Mats Rydström

Robust Sensor Network Positioning Based on Projections onto Circular and Hyperbolic Convex Sets (POCS)

We consider the problem of locating a signal-source node, using characteristic signals emitted by the node that are captured by a set of sensor nodes. This estimation problem has often been formulated as a weighted least-squares problem in the literature. Received signal strength and asynchronous time-of-arrival measurements, however, give rise to objective functions with multiple local minima and saddle-points, complicating the optimization process. Recently, the method of projection onto convex sets (POCS) was suggested as a means to estimate source position, when received signal strength measurements are available. POCS has been shown to be robust to local minima in the objective function, is of low complexity, and is possible to distribute over the sensor nodes in the network. The drawback of POCS, when convex sets bounded by circles are used, is its poor performance in locating source nodes outside the outer perimeter of sensor nodes. We propose an extension to the presented POCS algorithm, called hyperbolic POCS, that increases the performance of circular POCS, allows for positioning of source nodes outside the outer perimeter of sensor nodes, and is applicable also for the case of asynchronous time-of-arrival measurements

Wireless network positioning as a convex feasibility problem

In this semi-tutorial paper, the positioning problem is formulated as a convex feasibility problem (CFP). To solve the CFP for non-cooperative networks, we consider the well-known projection onto convex sets (POCS) technique and study its properties for positioning. We also study outer-approximation (OA) methods to solve CFP problems. We then show how the POCS estimate can be upper bounded by solving a non-convex optimization problem. Moreover, we introduce two techniques based on OA and POCS to solve the CFP for cooperative networks and obtain two new distributed algorithms. Simulation results show that the proposed algorithms are robust against non-line-of-sight conditions.

Positioning algorithms for cooperative networks in the presence of an unknown turn-around time

This paper addresses the problem of single node positioning in cooperative network using hybrid two-way time-of-arrival and time-difference-of-arrival where, the turn-around time at the target node is unknown. Considering the turn-around time as a nuisance parameter, the derived maximum likelihood estimator (MLE) brings a difficult global optimization problem due to local minima in the cost function of the MLE. To avoid drawbacks in solving the MLE, we obtain a linear two-step estimator using non-linear pre-processing which is algebraic and closed-form in each step. To compare different methods, Cramér-Rao lower bound (CRLB) is derived. Simulation results confirm that the proposed linear estimator attains the CRLB for sufficiently high signal-to-noise ratios.

A distributed positioning algorithm for cooperative active and passive sensors

The problem of positioning a target node is studied for wireless sensor networks with cooperative active and passive sensors. Two-way time-of-arrival and time-difference-of-arrival measurements made by both active and passive nodes are used to estimate the position of the target node. A maximum likelihood estimator (MLE) can be employed to solve the problem. Due to the nonlinear nature of the cost function in the MLE, an iterative search might converge to local minima which often results in large estimation errors. To avoid this drawback, we instead formulate the problem of positioning as finding the intersection of a number of convex sets derived from measurements. To obtain this intersection, we apply the projection onto convex sets approach, which is robust and can be implemented in a distributed manner. Simulations are performed to compare the performance of the MLE and the proposed method.

Robust distributed positioning algorithms for cooperative networks

The problem of positioning targets based on distance estimates is studied for cooperative wireless sensor networks when there is limited a priori information about measurements noise. To solve this problem, two different methods of positioning are considered: statistical and geometrical. Based on a geometric interpretation, we show that the positioning problem can be rendered as finding the intersection of a number of convex sets. To find this intersection, we propose two different methods based on projection onto convex sets and outer-approximation. In the statistical approach, a partly novel two-step linear estimator is proposed which can be expressed in a closed-form solution. We also propose a new constrained non-linear least squares algorithm based on constraints derived in the outer-approximation approach. Simulation results show that the geometrical methods are more robust against non-line-of-sight measurements than the statistical approaches while in dense networks with line-of-sight measurements statistical approaches outperform geometrical methods.

An Algorithm for Positioning Relays and Point Scatterers in Wireless Systems

In this letter, we discuss the problem of positioning an unknown number of amplify-and-forward relays or point scatterers using a wireless network of wideband transceivers. We propose to model this positioning problem as an assignment problem. In doing so, we avoid the problems associated with an exhaustive gridsearch approach, which has recently been proposed to solve the problem. Numerical simulations show promising results in terms of performance and robustness.

Hybrid TW-TOA/TDOA Positioning Algorithms for Cooperative Wireless Networks

The problem of positioning an unknown target is studied for a cooperative wireless sensor network using hybrid two-way time-of-arrival and time-difference-of-arrival measurements. A maximum likelihood estimator (MLE) can be employed to solve the problem. Due to the non-linear nature of the cost function in the MLE, a numerical method, e.g., an iterative search algorithm with a good initial point, should be taken to accurately estimate the target. To avoid drawbacks in a numerical method, we instead linearize the measurements and obtain a new two-step estimator that has a closed-form solution in each step. Simulation results confirm that the proposed linear estimator can attain Cramer-Rao lower bound for sufficiently high SNR

Ultra-wide band sensor networks for tracking point scatterers or relays

In this paper we investigate the possible benefits and drawbacks of using a previously proposed “soft” ranging algorithm for tracking moving objects in indoor scenarios. The objects considered in this work could be either passive point scatterers or active relays. A number of ultra-wide band transceivers are deployed throughout the environment, and they are used to estimate multi-path propagation delays, that constitute the input data to the tracking algorithm. The motivation behind this work is that “soft” ranging algorithm enables both estimation of ranging error on a measurement-by-measurement basis, and also a straight-forward approach to multiple-hypotheses testing (MHT). Thus, this type of ranging algorithm is likely to fit well into a filter-bank tracking system. Computer simulations based on Ray-tracing propagation data indicates that the proposed approach enables both more robust and also more accurate positioning than when a more conventional threshold-based technique is used for ranging.

Positioning of Node Using Plane Projection onto Convex Sets

We deal with positioning of node in wireless sensor network (WSN) using received signal strength (RSS) when there is no priori knowledge about path-loss exponent and transmission power. Since the RSS decreases on the average with distance, it carries some information about the distance to an unknown node. By ordering the RSSs, we conclude that there are some convex sets where the position of the unknown node can be found in the intersection of them. We introduce a plane projection onto convex sets (PPOCS) approach to solve the positioning problem. Simulation results show good performance for the new methods compared to other reduced complexity algorithms.

Clock-offset cancellation methods for positioning in asynchronous sensor networks

For most applications of wireless sensor networks, knowledge about the position of sensors relative to other sensors in the network, or to some global coordinate system, is a key ingredient to successful operation of the network. Estimation of relative node coordinates, based on measured time-of-flight between neighbouring nodes, has been suggested as a means to provide position-awareness in sensor networks where satellite based systems are not available. However, to directly measure the inter-node distances, based on RF or ultra-sound propagation delay, requires the nodes to be tightly synchronized in time. This is an assumption that is not easily justified in sensor networks operating under complexity, latency, power consumption or bandwidth constraints. Joint ML estimation of clock-offsets and node coordinates has been suggested, but, although this approach shows great promise in terms of coordinate estimation accuracy, it does not scale well as sensor networks grow in size. In this paper, we present two linear preprocessing operations that cancels the effect of unknown clock-offsets from the estimation problem. For both operations, we show that the Fisher information on node coordinates in the original data-set remains unchanged after preprocessing. Novel ML estimators of relative node coordinates are proposed, that are of significantly lower complexity, while their performance equals that of the joint ML estimator.