense
  g/tenseg.html
• See also Simon Guest for information about how to
  use symmetry in rigidity calculations.

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Part III Bar frameworks

• More stress matrices using the generic philosophy.

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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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Generic rigidity

• Theorem A bar framework is rigid at a generic
  configuration if and only if it is
  infinitesimally rigid at any configuration.
• This means that generic rigidity is a
  combinatorial property of the graph G only.
• Theorem (Laman 1972) Suppose that G has n
  vertices and e 2n-3 edges. Then G is
  generically rigid in the plane if and only if for
  every subgraph of G with n vertices and e edges
  e 2n-3.

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Algorithms

• Corollary (Lovasz and Yemini ) Given a graph
  G with n vertices and e edges, there is an
  algorithm to determine whether G is generic rigid
  in the plane in at most O(ne) steps.
  (Variations 2-tree condition, the pebble game,
  etc.)

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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 generically globally
  rigid in the plane

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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

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When is a graph G generically globally rigid in
Ed?

• Necessary conditions
• G must be vertex (d1)-connected. (This means
  d1 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 d2, these
  conditions are also sufficient.
• For d3, these conditions are not sufficient. (Me
  1991)

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Vertex connectivity

• If G is not (d1) vertex connected, then d1
  vertices separate G, and reflection of one of the
  components of G about the (hyper)-line through
  those d1 vertices violates global rigidity.

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Vertex connectivity

• If G is not (d1) vertex connected, then d1
  vertices separate G, and reflection of one of the
  components of G about the (hyper)-line through
  those d1 vertices violates global rigidity.

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When is a graph G generically globally rigid in
Ed?

• Sufficient condition
• For p generic, G(p) has an equilibrium stress w,
  and an associated stress matrix W with maximal
  rank n-d-1, where n is the number of vertices of
  G. (Me, to appear.)
• Unlike the necessary conditions above, this
  condition is numerical, and it involves solving
  linear equations, as well as computing the rank
  of the n-by-n matrix W.
• Conjecture If G is generically globally rigid
  in Ed, then either G is a simplex or it satisfies
  the condition above for some generic
  configuration p.

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The rigidity matrix

• For a graph G, the rigidity map f End -gt Ee is
  the function that assigns to each configuration p
  of n vertices in d-space, the squared lengths of
  edges of G, f(p)(. . ., pi - pj2, . . .),
  where e is the number of edges of G.
• The rigidity matrix R(p) df is the differntial
  of f.
• Fact The co-kernel of R(p) is the space of
  equilibrium stresses of G(p). I.e. w is an
  equilibrium stress if and only if wR(p) 0.

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Proof that the stress condition implies global
rigidity
The Tarski-Seidenberg theory of quantifier
elimination implies that the situation below
cannot happen, since the configuration p is
generic. So a neighborhood of p can be mapped to
a neighborhood of q by a diffeomorphism h, so
that fh f, and dfp dfq dhp. This implies
that any equilibrium stress for p is an
equilibrium stress for q. If the rank of the
associated stress matrix is maximal, this implies
that p and q are affine images of each other, and
ultimately that they are congruent.

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Henneberg transformations

• Suppose that G(p) is a rigid framework in the
  plane, with p generic, such that it has an
  equilibrium stress w whose stress matrix W has
  rank n-3. Then the following transformation adds
  one new vertex to G while the new framework has
  an equilibrium stress whose stress matrix has
  rank (n1)-3.

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Henneberg transformations

• Suppose that G(p) is a rigid framework in the
  plane, with p generic, such that it has an
  equilibrium stress w whose stress matrix W has
  rank n-3. Then the following transformation adds
  one new vertex to G while the new framework has
  an equilibrium stress whose stress matrix has
  rank (n1)-3.

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Henneberg transformations

• Then a perturbation of the new vertex creates a
  generic configuration for this transformed graph,
  as below.
• The rigidity matrix for all three frameworks has
  maximal rank, and so the stress space changes
  continuously, and the stress matrix remains
  maximal, and we get generic global rigidity for
  this generic configuration for this transformed
  graph.

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What graphs can be obtained from Henneberg
transformations?

• Conjecture Any redundantly rigid, vertex
  3-connected graph G can be obtained from a
  sequence of Henneberg operations and edge
  insertions, starting from K4, the complete graph
  on 4 vertices.
• For example

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What graphs can be obtained from Henneberg
transformations?

• Conjecture Any redundantly rigid, vertex
  3-connected graph G can be obtained from a
  sequence of Henneberg operations and edge
  insertions, starting from K4, the complete graph
  on 4 vertices.
• For example

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What graphs can be obtained from Henneberg
transformations?

• Conjecture Any redundantly rigid, vertex
  3-connected graph G can be obtained from a
  sequence of Henneberg operations and edge
  insertions, starting from K4, the complete graph
  on 4 vertices.
• For example

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What graphs can be obtained from Henneberg
transformations?

• Conjecture Any redundantly rigid, vertex
  3-connected graph G can be obtained from a
  sequence of Henneberg operations and edge
  insertions, starting from K4, the complete graph
  on 4 vertices.
• For example

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What graphs can be obtained from Henneberg
transformations?

• Conjecture Any redundantly rigid, vertex
  3-connected graph G can be obtained from a
  sequence of Henneberg operations and edge
  insertions, starting from K4, the complete graph
  on 4 vertices.
• For example

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Hungarian and student to the rescue

• Theorem (A. Berg, T. Jordan, 2002) Let G be a
  3-connected, generically redundantly rigid graph
  with 2n-2 edges, where n is the number of
  vertices of G. Then G can be obtained from K4 by
  a sequence of Henneberg transformations.

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Hungarian and colleague to the rescue

• Theorem (T. Jordan and W. Jackson, 2003) Let G
  be a 3-connected, generically redundantly rigid
  graph. Then G can be obtained from K4 by a
  sequence of Henneberg transformations and edge
  insertions.
• Corollary Hendricksons conjecture is true in
  the plane, and there is a polynomial time
  algorithm to test for generic global rigidity.

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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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An example of the Berg-Jordan Henneberg
transformations
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Whoops!

• Care must be taken as how to choose the
  operations of inverse Hennenberg operations.
• The graph to the right is not vertex 2-connected.

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Part IV Existence of realizations

• When does a graph have a realization with
  predermined edge lengths?

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The molecule problem

• Suppose that you given distance constraints on
  the edges lengths of a graph G, exact lengths, or
  upper and lower bounds. Determine a
  configuration that satisfies those constraints.

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A modified problem

• Suppose that instead that you are given a
  realization of the graph G in some possibly high
  dimensional Euclidean space EN. For what graphs
  can you ALWAYS be sure that there is a
  realization p in E3 with the same edge lengths?

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A modified problem

• Suppose that instead that you are given a
  realization of the graph G in some possibly high
  dimensional Euclidean space EN. For what graphs
  can you ALWAYS be sure that there is a
  realization p in E3 with the same edge lengths?
• Answer See Maria Sloughter. It is related to
  the idea of backbones.