On affine rigidity

We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is a generic property of the hypergraph.Then we prove that if a graph is is $(d+1)$-vertex-connected, then it must be "generically neighborhood affinely rigid" in $d$-dimensional space. This implies that if a graph is $(d+1)$-vertex-connected then any generic framework of its squared graph must be universally rigid. Our results, and affine rigidity more generally, have natural applications in point registration and localization, as well as connections to manifold learning.Comment: Updated abstrac

A Theoretical Analysis of the Conditions for Unambiguous Node Localization in Sensor Networks

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring distances or bearings to their neighbors. Distance information is the separation between two nodes connected by a sensing/communication link. Bearing is the angle between a sensing/communication link and the x-axis of a node's local coordinate system. We construct grounded graphs to model network localization and apply graph rigidity theory and parallel drawings to test the conditions for unique localizability and to construct uniquely localizable networks. We further investigate partially localizable networks

A Theoretical Analysis of the Conditions for Unambiguous Node Localization in Sensor Networks

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring distances or bearings to their neighbors. Distance information is the separation between two nodes connected by a sensing/communication link. Bearing is the angle between a sensing/communication link and the x-axis of a node's local coordinate system. We construct grounded graphs to model network localization and apply graph rigidity theory and parallel drawings to test the conditions for unique localizability and to construct uniquely localizable networks. We further investigate partially localizable networks

Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age

Simultaneous Localization and Mapping (SLAM)consists in the concurrent construction of a model of the environment (the map), and the estimation of the state of the robot moving within it. The SLAM community has made astonishing progress over the last 30 years, enabling large-scale real-world applications, and witnessing a steady transition of this technology to industry. We survey the current state of SLAM. We start by presenting what is now the de-facto standard formulation for SLAM. We then review related work, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers. This paper simultaneously serves as a position paper and tutorial to those who are users of SLAM. By looking at the published research with a critical eye, we delineate open challenges and new research issues, that still deserve careful scientific investigation. The paper also contains the authors' take on two questions that often animate discussions during robotics conferences: Do robots need SLAM? and Is SLAM solved

An Algorithmic Approach to Wireless Sensor Networks Localization Using Rigid Graphs

In this work estimating the position coordinates of Wireless Sensor Network nodes using the concept of rigid graphs is carried out in detail. The range based localization approaches use the distance information measured by the RSSI, which is prone to noise, due to effects of path loss, shadowing, and so forth. In this work, both the distance and the bearing information are used for localization using the trilateration technique. Rigid graph theory is employed to analyze the localizability, that is, whether the nodes of the WSN are uniquely localized. The WSN graph is divided into rigid patches by varying appropriately the communication power range of the WSN nodes and then localizing the patches by trilateration. The main advantage of localizing the network using rigid graph approach is that it overcomes the effect of noisy perturbed distance. Our approach gives a better performance compared to robust quads in terms of percentage of localizable nodes and computational complexity

An Algorithmic Approach to Wireless Sensor Networks Localization Using Rigid Graphs

In this work estimating the position coordinates of Wireless Sensor Network nodes using the concept of rigid graphs is carried out in detail. The range based localization approaches use the distance information measured by the RSSI, which is prone to noise, due to effects of path loss, shadowing, and so forth. In this work, both the distance and the bearing information are used for localization using the trilateration technique. Rigid graph theory is employed to analyze the localizability, that is, whether the nodes of the WSN are uniquely localized. The WSN graph is divided into rigid patches by varying appropriately the communication power range of the WSN nodes and then localizing the patches by trilateration. The main advantage of localizing the network using rigid graph approach is that it overcomes the effect of noisy perturbed distance. Our approach gives a better performance compared to robust quads in terms of percentage of localizable nodes and computational complexity