Rigidity

A. S. Morse

Yale University

Gif – sur - Yvette May 24, 2012

Supelec

EECI Graduate School in Control

Consider the problem of maintaining in a formation, a group of mobile autonomous agents

Focus mainly on the 2d problem

Think of agents as points in the plane

point setmotionin the plane

Rigid motion: means distances between all pairs of points are constant

Maintaining a formation of points …..with maintenance links

p5

p4

p3p2

p1

p6

p7

p8

p9

p10

p11

p = {p1, p2, …, p11}

L = {(1,2), (2,3), …, }

point formation Fp(L)

65

410

11

9

7

8

1

3 2

d9,6

d7,4

d6,5

d11,1

d5,4

d9,11

d10,11

d10,9

d1,2

distance graph

framework

p5

p4

p3p2

p1

p6

p7

p8

p9

p10

p11

p = {p1, p2, …, p11}

L = {(1,2), (2,3), …, }

point formation Fp(L)

distance graph

translationrotationreflection

65

410

11

9

7

8

1

3 2

d9,6

d7,4

d6,5

d11,1

d5,4

d9,11

d10,11

d10,9

d1,2

Fp = rigid if congruent to all “close by” formations with the same distance graph.

Euclidean transformation

congruent

Euclidean GroupSpecial SE(2)

minimally rigid{isostatic}

redundantly rigid non-rigid{flexible}

redundant link missing link

Fp = rigid if congruent to all “close by” formations with the same distance graph.

rigid means can’t be “continuously deformed”

The number of maintenance links in a minimally rigid n point formation in 2d is 2n - 3

Fp = rigid if congruent to all “close by” formations with the same distance graph.

Fp = generically rigid if all “close by” formations with the same graph are rigid.

G = rigid graph it is meant the graph of a generically rigid formation

Denseness: If G is a rigid graph, almost every formation with this graph is generically rigid.

so generic rigidity is a robust property

R(p) = rigidity matrix - a specially structured matrix depending linearly on p whose rank can be used to decide whether or not Fp is generically rigid.

Laman’s theorem {1970}: A combinatoric test for deciding whether or not a graph is rigid.

Three-dimensions: All of the preceding, with the exception of Laman’s theorem, extend to three dimensional space.

Constructing Generically Rigid Formations in Rd

Vertex addition: Add to a graph with at least d vertices, a new vertex v and d incident edges.

Edge splitting: Remove an edge (i, j) from the a graph with at least d +1 vertices and add a new vertex v and d +1 incident edges including edges (i, v) and (j,v).

Henneberg sequence {1896}: Any set of vertex adding and edge splittingoperations performed in sequence starting with a complete graph withd vertices

Every graph in a Henneberg sequence is minimally rigid.

Every rigid graph in R2 can be constructed using a Henneberg sequence

Applications

Splitting Formations

Merging Formations

Closing Ranks in Formations

CLOSING RANKS

Suppose that some agents stop functioning

CLOSING RANKS

Suppose that some agents stop functioningand drop out of formation along with incident links

CLOSING RANKS

Among adjacent agents,

Suppose that some agents stop functioningand drop out of formation along with incident links

CLOSING RANKS

Among adjacent agents, between which pairsshould communications be established to regaina rigid formation?

Suppose that some agents stop functioningand drop out of formation along with incident links

Among adjacent agents,

Can be solved using modified Henneberg sequences

Leader – Follower Constraints

Leader – Follower Constraints

2

1

31 follows 2 and 3

Leader – Follower Constraints

2

1

31 follows 2 and 3

Can cause problems

Fp = globally rigid if congruent to all formations with the same distance graph.

Fp = rigid if congruent to all “close by” formations with the same distance graph.

Globally rigid

Global rigidity is too “rigid” a property for vehicle formation maintenance

But there is a nice application of global rigidity in systems…………

a rigid formationAnother rigid formation with the same distance graph but not congruent to the first

Fp = globally rigid if congruent to all formations with the same distance graph.

shorter distance

{not complete}

Fp = rigid if congruent to all “close by” formations with the same distance graph.

1. Distance between some sensor pairs are known.

2. Some sensors’ positions in world coordinates are known.

Localization problem is to determine world coordinatesof each sensor in the network.

500m

Does there exist a unique solution to the problem?

Localization of a Network of Sensors in Fixed Positions

3. Thus so are the distances between them

Does there exist a unique solution to the problem?

Localization problem is to determine world coordinatesof each sensor in the network.

Localization of a Network of Sensors in Fixed Positions

Uniqueness is equivalent to this formation being globally rigid

Global rigidity settles the uniquenessquestion.

A polynomial time algorithm exists for testing for global rigidity in 2d.

Localization problem is NP hard

Nonetheless algorithms exist for {sequentially} localizing certaintypes of sensor networks in polynomial time

Localization of a Network of Sensors in Fixed Positions

More Precision

A point formation is rigid if for all possible motions of the formation’spoints which maintain all link lengths constant, the distances betweenall pairs of points remain constant .

A point formation {G , x} is generically rigid if it is rigid on an open subset contain x.

Generic rigidity depends only on the graph G – that is, on the distancegraph of the formation without the distance weights.

A multi-point x in R2n is a vector composed of n vectors x1 , x2 ... xn in R 2

A framework in R2 is a pair {G , x} consisting of a multipoint x 2 R2n and a simpleundirected graph G with n vertices.

no self-loops, no multiple loops

With understanding is that the edges of the graph are maintenance links, a point formation and a framework are one and the same.

A graph G is rigid if there is a multi-point x for which {G, x} is generically rigid

Almost all rigid frameworks are infinitesimally rigid - see Connelly notes for def.

Infinitesimally rigid frameworks can be characterized algebraically

Algebraic Conditions for Infinitesimal Rigidity in Rd

Distance constraints: ||xi – xj||2 = distanceij2, (i, j) 2 L

.(xi – xj)0(xi – xj) = 0, (i, j) 2 L

.

.Rm£nd(x)x = 0, m = |L|

x = column {x1, x2, …, xn}

{G, x} infinitesimally rigid iff dim(kernel R(x)) = 3 if d = 2

6 if d = 3

2n - 3 if d = 2 3n - 6 if d = 3

{G, x} infinitesimally rigid iff rank R(x)) =

G = {{1,2,...,n}, L}

For a minimally rigid framework in R2, m = 2n - 3

For a minimally rigid framework in R2, R(x) has linearly independent rows.

Graph-Theoretic Test for Generic Rigidity in R2

Generic rigidity of {G , x} depends only on G

Laman’s Theorem: G generically rigid in R2 iff there is a non-empty subset E ½ L of size |E| = 2n – 3 such that for all non-empty subsets S ½ E, |S| · 2|V(S)| where V(S) is the number of vertices which are end-points of the edges in S.

There is no similar result for graphs in R3

A graph is rigid in Rd if it is the graph of a generically rigid framework in Rd.

Constructing Rigid Graphs in Rd

Vertex addition: Add to a graph with at least d vertices, a new vertex v and d incident edges.

Edge splitting: Remove an edge (i, j) from the a graph with at least d +1 vertices and add a new vertex v and d +1 incident edges including edges (i, v) and (j,v).

A graph is minimally rigid if it is rigid and if it loses this property when any single edge is deleted.

Henneberg sequence: Any set of vertex adding and edge splittingoperations performed in sequence starting with a complete graph with d vertices

Every graph in a Henneberg sequence is minimally rigid.

Every rigid graph in R2 can be constructed using a Henneberg sequence

Vertex Addition in R2

Vertex addition: Add to a graph with at least 2 vertices, a new vertex v and 2 incident edges.

Edge splitting: Remove an edge (i, j) from the a graph with at least 3 vertices and add a new vertex v and 3 incident edges including edges (i, v) and (j,v).

Edge Splitting in R2