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Supelec EECI Graduate School in Control. Rigidity. A. S. Morse Yale University. Gif – sur - Yvette May 24, 2012. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A A A A A. - PowerPoint PPT Presentation
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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