A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions. It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid. In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements. Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation. It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations. Numerical simulations are provided to support the analysis.Comment: Accepted as a full paper by IEEE Transactions on Automatic Control. This is the final version before the official publication by IEE
Abstract-A multi-agent formation control task usually consists of two subtasks. The first is to steer the agents to form a desired geometric pattern and the second is to achieve desired collective maneuvers so that the centroid, orientation, scale, and other geometric parameters of the formation can be changed continuously. This paper proposes a novel affine formation maneuver control approach to achieve the two subtasks simultaneously. The proposed approach relies on stress matrices, which can be viewed as generalized graph Laplacian matrices with both positive and negative edge weights. The proposed control laws can track any target formation that is a time-varying affine transformation of a nominal configuration. The centroid, orientation, scales in different directions, and even geometric pattern of the formation can all be changed continuously. The desired formation maneuvers are only known by a small number of agents called leaders, and the rest agents called followers only need to follow the leaders. The proposed control laws are globally stable and do not require global reference frames if the required measurements can be measured in each agent's local reference frame.
This paper addresses the problem of bearing-based network localization, which aims to localize all the nodes in a static network given the locations of a subset of nodes termed anchors and inter-node bearings measured in a common reference frame. The contributions of the paper are twofold. Firstly, we propose necessary and sufficient conditions for network localizability with both algebraic and rigidity theoretic interpretations. The analysis of the localizability heavily relies on the recently developed bearing rigidity theory and a special matrix termed the bearing Laplacian. Secondly, we propose a linear distributed protocol for bearing-based network localization. The protocol can globally localize a network if and only if the network is localizable. The sensitivity of the protocol to constant measurement errors is also analyzed. One novelty of this work is that the localizability analysis and localization protocol are applicable to networks in arbitrary dimensional spaces.Distributed localization of sensor networks is a core problem in many multi-agent coordination tasks. Network localizability and distributed protocols are two fundamental problems for any network localization problems. Network localizability characterizes whether or not a network can be possibly localized given the anchor locations and interneighbor relative measurements, whereas distributed protocols are used for localizing the network in a distributed manner if the network is localizable. According to the types of the relative measurements used for localization, the existing works can be divided into three classes: distance-based, bearing-based, and position-based. Distance-based network localization has been studied extensively so far (see [1][2][3][4] and the references therein). The analysis of the localizability in distance-based network localization relies heavily on the distance rigidity theory. It has been shown that a network in an n-dimensional space can be uniquely localized if the network is globally rigid and has at least n + 1 anchors in a general position [1]. More recently, bearing-based network localization has also attracted extensive research attention [5][6][7][8][9][10][11]. The analysis of the localizability in bearing-based network localization relies on the analogous bearing rigidity theory [12][13][14][15]. Finally, position-based network localization, where the inter-neighbor distance and local bearing measurements are used together for network localization, has been studied in [16] by using a complex graph Laplacian.Although bearing-based network localization has been studied by many researchers, the two fundamental problems, network localizability and distributed protocols, have not yet been fully explored. It was shown in [7-10] that a network is localizable when the network is bearing rigid and has at least two anchors. This condition is, however, sufficient but not necessary when the number of anchors is greater than two [10, Cor 10]. A necessary and sufficient condition for network localizability was proposed...
This paper analytically characterizes optimal sensor placements for target localization and tracking in 2D and 3D. Three types of sensors are considered: bearing-only, range-only, and received-signal-strength. The optimal placement problems of the three sensor types are formulated as an identical parameter optimization problem and consequently analyzed in a unified framework.Recently developed frame theory is applied to the optimality analysis. We prove necessary and sufficient conditions for optimal placements in 2D and 3D. A number of important analytical properties of optimal placements are further explored. In order to verify the analytical analysis, we present a gradient control law that can numerically construct generic optimal placements.
Abstract-This paper studies distributed maneuver control of multi-agent formations in arbitrary dimensions. The objective is to control the translation and scale of the formation while maintaining the desired formation pattern. Unlike conventional approaches where the target formation is defined by relative positions or distances, we propose a novel bearing-based approach where the target formation is defined by inter-neighbor bearings. Since the bearings are invariant to the translation and scale of the formation, the bearing-based approach provides a simple solution to the problem of translational and scaling formation maneuver control. Linear formation control laws for double-integrator dynamics are proposed and the global formation stability is analyzed. This paper also studies bearing-based formation control in the presence of practical problems including input disturbances, acceleration saturation, and collision avoidance. The theoretical results are illustrated with numerical simulations.
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