Convex Position Estimation in Convex Position Estimation in Wireless Sensor NetworksWireless Sensor Networks

Lance DohertyLance Doherty

Kristofer S. J. PisterKristofer S. J. Pister

Laurent El GhaouiLaurent El Ghaoui

Presented by Nitya Kalathuru and Dharmen Shah

Thursday, 6 March 2002

Objective of the PaperObjective of the Paper

Given: position of solid Given: position of solid nodesnodes

Find: a possible solution Find: a possible solution for each open nodefor each open node

Subject to: proximity Subject to: proximity constraints imposed constraints imposed by known by known connectionsconnections

To estimate unknown node positions in a To estimate unknown node positions in a sensor network using connectivity constraints.sensor network using connectivity constraints.

What isWhat is• Proximity/ Connectivity ConstraintProximity/ Connectivity Constraint

- if one node can communicate with another - if one node can communicate with another

- these restrict the feasible set of unknown node - these restrict the feasible set of unknown node positionspositions

- these must form a convex set- these must form a convex set

• Convex SetConvex Set

- any two points in the set can be any two points in the set can be connected with a line contained connected with a line contained in the setin the set

- if the line if the line is not completely is not completely contained within the setcontained within the set, the set , the set is not convex.is not convex.

Convex

Non-Convex

IntroductionIntroductionWhere is the data coming from – location of the nodes?Where is the data coming from – location of the nodes?• Equip all nodes with GPS to know absolute position - costlyEquip all nodes with GPS to know absolute position - costly• Inferring positional information from connection-imposed Inferring positional information from connection-imposed

proximity constraints – very generalproximity constraints – very general

In the method proposedIn the method proposed• A few nodes have known positions – equipped with GPS or A few nodes have known positions – equipped with GPS or

placed deliberatelyplaced deliberately• Feasible solutions are obtained using convex optimizationFeasible solutions are obtained using convex optimization• Only planar networks are consideredOnly planar networks are considered• Requires centralized computationRequires centralized computation• Authors use math i.e., linear programming, to solve the Authors use math i.e., linear programming, to solve the

problemproblem

Mathematical FormulationMathematical Formulation• Linear Program (LP)Linear Program (LP)

MinimizeMinimize c cTTxxSubject toSubject to Ax < bAx < b

• Semidefinite Program (SDP) Semidefinite Program (SDP) MinimizeMinimize c cTTxxSubject toSubject to F (x) = FF (x) = F00 + x + x11FF11 +........+ x +........+ xnnFFnn < 0 < 0

Ax < bAx < bFFii = F = Fii T T

- a generalization of LP- a generalization of LP- sufficient to solve all numerical problems in this - sufficient to solve all numerical problems in this paperpaper- first inequality is called the Linear matrix inequality - first inequality is called the Linear matrix inequality

(LMI) (LMI)

c, A, B, Fn – matrices

x - vector

Convex Constraint Models for RF and Convex Constraint Models for RF and Optical Communication ModelsOptical Communication Models

• Connections as convex constraintsConnections as convex constraintsProvided that the network connectivity can be represented as a Provided that the network connectivity can be represented as a set of convex position constraints, the mathematical models set of convex position constraints, the mathematical models can be used to generate feasible positions for the nodes in the can be used to generate feasible positions for the nodes in the networknetwork

• Radial constraint Radial constraint

– – RF communicationRF communication- - RF TRF TXX of the node is configured of the node is configured

to have rotationally symmetric to have rotationally symmetric

range range

- Two types: fixed radial- Two types: fixed radial

variable radialvariable radial

Convex Constraint Models.. cont’dConvex Constraint Models.. cont’d

• Angular constraint - optical communicationAngular constraint - optical communication

- laser T- laser TXX and R and RXX rotate and scan through some angle rotate and scan through some angle

- R- RXX rotates and calculates the angle first roughly & then finely rotates and calculates the angle first roughly & then finely

- by observing this angle, an estimate of the relative angle & max.- by observing this angle, an estimate of the relative angle & max.

distance to the Tdistance to the TXX can be obtained can be obtained

- in 3-D this results in a cone for the feasible set- in 3-D this results in a cone for the feasible set

Convex Constraint Models.. cont’dConvex Constraint Models.. cont’d

• Combining Individual Combining Individual

ConstraintsConstraints- nodes are constrained by- nodes are constrained by connections to other nodesconnections to other nodes

- feasible region becomes smaller - feasible region becomes smaller with each added constraintwith each added constraint

- this is the mechanism used in - this is the mechanism used in position estimation position estimation

• Other convex constraintsOther convex constraints- quadrant detector scheme- quadrant detector scheme

- trapezoid - trapezoid

Summary of Constraints TypesSummary of Constraints Types

Simulation and ResultsSimulation and Results

Software tools Software tools Mat lab using Mosek Mat lab using Mosek optimization optimization

HardwareHardware AMD k-6 400 MHz processorsAMD k-6 400 MHz processors 64 Mb RAM64 Mb RAM

Simulation Test bed..Simulation Test bed..

Node positions in Node positions in 10R square region10R square region

200 Nodes 200 Nodes randomly placesrandomly places

10 Best-connected 10 Best-connected networks chosennetworks chosen

Performance Metrics..Performance Metrics..

Best estimation of node is in the Best estimation of node is in the Intersection of allowable regionsIntersection of allowable regions

Performance is calculated as the measure Performance is calculated as the measure of mean error of mean error

Mean error – provides a feasible setMean error – provides a feasible set

Method Adopted..Method Adopted.. Bounding of the feasible setBounding of the feasible set Use of a Rectangular region for best Use of a Rectangular region for best

estimation approximationestimation approximation

(1) (2)(1) (2)

(3)(3)

Centroid gives the best approximationCentroid gives the best approximation

Constraint Results..Constraint Results..

Radial and Angular constraintsRadial and Angular constraints ProcedureProcedure

1.1. Selecting Node 1 as known position (m=1)Selecting Node 1 as known position (m=1)2.2. Solve for remaining n-m positionsSolve for remaining n-m positions3.3. Computing the mean error for n-m unknown Computing the mean error for n-m unknown

positionspositions4.4. Increase m by 1Increase m by 15.5. Redo 2-4 until m=100Redo 2-4 until m=100

Results..Results..

Radial Radial constraintsconstraints Fixed radius and Fixed radius and

Variable radius Variable radius analysisanalysis

Variable radius Variable radius method is method is superior since it superior since it gives flexibility gives flexibility of distance of distance sensingsensing

Analysis contd..Analysis contd.. Radial constraints have the convex hull on Radial constraints have the convex hull on

the position of the unknown nodesthe position of the unknown nodes Best results can be obtained by placing the Best results can be obtained by placing the

nodes on the peripherynodes on the periphery Experiment resultsExperiment results

Using 4-known nodes at the corners, mean error Using 4-known nodes at the corners, mean error reduces from 2.4R to 1.2R (variable radius case)reduces from 2.4R to 1.2R (variable radius case)

Additional nodes at the centre of the edges Additional nodes at the centre of the edges reduces the mean error from 1.7R to 0.72R reduces the mean error from 1.7R to 0.72R

Analysis contd..Analysis contd.. Using Rectangular Using Rectangular

bounds on Radial bounds on Radial constraintsconstraints

Figure shows :Figure shows : 1,2,3 are known 1,2,3 are known

positionspositions 4 thru 7 are unknown 4 thru 7 are unknown

positionspositions 6 -> 1 and 3 ( two known 6 -> 1 and 3 ( two known

positions)positions) 5 -> 1 (one known 5 -> 1 (one known

position)position) 4 -> 2 (one known 4 -> 2 (one known

position with R=2)position with R=2) 7 -> 2 (one known 7 -> 2 (one known

position)position)

Inferences..Inferences.. Results are drawn by having Results are drawn by having

8 known node positions with 8 known node positions with variable radiusvariable radius

Hence we see that at least 8 Hence we see that at least 8 nodes are need to achieve nodes are need to achieve connection bound of 4Rconnection bound of 4R2 2

Two evidencesTwo evidences Nodes with dist R have Nodes with dist R have

rectangle area less than 4Rrectangle area less than 4R22 (centre nodes with high (centre nodes with high connectivity)connectivity)

Centroid approximation Centroid approximation reduces the error from reduces the error from 0.72R to 0.64R0.72R to 0.64R

Results Contd..Results Contd..

For Angular constraintsFor Angular constraints Two approachesTwo approaches

Using half angle of uncertaintyUsing half angle of uncertainty Varying the Varying the ΘΘ from from ΠΠ/4 to /4 to ΠΠ/10 and /10 and ΠΠ/100/100

Using distance to outer boundUsing distance to outer bound Varying the cone length such as Varying the cone length such as

R,2R,4R,10RR,2R,4R,10R

Graphs and Inferences..Graphs and Inferences..

Smaller individual constraints lead to better position Smaller individual constraints lead to better position estimates as seen from the graphestimates as seen from the graph

Larger cone length leads to worst results since it Larger cone length leads to worst results since it causes more divergencecauses more divergence

Analysis..Analysis..

On Angular On Angular constraintsconstraints Effect of increasing Effect of increasing

the node density the node density decreases the mean decreases the mean errorerror

Conclusions..Conclusions.. Sensor network positioning can be Sensor network positioning can be

formulated as a problem of LP or SDPformulated as a problem of LP or SDP Variable radius performs better Variable radius performs better Placement on known nodes should be on Placement on known nodes should be on

the boundary of the regionthe boundary of the region Rectangular bounding improves estimationRectangular bounding improves estimation For Angle constraints, decreasing For Angle constraints, decreasing

uncertainties reduces the mean erroruncertainties reduces the mean error Increasing the nodes density Increasing the nodes density

(Connectivity) increases the performance (Connectivity) increases the performance of estimation methodsof estimation methods

Applications and Future..Applications and Future..

Tracking through sensor networkTracking through sensor network Hierarchical solution for large networksHierarchical solution for large networks Implementing continuous distributionsImplementing continuous distributions Combination of angular and radial Combination of angular and radial

constraintsconstraints Erroneous data managementErroneous data management Modeling uncertainty in "known" Modeling uncertainty in "known"

positionspositions

The EndThe End

Questions ????Questions ????