Instructions

instructions

User Manual: Pdf

Open the PDF directly: View PDF PDF.
Page Count: 4

ProgrammingAssignmentWeek2
KalmanFilter⎼TargetTracking
1.Introduction
InthisassignmentyouwillbeimplementingaKalmanfilterforballtrackingin2Dspace.Imaginea
soccerrobotwantstopredicttheball’slocationconstantlyforplanningitsnextmotion.Thefigure
belowdepictsanexampleofballlocationmeasurements,locationestimates,andapredicted
locationatcertaintime.
2.Instructions
Phase1.DesignyourKFModel
1. DynamicsModel(=SystemModel)
a. Inordertohaveanestimateof2Dlocationandpredictafuturelocation,thestate
vector(s)shouldincludevelocity(v)aswellasposition(p)ofbothdimensions,x
andy.

Example:s=[pxvxpyvy]’ors=[pxpyvxvy]’
*Notethatthesymbols
isusedtoindicatethestatevector,whichisdenotedasxinthelecture
material.ThisistoavoidconfusionwiththenotationxoftheCartesiancoordinatex,y.
b. DefineyourtransitionmatrixA(Seep.3ofslides[221]),wheredt=0.033sec. 
c. Nowconsiderthetypeofthesystemnoisecovariance .ThemostsimpleformΣm
isadiagonalmatrix,withthefourvariances: =diag
([ , , , ])orΣmσpx
2 σpy
2 σvx
2σvy
2
=diag
([ , , , ]).Σmσpx
2 σvx
2σpy
2 σvy
2
Ingeneral,youdon’thavetorestricttheform,butadiagonalmatrixisagood
startingpointifyoudonothaveastrongreasonnottodoso.Howtosetthe
values?ItwillbeexplainedPhase2.
2. MeasurementModel
a. Themeasurementdataisreadilyinthe2Dlocationunit.
z=[zxzy]’=[pxpy]’
HencetheobservationmatrixCwillbea2by4matrixwithtwoonesandsix
zeros.PleasefigureoutwhatCisbyyourself!
b. Considerthetypeofthemeasurementnoisecovariance .ThemostsimpleΣo
formisadiagonalmatrix,withthetwovariances: =diag
([ , ]).WewillΣoσzx
2 σzy
2
seehowtosetthevaluesinPhase2.
3. Reviewthemodels[Advancedoption].
a. Ifyouhavechosen(block)diagonalformsforthenoisecovariances,thexandy
componentsarenowuncorrelated
.Thenyoucansimplifythemodelandhavetwo
independentidenticalfilters.Thesimplifiedfilterhass=[pv]’,2by2Aand ,z=Σm
p,thescalarvariance ,and1by2C.Σo
You only need to keep two instances of this filter, one for x, and the other for y.
This idea (separating uncorrelated systems) reduces the computational load, which
canbeeffectiveforhighdimensionalsystems.
*Notethatthisreductionisoptionalinthisassignment.IfyougotmoreconfusedbecauseofStep3.a,
thenjustignoreitandgoaheadtoPhase2.
b. Ifyouhavecorrelatedterms,thatisfine.Youwillkeepthemodelasitis.
Phase2.SetParameterValues
1. Inordertosettheinitialvaluesofthestateandstateerrorcovariance,youmayusethefirst
measurementastheposition,andsetzeroforthevelocity.Also,youmaysetthestate
covariancePtobeaverylargeidentitymatrix,whichimpliesalargeuncertaintyofthe
state.
2. Youalsoneedtosetthemodelnoiseparameters and .ThinkaboutaroughyetΣmΣo
reasonablerangeoftheparametersbyconsideringthephysicalunitsofthe
noise/uncertaintymodeloftheproblem.
Forexample,Ifthemeasurementnoiseisinmeters,thenitcouldrangefromO(0.01)to
O(0.1).Similarly,ifthedynamicsmodelincludespositionandvelocity,thentheyhaveunits
suchasmeterandmeter/sec.Becauseyoursystemisasmallballrollingonthefloor,the
velocitywillNOTbeaslargeas10meter/sec.
3. CoarseTests
a. Stillyouwillhavemanymanychoices.Nowyoucantestthenumbersofthe
reasonablerangeFIRSTCOARSELY.
b. Onegoodstartingpointistosetthesystemmodelnoisecovariancetohave
unreasonablylarge(forexample,1e^6),andthemeasurementnoisecovarianceto
besmallinthereasonablerange.ThenyourKFshouldjustfollowyour
measurements,unlessyouhaveabuginyourcode.
c. Afterthat,youcantestdifferentvaluesofthesystemmodelnoiseparameter.
4. FineTuning
a. OnceyourKFseemstowork,thenyoucanfinetunetheparameters.
b. Youcaneventrymorecomplexparameters,suchasnondiagonalcovariance
matrixoradaptive(nonconstant)covariances.WesuggestthatyoudothisONLY
AFTERyoursimplestKFworks.
5. Becareful,thevarianceorcovariancetakesasquaredform.
Phase3.Implementation
1. Youwillcompleteafunctionthattakesthestate,parameters,andsensordata(x,y)asinput,
andreturnsaballpositionprediction,possiblyusefulforgoalieapplications.Thesignature
ofthefunctionisgivenas:
function[predictx,predicty,state,param]=kalmanFilter(t,x,y,state,param,previous_t)
2. Pleasepassthestateandstatecovarianceasinput/outputarguments.Thusparam
should
havethefollowingfields: 
param
.P
Youcannameitdifferently,butmakesurethatyoukeeptrackofthestatecovarianceP.(If
youhavetwoindependentfilters,thenyoumayhaveparam
.Pxandparam
.Py).Youmay
useparam
toaddotherfieldsifyouneed.
3. Yourfunctionshouldreturnthepredictedxandypositionsoftheballatatime330msinto
thefuture.Witharateof30Hz,this330msintimerepresents10framesinthefuture.Inthe
kalmanFilterfunction,anaiveimplementationshowshowtouseinstantaneousvelocity
fromonlytwoframesandnostatetocalculatethefuturepositionoftheball,returnedas
predictxandpredicty.Youwillkeeptrackofthefourstatecomponents,comprisedofx
position,yposition,xvelocity,andyvelocity.Thisstatewillbefedintoeachfunctioncall.
4. example_test.misprovidedtohelpvisualizeyourresult.
Phase4.Evaluationandsubmission
Tosubmityourresulttoourserver,youneedtorunthescriptrunevalinyourMATLAB
commandwindow.Pleasespecifythepathwheretheencryptedtestdataarelocated.Ascript
willthenevaluateyourkalmanFilterfunctionandgenerateanoutputfile,
SubmissionFilter.mat,tobeuploadedtotheCourserawebUI.Youmaysubmityourresult
multipletimes,andwewillcountonlythehighestscoretowardsyourgrade.Thescorereflects
howcloseyourpredictedballpositionsaretotheobservedballpositions.
 

Navigation menu