Convex Position Estimation in Wireless Sensor Networks

1. Convex Position Estimation in Wireless Sensor Networks Lance Doherty Kristofer S. J. Pister Laurent El Ghaoui Presented by Nitya Kalathuru and Dharmen Shah Thursday, 6 March 2002

2. Objective of the Paper To estimate unknown node positions in a sensor network using connectivity constraints. Given: position of solid nodes Find: a possible solution for each open node Subject to: proximity constraints imposed by known connections

3. What is • Proximity/ Connectivity Constraint - if one node can communicate with another - these restrict the feasible set of unknown node positions - these must form a convex set • Convex Set • any two points in the set can be connected with a line contained in the set • if the line is not completely contained within the set, the set is not convex. Convex Non-Convex

4. Introduction Where is the data coming from – location of the nodes? • Equip all nodes with GPS to know absolute position - costly • Inferring positional information from connection-imposed proximity constraints – very general In the method proposed • A few nodes have known positions – equipped with GPS or placed deliberately • Feasible solutions are obtained using convex optimization • Only planar networks are considered • Requires centralized computation • Authors use math i.e., linear programming, to solve the problem

5. Mathematical Formulation c, A, B, Fn – matrices x - vector • Linear Program (LP) Minimize cTx Subject to Ax < b • Semidefinite Program (SDP) Minimize cTx Subject to F (x) = F0 + x1F1 +........+ xnFn < 0 Ax < b Fi = Fi T - a generalization of LP - sufficient to solve all numerical problems in this paper - first inequality is called the Linear matrix inequality (LMI)

6. Convex Constraint Models for RF and Optical Communication Models • Connections as convex constraints Provided that the network connectivity can be represented as a set of convex position constraints, the mathematical models can be used to generate feasible positions for the nodes in the network • Radial constraint – RF communication - RF TX of the node is configured to have rotationally symmetric range - Two types: fixed radial variable radial

7. Convex Constraint Models.. cont’d • Angular constraint - optical communication - laser TX and RX rotate and scan through some angle - RX rotates and calculates the angle first roughly & then finely - by observing this angle, an estimate of the relative angle & max. distance to the TX can be obtained - in 3-D this results in a cone for the feasible set

8. Convex Constraint Models.. cont’d • Other convex constraints - quadrant detector scheme - trapezoid • Combining Individual Constraints - nodes are constrained by connections to other nodes - feasible region becomes smaller with each added constraint - this is the mechanism used in position estimation

9. Summary of Constraints Types

10. Simulation and Results Software tools Mat lab using Mosek optimization Hardware AMD k-6 400 MHz processors 64 Mb RAM

11. Simulation Test bed.. • Node positions in 10R square region • 200 Nodes randomly places • 10 Best-connected networks chosen

12. Performance Metrics.. • Best estimation of node is in the Intersection of allowable regions • Performance is calculated as the measure of mean error • Mean error – provides a feasible set

13. Method Adopted.. • Bounding of the feasible set • Use of a Rectangular region for best estimation approximation (1) (2) (3) • Centroid gives the best approximation

14. Constraint Results.. • Radial and Angular constraints • Procedure • Selecting Node 1 as known position (m=1) • Solve for remaining n-m positions • Computing the mean error for n-m unknown positions • Increase m by 1 • Redo 2-4 until m=100

15. Results.. • Radial constraints • Fixed radius and Variable radius analysis • Variable radius method is superior since it gives flexibility of distance sensing

16. Analysis contd.. • Radial constraints have the convex hull on the position of the unknown nodes • Best results can be obtained by placing the nodes on the periphery • Experiment results • Using 4-known nodes at the corners, mean error reduces from 2.4R to 1.2R (variable radius case) • Additional nodes at the centre of the edges reduces the mean error from 1.7R to 0.72R

17. Using Rectangular bounds on Radial constraints Figure shows : 1,2,3 are known positions 4 thru 7 are unknown positions 6 -> 1 and 3 ( two known positions) 5 -> 1 (one known position) 4 -> 2 (one known position with R=2) 7 -> 2 (one known position) Analysis contd..

18. Results are drawn by having 8 known node positions with variable radius Hence we see that at least 8 nodes are need to achieve connection bound of 4R2 Two evidences Nodes with dist R have rectangle area less than 4R2 (centre nodes with high connectivity) Centroid approximation reduces the error from 0.72R to 0.64R Inferences..

19. Results Contd.. • For Angular constraints • Two approaches • Using half angle of uncertainty • Varying the Θ from Π/4 to Π/10 and Π/100 • Using distance to outer bound • Varying the cone length such as R,2R,4R,10R

20. Smaller individual constraints lead to better position estimates as seen from the graph Larger cone length leads to worst results since it causes more divergence Graphs and Inferences..

21. Analysis.. • On Angular constraints • Effect of increasing the node density decreases the mean error

22. Conclusions.. • Sensor network positioning can be formulated as a problem of LP or SDP • Variable radius performs better • Placement on known nodes should be on the boundary of the region • Rectangular bounding improves estimation • For Angle constraints, decreasing uncertainties reduces the mean error • Increasing the nodes density (Connectivity) increases the performance of estimation methods

23. Applications and Future.. • Tracking through sensor network • Hierarchical solution for large networks • Implementing continuous distributions • Combination of angular and radial constraints • Erroneous data management • Modeling uncertainty in "known" positions

24. The End Questions ????