A. S. Morse Yale University

1. IMA Short Course Distributed Optimization and Control Rigidity 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.: AAAAAAAAAAAAAAAAA

2. 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

3. motion point formation Fp(L) d11,1 p = {p1, p2, …, p11} d1,2 d10,11 L= {(1,2), (2,3), …, } d7,4 d9,11 d10,9 d5,4 p1 d9,6 d6,5 p11 1 p2 3 2 p3 p7 p10 4 11 p4 p8 10 7 p9 5 p5 6 p6 8 9 point set in the plane Rigid motion: means distances between all pairs of points are constant Maintaining a formation of points …..with maintenance links framework distance graph

4. Euclidean transformation point formation Fp(L) d11,1 p = {p1, p2, …, p11} d1,2 d10,11 L = {(1,2), (2,3), …, } d7,4 d9,11 d10,9 d5,4 p1 d9,6 d6,5 p11 1 p2 3 2 p3 p7 p10 4 11 p4 p8 10 7 p9 5 p5 6 p6 8 9 Fp= rigid if congruent to all “close by” formations with the same distance graph. Special Euclidean Group SE(2) translation rotation reflection congruent distance graph

5. missing link redundant link Fp= rigid if congruent to all “close by” formations with the same distance graph. rigid means can’t be “continuously deformed” minimally rigid {isostatic} redundantly rigid non-rigid {flexible} The number of maintenance links in a minimally rigid n point formation in 2d is 2n - 3

6. 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. 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. 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. Laman’s theorem {1970}: A combinatoric test for deciding whether or not a graph is rigid. Three-dimensions: All of the preceding, with theexception of Laman’s theorem, extend to three dimensional space.

7. 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 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 R2 can be constructed using a Henneberg sequence

8. Applications Splitting Formations Merging Formations Closing Ranks in Formations

9. CLOSING RANKS Suppose that some agents stop functioning

10. CLOSING RANKS Suppose that some agents stop functioning and drop out of formation along with incident links

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

12. CLOSING RANKS Suppose that some agents stop functioning and drop out of formation along with incident links Among adjacent agents, between which pairs should communications be established to regain a rigid formation? Among adjacent agents, Can be solved using modified Henneberg sequences

13. Leader – Follower Constraints

14. 2 1 follows 2 and 3 3 1 Leader – Follower Constraints

15. 2 1 follows 2 and 3 3 1 Leader – Follower Constraints Can cause problems

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

17. Globally rigid Fp= rigid if congruent to all “close by” formations with the same distance graph. Fp= globally rigid if congruent to all formations with the same distance graph. shorter distance Another rigid formation with the same distance graph but not congruent to the first a rigid formation {not complete} Global rigidity is too “rigid” a property for vehicle formation maintenance But there is a nice application of global rigidity in systems…………

18. Localization of a Network of Sensors in Fixed Positions Does there exist a unique solution to the problem? 1. Distance between some sensor pairs are known. 500m 2. Some sensors’ positions in world coordinates are known. 3. Thus so are the distances between them Localization problem is to determine world coordinates of each sensor in the network.

19. Localization of a Network of Sensors in Fixed Positions Does there exist a unique solution to the problem? Localization problem is to determine world coordinates of each sensor in the network.

20. 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

21. More Precision no self-loops, no multiple loops A multi-pointx in R2n is a vector composed of n vectors x1 , x2 ... xn in R2 A framework in R2 is a pair {G , x} consisting of a multipoint x2R2n and a simple undirected graph G with n vertices. With understanding is that the edges of the graph are maintenance links, a point formation and a framework are one and the same. 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 graph G is rigid if there is a multi-point x for which {G, x} is generically rigid - see Connelly notes for def. Almost all rigid frameworks are infinitesimally rigid Infinitesimally rigid frameworks can be characterized algebraically

22. . . (xi– xj)0(xi– xj) = 0, (i, j) 2L . Rm£nd(x)x= 0, m = |L| Algebraic Conditions for Infinitesimal Rigidity in Rd Distance constraints: ||xi– xj||2 = distanceij2, (i, j) 2L G = {{1,2,...,n}, L} x= column {x1, x2, …, xn} 3 if d = 2 6 if d = 3 {G, x} infinitesimally rigid iffdim(kernel R(x)) = 2n – 3 if d = 2 3n – 6 if d = 3 {G, x} infinitesimally rigid iff rank R(x)) = For a minimally rigid framework in R2, m = 2n - 3 For a minimally rigid framework in R2, R(x) has linearly independent rows.

23. Graph-Theoretic Test for Generic Rigidity in R2 Generic rigidity of {G , x} depends only on G A graph is rigid in Rd if it is the graph of a generically rigid framework in Rd. Laman’s Theorem: G generically rigid in R2 iff there is a non-empty subset E ½L of size |E| = 2n – 3 such that for all non-empty subsets S ½E, |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

24. Constructing Rigid Graphs in Rd A graph is minimally rigid if it is rigid and if it loses this property when any single edge is deleted. 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: 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 R2 can be constructed using a Henneberg sequence

25. Vertex Addition in R2 Vertex addition: Add to a graph with at least 2 vertices, a new vertex v and 2 incident edges.

26. Edge Splitting in R2 Edge splitting: Remove an edge (i, j) from the a graph with at least 3vertices and add a new vertex v and 3incident edges including edges (i, v) and (j,v).