A. S. Morse Yale University University of Minnesota June 4, 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A AA A A AAAA A IMA Short Course Distributed Optimization and Control Rigidity

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 set motion in the plane Rigid motion: means distances between all pairs of points are constant Maintaining a formation of points…..with maintenance links p5p5 p4p4p4p4 p3p3p3p3 p2p2 p1p1p1p1 p6p6p6p6 p7p7 p8p8p8p8 p9p9 p 10 p 11 p = {p 1, p 2, …, p 11 } = {(1,2), (2,3), …, } L = {(1,2), (2,3), …, } point formation F p ( L ) 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 framework

p5p5 p4p4 p3p3 p2p2 p1p1 p6p6 p7p7 p8p8 p9p9 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 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 = if congruent to all “close by” formations with the same graph. F p = rigid if congruent to all “close by” formations with the same distance graph. Euclidean transformation congruent Euclidean GroupSpecial SE(2)

minimally rigid {isostatic} redundantly rigidnon-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 2d is 2n - 3

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 F p 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 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 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 functioning and drop out of formation along with incident links

CLOSING RANKS Among adjacent agents, Suppose that some agents stop functioning and drop out of formation along with incident links

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 sequences

Leader – Follower Constraints

follows 2 and 3

Leader – Follower Constraints follows 2 and 3 Can cause problems

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.

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.

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 them

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 Positions

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 Positions

More Precision A point formation is rigid if for all possible motions of the formation’s points which maintain all link lengths constant, the distances between all 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 R 2n is a vector composed of n vectors x 1, x 2... x n in R 2 A framework in R 2 is a pair { G, x} consisting of a multipoint x 2 R 2n 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 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 R d Distance constraints: ||x i – x j || 2 = distance ij 2, (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, x 2, …, x n } G = {{1,2,...,n}, L } For a minimally rigid framework in R 2, m = 2n - 3 For a minimally rigid framework in R 2, R(x) has linearly independent rows. { G, x} infinitesimally rigid iff rank R(x)) = 2n – 3 if d = 2 3n – 6 if d = 3 { G, x} infinitesimally rigid iff dim(kernel R(x)) = 3 if d = 2 6 if d = 3

Graph-Theoretic Test for Generic Rigidity in R 2 Generic rigidity of { G, x} depends only on G Laman’s Theorem: G generically rigid in R 2 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 R 3 A graph is rigid in R d if it is the graph of a generically rigid framework in R d.

Constructing Rigid Graphs 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). 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 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 sequence

Vertex Addition in R 2 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 R 2