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Slide1
Rigidity
A. S. MorseYale University
Gif – sur - Yvette May 24, 2012
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAA
Supelec
EECI
Graduate School in ControlSlide2
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 planeSlide3
point set
motion
in the plane
Rigid motion
: means distances between all pairs of points are constant
Maintaining a formation of points
…..with
maintenance links
p
5
p
4
p
3
p
2
p
1
p
6
p
7
p
8
p
9
p
10
p
11
p
= {
p
1
,
p
2
, …,
p
11
}
L
= {(1,2), (2,3), …, }
point formation
F
p
(
L
)
6
5
4
10
11
9
7
8
1
3
2
d
9,6
d
7,4
d
6,5
d
11,1
d
5,4
d
9,11
d
10,11
d
10,9
d
1,2
distance graph
frameworkSlide4
p
5
p
4
p
3
p
2
p
1
p
6
p
7
p
8
p
9
p
10
p
11
p
= {
p
1
,
p
2
, …,
p
11
}
L
= {(1,2), (2,3), …, }
point formation
F
p
(
L
)
distance graph
translation
rotation
reflection
6
5
4
10
11
9
7
8
1
3
2
d
9,6
d
7,4
d
6,5
d
11,1
d
5,4
d
9,11
d
10,11
d
10,9
d
1,2
F
p
=
rigid
if congruent to all “close by” formations with the same
distance
graph.
Euclidean transformation
congruent
Euclidean Group
Special
SE(2)Slide5
minimally rigid
{isostatic}
redundantly rigid
non-rigid
{flexible}
redundant link
missing link
F
p
=
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 2
d
is 2
n
- 3Slide6
F
p
=
rigid
if congruent to all “close by” formations with the same
distance
graph.
F
p
=
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.Slide7
Constructing Generically Rigid Formations in
R
d
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 splitting
operations performed in sequence starting with a complete graph with
d
vertices
Every graph in a Henneberg sequence is minimally rigid.
Every rigid graph in
R
2
can be constructed using a Henneberg sequenceSlide8
Applications
Splitting Formations
Merging Formations
Closing Ranks in FormationsSlide9
CLOSING RANKS
Suppose that some agents stop functioningSlide10
CLOSING RANKS
Suppose that some agents stop functioning
and drop out of formation along with incident linksSlide11
CLOSING RANKS
Among adjacent agents,
Suppose that some agents stop functioning
and drop out of formation along with incident linksSlide12
CLOSING RANKS
Among adjacent agents, between which pairs
should communications be established to regain
a rigid formation?
Suppose that some agents stop functioning
and drop out of formation along with incident links
Among adjacent agents,
Can be solved using modified Henneberg sequencesSlide13
Leader – Follower ConstraintsSlide14
Leader – Follower Constraints
2
1
3
1 follows 2 and 3Slide15
Leader – Follower Constraints
2
1
3
1 follows 2 and 3
Can cause problemsSlide16
F
p
= globally rigid if congruent to all formations with the same distance graph.
F
p
= rigid if congruent to all “
close by
” formations with the same distance graph. Slide17
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 formation
Another rigid formation with the same distance
graph but not congruent to the first
F
p
=
globally rigid
if congruent to
all
formations with the same distance graph.
shorter distance
{not complete}
F
p
= rigid if congruent to all “
close by
” formations with the same distance graph. Slide18
1. Distance between
some sensor pairs
are known.
2. Some sensors’
positions in world
coordinates are known.
Localization problem is to
determine world coordinates
of 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
themSlide19
Does there exist a unique
solution to the problem?
Localization problem is to
determine world coordinates
of each sensor in the network.
Localization of a Network of Sensors in Fixed PositionsSlide20
Uniqueness is equivalent to this formation being
globally rigid
Global rigidity settles the uniqueness
question.
A polynomial time algorithm exists for
testing for global rigidity in 2d.
Localization problem is NP hard
Nonetheless algorithms exist for
{sequentially} localizing certain
types of sensor networks in
polynomial time
Localization of a Network of Sensors in Fixed PositionsSlide21
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 distance
graph
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 simple
undirected 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 rigidAlmost all rigid frameworks are infinitesimally rigid
- see Connelly notes for def.
Infinitesimally rigid frameworks can be characterized algebraicallySlide22
Algebraic Conditions
for Infinitesimal Rigidity in R
dDistance constraints: ||
xi – xj||2 = distanceij2, (i, j) 2 L
.
(
x
i
–
x
j
)
0
(
x
i
– x
j) = 0, (i, j
) 2 L
.
.
R
m
£
nd
(x)
x = 0, m = |
L|
x
= column {x
1, x2, …,
xn}
{G, x} infinitesimally rigid iff dim(kernel R(x)) =
3 if d = 26 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 R
2, m = 2n - 3
For a minimally rigid framework in R2, R
(x) has linearly independent rows. Slide23
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
| = 2
n
– 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.Slide24
Constructing
Rigid Graphs in R
dVertex 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 sequenceSlide25
Vertex Addition in R2
Vertex addition:
Add to a graph with at least
2
vertices
, a new
vertex
v
and
2
incident edges.Slide26
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
R
2