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Three-Dimensional Trajectory Generation for Flight Within an Obstacle Rich Environment

Permanent Link: http://ufdc.ufl.edu/UFE0024478/00001

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Title: Three-Dimensional Trajectory Generation for Flight Within an Obstacle Rich Environment
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Hurley, Ryan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aerial, air, airplane, autonomous, avoidance, bridge, building, city, dense, dimension, dimensional, dubins, hurley, lind, mav, micro, obstacle, path, planning, random, ryan, structure, three, trajectory, tree, two, uav, unmanned, urban, vehicle
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Autonomous flight through urban environments requires methods to generate trajectories that traverse the region and its associated obstacles. This thesis introduces the development of a 3-dimensional motion planning algorithm using a random dense tree (RDT) based on a set of motion primitives in cooperation with a 3-dimensional version of the Dubins car called the Dubins airplane. The motion primitives consist of 3-dimensional maneuvers formulated as combinations of turn segments and straight segments with an associated constant rate of climb. The resulting motion planner builds the trajectory generating RDT by pruning nodes that intersect 3-dimensional obstacles while connecting the remaining nodes with the motion primitives. Several examples of the motion planner are presented for cases with no obstacles, building-style obstacles arranged in an urban environment, and an urban environment that includes bridges. These examples demonstrate that feasible paths are computed as sub-optimal solutions to minimize the cost of flight time.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ryan Hurley.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024478:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024478/00001

Material Information

Title: Three-Dimensional Trajectory Generation for Flight Within an Obstacle Rich Environment
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Hurley, Ryan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aerial, air, airplane, autonomous, avoidance, bridge, building, city, dense, dimension, dimensional, dubins, hurley, lind, mav, micro, obstacle, path, planning, random, ryan, structure, three, trajectory, tree, two, uav, unmanned, urban, vehicle
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Autonomous flight through urban environments requires methods to generate trajectories that traverse the region and its associated obstacles. This thesis introduces the development of a 3-dimensional motion planning algorithm using a random dense tree (RDT) based on a set of motion primitives in cooperation with a 3-dimensional version of the Dubins car called the Dubins airplane. The motion primitives consist of 3-dimensional maneuvers formulated as combinations of turn segments and straight segments with an associated constant rate of climb. The resulting motion planner builds the trajectory generating RDT by pruning nodes that intersect 3-dimensional obstacles while connecting the remaining nodes with the motion primitives. Several examples of the motion planner are presented for cases with no obstacles, building-style obstacles arranged in an urban environment, and an urban environment that includes bridges. These examples demonstrate that feasible paths are computed as sub-optimal solutions to minimize the cost of flight time.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ryan Hurley.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024478:00001


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THREE-DIMENSIONALTRAJECTORYGENERATIONFORFLIGHTWITHI NAN OBSTACLERICHENVIRONMENT By RYANDONOVANHURLEY ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2009 1

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c r 2009RyanDonovanHurley 2

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DedicatedtoJessica 3

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ACKNOWLEDGMENTS IwouldrstliketoexpressmysincerestgratitudetoDr.Ric kLindfornotonly providingcouncilformyresearch,butforgivingmeachance topursueaprofessionthat wouldnothavebeenpossiblewithouthisassistance.Hisadv isehasalwaysbeenformy ownbenet,evenifIdidnotwanttobelieveitsometimes.Iwo uldalsoliketothankDr. JoeKehoeandAbePachikarafortheirassistanceinthisrese arch.Inaddition,Iwould liketothankDr.PrabirBarooahandDr.AnilRaofortakingth etimetositonmy committeeandreviewmyresearch. ThanksarealsoinordertomyfellowmembersoftheFlightCon trolLab.Icould notaskforabetterworkenvironmentorabettergroupoffrie nds.Thanksgooutto(in clockwiseorder)Michael,Baron,Sebastian,David,Stephe n,Brian,Rob,Sanketh,Dan, andDongfor(inrandomorder)impressingmywife,impressin gme,makingmelaughall thetime,alwaysbeingthebuttofthejoke,tweeting,alsoim pressingmywife,lettingme knowhowyoufeel,beingthegooddistraction,movingmeouto ftheway,andgrounding usall. Recognitionisalsoinordertoallofmyfriendsandfamilyfo ralltheencouragement theyhaveprovidedme.IwanttoespeciallythankmyparentsF redandJoanHurleyfor alwayslovingmenomatterwhatIdo.Theyneverwanttotakecr editbuttheymademe whoIamtoday.Additionally,Iwanttothankmyfatherandmot her-in-law,Brianand ChristineFarrar,fortheendlesssupporttheyhaveprovide dme.Godcouldnothavegiven meabetterfamily.Iloveyouall. Finally,tomymotivatorandspiritualguide,Iwanttothank mywifeJessicaforallof herphysicalandmentalsupport.Icouldnothavedoneanyoft hiswithouther. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1Motivation .................................... 10 1.2ProblemDescription .............................. 12 1.3ApproachOverview ............................... 13 1.4Contribution ................................... 13 2MOTIONPRIMITIVES ............................... 14 2.1Introduction ................................... 14 2.2Two-DimensionalDubinsCar ......................... 14 2.3Three-DimensionalDubinsAirplane ...................... 17 2.4Library ...................................... 19 3RANDOMDENSETREES ............................. 22 3.1Introduction ................................... 22 3.2Two-DimensionalRandomDenseTrees .................... 22 3.2.1Rapidly-ExploringRandomTrees(RRT) ............... 23 3.2.2Expansive-SpacesTrees(EST) ..................... 24 3.2.3Discussion ................................ 25 3.3Three-DimensionalRandomDenseTrees ................... 25 3.4Three-DimensionalRandomDenseTreeTrajectoryGenera tion ....... 28 4MOTIONPLANNING ................................ 32 4.1Introduction ................................... 32 4.2Model ...................................... 33 4.3Overview .................................... 34 4.3.1SelectaNode .............................. 34 4.3.2ExtendaBranch ............................ 34 4.3.3CheckforSolutions ........................... 34 4.4Discussion .................................... 35 5

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5EXAMPLES ..................................... 37 5.1Introduction ................................... 37 5.2Three-DimensionalDubinsAirplanePaths(NoObstacles ) ......... 37 5.3FullMotionPlanninginUrbanEnvironment ................. 38 5.4RandomDenseTreeAlgorithmModication ................. 42 5.5FullMotionPlanninginUrbanEnvironmentWithBridges ......... 44 6CONCLUSION .................................... 47 REFERENCES ....................................... 49 BIOGRAPHICALSKETCH ................................ 52 6

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LISTOFTABLES Table page 2-1Dubinscarprimitivesequences ........................... 15 2-2Dubinsairplaneclimbingprimitivesequences ................... 17 2-3Dubinsairplanelevelprimitivesequences ...................... 18 2-4Dubinsairplanedivingprimitivesequences ..................... 18 5-1Vehiclepropertiesforexamples ........................... 37 5-2Three-dimensionalobstaclespecicationsforurbanen vironmentexamples ... 39 5-3Motionplanningalgorithmresultscomparison ................... 44 5-4Tower,walkway,andbridgespecications ..................... 45 7

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LISTOFFIGURES Figure page 1-1MicroairvehiclefromtheUniversityofFloridaFlightC ontrolLaboratory ... 10 1-2Microairvehicletravelingthroughanurbanenvironmen t ............. 11 2-1Possibleandoptimal2-dimensionalDubinspathsolutio ns ............ 16 2-2Turn-straightsolutionsequences ........................... 16 2-3Possibleandoptimal3-dimensionalDubinsairplanesol utions .......... 19 2-4Turn-straightsolutionsequencesfor3-dimensionalex ample ............ 20 3-1Rapidly-exploringrandomtreealgorithm ...................... 24 3-2Expansive-spacestreealgorithm ........................... 25 3-3DierencesinexplorationstrategyfortheRRTalgorith mvs.theESTalgorithm 26 3-4Three-dimensionalrandomdensetree ........................ 27 3-5Growthof3-dimensionaldynamicallyconstrainedtreei ntounoccupiedspace .. 29 3-6Growthof3-dimensionaldynamicallyconstrainedtreei ntooccupiedspace ... 30 5-1Dubinsairplanepathsforobstaclefreeexample .................. 39 5-2Urbanenvironmentexamplerun#1results .................... 40 5-3Urbanenvironmentexamplerun#2results .................... 41 5-4Urbanenvironmentexamplerun#3results .................... 41 5-5Algorithmalterationexamplerun#1results ................... 43 5-6Algorithmalterationexamplerun#2results ................... 43 5-7Algorithmalterationexamplerun#3results ................... 44 5-8Randomdensetreeandsub-optimalpathsforexamplewith bridges ....... 46 8

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience THREE-DIMENSIONALTRAJECTORYGENERATIONFORFLIGHTWITHI NAN OBSTACLERICHENVIRONMENT By RyanDonovanHurley May2009 Chair:RichardC.LindJr.Major:AerospaceEngineering Autonomousrightthroughurbanenvironmentsrequiresmeth odstogenerate trajectoriesthattraversetheregionanditsassociatedob stacles.Thisthesisintroduces thedevelopmentofa3-dimensionalmotionplanningalgorit hmusingarandomdense tree(RDT)basedonasetofmotionprimitivesincooperation witha3-dimensional versionoftheDubinscarcalledtheDubinsairplane.Themot ionprimitivesconsist of3-dimensionalmaneuversformulatedascombinationsoft urnsegmentsandstraight segmentswithanassociatedconstantrateofclimb.Theresu ltingmotionplannerbuilds thetrajectorygeneratingRDTbypruningnodesthatinterse ct3-dimensionalobstacles whileconnectingtheremainingnodeswiththemotionprimit ives.Severalexamplesof themotionplannerarepresentedforcaseswithnoobstacles ,building-styleobstacles arrangedinanurbanenvironment,andanurbanenvironmentt hatincludesbridges. Theseexamplesdemonstratethatfeasiblepathsarecompute dassub-optimalsolutionsto minimizethecostofrighttime. 9

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CHAPTER1 INTRODUCTION 1.1Motivation Thematurationofmicroairvehicles(MAVs)hasintroduceda classofaircraftwith sucientagilitytoenablerightthroughurbanenvironment s.Anexampleofoneofthese MAVsisshowninFigure 1-1 .Techniquesformotionplanningarerequiredthatcan computetrajectoriesforsuchrightthatacknowledgethefu lly3-dimensional(3-D)nature ofthemissionandassociatedobstaclesthroughouttheregi on.Also,tighttoleranceson therightpathrequirethattheresultingpathisentirelyfe asibleandcanbeaccurately followedbythevehicle. Figure1-1.Amicroairvehicle(MAV)fromtheUniversityofF loridaFlightControl Laboratory. Suchclose-proximityrightpresentschallengesforpathpl anning.Inparticular,the vehiclewillneedtoaggressivelymaneuveramongthedenseo bstaclestoachievethe 10

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vantagesthatprovidetherequiredinformationasillustra tedinFigure 1-2 .Traditional approachesthatchoosetrajectoriesbasedonkinematicmod elsaresuspectduetotheir relianceonaninner-loopcontrollertotightlytrackthose trajectories.Assuch,thepath planningshouldberestrictedtoconsiderationoftrajecto riesthatareinherentlyfeasible accordingtothedynamicconstraintsofmaneuvering. Figure1-2.Microairvehicletravelingthroughanurbanenv ironment. Inclusionofdynamically-feasiblemotionsinaplannedtra jectoryistypicallytreated ineitheradirectoradecoupledfashion[ 1 ].Directplanningmethods,suchasoptimal control,considerarepresentationofthevehicledynamics intheformulationofthe planningproblemanddirectlysolveforoptimalsysteminpu ts.Alternatively,indirect methodsuseasimpliedmodelofvehiclemotiontoplanarefe rencepathandthen \smooth"thepathtosatisfydynamicsusingmethodssuchasf eedbackcontrol.Direct methodscomputeoptimaltrajectoriesbutareoftenintract ableforrealisticproblem descriptionswhereasindirectmethodsoftenexhibittract ablecomplexitypropertiesthat comeattheexpenseofoptimality. Researchershavefoundcleverwaystomanipulatethistrade othroughavariety oftechniquessuchthatdynamicscanbedirectlyincludedin theplanningprocess.For 11

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example,someresearchershaverecognizedthatsystemswhi chexhibitdierentialratness propertiesadmitsolutionsthatcanberepresentedparamet ricallyintermsofasetofrat outputsandtheirderivatives[ 2 { 4 ].Othershaveappliedmixed-integerlinearprogramming (MILP)tomodeldynamicconstraintsasasetofswitchingbou ndsonsystemvelocities andaccelerations[ 5 { 7 ].Frazzolietalintroducedaplanningtechniquethatutili zes a\sampled-dynamics"modelwhichemploysasetofdynamical ly-consistentmotion primitives[ 8 9 ].Additionally,recentadvancesinrandomizedplanningal lowtheuseof anyofthesetechniquesaslocaltrajectorygenerationmeth odsforgrowingaprobabilistic treeofactionstoexplorethesolutionspace[ 10 { 12 ]. Theconceptofmotionprimitivesisacentralthemeforsever aloftheseinvestigations intofeasible-pathmotionplanning.Acriticalfoundation wasestablishedbyDubinsfora 2-dimensional(2-D)car[ 13 ]andhassincebeenexpandedinto3-Dversions.Oneapproach hasbeendevelopedbutdoesnotdealwithconstraintsinthec limbrateorspecicvalues oftheseclimbratesasisthecasewithmotionprimitives[ 14 ].Acompleteanaloguetothe Dubinscarin3-Disbeingdevelopedtoaccountfortheshorte stpathbetweentwopoints withassociatedheadingconstraintswhichdecomposesthep roblemintodierentcasesof whichonlysomecanbesolved[ 15 ]. 1.2ProblemDescription Givenaninitiallocationandheadingin3-dimensionalspac eandthelocationsand dimensionsofallthe3-dimensionalobstaclesinanenviron ment,thetrajectorytoagoal locationandheadingisdesired.Theworkpresentedinthist hesiswilladdresstheproblem ofmotionplanningin3-dimensionalspace.Specically,mo tionplanningformissions involvingvehicleswith3-dimensionalmotionincloseprox imityto3-dimensionalobstacles isdeveloped.Thetrajectorymustbedynamicallyconstrain edtotheperformance attributesofthevehicleandmustnotintersectwithanyoft heobstacles. 12

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1.3ApproachOverview Thetopicof3-dimensionalmotionprimitivesisdiscussedi nChapter 2 .Both 2-dimensionaland3-dimensionalversionsofthe2-primiti veand3-primitivemotion sequencesarediscussedanddevelopedrespectively.The3primitivemotionsequences arereferredtoasDubinspathsandareusedinthedevelopmen tofthemotionplanner. Chapter 3 focusesonthesubjectofrandomdensetrees(RDTs).Thesetr eeswillbe utilizedfortrajectorygenerationthroughtheobstacleri chenvironment.Thetheory ofrandomizedmethodsforpathplanningisdiscussedfollow edbythetransitionof 2-dimensionalRDTsto3-dimensionalRDTs.InChapter 4 ,theconceptof3-dimensional motionprimitivesand3-dimensionalRDTsarefusedtogethe rintothemotionplanning algorithm.Anexplanationofhowthealgorithmfunctionsis presentedanddiscussedin detail.Chapter 5 utilizesthismotionplanningalgorithminfourdierentex amplesto demonstratetheperformanceandcharacteristicsofthealg orithm. 1.4Contribution TheimplementationoftheRDTasatrajectorygenerationtoo linthemotion planningalgorithmisthecontributionbeyondpreviouswor kintheareaof3-dimensional motionplanning[ 5 14 { 17 ].Thisproducesasearchthatprovidesbetteroptimality throughenvironmentswithobstaclesdenselyplacedthroug hout;specically,acondition tolimitthebranchesbetweennodesisdenedalongwithater minalconditionfortree growth.Thebranchlimitationactuallyseekstoplacemoreb rancheswithintheregionto increaseprobabilityofndingalocalminimumamongthenum eroussolutionsforpaths throughobstacles.Theterminalconditionnotesthattheex plorationcanchangefroma treegrowthtoadirectsolutiononceanoptimalpathtothego alcanbereachedwithout intersectinganyobstacles.Inthisway,theinitialcomput ationburdenmayslightly increaseduetotheextranodesandadditionalterminalchec k;however,theresultingpath willhavelowercost. 13

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CHAPTER2 MOTIONPRIMITIVES 2.1Introduction Theconceptofmotionprimitivesleadstoausefulframework forthesimplicationof complicateddynamicmodels.Thisframeworkinvolvescombi ningsequencesofcompatible primitivestorepresentmorecomplicatedtrajectories.As etofcompatibilityconditions aredetailedintheliterature[ 8 ].Thisresearchutilizespathcombinations,ormotion primitives,consistingofstraightpaths,turningpaths(b othleftandright),climbing paths,anddivingpaths.Motionprimitivemodeltheoryisde nedanddescribedinmore detailintheliterature[ 18 ].Theprogressionofresearchfrom2-dimensional(2-D)to 3-dimensional(3-D)motionprimitiveanalysisispresente dinthefollowingsections. 2.2Two-DimensionalDubinsCar AstandardmodelusingmotionprimitivesisknownastheDubi nscar[ 13 19 { 23 ]. Thissimple2-dimensionalvehiclemotionmodeloperatesin acongurationspace,or C -space,spannedbytwoEuclideanpositionvariables, p x and p y ,andanangledescribing heading, .Vehiclemovementisrestrictedtodrivingstraightorturn ingtoeitherthe leftorright.Thestraightmotionisconstrainedtoaconsta ntvelocitywhiletheturnsare constrainedtoaconstantvelocityandturnrate.Assuch,th emotionoftheDubinscaris describedbythedierentialsystemshowninEquation 2{1 266664 p x p y 377775 = 266664 cos sin 377775 (2{1) Thediscretesetofvaluesassumedbytheturnrate isshownbyEquation 2{2 2f 1 ; 0 ; +1 g =sec (2{2) 14

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TheDubinscarisanespeciallyinterestingmodelinthatacl osed-formsolution foroptimaltrajectorieshasbeenderived[ 13 ].Thissolutionnotesthepathfromany initialpoint( x o ;y o )andheading( o )toanynalpoint( x f ;y f )andheading( f )canbe expressedusingonlythepropersequenceofonly3primitive s.Thereare6combinationsof these3primitivesequencesandaredividedintoTurn-Strai ght-TurnandTurn-Turn-Turn categories,asdetailedinTable 2-1 .Sucharesultallowsstrategiesusingoptimalcontrolto bedirectlycomparedwithaglobalminimumforevaluatingte chniquesofpathplanning. Table2-1.Dubinscarprimitivesequences. Turn-Straight-TurnTurn-Turn-Turn Left-Straight-LeftRight-Left-Right Right-Straight-RightLeft-Right-Left Right-Straight-LeftLeft-Straight-Right Thetransformationsdescribingtheleftturn,rightturn,a ndstraightaheadmotions areshownasEquations 2{3 2{4 ,and 2{5 ,respectively.Theclosed-formexpressionsfor thetrimdurations arepresentedintheliterature[ 18 24 ]. L ( p x ;p y ; ; )=( p x +sin( + ) sin ;p y cos( + )+cos ; + )(2{3) R ( p x ;p y ; ; )=( p x sin( )+sin ;p y +cos( ) cos ; )(2{4) S ( p x ;p y ; ; )=( p x + cos ;y + sin ; )(2{5) Examplesofsolutionsfromaninitialpositionandheadingt oanalpositionand headingareshowninFigure 2-1 foraDubinscar.AllsixDubinscarcombinationsare utilizedandplotted.Theoptimalsolution,whichcorrespo ndstothelowesttraveltime,is highlighted. Inadditiontothe3-primitivesequenceslistedabove,ther eisalsoafamilyof 2-primitivesequencesutilizingonlytheTurn-Straightme thodology(Left-Straightand Right-Straight).Thiscombinationwillprovideapathfrom anyinitialpoint( x o ;y o )and heading( o )toanynalpoint( x f ;y f )butthenalheading( f )cannotbeguaranteed 15

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-200 0 200 400 -100 0 100 200 300 X (ft)Y (ft) Figure2-1.Possible(|)andoptimal( )2-DDubinspathsolutions. usingthe2-primitivesequenceinviewofthefactthatathir dprimitiveisnecessaryto providethedesiredheading.Examplesofsolutionsfromani nitialpositionandheadingto analpositionareshowninFigure 2-2 fora2-primitiveTurn-Straightsolutionsequence. Thereare3examplesofboththeLeft-StraightandRight-Str aightpaths,eachwitha dierentturnrate 0 5 10 -4 -2 0 2 4 6 X (ft)Y (ft) Figure2-2.Turn-straightsolutionsequences. 16

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2.3Three-DimensionalDubinsAirplane Utilizingthetheoryof2-dimensionalmotionprimitivesan dtheDubinscar,a 3-dimensionalversionofmotionprimitivesisdevelopedth atcreatesanaircraftversion oftheDubinscar.Thismodeloperatesina C -spacespannedbythreeEuclideanposition variables, p x p y ,and p z ,andanangledescribingheading, .Suchadevelopmentaddsa constantrateofchangeinaltitudetotheconstantrateofch angeofturnalreadyusedby the2-DDubinscar.TheresultingdynamicsaredescribedinE quation 2{6 266666664 p x p y p z 377777775 = 266666664 cos sin r 377777775 (2{6) Discretevaluesofturnrate, ,andclimbrate, r ,aredenedinEquations 2{7 and 2{8 2f 1 ; 0 ; +1 g =sec (2{7) r 2f 1 ; 0 ; +1 g ft=sec (2{8) Thissolutionnotesthepathfromanyinitialpoint( x o ;y o ;z o )andheading( o )toany nalpoint( x f ;y f ;z f )andheading( f )canbeexpressedusingonlythepropersequence ofonly3primitives.Thereare18combinationsofthese3pri mitivesequencesorganized into3groupsof6combinations.ThesearedetailedinTables 2-2 2-3 ,and 2-4 Table2-2.Dubinsairplaneclimbingprimitivesequences. Turn-Straight-TurnwhileClimbingTurn-Turn-TurnwhileC limbing (Left-Straight-Left)Climb(Right-Left-Right)Climb (Right-Straight-Right)Climb(Left-Right-Left)Climb (Right-Straight-Left)Climb(Left-Straight-Right)Climb 17

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Table2-3.Dubinsairplanelevelprimitivesequences. Turn-Straight-TurnwithLevelFlightTurn-Turn-Turnwith LevelFlight (Left-Straight-Left)Level(Right-Left-Right)Level (Right-Straight-Right)Level(Left-Right-Left)Level (Right-Straight-Left)Level(Left-Straight-Right)Level Table2-4.Dubinsairplanedivingprimitivesequences. Turn-Straight-TurnwhileDivingTurn-Turn-TurnwhileDiv ing (Left-Straight-Left)Dive(Right-Left-Right)Dive (Right-Straight-Right)Dive(Left-Right-Left)Dive (Right-Straight-Left)Dive(Left-Straight-Right)Dive Anoptimalpathcanbecomputedasasequenceofthesemotionp rimitivesto traversefromaninitialpositionandheadingtoanalposit ionandheading.Anexample scenarioisshowninFigure 2-3 thatdemonstratesalloftheTurn-Straight-Turnprimitive sequencesalongwiththeoptimalpathforclimbing,levelri ght,anddivingcases. Itmustbenotedthattherearescenarioswheretheplanecann otreachthegoal position.Thevehiclemodelhasaprescribedinabilitytotr aveldirectlyverticalor horizontalandthuscannotreachasetofnalconditionsdue tolimitationsonclimbrate andturnrate.Pachikarahasstudiedthisprobleminhisrese archandimplementationof thatworkcanbeusedtoaddressthisissue[ 25 ].Foralloftheresearchpresentedhere, theinitialpointandgoalpointaresucientlyseparatedin theXY-planesuchthatthe vehicle'sturnrateand/orclimbratedonotpreventthevehi clefromattainingthegoal conguration. Inadditiontothe3-primitiveDubinsairplanesequencesli stedabove,thereisalsoa familyof2-primitivesequencesutilizingonlytheTurn-St raightmethodology(Left-Straight andRight-Straight)justasinthe2-dimensionalcasebutin cludingtheconstantrateof changeinaltitude.Thiscombinationwillprovideapathfro manyinitialpoint( x o ;y o ;z o ) andheading( o )toanynalpoint( x f ;y f ;z f )butthenalheading( f )cannotbe 18

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0 200 400 -100 0 100 200 300 400 -500 0 500 East (ft) North (ft)Up (ft) Figure2-3.Possible(|)andoptimal( )3-DDubinsairplanesolutions. guaranteedusingthe2-primitivesequenceowingtothefact thatathirdprimitiveis necessarytoprovidethedesiredheading.Theseprimitives aregraphicallyshownin Figure 2-4 as9possibilities.Thehorizontalmotionhasonly3possibi litiesofgoing left-straight,right-straight,andstraightonly( =0).Thesehorizontalmotionsare coupledwithindependentverticalmotionthatallowslevel rightandclimbordive. 2.4Library Themotionprimitivesrepresentmaneuversthatthevehicle canperform.Since mostvehiclescanvarytheirratesofchange,itisreasonabl etodeneasetofparameters suchthatn= f 1 ;:::;! n g representsthesetofpossibleturnratesand= f r 1 ;:::r m g representsthesetofpossibleclimbrates. Asetofmotionprimitivescanthenbedenedthatrepresenta llpossiblemaneuvers. Eachelementinthissetisactuallyatrajectorydenedbyth etime-varyingvaluesof 19

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-200 0 200 0 100 200 -200 -100 0 100 200 East (ft) North (ft) Up (ft) Figure2-4.Turn-straightsolutionsequencesfor3-D. positionandorientationduringthemaneuver.Assuch,each elementisparametrizedby thedurationofthemaneuver, ,turnrate, ,climbrate, r ,toresultinsetof X X = 8>>>>>>><>>>>>>>: X ( ;!;r ): X = 266666664 p x (0: ) p y (0: ) p z (0: ) (0: ) 377777775 2 Eq 2 6 ;! 2 n ;r 2 9>>>>>>>=>>>>>>>; (2{9) Acriticalfeatureofthissetisthenotionoffeasibility.E ssentially,anymember, X 2X ,isconstrainedsotheevolutionofthetrajectorymustfoll owthedynamicsof Equation 2{6 .Theresultingsetisacollection,orlibrary,offeasiblem aneuversthatcan beachievedbythevehicle. Thecongurationswhichmaybereachedbythevehicleaftera singlemaneuvercan thusbeexpressedusingthisparameterizedset.Aninitialc ongurationofpositionand orientationcanbedenedas C (0) 2R 4 andtheensuingcongurationatanytime, t 2R 20

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canbedenedas C ( t ) 2R 4 asaresultofinitiatingsomemaneuver, X ( t;!;r )asin Equation 2{10 C ( t )= C (0)+ X ( t;!;r )(2{10) 21

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CHAPTER3 RANDOMDENSETREES 3.1Introduction Randomizedmethodsforpathplanninghavebeenformulatedt oconsidersystems withcomplicateddynamics.Thefundamentalfeatureofsuch methodsisalocalized approachthatconsiderssequentiallyexpandingintoasear chspacetorapidlyand ecientlyndsub-optimalsolutions.Avarietyofmethods, includingprobabilistic roadmapsandrandomdensetrees(RDTs),havebeendeveloped ;however,theuseof randomdensetreeswillbeadoptedduetoitsabilitytodirec tlyhandlemotionprimitives andgeneratefeasibletrajectoriesformodelsofrealistic vehicles[ 26 { 31 ]. 3.2Two-DimensionalRandomDenseTrees ThematerialinthissectionistakendirectlyfromSection3 .3of TrajectoryGenerationforEectiveSensingofaCloseProximityEnvironment byKehoe[ 18 ]. Randomdensetree(RDT)basedplannersprovideanalternati vetothebasic probabilisticroadmapmethodparadigmthatenablesecien tsolutionstodierentially constrainedproblems.Whiletheprobabilisticroadmapmet hodgeneratesaroadmapthat describestheconnectivityofmanycongurationstomanyot hercongurations,RDT methodsgenerateatreethatisrootedataspecicinitialco nditionandwhichdescribes connectivityofthisinitialconditiontoasmanyreachable congurationsaspossible. Algorithmicdetailsensureecientandrapidexplorationo fthespace.Adrawbackof RDTmethodsisthattheyaredesignedtosolveasingleplanni ngproblematatime. Thislimitationisincontrasttoprobabilisticroadmapmet hodplanningalgorithms,which establishanetworkthatspansthecongurationspace,or C -space,andcanbeusedmany timesformanydierentplanningtasks.Amajorbenetinthi stradeoisthatRDT methodscanoftenhandleproblemsinvolvingdynamicsystem s.Ingeneral,RDTmethods incrementallybuildasearchtreefromaninitialnodeinthr eemainsteps: 22

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1. NodeSelection: Anodefromtheexistingtreeisselectedasalocationtoadda branch.Selectionofaparticularnodeisusuallybasedonpr obabilisticcriteriathat mayrequireuseofavaliddistancemetric. 2. NodeExpansion: Alocalplanningmethodisusedtoextendafeasibletrajecto ry fromtheselectednode.Thelocalgoalforthistrajectorybr anchisdetermined probabilistically. 3. Evaluation: Thenewbranchisevaluatedaccordingtoperformancecriter iaand oftenforconnectiontothegoalconguration.Additionall y,thenewbranchmaybe subdividedintomultiplesegments,thusaddingseveralnew nodestotheexistingtree. AvarietyofRDT-basedplannershavebeendevelopedwithnum erousvariationson themainstepslistedpreviously,oftentooptimizeperform anceforaspecicapplication ortoaddressapathologicalcase[ 26 { 31 ].Twoalgorithms,theRapidly-exploringRandom Tree(RRT)algorithmandtheExpansiveSpacesTree(EST)alg orithms,demonstrate dierentcoreexplorationphilosophiesthroughthemanner inwhichnodesareselectedand expanded.Thesealgorithmsalsoserveasabasisformanyoft heexistingvariationsonthe generalmethod,andhenceproveusefulasdemonstrativeexa mples. 3.2.1Rapidly-ExploringRandomTrees(RRT) TheRRTalgorithmwasdevelopedbyLavalleandKunerspeci callytohandle problemsthatinvolvedynamicsanddierentialconstraint s[ 12 28 ].Thealgorithmbiases treegrowthtowardunexploredareasofthespaceandhencefo cusesonrapidexploration. Thenodeselectionstepisinitiatedwithasampledcongura tionthatischosenfroma uniformdistributionofthecongurationspace,or C -space.Adistancemetricisthenused todeterminetheclosestpointintheexistingtree.Duringt heexpansionstep,theselected nodeisextendedincrementally\toward"thesampledcongu rationusingalocalplanning method.Thisincrementalextensionisperformedtovarying degreesindierentversions ofthealgorithmandisultimatelyadesignparameter.Somev ersionsuseaxedstepsize, othersuseastepsizeproportionaltothedistancefromthes ample,whileothersattempt tocompletelyconnectthesampledcongurationtotheexist ingtree. 23

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Figures 3-1A and 3-1B depicttheRRTexpansionprocess.Bothimagesshowatree grownfromtherootnode, N 0 ,inatwo-dimensional C -spacethatcontainsobstacles. Figure 3-1A depictsthesamplingstep,inwhicharandomconguration, N rand ,isselected andthenearestnodeintheexistingtree, N near ,isdetermined.Figure 3-1B showsthe expansionstep,whereabranchisincrementallyextendedfr om N near toward N rand along thetrajectoryconnectingthetwocongurations.Anewnode N new ,isaddedatthe endofthenewbranch.Thealgorithmproceedsinthisfashion untilabranchofthetree reachesthegoalwithinsomespeciedtolerance. A B Figure3-1.RRTalgorithm.A)Samplingstep.B)Expansionst ep. 3.2.2Expansive-SpacesTrees(EST) TheESTalgorithmwasdevelopedbyHsuetalasaplanningmeth odtoaddress problemsinvolvinghigh-dimensional C -spacesandwaslateradoptedtohandlekinodynamic planningproblems[ 10 32 ].TheESTalgorithmexploresspaceinafundamentallydier ent waythantheRRTalgorithm.Specically,nodeselectionocc ursthroughtherandom selectionofanexistingnodeaccordingtoaprobabilitydis tributionthatisleftasadesign choice.Thisnodeisexpandedwithinalocalneighborhoodth atisdenedbyavalid distancemetric.Acongurationissampledrandomlyfromwi thinthisneighborhoodand alocalplanningmethodisusedtoconnecttheselectednodet othesampledconguration. Figures 3-2A and 3-2B depicttheESTexpansionprocess.Bothimagesshowatree grownfromtherootnode, N 0 ,inatwo-dimensional C -spacethatcontainsobstacles. Figure 3-2A depictsthenodeselectionstep,inwhichtheexpansionnode N exp ,isselected fromtheexistingnodes.Theneighborhoodof N exp isdenedhereusingaEuclidean 24

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distancemetricandisshownastheareawithinthedashedcir cleinFigure 3-2A Figure 3-2B showstheexpansionstep,wherearandomconguration, N rand isselected fromtheneighborhoodof N exp andthenatrajectoryisplannedfrom N exp to N rand .The algorithmproceedsinthisfashionuntilabranchofthetree reachesthegoalwithinsome speciedtolerance. A B Figure3-2.ESTalgorithm.A)Nodeselection.B)Samplingan dexpansion. 3.2.3Discussion Itisimportanttonotethefundamentaldierencebetweenth ewaysinwhichthe RRTandESTexplorethespace.Samplesfromemptyspacehavea tendencyto\pull" branchesoofthetreebuiltintheRRTalgorithm.Thus,thes paceisrapidlyspanned withcoarseresolution.Continuedsamplinghastheeectof improvingtheresolution ofthisexplorationwithoutappreciablychangingtheformo fthesolution.Thisconcept isdepictedinFigure 3-3A .Conversely,theESTselectsanoderandomlyandtendsto \push"branchesfromtheselectednodetowardemptyspaceas showninFigure 3-3B .A benettothis\pushing"tendencyisthattheshapeofthetre eiscontinuallyevolving suchthatexpansionisguidedbythenodesamplingdistribut ion.Awisechoiceofthis distributioncanfavorablyaectsolutionperformancequa lities;however,caremustbe takentoavoidbiasingexplorationtowardpreviouslyexplo redareas. 3.3Three-DimensionalRandomDenseTrees Theconceptofrandomtreesisalsousefulforconsideringa3 -dimensional(3-D) Dubinsairplane.Thebasictreestructureactuallyallowse xplorationintoanyn-dimensional 25

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0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) 0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) 0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) A 0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) 0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) 0 200 400 600 800 1000 0 200 400 600 800 1000 X (ft)Y (ft) B Figure3-3.DierencesinexplorationstrategyfortheRRTa lgorithmvs.theESTalgorithm.A)RRTexpansion.B)EST expansion.26

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spacesoa3-dimensionaltreeisarelativelystraightforwa rdextensionofthe2-dimensional (2-D)tree.Aswiththe2-dimensionalcase,boththeRRTandE STapproachescanbe modiedtobeimplementedinto3-dimensions.Fortheremain derofthisinvestigation,the pullexpansionphilosophyexhibitedbytheRRTalgorithmwi llbeutilized.Subsequentto experimentationwiththetwoexpansionphilosophies,theR RTapproachisselectedrather thantheESTapproachduetoitssuperioreciency,computat iontime,androbustness attributesinndingfeasiblesolutions. Thegrowthofthetreefollowsastandardalgorithmusingnod esandbranches.A setofnodesarerandomlyplacedintotheenvironmentwithin somelimitonrange.An exampleisshowninFigure 3-4 todemonstratea3-dimensionalRRTatiterationcountsof 100,250,and500. 0 500 1000 0 500 1000 0 500 1000 East (ft) North (ft) Up (ft) 0 500 1000 0 500 1000 0 500 1000 East (ft) North (ft) Up (ft) 0 500 1000 0 500 1000 0 500 1000 East (ft) North (ft) Up (ft) Figure3-4.Growthof3-DTreeafter100Iterations(topleft ),250Iterations(topright) and500Iterations(bottom) 27

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3.4Three-DimensionalRandomDenseTreeTrajectoryGenera tion Thenatureoftheexplorationandtheassociatedterminatio nconditionsofthe 3-dimensionalRDTdevelopedinSection 3.3 areissuesthatstillneedtobeaddressed. Thecombinationofthis3-DRDTandthedynamicmotionprimit ivesdescribedin Chapter 2 arethebasisforthedevelopmentofatrajectorygeneration algorithm. Tocombinetheideas,thenodesforthetreegrowtharerandom lyplacedintothe environmentwithinsomelimitonrangeplacedbyprobabilis ticassessmentofthevehicle velocity.Thevalidityofeachnodeisthenassessedbydeter miningifasequenceofturn maneuver, X t ,andstraightmaneuver, X s ,couldreachthatlocation, C ( t + ),fromthe previousnodelocation, C ( t ).Inotherwords,thetree'sbranchesarecreatedbyaseries of Turn-StraightmaneuverslikethosedetailedinSection 2.3 .SeveraloftheseTurn-Straight sequencesmayreachtherequirednewnodesothenalpath,or branch,isselectedby minimizingthetimerequiredfortravelasshowninEquation 3{1 min X t 2X X s 2X 1 + 2 (3{1) subjectto C ( t + 1 + 2 )= C ( t )+ X t ( 1 ;!;r )+ X s ( 2 ; 0 ;r ) ArepresentativeexampleisshowninFigure 3-5 todemonstrateatreebuiltuponthe motionprimitivesin X .Eachbranchconsistsofaturnmotionfollowedbystraightm otion witheachmotionobtainedasafeasiblemaneuverfromthelib rary.Thetotallengthof eachmaneuverisjointlydeterminedbytheminimizationofE quation 3{1 Pathplanninginto3-dimensionalspaceusingtreescanbeco nstrainedtoaccount forobstacles.Obviouslyanyresultingpathmustbefeasibl ebothinthesensethatthe 28

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0 200 400 600 0 200 400 600 0 100 200 300 East (ft) North (ft) Up (ft) 0 200 400 600 0 200 400 600 0 100 200 300 East (ft) North (ft) Up (ft) 0 200 400 600 0 200 400 600 0 100 200 300 East (ft) North (ft) Up (ft) Figure3-5.Growthof3-Ddynamicallyconstrainedrandomde nsetreeintounoccupied space. vehiclecanfollowthetrajectoryandinthesensethatthetr ajectorydoesnotintersectany obstacles. Apruningmethodisusedtoensureobstacleavoidance.Thism ethoddoesnot directlyconsiderthelocationoftheobstaclestooptimize treegrowth;rather,itsimply prunesnodesandbranchesthatliewithinanobstacle.Theno deselectionthusremains randomwithsomeofthenodesbeingeliminatedbyacheckonth enodelocationandthe obstaclelocations. Arepresentativeexpansionintoa3-Dspacewithobstaclesi sshowninFigure 3-6 Thetreegrowsandbranchesareformedalongpathsthatdonot intersectanyobstacles duetothepruningapproach.Theresultingpathisthusablet oavoidobstaclesandreach thegoalconguration. 29

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Figure3-6.Growthof3-Ddynamicallyconstrainedrandomde nsetreeintooccupied space. Thispruningnotesthatasetoflocations, O ,maybedenedthatencompasses theobstacles.ThedenitioninEquation 3{2 usesasimpleorthogonalpolyhedron approximationsuchthateachobstaclehaslimitsoneastran ge,[ x 1 ;x 2 ],northrange, [ y 1 ;y 2 ],andaltituderange,[ z 1 ;z 2 ],for k obstacles. O = 8>>>><>>>>: O : O = 266664 x 2 [ x i1 ;x i2 ] y 2 [ y i 1 ;y i 2 ] z 2 [ z i 1 ;z i 2 ] 377775 8 i 2 [1 ;k ] 9>>>>=>>>>; (3{2) 30

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Thegrowthofthetreeoccurssuchthatanode, C ( t + 1 + 2 ),asinEquation 3{1 is validifneitherthatnodenorapathtothatnodeintersectan yobstaclesasdescribedin Equation 3{3 : C ( t + 1 + 2 ) 2 = O 9 X 2X withX 2 = O (3{3) 31

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CHAPTER4 MOTIONPLANNING 4.1Introduction Motionplanningdescribestheprocessofdevelopingthetra nsformationofasystem fromaninitialcongurationtoaterminalorgoalcongurat ion.Systemcongurationsare denedasavector, ~x c ,onthecongurationspace,or C -space.The C -spaceencompasses thevariablesthatdescribethepositionandorientationof thebodycoordinateframe includingtheterminalorgoalconguration. Motionplanningproblemsforvehiclesystemsaretypically classiedaccordingto threemaincategories[ 33 ]:point-to-pointmotion,pathfollowing,andtrajectoryt racking. Inallofthesecases,thevehicleisrequiredtomovefromani nitialcongurationtoa goalconguration.Thedierencesoccurinhowthefunction describingthevehicle's motionisconstrained.Inthecaseofpoint-to-pointmotion ,therearenorestrictionson thetransitionalmotionsoccurringbetweenthebeginninga ndnalcongurationsbutjust thatnalcongurationisreached.Point-to-pointmotionp lanningissolvedviaaseries ofwaypoints.Forthepath-followingcase,thevehicleisin structedtoobeyacontinuous pathinthe C -spaceprescribedbysystemdierentialconstraintsandwh ichhastheinitial andgoalcongurationsasendpoints.Theplansforpathfoll owingmotionaredeveloped bywayofafunctiondenedonthe C -space.Thetrajectorytrackingcaseisidentical tothepath-followingcasewiththeadditionofatimerequir ement.Trajectorymotion plansaredenedaseitherasysteminputfunction, ~u ,thatisafunctionoftimeoras time-parameterizedfunctionsdenedonthe C -space. Inthissection,analgorithmformotionplanningisformula tedforgenerating guidancetrajectoriesforavehiclesystemgivenamodeloft hevehiclemotionanda knownenvironment.Theenvironmentinwhichthevehiclefun ctionsisoccupiedwith denselyplaced3-dimensional(3-D)obstaclesthatforceth evehicletotravelwithinclose proximitytotheobstacles.Thevehicle'spathisgenerated using2parts.Therstportion 32

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combinesthe3-Dmotionprimitiveswiththe3-Drapidly-exp loringrandomtree.This segmentofthepathwilltravelamongsttheobstacles.These condportionofthepath implementsthe3-DDubinspathtotravelfromtheendofatree branchtothegoalpoint. Thiscombinationwillproduceapproximateminimum-timetr ajectoriesfortheconstrained system. 4.2Model Themotionplannerdevelopedutilizesamotionprimitivemo delthatbehaves accordingtothedynamicsdescribedbyEquation 4{1 ,whichisanextensionofthe DubinsairplanemodelexaminedinSection 2.3 .Theconstanttranslationalvelocity, V ,is restricteddierentiallytoactforwardinthedirectionof thevehicleheading.Themotion oftheofthevehicleiscontrolledviatheinputstothedier entialsystem:theturnrate, ,andtheclimbrate, r .ThesystemdescribedbyEquation 4{1 admitstrimtrajectories thatbelongtothreefamilies:constantrateturns(leftand right),constantrateclimbsand dives,andstraightforwardmotion. 266666664 p x p y p z 377777775 = 266666664 V cos V sin r 377777775 (4{1) Here,trimtrajectoriesbehaveaccordingtothekinematicc onditionsshownin Equations 4{2 4{3 ,and 4{4 .Amotionprimitivemodelcanbeformedbyselectingaset oftrimprimitivesthatbehaveaccordingtothesecondition s. V =const. (4{2) p z = r =const. (4{3) = =const. (4{4) 33

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Thereareatotalof3 m (2 n +1)trimprimitivesselected.Thesetrimprimitives consistofconstant-rateturnsandconstant-rateclimbsat n dierentturnratesand m dierentclimbratesineachdirection.Inaddition,therei sastraight-aheadprimitive thatcorrespondsto =0and r =0,purelyclimbingandpurelydivingprimitives correspondingto =0,andpurelyturning(bothleftandright)primitivescorr esponding to r =0.ThissetisshownasEquations 4{5 and 4{6 .Thevelocity, V ,isheldxedover thesetofallprimitives. 2f 0 ; 1 ; 2 ; ; n gj i j max ;i =1 ; ;n (4{5) p z 2f 0 ; r 1 ; r 2 ; ; r m gj r i j p z ;max;i =1 ; ;m (4{6) 4.3Overview Thepathcreationalgorithmisexecutedusingaseriesofpri marysteps.Adetailed explanationofthesestepsarelocatedintheliterature[ 18 ]butaresummarizedasfollows: 4.3.1SelectaNode Apointisselectedfromthesubspaceofthefeasibilityspac ewhichisspannedbythe positionvariables.Anapproximate,obstacle-freedistan cemetricisusedtodeterminethe nearestnodeintheexistingtree.4.3.2ExtendaBranch Thesetof2 n uniquesolutionsonthepositionsubspaceareenumerated,e valuated, andpruned.Selectioncriteriaisusedtochooseabranchfro mthesetforadditiontothe solutiontree.4.3.3CheckforSolutions Thenewbranchissplitintoanintermediatesetofnodes.Ate achnode,acheckis presentedforobstacle-freeconnectiontothegoalcongur ationonthefull C -spaceusing theoptimal-controlsolution.Whenalowercostsolutionis found,thenewsolutionis 34

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addedtoasolutionlistandthecurrentupperboundisupdate d.Thisobstacle-freepath checknotesifadirectsolutionfromthecurrentnode, C ( t ),tothegoallocation, C g ,exists forthestandard3-sequenceDubinssolution.Thischeckcan bedescribedas: if 9 X 1 ;X 2 ;X 3 suchthat C g = C ( t )+ X 1 ( 1 ;! 1 ;r 1 )+ X 2 ( 2 ;! 2 ;r 2 )+ X 3 ( 3 ;! 3 ;r 3 ) then stoptreegrowth else choosenew C ( t + 1 + 2 )asinEquation 3{1 end 4.4Discussion Itshouldbeemphasizedthatthetreegrowthusesasequenceo ftwomaneuverswhile thenaldirectsolutionusesasequenceofthreemaneuvers. Thisdiscrepancynotesthat a3-sequencepathisoptimalforaDubinsvehiclebuta2-sequ encepathispreferable formaneuveringthroughobstacles.Essentially,the2-seq uencepathwilllimittherange betweennodesandthusforcethenodestolieclosertogether .Suchclosenodeswill addmorenodestothenalsolutionand,whileincurringasma llcomputationalcost, willresultinmoresub-optimalsolutionsfortheresulting path[ 16 ].Alocalminimum withlowercostisanticipatedusingthisalgorithmsincemo resub-optimalsolutionsare computed.Itisanticipatedthatfewernodeswillbemoreec ientforless-obstacle-dense environmentsbutmorenodeswillbemoreecientformore-ob stacle-denseenvironments. Also,toclarifytherecordingfunctionofthealgorithm,th eprocessaboveismodied toimproveupontheoptimalityofthesolution.Whilethetre egrowthiterationsare completed,pathsolutionswillbefound.Afterasolution(p ath)iscomputed,the solution'sassociatedcostfunction(inthiscasetheovera lltraveltime)iscomparedto anyprevioussolutioncosts.Whenalowercostsolution(sma llertotaltraveltime)is identied,thesolutionisrecordedasthe\mostfavorables olution".Thetreeisprescribed 35

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tocontinuetoexpandaxednumberofiterationsandadditio nalsolutionswillbefound andcheckedinthesamefashion.Afteralltheiterationsare complete,thenal\most favorablesolution"willbepresentedasthebestsub-optim alsolution. 36

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CHAPTER5 EXAMPLES 5.1Introduction Severalexamplesofmotionplanningforanaircraftthrough avarietyofenvironments arepresentedinthischapter.Asetofinitialconditionsan dnalconditionsarechosen foreachexampletorerectamissionoftraversingtheregion s.Intherstcase,an obstacle-freeenvironmentisinvestigatedandapathiscre atedusingonlya3-dimensional (3-D)DubinsAirplanepath.Inthesecondandthirdexamples ,thefullmotionplanning algorithmisutilizedforanurbanenvironmentwithamodic ationtotherandomdense tree(RDT)growthalgorithmbeingpresentedinthelatterex ample.Inthefourth andnalexample,thefullmotionplanningalgorithmisutil izedonceagainbutfor anurbanenvironmentthatincludesanelevatedbridgeandac overedwalkway.The dynamicpropertiesfortheformulationofthemotionprimit ivesarelistedinTable 5-1 andarebasedonmeasurementsfromaclassofmicroairvehicl esfromtheFlightControl LaboratoryattheUniversityofFlorida.Thesevehicleprop ertieswillbeusedforallofthe examplesinthischapter. Table5-1.Vehiclepropertiesforexamples. PropertyValue forwardvelocity40ft/s turnradius76ft maxclimbrate30ft/s 5.2Three-DimensionalDubinsAirplanePaths(NoObstacles ) Inthisexample,anobstaclefreeenvironmentisassumed.Si ncethereareno obstaclesthatneedtobeavoided,thetreeportionofthemot ionplanningalgorithm isnotnecessaryandonlythesecondportionofthemotionpla nningalgorithm,the 3-dimensionalDubinsairplanesequence,isimplemented. Thevehiclestartsatapositionof(0,0,0)andaheadingof90 o (dueEast)whileit isrequiredtoendatapositionof(400,400,100)andaheadin gof90 o .Theresultsof 37

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thesimulationarepresentedinFigure 5-1 .Thereare3pathspresented,eachwitha lowercostfunctionthantheoneprevious.Thecostfunction tobeminimizedisdened tobepurelythetimetotraversetheregion.Therstpathcre atedwillbeidentied astherst"best"solution.Asthesimulationrunsthrougha llofthepossibleDubins airplanepossibilities,thesimulationdenesapathasbei ngthe"best"ifithasasmaller overalltraveltimethanthe"best"pathcreatedbeforeit.R ecallingfromTable 2-2 thattherearesixtotalcombinationsofDubinsairplanepat hsforclimbing,onlythree areproducedhere.Theotherthreearenotpresentedbecause astheywereutilized inthealgorithm,theirtraveltimewasnotsmallerthanthep reviouslydened"best" path.Thealgorithmdecidednottoproducethepathandinste admovetothenext possibleDubinsairplanemotionprimitivecombinationlis tedinthealgorithm.Inthis example,the(Right-Straight-Right)Climbsequencepathi sdenedrstashavingatravel timeof36.9seconds.Proceedingthroughthesimulation,th e(Left-Straight-Left)Climb sequencepathisdenedashavingashorterpathtraveltimeo f27.9seconds.Finally, the(Left-Straight-Right)Climbsequencepathisdenedas havingtheshortestpath traveltimeofallat16.4seconds.Inspectionoftheplotscl earlyshowsthattheclimbing Left-Straight-Rightsequencehasthelowestcostfunction ofallasitisalmostastraight pathfromtheinitialcongurationtothegoalconguration 5.3FullMotionPlanninginUrbanEnvironment Inthisexample,thefullmotionplanningalgorithmisutili zedforanurban environment.Thevehiclestartsatapositionof(0,0,0)and aheadingof30 o whileitis requiredtoendatapositionof(500,500,200)andaheadingo f90 o .Thereare4obstacles, orbuildings,ofvarioussizesthatliethroughouttheregio nandmustbestrictlyavoided. ThedetailsoftheobstaclesarepresentedinTable 5-2 Thetreeportionofthemotionplanningalgorithmisutilize dformaneuveringthrough theobstaclerichportionoftheenvironment,followedbyth e3-dimensionalDubins airplanesequencefortravelingfromtheendofthebranches tothegoalpointandheading. 38

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0 200 400 0 200 400 0 50 100 East (ft) North (ft) Up (ft) 0 200 400 0 200 400 0 50 100 East (ft) North (ft) Up (ft) 0 200 400 0 200 400 0 50 100 East (ft) North (ft) Up (ft) Figure5-1.3-DDubinsairplanepathsusingRSR(topleft),L SL(topright),andLSR (bottom)motionprimitivesequences.RDTportionofmotion planningisnot neededbecausetherearenoobstacles. Table5-2.3-Dobstacledimensionsandlocationsforurbane nvironment.Allunitsinfeet. ObstacleCoordinatesofCenterdxdydz Cyan(200,100,50)240-160=80140-60=80100-0=100 Red(100,200,25)115-85=30215-185=3050-0=50 Green(100,300,25)115-85=30315-285=3050-0=50 Pink(300,300,100)340-260=80340-260=80200-0=200 Thealgorithmforthisexampleisrunthreetimestoattemptt olowerthecostofthe trajectoriesevenmorethanwhatcanbedonewithonerunofth ealgorithm.Whenthe algorithmisrunandtherandomdensetree(RDT)isgrown,all thepathsgeneratedmust utilizethistreeintheirmaterialization.Sincethetreei sgrownrandomlyinthespace, thereisachancethatrunningthealgorithmagainwillprodu ceaRDTthatwillprovide evenlowercosttrajectories. 39

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Thealgorithmgeneratesaninitialsub-optimalsolutionas acombinationof tree-expandedpathanddirect-solutionpath.Aftertheite rationcountreachesfruition, atotalof9pathsarecreated,eachwithadecreasedcostasme asuredbytotaltimeto traversefromtheinitialcongurationtothegoalcongura tion.Thecostforthenal solutionoftherstrunis21.9 s .Thevehicleapproachestheregionbutthenturnstothe lefttoavoidthelargecentralobstacle,climbs,andturnst otherightwhereaclearpathto thegoallocationandheadingcanbemadeusingthe3-DDubins airplane.Thenalpath isproducedafter15.8 s ofcomputingtimeandisshownalongwiththecompleteRDT associatedwithitinFigure 5-2 Figure5-2.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#1ofmotion planningalgorithminurbanenvironmentwithtime-travele dcostof21.9 s ThesimulationisrunagainwithanewRDT.Afteralloftheite rationsarecomplete, 10moresub-optimalsolutionsaregenerated,eachwithasma llertotaltraveltimethan theoneprior.Thetenthpathisdeterminedafter16.7 s ofcomputation.Thenalsolution resultsinapathshowninFigure 5-3 thatincludesacostof21.2 s .Thispathalsoturnsto thelefttoavoidthecenterobstaclebutclimbsatahigherra te,thenutilizestheDubins airplanetoreachthegoalconguration. Athirdandnalrunofthesimulation,withanothernewrando mdensetree, generatesonly5sub-optimalsolutionswithdecreasingtot alcost,thenalbeingfound afteronly9.6 s ofcomputationtime.Inthiscase,thenalsolutionrequire sonly18.8 s 40

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Figure5-3.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#2ofmotion planningalgorithminurbanenvironmentwithtime-travele dcostof21.2 s tofollowthepathshowninFigure 5-4 .Thissolutionperformsanalmoststraightpath towardthegoalpoint,turningslightlytothelefttoavoidt hecentralobstacle,allwhile implementingalargerclimbrate.Oncepasttheobstacle,th eoptimalportionofthe solutionisimplemented. Figure5-4.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#3ofmotion planningalgorithminurbanenvironmentwithtime-travele dcostof18.8 s Inthecaseofrunningthesethreesimulations,thethirdrun ofthesimulation providesthesolutionwiththelowestcostof18.8 s oftraveltime.Itappearsthatthis isduetotheutilizationofthelargerclimbrateandthefact thatthepathisalmost 41

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straightbetweentheinitialandgoalpoints.Itshouldbeno tedthatthehigherclimbrate implementedinthisnalrunisstilllowerthanthevehicles maximumclimbrate. 5.4RandomDenseTreeAlgorithmModication Inthisexample,thefullmotionplanningalgorithmisutili zedforthesamecircumstances inSection 5.3 (initialconguration,desiredgoalconguration,obstac lelocations,and obstacledimensions),butsomeoftheattributesoftheRDTd evelopmentarealtered. Thisalgorithmrunsfortwiceasmanyiterationsallowingfo rtheRDTtogrowlarger andwithmorebranches.Inaddition,therandomizednodepla cementisprescribedtobe inruencedbythegoalpointtoagreaterdegree.Thealgorith mforthisexampleisalso runthreetimeswiththepurposeofpotentiallyproducingaR DTthatwillyieldlowercost trajectories.Thesolutionstoallthesimulationsarepres entedasfollows. Uponcompletionoftherstrunofthealgorithm,atotalof4p athsarecreated,each withadecreasedcost.Thenalpathisdeterminedafter25.1 s ofcomputingtime.The costforthenalsolutionoftherstrunis19.6 s .Thevehicleturnsleftimmediatelyand climbsalongthewesternboundaryofthe C -spaceandthenturnsrightbetweenthetwo smallerobstacleswheretheoptimalportionofthealgorith misimplementedtocreatethe secondportionofthepath.ThisnalpathandthecompleteRD Tassociatedwithitare showninFigure 5-5 ThesimulationisrunasecondtimewithanewRDTandafterall oftheiterations arecompleted,11moresub-optimalsolutionsaregenerated ,eachwithasmallertotal traveltimethantheoneprior.Theeleventhandnalpathisd eterminedafter24.0 s of computationanditisshowninFigure 5-6 withatraveltimeof18.6 s .Thispathresembles thenalpathofthethirdruninSection 5.3 asitperformsanalmoststraightpath towardthegoalpoint,turningslightlytothelefttoavoidt hecentralobstacle,allwhile implementinganevenlargerclimbrate.Oncepasttheobstac le,theoptimalportionofthe solutionisimplemented. 42

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Figure5-5.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#1ofmotion planningalgorithmwithRDTformulationadjustmentinurba nenvironment withtime-traveledcostof19.6 s Figure5-6.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#2ofmotion planningalgorithmwithRDTformulationadjustmentinurba nenvironment withtime-traveledcostof18.6 s Thethirdandnalrunofthesimulation,withanothernewRDT ,alsogenerates11 sub-optimalsolutions,eachwithdecreasingtotalcost.47 .1 s ofcomputationtimeisneeded forthisrunofthealgorithmtondthenalsolution.Acosto f19.0 s isrequiredtofollow thenalpathshowninFigure 5-7 .Thepathgeneratedinthissolutionstartsclimbing tothenorth-northeastandtravelsovertherstsmallobsta cle,continuesclimbingwhile turningtotheeastandthenswitchestotheoptimalportiono fthesolution. Inthecaseofrunningthesethreesimulations,thesecondru nofthesimulation providesthesolutionwiththelowestcostof18.6 s oftraveltime.Similartothenalrun 43

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Figure5-7.FullygrownRDT(left)andnalsub-optimalpath (right)ofrun#3ofmotion planningalgorithmwithRDTformulationadjustmentinurba nenvironment withtime-traveledcostof19.0 s ofthealgorithminSection 5.3 ,theevenlargerclimbrateandgeneralheadingofthepath providesthefavorableperformance. Incomparison,therearesomesimilaritiesanddierencesi ntheperformanceofthese twoversionsofthealgorithm.ThesearepresentedinTable 5-3 .Thealteredalgorithm attainstheshortestoverallpathtraveltimeof18.6 s andashorteraveragepathtravel timeof19.07 s .Theaveragecomputingtimemorethandoubleswhenusingthe altered algorithmversestheoriginalalgorithmwhiletheperforma nceonlyimprovesslightly. Table5-3.Motionplanningalgorithmresultscomparison. ShortestOverallAverageOverallAverage AlgorithmTravelTimeTravelTimeComputationTime Original18.8 s 20.63 s 14.03 s Altered18.6 s 19.07 s 32.07 s 5.5FullMotionPlanninginUrbanEnvironmentWithBridges Inthisexample,theoriginalfullmotionplanningalgorith misutilizedforanurban environmentthatincludesacoveredwalkwayandanelevated bridge.Thevehiclestarts atapositionof(0,0,0)andaheadingof60 o whileitisrequiredtoendatapositionof (500,500,200)andaheadingof90 o .Theenvironmentconsistsof2largetowers,1small 44

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Table5-4.Tower,walkway,andbridgedimensionsandlocati ons.Allunitsinfeet. ObstacleCoordinatesofCenterdxdydz NorthTower(50,250,100)100-0=100300-200=100200-0=200 CoveredWalkway(175,275,25)250-100=150300-250=5050-0 =50 NortheastTower(275,275,100)300-250=50300-250=502000=200 ElevatedBridge(275,175,175)300-250=50250-100=150200 -150=50 EastTower(250,50,100)300-200=100100-0=100200-0=200 tower,acoveredwalkway,andanelevatedbridge.Theseobst aclesmustbestrictlyavoided justasintheotherexamples.Thedetailsoftheobstaclesar epresentedinTable 5-4 JustasintheexamplesinSections 5.3 and 5.4 ,thetreeportionofthemotion planningalgorithmisutilizedformaneuveringaroundtheo bstaclesandthe3-dimensional Dubinsairplanesequenceisutilizedfortravelingfromthe endofthebranchestothegoal pointandheading.Thedierencewiththiscaseisthatthepa thisencouragedtoryover thecoveredwalkwayand/orundertheelevatedbridge.Thisd emonstratesthatthemotion plannertreatstheobstaclesas3-dimensionalobjectsrath erthan2-dimensionalregions. Thealgorithmforthisexampleisonlyrunonceastheintenti onistodemonstratethe motionplanningcapabilitiesofthealgorithm,nottheperf ormanceasinSections 5.3 and 5.4 .Twoofthepathscreatedinthissimulation,alongwiththe nalRDT,arepresented inFigure 5-8 .Noticethatthetrajectorypathsgeneratedindeedhavethe abilitytory overobstacles,asinthecaseofthecoveredwalkway,andund erobstacles,asinthecaseof theelevatedbridge. 45

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Figure5-8.Themotionplanningalgorithmtreatstheobstac lesas3-dimensionalobjects ratherthan2-dimensionalregions.Generatedpathscangoo verthecovered walkway(topleft)andundertheelevatedbridge(topright) .ThenalRDT (bottom)isalsoshowngrowingover,under,andaroundtheob stacles. 46

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CHAPTER6 CONCLUSION Theworkpresentedinthisthesishasaddressedtheproblemo fmotionplanningin 3-dimensionalspace.Specically,motionplanningformis sionsinvolvingvehicleswith 3-dimensionalmotionincloseproximityto3-dimensionalo bstacleswasdeveloped.The topicof3-dimensionalmotionprimitiveswasaddressedinC hapter 2 withadevelopment ofa3-dimensionalversionoftheDubinscarcalledtheDubin sairplane.ThisDubins airplaneimplementedaconstantclimbrateinthemotionpri mitivedenitionstoexpand intothethirddimension.Inaddition,the2-primitiveturn -straightpathsegmentswere discussedanddevelopedfor3-dimensionspace.Chapter 3 focusesonthesubjectof randomdensetrees(RDTs).Thetheoryofrandomizedmethods forpathplanning wastoucheduponbriery,followedbytheconceptof2-dimens ionalRDTspresented byKehoe[ 18 ].ThegeneralideaofhowRDTsgrowwaspresentedandthenbot h rapidly-exploringrandomtreetheoryandexpansive-space streetheoryweredescribed anddiscussed.Followingtheintroductionof2-dimensiona lRDTtheory,3-dimensional RDTtheorywasintroducedutilizingtherapidly-exploring randomtreetheorybutaltered for3-dimensionalgrowth.Finallythetopicof3-dimension alRDTtrajectorygeneration wasdiscussedinfull. InChapter 4 ,theideasof3-dimensionalmotionprimitives,boththe2-p rimitive turn-straightcombinationsand3-primitiveDubinsairpla ne,andthe3-dimensionalRDT conceptswerebroughttogetherintothemotionplanningalg orithmthatwasdevelopedfor thisthesis.Anexplanationofhowthealgorithmfunctionsw aspresentedanddiscussedin detail.Thismotionplanningalgorithmwasthenputthrough 4examplesinChapter 5 Therstexamplewasforanobstacle-freeenvironmentandon lythesecondportion ofthemotionplanningalgorithmwasimplemented.Inthesec ondexample,thefull motionplanningalgorithmwasputtouseforavehicletravel ingthroughanobstacle-rich environment.Thethirdexamplewasconductedtoexhibitthe dierenceinperformance 47

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oftheoriginalRDTgrowthalgorithmandanalteredRDTgrowt halgorithmthatgrew fortwiceaslongandwasmoreinruencedbythegoalcongurat ion.Lastly,anexample waspresentedthatshowcasedthemotionplanningalgorithm 'sabilitytonotonlyplan trajectoriesaroundobstacles,butoverandunderthemaswe llinthecaseofaseriesof bridges. Thereareseveraldirectionsthatcanbeconsideredforfutu reworkusingthisresearch. Implementationofthepathparameterizationworkcomplete dbyPachikara[ 25 ]intothe motionplanningalgorithmpresentedinthisthesisisbeing employedforprogressionof thisresearch.Theextensionofthisinvestigationforusei nasensor-basedmotionplanning schemeisalogicaloneandwillalsobeinvestigated.Thecoo perationoftheeldofright controlswiththismotionplanningalgorithmisalsoapoten tialareaofinterestforfuture work. 48

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REFERENCES [1] Choset,H.,Lynch,K.,Hutchinson,S.,Kantor,G.,Burgard, W.,Kavraki,L.,and Thrun,S., PrinciplesofRobotMotion:Theory,Algorithms,andImplem entations MITPress,Cambridge,MA,2005. [2] Faiz,N.,Agrawal,S.,andMurray,R.,\TrajectoryPlanning ofDierentiallyFlat SystemswithDynamicsandInequalities," JournalofGuidance,Control,and Dynamics ,Vol.24,No.2,2001,pp.219-227. [3] Fliess,M.,Levine,J.,Martin,P.,andRouchon,P.\Flatne ssandDefectofNonlinear Systems:IntroductoryTheoryandExamples," InternationalJournalofControl ,Vol. 61,No.6,1995,pp.1327-1361. [4] VanNieuwsadt,M.J.,andMurray,R.M.,\Real-TimeTrajecto ryGenerationfor DiereniallyFlatSystems," InternationalJournalofRobustandNonlinearControl Vol.8,No.11,1998,pp.1995-1020. [5] Kuwata,Y.,andHow,J.,\ThreeDimensionalRecedingHorizo nControlforUAVs," Proceedingsofthe2004AIAAGuidance,Navigation,andCont rolConference Providence,RI,August2004. [6] Richards,A.,andHow,J.,\AircraftTrajectoryPlanningwi thCollisionAvoidance UsingMixedIntegerLinearProgramming," Proceedingsofthe2002IEEEAmerican ControlConference ,Anchorage,AK,May2002. [7] Schouwenaars,T.,How,J.,andFeron,E.,\RecedingHorizon PathPlanningwith ImplicitSafetyGuarantees," Proceedingsofthe2004IEEEAmericanControl Conference ,Boston,MA,June2004. [8] Frazzoli,E.,Dahleh,M.,andFeron,E.\Maneuver-BasedMot ionPlanningfor NonlinearSystemswithSymmetries," IEEETransactionsonRobotics ,Vol.21,No. 6,December2005. [9] Frazzoli,E.,Dahleh,M.,andFeron,E.,\Real-TimeMotionP lanningforAgile AutonomousVehicles," JournalofGuidance,Control,andDynamics ,Vol.25,No.1, January-February2002. [10] Hsu,D.,Latombe,J.C.,andMotwani,R.,\PathPlanninginEx pansive CongurationSpaces," ProceedingsoftheIEEEConferenceonRoboticsandAutomation ,1997. [11] Kavraki,L.E.,Svestka,P.,Latombe,J.C.,andOvermars,M. H.,"Probabilistic RoadmapsforPathPlanninginHigh-DimensionalCongurati onSpaces," IEEE TransactionsonRoboticsandAutomation ,Vol.12,No.4,August1996,pp.566-580. [12] LaValle,S.M.,andKuner,J.J.,\RandomizedKinodynamicP lanning," InternationalJournalofRoboticsResearch ,Vol.20,No.5,May2001,pp.378-400. 49

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[13] Dubins,L.,\OnCurvesofMinimalLengthwithaConstrainton AverageCurvature andwithPrescribedInitialandTerminalPositionsandTang ents," American JournalofMathematics ,Vol.79,No.1,1957,pp.497-516. [14] Shanmugavel,M.,Tsourdos,A.,Zbikowski,R.,andWhite,B. A.,\3DDubinsSets BasedCoordinatedPathPlanningforSwarmofUAVs" AIAAGuidance,Navigation, andControlConference ,Keystone,Colorado,2006. [15] Chitsaz,H.,andLaValle,S.M.,\OnTime-optimalPathsfort heDubinsAirplane" 2007IEEEConferenceonDecisionandControl NewOrleans,LA,2007. [16] Hwangbo,M.,Kuner,J.andKanade,T.,\EcientTwo-Phase3 DMotion PlanningforSmallFixed-WingUAVs," IEEEInternationalConferenceonRobotics andAutomation ,April2007,WeC12.1. [17] Sasiadek,J.,andDuleba,I.,"3DLocalTrajectoryPlannerf orUAV," Journalof IntelligentandRoboticSystems ,Vol.29,2000,191-210. [18] Kehoe,J.,"TrajectoryGenerationforEectiveSensinginC loseProximity Environments," Ph.D.Dissertation ,UniversityofFlorida,August2007. [19] Howlett,J.,Goodrich,M.,andMcLain,T.,\LearningReal-T ime A PathPlanner forSensingClosely-SpacedTargetsfromanAircraft," Proceedingsofthe2003AIAA Guidance,Navigation,andControlConference ,Austin,TX,August2003. [20] LeNy,J.,andFeron,E.,\AnApproximationAlgorithmfortheCurvature-ConstrainedTravelingSalesmanProblem," Proceedingsofthe 43 rd Annual AllertonConferenceonCommunications,Control,andCompu ting ,September2004. [21] McGee,T.G.andHedrick,J.K.,\OptimalPathPlanningwitha KinematicAirplane Model," JournalofGuidance,Control,andDynamics ,Vol.30,No.2,March-April 2007. [22] Tang,Z.andOzguner,U.,\MotionPlanningforMultitargetS urveillancewith MobileSensorAgents," IEEETransactionsonRobotics ,Vol.21,No.5,October 2005,pp.898-908. [23] Yang,G.,andKapila,V.,\OptimalPathPlanningforUnmanne dVehicleswith KinematicandTacticalConstraints," Proceedingsofthe 41 st IEEEConferenceon DecisionandControl ,LasVegas,NV,December2002. [24] Shkel,A.,andLumelsky,V.,\ClassicationoftheDubinsSe t," Roboticsand AutonomousSystems ,Vol.34,2001,pp.179-202. [25] Pachikara,A.,Kehoe,J.,Lind,R.,"APath-Parametrizatio nApproachusing TrajectoryPrimitivesfor3-DimensionalMotionPlanning, AIAAGuidance, NavigationandControlConference ,August2009. 50

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[26] Ferguson,D.,Nidhi,K.,andStentz,A.,\ReplanningwithRR Ts," Proceedingsof theIEEEInternationalConferenceonRoboticsandAutomati on ,May2006,pp. 1243-1248. [27] Kalisiak,M.,andVandePanne,M.,\RRT-Blossom:RRTwithaL ocalFlood-Fill Behavior," ProceedingsoftheIEEEInternationalConferenceonRoboti csand Automation ,May2006,pp.1237-1242. [28] Kuner,J.,andLaValle,S.,\RRTConnect:AnecientApproa chtoSingle-Query PathPlanning," ProceedingsoftheIEEEInternationalConferenceonRoboti csand Automation ,2000,pp.995-1001. [29] Melchior,N.,andSimmons,R.,\ParticleRRTforPathPlanni ngwithUncertainty," ProceedingsoftheIEEEInternationalConferenceonRoboti csandAutomation April2007,pp.1617-1624. [30] Phillips,J.,Bedrossian,N.,andKavraki,L.,\GuidedExpa nsiveSpacesTrees:A SearchStrategyforMotion-andCost-ConstrainedState-Sp aces," Proceedingsofthe 2004IEEEInternationalConferenceonRoboticsandAutomat ion ,NewOrleans,LA, April2004. [31] Strandberg,M.,\AugmentingRRT-PlannerswithLocalTrees ," Proceedingsofthe IEEEInternationalConferenceonRoboticsandAutomation ,Vol.4.,April2004,pp. 3258-3262. [32] Hsu,D.,Kindel,R.,Latombe,J.C.,andRock,S.,\Randomize dKinodynamic MotionPlanningwithMovingObstacles," InternationalJournalofRoboticsResearch ,Vol.21,No.3,2002,pp.233-255. [33] Laumond,J.P., RobotMotionPlanningandControl ,Onlinebook: < http://www.laas.fr/jpl/book.html > 51

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BIOGRAPHICALSKETCH RyanDonovanHurleywasbornin1982inFortMyers,FL.Hegrew upinFort MyersBeach,FLandgraduatedfromCypressLakeHighSchoolC enterfortheArtsin 2001.HethenattendedtheUniversityofFloridawhereherec eivedBachelorofScience degreesinbothaerospaceengineeringandmechanicalengin eeringinDecemberof2006. Thefollowingmonth,hebeganworkforAttractionsandEngin eeringServicesatWalt DisneyWorld.HeleftthecompanytoreturntotheDepartment ofMechanicaland AerospaceEngineeringattheUniversityofFloridainJanua ryof2008topursuegraduate school.HisresearchintheFlightControlLaboratoryhasfo cusedonthematurationof autonomy-enablingguidanceandcontroltechnologiesfors mallunmannedaerialvehicles (UAVs).HereceivedhisMasterofSciencedegreefromtheUni versityofFloridainthe springof2009inaerospaceengineering.Afterpursuinghis doctoraldegree,heplansto worktowardasuccessfulcareerincontrolsengineeringint heeldofaerospacetechnology. 52