Optimal RSS Threshold in Connectivity-Based Localization Schemes 115 $ Gianni Giorgetti Sandeep K.S. Gupta Gianfranco Manes ACM MSWiM - Vancouver October 28, 2008 IMPACT LAB http://impact.asu.edu Arizona State University 185 $ 300 $ 70 $
What is this about? Optimal RSS Threshold in Connectivity-Based Localization Schemes • Localization: the problem of locating devices and/or people • Localization based on proximity • We can reduce the error by optimal selection of one of the parameters
Remote Monitoring Applications Mesh sensor network Server Gateway (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ? (x, y) = ?
Why not to use GPS? Shopping List: Wireless Node - 115 $ x N Sensor Board - 185 $ x N NOT RELIABLE INDOORS GPS Board - 300 $ x N GPS Receiver - 70 $ x N Sometimes “good enough” is good enough
Collaborative Localization • Inputs: • A set of anchor nodes • In-network measurements • Output: • Node Coordinates d2 d1 d3 d4 d5 • RF-Based Approaches: • Scene analysis (Fingerprinting) • Range-Based (RSS, Interferometric) • Connectivity
Radio-Based, Range-Free Localization 3 HOPS What we like about connectivity: Easy to acquire Easy to communicate (binary value) Easy to process Reasonable accuracy 1 HOP 2 HOPS
Example – 49 node network Comm. Range = ~ 33 m Connectivity = 9 Avg. Error = 6 – 10 m (0.2 – 0.3 R)
Does it work indoors?
Why it doesn’t work… • Every node is in the radio range of every node • Nodes at different locations have the • same neighbor sets • Impossible to distinguish between • nodes at different locations • IDEA: TO REDUCE CONNECTIVITY • BY SETTING A TRESHOLD. • WHAT IS THE OPTIMAL VALUE?
1D Localization RSS1 RSS2 RSS3 RSS4 … 01010101 1 0 0 1 = -72 dBm Connectivity-Based Localization
Log-Normal Shadowing Model Path-Loss Exponent
Measurement Model Connectivity is a random variable Probability of detecting the nodes as “connected” Parameter Estimation Problem: We want to estimate d using observations C={0,1}. Is there a value Pth that will reduce the estimation error?
Fisher Information Fisher Information: measures the amount of information that a random variable carries about an unknown parameter Cramér-Rao bound: minimum theoretical estimation error 0 1 Large value of F Small Error Small value of F Large Error
What does F tell us? The available Fisher information: • Decreases with the distance • Decreases with the noise in the RSS data • Depends on how we set the threshold
-45 -50 -55 -60 -65 -70 -75 -80 -85 -90 -95 A toy problem There are two nodes (Node 1 and Node 2). You have to decide which one is closer using connectivity information. How do you set the threshold? 0 2 1 2 1 2 1 1 2 RSS [dBM] pdf
-45 -50 -55 -60 -65 -70 -75 -80 -85 -90 -95 Optimal Connectivity Threshold For a single device the optimal threshold is equal to the expected received power. (p = 0.5) 1 0 RSS [dBM] pdf
Network Localization • Fisher Information Matrix • Cramér-Rao Bound: Blind Nodes Anchors
CRB for 2D Network
CRB for 3D Network We cannot compute the CRB at runtime (it requires knowledge of the node positions) Using the CRB we can determine the optimal threshold
Optimal Connectivity • Setting the optimal threshold is equivalent to finding an optimal connectivity value. • Easier to deal with (it doesn’t depend on the hardware) • We investigated how this optimal connectivity value changes with different network parameters.
Invariance of the optimal conn. GOOD NEWS: The optimal connectivity doesn’t change with network scaling and with the propagation model parameters
Approximation and Simulations 2D • The Optimal Connectivity value increases with the network size. • We find a formula to approximate the optimal connectivity value
Case Studies Opt. Conn = 9.27 Opt. Conn = 11.1 (Pth = -53.3 dBm) (Pth = -34.3 dBm) http://www.eecs.umich.edu/~hero/localize/ http://www.eng.yale.edu/enalab/XYZ/data_set_1.htm Using the approximate formula we find:
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