1. 1 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks 17th Lecture 09.01.2007 Christian Schindelhauer

2. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-2 Goals of this chapter  Means for a node to determine its physical position (with respect to some coordinate system) or symbolic location  Using the help of –Anchor nodes that know their position –Directly adjacent –Over multiple hops  Using different means to determine distances/angles locally

3. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-3 Overview  Basic approaches  Trilateration  Multihop schemes

4. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-4 Trilateration  Assuming distances to three points with known location are exactly given  Solve system of equations (Pythagoras!) –(x i,y i ) : coordinates of anchor point i, r i distance to anchor i –(x u, y u ) : unknown coordinates of node –Subtracting eq. 3 from 1 & 2: –Rearranging terms gives a linear equation in (x u, y u )!

5. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-5 Trilateration as matrix equation  Rewriting as a matrix equation:  Example: (x 1, y 1 ) = (2,1), (x 2, y 2 ) = (5,4), (x 3, y 3 ) = (8,2), r 1 = 10 0.5, r 2 = 2, r 3 = 3 ! (x u,y u ) = (5,2)

6. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-6 Trilateration with distance errors  What if only distance estimation r i 0 = r i +  i available?  Use multiple anchors, overdetermined system of equations  Use (x u, y u ) that minimize mean square error, i.e,

7. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-7 Minimize mean square error  Look at square of the of Euclidean norm expression (note that for all vectors v)  Look at derivative (gradient) with respect to x, set it equal to 0: –Normal equation –Has unique solution (if A T A has full rank), which gives desired minimal mean square error  Essentially similar for angulation as well

8. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-8 Overview  Basic approaches  Trilateration  Multihop schemes

9. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-9 Multihop range estimation  How to estimate range to a node to which no direct radio communication exists? –No RSSI, TDoA, … –But: Multihop communication is possible  Idea 1: Count number of hops –assume length of one hop is known (DV-Hop, Niculescu et al.) –Start by counting hops between anchors, divide known distance  Idea 2: If range estimates between neighbors exist –use them to improve total length of route estimation in previous method (DV-Distance)

10. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-10 Iterative multilateration  Assume some nodes can hear at least three anchors (to perform triangulation), but not all  Idea: –let more and more nodes compute position estimates, spread position knowledge in the network  Problem: –Errors accumulate

11. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-11 Probabilistic position description  Similar idea to previous one, but accept problem that position of nodes is only probabilistically known –Represent this probability explicitly, use it to compute probabilities for further nodes

12. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-12 Mobile Beacon  Localization of Wireless Sensor Networks with a Mobile Beacon  Mihail L. Sichitiu and Vaidyanathan Ramadurai A mobile beacon is equipped with a positioning system The constraints imposed by the node being in the sensing distance of the mobile beacon. Use Bayesian probability distribution analysis based on RSSI (Received Signal Strength Indication)

13. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-13 One Measurement  From one position only the following probability distribution can be derived based on RSSI Distance Signal strength PDF

14. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-14 Bayes Theorem  Discretization  Bayes Theorem:  Proof –By definition –Combine equations

15. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-15 Interpretation  A: Node is in square x  B: RSSI is s  Pr[A|B]: Node is in square under the condition of RSSI is s  Pr[B|A]: RSSI is s if node is in square x  Pr[A] is chosen according to prior measurement (or uniform) –Iterate estimation using multiple measurements  Pr[B] is unknown but a constant –Sum over all squares to compute Pr[B]

16. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks09.01.2007 Lecture No. 17-16 Combining Mobile Beacons with Bayes Theorem 1 2 4 6 8

17. 17 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Thank you (and thanks go also to Holger Karl for providing some slides) Wireless Sensor Networks Christian Schindelhauer 17th Lecture 09.01.2007