Connectivity, Rigidity and Online Decentralized Maintenance Methods Antonio Franchi CNRS, LAAS, France, Europe 2015 IROS Workshop on ‘On-line decision-making in multi-robot coordination’ (DEMUR’15) Hamburg, Germany 12th October, 2015 Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 1 of 69
Table of Contents 1. Graphs, Matrices, and Eigenvalues 2. Connectivity vs Infinitesimal Rigidity 3. Maintenance Problems and Methods 4. Handling Multiple Objectives in Maintenance Problems 5. Applications Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 2 of 69
Partial State of the Art Partial list: • P. Yang, R.A. Freeman, G.J. Gordon, K.M. Lynch, S.S. Srinivasa, and R. Sukthankar, ”Decentralized estimation and control of graph connectivity for mobile sensor networks,” Automatica, vol. 46, no. 2. pp. 390–396, Feb. 2010. • G. Hollinger and S. Singh, ”Multirobot coordination with periodic connectivity: Theory and experiments,” IEEE Transactions on Robotics , 2012, • L. Sabattini, C. Secchi, N. Chopra, and A Gasparri. Distributed Control of Multirobot Systems With Global Connectivity Maintenance. Robotics, IEEE Transactions on Robotics, 29(5):1326-1332, 2013. • D. Carboni, R.K. Williams, A. Gasparri, G. Ulivi, and G.S. Sukhatme. Rigidity-Preserving Team Partitions in Multi-Agent Networks. IEEE Transactions on Cybernetics, pp 1-14, 2014. Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 3 of 69
Main Sources If you want to know more about what follows: • Robuffo Giordano, P., A. Franchi, C. Secchi, and H. H. B¨ ulthoff (2013). A Passivity-Based Decentralized Strategy for Generalized Connectivity Maintenance”. The International Journal of Robotics Research 32.3, pp. 299–323. • Zelazo, D., A. Franchi, H. H. B¨ ulthoff, and P. Robuffo Giordano (2014). Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems . The International Journal of Robotics Research 34.1, pp. 105–128. • Nestmeyer T., P. Robuffo Giordano, H. H. B¨ ulthoff, and A. Franchi, Decentralized Simultaneous Multi-target Exploration using a Connected Network of Multiple Robots . Under Review. Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 4 of 69
Graphs, Matrices, and Eigenvalues Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 5 of 69
Graph G = ( V , E ) is an undirected graph or simply graph (1,3) 3 1 (3,4) • V = { 1 , . . . , N } vertex set ( 1 , ( • E ⊂ ( V × V ) / ∼ edge set 1 , 2 4 ) (2,3) ) 4 • ∼ equivalence relation identifying ( i , j ) and ( j , i ) 2 A Graph models an Adjacency Structure [( i , j )] ∈ E ⇔ vertexes i and j are neighbors or adjacent • ( i , j ), i < j representative element of the equivalence class [( i , j )] [ V × V ] = { (1 , 2) , (1 , 3) , . . . , (1 , N ) , . . . , ( N − 1 , N ) } = { e 1 , e 2 , . . . e N − 1 , . . . , e N ( N − 1) / 2 } • [( i , i )] / ∈ E , ∀ i ∈ V ( no self-loops ) • N i = { j ∈ V | ( i , j ) ∈ E} set of neighbors of i Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 6 of 69
Incidence Matrix E ∈ R N × N ( N − 1) / 2 is the (full) incidence matrix of G ∀ e k = ( i , j ) ∈ [ V × V ]: Example: • E ik = − 1 and E jk = 1, if e k ∈ E (1,3) 3 1 (3,4) • E ik = 0 and E jk = 0, otherwise ( 1 (1,4) , 2 (2,3) ) 4 Matricial representation of a graph 2 1 1 1 0 0 0 − 1 0 0 1 0 0 E = 0 − 1 0 − 1 0 1 0 0 − 1 0 0 − 1 e 1 e 2 e 3 e 4 e 5 e 6 remember: { e 1 , e 2 , . . . , e N − 1 , . . . , e N ( N − 1) / 2 } = { (1 , 2) , (1 , 3) , . . . , (1 , N ) , . . . ( N − 1 , N ) } Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 7 of 69
Network of Robots in an Environment Assume N mobile robots moving in an environment: • x i ∈ R N x i -th robot configuration , i ∈ 1 . . . N • z ∈ R N z environment configuration Consider two maps R N x ∋ x i �→ v ( x i ) = v i ∈ R N v robot map v : R N x × R N x × R N z ∋ ( x i , x j , z ) �→ w ( x i , x j , z ) = w ij ∈ R ≥ 0 connection map w : with the properties • w ij = w ji (symmetry) • w ii = 0 example: what can those maps model? Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 8 of 69
Associated Graph and Framework The connection map w defines an associated graph G = ( V , E ), where • V = { 1 , 2 , . . . , N } • E = { e k = ( i , j ) | w ij > 0 } • the positive weight w ij is associated to each edge ( i , j ) ∈ E Both maps v and w define an associated framework ( G , v ) where • G is the associated graph • v i is associated to each vertex i ∈ V Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 9 of 69
Adjacency/Weight Matrix 0 w 12 . . . w 1 N w 12 0 . . . w 2 N ∈ R N × N is the adjacency (or weight ) matrix of G A = . . . ... . . . . . . w 1 N w 2 N · · · 0 Note that Example: • A ij = 0 if ( i , j ) / ∈ E (1,3) 3 1 (3,4) • A ij > 0 otherwise ( 1 , (1,4) Properties: 2 (2,3) ) 4 P.1 A = A ( x 1 , . . . , x N , z ) 2 P.2 A is square P.3 A ij = A ij (symmetric) 0 w 12 w 13 w 14 w 12 0 w 23 0 P.4 A ij = A ij ≥ 0 (nonnegative) A = w 13 w 23 0 w 34 P.5 A ii = 0 w 14 0 w 34 0 Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 10 of 69
Laplacian Matrix � n j =1 w 1 j − w 12 . . . − w 1 N � n − w 12 j =1 w j 2 . . . − w 2 N ∈ R N × N is the Laplacian matrix of G L = . . . ... . . . . . . � n − w 1 N − w 2 N · · · j =1 w jN Note that Example: • L = diag ( δ i ) − A , (1,3) 3 1 ( 3 where δ i = � n , 4 j =1 w ij ) ( 1 , (1,4) (degree of vertex i ) 2 (2,3) ) 4 2 Properties: � w 12 + w 13 + w 14 P.1 L = L ( x 1 , . . . , x N , z ) � − w 12 − w 13 − w 14 − w 12 w 12 + w 23 − w 23 0 L = P.2 L is square − w 13 − w 23 w 13 + w 23 + w 34 − w 34 − w 14 0 − w 34 w 14 + w 34 P.3 L ij = L ij (symmetric) Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 11 of 69
Connected Graph Connectivity G is connected if there is a path between every pair of vertices, i.e., ∀ i ∈ V and j ∈ V\ i , ∃ a path (sequence of adjacent edges) from i to j This is a combinatorial definition of connectivity 3 3 1 1 4 4 5 5 2 2 6 6 connected graph disconnected graph question: connectivity is a global property, what does it mean? and why it is global? Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 12 of 69
Importance of Connectivity What connectivity can model? • connected communication network • connected sensing network • connected control network • connected planning roadmap What connectivity is important for? • pass a message from any robot to any other robot • know the relative position between any two robots in a common frame • converge to a common point • share a common goal Related concepts • group, cohesiveness • aggregation • sharing Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 13 of 69
Spectrum of the Laplacian Matrix and Algebraic Connectivity Additional properties of L = diag ( δ i ) − A • L is positive semi-definite , i.e., all the eigenvalues are real and non-negative 0 ≤ λ 1 ≤ λ 2 ≤ . . . ≤ λ N • � n j =1 L ij = 0 ∀ i = 1 . . . N , i.e., L 1 = 0 , therefore � � T λ 1 = 0 and it is associated to the eigenvector 1 = 1 1 . . . 1 (Fiedler 1973) λ 2 > 0 if the graph G is connected and λ 2 = 0 otherwise Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 14 of 69
Spectrum of the Laplacian Matrix and Algebraic Connectivity λ 2 provides an algebraic definition of connectivity ⇒ λ 2 is called algebraic connectivity , connectivity eigenvalue , or Fiedler eigenvalue λ 2 = λ 2 ( x 1 , . . . , x N , z ) is a global quantity Example (if w ij ∈ { 0 , 1 } ): λ 2 = 4 λ 2 = 2 λ 2 = 0 . 58 λ 2 = 0 Antonio Franchi Connectivity, Rigidity and Online Decentralized Maintenance Methods – http://homepages.laas.fr/afranchi/robotics/ 15 of 69
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