Wireless Sensor Networks 17th Lecture

Transcription

1 Wireless Sensor Networks 17th Lecture Christian Schindelhauer 1

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 Wireless Sensor Networks Lecture No. 17-2

3 Overview Basic approaches Trilateration Multihop schemes Wireless Sensor Networks Lecture No. 17-3

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 )! Wireless Sensor Networks Lecture No. 17-4

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 = , r 2 = 2, r 3 = 3! (x u,y u ) = (5,2) Wireless Sensor Networks Lecture No. 17-5

6 Trilateration with distance errors What if only distance estimation r i 0 = r i + ε i available? Use multiple anchors, overdetermined system of equations Use (x u, y u ) that minimize mean square error, i.e, Wireless Sensor Networks Lecture No. 17-6

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 Wireless Sensor Networks Lecture No. 17-7

8 Overview Basic approaches Trilateration Multihop schemes Wireless Sensor Networks Lecture No. 17-8

9 Multihop range estimation How to estimate range to a node to which no direct radio communication exists? B 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) X C A Wireless Sensor Networks Lecture No. 17-9

10 Iterative multilateration Assume some nodes can hear at least three anchors (to perform triangulation), but not all I: (18,20) II: A B (?,?) (?,?) (18,20) A B (12,14) (?,?) Idea: (2,10) (2,10) let more and more nodes compute position estimates, spread position knowledge in the network Problem: Errors accumulate III: (8,0) (2,10) C (?,?) (18,20) A B (12,14) (38,5) (30,12) IV: (8,0) (2,10) C (?,?) (18,20) A B (12,14) (38,5) (30,12) (38,5) (38,5) (8,0) C (?,?) (8,0) C (22,2) Wireless Sensor Networks Lecture No

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 Wireless Sensor Networks Lecture No

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) Wireless Sensor Networks Lecture No

13 One Measurement From one position only the following probability distribution can be derived based on RSSI PDF Distance Signal strength Wireless Sensor Networks Lecture No

14 Bayes Theorem Discretization Bayes Theorem: Proof By definition Combine equations Wireless Sensor Networks Lecture No

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] Wireless Sensor Networks Lecture No

16 Combining Mobile Beacons with Bayes Theorem Wireless Sensor Networks Lecture No

17 Thank you (and thanks go also to Holger Karl for providing some slides) Wireless Sensor Networks Christian Schindelhauer 17th Lecture