Lesson04 汽车底盘课程 Vehicle Axis Systems &Coordinate

ME452AutomotiveVehicleDynamicsVehicleAxisSystems&CoordinateTransformationsSeanBrennanDepartmentofMechanical&[e-mail]©2004SNB,©2000,UIUCBOT,AllRightsReservedME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide1of23VehicleAxisSystemsSAEDefinitionRef.Milliken&Milliken,RaceCarVehicleDynamics,SAEPress1995,p.116ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide2of23VehicleAxisSystemsLet’sfirstconsiderbodyfixedcoordinates.Thecoordinatesystemwillbearighthandedcoordinatesystemmeaningthatallrelationswillbeclockwiseabouttheaxisofrotation.Theaxesofmotionforthevehicleare:zverticalyawxlongitudinalrollylateralpitchvehiclec.g.ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide3of23VehicleAxisSystemsThevariablesthatwe’llbeusingareasfollows.(usually)x=longitudinalpositiony=lateralpositionz=verticalpositionXYZbodyfixedearthfixedME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide4of23VehicleAxisSystemsU=longitudinalvelocityV=lateralvelocityW=verticalvelocityp=rollrate=rollangleq=pitchrate=pitchangler=yawrate=yawanglebodyfixedbodyfixedME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide5of23VehicleAxisSystemsInadditiontoaxesfixedtothevehicle,thereareaxesthatarefixedtoaninertialpoint.Thesearecalledearth-fixedaxes.yxXXoYoY2-DcaseXYneglectingzearthfixedME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide6of23CoordinateTransformationTherelationshipbetweenearthfixedandbodyfixedcoordinatesconsistsofthreerotationsandonetranslationforthegeneralcase.ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide7of23CoordinateTransformationOrientationTransformationin2-Dxo=x1coso–y1sinoyo=y1coso+x1sinoor11cossinsincosyxyxooooooX0Y0PX1Y1x0x1y0y100ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide8of23CoordinateTransformationTogofromearthfixedcoordinatestobodyfixedcoordinatesinvolvesthreerotations,(let’sneglecttranslationfornow)1st,rotateabouttheyaw(Z)axis2nd,rotateaboutthepitch(Y)axis3rd,rotateabouttheroll(X)axisThentheearthfixedcoordinateswillbealignedwiththebody-fixedcoordinates.ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide9of23CoordinateTransformationIfpositionintheearth-fixedcoordinatesystemsisdefinedbya31vectorThentherotationisthesameasmultiplicationbyatransformationmatrixconsistingoffunctionsoftherotationangle.XYZME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide10of23CoordinateTransformatione.g.Torotatetheaxesabouttheyaw(Z)axisgivesTorotatetheaxesaboutthepitchaxis(Y)XoYoZo=cos-sinOsincosOOO1X1Y1Z1oldRotationMatrixR1newX1Y1Z1=cosO-sinO1OsinOcosX2Y2Z2originalrotatedaboutzRotationMatrixR2newME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide11of23CoordinateTransformationTorotatetheaxesabouttherollaxisgive(X)X2Y2Z2=1OOOcos-sinOsincosX3Y3Z3originalrotatedaboutzthenyRotationMatrixR3newME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide12of23CoordinateTransformationTheoriginalcoordinates[Xo,Yo,Zo]Taretheearthfixedcoordinates.Thenewcoordinates[X3,Y3,Z3]Tareveryclosetothebodyfixedcoordinate.Theonlydifferenceisatranslation.PP]Z,Y,X[]z,y,x[TT333ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide13of23CoordinateTransformationTherelationshipbetweenearthfixedandbodyfixedcoordinatesis:Notethattheserotationsarenotuniqueandchangingtheorderoftherotationwillgivedifferenttransformations.Thisisduetothefactthat,ingeneralformatrices.XYZ=R1R2R3xyzearthfixedbodyfixed+displacementvectorearthfixedABBAME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide14of23CoordinateTransformationsWedefinetheorderusedabovetobetheconventionthatwe’llbeworkingin.(yaw,pitch,roll)Notealsothatwhiletheangulardisplacementsarenoncommutable,theangularvelocitiesarecommutable.R1R2R3ME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide15of23CoordinateTransformationTheprocesscanbereversedandthepositioninbodyfixedcoordinatescanbefoundrelativetoearthfixedcoordinates.xyz=(R1R2R3)-1XYZbodyfixedearthfixed;TRR111sincethetransformationisorthogonalME452-AutomotiveVehicleDynamics©2004,SeanBrennan,PSU,©2000,AndrewAlleyne,UIUCLecture4-VehicleAxisSystemsSlide16of23BodyFixedandEarthFixedCoordinatesMostoftheworkinthisclasswill