Laman graphs are generically bearing rigid in arbitrary dimensions Shiyu Zhao 1 , Zhiyong Sun 2 , Daniel Zelazo 3 , Minh-Hoang Trinh 4 , and Hyo-Sung Ahn 4 1 University of Sheffield, UK 2 Australian National University, Australia 3 Technion - Israel Institute of Technology, Israel 4 Gwangju Institute of Science and Technology, Korea December 2017
What is bearing rigidity? Revisit distance rigidity: ⋄ If we fix the length of each edge in a network, can the geometric pattern of the network be uniquely determined? 1 / 13
What is bearing rigidity? Revisit distance rigidity: ⋄ If we fix the length of each edge in a network, can the geometric pattern of the network be uniquely determined? Bearing rigidity: ⋄ If we fix the bearing of each edge in a network, can the geometric pattern of the network be uniquely determined? Loose definition: a network bearing rigid if its bearings can uniquely determine its geometric pattern. 1 / 13
Why study bearing rigidity? ⋄ Initially: computer-aided graphical drawing [Servatius and Whiteley, 1999] 2 / 13
Why study bearing rigidity? ⋄ Initially: computer-aided graphical drawing [Servatius and Whiteley, 1999] ⋄ In recent years: Formation control and network localization [Eren et al., 2003, Bishop, 2011, Eren, 2012, Zelazo et al., 2014, Zhao and Zelazo, 2016a] 2 / 13
Why study bearing rigidity? ⋄ Initially: computer-aided graphical drawing [Servatius and Whiteley, 1999] ⋄ In recent years: Formation control and network localization [Eren et al., 2003, Bishop, 2011, Eren, 2012, Zelazo et al., 2014, Zhao and Zelazo, 2016a] ⋄ Network localization: Anchor Follower: initial estimate 120 Follower: final estiamte 100 80 z (m) 60 40 20 120 100 80 60 20 40 60 40 20 80 120 100 y (m) x (m) ⋄ Formation control: Initial position Final position 10 8 6 4 2 0 10 5 0 5 0 10 2 / 13
Two key problems in bearing rigidity theory • How to determine the bearing rigidity of a given network? • How to construct a bearing rigid network from scratch? 3 / 13
Notations for Bearing Rigidity ⋄ Notations: • Graph: G = ( V , E ) where V = { 1 , . . . , n } and E ⊆ V × V • Configuration: p i ∈ R d with i ∈ V and p = [ p T 1 , . . . , p T n ] T . • Network: graph+configuration ⋄ Bearing: p j − p i g ij = ∀ ( i, j ) ∈ E . � p j − p i � Example: 3 g 31 g 32 g 13 g 23 1 2 g 12 g 21 ⋄ An orthogonal projection matrix: P g ij = I d − g ij g T ij , 4 / 13
Notations for Bearing Rigidity ⋄ Notations: • Graph: G = ( V , E ) where V = { 1 , . . . , n } and E ⊆ V × V • Configuration: p i ∈ R d with i ∈ V and p = [ p T 1 , . . . , p T n ] T . • Network: graph+configuration ⋄ Bearing: p j − p i g ij = ∀ ( i, j ) ∈ E . � p j − p i � Example: 3 g 31 g 32 g 13 g 23 1 2 g 12 g 21 ⋄ An orthogonal projection matrix: P g ij = I d − g ij g T ij , 4 / 13
Notations for Bearing Rigidity ⋄ Notations: • Graph: G = ( V , E ) where V = { 1 , . . . , n } and E ⊆ V × V • Configuration: p i ∈ R d with i ∈ V and p = [ p T 1 , . . . , p T n ] T . • Network: graph+configuration ⋄ Bearing: p j − p i g ij = ∀ ( i, j ) ∈ E . � p j − p i � Example: 3 g 31 g 32 g 13 g 23 1 2 g 12 g 21 ⋄ An orthogonal projection matrix: P g ij = I d − g ij g T ij , 4 / 13
Notations for Bearing Rigidity ⋄ Properties: g ij x 0 P gij x • P g ij is symmetric positive semi-definite and P 2 g ij = P g ij • Null( P g ij ) = span { g ij } ⇐ ⇒ P g ij x = 0 iff x � g ij (important) ⋄ Bearing Laplacian: B ∈ R dn × dn with the ij th subblock matrix as 0 d × d , i � = j, ( i, j ) / ∈ E [ B ] ij = − P g ij , i � = j, ( i, j ) ∈ E � j ∈N i P g ij , i ∈ V Example: 3 g 31 g 32 − P g 12 − P g 13 P g 12 + P g 13 B = − P g 21 P g 21 + P g 23 − P g 23 g 13 g 23 − P g 31 − P g 32 P g 31 + P g 32 1 2 g 12 g 21 5 / 13
Notations for Bearing Rigidity ⋄ Properties: g ij x 0 P gij x • P g ij is symmetric positive semi-definite and P 2 g ij = P g ij • Null( P g ij ) = span { g ij } ⇐ ⇒ P g ij x = 0 iff x � g ij (important) ⋄ Bearing Laplacian: B ∈ R dn × dn with the ij th subblock matrix as 0 d × d , i � = j, ( i, j ) / ∈ E [ B ] ij = − P g ij , i � = j, ( i, j ) ∈ E � j ∈N i P g ij , i ∈ V Example: 3 g 31 g 32 − P g 12 − P g 13 P g 12 + P g 13 B = − P g 21 P g 21 + P g 23 − P g 23 g 13 g 23 − P g 31 − P g 32 P g 31 + P g 32 1 2 g 12 g 21 5 / 13
Notations for Bearing Rigidity ⋄ Properties: g ij x 0 P gij x • P g ij is symmetric positive semi-definite and P 2 g ij = P g ij • Null( P g ij ) = span { g ij } ⇐ ⇒ P g ij x = 0 iff x � g ij (important) ⋄ Bearing Laplacian: B ∈ R dn × dn with the ij th subblock matrix as 0 d × d , i � = j, ( i, j ) / ∈ E [ B ] ij = − P g ij , i � = j, ( i, j ) ∈ E � j ∈N i P g ij , i ∈ V Example: 3 g 31 g 32 − P g 12 − P g 13 P g 12 + P g 13 B = − P g 21 P g 21 + P g 23 − P g 23 g 13 g 23 − P g 31 − P g 32 P g 31 + P g 32 1 2 g 12 g 21 5 / 13
Examine the bearing rigidity of a given network Condition for Bearing Rigidity [Zhao and Zelazo, 2016b] A network is bearing rigid if and only if rank( B ) = dn − d − 1 Proof. g 1 . ∈ R dm . f ( p ) � . . g m R ( p ) � ∂f ( p ) ∈ R dm × dn . ∂p d f ( p ) = R ( p ) dp Trivial motions: translation and scaling 6 / 13
Examine the bearing rigidity of a given network Condition for Bearing Rigidity [Zhao and Zelazo, 2016b] ⋄ Examples of bearing rigid networks: A network is bearing rigid if and only if rank( B ) = dn − d − 1 Proof. g 1 (e) (f) (g) (h) . ∈ R dm . f ( p ) � . . ⋄ Examples of networks that are not g m bearing rigid: R ( p ) � ∂f ( p ) ∈ R dm × dn . ∂p (a) (b) (c) (d) d f ( p ) = R ( p ) dp Trivial motions: translation and scaling 6 / 13
Construction of bearing rigid networks ⋄ Importance: construct sensor networks and formation ⋄ Need to design graph G and configuration p 7 / 13
Construction of bearing rigid networks ⋄ Importance: construct sensor networks and formation ⋄ Need to design graph G and configuration p ⋄ Graph VS configuration: 7 / 13
Construction of bearing rigid networks ⋄ Importance: construct sensor networks and formation ⋄ Need to design graph G and configuration p ⋄ Graph VS configuration: ⋄ Intuitively, it seems configuration is not that important. Is it true? 7 / 13
Construction of bearing rigid networks Definition ( Generically Bearing Rigid Graphs ) A graph G is generically bearing rigid in R d if there exists at least one configuration p in R d such that ( G , p ) is bearing rigid. Lemma ( Density of Generical Bearing Rigid Graphs ) If G is generically bearing rigid in R d , then ( G , p ) is bearing rigid for almost all p in R d in the sense that the set of p where ( G , p ) is not bearing rigid is of measure zero. Moreover, for any configuration p 0 and any small constant ǫ > 0 , there always exists a configuration p such that ( G , p ) is bearing rigid and � p − p 0 � < ǫ . Summary: • If a graph is generically bearing rigid, then for any almost all configurations the corresponding network is bearing rigid. • If a graph is not generically bearing rigid, by definition for any configuration the corresponding network is not bearing rigid. 8 / 13
Construction of bearing rigid networks Definition ( Generically Bearing Rigid Graphs ) A graph G is generically bearing rigid in R d if there exists at least one configuration p in R d such that ( G , p ) is bearing rigid. Lemma ( Density of Generical Bearing Rigid Graphs ) If G is generically bearing rigid in R d , then ( G , p ) is bearing rigid for almost all p in R d in the sense that the set of p where ( G , p ) is not bearing rigid is of measure zero. Moreover, for any configuration p 0 and any small constant ǫ > 0 , there always exists a configuration p such that ( G , p ) is bearing rigid and � p − p 0 � < ǫ . Summary: • If a graph is generically bearing rigid, then for any almost all configurations the corresponding network is bearing rigid. • If a graph is not generically bearing rigid, by definition for any configuration the corresponding network is not bearing rigid. 8 / 13
Construction of bearing rigid networks Definition ( Generically Bearing Rigid Graphs ) A graph G is generically bearing rigid in R d if there exists at least one configuration p in R d such that ( G , p ) is bearing rigid. Lemma ( Density of Generical Bearing Rigid Graphs ) If G is generically bearing rigid in R d , then ( G , p ) is bearing rigid for almost all p in R d in the sense that the set of p where ( G , p ) is not bearing rigid is of measure zero. Moreover, for any configuration p 0 and any small constant ǫ > 0 , there always exists a configuration p such that ( G , p ) is bearing rigid and � p − p 0 � < ǫ . Summary: • If a graph is generically bearing rigid, then for any almost all configurations the corresponding network is bearing rigid. • If a graph is not generically bearing rigid, by definition for any configuration the corresponding network is not bearing rigid. 8 / 13
Recommend
More recommend
