Chapter 4 Attitude Determination - Virginia Tech

1 Chapter4 AttitudeDeterminationEssentiallyall control systemsrequiretwo types of hardware components: usedto senseor measurethe stateof the system ,andactuatorsare usedto adjustthe stateof the example,a typicalthermostatusedto control room temperaturehas a thermocouple( sensor )and a connectiontothe furnace(actuator).Thecontrol systemcomparesthe referencetemperaturetothe measuredtemperatureand eitherturnsthe furnaceon or o , dependingon thesignof the di erencebetween the two. A spacecraftattitudedeterminationandcontro l systemtypicallyusesa variety of sensorsand by threeor moreattitudevariables,the di erencebetween the desiredandmeasuredstatesis slightly morecomplicatedthanfor a thermostat,or even for thepositionof the satellitein ,the mathematicalanalysisof attitudedeterminationis complicatedby the factthatattitudedeterminationis necessarilyeitherunderdeterminedor this Chapter ,we developthe basicconceptsand tools for attitudedetermina-tion, focus hereon staticattitudedetermination.

2 Wheretimeis not in-volved in the dynamicattitudedeterminationis treatedin combinationof sensorsandmathematicalmodelstocollectvec torcomponents in the body are usedin one of severaldi erent algorithmsto determinethe Attitude ,typicallyin the formof a quaternion,Eulerangles,or a takes atleasttwo vectorsto estimatethe example,an attitudedeterminationsystemmight use a sun vector,^sand a magnetic eldvector^m. A sun sensormea-suresthe components of^sin the body frame,sb, whilea mathematicalmodel of theSun'sapparent motionrelative to the spacecraftis usedto determinethe components4-1 Copyright Chris Hall March 18, 20034-2 CHAPTER4.

3 ATTITUDEDETERMINATIONin the inertialframe,si. Similarly, a magnetometermeasuresthe components of^min the body frame,mb, whilea mathematicalmodel of the Earth'smagnetic eldrelative to the spacecraftis usedto determinethe components in the inertialframe,mi. An attitudedeterminationalgorithmis thenusedto nda rotationmatrixRbisuch thatsb=Rbisiandmb=Rbimi( )Theattitudedeterminationanalystneedsto understandhow varioussensorsmeasurethe body-framecomponents, how mathematicalmodelsare usedto determinetheinertial-framecomponents, and how standardattitudedeterminationalgorithmsa reusedto Overdetermined?

4 In the previoussectionwe make claimthatat leasttwo vectorsare requiredto de-terminethe takes threeindependent parametersto determinethe Attitude ,and thata unitvectoris actuallyonlytwo parametersbecauseof theunitvectorconstraint. Thereforewe requirethreescalarsto determinethe the requirement is for morethanone and less thantwo thus uniquein thatone measurement is not enough, , the problemis underdetermined,andtwo measurements is too many, , theproblemis thisobservationis thatallattitudedeterminationalgorithmsar e two basicclassesof rst classmakesabsolutemeasure-ments, whereasthe secondclassmakesrelativemeasurements.

5 Absolutemeasurementsensorsare basedon the factthatknowingthe positionof a spacecraftin its orbitmakes it possibleto computethe vectordirections,withrespect to an inertialframe,of certainastronomicalobjects,and of the forcelinesof the Earth'smagnetic sensorsmeasurethesedirectionswithrespect to a spacecraft-or body- xedreferenceframe,and by comparingthe measurements withthe knownreferencedirectionsin an inertialreferenceframe,are ableto determine(at leastapproximately)the relative orientationof the body framewithrespect to the are usedin the staticattitudedeterminationalgorithmsdev eloped in this measurement sensorsbelongto the classof gyroscopicinstruments, in-cludingtherate gyroandtheintegratinggyro.

6 Classically, theseinstruments havebeen implemented as spinningdisksmounted on gimbals;however, moderntechnol-ogy has brought such marvels asringlasergyros, ber opticgyros, andhemisphericalCopyright Chris Hall March 18, Relative measurement sensorsare usedin the dynamicattitudedeterminationalgorithmsde veloped in SensorsWe beginwithsunsensorsbecauseof theirrelative simplicity andthe factthatvirtuallyall spacecraftuse sun sensorsof sometype. Thesun is a usefulreferencedirectionbecauseof its brightnessrelative to otherastronomicalobjects,anditsrelativel y smallapparent radiusas viewed by a spacecraftnearthe ,mostsatellitesuse solarpower, andso needto make surethatsolarpanelsare orientedcorrectlywithrespect to the sensitive instruments thatmustnot be exposedto directsunlight.

7 For all thesereasons,sun sensorsare importantcomponents in spacecraftattitudedeterminationand control systems.^n^s^n^s^t 0 0+ 0 photocell(a)two-cellsensor(b) :Photocellsfor SunSensors.(a) Singlephotocell.(b) Pair of photocellsfor measurement of in^n ^ of a sun sensoris to providean approximateunitvector,withrespectto the body referenceframe,thatpoints towardsthe denotethis vectorby^s, which can be writtenas^s=siTf^ig=sbTf^bg( )If the positionof the spacecraftin its orbitis known, alongwiththe positionof theEarthin its orbit,thensiis algorithmto computesiis given at the endof this types of sun sensorsare available:analoganddigital.

8 Analogsun sensorsarebasedon photocellswhosecurrent outputis proportionalto the cosineof the angle between the directionto the sun and the normalto the photocell ( ).Thatis, the current outputis given byI( ) =I(0) cos ( )Copyright Chris Hall March 18, 20034-4 CHAPTER4. ATTITUDEDETERMINATION fromwhich can be unitnormalof the photocell by^n,we see that^s ^n= cos ( )However, knowing does not provideenoughinformationto determine^scompletely,sincethe component of^sperpendicularto^ , sun sen-sorscombinefoursuch photocellsto providethe speci csun sensorsare includedin , 2, and determinethe anglein a speci cplane,one normallyusestwo photocellstiltedat an angle 0withrespect to the normal^nof the sun sensor (see the seconddiagramin ).

9 Thisarrangement gives the anglebetween the sun sensornormal,^nandthe projectionof the sun vector^sonto the^n ^ two photocellsgeneratecurrentsI1( ) =I(0) cos ( 0 )( )I2( ) =I(0) cos ( 0+ )( )Takingthe di erenceof thesetwo expressions,we obtain I=I2 I1( )=I(0) [cos ( 0+ ) cos ( 0 )]( )= 2I(0) sin 0sin ( )=Csin ( )whereC= 2I(0) sin 0is a constant thatdependson the electricalcharacteristicsofthe photocellsand the geometricalarrangement of the two appropriatelyarrangedpairsof photocells,we obtainthe geometryshown in In this picture,^n1is the normalvectorfor the rstpairof photo-cells,and^n2is the normalvectorfor the ^tvectoris chosento de nethe two planesof the photocell pairs; ,^n1and^tare as shown in (b)forone pair,and^n2and^tare for the ^n1;^n2;^tocomprisethe threeunitvectorsof a framedenotedbyFs(sfor sun sensor ).

10 Thespacecraftdesignerdeterminesthe orientationof thisframewithrespect to the body frame;thus theorientationmatrixRbsis providethe components of thesun vectorin the sun sensorframe,ss, and the matrixRbsprovidesthe componentsin the body frame,sb= two measuredangles 1and 2determinessas follows. We want componentsof a unitvector,but it is easiestto beginby lettingthe component in the^n1directionbe equalto geometryof the arrangement impliesthatthe componentsin the^n2and^tdirectionsare tan 1=tan 2, and tan 1, respectively. Denotingthecomponents of this non-unitvectorbys s= [1 tan 1=tan 2tan 1]T( )Copyright Chris Hall March 18, ^n1^s^n2^t 1 : Geometryfor a four-photocell sun sensorwe obtainthe sun vectorcomponents by normalizingthis vector:ss=[1 tan 1=tan 2tan 1]Tqs sTs s( )Thus, usinga four-photocell sun sensorleadsdirectlyto the calculationof a unitsun vectorexpressedin the sun sensorframe,Fs.