We present two algorithms for indoor positioning estimation in peer-to-peer networks. The setup is a network of two types of devices: reference devices with a known location and blindfolded devices that can determine distances to reference devices and each other. From this information the blindfolded devices try to estimate their positions. A typical scenario is navigation inside a shopping mall where devices in the parking lot can make contact with GPS satellites, whereas devices inside the building make contact with each other, devices on the parking lot, and devices fixed to the building. The devices can measure their in-between distances, with some measurement error, and exchange positioning information. However, other devices might only know their position with some error. We present two algorithms for positioning estimation in such a peer-to-peer network. The first one is purely geometric and is based on Euclidean geometry and intersecting spheres. We rewrite the information to a linear system, which is typically overdetermined. We use least squares to ??nd the best estimate for a device its position. The second approach can be considered as a probabilistic version of the geometric approach. We estimate the probability density function that a device is located at a position given a probability density function for the positions of the other devices in the network, and a probability density function of the measured distances. First we study the case with a distance measurement to a single other user, then we focus on multiple other users. We give an approximation algorithm that is the probabilistic analogue of the intersecting spheres method. We show some simulated results where ambiguous data lead to well defined probability distributions for the position of a device. We conclude with some open questions.
|Title of host publication||Proceedings of the 72nd European Study Group Mathematics with Industry (SWI 2010, Amsterdam, The Netherlands, January 25-29, 2010)|
|Place of Publication||Amsterdam|
|Publisher||Centrum voor Wiskunde en Informatica|
|Publication status||Published - 2010|