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BuildingLEGORobotsFor
FIRSTLEGOLeagueVersion:1.0
Sept.23,2002
ByDeanHystad
www.hightechkids.org
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AbouttheAuthorandThisDocumentDeanHystadisafirstclassLEGOfanatic.DeanisanengineerforMTSSystemslocatedinEdenPrairie,Minnesota.MTSSystemsCorporationisoneoftheworldsleadingsuppliersofmechanicaltestingandsimulationequipment,makingeverythingfrom
earthquakesimulatorstoamusementparkrides.AtMTS,Deanwritescontrolsoftwareforarangeoflargeroboticsystems.Asyoucanseeinthisbook,DeanbringsthatexpertisetohispassionforLEGO.DeanhasjudgedatFLLeventsforseveralyearsandiscurrentlyworkingasacontributingauthoronasoon-to-bepublishedbookonLEGOMindstorms.Deancanbereachedatdean.hystad@mts.com.
Thisbookisasignificantpieceofwork,andfranklyisaweinspiringtome.Itisanexcellentmixoftheoryandpractice.Someofthetextmaybebeyondmiddleschoolyouth,buttheaccompanyinglabsandpresentationmaterialshouldhelpcoachesunderstandtheconceptsandpresentkeyideastotheirteams.ThebookcanalsoprovideausefultextforthoseofyouwantingtoincorporateFLLintohighschoolclasses.Ihope
youfindituseful.
ThisbookfitswellwithINSciTE'smissiontoadvanceinnovativeprogramsthatprovideanenvironmentwherekids,educators,andthetechnicalcommunitycometogethertocultivatelifelonglearninginscience,mathandtechnology.
[email protected],2002
CopyrightandTrademarkNotice2002INSciTEinagreementwith,andpermissionfromFIRSTandtheLEGOGroup.Thisdocumentis
developedbyINSciTEandisnotanofficialFLLdocumentfromFIRSTandtheLEGOGroup.Thisdocumentmaybefreelycopiedanddistributed,electronicallyorotherwise,initsentiretyonly,andonlyifusedinconjunctionwithFIRSTLEGOLeague.Anyuse,reproduction,orduplicationofthismanual
forpurposesotherthandirectlyrelatedtoFIRSTLEGOLeagueisstrictlyprohibitedwithoutspecificwrittenpermissionfromINSciTE.
LEGO,ROBOLAB,andMINDSTORMSaretrademarksoftheLEGOGroupusedherewith
specialpermission.FIRSTLEGOLeagueisatrademarkownedbyFIRST(ForInspirationandRecognitionofScienceandTechnology)andtheLEGOGroupusedherewithspecialpermission.INSciTEisatrademarkofInnovationsinScienceandTechnology
Education.
INSciTEPOBox41221
Plymouth,MN55441
www.hightechkids.org
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TableofContents1 Structures ................................................................................................................1-1
1.1 Bricks,Plates,andBeams...............................................................................1-1
1.1.1 Bricks......................................................................................................1-11.1.2 Plates.......................................................................................................1-21.1.3 Beams .....................................................................................................1-31.1.4 AxlesandPins.........................................................................................1-31.1.5 LEGOVocabulary ..................................................................................1-4
1.2 BuildingaFrame ............................................................................................1-51.2.1 LEGOGeometry .....................................................................................1-7
1.3 SNOT..............................................................................................................1-92 Gears.....................................................................................................................2-12
2.1 Spur Gears ....................................................................................................2-122.1.1 Gear Spacing.........................................................................................2-12
2.1.2 Gear Ratio.............................................................................................2-162.1.3 Torque...................................................................................................2-182.1.4 Speed.....................................................................................................2-202.1.5 Gear Trains............................................................................................2-212.1.6 ClutchGear ...........................................................................................2-23
2.2 CrownGear...................................................................................................2-242.3 BevelGear ....................................................................................................2-242.4 WormGear ...................................................................................................2-25
2.4.1 DirectionalTransmission ......................................................................2-272.5 Differential....................................................................................................2-27
2.5.1 RatchetSplitter......................................................................................2-29
2.6 Gear
Rack .....................................................................................................2-302.7 Pulleys...........................................................................................................2-312.7.1 Torque...................................................................................................2-32
2.8 Reinforcinggeartrains..................................................................................2-332.9 Backlash........................................................................................................2-34
3 Wheels ..................................................................................................................3-373.1 Sizes..............................................................................................................3-37
3.1.1 Speed.....................................................................................................3-373.1.2 Force .....................................................................................................3-39
3.2 Treads ...........................................................................................................3-413.3 Balance .........................................................................................................3-42
3.3.1 Finding
the
Center
Of
Gravity...............................................................3-423.3.2 Inertia ....................................................................................................3-433.4 WheelLoadingandFriction..........................................................................3-46
4 LegoElectronics ...................................................................................................4-484.1 RCXBrick ....................................................................................................4-48
4.1.1 Firmware...............................................................................................4-484.1.2 Programming ........................................................................................4-49
4.2 Motors...........................................................................................................4-51
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4.2.1 Modes....................................................................................................4-524.2.2 Attaching...............................................................................................4-53
4.3 TouchSensor ................................................................................................4-544.3.1 Bumpers................................................................................................4-554.3.2 LimitandPositionSwitches..................................................................4-57
4.3.3 Rotation
sensor......................................................................................4-574.4 LightSensor..................................................................................................4-584.4.1 Experiment#1,Color ............................................................................4-594.4.2 Experiment#2,AmbientLight..............................................................4-64
4.5 RotationSensor.............................................................................................4-664.5.1 Resolution .............................................................................................4-664.5.2 Internals ................................................................................................4-674.5.3 CountingErrors.....................................................................................4-68
4.6 Sensor Stacking.............................................................................................4-695 RobotDrives .........................................................................................................5-70
5.1 DifferentialDrive..........................................................................................5-70
5.1.1 Casters...................................................................................................5-715.1.2 WheelConfiguration.............................................................................5-735.1.3 Steering .................................................................................................5-745.1.4 SteeringMadeEasier ............................................................................5-765.1.5 StraightLineTravel ..............................................................................5-765.1.6 DifferentialSkid....................................................................................5-78
5.2 Steeringdrive................................................................................................5-795.2.1 Turning..................................................................................................5-805.2.2 TricycleDrive .......................................................................................5-82
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FiguresFigure1-1. BasicLEGOBrick ......................................................................................1-1Figure1-2. BrickDimensions........................................................................................1-1Figure1-3. ThreePlates=OneBrickhigh ....................................................................1-2Figure1-4. SimpleGearboxUsingTechnicPlates ........................................................1-2
Figure1-5. TechnicBeams............................................................................................1-3Figure1-6. PinsandAxles.............................................................................................1-3Figure1-7. StudsontheSideofaBeam?......................................................................1-4Figure1-8. ATypicalDrivenWheelAssembly.............................................................1-4Figure1-9. A16LTechnicPin ......................................................................................1-4Figure1-10. CommonTechnicPieces...........................................................................1-5Figure1-11. SimpleFrame ............................................................................................1-5Figure1-12. ImprovedFrame........................................................................................1-6Figure1-13. CrossBracedFrame ..................................................................................1-6Figure1-14. SnapOnConnectionsareWeakinTension ..............................................1-7Figure1-15. CrossBracing ............................................................................................1-7
Figure1-16. TwoCrossBracingChoices ......................................................................1-8Figure1-17. ATallFrame .............................................................................................1-8Figure1-18. DiagonalCrossBracing.............................................................................1-9Figure1-19. OneofJenniferClark'sIncredibleCreations. YesitsLEGO. ...............1-10Figure1-20. Turning90degrees. StudsOut ...............................................................1-10Figure1-21. Turning90degrees. StudsIn..................................................................1-11Figure1-22. UpsideDown...........................................................................................1-11Figure1-23. ExtendingBeamsUsingPinsandPlates .................................................1-11Figure2-1 LegoSpurGears.........................................................................................2-12Figure2-2. StudGearSpacing.....................................................................................2-13Figure2-3. HalfStudSpacingUsing2Holed1x2Beam ..........................................2-14
Figure
2-4.
Vertical
Gear
Spacing ...............................................................................2-14Figure2-5. CircumventingVerticalGearSpacingRestrictions...................................2-15Figure2-6. DiagonalGearSpacing..............................................................................2-15Figure2-7.CircumventingDiagonalGearSpacingRestrictions ..................................2-16Figure2-8. 3:1GearRatio ...........................................................................................2-17Figure2-9. Torque=ForcexDistance ........................................................................2-18Figure2-10. TheRatiooftheTorquesisEqualtotheRatiooftheRadii....................2-19Figure2-11. RatioofAngularVelocitiesisEqualtoInverseRatiooftheRadii .........2-20Figure2-12. Multi-stageGearTrain ............................................................................2-21Figure2-13. IdlerGear ................................................................................................2-23Figure2-14. UsingClutchGeartoLimitForces..........................................................2-23
Figure
2-15.
Crown
Gear.............................................................................................2-24Figure2-16. BevelGear...............................................................................................2-24Figure2-17. ASmallWheelBuiltfromTwo12tBevelGears....................................2-25Figure2-18. WormGear..............................................................................................2-25Figure2-19. UsingtheWormGear'sSelfLockingFeature.........................................2-26Figure2-20. LEGOLeadScrew ..................................................................................2-26Figure2-21. DirectionalTransmission.........................................................................2-27Figure2-22. LEGODifferential...................................................................................2-27
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Figure2-23. DuringTurnstheWheelsCoverDifferentDistances..............................2-28Figure2-24. UsingDifferentialtoCalculateAverageRotation...................................2-29Figure2-25. UsingaDifferentialtoCalculateDifferenceinRotation ........................2-29Figure2-26.RatchetSplitter.........................................................................................2-30Figure2-27. LEGOGearRackandPinion ..................................................................2-30
Figure
2-28.
Pulleys
and
Belts .....................................................................................2-31Figure2-29. LegoPulleys............................................................................................2-31Figure2-30. TwowaystoIncreaseTorqueCapacity...................................................2-32Figure2-31. UsingPulleystoLimitTorque ................................................................2-33Figure2-32. Forcesongears........................................................................................2-33Figure2-33. GearboxesDontHavetobeBigtobeStrong ........................................2-34Figure2-34. Backlashiscausedbypoorgearmeshing ...............................................2-34Figure2-35. Preloadingageartrainwitharubberband..............................................2-35Figure2-36. SplitgearmadefromLEGO....................................................................2-36Figure3-1. LegoWheelsandTires..............................................................................3-37Figure3-2. AVeryFastTractor...................................................................................3-39
Figure
3-3.
Force
=
Torque
/
Radius............................................................................3-40Figure3-4. TrackedRobot...........................................................................................3-41Figure3-5. MarioFerrari'sJohnny5............................................................................3-41Figure3-6. Wheelbase .................................................................................................3-42Figure3-7. FindingCGUsingBalanceMethod ..........................................................3-43Figure3-8. ModifiedBalanceMethod.........................................................................3-43Figure3-9. AnInclineMovestheEffectiveCG ..........................................................3-44Figure3-10. TurningGeneratesForcesandMoments.................................................3-45Figure3-11. FIRSTTeam254'sRobot"CheesyPoofs"DoesaVictoryWheelie .......3-45Figure3-12. CantileveredandFullySupportedWheels ..............................................3-46Figure3-13. WheelLoading ........................................................................................3-46Figure4-1. TheRCXProgrammableBrick .................................................................4-48Figure4-2. RCXCodeScreenshot...............................................................................4-50Figure4-3.ROBOLABScreenshot ..............................................................................4-50Figure4-4. 9VoltGearedMotor .................................................................................4-51Figure4-5. PWMDutyCycles ....................................................................................4-52Figure4-6. UsingaPulleytoIncreaseDrag ................................................................4-53Figure4-7. UsingColorCodingtoDocumentProperConnectorOrientation.............4-53Figure4-8. StrengtheningMotorMountswithCrossBracing.....................................4-53Figure4-9. MotorMountUsingRails..........................................................................4-54Figure4-10. WiringtheTouchSensor.........................................................................4-55Figure4-11. ASimpleBumper....................................................................................4-55Figure4-12. ABumperthatUsestheRotationSensor ................................................4-55Figure4-13. ANormallyClosedBumperDesign........................................................4-56Figure4-14. ImprovedNormallyOpenBumper..........................................................4-56Figure4-15. PositionandLimitSwitches....................................................................4-57Figure4-16. ATouchRotationSensor ........................................................................4-58Figure4-17. TheLightSensor .....................................................................................4-58Figure4-18. LightSensorColorExperiment...............................................................4-60Figure4-19. VisibleLightSpectrum ...........................................................................4-61
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Figure4-20.TheLightSensorSeesDifferentlythanOurEyes....................................4-61Figure4-21. LightColors ............................................................................................4-62Figure4-22. ColorExperimentSensorReadings.........................................................4-63Figure4-23. AmbientLightExperiment......................................................................4-64Figure4-24. LightSensorReadingsforGrayLEGOBrick.........................................4-65
Figure
4-25.
Rotation
Sensor .......................................................................................4-66Figure4-26. UsingGearReductiontoIncreaseResolution.........................................4-67Figure4-27. RotationSensorInternals. DON'TDOTHIS!!! .....................................4-67Figure4-28.HomemadeRotationSensorMadeFromLEGOParts.............................4-68Figure4-29. AHomingSwitchtoResettheRotationSensor......................................4-68Figure5-1. ADifferentialDriveRobot........................................................................5-71Figure5-2. SwivelCasters...........................................................................................5-71Figure5-3. SwivelCastersareSelfAligning...............................................................5-72Figure5-4. CastersGenerateSteeringForcesWhileAligning ....................................5-72Figure5-5. DiamondShapedWheelLayout................................................................5-73Figure5-6. TriangleShapedWheelLayout .................................................................5-74
Figure
5-7.
Turning
Radius
is
Determined
Wheelbase
and
Relative
Speed.................5-74Figure5-8. PivotingAboutaWheel ............................................................................5-76Figure5-9. SimpleLEGOSlipLimiter........................................................................5-77Figure5-10. ALockingDifferential ............................................................................5-78Figure5-11. APairofDifferentialSkidRobots ..........................................................5-79Figure5-12. Twosteeredwheelrobots........................................................................5-80Figure5-13. PathPlanningisHarderforNon-HolonomicRobots ..............................5-80Figure5-14. AckermanSteeringMinimizesGeometryInducedWheelSkid..............5-81Figure5-15. TurningRadiusisDeterminedbyWheelBaseandSteeringAngle ........5-81Figure5-16. TricycleDrive(Left)andSteeringDrive(Right)RobotsLookSimilar..5-82Figure5-17. AnySteerAngleisPossiblewithaTricycleDrive .................................5-83
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TablesTable1-1. CalculatingDiagonalLengths ......................................................................1-9Table2-1. SpurGearSizes ..........................................................................................2-13Table2-2. DiagonalGearSpacing...............................................................................2-16Table2-3. GearRatiosforLEGOSpurGears .............................................................2-18
Table2-4. Measuredpulleydiameters .........................................................................2-31Table2-5.PulleyRatios................................................................................................2-32Table4-1. LightSensorReadingsforPeanutM&Ms ..................................................4-59Table4-2. LightSensorReadingsforColorExperiment.............................................4-60Table4-3. SensorReadingsforAmbientLightExperiment ........................................4-65Table4-4. SensorReadingsforTwoStackedSensors.................................................4-69
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1-1
1 Structures1.1 Bricks,Plates,andBeamsBricks,plates,andbeamsarenotasglamorousastheRCXbrick,motors,andsensors.ButtheyarethefundamentalcomponentsthatareusedtobuildtheframethatsupportstheRCX,counteractsthemotorsforces,andholdsthesensorspreciselyinplace. MastersomeLEGObuildingfundamentalsfirst,andyourteamwillhavesuccess. Ignorethem,andyoullspendmoretimerepairingyourrobotthanyoudidbuildingit.
1.1.1 BricksThisisaLEGObuildingbrick. Littlehaschangedsinceitsintroductionin1949.AccordingtoLEGO,theyhaveproduced320billionbricks1sincethattime. Thatsapproximately52bricksperpersonlivingtoday.
Figure1-1. BasicLEGOBrickLEGObricksaremadeoutofABSplastic. Theyareinjectionmoldedtoveryexactingtolerances(0.002mm)2. Thetopofthebrickiscoveredwithcylindricalplasticbumpscalledstuds. Thebottomofthebrickhascylindricalholesortubes. Whenyousnaptwo
brickstogether,thetubesdeformslightlyaroundthestuds,lockingthetwofirmlytogether.
1.1.1.1DimensionsThecommonpracticeistorefertobricksusingtheirdimensions:width,length,andheight(thoughheightisoftenleftoffwhenreferringtostandardsizedbricks). Whendoingthis,thewidthandlengthdimensionsaregiveninstuds. Thepiecebelowisa2x4
brick.
Figure1-2. BrickDimensionsLEGObricksarebasedonthemetricsystem. The2x4brickaboveis16mmwide,32mmlong,and9.6mmhigh(ignoringthestudsontop). Thatworksoutto1stud=8mm.
1Fromwww.lego.com/eng/info/history
2FromJinSatosLegoMindstormsTheMastersTechnique
24
1
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1-2
Italsomeansthatbricksare1.2studshigh. Thisasymmetrycanleadtodesignandbuildingdifficultiesaswillbediscussedlater.
Question: What is the smallest sized cube that can be made out ofLEGO bricks?
1.1.2 PlatesPlatesareessentiallyshortbricks. Theyare1/3theheightofstandardbricks--3.2mmor0.4studs. Platesusethesamenamingconventionasbricks.
Figure1-3. ThreePlates=OneBrickhighSomeplateshavethroughholesalignedwiththebacksidetubes. TheyarereferredtoasTechnicplates,orlessobscurely,plateswithholes. Theholesacceptaxlesandconnector
pinsandmaketheTechnicplatesmuchmoreuseful.
Figure1-4. SimpleGearboxUsingTechnicPlatesConstruction Note: Use normal plates when you dont need the through
holes. Save the Technic plates for where they are needed.
Question: What is the smallest sized cube that can be made out ofbricks and plates?
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1-3
1.1.3 BeamsIn1977,LEGOintroducedTechnic3,aseriesofcomplexmodelsforolderchildrento
build. CentraltoTechnicarethenewbeamswhichare1xbrickswithholesintheirsides. Theholesarespacedatone-studintervalsandcenteredbetweenthestudsonthetopofthebeam. Thebeamscanbestackedontopofeachotherjustlikebricks. In
addition,connectorpinscanbeplacedinthesideholesallowingthebeamstobeassembledsidebyside. Thenumberofassemblytechniquesavailableusingthenew
partsisstaggering.
Figure1-5. TechnicBeams1.1.4 AxlesandPinsTheRISkitcomessuppliedwithawidevarietyofpinsandaxlesforconnectingTechnic
beamstogether. ThemostcommonlyusedoftheseistheblackTechnicpinwithfriction,orfrictionpin. Thefrictionpinhassmall,raisedridgesthatmakeitlocktightlyintheholesofaTechnicbeamprovidingaverystrongconnection. Alongversionofthefrictionpincanbeusedtopinthreebeamstogether. ThedoublepinworkswellwiththegoofytransparentblueconnectorblockthatcomesinthenewerRISsetsandhasanaxleholethatcansometimesbeuseful.
Figure1-6. PinsandAxlesSlightlylesscommonisthegrayTechnicpin. Similarinappearancetothefrictionpin,itlackstheridgesandhasaslightlyloosefit. TheTechnicpinisagoodchoiceforpivotsorhinges. Theshortpostofthe TechnicpinfitsnicelyinthehalfdeepholesinthesideoftheRCX,anditisoftenusedtoattachbeamstothesideoftheprogrammable
3Fromwww.lego.com/eng/info/history
A
B
C
D
E
F
G
H
I
J
K
L
A. PinwithFriction G. AxlePinB. 3LDoublePin H. 6LAxleC. LongPinwithFriction I. 3LPinwithStudD. TechnicPin J. FullBushingE. HalfPin K. HalfBushing
F.
Pin L. Long
Pin
with
Stop
Bushing
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1-4
brick. Theshortpostofthe TechnicpinisactuallythesamesizeasaLEGOstudandcanbeusedtomimicstudscomingoutofthesideofabeam.
Figure1-7. StudsontheSideofaBeam?Axlesarelongrodsthathavea + shapedcrosssection. TheyslideeasilythroughtheholesinTechnicbeams,buttheyfittightlyinthecross-holesfoundinwheels,gears,
bushings,andotherTechnicelements. Axlesareavailableinevenstudlengthsstartingat2andgoingupto12. Therearealso3and5studlongaxles. Itiscommontouseshorthandwhenreferringtoaxles,describingtheaxleusingitslengthfollowedby L .
Thus
a
four
stud
long
axle
is
a
4L
axle.
Figure1-8. ATypicalDrivenWheelAssemblyThereareafewoddballpartsthatdontreallyfitintheaxleorpincategory. ThefirstistheaptlynamedaxlepinwhichishalfTechnicpinandhalfaxle. Itismostcommonlyusedtoattachgearstothesidesofbeams. Anotheristhe3Laxlewithstud,whichisathreestudlongTechnicaxlewithastudononeend. TrystickingthestudintheholeofaTechnicbeam. Theconnectionissurprisinglystrong. ThelongTechnicpinwithstop
bushingismyfavoritepart. IuseitwheneverIneedaremovablepinconnectionbecausethebushingontheendiseasytograsp. Thecross-holeinthestopbushingacceptsanaxleandcanbeusedtomakesomeinterestingparts.
Figure1-9. A16LTechnicPin1.1.5 LEGOVocabularyTheRISkitisfilledwithallkindsofinterestingplasticpartsthatdontlookmuchlikeanythingwithwhichyouareusedtobuilding. Lookhardandyoullnotfindonescrew,
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1-5
nut,bolt,ornail. Instead,youhavetheblueholey-thingandthelittlegraywhatchamabobwiththeaxleinit.Toaidincommunicationandfacilitatetheexchangeofideas,theLEGOcommunityhascomeupwithnamesforeachthingamajiganddoohickey. Mostofthenameswere
providedbyLEGO--extractedfrommarketingorpackagingliterature. Butmanypart
monikers
originated
in
the
user
community.
Figure
1-10
lists
the
names
of
some
of
theTechnicpartsincludedintheRISkit.
Figure1-10. CommonTechnicPiecesLinks: Jim Hughes maintains Technica, a beautiful website that has acomplete Technic parts registry with pictures and an interesting history of
LEGO. His URL is w3.one.net/~hughesj/technica/technica.html.
1.2 BuildingaFrameArobotneedssomesortofframe. Theframegivestherobotitsshape. Itprovides
mountingpointsforsensorsandreactstheforcesgeneratedbymotorsandgears. Itislikeourskeleton,whichgivesusourshape,supportsourorgans,andreactstheforcesgeneratedbyourmuscles. Agoodframeisstrong,lightweight,andholdstogetherevenaftermuchuse.
Figure1-11. SimpleFrameFigure1-11showsasimpleframemadeoutofbeamsand1x8plates. Itsstrong,lightweight,andthedimensionsareappropriateforthebaseofarobotplatform. Butitis
A
B
C
D
E
F
G
H
I
A. AxleJointerPerpendicular F. Beam1x2withAxleholeB. AngleConnector#1 G. AxleJoinerC. ConnectorwithAxlehole H. Liftarm1x3D. PoleReverserHandle I. Liftarm1x6E. ConnectorBlock3x2x2
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1-6
notveryrigid. Agentlepushonopposingcornerscausestheframetotwistoutofshape.Eventuallythecornerconnectionsworkloose,andtheframefallsapart.
Theproblemisthattheplatesdonotlockthecornersatrightangles. Thereisasmallamountofclearancebetweentheendsofthe1x6beamsandthesidesofthe1x12
beams.
This
allows
the
studs
to
act
as
hinges.
Replacing
one
or
more
of
the
1
x
8
plateswith2x8platesmakestheframemuchmorerigid.
Figure1-12. ImprovedFrameTheimprovedframeismuchstiffer. Pushingonthecornerscausesittoflexhardlyatall.The2x8platefirmlylockstheshortandlongbeamstogetheratrightangles. Thisframeisadequateformanyapplications,butitcanbemadeevenstronger.
Figure1-13. CrossBracedFrameTheframeinFigure1-13usescrossbracingtoholdittogether. The1x3liftarms
preventtheframefrombeingpulledapart. CrossbracingisausefultechniqueforbuildingverystrongLEGOstructures. TheconnectionsbetweenLEGObricksareverystrongincompression(aforcepushingthebrickstogether),andinshear(asideways
forcetryingtoslidethebricksacrosseachother),buttheyarerelativelyweakintension(aforcepullingthebricksapart). Whencrossbracing,wereinforceaconnectionwhichisweakintensionbyaddingcomponentsthatwillbeinshear. Inthiscase,the1x3liftarmsandaxlepinsareinshearwhentryingtopullthebeamsapart.
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Figure1-14. SnapOnConnectionsareWeakinTension1.2.1 LEGOGeometryA1x6Technicbeamis1studwide,6studslong,and1.2studshigh. Bricksandbeamsnotbeinganintegernumberofstudshighcausesproblemswhentryingtodocross
bracing. ThiscanbeseenintheleftassemblyinFigure1-15. Thesecondholeinthe
verticalbeamdoesnotlineupwiththeholeinthehorizontalbeam.
Figure1-15. CrossBracingIntheassemblyontheright,thethirdholeintheverticalbeamalignsperfectlywiththeholeinthehorizontalbeam. Thisisbecausethebeamandthetwoplatesadduptoexactlytwostudstall(1.2+2*0.4=2). Youwillfindthatcrossbracingonlyworksoutwhentheverticalholespacingisdivisiblebytwo.
Question: What is the shortest stack of beams (no plates allowed)that will align with holes in a vertical beam?
Construction Note: The two assemblies below are the same height, but the
one on the right allows for locking the beam at an intermediate point and has
better spacing for vertical meshing of LEGO gears.
Compression Shear Tension
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1-8
Figure1-16. TwoCrossBracingChoicesThissametechniquecanbeusedtobuildtallframesthatarelightweightandstrong.Figure1-17showsatallframethatusesfrictionpinsatthecornerstoattachtheverticalandhorizontalmembers.Noticethatthe1-2-1(1Beam-2Plates-1Beam)techniqueisusedtogettheproperspacing.
Figure1-17. ATallFrame1.2.1.1DiagonalBracingItsveryeasytogetlockedintothemindsetthathorizontalandverticalaretheonlywaystobuild. Thegrid-likenatureoftheLEGOpiecesreinforcesthisthinking. Butdiagonalconnectionsarepossibleaswell.
Diagonalbracingistrickiertoimplementthanperpendicularcrossbracing. Crossbracingcanbeusedonanyassemblywherethedimensionisevenlydivisiblebytwostuds,butitcantbeusedanywhereelse. Withdiagonalbracingtherearemoresolutions,
buttheirderivationsarenotasobvious.
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1-9
Figure1-18. DiagonalCrossBracingYoucanfinddiagonalbracingsolutionsthroughexperimentation. Laythediagonal
braceonthepart,andmoveitaboutuntilyoufindaplacewheretheholeslineup. Itcan
alsobedoneanalyticallyusingthePythagoreanTheoremforrighttriangles.Thesumofthesquaresofthelegsofarighttriangleequalsthesquareofthehypotenuse. Thisisoftenwrittenas 22 BAC += . Thetwolegsarethebase(widthacrossthebottom)andtheheight. Thehypotenuseisthediagonalbeam. Diagonalbracingispossibleifthehypotenuseisclosetoanintegernumber(lessthan0.05studsdifference).
Table1-1. CalculatingDiagonalLengthsBase(A) Height(B) Hypotenuse Comments4 4 5.65 Doesntfit3 4 5 Perfectfit1.5 4.8 5.03 Fits,butalittletight
6 8 10 Perfectfit
1.3 SNOTSNOTisalltherageintheLEGObuildingcommunitytoday. SNOT,standingforStuds
NotOnTop,isawayofbuildingwithLEGOthatdoesnotalwaysutilizethetraditionalclick-fitassemblytechniques. Thiscanbeverytrickyandrequiresgreatimagination,buttheresultingmodelsareverybeautifulandquiterealistic.
3
4
1.5
4.8 58
6
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Figure1-19. OneofJenniferClark'sIncredibleCreations. YesitsLEGO.Luckily,TechnicismoreflexibletobuildwiththanotherLEGO. Butitisstilldifficultsometimestofigureoutawaytoattachthemotorwhereyouwantitortopositionthesensorinjusttherightplace. TryingtodiscovernewwaystoputLEGOtogetherisachallengingactivitythatcanbealotoffun. Herearesomeideastogetthecreativejuicesflowing.
Figure1-20. Turning90degrees. StudsOut
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Figure1-21. Turning90degrees. StudsIn
Figure1-22. UpsideDown
Figure1-23. ExtendingBeamsUsingPinsandPlates
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2 GearsEventuallyyouwillwanttomakeyourrobotmove. Afterall,thisisnottheFLLSculptureCompetition! The9voltmotorsprovidethemotivepower,buttheymaynotrunattherightspeedorbepowerfulenough. Itmayalsobetoodifficulttopositionthe
motorswheretheycanbedirectlyattachedtothewheels. Alltheseproblemscanbesolvedusinggears.
Gearsaregenerallyusedforoneofthefollowingreasons:1. Totransmittorquefromoneaxletoanother2. Toincreaseordecreasethespeedofrotation3. Toreversethedirectionofrotation4. Tomoverotationalmotiontoadifferentaxis5. Tochangerotarymotiontolinearmotion6. Tokeeptherotationoftwoaxlessynchronized
2.1 SpurGearsAspurgearisusedwhenshaftsmustrotateinthesameplane.Inaspurgeartheteetharestraightandparalleltotheshaft.Theyarebyfarthemostcommontypeofgears,andtheyarewhatmostpeoplepicturewhenyoumentiongears. LEGOincludesfourdifferentsizedspurgearsintheRoboticsInventionSystem.
Figure2-1 LegoSpurGearsGearsarenormallyreferredtobytheirtypeandthenumberofteeth. Take,forexample,an8toothspurgear. Sometimesakindofshorthandnotationisusedwhere tooth isreplacedwith t andthetypeisnotspecifiedforspurgears(becausetheyaresocommon). Forexample,a40toothspurgearwouldbereferredtoasa40tgear.
2.1.1 GearSpacingSizesofLEGOspurgearsareshowninTable2-1. Itisinterestingtonotethattheratiooftheradiiisequaltotheratioofthetoothcount(8/24=0.5/1.5=1/3). Thisisbecauseallthedifferentsizedspurgearshavethesamesizedteeth--eventhelittle8tgearwithitsinvoluteprofilegearteeth. Havingthesamesizedteethallowsthegearstomesh
properly.
24 40168
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Table2-1. SpurGearSizesTeeth 8 16 24 40Radius(studs) 0.5 1 1.5 2.5
Knowingtheradiusofagearandthenumberofteeth,wecancalculatethesizeofeach
tooth. Thisinformationcanbeusefultoknowwhenusingaspurgearwithagearrack.
Circumference =2xPixRadiusCircumference =ToothSizexToothCountSizexCount =2xPixRadiusSize =2xPixRadius/Count16tooth =2xPix1/16
=Pi/8studs=0.392studsor3.14mm
Checkitoutforyourself. Theteethforeachspurgearreallydoevaluatetothesame
size!
2.1.1.1HorizontalGearSpacingUsingtheinformationfromFigure2-2,wecancalculatethegearspacingrequiredfor
propermeshing. Thedistancebetweenthetwoaxlesisequaltothesumofthetworadii.
8t 16t 24t 40t
8t 1.0studs 1.5studs 2.0studs 3.0studs
16t 1.5studs 2.0studs 2.5studs 3.5studs24t 2.0studs 2.5studs 3.0studs 4.0studs
40t 3.0studs 3.5studs 4.0studs 5.0studs
Figure2-2. StudGearSpacingCombinationsusingeight,twenty-four,andfortytoothgearsarestraightforwardtosetup. Sixteentoothgearsareeasytousewithothersixteentoothgears,butrequirehalfstudspacingtomeshwith8t,24t,or40tgears. Atwoholed1x2Technicbrickcanbeusedtogethalfstudspacing.
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Figure2-3. HalfStudSpacingUsing2Holed1x2BeamConstruction Notes: Gear spacing of two or three studs provides the
maximum number of gear combinations. Two stud spacing also works well in
vertical layouts.
2.1.1.2VerticalSpacingItisdifficulttomeshgearslaidoutonaverticalaxis. Thegearspacingtableshowsthat
propergearmeshingisachievedathalfstudintervalsstartingat1studandgoingupto5
studs. Forgoodverticalgearmeshing,weneedtobuildastructureusing1.2studhighbeamsand0.4studhighplatesthatprovidesthecorrectspacing. Itworksoutthat2and4studsspacingaretheonlysolutions.
8t 16t 24t 40t8t 2.0studs
16t 2.0studs24t 2.0studs 4.0studs40t 4.0studs
Figure2-4. VerticalGearSpacingThespacingdoesnthavetobeperfectforthegearstomesh. Anythingwithin0.08studsworksfairlywell. The8toothand16toothgearsalmostfitat1.6studspacing(idealis
1.5studs),buttheydontmeshsecurely,anditispossibleforthegearstoslip.
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Figure2-5. CircumventingVerticalGearSpacingRestrictionsFigure2-5showsonewaytogetaroundverticalgearspacingrestrictions. Themiddlegearisheldinplacebyanaxlepinintheverticalbeam. Thetopandbottomgearsare
spacedsixstudsapartallowingthemtofitnicelyonaxlespassingthroughthehorizontalbeams.
2.1.1.3DiagonalgearspacingDiagonalgearspacingistrickiertocalculatethanperpendicularorverticalgearspacing.Aswasdonetocalculatediagonalcrossbracingsolutions,Pythagorasstheoremforrighttrianglesisusedtocomputethediagonal(hypotenuse). ThisvalueisthencomparedtotheidealgearspacinginformationfromthetableinFigure2-2.
Figure2-6. DiagonalGearSpacingTable2-2showsthepossiblelayoutsfor8,16,24,and40toothspurgears. Verticaldimensionsaredisplayedinthelefthandcolumn,horizontaldimensionsinthetoprow.Attheintersectionofeachrowandcolumnisthecalculateddiagonalgearspacing. Theentryisleftblankifthisdistancedoesnotmatchthespacingofanyoftheavailablegearcombinations. Acceptablegearmeshesmustbewithin0.08studsoftheidealvalue.Excessivebindingofthegearsoccursifthespacingistoosmall. Ifthespacingistoogreat,thegeartrainwillhavelargeamountsofbacklash,andslippagemayoccur.
2.8
2
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Table2-2. DiagonalGearSpacing
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 1 1.5 2 2.5 3 3.5 4 5
1.2 1.56 1.921.6 2.56 2.97
2 2 2.06 2.5 4.03 4.92
2.4 2.45 3.47
2.8 2.97 3.44
3.2 3.53 4.06
3.6 5.02
4 4 4.03 5.00
4.4 5.06
4.8 5.03
Figure2-7.CircumventingDiagonalGearSpacingRestrictionsFigure2-7showsonewaytogetarounddiagonalgearspacingrestrictions. Three
perpendicularaxlejoinersareusedtoholdthegearsinplace. Themiddle24toothgearisattachedusinganaxlepin. Thetwo8toothgearsarespaced4.1studsaparttofitonaxlespassingthroughthehorizontalbeams. Theextra0.1studsspaceisdividedevenly
betweenthetwogearmeshes.
2.1.2 GearRatioGearratioishowmuchtheoutputshaftofagearboxturnsforagivenrotationofthe
inputshaft. InFigure2-8,wehaveagearboxconsistingoftwogears:an8tgearontheinputshaftanda24tgearontheoutputshaft. Ifthe8tgearrotatesonefullrevolutiontheneightofitsteethwouldpassthroughthestartingline. Becausethetwogearsaremeshed,eightofthe24tgearsteethwouldalsopassthestartingline.Sincetheteethareevenlydistributedaroundthecircumferenceofthegear,the24tgearturns8/24thsor1/3ofarevolution.
HorizontalSpacing
VerticalSpacing
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Figure2-8. 3:1GearRatioUsingtherotationinformationwecancalculatethegearratio. Followingpopularconvention,itisexpressedasaratioofwholenumbers.
GearRatio =1:1/3=3:1
The3:1gearratiotellsusthattheinputshaft(attachedtothe8tgear)hastocompletethreefullrevolutionsfortheoutputshaft(attachedtothe24tgear)torotateallthewayaroundjustonce. Usinggearstoslowdownrateofrotationordecreasetheamountofrotationiscalledgearingdown. Ifweweretoswitchthe8tand24tgearsaround,theoutputshaftwouldspinthreerevolutionsforeachrevolutionoftheinputshaft. Thisisgearingup,andthegearratiowouldbe1:3.
Youmayhavenoticedthatthegearratioistheinverseoftheratioofthenumberofgearteeth. Thereasonforthisiseasiertoseeifwerecalculatethegearratiousinganinputshaftrotationofonly1tooth. Inthecaseofthe8tgeardrivingthe24tgear,theinputshaftwouldturn1/8ofarevolutionandtheoutputshaft1/24ofarevolution.
GearRatio =1/8:1/24=24:8=3:1
Usinggeartoothcountstodirectlycalculategearratiosiseasierandfasterthancalculatingtherotationsfirstandthenusingthesevaluestoderivethegearratio.
1rev8teeth
1/3rev8teeth
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Table2-3. GearRatiosforLEGOSpurGears
8t 16t 24t 40t
8t 1:1 2:1 3:1 5:116t 1:2 1:1 3:2 5:2
24t 1:3 2:3 1:1 5:340t 1:5 2:5 3:5 1:1
Links: Ted Cochran has a nice treatment of gear ratios with Minnesota FLLexamples at www.hightechkids.org/fll/coaching/Gears/gears.htm.
2.1.3 TorqueTorqueisaforcethattendstorotateorturnthings. Yougenerateatorqueanytimeyouapplyaforcetothehandleofawrench.Thisforcecreatesatorqueonthenut,whichtendstoturnthenut. Ifthenutistootight,youeitherpullharder(moreforce),orgeta
longerhandledwrench(moredistance).
Figure2-9. Torque=ForcexDistanceFromthewrenchexample,weseethattorqueisaproductofforceanddistance. Theunitsusedwhenmeasuringtorquereflectthis. IntheU.S.wemeasuretorqueinfoot-
pounds(ft-lbs).Newton-meters(Nm)istheunitoftorqueinthemetricsystem.
Distance
TorqueForce
16:8or
2:124:8
or3:1
40:8or
5:1
OutputShaftorDrivenGear
Inpu
tShaftor
DrivingGear
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Gearsoperatebytransmittingforcesattheteethofthegear. InFigure2-10,the24tgearisgeneratingaforceagainsttheteethofthe40tgear. Theforce(f)isequaltothetorque(t1)appliedtothe24tgeardividedbytheradius(r1). Theforcetransmittedbyagearisinverselyproportionaltothegearsradius.Thelargertheradiusofthegear,thelessforceitwillgenerateforagiventorque.
Theforceagainsttheteethofthe40tgearcreatesatorque(t2)equaltotheforce(f)timestheradius(r2). Thetorquecreatedbyapplyingaforcetoagearisproportionaltothegearsradius. Applyingaforcetoalargegearwillcreatemoretorquethanapplyingthesameforcetoasmallgear.
Figure2-10. TheRatiooftheTorquesisEqualtotheRatiooftheRadiiThetorqueavailableattheaxleofdrivengear(40t)canbeexpressedasafunctionofthetorqueturningthedrivinggear(24t)andthetworadii;t2=t1xr2/r1. Asmallgeardrivingalargergearwillamplifytorque. Therewillbemoretorqueavailableattheshaftofthelargergearthanissuppliedtotheshaftofthesmallergear.
Fromthediscussionearlier,weknowthatthegearratioistheinverseoftheratioofthegearsradii. Thisallowsustousethegearratiotocalculatethetorqueamplificationofagearsystem. Usingthegearratiofortheexampleabove:
GearRatio=r2:r1=r2/r1=5/3
t2 =t1xr2/r1=t1xGearRatio=t1x5/3
TheTechnicgearmotorsuppliedwiththeRISkitcangenerateatorqueofabout9Ncm
withfreshbatteries. Intheexampleabove,howmuchtorquewouldbeavailableattheshaftofthe40tgearifweattachedthemotordirectlytotheshaftofthe24tgear?
t2 =t1x5/3=9Ncmx5/3=15Ncm
r1
r2
t2
t1
f
f =t1/r1t2 =fxr2t2 =(t1/r1)xr2t2/t1=r2/r1
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2.1.4 SpeedWhenusingsimplemachineslikegears(oranykindofmachineforthatmatter)younevergetsomethingfornothing. Inourpriorexample,weusedgearstoincreasetorque.Whatwetradedtogetthetorqueincreasewasspeed. Theoutputshaftmayturnstronger,
butitalsoturnsslower.
Ifwemeasuretheanglesinradians(1radian=180degrees/Pi=1revolution/(2xPi)),thetoothvelocityofagear(v)isequaltotheangularvelocity()timestheradius(r).Thereisaproportionalrelationshipbetweentheradiusandthetoothvelocity. Atagivenangularvelocity,theteethofalargergearwilltravelfasterthantheteethofasmallergear.
Figure2-11. RatioofAngularVelocitiesisEqualtoInverseRatiooftheRadiiWhentwogearsaremeshed,thetoothvelocityforeachgearisthesame. Intheexampleabove,whatistheangularvelocityofthe40tgearifthe24tgearisspinningat180rpm?
r1 =1.5studs=12mmr2 =2.5studs=20mm1 =180rpm
=180revolutions/minutex1minute/60seconds=3revolutions/second=3revolutions/secondx2Piradians/revolution=6Piradians/second
=18.85radians/secondv =1xr1
=18.85radians/secondx12mm/radian=226.19mm/second
2 =v/r2=(226.19mm/second)/(20mm/radian)=11.31radians/second=1.8revolutions/second=108rpm
Thelargergearspinsmoreslowlythanthesmallergear. Theratiooftheangularvelocitiesisequaltotheinverseoftheratiooftheradii. Knowingthiswecancalculatetheangularvelocityusingtheradiusratiodirectly.
r2
r1
1
2
v
v =1xr12 =v/r22 =(1xr1)/r2
=1xr1/r22/1=r1/r2
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2 =1xr1/r2=180rpmx1.5studs/2.5studs=108rpm
Just
as
with
torque,
the
gear
ratio
can
be
used
in
place
of
the
ratio
of
the
radii.
GearRatio=r2:r1=r2/r1=5/3
2 =1xr1/r2=1x(1/GearRatio)=1/GearRatio=180rpm/(5/3)=180rpmx3/5=108rpm
Links: How Stuff Works, one of the best sites on the web, has a wonderfularticle about gears, gear ratios, torque, and speed. Check it out at
www.howstuffworks.lycozone.com.
2.1.5 GearTrainsIfanumberofgearsarecascadedsothatseveralmeshesrelateaninputtoanoutput,ageartrainisformed.Usingfour40tandfour8tgears,itispossibletocreateagearratioof625:1.
Figure2-12. Multi-stageGearTrainThegeartrainaboveconsistsoffourstages,eachhavinga5:1gearratio.Tocalculatetheoverallgearreduction,westartwiththegearratiobetweenAandB,multiplythattimestheratiobetweenBandC,andsoonuntilwegettoE.
A C E DB
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AB =5:1BC =5:1CD =5:1DE =5:1AE =ABxBCxCDxDE
=5:1
x5:1
x5:1
x5:1=5x5x5x5:1x1x1x1
=625:1The9vgearedmotorcangenerateabout0.06ft-lbsoftorquewithfreshbatteries. IfwehookedoneuptoshaftAandturnediton,thetorqueamplificationsshouldproduce37.5ft-lbsoftorqueatshaftE. Thatsenoughtorquetotightenthelugnutsonmycarswheels.Duetofrictioninthegeartrain,theactualavailabletorquewillbemuchlessthan37.5ft-lbs,butitwillbestrongenoughtosnapaxlesandbreakgearteeth. Becarefulwhenusinglotsofgearreduction.
As
mentioned
before,
any
increase
in
torque
is
accompanied
by
a
corresponding
decreaseinrotationspeed.Ifwetookourmotor,hookedituptoshaftAandgotitspinningat200revolutionsperminute(rpm),shaftEwouldspinat0.32rpmorabout1revolutionevery3minutes. Ifinstead,weattachedthemotortoshaftE,thenshaftAshouldspinat125,000rpm. Thatwouldhavetheteethofthelast40tgeartravelingat585mph!Fortunately,thelargeamountoffrictioninthegeartrainpreventstheweak9vmotorfromgeneratingsuchdangerousspeeds. Whengearinguptoincreasespeed,torqueamplificationmagnifiesthefrictionforcesofthelaterstages. Withagearratioof1:625,itisunlikelythatthemotorispowerfulenoughtospinthegearsatall.
Question: For the example above, what would the gear ratio be if the
8 tooth gears were replaced with 24 tooth gears?
2.1.5.1IdlerGearThe24toothgearinFigure2-13isanidlergear. Anidlergeardoesnotaffectthegearratioofageartrain. ThegearratioforACisthesameasitwouldbeifthe24toothgearwereleftout. Idlergearsarequitecommoninmachineswheretheyareusedtoconnectdistantaxles. Theyarealsousedtochangethedirectionofrotationoftheoutputshaft.
AB =3:1BC =1:3AC =ABxBC
=3:1x1:3=3x1:1x3=1:1
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Figure2-13. IdlerGear2.1.6 ClutchGearThefunnylookingwhite24toothspurgearwiththewritingonitsfaceistheclutchgear.Theclutchgearisspecialinthatthegearteethareabletorotateabouttheshaft. Ithasaninternalclutchmechanismthatstartstoslipwhenitsmaximumratedtorqueisexceeded.Theclutchgearisusedtolimitthetorqueofagearedsystem,savingmotorsand
preventingyourrobotfromtearingitselfapart.
Theclutchgearhas2.5.5Ncm stampedonitsface. Thisisthetorqueratingoftheclutch.NcmstandsforNewtonCentimeter,aunitoftorque(torqueisaproductofforceanddistance,centimeterisaunitofdistance,andNewtonisaunitofforce). Theclutchgearcantransmitamaximumtorqueoffrom2.5to5Ncm(0.018to0.037ftlbsor0.22to0.44inlbs).
Figure2-14. UsingClutchGeartoLimitForcesInFigure2-14,theclutchgearisusedtolimittheforceoftheliftarmpressingagainsttheconnectorpegstops. Withouttheclutchgear,wewouldruntheriskofstallingamotorordamagingtheassembly. Usingtheinformationwehaveabouttheclutchgear,
gearratios,anddistance,itspossibletocalculatethemaximumforcetheliftarmcangeneratetopushagainsttheconnectorpeg.
Amaxtorque =5NcmGearratioA:B =5:3Bmaxtorque =AtorquexGearRatio
=5Ncmx5/3=8.33Ncm
24mmBA
B
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DistanceBtoPin=3studs=24mmor2.4cm
MaxforceatPin =Btorque/Distance=8.33Ncm/2.4cm=3.47Nor0.78lbs
2.2 CrownGearThecrowngearhasteeththatareraisedononesideandroundedoffontheother.Thisgivesitacrown-likeappearance. Thecrowngearisusedwhentheshaftstobeturnedmeetatanangle--usuallyarightangle. Thecrowngearcanbemeshedtospurgearsandwormgears,butitdoesntmeshwellwithothercrowngears. Thecrowngearcanalsobeusedinplaceofa24toothspurgear.
Figure2-15. CrownGearTreatthecrowngearjustlikethe24tspurgearwhencomputinggearreduction. Forthegearboxabove,ifthe8tgearisattachedtotheinputshaft,andthe24tspurgeartotheoutputshaft,thegearratiois:
GearRatio =24:8x24:24=
3:1
x1:1=3:1
2.3 BevelGearThebevelgearhasteeththatslopealongonesurfaceofthedisc.Itisusedwhentheshaftstobeturnedmeetatanangle--usuallyarightangle. Ithaslessfrictionthanthecrowngear,butitcanonlymeshwithanotherbevelgear. LEGOproduces12tooth,14tooth,and20toothbevelgears. Unfortunately,onlythe12toothbevelgearisincludedintheRISkit.
Figure2-16. BevelGear
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Theold14toothbevelgearshaveaproblemwiththeirthinfacetedteethbreaking.The12toothbevelgearhasacircularbackingplatethatstrengthensthegearteethtopreventthisfromhappening. Thebackingplategivesthegearasmoothcircularoutlineandallowsittobeusedasaverysmallwheel.
Figure2-17. ASmallWheelBuiltfromTwo12tBevelGearsQuestion: A wheel made out of bevel gears would not have muchtraction. For what applications could this be a benefit?
2.4 WormGearAwormgearisascrewthatusuallyturnsalongaspurgear.Motionistransmitted
betweenshaftsthatareatrightangles.Ifyouwanttocreateahighgearratio,nothingbeatsawormgear.Eachtimetheshaftspinsonerevolution,thespurgearmovesonetoothforward.Ifthespurgearhas24teeth,youhavea24:1gearratioinaverysmall
package.
Figure2-18. WormGearTheefficiencyofawormgearsystemismuchlowerthanthatofnormalmeshesbecausethewormworksprimarilybysliding,thusincreasingfrictionallosses. Thishasan
unusualsideeffectinthatthewormgearisasymmetricandself-locking. Youcanturntheinputshafttodrivetheoutputshaft,butyoucannotturntheoutputshafttodrivetheinputshaft.
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Figure2-19. UsingtheWormGear'sSelfLockingFeatureThemechanisminFigure2-19makesuseofthewormgearsself-lockingfeaturetoholdthebucketinplace. Theaxlethroughthewormgearcanbeturnedtoraiseorlowerthe
bucket. Onceinplace,notorqueisrequiredtomaintaintheposition.
Thewormgearcanalsobeusedtoimplementaleadscrew. Leadscrewsconvertrotarymotiontolinearmotion. Aleadscrewconsistsofathreadedrodandacapturednut.Spinningtherodcausesthenuttomovealongthelengthoftherod. Leadscrewsareusedinmanydevices. OneofthemostvisibleexamplesistheGeniegaragedooropener.
Figure2-20. LEGOLeadScrewFigure2-20isaLEGOimplementationofaleadscrew. Thethreadedrodisconstructedofmultiplewormgearsonthecenteraxle. Thehalfbushingsontheouteraxlesmakeupthecapturednut. Spinningthecenteraxlecausestheouteraxlestomoveinorout.
Construction Note: Be careful when constructing a threaded rod out of
worm gears. You can put two worm gears on an axle in four different ways,
but only one results in a continuous thread.
Good Bad
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2.4.1 DirectionalTransmissionOneofthecleverestusesIhaveseenforawormgearisthedirectionaltransmission. Adirectionaltransmissionletsyouuseoneoutputporttoperformtwofunctions. Ithasoneinputshaftandtwooutputshafts. Awormgearslidesalongtheinputshaftandengagesspurgearsontheoutputshafts. Whichgearisengagedisdependentupontherotation
directionoftheinputshaft.
Figure2-21. DirectionalTransmissionThe
directional
transmission
works
because
of
friction.
When
you
spin
the
input
shaft,thewormgearexertsaforceagainstoneofthespurgears. Thespurgearresiststurningbecauseoffrictionandpushesbackagainstthewormgear. Iffreetodoso,thewormgearslidesalongtheinputshaft,engagingtheotherspurgearandeventuallyrunningintothesupportbeam.Nowtheforcerequiredtoslidethewormgearexceedstheforcerequiredtoturnthespurgear,andthespurgearbeginsturning. Reversingrotationoftheinputshaftcausesthewholesequencetooccuragainbutintheoppositedirection.
2.5 DifferentialAdifferentialisadevicethattakesatorqueappliedtoitshousingandevenlydistributesittotwooutputshafts,allowingeachoutputtospinatadifferentspeed. Differentialsare
foundonallmoderncarsandtrucks. All-wheeldrivecarsliketheAudiQuattrocanhavethreedifferentials;onebetweenthefronttires,onebetweenthereartires,andonebetweenthefrontandreardifferentials. LEGOprovidesadifferentialwiththeRISkit.
Figure2-22. LEGODifferentialCarwheelsspinatdifferentspeedswhenturning.Eachwheeltravelsadifferentdistancethroughtheturnwiththeinsidewheelstravelingashorterdistancethantheoutsidewheels.Sincespeed=distance/time,thewheelsthattravelashorterdistancetravelatalowerspeed.
Thisisnotaproblemforwheelsthatcanspinindependently.Butinmostvehicles,thedrivenwheelsarelinkedtogethersothattheycanbepoweredbyasinglemotor.Without
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adifferential,thewheelswouldbelockedtogether,forcedtospinatthesamespeed.Thiswouldmaketurningdifficult.Forthecartobeabletoturn,onewheelwouldhavetoslip,whichcanrequiregreatforce.
Figure2-23. DuringTurnstheWheelsCoverDifferentDistancesThedifferentialisamechanicalcalculator. Itcomputestheaveragerotationspeedofthetwoinputaxlesandspinsthedifferentialhousingatthisrate. Thisisaninteresting
propertythatcanbeusedinmanyways. InFigure2-24,thedifferentialisusedto
calculatetheaveragerotationspeedofthetwowheels. Attachingarotationsensortothedifferentialhousingwouldprovideanaccuratemeasureoftraveldistance.
Wheel1 Wheel2 Housing100rpm 100rpm (100+100)/2=100rpm70rpm 50rpm (70+50)/2=60rpm
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80rpm -80rpm (8080)/2=0rpm
Figure2-24. UsingDifferentialtoCalculateAverageRotationTheplatformbelowusesthedifferentialtomeasurethedifferenceinrotationspeed
betweenthetwoaxles.Noticetheextragearbetweenthedifferentialandthewheelon
the
right
side.
It
reverses
the
direction
of
rotation
for
the
right
differential
shaft.
Arotationsensorcouldbeusedheretomeasureturning.
Wheel1 Wheel2 Housing100rpm 100rpm (100-100)/2=0rpm70rpm 50rpm (70-50)/2=10rpm80rpm -80rpm (80+80)/2=80rpm
Figure2-25. UsingaDifferentialtoCalculateDifferenceinRotation
2.5.1 RatchetSplitterAnintriguinguseforthedifferentialisaspartofaratchetsplitter. Aratchetsplitterletsyouperformtwodifferentfunctionswithasinglemotor,justlikethedirectionaltransmission. Butinsteadofusingfrictiontoswitchgears,theratchetsplitterusesaratchetingmechanismtopreventanaxlefromspinninginbothdirections.
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Figure2-26.RatchetSplitterTheassemblyinFigure2-26ispartofadifferentialdrivecarIbuiltthatusedasinglemotortoprovidebothlocomotionandsteering. Theperpendicularaxlejoinerand24tgearisaratchetingmechanismthatpreventstherightaxlefromrotatingcounterclockwise. Whenthemotorturnsclockwise,theleftandrightaxlesturninthesamedirectionandpropelthevehicleforward. Whenthemotorturnscounterclockwise,theratchetlockstherightwheel,andthevehicleturnstotherightasitbacksup.
Question: What would happen if a second ratchet was added toprevent the left axle from turning clockwise?
2.6 GearRackThegearracklookslikeaspurgearlaidoutflat. Itisusuallyusedinconjunctionwitha
spurgear(whichisreferredtoasthepinion). Rackandpiniongearsareusedtoconvertrotationintolinearmotion.Anexampleofthisisthesteeringsystemonmanycars.Thesteeringwheelrotatesagearthatengagestherack.Asthegearturns,itslidestherackeithertotherightorleftdependingonwhichwayyouturnthesteeringwheel. Themotionistransmittedvialinkagestothefrontwheelscausingthemtopivot.
Figure2-27. LEGOGearRackandPinionOneofthetrickierproblemsencounteredwhenusingthegearrackishowtoprovideasmoothsurfaceuponwhichtherackcanslide. MostLEGOcarmodelssolvethis
problemwithtiles,whichareplateswithoutstudsontop. Figure2-27usestwo1x5liftarmstoprovideasmoothsurface. AnupsidedownTechnicbeamisanothercommonlyusedsolution.
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2.7 PulleysApulleyisawheelwithagrooveaboutitsdiameter. Thegroove,calledtherace,acceptsabeltwhichattachesthepulleytootherpulleys. Asthepulleyrotates,frictionforcespullonthebeltputtingitintension. Thebelttransmitstheforcetotheotherpulleycausingittorotate.
Figure2-28. PulleysandBeltsLEGOincludesfoursizesofpulleysintheRIS--thehalfbushing,thesmallpulley,the
pulleywheel,andthelargepulley.Pulleysareconnectedusingbeltsthatrestinthepulleysraces. LEGObeltsarecolorcoded:small(white),medium(blue),andlarge(yellow). Theblackrubberbandscanbeusedasbelts,buttheyaremoreelasticthantheLEGObelts,andtheirrectangularcrosssectiondoesntfitwellinapulleysrace.
Figure2-29. LegoPulleysWithfourdifferentsizedpulleys,itispossibleto gear upand gear down. Thereductionratiowithpulleysisdeterminedbytheratiobetweentheirdiameters. Thetrickisdeterminingexactlywheretomeasurethediameter. Table2-4showspulleydiametersmeasuredatthebottomoftherace. Theresultingreductionratiosareclosetothevalues
determinedexperimentallyinTable2-5.
Table2-4. Measuredpulleydiameters
DiameterHalfBushing
SmallPulley
MediumPulley
LargePulley
Millimeters 5.7 8.7 22 34.5Studs 0.71 1.09 2.75 4.3
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Table2-5.PulleyRatiosHalfBushing
SmallPulley
MediumPulley
LargePulley
HalfBushing 1:1 3:2 7:2 6:1SmallPulley 2:3 1:1 7:3 4:1MediumPulley 2:7 3:7 1:1 5:3LargePulley 1:6 1:4 3:5 1:1
2.7.1 TorquePulleysmaybeusedinplaceofgearsinmanyapplications. Sincetherearenoteethtomesh,placementismuchmoreforgiving. Butbecauseithasnoteeth,apulleycannotbeusedtotransmithightorques. Thebeltwillslipfirst. Determiningwhenandhowmuchthebeltwillslipisdifficult. Itdependsonmanyfactorssuchasthesizeofthepulleys,thetensionofthebelt,andthefrictionbetweenthepulleyandthebelt.
Youcanincreasethetorquecapacityofapulleysystembyincreasingthereductionratioorbyincreasingthefrictionbetweenthepulleysandthebelt. Increasingthereductionratiomaynotbedesirablebecauseithasanassociateddecreaseinspeed. Frictioncanbeincreasedeitherbyusingatighterbeltorbyincreasingthecontactareabetweenthe
pulleysandthebelt.
Figure2-30. TwowaystoIncreaseTorqueCapacityQuestion: How else can we increase the friction between the pulleysand the belt?
Someenterprisingrobotdesignersusebeltslippageasatorquelimitingdevice,similartotheclutchgear. Becauseslippageisdifficulttopredict,itisbesttouseexperimentationtofindthebeltandpulleycombinationthatwillslipatthedesiredtorque.
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Figure2-31. UsingPulleystoLimitTorqueConstruction Notes: The tank tread and sprocket can be used like a pulley
and belt system. The treads teeth prevent slipping even if a large torque is
applied to the sprocket.
2.8 ReinforcinggeartrainsThecomponentsofthegeartrainarelikelytoexperiencethelargestforcesofanythinginyourrobot. Themotorwillstallandwheelswillslipbeforeforcesgettoooutofhand.Butwithenoughgearreduction,itiseasytogenerateforcesthatwilltearagearboxapart.
Figure2-32. ForcesongearsDifferenttypesofgearsrequiredifferentkindsofsupport. Inspurgears,forcesareappliedperpendiculartotheaxles. Unlesstheaxlesareproperlysupported,theywillbe
pushedapartwhenhightorquesareapplied. Supportingtheaxlesisnotaprobleminhorizontallayouts,butverticalanddiagonallayoutsmayrequireadditionalbracing.
Crown
and
bevel
gears
have
forces
directed
perpendicular
and
parallel
to
their
axles.
Aswithspurgears,thesupportingstructurewillprobablyrequirebracingtoholdtheaxlesinplace. Thiscanbedifficultattimesbecausethesupportsareperpendiculartoeachother.Crownandbevelgearsalsorequireasolidbackingtopreventthemfromslidingontheiraxles.
Themostdifficultproblemwiththewormgearislockingthewormscrewinplaceonitsaxle. Thisisexacerbatedbythegearsoddlength(its15.5mmlonginsteadofthe16mm
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requiredforaperfectfit). Whenthewormgearisusedwithaspurgear,thespurgearsaxlemustbewellsupported.
Figure2-33. GearboxesDontHavetobeBigtobeStrong
2.9 BacklashThebacklashofageartrainistheamounttheinputshaftcanrotatewithoutmovingtheoutputshaft. Backlashiscausedbythegearsnotmeshingperfectly(seeFigure2-34). Inthisexample,whengearAreversesrotation,thetoothongearBgoesfrombeingloadedontheleftsidetobeingloadedontherightside. Becauseofthegapbetweentheteeth,AwillbeabletorotateasmallamountbeforeBnoticesthechangeindirection.
Figure2-34. BacklashiscausedbypoorgearmeshingBacklashintroducesdiscontinuity,uncertainty,andimpactinmechanicalsystems. Thismakesaccuratecontroldifficult. Positioningaccuracyisalsocompromiseddueto
backlash. Alargeamountofbacklashmakesarobotfeelsloppyandgivesanoverallimpressionofpoorengineering(youwanttoimpressthosejudges).
Inindustry,backlashisreducedbyusingspeciallymatedgearsthataredesignedtomeshperfectly. LEGOgearsaredesignedtoworkwithawidevarietyofgearsofdifferentsizesanddesigns. Thislimitshowperfectlytheycanfittogether. Luckily,thereareotherwaystominimizebacklashanditseffects.
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ReducingBacklash:1. Placerotationsensorsneartheoutputshaft. Thisminimizestheeffectbacklash
hasonthesensorreadings. Unfortunately,italsolimitsyourabilitytoincreasesensorresolutionbygearingup.
2. Uselargegears. Backlashislessnoticeableinlargergears.3.
Minimize
the
number
of
gears
in
a
gear
train.
The
larger
the
number
of
meshes,thegreaterthebacklash.4. Backlashincreaseswhenyougearupanddecreaseswhenyougeardown.5. Whenlayingoutgearsdiagonally,keepgearspacingneartheidealvalues.
Backlashincreasesifthegearsaretoofarapart.6. Becarefulwhenusingthewormgear. Thewormgearsoddsize(1.94studs
long)makesitdifficulttosupport. Multiplehalfbushingscanbeusedtoholdthewormgearinplaceforlightlyloadedapplications.
Aninterestingalternativeistopreventbacklashfromoccurring. Youcantaketheplayoutofageartrainbyapplyingatorquetotheoutputshaft,tradingsomeinputtorquefor
more
precision.
If
the
torque
is
great
enough,
it
prevents
backlash
from
happening
whenchangingdirection. InFigure2-35,therubberbandissupplyingtheforceusedtopreloadthegeartrain.
Figure2-35. PreloadingageartrainwitharubberbandSomehighprecisionmachinesuseasplitgeartominimizebacklash. Asplitgearlookslikeaspurgearthathasbeencutthroughthemiddleaboutitscircumference.Springsare
placedinchannelsbetweentwogears.Thespringspushthegearsapartradially. Whenmeshedtoanothergear,thespringiscompressedprovidingapreloadforce.
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Figure2-36. SplitgearmadefromLEGOFigure2-36showsasplitgearmadeoutofLEGOparts. Itusestheaxlesinsteadof
springs
to
provide
the
preload
force.
When
assembling,
the
40t
gears
are
twisted
onetoothrelativetoeachotherbeforemeshingthemtothe8tgears.
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3 WheelsFLLchallengesinvolvedmovingaroundandpickingupthings,movingaroundanddroppingthings,movingaroundandtriggeringthings. Allthismovingaroundrequiressomesortofmobileplatform,andmostofthemobileplatformsusewheels.
Wheelssupporttheweightoftherobotandtransmitthepowerofthemotorstotheground.Whatwheelsyouusewillaffectyourrobotsspeed,power,accuracy,andabilitytohandlevariationsinterrain.Wheelchoiceswillhaveaprofoundeffectonyourrobotssuccessorfailure.
3.1 SizesLEGOhassuppliedawidevarietyofwheelsandtiresintheRISkit. Therearethreesizesofsolidrubbertireswhichallfitonthesameplasticwheelandthreesizesof
balloontiresthatfitondifferentlysizedhubs. Dimensionsinmillimetersarestampedonthesidewalloftheballoontires. Dimensionsforthesolidtiresarenotavailable,
therefore,approximatedimensionsaresuppliedinthefigurebelow.
Figure3-1. LegoWheelsandTires
The
diameter
of
the
wheels
chosen
will,
in
a
large
part,
determine
how
fast
and
powerfulyourrobotis. Largewheelswillmakearobotrunfasterbutdecreaseitstowingcapacity.Smallwheelswillgiveyoumorepowerbutwithacorrespondingdecreaseinspeed.
3.1.1 SpeedAsyoudrivedownthehighwaythespeedyourcartravelsisdependentupontherotationalspeedoftheengine,thegearratioofthetransmission,andthediameterofthetires. Theenginespeedandgearratiodefinehowfastthedrivewheelsarespinning
SmallSolid24mmx7mm
MediumSolid30mmx10.7mm LargeSolid
43mmx10.7mm
LargeBalloon81.6mmx15mm
SmallBalloon30.4mmx14
MediumBalloon49.6mmx28mm
PulleyWheel30mmx4mm
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(theirangularvelocity). Theangularvelocitycanthenbeconvertedtolinearvelocityusingtheequationforthecircumferenceofacircle(distancearoundthecircle).
IamdrivingdownthehighwayinmytrustyCamrywiththebrokenspeedometercablewhenIseeapolicecarwithitslightson. Aquicklookattheinstrumentsshowstheengineisat800rpmandthatIaminthirdgear. AmIgoingtogetaticket?
Iknow(becauseIdiligentlyreadtheownersmanual)thatthegearratioformyautomatictransmissionis1:1inthirdgear. Ialsoknowthatthetiresare24.6 indiameter. Sowhatisthespeed?
=EngineRPMxGearRatio
=800rpmx1:1=800rpm
v =xPixd=800rpmx3.14x24.6inches=61826inchesperminute
=58.5mph!I think Im OK
Thesameequationscanbeusedtocalculatetheexpectedtravelspeedofyourrobot.Forexample,letsusetherobotshowninFigure3-2. Ithasone9vmotorthatisgearedupby
a
factor
of
3:1
(a
24
tooth
crown
gear
driving
an
8
tooth
spur
gear).
It
uses
the
big
81.6
x15balloontirestomakeitgoreallyfast. Themotorshouldbeabletoreach300rpmforalightlyloadedstructurelikethis. Whatistheexpectedspeed?
=MotorRPMxGearRatio=300rpmx3:1=900rpm
v =xPixd
vd
v=xPi x d
Circumference
Diameter 1Revolution
Circumference=PixDiameter
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=900rpmx3.14x81.6mm
=230,601mmperminuteor8.7mph!Wow!!!
Figure3-2. AVeryFastTractorQuestion: For the example above, what would the travel speed be ifwe changed the gear ratio by putting the 8 tooth gear on the motor and
the 24 tooth crown gear on the axle?
3.1.2 ForceInourdiscussionofgears,wesawthatthereisarelationshipbetweenforce,torque,and
radius--thesamerelationshipsholdtrueforwheels.Whenyouusebigwheelstoincreasespeed,youhavetogiveupsomething,andthatsomethingisforce. Arobotwithbigwheelscannotpullasmuchasarobotwithsmallwheels.
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Figure3-3. Force=Torque/RadiusTherelationshipbetweenforce,distance,andtorqueworksagainstuswhenitcomestowheels. Largewheelshaveabiggerradius(leverlength)andwillgeneratelessforceforagiventorquethansmallwheelswould. Lessforcemeanslessacceleration. Thecarwillgofaster,butitwilltakelongertoreachthatspeed. Using9Ncmasanestimatefortheavailablemotortorque,letsdeterminethewheelforcesforthevehicleinFigure3-2.
Torque =MotorTorquexGearRatio=9Ncmx1:3=3Ncm
Radius =Diameter/2=81.6mm/2=
40.8
mm
or
4.08
cmForce =WheelTorque/Radius
=3Ncm/4.08cm=0.735N=0.1653lbsor2.64oz!Weak!!!
Forcomparison,ifweusedthesmallsolidwheels(24mm),thetractorwouldtravelat2.5mphwiththewheelssupplyingupto9ozofforce. Switchingthegearratiofrom3:1to1:3wouldhavethetractorgoingonly mph,butitwouldbecapableofdeliveringawhopping5lbsofforce.Itsunlikelythatthetireswouldhaveenoughtractiontotransmitthismuchforcetotheground,consequently,someslippingwouldresult.
Question: If big wheels are less efficient for towing, why do tractorshave big wheels?
Force
Torque
Radius
Radius
Torque
Force
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3.2 TreadsTrackedrobotshavebeenverypopularinFLLcompetitions. Theyareeasytobuild,andthekidsthinktheylookcool. TheRISkitcomeswithtwotanktreadsandfoursprocketwheels. Thesprocketwheelshavearoundaxleholeallowingthemtospinfreely.Drivingthetreadrequiressomeadditionalmechanismtolockthesprocketwheelstotheir
axle. Thisisusuallydonewitha16toothspurgear.
Figure3-4. TrackedRobotTreadshavegoodtractioninroughterrainand,ifwellsupported,cancrosssmallravinesthatwouldstopawheeledvehicle. Trackedrobotstendtobewideandlowtotheground,givingthemexcellentstability. Theyarealsoveryagile. Atrackedrobotcanturninplacebyspinningitstreadsinoppositedirections.
Unfortunately,treadshaveanumberofdisadvantages. Theyhavefairlypoortractiononsmoothsurfaces. Theslippingtreadsmakeitdifficulttousedeadreckoningnavigation.
Also
there
is
a
lot
of
power
loss
due
to
the
continuous
bending
and
straightening
of
thetreadsastheyrollaroundthesprocketwheels.
Figure3-5. MarioFerrari'sJohnny54
4Fromwww.marioferrari.org/lego.html
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OnlyonesizetrackisincludedintheRISkit. Forthetracknottoslip,thespacingbetweenthesprocketsmustbelargeenoughtomaintainalighttension(about12studs).Thewheelbasecanbeshortenedifadditionalsprocketsareusedtochangetheshapeofthetrack. Intheexampleabove,twopulleywheelsmakeuptheadditionalsprocketthatgivesJohnny5hisdistinctivedrivetrain.
3.3 BalanceProperbalanceisveryimportantinrobotdesign. Foryourrobottomoveinapredictableandrepeatablemanner,allwheelsmustbeincontactwiththegroundatalltimes,andtheweightcarriedbyeachwheelmustbeconsistent. Improperbalancewillresultinarobotthattipsovereasilyorliftsitswheelswhenacceleratingorturning.
Balanceisdependentupontwofactors:wheelbaseandcenterofgravity. Centerofgravity(CG)isthepointwithinyourrobotwherethereareequalamountsofmassaboveandbelow,equalamountstotheleftandright,andequalamountsforeandaft. Forstabilitycalculations,wepretendthatallthemassofyourrobotisconcentratedinthis
tinyspot. Thewheelbaseistheregionoutlinedwhenyouplayconnectthedotswiththepointswhereyourrobotswheelstouchtheground.
Figure3-6. WheelbaseTomaintainbalance,theCGmustremainwellinsidethewheelbase. Ifitisoutsidethewheelbase,therobotwilltipover. TheclosertheCGistothecenterofthewheelbase,themorestabletherobotbecomes.
3.3.1 FindingtheCenterOfGravityDeterminingyourrobotsCGisaninformativeandeducationalactivity. Therearemanywaysthiscanbedone,rangingfromthedryandboring(summingthemomentofinertiaforeachcomponentanddividingbythetotalmass)totheslightlyfrightening(hangingtherobotfromastring). AfairlysafeandeasywaytolocatetheCGisthebalancemethod.
TodeterminetheCGusingthebalancemethodyouneedtolocatethebalancepointonthelateral,longitudinal,andverticalaxes. EachbalancepointdefinesaplaneuponwhichtheCGresides. TheCGislocatedattheintersectionofthethreeplanes.
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Figure3-7. FindingCGUsingBalanceMethodTofindthebalancepoints,youneedafulcrumthatwillsupporttheweightoftherobotwhilestillallowingittotipeasily. A2x4curvedtopbrickisthefulcrumintheexample
above.
Place
the
robot
on
the
fulcrum
such
that
the
fulcrum
is
parallel
to
the
balanceplaneyouaretryingtolocate. Slowlyadjustthepositionofthefulcrumuntiltherobotbalances. Thisisthebalancepoint. RepeatfortheothertwoaxestofindtheCG.
FindingtheverticalcomponentoftheCGcanbedonethesameway,butsometimesyourrobotsdesignwillallowthis. IfIcantbalancethefront,rear,orsidesoftherobotonafulcrum,Imodifythebalancemethodtouseapartoftherobotasthefulcrum. Carefullytilttherobotupuntilitfeelslikeitisbalanced.Nowuseacarpenterssquare(abookortissueboxwillworkaswell)toprojectaplaneupverticallyfromthefulcrum. TheCGisthepointwherethisplaneintersectsthelateralandlongitudinalbalanceplanes.
Figure3-8. ModifiedBalanceMethod3.3.2 InertiaDeterminingtheverticalcomponentofthecenterofgravityismoredifficultthanfindingtheleft/rightcenterorthefore/aftcenter. Sowhybotherdoingit? Afterall,ifyouknow
LongitudinalBalancePlane
CGisonthislineLateralBalancePlane
CenterofGravity
LateralandLongitudinalBalancePlanesIntersection
VerticalLine
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thattheCGisnearthecenterofthewheelbase,thentherobotisguaranteedtobestable,right? Well,notalways. Figure3-9showshowclimbinganinclinemovestheeffectiveCG. TherobotwiththelowerCGisstillstable,butclimbingtheinclinemovestheCG
behindtherearwheelsontherobotwiththehigherCG,causingittotipoverbackwards.
Figure3-9. AnInclineMovestheEffectiveCGAmoreimportantreasonforknowingtheverticalCGistobeabletopredicttheeffectsthataccelerationwillhaveonyourrobot. Thewidelypublicized(andoverhyped)
problemswiththeSuzukiSidekickandothersmallSUVshavemadethepublicawarethatsmallwheelbasevehicleswithhighcentersofgravityaremoresusceptibletoroll-oversthanvehicleswithlargewheelbasesandlowcentersofgravity. Figure3-9showshowthismayoccuronanincline,buthowdoesithappenonaflatsurfacewhenturning?
InhisPrincipia,SirIsaacNewtonwasamongthefirsttodocumentobservationsofapropertycommontoallmatterwhichwenowcallInertia. Hestatedthatbodiesatrestliketostayatrest,andbodiesinmotionliketostayinmotiongoingthesamerateinastraightline. Tomovearestingbodyoralterabodysmotionrequirestheapplicationof
anexternalforce. ThisissummedupnicelyintheequationForce=massxacceleration,orF=ma.
Whenyouaredrivingdowntheroadandturnthesteeringwheel,thedirectionyourcaristravelingchanges. Theroadpushesagainstthetiresofyourcarcausingittoacceleratesideways. Inertiadictatesthatyourcarandeverythinginsideitresistthisaccelerationwithaforceequaltotheirmasstimestheacceleration. TheinertialforceislocatedatthecarsCGandispointinginthedirectionoppositetheacceleration. BecauseCGisnotdownontheroadwherethewheelforceisapplied,thisinertialforcecreatesatorque(ormoment)whichtriestotipthecarover.
Luckily,theweightofthecarisgeneratinganopposingmomentthatworkstopreventitfromtippingover. Alliswellaslongasthemomentduetogravityisgreaterthanthemomentduetotheinertialforce. Widewheelbasecarswithlowercentersofgravityaresaferbecausetheinertialforcesrequiredtorollthecaroveraremuchgreater. Thetiresareincapableofgeneratingsuchlargeforces,therefore,thecarskidsinsteadofrollingover.
CG
CG
BalancedOK OhOh!TipsOver
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Figure3-10. TurningGeneratesForcesandMomentsInertialforcesarealsothecauseofthe wheelies towhichsomanyFLLrobotsare
prone. Whenyourrobotisstoppedandyouturnonthemotors,theyinitiallygeneratetheirmaximumtorque. Iftheresultingaccelerationislargeenough,theinertiageneratedmomentwilltiptherobotbackwards,pullingthefrontwheelsofftheground.Luckilythetorqueoutputfromthemotordropsoffsignificantlyoncethewheelsstartturning,stoppingthewheeliealmostasquicklyasitstarted.
Figure3-11. FIRSTTeam254'sRobot"CheesyPoofs"DoesaVictoryWheelieWheeliesshouldbeavoidedbecauseitisdifficulttocontrolarobotpredictablywithoutallofitswheelsontheground. Fromtheturningdiscussion,wesawthatthelikelihoodoftippingisreducedbyhavingalowercenterofgravityorawiderwheelbase. Thisis
also
true
for
wheelies,
which
are
less
likely
to
happen
if
the
center
of
gravity
is
low
andwellaheadofthedrivenwheels. YoucanmovetheCGforwardbyrepositioningcomponentsorbyaddingextraweighttothefrontoftherobot. Asimilareffectisachievedbymovingthedrivenwheelstowardstheback.
Question: What else can be done to prevent wheelies besides movingthe CG or the wheels?
a
WeightD2
CGF2=mxa=F
F
M1=WeightxD1M2=ForcexD2
RolloveroccurswhenM2>M1
D1
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3.4 WheelLoadingandFrictionTheweightoftheRCXbrick,batteries,threemotors,alightsensor,andarotationsensoriscloseto1pound. Addtothisaframe,motorsupports,gears,andsomesortofmanipulator,andatypicalFLLrobot(ifthereissuchathing)issupporting1.5to2
poundsonitswheels. Dependingontherelativeplacementofthewheelsandsupporting
framecomponents,theforceontheaxlesmaybemanytimesgreater. Thesehighforcescancausemuchofthemotorpowertobewastedtryingtoovercometheresultingfriction. Whatisgeneratingtheseforces,andhowcantheybeminimized?
Figure3-12. CantileveredandFullySupportedWheelsMostwheeledvehiclesusesomesortofcantileveredassemblytoattachthewheelstothevehicle. Supportedononlyoneside,theaxleactslikealever,generatingamomentthathastobereactedtobytheframe. Amomentistheproductofaforce(theweightbeingcarriedbythewheel)anditsperpendiculardistancefromitsaxis(theplacewheretheaxleissupportedbytheframe). Thefartherthewheelisfromtheframe,thelargerthemoment.
Thewaythattheframereactsthemomenthasagreateffectontheamountoffriction.
Where
the
axle
passes
through
the
frame
it
does
not
rest
against
the
top
of
the
channel,butinsteaditistilted,contactingtheframeatonlytwopoints. Thefartherapartthetwocontactpointsare,thelowertheforcesexertedagainsteach.
Figure3-13. WheelLoadingDistancea Distance b ForceA ForceB FrictionForces1stud 2.5studs 2.5W -3.5W 6W1stud 1.5studs 1.5W -2.5W 4W4studs 1.5studs 0.375W -1.375W 1.75W
W
A A A
B B B
aa a b b b
W W
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Figure3-13andthecorrespondingtabledemonstratetherelationshipbetweenwheelplacementandloading. W istheweightofthemodelsupportedbythewheel. A and B areforcesexertedagainsttheaxlebytheframe. a isthedistancebetweenthetwocontactpointsontheframe. b isthedistancefromtheframetothewheel.
To
calculate
the
values
of
forces
A
and
B
we
need
to
use
what
we
have
learned
aboutforcesandmoments. FromNewtonsequation,F=ma,weknowthatabodywillaccelerateifacteduponbyaforce. Sinceourwheelisnotacceleratingupordown,wecandeducethattheforcesA,B,andWmustallcancelout;theirsumisequaltozero.Forourpurposeshere,wewillsayforcespushinguparepositiveandforcespushingdownarenegative.
A-B+W=0B=A+W
Amomentisjusttheproductofaforceactingatadistance,anditfollowsthesamerules
as
regular
forces.
A
body
acted
on
by
a
moment
will
experience
an
angular
acceleration.Iftheangularaccelerationofabodyiszero,thesumofthemomentsaboutanypointonthatbodymustalsobezero. Tomaketheequationseasiertoworkwith,Iamgoingto
pickthepointwhereforceBisapplied. Here,wewillusetheconventionthatamomentthatwantstocauseaclockwiserotationispositive,andacounterclockwiserotationisnegative.
Axa+Bx0Wxb=0Axa=WxbA=b/axW
Nowwecanstuffinsomenumbers. ThevaluesbelowarefromtheleftmostexampleinFigure3-13.
a =1studb =2.5studsA =b/axW
=2.5xWB =A+W
=2.5xW+W=3.5xW
Theimportantthingtotakeawayfromthisdiscussionisthatyoucanreducefrictionby:
1. Placingthewheelsclosetotheframe2. Spreadingouttheaxlesupportstoreducetheforcesrequiredtoreactthemoment
fromthewheel.
Question: What would the table entry look like for the fullysupported wheel assembly shown in Figure 3- 1 2?
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4 LegoElectronics4.1 RCXBrickTheRCXisyourrobotsbrain,atinycomputershapedlikeaLEGObrick. AtthecoreoftheRCXisaHitachiH88bitmicrocontrollerrunningat5to20MHzwith32KofRAM.Thismayseemanemicwhencomparedtomoderndesktopcomputerswith2GHz32bit
processorsand512Mbytesofmemory,butitsaspowerfulastheAppleIIcomputerIlearnedtoprogramonandsignificantlymorepowerfulthanthecomputersthatsentmentothemoon.
Themicrocontrollerisusedtocontrolthreevoltageoutputs,threesensorinputs,andaninfraredserialcommunicationsport. Snap-onwireleadsareusedtoconnecttheRCXtomotors,lamps,touchsensors,lightsensors,rotationsensors,etc TheserialcommunicationportisusedtodownloadprogramsfromaPCandcanbeusedtocommunicatebetweentwoRCXs.
TheRCXispoweredby6AAbatteries. OlderversionsoftheRCXalsohada9VDCinputplugforusewithanoptionalACadapter. IfyouhavethismodelofRCX,thetransformerwillpayforitselfinsavedbatterycosts. Havingtwoorthreesetsofgoodqualityrechargeablebatteriesisalsoagoodidea. TherechargeablealkalinebatteriesseemtoworkbestwiththeRCX.
Construction Note: When building your robot, remember that you will have
to occasionally remove the RCX to replace the batteries. Design accordingly.
Figure4-1. TheRCXProgrammableBrick4.1.1 FirmwareTheRCXcontains32Kofbattery-backedRAM. Mostofthisisusedupbythefirmwarethatyouhavetoinstallpriortoloadingyourownrobotprograms. Anadditional6Kisreservedtostoreuptofiveuserprograms. Thismaynotseemlikemuchspace,butmost
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RCXprogramsareonlyafewhundredbyteslong. Theremainingmemoryisusedforrunningprograms.
WhenaprogramiswrittenfortheRCX,itdoesnotcontainnativecodefortheCPU.Instead,itismadeupofspecialbytecodeswhichareinterpretedbythefirmware. ThisissimilartothewayJavaprogramsonthewebareinterpretedbytheJavaVirtualMachinerunninginyourbrowser. UsingabytecodeinterpreterallowstheRCXtomaintainareliableenvironmentforexecutinguserprograms. Yourprogrammaynotdowhatyouwant,butitwontcrashtheRCX.
BecausethefirmwareisstoredinRAM,itiserasedifyourRCXremainswithoutpowerformorethanafewseconds. AcapacitorintheRCXpowersthememorylongenoughforyoutochangethebatteries(15-30seconds). IfafterchangingbatteriesyourRCXnolongerfunctionsproperly,youmayhavelostyourfirmware. YoucantellifthefirmwareisloadedbylookingattheLCDdisplay. Ifallyoucanseeistheprogramnumberandtheminifigicon(eitherstandingorwalkingasinFigure4-1),thenthefirmwareisnotloaded. ThiswillbetheconditionofyourRCXbrickwhenyoureceiveyourkit.
Construction Note: Whenever it is turned on, the RCX is receptive to
messages coming in on its infrared serial port. With all the activity and
congestion at competitions, it is quite possible that another team could
unintentionally overwrite the programs stored in your RCX. Care should be
taken to block the IR port when not in use.
4.1.2 ProgrammingTheRCXisprogrammedusingspecialsoftwarethatresidesonyourPC. Theprogramis
translated
into
bytecodes
and
transferred
to
the
RCX
via
an
infrared
serial
link
(using
theIRtower). Thismethodofprogramming--writingcodeononetypeofcomputertoberunonanothertypeofcomputer--iscalledcrossdevelopment. Acomputerwithoutakeyboardandmonitor,liketheRCX,iscalledanembeddedsystem. ThatmeansallFLLersareembeddedsystemsdevelopers,familiarwithcrossdevelopmenttools. Thisisimpressivestufftoputonyourresume.
UserprogramscanbewritteneitherinRCXCodeorROBOLAB. RCXCodeisthegraphicalprogrammingtoolthatLEGOsuppliestoprogramtheRCX. Itistargetedto
peoplewithnoprogrammingexperience. Programsarewrittenbypickinginstructionblocksandsnappingthemtogether,muchlikeu