Localization. Theobjectiveof the localizationproblemin$2$Dis to infer, froma set of range

measurements, thepositionof anobjectasaccuratelyaspossible. Theproblemwould, for example,

arise inmobilephonetrackingwherewewanttofindthebestestimateof the locationofacellphone

userbasedonthemeasurementsofstrengthofasignalemittedfromthecellphonetoanumberofbase

stations$($accesspoints$).$

Thefigurebelowshowsthesetup.Thepositionsofthreeanchorpointsaregivenas:

p$1=$

$[0\mathrm{.}46$

$0\mathrm{.}03$

$]$

$,$ p$2=$

$[0\mathrm{.}19$

$0\mathrm{.}54$

$]$

$,$ p$3=$

$[0\mathrm{.}48$

$0\mathrm{.}49$

$]$

$,$

andthemeasurementsfromthesepointsareestimatedas:

d$=[0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}85],$

If themeasurementshadperfectaccuracythentheshadedareainthefiguresbelowwouldbeasingle

point. Ingeneralhowever, thenoise inthemeasurementsmeans the intersectionisnot reducedtoa

uniquepointandwewouldliketofindthe$$center$$of theshadedareawhereanobject is located.We

alsowouldliketoknowtheuncertaintyinourestimatedposition.

Wecanformulatetheproblemasa$($convex$)$optimizationproblemtofindthelargestdiskthatiscontained

intheshadedarea. Theoptimal centerwouldbeourestimatedlocationandtheoptimal radius isa

measureofuncertaintyofourestimate.