We consider the problem of localizing wireless devices in an ad-hoc network embedded in a d-dimensional Euclidean space. Obtaining a good estimation of where wireless devices are located is crucial in wireless network applications including environment monitoring, geographic routing and topology control. When the positions of the devices are unknown and only local distance information is given, we need to infer the positions from these local distance measurements. This problem is particularly challenging when we only have access to measurements that have limited accuracy and are incomplete. We consider the extreme case of this limitation on the available information, namely only the connectivity information is available, i.e., we only know whether a pair of nodes is within a fixed detection range of each other or not, and no information is known about how far apart they are. Further, to account for detection failures, we assume that even if a pair of devices is within the detection range, it fails to detect the presence of one another with some probability and this probability of failure depends on how far apart those devices are. Given this limited information, we investigate the performance of a centralized positioning algorithm MDS-MAP introduced by , and a distributed positioning algorithm, introduced by , called HOP-TERRAIN. In particular, for a network consisting of n devices positioned randomly, we provide a bound on the resulting error for both algorithms. We show that the error is bounded, decreasing at a rate that is proportional to R Rc, where Rc is the critical detection range when the resulting random network starts to be connected, and R is the detection range of each device.
Perhaps a more practical and interesting case is when there is no central infrastructure. @cite_34 identifies a common three-phase structure of three, popular, distributed sensor-localization algorithms, namely robust positioning @cite_8 , ad-hoc positioning @cite_13 and N-hop multilateration @cite_7 . Table illustrates the structure of these algorithms. In the first phase, nodes share information to collectively determine the distances from each of the nodes to a number of anchors. Anchors are special nodes with a priori knowledge of their own position in some global coordinate system. In the second phase, nodes determine their position based on the estimated distances to the anchors provided by the first phase and the known positions of the anchors. In the last phase, the initial estimated positions are iteratively refined. It is empirically demonstrated that these simple three-phase distributed sensor-localization algorithms are robust and energy-efficient @cite_34 . However, depending on which method is used in each phase, there are different tradeoffs between localization accuracy, computation complexity and power requirements. In @cite_39 , a distributed algorithm-called the Gradient algorithm- was proposed; it is similar to ad-hoc positioning @cite_13 but uses a different method for estimating the average distance per hop.
@cite_7 @cite_8 @cite_39 @cite_34 @cite_13
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This paper studies the problem of determining the node locations in ad-hoc sensor networks. We compare three distributed localization algorithms (Ad-hoc positioning, Robust positioning, and N-hop multilateration) on a single simulation platform. The algorithms share a common, three-phase structure: (1) determine node-anchor distances, (2) compute node positions, and (3) optionally refine the positions through an iterative procedure. We present a detailed analysis comparing the various alternatives for each phase, as well as a head-to-head comparison of the complete algorithms. The main conclusion is that no single algorithm performs best; which algorithm is to be preferred depends on the conditions (range errors, connectivity, anchor fraction, etc.). In each case, however, there is significant room for improving accuracy and or increasing coverage. The localization of sensor nodes is a fundamental problem in sensor networks and can be implemented using powerful and expensive beacons. Beacons, the fewer the better, can acquire their position knowledge either from GPS devices or by virtue of being manually placed. In this paper, we propose a distributed method to localization of sensor nodes using a single moving beacon, where sensor nodes compute their position estimate based on the range-free technique. Two parameters are critical to the location accuracy of sensor nodes: the radio transmission range of the beacon and how often the beacon broadcasts its position. Theoretical analysis shows that these two parameters determine the upper bound of the estimation error when the traverse route of the beacon is a straight line. We extend the position estimate when the traverse route of the beacon is randomly chosen in a real-world situation, where the radio irregularity might cause a node to miss some crucial coordinate information from the beacon. We further point out that the movement pattern of the beacon plays a pivotal role in the localization task for sensors. To minimize estimation errors, sensor nodes can carry out a variety of algorithms in accordance with the movement of the beacon. Simulation results compare variants of the distributed method in a variety of testing environments. Real experiments show that the proposed method is feasible and can estimate the location of sensor nodes accurately, given a single moving beacon. This paper describes a distributed, linear-time algorithm for localizing sensor network nodes in the presence of range measurement noise and demonstrates the algorithm on a physical network. We introduce the probabilistic notion of robust quadrilaterals as a way to avoid flip ambiguities that otherwise corrupt localization computations. We formulate the localization problem as a two-dimensional graph realization problem: given a planar graph with approximately known edge lengths, recover the Euclidean position of each vertex up to a global rotation and translation. This formulation is applicable to the localization of sensor networks in which each node can estimate the distance to each of its neighbors, but no absolute position reference such as GPS or fixed anchor nodes is available. We implemented the algorithm on a physical sensor network and empirically assessed its accuracy and performance. Also, in simulation, we demonstrate that the algorithm scales to large networks and handles real-world deployment geometries. Finally, we show how the algorithm supports localization of mobile nodes. This paper focuses on the principled study of distance reconstruction for distance-based node localization. We address an important issue in node localization by showing that a highly incomplete set of inter-node distance measurements obtained in ad-hoc node deployments carries sufficient information for the accurate reconstruction of the missing distances, even in the presence of noise and sensor node failures. We provide an efficient and provably accurate algorithm for this reconstruction, and we show that the resulting error is bounded, decreasing at a rate that is inversely proportional to radicn, the square root of the number of nodes in the region of deployment. Although this result is applicable to many localization schemes, in this paper we illustrate its use in conjunction with the popular multidimensional scaling algorithm. Our analysis reveals valuable insights and key factors to consider during the sensor network setup phase, to improve the quality of the position estimates