Bearing Rigidity Theory and its Applications for Control and Estimation of Network Systems Shiyu Zhao Department of Automatic Control and Systems Engineering University of Sheffield, UK Nanyang Technological University, Jan 2018
My research interests Networked unmanned aerial vehicle (UAV) systems • Low-level: guidance, navigation, and flight control of single UAVs • High-level: air traffic control, distributed control and estimation over multiple UAVs • Application of vision sensing: vision-based guidance, navigation, and coordination control 1 / 36
Research motivation Vision-based formation control of UAVs Two problems: formation control and vision sensing 1) formation control: (a) Initial formation (b) trajectory (c) Final Mature, require relative position measurements 2 / 36
Research motivation 2) vision sensing Step 1: recognition and tracking x 3D Point y z Camera frame Image Plane Step 2: position estimation from bearings Challenge: it is difficult to obtain accurate distance or relative position 3 / 36
Research motivation ⋄ Idea: formation control merely using bearing-only measurements ⋄ Advantages: In practice, reduce the complexity of vision system. In theory, prove that distance information is redundant. ⋄ Challenges: • Nonlinear system (linear if position feedback is available) • A relatively new topic that had not been studied ⋄ The focus of this talk: bearing-only formation control and related topics ⋄ Vision sensing: ongoing research 4 / 36
Outline 1 Bearing rigidity theory 2 Bearing-only formation control 3 Bearing-based network localization 4 Bearing-based formation control 5 Formation control with motion constraints 6 Affine formation maneuver control 5 / 36
Outline 1 Bearing rigidity theory 2 Bearing-only formation control 3 Bearing-based network localization 4 Bearing-based formation control 5 Formation control with motion constraints 6 Affine formation maneuver control 6 / 36
Bearing rigidity theory - Motivation With bearing feedback, we control inter-agent bearings [1,0] 2 (-1,1) (1,1) [1,-1] 2 2 [0,1] 2 2 [0,1] (-1,-1) (1,-1) 2 [1,0] Question: when bearings can determine a unique formation shape? (a) Desired (b) Resut (a) Desired (b) Result formation formation 7 / 36
Bearing rigidity theory - Necessary notations ⋄ 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 vector: p j − p i g ij = ∀ ( i, j ) ∈ E . � p j − p i � ⋄ An orthogonal projection matrix: P g ij = I d − g ij g T ij , x y P x y • 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) 8 / 36
Bearing rigidity theory - Two problems Two problems in the bearing rigidity theory • How to determine the bearing rigidity of a given network? • How to construct a bearing rigid network from scratch? (a) (b) (c) (d) ⋄ Definition of bearing rigidity: shape can be uniquely determined by bearings ⋄ Mathematical tool 1: bearing rigidity matrix ⋄ Mathematical tool 2: bearing Laplacian matrix 9 / 36
Bearing rigidity theory - Bearing rigidity matrix ⋄ Mathematical tool 1: bearing rigidity matrix Examples: g 1 . ∈ R dm f ( p ) � . . g m R ( p ) � ∂f ( p ) (a) (b) (c) (d) ∈ R dm × dn ∂p d f ( p ) = R ( p ) dp Trivial motions: translation and scaling (a) (b) (c) (d) Condition for Bearing Rigidity A network is bearing rigid if and only if rank( R ) = dn − d − 1 . Reference: S. Zhao and D. Zelazo, “Bearing rigidity and almost global bearing-only formation stabilization,”, IEEE Transactions on Automatic Control , vol. 61, no. 5, pp. 1255-1268, 2016. 10 / 36
Bearing rigidity theory - Bearing Laplacian matrix ⋄ Mathematical tool 2: bearing Laplacian matrix ⋄ 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 Condition for Bearing Rigidity A network is bearing rigid if and only if rank( B ) = dn − d − 1 . Reference: S. Zhao and D. Zelazo, ”Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions,” Automatica, vol. 69, pp. 334-341, 2016. 11 / 36
Bearing rigidity theory - Construction of networks Construction of bearing rigid networks Definition ( Laman Graphs ) A graph G = ( V , E ) is Laman if |E| = 2 |V| − 3 and every subset of k ≥ 2 vertices spans at most 2 k − 3 edges. ⋄ Why consider Laman graphs: (i) favorable since edges distribute evenly in a Laman graph; (ii) widely used in, for example, distance rigidity; (iii) can be constructed by Henneberg Construction. Definition ( Henneberg Construction ) Given a graph G = ( V , E ) , a new graph G ′ = ( V ′ , E ′ ) is formed by adding a new vertex v to G and performing one of the following two operations: (a) Vertex addition : connect vertex v to any two existing vertices i, j ∈ V . In this case, V ′ = V ∪ { v } and E ′ = E ∪ { ( v, i ) , ( v, j ) } . (b) Edge splitting : consider three vertices i, j, k ∈ V with ( i, j ) ∈ E and connect vertex v to i, j, k and delete ( i, j ) . In this case, V ′ = V ∪ { v } and E ′ = E ∪ { ( v, i ) , ( v, j ) , ( v, k ) } \ { ( i, j ) } . 12 / 36
Bearing rigidity theory - Construction of networks Two operations in Henneberg construction: v v i i k j j G G (a) Vertex addition (b) Edge splitting Main Result: Laman graphs are generically bearing rigid in arbitrary dimensions. 3 4 3 4 3 4 3 3 4 3 4 1 5 1 5 1 5 1 1 1 5 2 2 8 2 2 2 2 6 7 6 7 6 Step 1: vertex Step 2: edge Step 3: edge Step 4: edge Step 5: edge Step 6: edge addition splitting splitting splitting splitting splitting Reference: S. Zhao, Z. Sun, D. Zelazo, M. H. Trinh, and H.-S. Ahn, “Laman graphs are generically bearing rigid in arbitrary dimensions,” in Proceedings of the 56th IEEE Conference on Decision and Control, (Melbourne, Australia), December 2017. accepted 13 / 36
Outline 1 Bearing rigidity theory 2 Bearing-only formation control 3 Bearing-based network localization 4 Bearing-based formation control 5 Formation control with motion constraints 6 Affine formation maneuver control 14 / 36
Bearing-only formation control - Control law Nonlinear bearing-only formation control law � P g ij ( t ) g ∗ p i ( t ) = − ˙ ij , i = 1 , . . . , n j ∈N i • p i ( t ) : position of agent i • P g ij ( t ) = I d − g ij ( t )( g ij ( t )) T • g ij ( t ) : bearing between agents i and j at time t • g ∗ ij : desired bearing between agents i and j p j 2 g ij − P g ij g ∗ 1 2 ij g ∗ ij p i 1 P g ij g ∗ ij Figure: The simplest simulation example. Figure: The geometric interpretation of the Show video 15 / 36 control law.
Bearing-only formation control - Stability analysis Centroid and Scale Invariance Almost global convergence • Centroid of the formation • Two isolated equilibriums: one stable, one unstable n p � 1 � ¯ p i n 2 i =1 3 1 • Scale of the formation 1 3 � n � � 1 � � 2 s � p � 2 . � p i − ¯ n Figure: The solid one is the target formation. i =1 Reference: S. Zhao and D. Zelazo, “Bearing rigidity and almost global bearing-only formation stabilization,” IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1255-1268, 2016. (a) Initial (b) trajectory (c) Final formation 16 / 36
Outline 1 Bearing rigidity theory 2 Bearing-only formation control 3 Bearing-based network localization 4 Bearing-based formation control 5 Formation control with motion constraints 6 Affine formation maneuver control 17 / 36
Bearing-based network localization Distributed network localization Given the inter-node bearings and some anchors, how to localize the network? 10 5 z (m) 0 -5 -10 10 -10 0 0 y (m) x (m) 10 -10 Two key problems • Localizability: whether a network can be possibly localized? • Localization algorithm: if a network can be localized, how to localize it? 18 / 36
Bearing-based network localization - Localizability Not all networks are localizable: From bearing rigidity to network localizability: Add anchor/leader (a) infinitesimally (b) localizable bearing rigid Observations: • bearing rigidity + two anchors = ⇒ localizability • bearing rigidity is sufficient but not necessary for localizability (a) (b) 19 / 36
Bearing-based network localization - Localizability Bearing Laplacian : B ∈ R dn × dn and the ij th subblock matrix of B is � j ∈N i P g ij , i ∈ V . [ B ] ij = − P g ij , i � = j, ( i, j ) ∈ E , 0 d × d , i � = j, ( i, j ) / ∈ E , Bearing Laplacian is a matrix-weighted Laplacian matrix . The bearing Laplacian B can be partitioned into � B aa � B af B = B fa B ff Necessary and sufficient condition A network is localizable if and only if B ff is nonsingular Examples: (a) (b) (c) (d) (e) 20 / 36
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