Angle-based shape determination theory of planar graphs with application to formation stabilization

Jing, Gangshan; Zhang, Guofeng; Lee, Heung Wing Joseph; Wang, Long

doi:10.1016/j.automatica.2019.03.026

Cited by 68 publications

(108 citation statements)

References 43 publications

(121 reference statements)

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“…In [19], angle rigidity theory is proposed to study what kind of geometric shapes can be uniquely determined by angles subtended in the graph only. Similar to distance rigidity theory in range-based SNL and bearing rigidity theory in bearingbased SNL, angle rigidity theory plays an important role in solving ASNL.…”

Section: Preliminaries Of Graph Theory a Angle Rigidity Theorymentioning

confidence: 99%

“…As a result, different sensors are allowed to have different definitions about the rotational direction. In this subsection, we will briefly review several definitions regarding angle rigidity theory proposed in [19] that will be used later.…”

Section: Preliminaries Of Graph Theory a Angle Rigidity Theorymentioning

confidence: 99%

“…In bearing-based and range-based SNL, bearing rigidity theory [15] and distance rigidity theory [18] are employed to give conditions for localizability, respectively. In [19], the authors proposed an angle-based shape determination approach (namely, angle rigidity theory), where the number of edges required for shape determination is the same as that for infinitesimal bearing rigidity, which is necessary for localizability of bearing-based SNL. We notice that an angle between two edges joining one sensor can always be computed by bearings corresponding to these two edges measured in the local coordinate system of the sensor, hence is independent of the global coordinate frame.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we propose the concept of angle fixability based on angle rigidity theory in [19], [23] to characterize the property of a network that can be determined by angles uniquely up to translations, rotations, scalings and reflections. By establishing connections between angle fixability and angle localizability, the results on angle fixability are applied to ASNL problems.…”

Section: Introductionmentioning

confidence: 99%

“…A configuration p = (p 1 , · · · , p n ) ∈ R 2n is generic if its 2n coordinates are algebraically independent[19].…”

mentioning

confidence: 99%

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Angle Fixability and Angle-Based Sensor Network Localization

Jing

Wan

Dai

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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This paper studies angle-based sensor network localization (ASNL) in a plane, which is to determine locations of all sensors in a sensor network, given locations of partial sensors (called anchors) and angle constraints based on bearings measured in the local coordinate frame of each sensor. We firstly show that a framework with a non-degenerate bilateration ordering must be angle fixable, which implies that it can be uniquely determined by angles between edges up to translation, rotation, scaling, and reflection. Then we prove that an ASNL problem has a unique solution if and only if its grounded framework is angle fixable and anchors are not all collinear. Subsequently, ASNL is solved under centralized and distributed approaches, respectively. The noise-free centralized ASNL is formulated as a rank-constrained optimization problem, and shown to be equivalent to a linear semi-definite program (SDP) when the grounded framework is acute-triangulated. In order to deal with large and sparse ASNL, a decomposition strategy is proposed to recast the centralized ASNL as an SDP with semi-definite constraints on multiple submatrices in reduced sizes. The centralized ASNL considering noise is investigated via a maximum likelihood formulation, which can be formulated as a rank-constrained SDP with multiple semi-definite constraints. A distributed protocol based on inter-sensor communications is also proposed, which solves ASNL in finite time when the grounded framework has a non-degenerate bilateration ordering. Finally, simulation examples are provided to validate effectiveness of our theoretical analysis.

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Section: Preliminaries Of Graph Theory a Angle Rigidity Theorymentioning

confidence: 99%

Section: Preliminaries Of Graph Theory a Angle Rigidity Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…A configuration p = (p 1 , · · · , p n ) ∈ R 2n is generic if its 2n coordinates are algebraically independent[19].…”

mentioning

confidence: 99%

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Angle Fixability and Angle-Based Sensor Network Localization

Jing

Wan

Dai

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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Synthesis of a linear controller for a control system with interval parameters on a base of D‐partition in vertices of a coefficient's polyhedron of characteristic polynomial

Gayvoronskiy

Ezangina

2022

Asian Journal of Control

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The article deals with the technique developed for the parametric design of a robust controller which ensures the desired robust quality indicators in the system with interval parameters. This technique is based on the construction of the D‐partition in the space of controller parameters at vertices of the coefficient polyhedron of a characteristic polynomial. Grounds for this are the fact that the interval system has its minimum degree of stability and maximum degree of oscillation at certain vertices of the specified polyhedron. To determine the minimum sets of those vertices, a new technique has been developed. The developed technique is based on inequalities obtained for outlet angles of the edge branches coming out from system poles, which determine the root indices of robust quality. Applying the developed technique for the vertex D‐partition, the design of the PI controller of the stabilization system for a tethered underwater vehicle has been carried out. The results of the design confirm the efficiency of the developed approach.

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Learning‐based containment control in the absence of leaders' information

Zhao,

Fan,

Guan

2023

Engineering Reports

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This article considers the containment control problem of continuous‐time dynamic agents without leaders' state information. The leaders have first‐ and second‐order dynamics, respectively, while first‐order dynamics always govern the followers. Each agent has inherent nonlinear dynamics and can only measure the output information of its neighbors. The output of each leader is expressed as the product of an unknown coefficient and a position‐like state, while the output of each follower is equal to its position‐like state. To stabilize the position‐like states of the followers to the convex hull spanned by leaders, the unknown coefficients are asymptotically tracked by leveraging reinforcement learning based on the inherent dynamics and the output information. Two distributed learning‐based containment protocols are proposed, respectively. It is proved that if the directed communication topology has a spanning forest and certain conditions in terms of the inherent nonlinear dynamics are satisfied, then the proposed controllers with proper control gains solve the containment control problem asymptotically under arbitrary initial states. An exciting conclusion is that the learning algorithms' convergence rate plays an important role in achieving containment control. Numerical simulations are performed to validate the effectiveness of the obtained theoretical results.

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Angle-based shape determination theory of planar graphs with application to formation stabilization

Cited by 68 publications

References 43 publications

Angle Fixability and Angle-Based Sensor Network Localization

Angle Fixability and Angle-Based Sensor Network Localization

Synthesis of a linear controller for a control system with interval parameters on a base of D‐partition in vertices of a coefficient's polyhedron of characteristic polynomial

Learning‐based containment control in the absence of leaders' information

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