惯性导航原理的理解.docx

惯性导航原理的理解.docx

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惯性导航原理的理解

Part1.Accelerometer

Tounderstandthisunitwe'llstartwiththeaccelerometer.Whenthinkingaboutaccelerometersitisoftenusefultoimageaboxinshapeofacubewithaballinsideit.Youmayimaginesomethingelselikeacookieoradonut,butI'llimagineaball:

了解本单元,我们从加速度计开始。

当考虑加速度计通常想象成一个立方体形状的盒子里面有一个球很有助于理解。

你也可以想像别的像饼干或者一个甜甜圈,但我会想象一个球:

Ifwetakethisboxinaplacewithnogravitationfieldsorforthatmatterwithnootherfieldsthatmightaffecttheball'sposition–theballwillsimplyfloatinthemiddleofthebox.Youcanimaginetheboxisinouter-spacefar-farawayfromanycosmicbodies,orifsuchaplaceishardtofindimagineatleastaspacecraftorbitingaroundtheplanetwhereeverythingisinweightlessstate.Fromthepictureaboveyoucanseethatweassigntoeachaxisapairofwalls（weremovedthewallY+sowecanlookinsidethebox）.Imaginethateachwallispressuresensitive.Ifwemovesuddenlytheboxtotheleft（weaccelerateitwithacceleration1g=9.8m/s^2）,theballwillhitthewallX-.Wethenmeasurethepressureforcethattheballappliestothewallandoutputavalueof-1gontheXaxis.

Pleasenotethattheaccelerometerwillactuallydetectaforcethatisdirectedintheoppositedirectionfromtheaccelerationvector.Thisforceisoftencalled InertialForceorFictitiousForce .Onethingyoushouldlearnfromthisisthatanaccelerometermeasuresaccelerationindirectlythroughaforcethatisappliedtooneofit'swalls（accordingtoourmodel,itmightbeaspringorsomethingelseinreallifeaccelerometers）.Thisforcecanbecausedbytheacceleration,butaswe'llseeinthenextexampleitisnotalwayscausedbyacceleration.

IfwetakeourmodelandputitonEarththeballwillfallontheZ-wallandwillapplyaforceof1gonthebottomwall,asshowninthepicturebelow:

Inthiscasetheboxisn'tmovingbutwestillgetareadingof-1gontheZaxis.Thepressurethattheballhasappliedonthewallwascausedbyagravitationforce.Intheoryitcouldbeadifferenttypeofforce–forexample,ifyouimaginethatourballismetallic,placingamagnetnexttotheboxcouldmovetheballsoithitsanotherwall.Thiswassaidjusttoprovethatinessenceaccelerometermeasuresforcenotacceleration.Itjusthappensthataccelerationcausesaninertialforcethatiscapturedbytheforcedetectionmechanismoftheaccelerometer.

WhilethismodelisnotexactlyhowaMEMSsensorisconstructeditisoftenusefulinsolvingaccelerometerrelatedproblems.Thereareactuallysimilarsensorsthathavemetallicballsinside,theyarecalledtiltswitches,howevertheyaremoreprimitiveandusuallytheycanonlytellifthedeviceisinclinedwithinsomerangeornot,nottheextentofinclination.

Sofarwehaveanalyzedtheaccelerometeroutputonasingleaxisandthisisallyou'llgetwithasingleaxisaccelerometers.Therealvalueoftriaxialaccelerometerscomesfromthefactthattheycandetectinertialforcesonallthreeaxes.Let'sgobacktoourboxmodel,andlet'srotatethebox45degreestotheright.Theballwilltouch2wallsnow:

Z-andX-asshowninthepicturebelow:

Thevaluesof0.71arenotarbitrary,theyareactuallyanapproximationforSQRT（1/2）.Thiswillbecomemoreclearasweintroduceournextmodelfortheaccelerometer.

Inthepreviousmodelwehavefixedthegravitationforceandrotatedourimaginarybox.Inlast2exampleswehaveanalyzedtheoutputin2differentboxpositions,whiletheforcevectorremainedconstant.Whilethiswasusefulinunderstandinghowtheaccelerometerinteractswithoutsideforces,itismorepracticaltoperformcalculationsifwefixthecoordinatesystemtotheaxesoftheaccelerometerandimaginethattheforcevectorrotatesaroundus.

Pleasehavealookatthemodelabove,Ipreservedthecolorsoftheaxessoyoucanmakeamentaltransitionfromthepreviousmodeltothenewone.Justimaginethateachaxisinthenewmodelisperpendiculartotherespectivefacesoftheboxinthepreviousmodel.ThevectorRistheforcevectorthattheaccelerometerismeasuring（itcouldbeeitherthegravitationforceortheinertialforcefromtheexamplesaboveoracombinationofboth）.Rx,Ry,RzareprojectionoftheRvectorontheX,Y,Zaxes.Pleasenoticethefollowingrelation:

R^2=Rx^2+Ry^2+Rz^2     （Eq.1）

whichisbasicallytheequivalentofthe Pythagoreantheoremin3D.

RememberthatalittlebitearlierItoldyouthatthevaluesofSQRT（1/2）~0.71arenotrandom.Ifyouplugthemintheformulaabove,afterrecallingthatourgravitationforcewas1gwecanverifythat:

1^2=（-SQRT（1/2））^2+0^2+（-SQRT（1/2））^2

simplybysubstitutingR=1,Rx=-SQRT（1/2）,Ry=0,Rz=-SQRT（1/2）in Eq.1

Afteralongpreambleoftheorywe'regettingclosertoreallifeaccelerometers.ThevaluesRx,Ry,Rzareactuallylinearlyrelatedtothevaluesthatyourreal-lifeaccelerometerwilloutputandthatyoucanuseforperformingvariouscalculations.

Beforewegettherelet'stalkalittleaboutthewayaccelerometerswilldeliverthisinformationtous.Mostaccelerometerswillfallintwocategories（大多受加速度计可以分为以下两类）:

digitalandanalog.Digitalaccelerometerswillgiveyouinformationusingaserialprotocol（协议）likeI2C,SPIorUSART,whileanalogaccelerometerswilloutputavoltagelevelwithinapredefinedrangethatyouhavetoconverttoadigitalvalueusinganADC（analogtodigitalconverter）module.IwillnotgointomuchdetailabouthowADCworks,partlybecauseitissuchanextensivetopicandpartlybecauseitisdifferentfromoneplatformtoanother.Somemicrocontrollerwillhaveabuilt-inADCmodulessomeofthemwillneedexternalcomponentsinordertoperformtheADCconversions.NomatterwhattypeofADCmoduleyouuseyou'llendupwithavalueinacertainrange.Forexamplea10-bitADCmodulewilloutputavalueintherangeof0..1023,notethat1023=2^10-1.A12-bitADCmodulewilloutputavalueintherangeof0..4095,notethat4095=2^12-1.

Let'smoveonbyconsideringasimpleexample,supposeour10bitADCmodulegaveusthefollowingvaluesforthethreeaccelerometerchannels（axes）:

AdcRx=586

AdcRy=630

AdcRz=561

EachADCmodulewillhaveareferencevoltage,let'sassumeinourexampleitis3.3V.Toconverta10bitadcvaluetovoltageweusethefollowingformula:

VoltsRx=AdcRx*Vref/1023

Aquicknotehere:

thatfor8bitADCthelastdividerwouldbe255=2^8-1,andfor12bitADClastdividerwouldbe4095=2^12-1.

Applyingthisformulatoall3channelsweget:

VoltsRx=586*3.3V/1023=~1.89V（weroundallresultsto2decimalpoints）

VoltsRy=630*3.3V/1023=~2.03V

VoltsRz=561*3.3V/1023=~1.81V

Eachaccelerometerhasazero-gvoltagelevel,youcanfinditinspecs,thisisthevoltagethatcorrespondsto0g.Togetasignedvoltagevalueweneedtocalculatetheshiftfromthislevel.Let'ssayour0gvoltagelevelisVzeroG=1.65V.Wecalculatethevoltageshiftsfromzero-gvoltageasfollows:

:

DeltaVoltsRx=1.89V–1.65V=0.24V

DeltaVoltsRy=2.03V–1.65V=0.38V

DeltaVoltsRz=1.81V–1.65V=0.16V

WenowhaveouraccelerometerreadingsinVolts,it'sstillnoting（9.8m/s^2）,todothefinalconversionweapplytheaccelerometersensitivity,usuallyexpressedinmV/g.LetssayourSensitivity=478.5mV/g=0.4785V/g.Sensitivityvaluescanbefoundinaccelerometerspecifications（规格）.Togetthefinalforcevaluesexpressedingweusethefollowingformula:

Rx=DeltaVoltsRx/Sensitivity

Rx=0.24V/0.4785V/g=~0.5g

Ry=0.38V/0.4785V/g=~0.79g

Rz=0.16V/0.4785V/g=~0.33g

Wecouldofcoursecombineallstepsinoneformula,butIwentthroughallthestepstomakeitclearhowyougofromADCreadingstoaforcevectorcomponentexpresseding.

Rx=（AdcRx*Vref/1023–VzeroG）/Sensitivity （Eq.2）

Ry=（AdcRy*Vref/1023–VzeroG）/Sensitivity

Rz=（AdcRz*Vref/1023–VzeroG）/Sensitivity

Wenowhaveall3componentsthatdefineourinertialforcevector,ifthedeviceisnotsubjecttootherforcesotherthangravitation,wecanassumethisisthedirectionofourgravitationforcevector.IfyouwanttocalculateinclinationofdevicerelativetothegroundyoucancalculatetheanglebetweenthisvectorandZaxis.Ifyouarealsointerestedinper-axisdirectionofinclinationyoucansplitthisresultinto2components:

inclinationontheXandYaxisthatcanbecalculatedastheanglebetweengravitationvectorandX/Yaxes.Calculatingtheseanglesismoresimplethanyoumightthink,nowthatwehavecalculatedthevaluesforRx,RyandRz.Let'sgobacktoourlastaccelerometermodelanddosomeadditionalnotations:

TheanglesthatweareinterestedinaretheanglesbetweenX,Y,ZaxesandtheforcevectorR.We'lldefinetheseanglesasAxr,Ayr,Azr.Youcannoticefromtheright-angletriangleformedbyRandRxthat:

cos（Axr）=Rx/R,andsimilarly:

cos（Ayr）=Ry/R

cos（Azr）=Rz/R

Wecandeductfrom Eq.1 thatR=SQRT（Rx^2+Ry^2+Rz^2）.

Wecanfindnowouranglesbyusingarccos（）function（theinversecos（）function）:

Axr=arccos（Rx/R）

Ayr=arccos（Ry/R）

Azr=arccos（Rz/R）

We'vegonealongwaytoexplaintheaccelerometermodel,justtocomeuptotheseformulas.Dependingonyourapplicationsyoumightwanttouseanyintermediateformulasthatwehavederived.We'llalsointroducethegyroscopemodelsoon,andwe'llseehowaccelerometerandgyroscopedatacanbecombinedtoprovideevenmoreaccurateinclinationestimations.

Butbeforewedothatlet'sdosomemoreusefulnotations:

cosX=cos（Axr）=Rx/R

cosY=cos（Ayr）=Ry/R

cosZ=cos（Azr）=Rz/R

Thistripletisoftencalled DirectionCosine ,