Conditions For Localizability Research Articles

Understanding Node Localizability in Barycentric Linear Localization

The barycentric linear localization (BLL) methods provide a lightweight, distributed way to calculate locations for resource-limited IoT devices. A crucial requirement for BLL is that the nodes participating in the iterative location propagation are localizable. Otherwise, the unlocalizable nodes will continuously pose error information in the location propagation process, making even the theoretically localizable nodes converge to the wrong locations. However, the research on node localizability in BLL is much lacked, greatly limiting the application scope of BLL. In specific, BLL node localizability is detected on a generated graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> . For any node, its neighbors appear in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> only when the neighbors can form triangle(s), so that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{G^A}$</tex-math> </inline-formula> is much sparser than the original <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal G$</tex-math> </inline-formula> . Thus, the node localizability condition in BLL is harder to be satisfied than that in traditional localization methods. Moreover, the distributed algorithm to detect BLL localizable nodes is still open. This paper thoroughly investigates the node localizability conditions and distributed localizable node detection algorithms in BLL. At first, an efficient and fully distributed Negative Edge Inference (NEI) algorithm is proposed for each node to infer implicit edges in its neighborhood. NEI strengthens the distance graph by revealing more distance constraints so that enables more neighboring triangles. Then a new sufficient condition, i.e., the recursive three disjoint path condition (Recursive-3DP) on the strengthened distance graph is proposed to identify BLL localizable nodes much more accurately. Secondly, a distributed Path Extension and Pruning (PEP) algorithm is proposed for distributed localizable node detection. PEP is proved to detect all the theoretically Recursive-3DP nodes in the strengthened distance graph. A Fast-PEP algorithm is further proposed, which misses very limited Recursive-3DP nodes while bringing significant improvement in efficiency. PEP and Fast-PEP guarantee to identify BLL localizable nodes in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2H$</tex-math> </inline-formula> rounds, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$</tex-math> </inline-formula> is the maximum hop number of the node disjoint paths. Finally, by using NEI and PEP (Fast-PEP), a localizability-aware BLL (LABEL) method is proposed, which correctly identifies localizable nodes and guarantees their correct location convergence. Extensive analysis and experiments show the advantages in localizability and location accuracy of the proposed schemes over the state-of-the-art methods.

A cluster positioning architecture and relative positioning algorithm based on pigeon flock bionics

Unmanned clusters can realize collaborative work, flexible configuration, and efficient operation, which has become an important development trend of unmanned platforms. Cluster positioning is important for ensuring the normal operation of unmanned clusters. The existing solutions have some problems such as requiring external system assistance, high system complexity, poor architecture scalability, and accumulation of positioning errors over time. Without the aid of the information outside the cluster, we plan to construct the relative position relationship with north alignment to adopt formation control and achieve robust cluster relative positioning. Based on the idea of bionics, this paper proposes a cluster robust hierarchical positioning architecture by analyzing the autonomous behavior of pigeon flocks. We divide the clusters into follower clusters, core clusters, and leader nodes, which can realize flexible networking and cluster expansion. Aiming at the core cluster that is the most critical to relative positioning in the architecture, we propose a cluster relative positioning algorithm based on spatiotemporal correlation information. With the design idea of low cost and large-scale application, the algorithm uses intra-cluster ranging and the inertial navigation motion vector to construct the positioning equation and solves it through the Multidimensional Scaling (MDS) and Multiple Objective Particle Swarm Optimization (MOPSO) algorithms. The cluster formation is abstracted as a mixed direction-distance graph and the graph rigidity theory is used to analyze localizability conditions of the algorithm. We designed the cluster positioning simulation software and conducted localizability tests and positioning accuracy tests in different scenarios. Compared with the relative positioning algorithm based on Extended Kalman Filter (EKF), the algorithm proposed in this paper has more relaxed positioning conditions and can adapt to a variety of scenarios. It also has higher relative positioning accuracy, and the error does not accumulate over time.

Triangular angle rigidity for distributed localization in 2D

Recent advances in sensing technology have enabled sensor nodes to measure interior angles with respect to their neighboring nodes. However, it is unknown which combination of angle measurements is necessary to make a sensor network localizable, and it is also unidentified if there is a distributed localization algorithm whose required communication only consists of the sensor nodes’ measured angles and estimated positions. Motivated by these two challenging problems, this paper develops triangular angle rigidity for those networks consisting of a set of nodes and triangular angle constraints in 2D. First, we transfer the geometric constraint of each triangle into an angle-induced linear constraint. Based on the linear constraint, we show that different from angle rigidity, triangular angle rigidity implies global triangular angle rigidity. More importantly, inspired by Laman’s theorem, we propose a topological, necessary and sufficient condition to check generic triangular angle rigidity. Based on the results on triangular angle rigidity, both algebraic and topological localizability conditions are developed, which are necessary and sufficient when the number of anchor nodes in the network is two. Both continuous and discrete localization algorithms are proposed, in which only measured angles and estimated positions are communicated among the sensor nodes. Finally, a simulation example with 32 sensor nodes is used to validate the effectiveness of the proposed approaches.

3-D Network Localization Using Angle Measurements and Reduced Communication

Other than the relative position-based and bearing-based localization with aligned coordinate frames on all sensor nodes, the existing 3D network localization algorithms using distance/bearing/angle measurements without aligned coordinate frames usually require each node to have at least four neighbors, which poses large communication burden for the network system. To reduce the communication burden in each iteration, this paper studies the 3D network localization problem where each node is allowed to have angle measurements with respect to only <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">three</i> neighbors. Firstly, we use a tetrahedral angularity to describe the network consisting of a set of nodes and tetrahedra among the nodes. Given that at least one node in each tetrahedron has the knowledge of the direction of the global <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Z$</tex-math></inline-formula> -axis, the geometric constraint of each tetrahedron is described by an angle-induced linear constraint whose coefficient matrices are only related to the interior angles among the four nodes in the tetrahedron and the angles between the global Z-axis and the inter-node edges. Secondly, both algebraic and topological localizability conditions are derived based on the coefficient matrices of the tetrahedra’s angle-induced linear constraints. Moreover, a distributed method is also presented to check the network localizability. Lastly, distributed localization laws are designed under the cases of continuous communication, aperiodic communication, and aperiodic communication on jointly localizable angularities, respectively. Simulation examples are provided to validate the effectiveness and advantages of the proposed localization laws.

Analysis of Noisy Bearing-Only Network Localization

Graph theory has been used to characterize the solvability of the sensor network localization problem with ideal (i.e., precisely known) bearing-only measurements between certain pairs of sensors and a limited amount of information about the position of certain nodes, i.e., anchors. In practice, however, bearing measurements will never be exact, and the equations whose solutions deliver sensor positions in the noiseless case may no longer have a solution. This technical brief argues that if the same conditions for localizability that exist in the noiseless case are satisfied and the bearing measurement errors are small enough (as will be formalized later in the technical brief), then the network will be approximately localizable, i.e., sensor position estimates can be found which are near the correct values. In particular, a bound on the position errors is found in terms of a bound on the bearing errors. Later, this bound is used to propose a method to select anchors to minimize the effect of noisy bearing measurements on the localization solution.

Multiple classical limits in relativistic and nonrelativistic quantum mechanics

The existence of a classical limit describing the interacting particles in a second-quantized theory of identical particles with bosonic symmetry is proved. This limit exists in addition to the previously established classical limit with a classical field behavior, showing that the limit ℏ→0 of the theory is not unique. An analogous result is valid for a free massive scalar field: two distinct classical limits are proved to exist, describing a system of particles or a classical field. The introduction of local operators in order to represent kinematical properties of interest is shown to break the permutation symmetry under some localizability conditions, allowing the study of individual particle properties.

Friedrich Hund and Chemistry

AbstractFriedrich Hund, one of the few physicists still alive of the generation that pioneered quantum mechanics, celebrates his 100th birthday this year. This is an occasion to recall how much chemistry owes to him. The important contributions of Hund to theoretical chemistry will not be presented in a primarily historical context—the cooperation of Hund and Mulliken during the development of MO theory will, however, be duly discussed—but the stress will be on the impact of the concepts introduced by Hund on theoretical chemistry today. Nearly all methods of quantum chemical calculations presently in use have a first step based on Hund's MO theory. Hund recognized that use of delocalized molecular orbitals does not necesarily conflict with a description of a molecule in terms of localized bonds. Moreover, he formulated conditions for localizability that are still pertinent today. The classification of the spectra of the spectra diatomic molecules by Hund in 1925 was virtually definitive. Recently Hund's rules, which, among other things, make predictions about the energetic ordering of states with different spin multiplicities within the same electronic configuration, have aroused renewed interest. The concept of atomic states of natural and unnatural parity plays a key role in this context. Hund's first tow rules for atomic states have been found to be special cases of more general rules. Molecules can violate Hund's first rule though spin polarization and though the preference of different equilibrium geometries of the singlet and triplet states.

Graph-theoretic approach to controllability and localizability of decentralized control systems

The controllability and the localizability problems are considered under the decentralized information structure using some concepts from graph theory. First of all, the information structure graph is introduced for a decentralized control system based upon local output controllability, matrices between stations when a local output feedback law is applied, and some invariant properties of the graph with respect to local output feedback laws are driven. Then by using these results, necessary and sufficient conditions of controllability under decentralized information structure are obtained. Also obtained are several conditions for localizability, of decentralized control systems, which is newly introduced in the present paper as an extended concept of a class of disturbance localization and decoupling for centralized systems.