Distributed motion coordination of robotic networks Lecture 2 – models and complexity notions Jorge Cort´ es Applied Mathematics and Statistics Baskin School of Engineering University of California at Santa Cruz http://www.ams.ucsc.edu/˜jcortes Summer School on Geometry Mechanics and Control Centro Internacional de Encuentros Matem´ aticos Castro Urdiales, June 25-29, 2007 ,
Roadmap Lecture 1: Introduction, examples, and preliminary notions Lecture 2: Models for cooperative robotic networks Lecture 3: Rendezvous Lecture 4: Deployment Lecture 5: Agreement ,
Today Cooperative robotic network model 1 Proximity graphs as interaction topology 2 Control and communication laws , coordination tasks 3 Complexity notions 4 Analysis of agree and pursue coordination algorithm ,
Modeling theme Broad aim: optimal trade-offs in sensing, control, communication Given two coordination algorithms that achieve the same task, which one is better? Specific objectives formalize execution of coordination algorithms characterize performance, costs – complexity rigorously combine strategies to achieve more complex tasks Long standing tradition in the theory of distributed algorithms and parallel computing – but networks have fixed topology ,
Outline 1 Proximity graphs 2 Model for cooperative robotic networks Control and communication laws Coordination tasks Complexity notions 3 Agree and pursue coordination law: complexity analysis ,
Proximity graphs Proximity graph graph whose vertex set is a set of distinct points and whose edge set is a function of the relative locations of the point set Appear in computational geometry and topology control of wireless networks ,
Proximity graphs Proximity graph graph whose vertex set is a set of distinct points and whose edge set is a function of the relative locations of the point set Appear in computational geometry and topology control of wireless networks Definition (Proximity graph) Let X be a d -dimensional space chosen among R d , S d , and R d 1 × S d 2 , with d 1 + d 2 = d . Let G ( X ) be the set of all undirected graphs whose vertex set is an element of F ( X ) (finite subsets of X ) A proximity graph G : F ( X ) → G ( X ) associates to P = { p 1 , . . . , p n } ⊂ X an undirected graph with vertex set P and edge set E G ( P ) ⊆ { ( p, q ) ∈ P × P | p � = q } . ,
Examples of proximity graphs On ( R d , dist 2 ) , ( S d , dist g ) , or ( R d 1 × S d 2 , (dist 2 , dist g )) 1 the r -disk graph G disk ( r ) , for r ∈ R > 0 , with ( p i , p j ) ∈ E G disk ( r ) ( P ) if dist( p i , p j ) ≤ r 2 the Delaunay graph G D , with ( p i , p j ) ∈ E G D ( P ) if V i ( P ) ∩ V j ( P ) � = ∅ Definition 3 the r -limited Delaunay graph G LD ( r ) , for r ∈ R > 0 , with ( p i , p j ) ∈ E G LD ( r ) ( P ) if V i, r 2 ( P ) ∩ V j, r 2 ( P ) � = ∅ Definition 4 the relative neighborhood graph G RN , with ( p i , p j ) ∈ E G RN ( P ) if p k �∈ B ( p i , dist( p i , p j )) ∩ B ( p j , dist( p i , p j )) for all p k ∈ P G disk ( r ) G D G LD ( r ) G RN ,
More examples of proximity graphs on Euclidean space 1 the Gabriel graph G G , with ( p i , p j ) ∈ E G G ( P ) if � p i + p j , dist( p i ,p j ) p k �∈ B � for all p k ∈ P 2 2 2 the Euclidean minimum spanning tree G EMST , that assigns to each P a minimum-weight spanning tree of the complete weighted digraph ( P , { ( p, q ) ∈ P × P | p � = q } , A ) , with weighted adjacency matrix a ij = � p i − p j � 2 , for i, j ∈ { 1 , . . . , n } 3 given a simple polygon Q in R 2 , the visibility graph G vis,Q , with ( p i , p j ) ∈ E G vis,Q ( P ) if the closed segment [ p i , p j ] from p i to p j is contained in Q G G G EMST G vis,Q ,
Set of neighbors map To each proximity graph G , each p ∈ X and each P = { p 1 , . . . , p n } ∈ F ( X ) associate set of neighbors map N G ,p : F ( X ) → F ( X ) defined by N G ,p ( P ) = { q ∈ P | ( p, q ) ∈ E G ( P ∪ { p } ) } Typically, p is a point in P , but this works for any p ∈ X When does a proximity graph provide sufficient information to compute another proximity graph? ,
Spatially distributed graphs E.g., if a node knows position of its neighbors in the complete graph, then it can compute its neighbors with respect to any proximity graph Formally, given G 1 and G 2 , 1 G 1 is a subgraph of G 2 , denoted G 1 ⊂ G 2 , if G 1 ( P ) is a subgraph of G 2 ( P ) for all P ∈ F ( X ) 2 G 1 is spatially distributed over G 2 if, for all p ∈ P , � � N G 1 ,p ( P ) = N G 1 ,p N G 2 ,p ( P ) , that is, any node equipped with the location of its neighbors with respect to G 2 can compute its set of neighbors with respect to G 1 ,
Spatially distributed graphs E.g., if a node knows position of its neighbors in the complete graph, then it can compute its neighbors with respect to any proximity graph Formally, given G 1 and G 2 , 1 G 1 is a subgraph of G 2 , denoted G 1 ⊂ G 2 , if G 1 ( P ) is a subgraph of G 2 ( P ) for all P ∈ F ( X ) 2 G 1 is spatially distributed over G 2 if, for all p ∈ P , � � N G 1 ,p ( P ) = N G 1 ,p N G 2 ,p ( P ) , that is, any node equipped with the location of its neighbors with respect to G 2 can compute its set of neighbors with respect to G 1 G 1 spatially distributed over G 2 = ⇒ G 1 ⊂ G 2 Converse not true: G D ∩ G disk ( r ) ⊂ G disk , but G D ∩ G disk ( r ) not spatially distributed over G disk ( r ) Illustration ,
Inclusion relationships among proximity graphs Theorem For r ∈ R > 0 , the following statements hold: 1 G EMST ⊂ G RN ⊂ G G ⊂ G D ; 2 G G ∩ G disk ( r ) ⊂ G LD ( r ) ⊂ G D ∩ G disk ( r ) 3 G RN ∩ G disk ( r ) , G G ∩ G disk ( r ) , and G LD ( r ) are spatially distributed over G disk ( r ) The inclusion G LD ( r ) ⊂ G D ∩ G disk ( r ) is in general strict Since G EMST is by definition connected, (1) implies that G RN , G G and G D are connected ,
Connectivity properties of G disk ( r ) Theorem For r ∈ R > 0 , the following statements hold: 1 G EMST ⊂ G disk ( r ) if and only if G disk ( r ) is connected; 2 G EMST ∩ G disk ( r ) , G RN ∩ G disk ( r ) , G G ∩ G disk ( r ) and G LD ( r ) have the same connected components as G disk ( r ) (i.e., for all point sets P ∈ F ( R d ) , all graphs have the same number of connected components consisting of the same vertices). ,
Proximity graphs over tuples Proximity graphs are defined for sets of distinct points We are interested in tuples – might contain coincident points Let i F : X n → F ( X ) be the natural immersion of X n into F ( X ) P = ( p 1 , . . . , p n ) �→ P = { p 1 , . . . , p n } Given a proximity graph G , define 1 G = G ◦ i F : X n → G ( X ) 2 The set of neighbors map N G : X n → F ( X ) is defined by N G ,p ( p 1 , . . . , p n ) = N G ,p ( i F ( p 1 , . . . , p n )) According to this definition, coincident points in the tuple ( p 1 , . . . , p n ) will have the same set of neighbors ,
Spatially distributed maps Given a set Y and a proximity graph G , a map T : X n → Y n is spatially distributed over G if ∃ a map ˜ T : X × F ( X ) → Y such that for all ( p 1 , . . . , p n ) ∈ X n and for all j ∈ { 1 , . . . , n } , T j ( p 1 , . . . , p n ) = ˜ T ( p j , N G ,p j ( p 1 , . . . , p n )) , where T j denotes the j th-component of T Equivalently, the j th component of a spatially distributed map at ( p 1 , . . . , p n ) can be computed with only the knowledge of the vertex p j and the neighboring vertices in the undirected graph G ( P ) ,
Outline 1 Proximity graphs 2 Model for cooperative robotic networks Control and communication laws Coordination tasks Complexity notions 3 Agree and pursue coordination law: complexity analysis ,
The physical components of a robotic network A synchronous robotic network as a group of robots with the ability to exchange messages according to a geometric communication topology, perform local computations and control their motion ,
The physical components of a robotic network A synchronous robotic network as a group of robots with the ability to exchange messages according to a geometric communication topology, perform local computations and control their motion Mobile robot: continuous-time continuous-space dynamical system, that is, tuple ( X, U, X 0 , f ) 1 X is d -dimensional space chosen among R d , S d , and the Cartesian products R d 1 × S d 2 , for some d 1 + d 2 = d , called the state space ; 2 U is a compact subset of R m containing 0 n , called the input space ; 3 X 0 is a subset of X , called the set of allowable initial states ; 4 f : X × U → R d is a smooth control vector field on X , that is, f determines the robot motion x : R ≥ 0 → X via x ( t ) = f ( x ( t ) , u ( t )) , ˙ subject to the control u : R ≥ 0 → U ,
Synchronous robotic network Definition (Robotic network) The physical components of a uniform robotic network S consist of a tuple ( I, R , E ) , where 1 I = { 1 , . . . , n } ; I is called the set of unique identifiers (UIDs) ; 2 R = { R [ i ] } i ∈ I = { ( X, U, X 0 , f ) } i ∈ I is a set of mobile robots; 3 E is a map from X n to the subsets of I × I ; this map is called the communication edge map . The map x �→ ( I, E ( x )) models the topology of the communication service among the robots. As communication graphs, we will adopt the proximity graph determined by network capabilities ,
Recommend
More recommend
