FIRST LEGO LEAGE Building Tutorial

8/8/2019 FIRST LEGO LEAGE Building Tutorial

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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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1-7

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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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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1-10

Figure1-19. OneofJenniferClark'sIncredibleCreations. YesitsLEGO.Luckily,TechnicismoreflexibletobuildwiththanotherLEGO. Butitisstilldifficultsometimestofigureoutawaytoattachthemotorwhereyouwantitortopositionthesensorinjusttherightplace. TryingtodiscovernewwaystoputLEGOtogetherisachallengingactivitythatcanbealotoffun. Herearesomeideastogetthecreativejuicesflowing.

Figure1-20. Turning90degrees. StudsOut


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1-11

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

3.44


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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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2-34

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

FIRST LEGO LEAGE Building Tutorial

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