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Trajectory Planning for Effective Close-Proximity Sensing with Agile Vehicles

Permanent Link: http://ufdc.ufl.edu/UFE0042697/00001

Material Information

Title: Trajectory Planning for Effective Close-Proximity Sensing with Agile Vehicles
Physical Description: 1 online resource (231 p.)
Language: english
Creator: JOHNSON,BARON
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ALPHA -- PATH -- PLANNING -- RDT -- RRT -- SENSING -- SENSOR -- SURROGATE -- TRAJECTORY -- UAS -- UAV -- UNMANNED
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Unmanned systems are expected to continue growing in usefulness for surveillance missions. Advancing technology in vehicle maneuverability and miniature control systems is allowing new sensing missions to be considered where the vehicle operates in close proximity to the targets it is sensing. This presents challenges not present in stand-off sensing missions commonly performed by unmanned systems. Vehicle motion is directly linked to sensing quality and thus must be considered in the mission-planning phase to ensure adequate sensing is performed. This dissertation presents a methodology for generating kinematically feasible trajectories through cluttered environments which satisfy sensing effectiveness requirements for multiple targets. Vehicles carrying a single line-of-sight (LOS) sensor are considered and the coupling between vehicle motion and sensor orientation is explicitly addressed. Algorithms are introduced which improve upon the required path time while preserving the sensing effectiveness. Surrogate modeling is also introduced as a method to improve trajectories in terms of any specified cost function. The sensor-based path planning framework is adapted for a highly agile unmanned aircraft capable of flying at high angles-of-attack and the results are presented as an example of the usefulness of these trajectory planning techniques. The inclusion of the unique high angle-of-attack flight capability is shown to provide improvements in both the sensing effectiveness and the overall path 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 BARON JOHNSON.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042697:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042697/00001

Material Information

Title: Trajectory Planning for Effective Close-Proximity Sensing with Agile Vehicles
Physical Description: 1 online resource (231 p.)
Language: english
Creator: JOHNSON,BARON
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ALPHA -- PATH -- PLANNING -- RDT -- RRT -- SENSING -- SENSOR -- SURROGATE -- TRAJECTORY -- UAS -- UAV -- UNMANNED
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Unmanned systems are expected to continue growing in usefulness for surveillance missions. Advancing technology in vehicle maneuverability and miniature control systems is allowing new sensing missions to be considered where the vehicle operates in close proximity to the targets it is sensing. This presents challenges not present in stand-off sensing missions commonly performed by unmanned systems. Vehicle motion is directly linked to sensing quality and thus must be considered in the mission-planning phase to ensure adequate sensing is performed. This dissertation presents a methodology for generating kinematically feasible trajectories through cluttered environments which satisfy sensing effectiveness requirements for multiple targets. Vehicles carrying a single line-of-sight (LOS) sensor are considered and the coupling between vehicle motion and sensor orientation is explicitly addressed. Algorithms are introduced which improve upon the required path time while preserving the sensing effectiveness. Surrogate modeling is also introduced as a method to improve trajectories in terms of any specified cost function. The sensor-based path planning framework is adapted for a highly agile unmanned aircraft capable of flying at high angles-of-attack and the results are presented as an example of the usefulness of these trajectory planning techniques. The inclusion of the unique high angle-of-attack flight capability is shown to provide improvements in both the sensing effectiveness and the overall path 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 BARON JOHNSON.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042697:00001


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IwouldliketoextendthankstoanumberofsupportiveandinspiringpeoplewithwhomI'vehadthehonor,privilege,andpleasuretoworkwithduringmytimeattheUniversityofFlorida.Firstofall,thankstomyadvisor,Dr.RickLind,forprovidingmetheopportunitytostudyunderhimduringgraduateschoolintheFlightControlLab.Hisguidance,suggestions,andcritiqueshaveproveninvaluable,perhapsevenasmuchashisadviceoncakesandtheiraccompanyingbeverages.ThankstoDr.PeterIfjuforofferingmethemostexcitingandfunjobIcouldhaveeverimaginedasanundergraduateintheMicroAirVehicleLab.TheexperienceIgainedinthatpositioninspiredmanyaspectsofmycareerpath.AnumberoffellowstudentshavealsosupportedmeinmytimeatUF,eitherdirectlyorindirectly.ScottBowmanalwaysprovidedagreatdealofsupportandmadegreatcontributionstomyunderstandingofelectronicsystems,particularlyintheeldofshootingbottlerocketsfromRCplanes.CarloFrancishelpedmybuddingcareerwithRChelicoptersinmanywaysandprovidedimmeasurablecomicreliefonourtrips.MujahidAbdulrahimhelpedinspiremygraduateschoolpathinbetweenhiswittysatiricalcomments(somehowmadewithastraightface).FellowmembersoftheFlightControlandMicroAirVehicleLabs,includingDanielGrant,DongTran,RyanHurley,BrianRoberts,RobertLove,SankethBhat,AbePachikara,StephenSorley,JudBabcock,JosCocquyt,FrankBoria,DanClaxton,andKyuhoLee,providedworkingenvironmentslikenoother.OtherstudentswithwhomIhadthepleasureofcollaboratingincludeAdamWattsandPJJonesoftheFloridaCooperativeFish&WildlifeResearchUnitandJohnPerryoftheGeomaticsdepartment.Finally,MichaelMortonwasthebestcolleague,classmate,roommate,andfriendforwhichIcouldhavehopedandwastakenfar,fartooearly.Itrulycherishthetimeweenjoyedtogether. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 17 CHAPTER 1INTRODUCTION ................................... 18 1.1Motivation .................................... 18 1.2ProblemStatement ............................... 19 1.3ResearchPlan ................................. 20 1.3.1DissertationOutline .......................... 20 1.3.2Contributions .............................. 21 1.3.3Publications ............................... 23 2MOTIONPLANNING ................................. 25 2.1PathConstraints ................................ 25 2.1.1DifferentialConstraints ......................... 25 2.1.2ObstacleConstraints .......................... 26 2.2MotionModels ................................. 28 2.2.1ModelingwithMotionPrimitives .................... 28 2.2.2MotionPlanningwithMotionPrimitives ................ 29 2.2.3TheDubinsCarKinematicMotionModel ............... 30 2.33-DimensionalMotionPlanning ........................ 38 2.3.12-ElementTrajectoryPrimitives .................... 39 2.3.23-ElementTrajectoryPrimitives .................... 43 3RANDOMIZEDSAMPLING-BASEDTRAJECTORYPLANNING ........ 46 3.1ProbabilisticRoadmap(PRM)Methods ................... 46 3.2RandomDenseTree(RDT)Methods ..................... 48 3.2.1Rapidly-ExploringRandomTrees(RRT) ............... 49 3.2.2ExpansiveSpacesTrees(EST) .................... 50 3.2.3Discussion ................................ 51 3.3RDT-BasedTrajectoryPlannerforPlanar-MotionVehicle .......... 53 3.3.1Overview ................................ 53 3.3.2NewNodeSelection .......................... 53 3.3.3BranchExpansion ........................... 56 3.3.4SolutionCheck ............................. 57 3.3.5Example ................................. 58 5

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................ 61 3.4.1Overview ................................ 61 3.4.2DistanceMetric ............................. 62 3.4.3Examples ................................ 64 3.4.3.1Withoutterminalheadingorightpathangleconstraints 65 3.4.3.2Withterminalheadingandightpathangleconstraints 67 4SENSORPLANNING ................................ 70 4.1RemoteSensorTechnologies ......................... 70 4.1.1ComputerVision ............................ 70 4.1.2Radar .................................. 71 4.1.3Sonar .................................. 73 4.1.4Lidar/Ladar ............................... 73 4.2ModelingtheSensingTask .......................... 74 4.2.1SensingGeometry ........................... 74 4.2.2Visibility ................................. 77 4.2.3ProximityEffects ............................ 82 4.3SensorEffectivenessMetrics ......................... 84 4.3.1Formulation ............................... 84 4.3.2TheQualitySet ............................. 85 4.3.3SensingMissionEffectiveness .................... 86 4.3.4ExampleContrivedMetrics ...................... 86 4.4PathPlanningforSensorEffectiveness ................... 88 4.4.1SensingTrajectoryPrimitives ..................... 89 4.4.2PathPlanningBetweenSensingTrajectoryPrimitives ........ 91 4.4.2.1Multiple-goaltree-growth .................. 91 4.4.2.2TravelingSalespersonProblem(TSP)formulation .... 92 5MULTI-RESOLUTIONPATHIMPROVEMENT ................... 100 5.1OrderReduction ................................ 101 5.1.1Approach ................................ 101 5.1.2Examples ................................ 103 5.1.3MonteCarloSimulations ........................ 106 5.1.3.12-Denvironment ....................... 106 5.1.3.22-DMonteCarlosimulationwitharticialintermediatewaypoints ........................... 106 5.1.3.33-Denvironment ....................... 110 5.1.3.43-DMonteCarlosimulationwitharticialintermediatewaypoints ........................... 111 5.1.3.5Summary ........................... 114 5.2OrderIncrease ................................. 114 5.2.12-DOrderIncrease ........................... 115 5.2.23-DOrderIncrease ........................... 117 5.3SurrogateModeling .............................. 119 6

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..................... 121 5.3.1.12-Dimensional ........................ 121 5.3.1.23-Dimensional ........................ 123 5.3.2EnvironmentswithObstacles ..................... 126 5.3.2.12-Dimensional ........................ 127 5.3.2.23-Dimensional ........................ 128 5.3.3LocalDesignRegions ......................... 131 5.3.3.12-Dimensional ........................ 132 5.3.3.23-Dimensional ........................ 133 5.3.3.3Summary ........................... 137 5.3.4CombinatorialCostFunction ..................... 137 5.3.5CooperativePathPlanning ....................... 138 5.4CombinatorialTrajectoryPlanningExample ................. 141 5.4.1Overview ................................ 141 5.4.2Approach ................................ 141 5.4.3RegionA-PathLength ........................ 142 5.4.4RegionB-SensingEffectiveness ................... 143 5.4.5CompleteTrajectory .......................... 145 6EXAMPLE-SENSORPLANNINGWITHPATHIMPROVEMENT ........ 147 6.1SensingTrajectoryPrimitives ......................... 148 6.2Non-SensingTrajectoryGeneration ...................... 149 6.3Results ..................................... 149 7HIGHANGLE-OF-ATTACKPATHPLANNING ................... 152 7.1HighAngle-of-AttackFlight .......................... 152 7.1.1Motivation ................................ 153 7.1.2PreviousResearch ........................... 154 7.2ExperimentalSetup .............................. 156 7.2.1Aircraft .................................. 156 7.2.2VerticalTail ............................... 158 7.2.3Avionics ................................. 160 7.3FlightTesting .................................. 164 7.4SystemIdentication .............................. 166 7.4.1Procedure ................................ 166 7.4.2DoubletModeling ............................ 169 7.4.2.1Longitudinal ......................... 169 7.4.2.2Lateral ............................ 171 7.4.2.3Directional .......................... 172 7.4.3Steady-StateFlight ........................... 174 7.4.3.1Longitudinal ......................... 174 7.4.3.2Lateral ............................ 176 7.4.3.3Directional .......................... 178 7.5VerticalTailRelationshiptoWingRock .................... 180 7

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............................... 180 7.5.2InvertedTail ............................... 185 7.5.3Parameterization ............................ 185 7.6PathPlanning .................................. 191 7.6.1TrajectoryPrimitives .......................... 192 7.6.2PathPlanningforTime ......................... 195 7.6.3SensorPlanning ............................ 198 7.6.3.1HighAngle-of-AttackAttitude ................ 198 7.6.3.2WingRockAugmentation .................. 199 7.6.3.3SensingTrajectoryPrimitives ................ 200 7.6.3.4PathsforTime ........................ 205 7.6.4Example ................................. 206 7.6.5Summary ................................ 210 8CONCLUSION .................................... 213 8.1Summary .................................... 213 8.2FutureWork ................................... 214 REFERENCES ....................................... 217 BIOGRAPHICALSKETCH ................................ 231 8

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Table page 1-1Publicationsresultingfromthisresearch. ..................... 24 3-1Solutionpathresults. ................................. 60 3-2Example3-DRRTtrajectorysolutionswithoutterminalconstraints. ....... 66 3-3Example3-DRRTtrajectorysolutionswithterminalconstraints. ......... 69 4-1PathlengthsresultingfromEuclideanandtree-basedTSPsolutions. ...... 99 5-1Performanceoforderreductionalgorithmin2-Dand3-Dexamples. ...... 104 5-2Performanceoforderreductionalgorithmin3-Dexamplewithvariedheightobstacles. ....................................... 105 5-3NumberofnoderesultsoforderreductionMonteCarlosimulationwithintermediateconguration. ..................................... 107 5-4PathtimeresultsoforderreductionMonteCarlosimulationwithintermediateconguration. ..................................... 109 5-5Numberofnoderesultsof3-DorderreductionMonteCarlosimulationwithintermediatewaypoints. ............................... 111 5-6Pathlengthresultsof3-DorderreductionMonteCarlosimulationwithintermediatewaypoints. ....................................... 112 5-7Resultsoforderincreasealgorithm2-Dexample. ................. 116 5-8Resultsoforderincreasealgorithm3-Dexample. ................. 117 5-9CombinatorialexampleperformanceresultsofeachstepoftheregionAtrajectoryplanningprocess. .................................. 143 6-1Environmentparametersforsensorplanningexample. .............. 148 6-2Sensorandvehicleparametersforsensorplanningexample. .......... 148 6-3Bestinstantaneoussensingqualityforexampletargets. ............. 149 6-4Pathlengthsfromeachstepinthesensor-basedtrajectoryplanningexample. 150 7-1MiniShowTimespecications. ........................... 157 7-2MiniShowTimecomponents. ............................ 157 7-3Verticaltailspecications. .............................. 159 7-4Sizeandmassofavionics. ............................. 161 9

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............................. 162 7-6TechnicalspecicationsofIMUsensors. ...................... 162 7-7IMUrawoutputmultipliersandresultingunits. ................... 164 7-8Pathtimeresultsofhighangle-of-attackMonteCarlosimulations. ....... 197 7-9Targetparametersforhighangle-of-attackMonteCarlosensingsimulations. 201 7-10Vehicleparametersforhighangle-of-attackMonteCarlosensingsimulations. 202 7-11Sensorparametersforhighangle-of-attackMonteCarlosensingsimulations. 202 7-12ResultsofMonteCarlosensingprimitivesimulationfortargetwithupwardnormalvector. .................................... 203 7-13ResultsofMonteCarlosensingprimitivesimulationfortargetwithsidewaysnormalvector. .................................... 204 7-14ResultsofMonteCarlosensingprimitivesimulationfortargetwithdownwardnormalvector. .................................... 205 7-15Environmentparametersforhighangle-of-attacksensorplanningexample. .. 207 7-16Vehicleparametersforhighangle-of-attacksensorplanningexample. ..... 207 7-17Sensorparametersforhighangle-of-attacksensorplanningexample. ..... 208 7-18Sensingeffectivenessforselectedsensingtrajectoryprimitives. ......... 208 7-19Pathtimeandimprovementofeachsegmentofhighangle-of-attacksensingtrajectory. ....................................... 211 7-20Pathtimeandimprovementoftotalhighangle-of-attacksensingtrajectory. .. 211 10

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Figure page 2-1Vertex-anglesumcollisiondetectiontechnique. .................. 27 2-2Obstacleexpansionforpointwisecollisiondetection. ............... 28 2-3Dubinsreferenceframe. ............................... 33 2-42-elementturn-straighttrajectoryprimitives. .................... 35 2-53-elementDubinspathtrajectoryprimitives. .................... 38 2-6Uniqueplanedenedbyinitialposition,vectorofinitialmotion,andterminalposition. ........................................ 40 2-73-D2-elementturn-straighttrajectoryprimitives. ................. 42 2-8Determinationofinitialrotationfor3-D4-elementtrajectoryprimitive. ...... 45 2-9Resulting3-D4-elementtrajectoryprimitive. .................... 45 3-1ThePRMalgorithm. ................................. 48 3-2TheRRTalgorithm. ................................. 50 3-3TheESTalgorithm. ................................. 51 3-4Treegrowthafter100,250,and500iterations. .................. 52 3-5Distancemetrics. ................................... 55 3-6Minimumcostnodeselection ............................ 56 3-7Branchextensionstepwithpruning. ........................ 57 3-8Thenewbranchsubdividedtoaseriesofintermediatenodesafterpruning. .. 58 3-9Exampleplanningenvironment. ........................... 59 3-10Treegrowthprogression. .............................. 60 3-11Progressionofsolutionpath. ............................ 61 3-12Progressionofbestsolutionin3-Dexamplewithterminalconstraints. ..... 64 3-13Example3-DRRTgrowththroughenvironmentwithobstacles. ......... 65 3-14Progressionofbestsolutionin3-DRRTexample. ................ 66 3-15Example3-DRRTgrowththroughenvironmentwithobstacles. ......... 67 3-16Progressionofbestsolutionin3-DRRTexamplewithterminalconstraints. .. 68 11

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................. 71 4-2Geometryofbody,sensor,andinertialreferenceframessensingtargetti. ... 74 4-3Typicalsensinggeometrywithvisibilityparametersidentied. .......... 78 4-4Imageplanevelocitymagnitudeduetounitsensormotion ............ 80 4-5Imageplanevelocityduetosimultaneoussensorroll,pitch,andyaw. ...... 81 4-6Downwardpointingsensorcoverageduringbanking ............... 83 4-7Visibilityparameterefciencyfunctions ...................... 89 4-8Pre-andpost-primitivecollisionfeasibilityconstraint. ............... 90 4-9Theenvironmentforthetree-basedTSPexample. ................ 93 4-10Sensingtrajectoryprimitivesfoundtoprovidebestsensingoftargets. ..... 94 4-11Visitationsitesfortravelingsalespersonproblem. ................. 94 4-12SuggestedroutefromEuclideanTSPsolution. .................. 95 4-13Besttree-basedpathsgrowntoeachsensingtrajectoryprimitive. ........ 96 4-14Besttree-basedpathsgrownbetweeneachsetofsensingtrajectoryprimitives. 96 4-15Besttree-basedpathsgrownfromsensingtrajectoryprimitivestonalcongurationoftrajectory. ...................................... 97 4-16Tree-basedTSPdistancemetricmatrixandsolutionhistory. ........... 97 4-17Suggestedroutefromtree-basedTSPsolution. .................. 98 4-18Completetrajectoryfromtree-basedTSPsolution. ................ 99 5-1Reductionofintermediatecongurations. ..................... 103 5-2Reductionofintermediatecongurations. ..................... 104 5-3Reductionofintermediatecongurationswithvariedheightobstacles. ..... 105 5-4EnvironmentfororderreductionMonteCarlosimulation. ............. 106 5-5Exampleoftree-basedtrajectorywithintermediatecongurationsandreducedordertrajectory. .................................... 107 5-6NumberofnoderesultsoforderreductionMonteCarlosimulationwithintermediateconguration. ..................................... 108 5-7PathtimeresultsoforderreductionMonteCarlosimulationwithintermediateconguration. ..................................... 109 12

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.......... 110 5-9Numberofnoderesultsof3-DorderreductionMonteCarlosimulationwithintermediatecongurations. ............................. 112 5-10Pathtimeresultsof3-DorderreductionMonteCarlosimulationwithintermediatecongurations. .................................... 113 5-11Samplepathfrom3-DMonteCarlosimulationwithintermediatecongurations. 114 5-12Stepsofthe2-Dorderincreasealgorithm. ..................... 116 5-13Stepsofthe3-Dorderincreasealgorithm. ..................... 118 5-14Exampleoforderexpansionalgorithmwithobstaclesofdifferentheights. ... 119 5-15Examplesoforderexpansionwithwaypointplanetiltedfromlevel. ....... 120 5-16Randomlysampledpathswithsingleintermediatewaypointinobstacle-freeenvironment. ..................................... 122 5-17Surrogatemodelofpathtimeforsinglewaypointinobstacle-freeenvironment. 122 5-18Expectedimprovementmodelforsinglewaypointlocationinobstacle-freeenvironment. ..................................... 123 5-19Randomlysampledseedingdatafor3-Dobstacle-freeexample. ........ 124 5-20Surrogatemodelregionsofminimumpathlengthcost. .............. 125 5-21Regionsofmaximumpredictedmodelimprovement. ............... 125 5-22Isometricviewsofsurrogatemodelcontoursofminimumpathlengthcost. ... 126 5-23Topviewofsurrogatemodelcontoursofminimumpathlengthcost. ...... 127 5-24Randomlysampledpathswithsingleintermediatewaypointinenvironmentwithobstacles. .................................... 128 5-25Surrogatemodeloftimecostfunctionforsinglewaypointlocationinenvironmentwithobstacles. .................................... 128 5-26Expectedimprovementmodelforsinglewaypointlocationinenvironmentwithobstacles. ....................................... 129 5-27Randomlysampledwaypointlocationswithcolorindicatingresultingpathlengthcostfunction. ..................................... 129 5-283-Dsurrogatemodelregionsofminimumpathlengthcostforintermediatewaypointlocationinenvironmentwithobstacles. ................. 130 5-29Regionsofmaximumpredictedmodelimprovementwithobstacles. ...... 130 13

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...................................... 131 5-31Reductionofintermediatecongurations. ..................... 132 5-32Localsurrogatemodelsforindividualwaypointimprovement. .......... 133 5-33Expectedimprovementmodelswithinlocaldesignregionsforindividualwaypointimprovement. ..................................... 134 5-34Reductionofintermediatecongurations. ..................... 134 5-35Localsurrogatemodelseedingwaypoints. ..................... 135 5-36Localsurrogatemodelregionsofminimumpathlengthcost. .......... 136 5-37Localsurrogatemodelregionsofmaximumpathlengthcost. .......... 136 5-38Localregionsofmaximumpredictedmodelimprovement. ............ 137 5-39Environmentforcombinatorialcostfunctionsensingmission. .......... 138 5-40Surrogatemodelsofcombinatorialcostfunction. ................. 139 5-41TwopathsandmosteffectivesensorLOSforseparatevehicleorseparatemissionstominimizethecostfunctionofsensingbothtargets. ......... 140 5-42SinglepathandmosteffectivesensorLOSestimatedtominimizesensingcostfunctionofbothtargets. ............................ 140 5-43Environmentusedincombinatorialexample. ................... 142 5-44Environmentusedincombinatorialexampledividedintoregions. ........ 143 5-45ResultingpathsthroughregionAfromeachstepofthecombinatorialexample. 144 5-46RandomlysampledpathwithsingleintermediatewaypointforsensingtargetinthelocalenvironmentofregionB. ........................ 145 5-47SurrogatemodelofsensingqualitycostfunctionthroughoutregionB. ..... 145 5-48Thetotaltrajectoryresultingfrmothecombinatorialplanningexample. ..... 146 6-1Theenvironmentforthesensor-basedtrajectoryplanningexample. ...... 147 6-2Sensingtrajectoryprimitivesampling. ....................... 149 6-3Resultingpathsfromsensor-basedtrajectoryplanningexample. ........ 150 7-1Graphicalrepresentationofangle-of-attack. .................... 152 7-2F/A-18HARVandX-29. ............................... 154 14

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................. 155 7-4MiniShowTimeaircraft.Photocourtesyofauthor. ................ 157 7-5Verticaltails. ..................................... 159 7-6Interchangeableverticaltailmountedonfuselage.Photocourtesyofauthor. 160 7-7Airbornesensors:IMU,GPSreceiver,andightdatarecorder. ......... 161 7-8Avionicsmountedundercanopy. .......................... 162 7-9WiringdiagramofMiniShowTimeaircraftfordatacollection.Photocourtesyofauthor. ....................................... 163 7-10MiniShowTimeinuprighthighangle-of-attackight.Photocourtesyofauthor. 165 7-11MiniShowTimeininvertedhighangle-of-attackight.Photocourtesyofauthor. 166 7-12Elevatorandpitchrateduringdoublets. ...................... 169 7-13Individualcontributionstoresponsefromlongitudinalmodel. .......... 170 7-14Aileronandrollrateduringdoublets. ........................ 171 7-15Individualcontributionstoresponsefromlateralmodel. ............. 172 7-16Rudderandyawrateduringdoublets. ....................... 173 7-17Individualcontributionstoresponsefromdirectionalmodel. ........... 174 7-18Elevatorandpitchrateduringsteadyight. .................... 175 7-19Uncommandedpitchrate. .............................. 175 7-20PSDofmeasuredpitchrate,uncommandedpitchrate,andelevatorinput. ... 176 7-21Aileronandrollrateduringsteadyight. ...................... 177 7-22Uncommandedrollrate. ............................... 177 7-23PSDofmeasuredrollrate,uncommandedrollrate,andaileroninput. ..... 178 7-24Rudderandyawrateduringsteadyight. ..................... 179 7-25Uncommandedyawrate. .............................. 179 7-26PSDofmeasuredyawrate,uncommandedyawrate,andrudderinput. .... 180 7-27Timeresponsesinuprightconguration. ...................... 181 7-28PSDinuprightcongurationofrollrateandailerondeection. .......... 182 15

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.............. 183 7-30Wavelettransformsofailerondeectioninuprightconguration. ........ 184 7-31Timeresponsesininvertedconguration. ..................... 186 7-32PSDininvertedcongurationofrollrateandailerondeection. ......... 187 7-33Wavelettransformsofrollrateininvertedconguration. ............. 188 7-34Wavelettransformsofailerondeectionininvertedconguration. ........ 189 7-35Meanrollratepeakmagnitudesandpeak-to-peakfrequencies. ......... 190 7-36UpperandlowerboundsofwingrockfrequencyfromPSDsandwavelets. ... 190 7-37Meanwingrockbandwidths. ............................ 191 7-38Conditionalrulesestablishedforhighangle-of-attacktrajectoryprimitives. ... 194 7-39Environmentforhighangle-of-attackMonteCarlosimulations. ......... 195 7-40Exampletrajectoriesfromhighangle-of-attackMonteCarlosimulations. .... 196 7-41Histogramsofpathtimesfromhighangle-of-attackMonteCarlosimulations. 197 7-42Imageplanevelocityaugmentationtoaccountforwingrock. .......... 200 7-43Environmentforhighangle-of-attackMonteCarlosensingsimulations. ..... 201 7-44Environmentforhighangle-of-attacksensorplanningexample. ......... 207 7-45Sensingprimitiveselection. ............................. 208 7-46Tree-basedsensingtrajectory. ........................... 209 7-47Reduced-ordersensingtrajectory. ......................... 210 7-48Increased-ordersensingtrajectory. ......................... 211 16

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1 3 ].Integrationofunmannedaircraftintothenationalairspacesystemwillleadtogreatlyexpandedcivilianusesofunmannedaircraftaswell[ 3 6 ].Theexpansionofunmannedsystemusewillleadtonewapplicationsforsuchsystems,thoughISRislikelytoremainaprimaryapplicationofunmannedsystems.Thecapabilityofunmannedaircraftisgrowing,whichwillexpandtheirapplicationsbeyondthereplacementofmannedmissionsandintonewmissionsmadepossiblebysuchadvancesincapability[ 7 ].Electronicightcontrolsystemscapableofcontrollingandnavigatingsmallunmannedaircrafthavebecomesmallerandmoresophisticated[ 8 10 ].Additionally,alternativemethodsofcontrolandnavigationhavebeenadvancing.Onesuchalternativemethodthatpotentiallycouldmakesmallerandlightervehiclescapableofperformingusefulmissionsisvision-basedcontrolandnavigation[ 11 15 ].Aircraftdesignitselfisalsoadvancing,suchaswithdesignandcontrolofmorphingaircraftwhichenhancemaneuverability[ 16 19 ].Anumberofpreviouslyusedapproachesforplanningmissionsforsensingarenotappropriateforcloser-rangesensingmissionsmadepossiblethroughtheadvancementofsmallerandmoremaneuverableaircraft.Acommonassumptionusedintrajectory 18

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20 22 ].Acommonapproachistodividetheenvironmentintocells,someofwhichrequirevisitationtobeconsideredappropriatelysensed[ 23 25 ].Previousstudieshaveattemptedtoaccountforsensorfootprintbuthavefailedtoaccountforthecoupledbehaviorbetweenthevehicleandsensor[ 25 28 ].Additionally,manyplanningapproachesusesimplieddynamics,suchasapointmassmodel[ 27 30 ],whicharenotappropriatetoconsiderthevehicle/sensororientation.LittleattentionhasbeenpaidtothecouplingbetweenvehiclemotionandsensorqualitybecausecurrentISRusesforunmannedaircraftarefromhighaltitudeandcoverrelativelylargeareas.Thisdissertationaimstoimprovetheprocessoftrajectoryplanningforunmannedsystemsnavigatingclutteredenvironments.Sensingrequirementswithinsuchanenvironmentposeauniquechallengeandrequiretheconsiderationofsensingqualityandvehicularmotionrequirements.Methodstoimprovethetimeand/orsensingqualityoftree-basedplanningmethodsarepresented.Examplesarepresentedwhichdemonstrateusingsuchtechniquesalongwithuniquevehiclecapabilities. 19

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1.3.1DissertationOutlineChapter 1 servesasanintroductiontotheresearch.Theproblemhasbeenmotivatedbythegrowthofunmannedsystemsalongwithadvancesinthecapabilitiesofsuchsystems.ThecapabilitiesandgrowthofunmannedsystemsprovidenewpotentialforISRmissionswhilealsoposingnewchallengeswhichtoaddress.Chapter 2 presentsnecessarybackgroundmaterialtomotionplanningfordynamically-constrainedsystems.Modelsimplicationsarediscussedwhichbetterfacilitatemotionplanning.Asetof2-Dplanarvehiclemaneuversaredenedwhichcanbeusedwithinatrajectoryplanningframework.Finally,the2-Dplanarmaneuversaremodiedtodescribe3-Dmaneuverswhichcanbeutilizedwithina3-Dtrajectoryplanningframework.Chapter 3 presentstheapplicationofrandomizedsampling-basedtrajectoryplanningappliedtodynamicalsystems.Anexistingframeworkoftherandomdensetree(RDT)algorithmismodiedtoaccountforthedynamically-constrainedvehiclemodel.TheRDTframeworkisalsoexpandedto3-Dthroughutilizationofthe3-DmaneuversdevelopedinChapter 2 .Chapter 4 providesanoverviewofremotesensortechnologiesanddevelopsthegeometricrelationshipsinvolvingaxedsensoronavehicle.Metricsofsensingeffectivenessarethenintroduced,whichareusedtodeterminethevalueofagivenpathintermsofsensingspecictargets.Aframeworkispresentedwhichrandomlysamplesalocalregionaroundeachtargetofsensingimportanceinordertocreatefeasibletrajectorysegmentswhicheffectivelysensethetargets.Thesensingsegmentsarethen 20

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5 introducesseveralmethodsforimprovingtree-basedtrajectories.Anorderreductionalgorithmattemptstosimplifythepathwhilepotentiallyreducingthetime/distanceofthepath.Anorderincreasealgorithmattemptstomodifyandaddintermediatewaypointsinordertoagainimprovethetime/distanceofthepath.Surrogatemodelingisthenintroducedasatechniquetomodifyexistingwaypointstoimproveagivencostfunction,whichcouldagainbedistance/timebutcouldalsobesensingeffectivenessorevenacombinationofmetrics.Chapter 6 presentsanexampleofamissionutilizingthesensorplanningframeworkpresentedinChapter 4 aswellaspathimprovementtechniquespresentedinChapter 5 .Thisframeworkisshowntoachieverelativelyshortfeasibletrajectoriesthroughaclutteredenvironmentwhilealsoeffectivelysensingmultipletargets.Chapter 7 presentsanexampleoftrajectoryplanningforavehiclecapableofyingathighangle-of-attack.Wingrockisidentiedasacharacteristicofhighangle-of-attackightthroughfrequencyandtime-frequencyanalysistechniques.Conditionalrulesgoverningwhenhighangle-of-attackightisfeasiblearedeveloped.Theserulesleadtoadditionaltrajectoryprimitiveswhichcanbemadeavailabletothesensor-planningframeworkdevelopedinChapter 4 .Theinclusionofsuchhighangle-of-attackmaneuversisshowntobeeffectiveatimprovingboththesensingeffectivenessaswellasthepathtimeasdeterminedbyRDTgrowth.Finally,Chapter 8 providesabriefsummaryofthedissertationprojectalongwithadiscussionoffutureextensionsofthiswork. 21

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Closed-formsolutionstoconvertplanartrajectoryprimitivesto3-dimensionalmaneuversarepresented. 3-dimensionalrandomdensetreepathplanningisintroducedinakinematically-feasiblemannerensuringcontinuitythroughoutthetrajectory. Anorderdecreasealgorithmisintroducedwhichattemptstosimplifythepathwhilepotentiallyreducingpathtime/distance. Anorderincreasealgorithmisintroducedwhichattemptstomodifyandaddintermediatewaypointstoreducepathtime/distance. Surrogatemodelingisintroducedasamethodtodetermineinitialwaypointlocationfromrandomsampleseeding. Surrogatemodelingispresentedasamethodtoreneexistingwaypointlocationbasedonanygivencostfunction,suchassensingeffectiveness. Surrogatemodelingispresentedasamethodtodeterminemultiple-vehicleusagetominimizecost. Therollrateandailerondeectionarelinearlyrelatedinhighangle-of-attackight,whileotheraxesexhibitcomplicatednonlinearbehavior. Uncommandedwingrockispresentonlywhenverticaltailisinuprightcongurationandnotwhenverticaltailisininvertedconguration. Wingrockisfoundtobeindependentofverticaltailsize. 22

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Wingrockisfoundtobehaveasanarrow-bandphenomenonwithuctuationsintimewithinabroaderfrequencyrange. Imageplanevelocityisintroducedasasensingeffectivenessmetrictoaddressblurringduetorelativemotionbetweenthesensorandtarget. Aframeworkispresentedwhichgenerateslowtime/distancetrajectoriesthroughclutteredenvironmentswhileachievingrequiredsensingeffectivenessofmultipletargets. Theuniqueoperatingcharacteristicsofhighangle-of-attackightarepresentedandincorporatedwithinsensor-basedplanning. 1.3.2 haveyieldedpublishableresults,bothforprofessionalconferencesandjournals.Thealreadyproducedandexpectedpublicationsrelatedtothisresearchareasfollows: 1. Johnson,B.andLind,R.,HighAngle-of-AttackTrajectoryPlanningforEffectiveClose-ProximitySensing,AIAAJournalofGuidance,Control,andDynamics[ToBeSubmitted]. 2. Johnson,B.andLind,R.,HighAngle-of-AttackTrajectoryPlanningforSensingEffectiveness,50thAIAAAerospaceSciencesMeeting[ToBeSubmitted]. 3. Johnson,B.andLind,R.,ImprovingTree-BasedTrajectoriesThroughOrderReduction/ExpansionandSurrogateModels,AIAAGuidance,Navigation,andControlConference,AIAAPaper2010-8020,2010. 4. Johnson,B.andLind,R.,ImprovementAlgorithmsforRandomly-SampledTree-BasedTrajectoriesThroughClutteredEnvironments,ASCEJournalofAerospaceEngineering[Submitted]. 5. Johnson,B.andLind,R.,WaypointSelectionwithSurrogateModeling,TechnicalNotes,AIAAJournalofGuidance,Control,andDynamics[ToBeSubmitted]. 6. Johnson,B.andLind,R.,CharacterizingWingRockwithVariationsinSizeandCongurationofVerticalTail,AIAAJournalofAircraft,Vol.47,No.2,March-April2010,pp.567-576. 23

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Perry,J.,Mohamed,A.,Johnson,B.,andLind,R.,EstimatingAngleofAttackandSideslipUnderHighDynamicsonSmallUAVs,Proceedingsofthe21stInterna-tionalTechnicalMeetingoftheSatelliteDivisionoftheInstituteofNavigationIONGNSS2008,pp.1165-1173,2008. 8. Johnson,B.andLind,R.,HighAngle-of-AttackFlightDynamicsofSmallUAVs,47thAIAAAerospaceSciencesMeeting,AIAAPaper2009-0062,2009. 9. Johnson,B.andLind,R.,CharacterizingWingRockasaFunctionofSizeandCongurationofVerticalTail,AIAAAtmosphericFlightMechanicsConference,AIAAPaper2009-6151,2009. 10. Johnson,B.andLind,R.,-DimensionalTree-BasedTrajectoryPlanningwithHighlyManeuverableVehicles,49thAIAAAerospaceSciencesMeeting,AIAAPaper2011-1286,2011. 11. Johnson,B.andLind,R.,TrajectoryPlanningforTimeandSensingwithHighAngle-of-AttackCapability,AIAAJournalofGuidance,Control,andDynamics[ToBeSubmitted].Thepublicationslistedabovecorrelatewithsectionsofthisdissertation.Table 1-1 presentstherelationshipsbetweentopics,dissertationsections,andresultingpublications. Table1-1. Publicationsresultingfromthisresearch. TopicDissertationSectionsPublications SensorPlanning 4.1 4.4 1 2 ImprovingTree-BasedTrajectories 5.1 5.4 3 4 5 HighAngle-of-AttackFlightandWingRock 7.1 7.5 6 7 8 9 HighAngle-of-AttackPathPlanning 7.6 1 2 10 11 24

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2.1 describespathconstraintsduetodynamicmotionandobstaclesorno-travelregionsintheenvironment.Section 2.2 describesmodelsimplicationsandtechniquestodeterminemaneuversequencesfrompointtopoint.Section 2.3 thenexpandsthemotionmodelto3-dimensionalmaneuversforusebynon-planarvehiclessuchasaircraftandsubmersibles.Theresultingkinematicmodelproducesaseriesofmaneuverswhichcanconsistofeithertwoorthreemotionprimitivestotransitionavehiclefromoneconguration(positionandorientation)toanewpositionorconguration.Themodelrepresentsatradeoffbetweendynamicfeasibilityandcomputationalcost. 25

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Constraintsdirectlyimposedduringtrajectoryplanning 2. Trajectoryplannedunderrelaxedconstraints,thenfullconstraintsappliedaftertheplanningprocessTherstapproachisclearlythemostdesirableandreliablemethod;however,directsolutionsareoftenproblematicwhendealingwiththecomplexityofpracticalapplications.Thesecondmethodcanbeeffectivebutalsodoesnotensuresatisfactionofallconstraintsand/oranyparticulartrajectorytrackingperformance.Anexamplewouldbeaseriesofpoint-to-pointwaypointswhichcanquicklybegeneratedthroughanenvironment.Thepathcanthenberecreatedsubjecttovehicledynamics,butthismayleadtocollisionswithobstaclesintheenvironmentorunsatisfactorydeviationsfromtheplannedtrajectory.Suchanapproachtothisclassofproblemsis`smoothing'adynamicallyinfeasibletrajectorywithmotionprimitives[ 31 ].Thisdissertationisprimarilyconcernedwiththerstmethodoftrajectoryplanningwithdifferentialconstraints.However,someaspectsofthealgorithmsoccassionallyemployprinciplesofindirecttrajectoryplanningmethod. 26

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2-1 .Theinterioranglesbetweeneachpairofadjacentvectors,~iand~j,denotedij,canthenbecomputedwiththedotproductasshowninEquation 2 Vertex-anglesumcollisiondetectiontechnique. 2 2-2A .Obstacleboundariescanbeexpandedbeyondtheactualboundariestoreducethechanceofsuchsituations,asshowninFigure 2-2B .Suchexpansion 27

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BFigure2-2. Obstacleexpansionforpointwisecollisiondetection.A)Satisfactionofpointwiseobstacleconstraintsbyaninfeasibletrajectory.B)Satisfactionofpointwiseexpandedobstacleconstraints. 2.2.1ModelingwithMotionPrimitivesMotionprimitivesprovideaconvenientkinematicmodeltorepresentpathsassociatedwithdynamicconstraints.Motionprimitivesconsistofmaneuversegmentsandtrimsegments.Thestatevectorresultingfromanite-durationmaneuvermotionprimitivecanberepresentedbythetransformation,GM,whichistherelativestatedisplacementduetothemaneuver,~m:C!C,translatedbytheinitialstatevector~x0androtatedbyR(~x0),asshowninEquations 2a and 2b 28

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2a and 2b ,whererepresentsthetimeduration. 2 .InherentinEquation 2 istheassumptionofwings-levelightwhennoturnisbeingperformed. g(2) 32 ].Acompletemodelwouldberepresentedbyatrajectorywhichalternatesbetweenmaneuverandtrimmotionprimitives,witheachmaneuverservingasatransitionbetweentrimstates.Thismodel 29

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2a and 2b ,whereGrepresentsthetotalstatedisplacementduetothesequenceoftransformations. 22 28 30 33 35 ].Thetypicalformulationofthismodeliswellrepresentedbytheuseofmotionprimitivesandhasmotionsimilartothatofacar-likevehicle.Motionprimitivesarethenappliedinsuccessiontocreatetrajectoryprimitives.ThemotionisplanarandoperateswithintheC-spacespannedbytwoEuclideanpositionvariables,px2Randpy2R,andaheadingangle,2R.TheDubinscarmoveswithunitforwardvelocityandisconstrainedtoturnratesofeitherpositiveornegativeunityorzero,whichallowsthemotiontobedescribedbythedifferentialsystemshowninEquation 2 30

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2b aregivenbyEquations 2a to 2c forturningmotionsandEquations 2a to 2c forstraightmotions.ItshouldbenotedthatEquation 2c isill-denedwhen!=0;hence,itisonlyvalidwhenturning,orwhen!=1. 31

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36 40 ].Aparticularalgebraicmethodisutilizedinthisstudy[ 41 ].Inthismethod,areferenceframe,D,isdenedwhichplacestheinitialposition(px0,py0)attheoriginandthenalposition(pxf,pyf)adistancedalongtheXDaxis.Theinitialandnalheadings,1and2,respectively,aremeasuredfromtheXDaxis.TheDcoordinatesystemisdepictedinFigure 2-3 ,andthetransformedinitialandterminalcongurationsaregivenbyEquations 2 and 2 ,respectively.Itshouldbenotedthattheterminalheading,2,inEquation 2 isspeciedfor3-sequencetrajectoryprimitivesbutisunconstrainedfor2-sequencetrajectoryprimitives. 32

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Dubinsreferenceframe. ThetransformationsdescribingtheL,R,andSmotionprimitivescanthenberepresentedwithintheDreferenceframeasshowninEquations 2a 2b ,and 2c ,respectively,wherethecoordinatesareintheorder(XD,YD,).Thesethreetransformationscanthenbeappliedconsecutivelytodescribetrajectoryprimitivesconsistingofaseriesofmotionprimitives. 2a and 2b fortheLSsequenceandEquations 2a and 2b fortheRSsequence,whererepresentsanunspeciedterminalheading[ 42 ].Theonlyinstanceinwhichoneofthese`turn-straight'trajectoryprimitiveswillbeinfeasiblewithintheobstacle-freeDreferenceframeiswhentheinitialandterminalpositionsliewithinoneturningdiameterofeachother,whichisasimpleconditionforwhichtocheck. 33

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2. 2 2-4A and 2-4B depicttheLSandRStrajectoryprimitives,respectively,foraparticularinitialcongurationandterminalposition.TheL,R,andStransformationscanalsobeappliedinsuccessiontoconstructthe3-elementtrajectoryprimitives.Inthisfamilyofprimitivestherearesixpossiblesequences:four`turn-straight-turn'andtwo`turn-turn-turn'sequences.3-elementtrajectoryprimitivesaregenerallyfunctionsofthethreevelocities,turnrates,and 34

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BFigure2-4. 2-elementturn-straighttrajectoryprimitives:A)left-straightsequenceandB)right-straightsequence. timedurations.However,asisthecasewiththe2-elementtrajectoryprimitives,thevelocityismaintainedconstantthroughouttheprimitive,meaningV=V1=V2=V3.Additionally,theturnrateforeachsegmentistypicallyconstrainedtothevehicle'smaximumturnrate,exceptforstraighttrimsegmentsinwhich!iszero,meaninggenerally!=!1=!2=!3.Thegeneralrepresentationfora3-elementtrajectoryprimitive,T3,canbeseeninEquation 2 2 2 2 2 2 ,and 2 ,respectively[ 41 ].Ifasequenceisinfeasible,atleastonetrimdurationsolutionwillresultinacomplexvalue. 35

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2. 3. 4. 36

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86d2+2cos(12)+2d(sin1sin2) 6. 86d2+2cos(12)+2d(sin1+sin2) 2-5 .ThenalturningsegmentoftheRSLandLSRtrajectoryprimitivesseeninFigures 2-5C and 2-5D areshortindurationandthereforedifculttoperceive.These3-elementtrajectoryprimitivesserveasausefulframeworkforsolvingminimum-timetrajectoryproblems.Thetotaltimedurationsofallfeasiblesequencescanbecomparedtondtheminimum-timetrajectory.Dubinsdeterminedthatforagivenconstantforwardspeedandif!representsthevehicle'smaximumturnrate,theoptimalminimum-timetrajectoryisdescribedbyoneofthesixtrajectoryprimitivesintheset[ 34 ].SuchpathsarethereforereferredtoasDubinspaths.Thismethodofdeterminingtheminimum-timepathislikelytonotsucceedwhentheenvironmentisobstacle-rich.However,itcanbeusedasalocalizedmethodofpathplanningwithinalargerframework,whichwillbediscussedlater.Trajectoryprimitivesof 37

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B C D E FFigure2-5. 3-elementDubinspathtrajectoryprimitives:A)LSL,B)RSR,C)RSL,D)LSR,E)RLR,andF)LRL. lengthfourorgreaterexistwhichsatisfyinitialandterminalcongurationrequirements.Thesesolutions,however,aregenerallynon-uniqueandnon-optimal. 43 46 ].Anothermethodwhichlosesoptimalitybutmaintainspathcontinuityconsistsofaddinganarcmotionfromtheinitialcongurationtoacongurationwhichliesonaplanecommontothegoalconguration.TheoptimalDubinscarpathonsuchatiltedplaneisthencomputedtoreachtheterminalconguration[ 47 48 ].YetanotherapproachiterativelyaddspitchingmaneuverstoboththebeginningandtheendofaDubinscarsolutiontoreachtherequiredterminalconguration,includingightpath 38

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49 ].OthermethodsofextendingthisapproachtakeadifferentapproachfromtheDubinscarformulation[ 50 51 ]orinvolvesmoothingtrajectoriestoensure3-dimensionaldynamicfeasibility[ 52 ].Thisdissertationpresentsanadaptationof2-elementturn-straighttrajectoryprimitivesinto3-Dtrajectoryprimitiveswithunconstrainedterminalheadingandightpathangle.Additionally,anadaptationof3-elementtrajectoryprimitivesispresentedwhichaddsanadditionalrotationelementtocreate3-dimensionaltrajectoryprimitiveswhichsatisfyinitialandterminalheadingandightpathangleconstraints.Thisapproachcreates3-Dtrajectorieswhicharefeasiblethankstocontinuousheadingandightpathangleswithoutaddingiterativeprocesses. 2-6 ,assumingthetwopointsandvectorarenotcollinear.A2-elementtrajectoryprimitivecanbeperformedonthisplaneandsatisfytheinitialposition,heading,andightpathangleconstraintsalongwiththeterminalpositionconstraint.The3-D2-elementtrajectoryprimitivesdonotsatisfytheterminalheadingorightpathangleconstraints.Thesolutionofthe3-D2-elementtrajectoryprimitivelyingontheuniquelydenedplaneshowninFigure 2-6 consistsofdeningtheDubinscoordinatesystem,asshowninFigure 2-3 ,ontheplaneandsolvingfora2-Dturn-straighttrajectoryprimitivewithintheDubinscoordinatesystem.Theleft-straightsolutionintheDubinscoordinatesystemaregiveninEquations 2a and 2b andtheright-straightsolutionintheDubinscoordinatesystemaregiveninEquations 2a and 2b .TheformulationoftherequiredtransformationofthetrajectoryprimitivefromtheDubinscoordinatesystembeginsbydeningasetofthreevectorsinboththe3-Dglobalcoordinatesystem(G)aswellasthe2-DDubinscoordinatesystem(D). 39

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Uniqueplanedenedbyinitialposition(),vectorofinitialmotion(!),andterminalposition(). Therstvectorintheglobalcoordinatesystem,VG12R3,isthenormalizedlineextendingfromtheinitialposition,PGi=[PGix,PGiy,PGiz]T2R3,tothenalposition,PGf=[PGfx,PGfy,PGfz]T2R3.Likewise,therstvectorintheDubinscoordinatesystem,VD12R3,isthenormalizedlineextendingfromtheinitialposition,PDi=[PDix,PDiy,PDiz]T2R3,tothenalposition,PDf=[PDfx,PDfy,PDfz]T2R3.TherstvectoriscalculatedfortheglobalandDubinscoordinatesystemswithEquations 2a and 2b ,respectively. 2 .Thesecondvector 40

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2a and 2b ,respectively. 2a and 2b 2a and 2b 41

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2a .Thetransformationmatrixtotransformthe2-Dsolutionbacktothe3-Dglobalcoordinatesystem,TD!G2R44iscalculatedwithEquation 2b 2-7 .ThetwopossiblesolutionsintheDubinscoordinatesystemareshowninFigure 2-7A andthecorrespondingtransformedsolutionsinthe3-DenvironmentareshowninFigure 2-7B BFigure2-7. 3-D2-elementturn-straighttrajectoryprimitivesinA)DubinsreferenceframeandB)3-Dglobalreferenceframe,withinitialposition()andnalposition(x).

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47 48 ].Thisapproachperformsaninitialrotationwhichthenachievescoplanaritybetweenthevectorofmotionandtherequirednalvectorofmotion,whichisuniquelydenedbyheadingandightpathangle.Theinitialrotationisdeterminedbyconsideringaseriesofplaneswhichcontaintheinitialvectorofmotionandarerotatedaboutthevectorofmotionbyvariousamounts.Therequirednalvectorofmotionisextendedinbothdirectionsandtheintersectionpointsbetweenthislineandthevariousinitialrotationplanesarecalculatedasafunctionofinitialposition,Pi,nalvectorofmotion,Vf,andthenormalvectortotheplane,N,asshowninEquation 2 43

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2a and 2a ,respectively.Suchrotationsalignthevectorofmotionwiththeintersectionpointswhichguaranteescoplanaritybetweenthevectorofmotionandtherequiredterminalvectorofmotionbecausethevectorsintersectatacommonpoint.Thiscommonplaneisthemaneuverplaneonwhichaplanar3-elementtrajectoryprimitivecanbeperformed.The3-elementtrajectoryprimitivescanbecalculatedintheDubinscoordinatesystemwithEquations 2 2 ,andthenrotatedintothe3-DenvironmentwiththetransformationcalculatedwithEquation 2b .Theresulting4-elementtrajectoryprimitiveconsistsofaninitialrotationfollowedbyaplanar3-elementtrajectoryprimitive.The6potential3-elementplanartrajectoryprimitives,alongwiththe2potentialinitialrotationdirections,resultsin12potentialtrajectoryprimitivesforeachinitialrotationplanetested.Anexampleofa4-elementtrajectoryprimitiveispresented.Fourtotalinitialrotationplanesareexamined,orientedlevel,vertical,andat45.Theterminalvectorofmotionisextendedandtheintersectionpointswitheachinitialrotationplanearecalculated.TheinitialrotationplanesandintersectionpointsareshowninFigure 2-8A .TheinitialrotationarcwithinoneoftheinitialrotationplanesisthencalculatedasshowninFigure 2-8B .Aftertheinitialrotation,showninFigure 2-8 ,theremainderofthemaneuverisaplanar3-elementtrajectoryprimitive.Theresulting4-elementtrajectoryprimitiveisshowninFigure 2-9 withtheinitialrotationandmaneuverplanesshown. 44

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BFigure2-8. Determinationofinitialrotationfor3-D4-elementtrajectoryprimitive:A)initialrotationplanesatlevel,vertical,and45withextensionofterminalvectorofmotion(),andplaneintersectionpoints(*),andB)resultinginitialrotationplanewithinitialrotationmaneuver(). Figure2-9. Resulting3-D4-elementtrajectoryprimitivewithinitialrotationplaneandmaneuverplanevisible. 45

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3.3 .Theapproachisexpandedto3-dimensionalenvironmentsandvehiclesthatarenotrestrictedtoplanarmotioninSection 3.4 53 ].APRMalgorithmprobabilisticallyconstructsaroadmapoftheC-spacethroughsamplingandconnectingcongurations.ThePRMplannerhasbeenshowntobecompleteinaprobabilisticsense[ 54 55 ], 46

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1. 56 58 ].Complexsystemsthatoperateincomplexenvironmentsoftenrequiretheuseofheuristic-basedsamplingmethodsthatbiassamplingtowardsareasoftheC-spacedeemedimportantfortheparticularsystem,environment,ortask. 2. 3. 4. 59 60 ].AnexampleofthePRMprocessisshowninFigure 3-1 ina2-dimensionalC-spacewithobstacles.Eachpointdepictedrepresentsasampledcongurationandeachlinerepresentsafeasibletrajectorybetweenthecongurationsateachendpoint.Figure 3-1A depictsthesamplingstep,Figure 3-1B depictsthelocalplanningstep(aftercompletingthenearestneighborstep),andFigure 3-1C depictsthequerystepandultimatesolution.PRMmethodsservetoeffectivelysimplifyaplanningprobleminacomplexC-spacetoasimpliedgraphsearchproblem.However,differentially-constrainedsystemsorthosewithhigh-dimensionalC-spacescancreatedifcultywithPRMmethods.Foradifferentially-constrainedsystem,samplingoftheC-spaceisnotsufcient;instead,thestate-spacemustbesampled,whichincreasesthedimensionoftheproblemandthereforerequiresmorelocalsolutionstoachievesimilarnetworkconnectivity. 47

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B CFigure3-1. ThePRMalgorithm:A)Samplingstep,B)nearestneighborandlocalplanningsteps,andC)querystepandsolution. 1. 2. 3. 48

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61 66 ].Twoprimaryalgorithmswhichserveasthebasisformanyoftheexistingvariationsonthegeneralmethodaretherapidly-exploringrandomtree(RRT)expansivespacestree(EST)algorithms.Thesebothexhibitdifferentcorephilosophiesinthemannerinwhichnodesareselectedandexpanded. 50 63 ].Themethodfocusesonrapidexplorationofthedesignspacebybiasingtreegrowthtowardunexploredareas.Therststepisnodeselection,inwhichasampledcongurationischosenfromauniformdistributionoftheC-spaceandadistancemetricisusedtodeterminethenearestexistingnode.Thesecondstepisnodeexpansion,inwhichthenearestnodeisextendedtowardsthesampledcongurationutilizingalocalplanningmethod.Thisdegreeofthisextensionisadesignchoice:somealgorithmsuseaxedstepsize,someusestepsizesproportionaltothedistancemetric,andothersattempttocompletelyconnectthesampledcongurationtothenearestnode.TheRRTprocessisshowninFigure 3-2 withapartially-growntreeina2-dimensionalC-spacewithobstacles.Thesamplingstep,whichselectsarandomposition,Nrand2C2,andthenodeselectionstep,whichselectsthenearestexistingconguration,Nnear,areshowninFigure 3-2A .Theexpansionstep,whichcreatesanewbranchtoanewconguration,Nnew,isshowninFigure 3-2B .NnewmaycoincidewithNrand,ormayliesomepartialdistancebetweenNnearandNrand,whichisthescenariodepictedinFigure 3-2B .Thisseriesofstepsisrepeateduntilthespeciedstoppingcriteriaismet. 49

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TheRRTalgorithm:A)SamplingandnodeselectionstepsandB)expansionstep. 67 68 ].Inthenodeselectionstep,theESTalgorithmselectsanexistingnodetoexpand(insteadofselectinganewcongurationandndingthenearestexistingnode,asintheRRTalgorithm).Theexistingnodetobeexpandedisselectedbyaccordingtoaprobabilitydistribution,whichcanbechosenasadesignchoice.Avaliddistancemetricisthenusedtodenealocalregionsurroundingtheselectednode,andacongurationisrandomlysampledfromwithinthislocalregion.Alocalplanningmethodisthenusedtoconnectthenodechosenforexpansiontotherandomlysampledcongurationwithinthelocalregion.TheESTprocessisshowninFigure 3-3 withapartially-growntreeina2-dimensionalC-spacewithobstacles.Thenodeselectionstep,whichselectstheexpansionnode,Nexp,isshowninFigure 3-3A .AlsoshowninFigure 3-3A isthedenitionofthelocalregionsurroundingNexp,whichisdepictedwiththedashedcircledenedsimplybyanarbitraryEuclideandistancemetricfromNexp,andtherandomsamplingofaconguration,Nrand,withinthelocalregion.Figure 3-3B depictstheexpansionstep,in 50

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TheESTalgorithm:A)NodeselectionandsamplingstepsandB)expansionstep. 69 ].AnexampletreegrowthpatternofanRDTandESTina2-dimensionaldesignspacespanningR2withnodifferentialconstraintsisshowninFigures 3-4A and 3-4B ,respectively,after100,250,and500iterations.Bothtreesarerootedat(0,0),andthedesignspacespansfrom0to10ineachdirection.ThedistributionofsampledcongurationswillaffecttheexplorationbehaviorofanRRT.Additionally,thedistancemetricusedtoidentifythenearestnodehas 51

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BFigure3-4. Treegrowthafter100,250,and500iterations:A)RRTandB)EST. beenshowntosignicantlyaffecttheperformanceofthealgorithm[ 50 70 ].Utilizingtheexactcost-to-govaluefordynamicsystemswithconstraintsmayrequiretoomuchcomputationtocheckall(oratleastallnearby)existingnodes.Therefore,anapproximatedistancemetricmaybeusedforcomputationalease;however,itispossibletothenchooseanodewhichcannotfeasiblyreachthelocalgoal.TheabilitytoselectthedistributionfromwhichexpansionnodesarechosenprovidesexibilityintheESTmethod.TheGuidedEST(GEST)isamodiedESTschemeinwhicheachnodeisassignedanexpansionperformanceweightbasedonproximitytoothernodes,costtoreachgoal,numberofotherbranchesemanatingfromthenode,andothermetrics[ 65 71 ].BasingthesamplingonexpansionperformanceandutilizingthepushmodelenablestheGESTtomorerapidlyimprovethesolution. 52

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3.3.1OverviewThealgorithmpresentedinthissectionisbasedontheRRTalgorithmpresentedinSection 3.2.1 .TheESTalgorithmdoesprovidebenets,suchasimprovedperformanceofresultingtrajectories;however,theRRTalgorithmisselectedforitsefciencyandrobustnessproperties[ 42 ].TheprimarystepsoftheRRTalgorithm,whichareexplainedindetailinthefollowingsections,aresummarizedasfollows: 1. 2. 3. 53

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56 ].Inthisalgorithm,twodifferentdistributionsareutilized:onewhichuniformlysamplestheP-spaceandonewhichisweightedtowardsthegoalpositionwithaGaussiandistribution.Eachdistributionisassignedaprobabilitythatitwillbeutilizedinthecurrentsamplingstep,whichisadesignchoicethatcanbeusedtotuneperformance.Therefore,eachsampleprocedurerstselectswhichdistributiontosamplefrom,andthensamplesPsampfromtheappropriatedistribution.Oncethesampledpoint,Psamp,ischosen,thebestexistingnodefromwhichtoextendabranchmustbedetermined.AnintuitivedistancemetricwouldbetheEuclidean2-norm,whichcapturesthestraightlinedistancebetweenpositions.However,suchametricwouldfailtoaccountforthepathlengthduetodifferentially-constrainedmotionandtheheadingchangethatisoftenrequiredbeforethevehiclecanproceedtowardsPsampinastraightline.Forexample,thepathlengthtoapointdirectlyinfrontofthevehiclewouldbeshorterthantoapointlocatedthesamedistancetothesideofthevehicle.Theidealdistancemetricwouldbetheoptimalcost-to-gowithobstacleavoidance.However,thisapproachaloneisoftenontheorderofdifcultyastheoriginalplanningproblemitself[ 57 ].Theapproachtakenistousetheminimumobstacle-free`turn-straight'trajectoryasthedistancemetric.Thisminimumoccurswhentheturningsegmentisexecutedatthevehicle'smaximumrateofturnandpossibleobstaclecollisionsarenotconsidered.Amorecompletesetofsolutionsandobstacle 54

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3-5A presentsadepictionoftheobstacle-free`turn-straight'distancemetric,basedonavehiclewithinitialcongurationof(px,py,)=(0,0,0)andvelocityandturnrateofunity.Forcomparisonpurposes,Figure 3-5B presentsadepictionoftheEuclidean2-normforthesameinitialvehiclelocation.ComparingFigures 3-5A and 3-5B revealsthesignicanceofincludingthedifferentialmotionconstraintsinthedistancemetric.Itshouldbenotedthatthecircularregionsimmediatelytotheleftandrightofthevehicle'sinitialpositioninFigure 3-5A areregionswhichareunreachableduetothevehicle'sturningradiusconstraint.SuchaconstraintdoesnotaffecttheEuclideandistancefunctioninFigure 3-5B BFigure3-5. Distancemetrics:A)obstacle-freeturn-straightcost-to-go,B)Euclideandistancefunction. ComputationofthetwodistancemetricsisperformedfromeachnodetoPsampusingEquations 2a 2b 2a ,and 2b .Somesolutionsmaybeinfeasible,whichisindicatedbyreturningcomplexvaluesfortrimdurations.Thus,thereareamaximumof2nsolutionbranchestoevaluate.TheprocessofevaluatingthecostfunctionfromeachexistingnodeisdepictedinFigure 3-6 55

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BFigure3-6. Minimumcostnodeselection:A)sampledpoint(+)andexistingnodes(),B)`turn-straight'pathsfromeachnode. Thefactthatthenumberofexistingnodesgrowswitheveryiteration,requiringmoreandmoredistancemetriccomputationsateachiteration,demonstratesadrawbackoftheRRTmethod.Eventhoughthedistancemetriccalculationsarecomputationallycheap,aninnitenumberofexistingnodeswouldeventuallyexistinthelimit.However,theRRTalgorithmdoestendtoconvergepriortothedistancemetricsbecomingcomputationallyburdensome,thoughthatcannotbeguaranteedforallproblems[ 42 ]. 2.1.2 .Theresolutionofsuchapointwisetrajectoryrepresentsatrade-offbetweenadequatelyrepresentingthecontinuoustrajectoryandexcessivecomputationalburden.Ifacollisionisdetectedthebranchisthenpruned,orshortened,priortotherstcollision.Thebranchisprunedsuchthattheendofthebranchliesadistanceofoneminimumturnradiusfromthecollisionpoint.Thisensuresthatafuturebranchcanbe 56

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3-7 BFigure3-7. Branchextensionstepwithpruning:A)thenewbranchisexaminedforpointwisecollisions,B)thenewbranchisprunedpriortotherstcollision. 3-7 beingsubdividedintointermediatenodesisdepictedinFigure 3-8 .Eachofthenodesfromthenewly-addedbrancharethentestedforconnectivitytothespeciedgoalconguration.Inordertosatisfytherequiredheadingandpositionofthegoalconguration,3-elementtrajectoryprimitivesmustbeusedasthenalsegmentofanytrajectory.ThesixpossibletrajectoryprimitiveswhichcanbeusedfortestingconnectivitytothegoalcongurationarediscussedinSection 2.2.3 anddescribedbyEquations 2 2 2 2 2 ,and 2 57

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BFigure3-8. Thenewbranchsubdividedtoaseriesofintermediatenodesafterpruning:A)newbranchasaddedandB)newbranchwithintermediatenodes(x). Eachpossibletrajectoryprimitivethatcanreachthegoalcongurationmustbetestedforobstaclecollisions.Oftheremainingprimitiveswhicharefeasible(ifthereareany),theonewiththelowestcostfunctionischosenasthebestpossiblesolutionprimitiveoriginatingfromthenodeinquestion.Thecostfunctionofthechosensolutionprimitiveisaddedtothecumulativecosttoreachitsoriginationnode.Thissumrepresentsthetotalcostprovidedbythegiventrajectorytoconnecttheinitialandterminalcongurationswhileavoidingallobstacles.Whenthespeciedstoppingcriteriaismet,theoveralltrajectorysolutionwhichprovidesthelowesttotalcostischosenasthebestsolution.Ifthestoppingcriteriahasnotbeenmetafterthisstep,thealgorithmreturnstonewnodeselection,Step 1 ,andrepeatstheprocesstoaddanewbranchandcheckforasolution. 3 3.3 .Thisproblemisbasedonakinematicmodelofaplanarvehiclewhichtravelswithconstantforwardvelocity,V=1unit/s,andexecutesturnsataconstantrate,_=!=1rad/s.Assuch,thevehicle 58

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3-9 ,whichspansfrom0to10inboththeXandYdirectionsandcontainspolygonalobstacles.Allsampledpointsarefromwithinthesebounds.Theinitialcongurationis(px,0,py,0,0)=(0,0,90)andtherequirednalcongurationsis(px,f,py,f,f)=(10,10,90).Theinitialandterminalheadingof90isalignedwiththedirectionoftheY-axis.AsdiscussedinSection 3.3 ,positionsaresampledfromwithintheP-space,consistingofpxandpypositionvariables.2-element`turn-straight'trajectoryprimitivesareusedtoconnectbranchestosampledpoints.3-elementtrajectoryprimitivesarethenusedtotestthereachabilityofthegoalcongurationfromeachnode. Figure3-9. Exampleplanningenvironment. Thestoppingcriteriaspeciedforthisexampleisthreesuccessiveimprovementsuponthepathtime,includingtherstachievedsolution.Thealgorithmwillndtherstsuccessfulpathandthencontinuerunninguntiltwomoresuccessfulpathsarefound, 59

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3.3.2 .TheprogressionoftreegrowthisshowninFigure 3-10 ,witheachtreeproducinganewbestsolution.TheevolutionofthebestsolutionisshowninFigure 3-11 andthecorrespondingpathtimesarepresentedinTable 3-1 .ThetreeshowninFigure 3-10A generatedthesolutionshowninFigure 3-11A andlikewiseforFigures 3-10B and 3-11B andFigures 3-10C and 3-11C B CFigure3-10. Treegrowthprogression:A)after8iterations,B)after9iterations,C)after33iterations. Table3-1. Solutionpathresults. IterationPathTime(s) 821.15915.193314.91 Therstsolutionisfoundafteronly8iterations,andtherstimprovementuponthissolutionisfoundwithonlyasinglefurtheriteration.ThisdemonstratestheabilityoftheRDT-basedplannertondandimproveuponfeasiblesolutiontrajectories.Thenal 60

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B CFigure3-11. Progressionofsolutionpath:A)after8iterations,B)after9iterations,C)after33iterations. solutionimprovementoccursafter33iterations.Atthetimeofndingthenalsolution,atotalof10successfulpathshavebeenfound.Thenalsolutionprovidesonlyaminorreductioninpathtimeoverthepreviousbestsolutionfoundafter9iterationssoclearlythereisariskofdiminishingreturnsonpathtimeimprovement.However,duetothenatureofthealgorithmitisdifculttopredicthowmuchfurtherimprovementisavailableandhowlongitmaytaketoachieve.Thereforethechoiceofstoppingcriteriashouldbeconsideredproblem-specic. 3.4.1OverviewThesolutionmethodintroducedinSection 3.3 canbeutilizedin3-dimensionalplanningproblemswiththeuseof3-Dtrajectoryprimitives,introducedinSection 2.3 .ThisapproachutilizesthelowcomputationalcostofboththeRRTalgorithmandtheclosed-formsolutionoftrajectoryprimitivestoprovideacomputationallyinexpensive3Dtrajectoryplannerthatcanaccountforvehiclekinematicconstraintsaswellasalargenumberofobstaclesorno-travelregions.AsbranchesareaddedtotheRRTgrowth,eachnewsolutionischeckedforfeasibleconnectivitywiththegoalposition.Therstfeasiblesolutionpathtothegoal 61

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3-12A .ThenextdistancemetrictestedistheEuclidean,orstraightline,distance,whichresultsinacomputationtimeof5.8s.Theresultingtreegrowth,asseenin 62

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3-12B ,containsalargenumberoflongturningsegmentsbetween180and360.Thischaracteristicofthetreegrowthislargelyattributabletothefactthat61.4%ofthenearestbranchselectionsareclosertothesampledwaypointthanoneturningdiameterofthevehicle.SuchclosebranchesarethenselectedbasedpurelyontheEuclideandistancedespitethefactthatlessdirectpathsmustbeutilizedtoreachsomecloseregionsthatarenotdirectlyinfrontofthevehicle.ThethirddistancemetricattemptstoaddressthecloseproximityissuebyaugmentingtheEuclideandistanceandstipulatingthatthenearestnodemustlieatleastoneturningdiameterawayfromthesampledwaypoint.Theresultingtreegrowth,showninFigure 3-12C ,requiresacomputationtimeof6.1sandhasfarfewerlongturningsegmentsascomparedtothetreegrowthinFigure 3-12B .However,therearestillanumberofturnsontheorderofapproximately180.Thefourthdistancemetricaugmentsthethirdbystipulatingthatthechangeofdirectiontoreachthesampledwaypointmustbenogreaterthan90.Theresultingtreegrowth,showninFigure 3-12D ,requiresacomputationtimeof6.4sandhasveryfewlongturningsegments.ThisdistancemetricintuitivelyprovidesmoredirecttreegrowthpatternsthanlessaugmentedEuclideandistancemetrics,asevidentinFigure 3-12D .ThecomputationtimerequiredbyalltheEuclideandistancemetricsinFigures 3-12B 3-12D isdrasticallylowerthanthecomputationtimerequiredtocomputethepathlengthofthetrajectoryprimitivestoconnecteachbranchtothesampledwaypoint,asseeninFigure 3-12A .TheEuclideandistancemetricwithconstraintsontheproximityandrequiredchangeindirection,showninFigure 3-12D ,providesthemostdirecttreegrowthwithoutaconsiderableincreaseincomputationtimeoverthepureEuclideandistance.Thisdistancemetricischosenforusewith3-DRRT-basedplannersinthisstudyduetothetreegrowthpropertiesandthecomputationtimeofmorethananorderofmagnitudelessthanusingthetruepathlengthasthedistancemetric. 63

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B C DFigure3-12. RDTgrowthof200attemptedbrancheswithvariousdistancemetricsfornodeselection:A)pathlength(147s),B)Euclideandistance(5.8s),C)Euclideandistancegreaterthanoneturndiameter(6.1s),andD)Euclideandistancegreaterthanoneturndiameteranddirectionchangelessthan90(6.4s). 64

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3.3 isperformedand3-D2-elementturn-straighttrajectoryprimitivesareutilizedforeverybranchgrowthaswellasforcheckingconnectivitytotheterminalposition.The3-D2-elementtrajectoryprimitivesaredescribedinSection 2.3 .Thedistancemetricutilizedtoselecttheexistingnodeforexpansionwasstraight-linedistancewiththerequirementsthatitmustlieatleastoneturndiameterfromtherandomlysampledpointandtherequireddirectionchangemustbelessthan90,asdescribedinSection 3.4.2 .TheRRTgrowthofthisexamplethroughthe3-Dobstacle-ladenenvironmentisshowninFigure 3-13 Figure3-13. Example3-DRRTgrowththroughenvironmentwithobstacles. Eachnewlyaddedbranchischeckedforfeasibleconnectivitywiththeterminalposition.Ifafeasiblesolutiontrajectoryexists,thetotaltrajectorytimeiscomparedtothepreviously-foundsolutions.Ifthenewly-foundsolutionyieldsashortertrajectorytimethananypreviously-foundsolutionsitisretainedasthecurrentbestsolution.Asthetreegrowthprogressesthroughtheenvironmentandbecomesmoredense,moresolutions 65

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3-14 B CFigure3-14. Progressionofbestsolutionin3-DRRTexamplewithoutterminalheadingorightpathangleconstraints:A)bestsolution1,B)bestsolution2,andC)bestsolution3. Thetotalnumberofbranchiterations,totalnumberoffeasiblesolutions,pathlength,improvementuponthepreviousbestpathtime,andtherequiredcomputationaltimeforeachnewbestsolutionarepresentedinTable 3-2 .Notethatatthetimeofeachbestsolutionmorefeasiblesolutionshavebeenfound.However,theyarenotfeaturediftheydonotimproveuponthepreviousbesttotalpathtime. Table3-2. Example3-DRRTtrajectorysolutionswithoutterminalconstraints. BestSolutionBestSolutionBestSolution123 BranchIteration3662TotalSolutions1339PathLength20.9419.7418.77ImprovementFromPreviousBestN/A5.73%4.91%TotalComputationalTime0.235s0.611s5.594s Thisexampledemonstratesthecapabilityofthisalgorithmtorapidlyproducefeasibletrajectoriesthroughobstacle-richenvironments.Furtheriterationscanbeutilizedtocontinueimprovingthetrajectoryascomputationalresourcesandtimepermit. 66

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3.3 isperformedand3-D2-elementturn-straighttrajectoryprimitivesareutilizedforeverybranchgrowth.Eachbranchischeckedforfeasibleconnectiontotheterminalcongurationwith4-elementtrajectoryprimitivesdescribedinSection 2.3 .Thedistancemetricutilizedtoselecttheexistingnodeforexpansionwasstraight-linedistancewiththerequirementsthatitmustlieatleastoneturndiameterfromtherandomlysampledpointandtherequireddirectionchangemustbelessthan90,asdescribedinSection 3.4.2 .TheRRTgrowthofthisexamplethroughthe3-Dobstacle-ladenenvironmentisshowninFigure 3-15 Figure3-15. Example3-DRRTgrowththroughenvironmentwithobstacles. Eachnewlyaddedbranchischeckedforfeasibleconnectivitywiththeterminalconguration.Ifafeasiblesolutiontrajectoryexists,thetotalpathlengthiscomparedtothepreviously-foundsolutions.Ifthenewly-foundsolutionyieldsashorterpathlengththananypreviously-foundsolutionsitisretainedasthecurrentbestsolution.Asthetreegrowthprogressesthroughtheenvironmentandbecomesmoredense,moresolutions 67

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3-16 B C DFigure3-16. Progressionofbestsolutionin3-Dexamplewithterminalconstraints:A)bestsolution1,B)bestsolution2,C)bestsolution3,andD)bestsolution4. Thetotalnumberofbranchiterations,totalnumberoffeasiblesolutions,pathlength,improvementuponthepreviousbestpathtime,andtherequiredcomputationaltimeforeachnewbestsolutionarepresentedinTable 3-3 .Notethatatthetimeofeachbestsolutionmorefeasiblesolutionshavebeenfound.However,theyarenotfeaturediftheydonotimproveuponthepreviousbesttotalpathtime. 68

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Example3-DRRTtrajectorysolutionswithterminalconstraints. BestSolutionBestSolutionBestSolutionBestSolution1234 BranchIteration271155TotalSolutions14737PathLength24.3220.6019.0118.98ImprovementFromPreviousBestN/A15.30%7.72%0.16%TotalComputationalTime1.140s12.234s15.029s152.828s Thecomputationaltimesforsolutionswithterminalconstraints,presentedinTable 3-3 ,aresignicantlygreaterthanthoseforthecasewithoutterminalconstraints,presentedinTable 3-2 .Theprimarydifferencebetweenthecomputationsofeachcaseisthecheckforgoalconnectivityateachbranchaddition.Withoutterminalconstraintstherearetwopotentialprimitivestoreachthegoalconguration:`left-straight'and`right-straight'.However,withterminalconstraintsthereare12potentialsolutionsforeachofthefollowinginitialrotationplanestested:level,vertical,and45.Therefore,eachbranchadditionmusttestthefeasibilityof48trajectoryprimitivesthatmayconnecttothegoalconguration.Thecalculationofthetrajectoryprimitivesisnotasignicantburdenonthecomputationaltime,though,becausetheyareallclosed-formalgebraicsolutions.Thesignicantincreaseincomputationaltimearisesfromthecollisioncheckprocessforeachpotentialsolution.Thealgorithmcouldbemademoreefcientbytestingonlycertainsetsofprimitivesthatarelikelytoyieldimprovementstothepriorbestsolution.Additionally,thecollisioncheckprocesscouldbemademoreefcientbyperformingacoarsecheckfollowedbycheckswithnerresolutionsinordertomorerapidlyrecognizethataprimitivewillcollidewithanobstacle. 69

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4.1 discussescommonlyusedremotesensingtechnologieswhichcouldbecarriedaboardavehicleconsideredinthisstudy.Section 4.2 formulatesthemathematicalrelationshipsofthesensingtask.Thegeometryofthevariousreferenceframesintheproblemareexpressed,aswellasthevisibilitysetandproximityeffectsduetoclose-rangesensing.Section 4.3 discussessensoreffectivenessmetricsbasedonparameterssuchasincidenceangleandrangeandpresentsasetofrepresentativeeffectivenessmetricstobeusedinexamples.Section 4.4 presentsaframeworkforplanningtrajectoriesthroughclutteredenvironmentswhileachievinghighsensingeffectivenessformultipletargets. 70

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Projectivegeometryofthecollinearitycondition. geometry,asshowninFigure 4-1 withtheassumptionofthecollinearitycondition[ 75 ].Thisprojectioncausesthelossofdepthinformationinasingleframe[ 76 ].Multipleoverlappingimagescantheoreticallyovercomethelossofdepthinformationifcameramotionisknown.Themovementof`featurepoints'withintheimageplanecanbeusedwith`structurefrommotion'(SFM)techniquestoreconstruct3-dimensionalenvironments[ 77 ].Attentionhasbeenplacedtowardssuchmethodsforvehiclenavigationandcontrol[ 13 15 78 81 ].Inmanysensingmissions,however,nosuchreconstructionisrequired.Manytargetsareidentiablesimplyby2-dimensionalvisible-lightimages,providedtheconditionsatcapturearesufcient.Conditionswhichcandegradetargetidenticationcapabilitiesincludegeometricfactors,suchasanglebetweenthefaceofthetargetandthesensor,distancetotarget,andradiallocationoftargetwithintheeld-of-view(FOV).Videoisparticularlysensitivetorelativemotionbetweenthesensorandtargetwhichcanbeduetovehiclespeed,angularrotations,orchangesindirection.[ 82 84 ].Therelativespeedofthecameraandtargetisjudgedintheprojectedimageplane. 71

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85 ].Ultrawideband(UWB)radarsystemshaveaddressedtherangeresolutionbyemittingawiderangeoffrequenciessimultaneously.Thebearingresolutionismoredifculttoaddresswithoutexceptionallylargeantennae.Forthispurpose,synthetic-apertureradar(SAR)processingtechniqueshavebeendeveloped.SARutilizesmultiplemeasurementsmadefromasmallerantennaonamovingplatformtocreateanimageasiftheenvironmentwassensedbyasinglepulsefromalargerantenna.UWBandSARradarsystemsallowalargenumberofapplicationsforradar.PulsedUWBsystems,whichpulseshort-durationenergyathighandlowfrequencies,havethecapabilitytopenetratefoliageandsoilwhilealsoresolvingsmalltargets[ 86 89 ].SuchcapabilitymakesUWBradarusefulforanumberofapplicationswithconcealedtargets[ 90 91 ].Predictedadvancesintechnologyforeseeradarsystemseventuallyhavingthecapabilitytoimageatframeratescomparabletovisible-lightsensors[ 89 92 ].Motionofthesensingplatformcanhaveeffectsonthesensingqualityofradarsystems.Scanningsystems,whichscanbackandforthtocoveralargearea,requireaconstantplatformvelocityandattitudeinordertoentirelycoverthespeciedareacompletelyandwithproperresolutionthroughout.SARradarsystems,whilerequiringmotionoftheantenna,aresensitivetochangesofvelocityandorientationaswell.Vehiclemotion,suchasrollingorpitching,canaffectthesequenceofimagescapturedbytheradarpriortoSARprocessing[ 93 ].Themostcommonwayofdealingwithsuchextraneousmotionsistoactivelygimbaltheantennatonegatetheeffectsofplatformmotion[ 94 ].Accuratepositionandattitudeinformationmustbeprovidedbyaninertialnavigationsystem(INS)inordertocompensateforvehiclemotioneitherwithactivelygimbaledantennaeorthroughpost-processing.Itisdifculttoachievehighaccuracy 72

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95 98 ].Theresultingdatacanbeusedforavarietyofapplications,includingnavigation,objectdetection,andmapping.MappingandimagingapplicationsutilizeprocessingsimilarlytothatofSARtocreatesyntheticaperturesonar(SAS)[ 99 101 ].Motionofthesensorplatformcanaffectsonarsimilarlytoradar.Scanningsystemsrequireconstantforwardvelocityandattitudetoensurecoverageandresolution,whileSASsystemsmustsomehowcompensatefortheplatformmotionsthatarenotprescribed,eitherthroughactivelycompensatingwithgimbaledantannaeorinpost-processing[ 101 102 ].Again,compensatingforplatformmotionrequiresaccurateINSsolutions,whichcanbedifculttoobtainonsmallvehicles. 73

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Geometryofbody,sensor,andinertialreferenceframessensingtargetti. intensityforanentire2-dimensionalimageoftheenvironment[ 103 104 ].Itisexpectedthatsuchsystemswillbeabletoaccomplishframeratescomparabletovideocameras,withalsothepotentialofbeingsmaller,lighter,andlesscomplicatedsensors[ 103 105 ].TheNATOResearchandTechnologyOrganizationpredictsthatsuchsystemswillyield`mega-pixel'imagingfromarangeof10kmatframeratesof30Hzorfasterwithinadecade[ 106 ].Scanninglidarsystemsareagainsusceptibletoanychangesintheplatform'svelocityoranyotherextraneousmovementofthevehicle,whichcancause`imagesmear'[ 107 108 ].FPAlidarsystemsmaynotbeassusceptibletoplatformmotion;however,inordertoaccuratelyresolveabsolutepositionsofobjectswithintheimage,accurateINSsolutionsmustbeavailable. 4.2.1SensingGeometryThegeneralsensingcongurationconsideredinthisstudyisasingle`line-of-sight'(LOS)sensorrigidlymountedtoavehicle,asdepictedinFigure 4-2 74

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4 ,whichalsovaryintimeduetothevehicle'smovements.TheEuleranglesinEquation 4 ,B,B,andB,correspondtothevehicle'sroll,pitch,andyawangles,respectively.Everyparameterhasatimedependencysothenotationwillbeomittedforconciseness. 4 4 75

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4 4 ,andthetransformation,TES,canthenbeexpressedintermsoftheseparameters,asinEquation 4 4 4 4 ,where^ns,irepresents^niintermsofthebasisvectorsofS. 76

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4.2.1 .Sensorandtask-specicparameterscanbeincludedinsuchaformulationtochangethevisibilityspaceappropriately.Suchcharacterizationofvisibilityhasbeenstudiedintheeldofrobotics,primarilyforcomputervisiontasks[ 109 111 ].Inthegeneralformulationpresentedhere,thefollowingvariablesareprojectedontothevisibilityspace:

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4 ,issimplythenormoftherelativepositionvectorbetweenthesensorandtarget.Variationsinrangeimpacttheintensityandresolutionofcollecteddata. 4 ,istheanglebetweentheLOSandthetargetsurfacenormal.Highincidenceanglescanresultindistortionandaregenerallyundesirableinsensingmissions. ri(4)TheFOVangle,describedbyEquation 4 ,istheanglebetweentheLOSandthecenteroftheFOVrepresentedby^s3.Thisanglerepresentstheradialpositionofthetargetinthesensor'sFOVandcanbeaconsiderationinsensingduetolensdistortion. ri(4)Therange,incidenceangle,andFOVangleareidentiedinthesensinggeometryshowninFigure 4-3 Typicalsensinggeometrywithvisibilityparametersidentied. 78

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4a and 4b ,respectively[ 76 112 114 ]andthetotalimageplanevelocityoftheithtarget,i,isthendescribedbyEquation 4 ricosf,iVs,xuti ricosf,iVs,yvti 4-4A 4-4B 4-4C ,and 4-4D ,respectively.Theimageplanevelocityduetox-andy-velocityareconstantthroughouttheimageplane;hence,theseplotsareomitted.Theimageplanevelocitymagnitudethroughouttheimageplaneduetosimultaneousrolling,pitching,andyawingofthesensorisshowninFigure 4-5 .Allsensorangularratesareofunitmagnitude.Necessaryconditionsforvisibilitycanbedenedinabinarysensewithsimpleboundsplacedonthevisibilityparameters.TheboundsonrangeandFOVarebasedontheassumptionsthatthesensorhasaniterangeandniteFOV(i.e.isnotomnidirectional),asshowninEquations 4 and 4 ,respectively. 79

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B C DFigure4-4. Imageplanevelocitymagnitudeduetounitsensormotion:A)rollrate,B)pitchrate,C)yawrate,andD)z-velocity. 4 80

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Imageplanevelocityduetosimultaneousunitsensorroll,pitch,andyawmotions. tohaveoccuredsuccessfullyonlyifthetargetvelocityintheimageplaneisbelowaspeciedmaximumvalue,asshowninEquation 4 2.1.2 .Thevisibilityset,orthecollectionofsensorcongurationswhichyieldvisibility,oftheithtargetistheregionwhichsatisesallconstraints.ThevisibilitysetfortheithtargetcanberepresentedbyEquation 4 81

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4.2.2 .AssumingthesensorrequiresLOS,ifanobstacleorsurfaceintersectsthesensor'sLOSnothingbeyondtheobstacleorsurfaceisvisible.Obstaclesthatarevisibletheneffectivelycastshadowsofnon-visiblepoints,andthecloserthesensoristotheoccludingsurface,thelargertheareaofnon-visibility.Assuch,clutteredenvironmentsmaysignicantlylimitViforsometargets,requiringcarefulsensorplacementtoensureeffectivesensing.Anothermajorissueisthecouplingbetweenvehicleandsensormotion.Assumingnon-gimbaledsensors,anychangeinvehiclecongurationdirectlyaffectsthesensorconguration.Theimpactofsuchcouplingvariesbytheproximityofthesensedenvironment.Instand-offsensing,theangularFOVprojectioncoversalargearea.Thus,signicantoverlapoftenexistsbetweensubsequentsensing`snapshots,'suchastheindividualframesofavideo.Inclose-proximitysensing,though,thesensorprojectioncoverslessarea,generallyresultinginlessoverlap.Thislackofoverlapmakesclose-proximitysensingparticularlyvulnerabletovehiclemotions.Asanexample,consideranaircraftcarryingadownward-lookingsensorthatmustmakeacoursecorrectionbybankingbrieyandreturningtolevel.Inthestand-offcase,theoverlapmaybesufcientsuchthatnosignicantarearemainsun-sensedduringthebriefbankingperiod,ascomparedtomaintainingthedownwardsensororientationthroughouttheprocess.However,intheclose-proximitycase,thelackofoverlap 82

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4-6 BFigure4-6. Downwardpointingsensorcoverageduringbanking:A)stand-offsensingwithlargeprojectionareacoversentireightpathduringbanking,B)close-proximitysensingwithsmallerprojectionareafailstocovertheentireightpathduringbanking. AthirdsignicantissuewiththescaleofsensingmissionsdealswiththevariationofrangeandincidenceanglesseenacrosstheFOV.Whenthesensorislocatedatastand-offdistance,littlevariationoccursintherangerelativetothesensingdistance.However,whenlocatedatclose-range,therangerelativetosensingdistancecanvarysignicantly.Additionally,theincidenceangleofasensedsurfacewouldvarylittleinthestand-offcase,whilepotentiallyvaryingalargeamountintheclose-proximitycase.Analissuewithclose-proximitysensingoccursduetorelativemotionofthetargetintheprojectedimageplane.Theimageplaneforagivensensorsuchasacameraisconstantandbaseduponthefocallength,f.Foragivenrelativemotionbetweenthesensorandtarget,theimageplanevelocityisinverselyproportionaltothesensingrange.Therefore,close-rangesensingismoresusceptibletoblurringfromvehiclemotionthanstand-offsensing.Furthermore,thenatureofclose-proximitysensingmissionsissuchthatmorerapidandaggressivemaneuversaretypicallyrequiredto 83

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4.3 4.2.2 establishedadenitionofvisibilityforagenericLOSsensor,suchthattheithtargetisvisibleonlyforsensorcongurationsthatliewithinVi.ThisdenitioncanbeexpressedasabinaryvariableasexpressedinEquation 4 4 representsanecessaryconditionforeffectivesensing;however,itignoresthequalityorusefulnessofthedatacollected.Thisomissionmaynotimpactstand-offsensingsignicantly,butinclose-proximitysensingthedataqualitymayvarysubstantiallyoverVi.Aframeworkcanbeusedwhichincludessuchcharacteristicsandtakesintoaccountthesensor-specicqualityofthecollecteddata. 4.2.2 .Thisfunctionisthenmappedtoaunitinterval,asshowninEquation 4 ,andrepresentsametricoftheeffectivenesswithwhichtheithtargethasbeensensed. 84

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4 4 to 4 4 to 4 lieoutsidetherangeofspeciedacceptablevaluesinEquations 4 to 4 ,thesensoreffectivenessmetricqimustequatetozero. 4 .TheactualformulationofQidependsuponthespecicformulationofqi,whichisspecictoboththesensorandmission.Thisformulationofasetofvariablecombinationsthatachievethedesiredvalueofqi,Qi,providesatargetsetformotionplanningproblemsofallowablesensorcongurations. 85

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4 4 .AresultingvalueofQmax,iindicatesthattargetiwassensedwiththateffectivenessatleastonceduringthetimeinterval.ThisisanintuitiveandsimpleformulationofQi,butothereffectivenessformulationscanbeusedasthemissionandsensordictate. 86

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4 4 ensuresthatthesensoreffectivenessequatestozeroifthesensorcongurationdoesnotliewithinthevisibilityset.Eachoftheefciencyfunctionsrangesfrom0to1;thereforethelargestproductattainablebyEquation 4 is1,whichrepresentsoptimalsensinginallvisibilityparameters.Representativecontrivedefciencyfunctionsarepresentedforrange,FOVangle,andincidenceangleinEquations 4 4 ,and 4 ,respectively.Theseefciencyfunctionshavebeenusedforsimilarsensing-basedmotionplanningsimulationsinthepast[ 42 ]. 1+(ri=rmax)40.7exp400r2i 1+(0.75f,i=f,max)4:f,if,max0:f,i>f,max(4) 1+(i=max)41:imax0:i>max(4)Anadditionalefciencyfunctionforimageplanevelocityisaddedtotheset,aspresentedinEquation 4 1+(i=max)31:imax0:i>max(4) 87

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4 4-7 .Theefciencyfunctionrelatedtorange,presentedinFigure 4-7A ,exhibitslossesatbothshortrangeduetopoorspatialcoverageandlongrangeduetopoorresolution.TheefciencyfunctionrelatedtoFOVangle,presentedinFigure 4-7B ,exhibitsonlyslightlossesthroughouttherangeofFOVangleswithinthevisibilitysetandasuddenlossinefciencyatthemaximumFOVangle.Thisistypicalofradialdistortionduetolenses,asseencommonlyincamerasandothersimilarsensors.Theefciencyfunctionrelatedtoincidenceangle,presentedinFigure 4-7C ,exhibitslossesasthesensorLOSapproachesnormalcywiththetargetnormalvector.Theefciencyfunctionrelatedtoimageplanevelocity,presentedinFigure 4-7D ,exhibitslossesduetoblurringasitapproachesthemaximumallowableimageplanevelocity. 4.3 .Theselectionofsuchsensingtrajectoryprimitivesisbasedentirelyuponsensingeffectivenessandfeasibilitywithoutregardtopathlength,time,oranyothercostfunction.Thepathplanningapproachthengrowsrandomdensetrees,asdescribedinSections 3.3 3.4 ,inordertodeterminetheorderinwhichthesensingtrajectoryprimitiveswillbeperformedandthepathswhichreacheachsensingtrajectoryprimitiveintheproperorder.Thepathsbetweenthesensingtrajectoryprimitivesaregrownbasedentirelyonpathlengthortime.Thenalpaththenalternatesbetween 88

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B C DFigure4-7. Visibilityparameterefciencyfunctions:A)range,B)FOVangle,C)incidenceangle,andD)imageplanevelocity portionswhichprovidelowpathlength/timeandportionswhichprovidehighsensingeffectiveness. 2.3.1 .Somerandomlysampledparameterswillyieldinfeasibletrajectoryprimitiveswhicharenotconsideredforsensing.Analpositionwhichislocatedwithinaminimumturn 89

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BFigure4-8. Pre-andpost-primitivecollisionfeasibilityconstraintwithtrajectoryprimitive()andextensionsofoneturnradius():A)InfeasibletrajectoryprimitiveandB)feasibletrajectoryprimitive. radiusoftheinitialpositionmaybeunreachable.Trajectoryprimitiveswhichresultincollisionswithobstacles,asdescribedinSection 2.1.2 ,areinfeasible.Oneadditionalconstraintisplacedontherandomlysampledsensingtrajectoryprimitivestoensureconnectivityispossibletoandfromtheinitialandnalconguration,respectively.Alinewithalengthofoneturnradiusisextendedfromtheinitialpositioninthedirectionoppositeoftheinitialvectorofmotionandcheckedforobstaclecollisions.Likewise,alinewithalengthofoneturnradiusisextendedfromthenalpositioninthedirectionofthenalvectorofmotionandcheckedforobstaclecollisions.Iftheselinescollidewithanobstacleitmaynotbepossibletofeasiblyconnectthistrajectoryprimitivetoandfromotherportionsoftheoveralltrajectory.Figure 4-8 depictsthisadditionalfeasibilityconstraint.Feasibletrajectoryprimitivesthatarerandomlysampledarethenexaminedforsensingeffectiveness.ThesensingeffectivenessparametersfromSection 4.3 areutilizedtocalculatethemaximuminstantaneoussensingeffectivenessexperiencedatanypointalongeachtrajectoryprimitive,qc,i.Thenumberofprimitivessampledwhichprovideasensingeffectivenessgreaterthanaspeciedthresholdcanbeutilizedasa 90

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4.4.1 producestrajectoryprimitiveswhicheffectivelysenseeachtarget.Theentiretrajectoryisnowgeneratedwhichoriginatesattheinitialconguration,travelstoandperformseachsensingprimitive,andterminatesatthenalconguration.Thetrajectoriesbetweenthesensingprimitives,initialconguration,andnalcongurationaregeneratedwithrandomdensetrees,asdescribedinSections 3.3 3.4 .Theorderinwhicheachsensingprimitiveisreachedandperformedcanbedeterminedwithmultipleapproaches.Twosuchapproachestodeterminetheorderofvisitationarepresented:multiple-goaltree-growthandthetravelingsalespersonproblem(TSP)formulation. 3.3 3.4 .Themodicationistheadditionofmultiplegoalcongurationswhichthetreeisattemptingtoreach.Connectivitytoeachgoalcongurationisattemptedatthetimeofeachnewbranchaddition.Eachfeasiblesolutionisstoredregardlessofwhichgoalisreached.Multiple-goaltreesarethenutilizedsequentiallytodeterminethetotaltrajectoryandtheorderinwhichthesensingprimitivesareperformedasfollows: 91

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2. 3. 92

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115 120 ].Thesimplestapproach,calledthe`alternatingalgorithm',rstdeterminestheorderofvisitationwiththeETSPsolution.ThealgorithmthencreatesapathwithalternatingstraightsegmentsandDubinspaths.Thisapproachisfastandstraightforwardbutlosestheguaranteeofoptimality.Inmanyenvironmentsandsetsofvisitationsitesthisalgorithmmaystillproduceoptimalornear-optimaltrajectories.However,ithasbeenfoundthattheinclusionofobstaclessignicantlyalterstheperformanceoftheETSPalgorithmwithrespecttotheactualpathlengthstosatisfytheorderofvisitation[ 121 ].ATSPformulationispresentedwhichdeterminestheorderinwhichthesensingtrajectoryprimitivesareperformedbasedonthepathlengthsfoundbygrowingrandomdensetreesbetweenthepossiblesites.TheapproachisdemonstratedwiththeenvironmentandtargetspresentedinFigure 4-9 BFigure4-9. Theenvironmentforthetree-basedTSPexamplewithtargets(4):A)isometricviewandB)topview. SensingtrajectoryprimitivesaresampledasdescribedinSection 4.4.1 .Thetrajectoryprimitiveswhichyieldthebestsensingeffectivenessarechosenand 93

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4-10 .Thestart/nishpointandthestartingandendingpositionsofeachsensingprimitivecreatethecollectionofrequiredvisitationsites.ThesevisitationsitesareshownandlabeledinFigure 4-11 Figure4-10. Sensingtrajectoryprimitives()betweenstartingpoints( 1234567Figure4-11. Visitationsites(numbered)fortravelingsalespersonproblem(TSP)formulationcorrespondingtosensingtrajectoryprimitivestartingpoints( Thedistancemetricsaremodiedtoensurethattheresultingpathvisitsthestartandendpointsofthesensingprimitivesinaproperorder.Whenthepathreachesthestartingpositionofasensingprimitivetheonlypropersitetovisitnextistheendingpositionofthesamesensingprimitive.Thereforethedistancemetricfromthestartingpositiontotheendingpositionofasensingprimitiveissetarbitrarilylowinorderto 94

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4-12 .TheEuclideandistanceresultingfromthisorderofvisitationis28.83. Figure4-12. SuggestedroutefromEuclideanTSPsolution:1!2!3!4!5!6!7!1. Thedistancemetricsoftheproperpathsaremodiedforthetree-basedTSPsolution.RandomdensetreesaregrownasdescribedinSections 3.3 3.4 foreachproperpath.Thenumberoftreesgrownforeachpathandthestoppingcriteriaforeachmaybechosenbytheuserforthespecicproblem.Thebestpathsfound 95

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4-13 .ThebestpathsfoundbetweentheendingcongurationofeachsensingprimitiveandthestartingcongurationofeachdifferentsensingprimitiveareshowninFigure 4-14 .Thebestpathsfoundfromtheendingcongurationofeachsensingprimitiveandreachingthestart/nishpointareshowninFigure 4-15 BFigure4-13. Besttree-basedpaths()grownfrominitialcongurationoftrajectorytostartingconguration( BFigure4-14. Besttree-basedpaths()grownbetweenendingcongurations( 96

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BFigure4-15. Besttree-basedpaths()grownfromtheendingconguration( Thepathlengthsoftheshortesttree-basedpathsfoundforeachproperpathareutilizedasthedistancemetrics.Agraphicrepresentationofthedistancemetricmatrixforthetree-basedTSPproblemisshowninFigure 4-16A .TheTSPsolutionpathisfoundwiththenewtree-baseddistancemetricsusingageneticalgorithm.Thesolutionhistoryofthetree-basedTSPproblemisshowninFigure 4-16B BFigure4-16. Tree-basedTSPformulation:A)graphicrepresentationofdistancemetricmatrixmodiedtoaccountforimproperandforcedpaths,B)TSPsolutionhistory. 97

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4-17 .TheEuclideandistanceresultingfromthisorderofvisitationis29.64.ThisEuclideandistanceishigherthanthatoftheorderofvisitationchosenbytheETSPwhichindicateswhythisorderofvisitationwasnotchosenbytheETSP. Figure4-17. Suggestedroutefromtree-basedTSPsolution:1!4!5!6!7!2!3!1. Utilizingthebesttree-basedpaths,showninFigures 4-13 4-15 ,andtheorderofvisitationdeterminedbytheTSPsolution,showninFigure 4-17 ,producesthenaltrajectoryshowninFigure 4-18 .Theactualpathlengthofthistrajectoryis54.41.TheEuclideanandactualpathlengthsforboththeEuclideanandtree-basedTSPsolutionsarepresentedinTable 4-1 .Theorderofvisitationsuggestedbythetree-basedTSPsolutionproducesahigherEuclideandistance,whichiswhythatorderwasnotchosenastheETSPsolution.However,theactualpathlengthresultingfromtheorderofvisitationsuggestedbythetree-basedTSPsolutionissignicantlylowerthanthatresultingfromtheorderofvisitationsuggestedbytheETSPsolution.This 98

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BFigure4-18. Completetrajectoryfromtree-basedTSPsolutionwithsensingtrajectoryprimitives( demonstratesthattheinclusionofvehicleandobstacleconstraintswithinthecalculationofthedistancemetricsisabenecialadditiontotheTSPproblemformulation.Thetree-basedTSPformulationservesasaworthwhiletoolwithinthesensorplanningframework.However,itisacombinatorialapproach;asthenumberofvisitationsitesgrowsthecomputationalrequirementswillgrowmorerapidly.Thereforethisapproachmaynotbepracticaltoincludewithinthesensorplanningframeworkwithproblemswhichhavelargenumbersoftargetstosense. Table4-1. PathlengthsresultingfromEuclideanandtree-basedTSPsolutions. TSPSolutionMethodEuclideanPathLengthActualPathLength EuclideanTSP28.8371.45Tree-BasedTSP29.6454.41 99

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5.1 presentsanalgorithmwhichattemptstosimplifyatrajectorybyreducingthenumberofintermediatewaypoints.Thoughnotguaranteed,thistechniqueisshowntobepowerfulinalsoreducingtheoveralltrajectorytime/distance.Section 5.2 introducesanalgorithmwhichattemptstomodifyandaddintermediatewaypointsinamannerthatwillreducetheoveralltrajectorytime/distance.Thisalgorithmisbasedoncomputationalgeometryandthepremisethatnecessaryturnsinthetrajectoryshouldbeperformedasclosetoobstaclesaspossibletominimizethetime/distance[ 122 ].Section 5.3 introducessurrogatemodelingasanapproachtodeterminewaypointplacementbasedonasetofseedingdataprovidedbytherandomizedtrajectoryplanningprocess.Surrogatemodelingprovidesanestimateofthecostfunctionthroughoutthedesignspaceaswellasanestimateofthepotentialmodelimprovement.Thisinformationcanbeusedtoguidefutureiterationsoftheplanningalgorithmtoguidethechoicesawayfromthepurelyrandomprocessandmakefutureiterationspotentiallymoreuseful.Anotheraspectofsurrogatemodelingthatisappealingisthatanycostfunctioncanbeused,wheremanyplanningalgorithmsarebasedpurelyontrajectorytime/distance.Finally,Section 5.4 presentsanexampleofacombinatorialtrajectoryplanningapproachforahypotheticalmissionwithcompetingobjectives.Anapproachispresentedwhichutilizestheorderreductionandincreasetechniquestoimproveuponatree-basedtrajectorythrougharegionofthedesignspaceinwhichpathlengthisthepriority.Theapproachthenutilizessurrogatemodelingtoplanthetrajectory 100

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5 5 101

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5 .ThisprovidesthefarthestcongurationalongtheoriginalRRT-basedtrajectorywhichcanbereachedbyasingletrajectoryprimitive.Onceasuccessfulconnectionisfound,thecongurationoftheRRT-basedtrajectorythatisreached,cn1,becomesthenextcongurationnodeofthereduced-ordertrajectory,j+1,asdescribedbyEquation 5 TheprocessdescribedbyEquations 5 and 5 isrepeateduntilthenalcongurationoftheRRT-basedtrajectory,cm,isreached.Thiscompletesthesetofreduced-ordercongurationnodesandtrajectoryprimitives,,whichisrepresentedbyEquation 5 5 102

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Reductionofintermediatecongurations:originalpath(),originalnodes(),reduced-orderpath(),andreduced-ordernodes(+). 5 and 5 ,respectively. 5-1 fora2-Denvironmentwithobstaclesandrequiredinitialandterminalheadingsof45.Anexampleoftheorderreductionalgorithmappliedtoa3-DtrajectoryispresentedinFigure 5-2 .The3-Dexampletrajectoryhasaninitialandterminalheadingof45,aninitialightpathangleof0,andaterminalightpathangleof-35.3.Theobstaclesareplacedidenticallytothe2-Dexampleandarefullheight. 103

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BFigure5-2. Reductionofintermediatecongurationswithoriginalpath(),originalnodes(),reduced-orderpath(),andreduced-ordernodes(+):A)isometricviewandB)topview. Table5-1. Performanceoforderreductionalgorithmin2-Dand3-Dexamples. 2-D3-DNodesDistanceNodesDistance Tree-BasedPath1025.781032.44Reduced-OrderPath415.95417.30Change-6(-60%)-9.83(-38.13%)-6(-60%)-15.14(-46.65%) TheperformanceoftheorderreductionalgorithmintermsofnumberofnodesandpathlengthisshowninTable 5-1 forboththe2-DexampleshowninFigure 5-1 andthe3-DexampleshowninFigure 5-2 .Theseexamplesindicatethattheorderreductionalgorithmseemstobeeffectiveatreducingboththenumberofintermediatecongurationsandthepathlength.Replicatingthepathwithasmanystraightsegmentsaspossiblewillreducethelengthandthereforetimeofthetrajectory.Also,ndingthefarthestcongurationonthetrajectorywhichisfeasibleanddoesnotcollidewithanobstacletendstocreatepathsthatpassbyobstaclesfairlyclosely,whichtendstoreducetheoveralllengthandtimeofthereduced-ordertrajectory.Theheightofsomeoftheobstaclesinthe3-Dorderreductionexamplearevariedinordertohighlightthe3-Dnatureofthealgorithm.Therstobstacle,centeredat(2.75,4),iselevatedabovethegroundsurfaceandthenalobstacle,centered 104

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BFigure5-3. Reductionofintermediatecongurationswithvariedheightobstacleswithoriginalpath(),originalnodes(),reduced-orderpath(),andreduced-ordernodes(+):A)isometricviewandB)topview. Table5-2. Performanceoforderreductionalgorithmin3-Dexamplewithvariedheightobstacles. NodesDistance Tree-BasedPath1032.44Reduced-OrderPath316.03Change-7(-70%)-16.41(-50.59%) at(8,8.5),isshortened.Theoriginal3-Dtree-basedtrajectoryisutilizedonceagain.Theresultingreduced-orderpath,showninFigure 5-3 ,nowtraversesundertherstobstacleandabovethenalobstacle,resultinginareduced-orderpathwithonlyasingleintermediatenode.TheperformanceoftheorderreductionalgorithmwithobstaclesofvariedheightispresentedinTable 5-2 .Thereduced-orderpathisshorterandhasfewerintermediatecongurationsthanthereduced-orderpathintheoriginal3-Dexamplewithfullheightobstacles.Thisexamplehighlightstheabilityoftheorderreductionalgorithmtoexploitthe3-Dnatureofanenvironmentandndatrajectorythatbettertraversestheenvironmentthantheoriginaltree-basedtrajectory. 105

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EnvironmentfororderreductionMonteCarlosimulationwithinitiallocation(X),terminallocation(+),requiredheadings(!),andobstacles(). 5-4 .TheRRTmethodisusedtogrowatreewhichfeasiblyconnectsinitialandterminalcongurationsforeachiteration.Theorder-reductionalgorithmdescribedpreviouslyisthenappliedtothepathtocreateareduced-orderversionofeachpath. 106

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Exampleoftree-basedtrajectorywithintermediateconguration()andreducedordertrajectory()throughobstacles(). Table5-3. NumberofnoderesultsoforderreductionMonteCarlosimulationwithintermediateconguration. MeanMedianSt.Dev.Min.Max. OriginalNodes22.52215.271436ReducedNodes7.2981.30510NodeChange-15.23-145.81-8-29NodeChange-65.68%-65.22%10.19%-50.00%-85.29% environmentdepictedinFigure 5-4 isutilizedinthissimulation,aswellastheinitialandterminalcongurations.Therequiredintermediatecongurationis(200,600,90).Resultsofthissimulationafter32iterationsarepresentedhere.Alargernumberofiterationscouldbeperformed,howevertheresultsappeartobefairlyconsistentandconclusive.Anexampleofatree-basedtrajectorywithintermediatecongurationsandthesubsequentreducedordertrajectoryispresentedinFigure 5-5 .TheperformanceoftheorderreductionalgorithmintermsofnodereductionisdepictedinFigure 5-6 .Figures 5-6A and 5-6B presentthenumbersofnodesintheRRT-generatedandreduced-orderpaths,respectively,whileFigures 5-6C and 5-6D presentthechangeinorderresultingfromthealgorithmintermsofactualnumbersofnodesandpercentages,respectively.ThestatisticaldataregardingthepathorderresultingfromthissimulationissummarizedinTable 5-3 107

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B C DFigure5-6. HistogramsofnumberofnoderesultsoforderreductionMonteCarlosimulationwithintermediateconguration:A)RRTnodes,B)reducedpathnodes,C)nodechangeduetoorderreduction,andD)nodepercentagechangeduetoorderreduction. TheperformanceoftheorderreductionalgorithmintermsofpathtimeisdepictedinFigure 5-7 .Figures 5-7A and 5-7B presentthepathtimesoftheRRT-generatedandreduced-orderpaths,respectively,whileFigures 5-7C and 5-7D presentthechangeinpathtimeresultingfromthealgorithmintermsofactualtimeandpercentages,respectively.ThestatisticaldataregardingthepathtimeresultingfromthissimulationissummarizedinTable 5-4 .ThedatafromthisMonteCarlosimulationdemonstratesthattheorderreductionalgorithmisquiteeffectiveatbothreducingthenumberofintermediatecongurations 108

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B C DFigure5-7. HistogramsofpathtimeresultsoforderreductionMonteCarlosimulationwithintermediateconguration:A)RRTtimes,B)reducedpathtimes,C)timechangeduetoorderreduction,andD)timepercentagechangeduetoorderreduction. Table5-4. PathtimeresultsoforderreductionMonteCarlosimulationwithintermediateconguration. MeanMedianSt.Dev.Min.Max. OriginalTime(s)3744.33480.4792.542776.85519.3ReducedTime(s)1391.91372.2155.321115.61786.8TimeChange(s)-2352.4-2136.7690.14-1527.4-4025.4TimeChange-61.96%-61.11%5.07%-53.00%-73.87% 109

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5-8 Figure5-8. Environmentfor3-DorderreductionMonteCarlosimulationwithinitiallocation(),terminallocation(x),andobstacles(). 110

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5-8 isusedonceagainforthisMonteCarlosimulation,alongwiththesameinitialandterminalcongurationsfromthepreviousexample.Therstintermediatecongurationiscomprisedofaposition,heading,andightpathangleof(6,2,4),-45,and-19.47,respectively,andthesecondintermediatecongurationiscomprisedofaposition,heading,andightpathangleof(3,7,8),135,and19.47,respectively.TheperformanceoftheorderreductionalgorithmintermsofnumberofnodesisdepictedinFigure 5-9 .Figures 5-9A and 5-9B presentthenumberofnodesoftheRRT-basedpathsandthereduced-orderpaths,respectively.Figures 5-9C and 5-9D presentthechangeinnumberofnodesduetotheorderreductionalgorithm,inactualchangeandpercentagechange,respectively.ThestatisticaldataregardingnumberofnodesissummarizedinTable 5-5 Table5-5. Numberofnoderesultsof3-DorderreductionMonteCarlosimulationwithintermediatewaypoints. MeanMedianSt.Dev.Min.Max. OriginalNodes7.3072.38416ReducedNodes3.1630.4424NodeChange-4.14-42.370-13NodeChange-52.68%-57.14%15.41%0.00%-81.25% TheperformanceoftheorderreductionalgorithmintermsofpathlengthisdepictedinFigure 5-10 .Figures 5-10A and 5-10B presentthepathlengthsoftheRRT-basedpathsandthereduced-orderpaths,respectively.Figures 5-10C and 5-10D present 111

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B C DFigure5-9. Histogramsofnumberofnoderesultsof3-DorderreductionMonteCarlosimulationwithintermediatecongurations:A)RRTnodes,B)reducedpathnodes,C)nodechangeduetoorderreduction,andD)nodepercentagechangeduetoorderreduction. thechangeinpathlengthduetotheorderreductionalgorithm,inactualchangeandpercentagechange,respectively.ThestatisticaldataregardingnumberofnodesissummarizedinTable 5-6 Table5-6. Pathlengthresultsof3-DorderreductionMonteCarlosimulationwithintermediatewaypoints. MeanMedianSt.Dev.Min.Max. OriginalDistance58.6258.019.6338.7583.14ReducedDistance25.3524.964.7417.5739.57DistanceChange-33.53-33.119.76-10.55-59.55DistanceChange-56.04%-56.31%10.02%-25.84%-74.44% 112

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B C DFigure5-10. Histogramsofpathtimeresultsof3-DorderreductionMonteCarlosimulationwithintermediatecongurations:A)RRTtimes,B)reducedpathtimes,C)timechangeduetoorderreduction,andD)timepercentagechangeduetoorderreduction. AnexampleofapathfromtheMonteCarlosimulationispresentedinFigure 5-11 withobstaclesremovedforclarity.Figure 5-11A presentstheRRT-basedtrajectorythatisrequiredtoreachtwointermediatecongurations.Figure 5-11B thenpresentsthetrajectoryafterapplyingtheorderreductionalgorithm.ThisMonteCarlosimulationdemonstratesthattheorderreductionalgorithmisquiteeffectiveatimprovingpoortree-basedtrajectories.Theorderreductionalgorithmdoesnotresultinanycaseswherethenumberofnodesorpathlengthisincreased. 113

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BFigure5-11. Exampletrajectoryfrom3-DMonteCarlosimulationwithintermediatecongurations:A)RRT-basedtrajectoryandB)reduced-ordertrajectory. Themeannodereductionandpathlengthreductionarebothover50%,asshowninTables 5-5 and 5-6 114

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5.1 ,althoughpriororderreductionisnotnecessary.Theconstraintonnodeplacementpresentintheorderreductionalgorithm,presentedinSection 5.1 ,isremovedintheorderincreasealgorithm,allowingnodestobeplacedanywherethatmaintainskinematicandobstacleconstraints.Atechniqueispresentedwhichisshowntobeeffectiveatreducingthetimeofanexistingpaththroughanenvironmentwithobstaclesinboth2-Dand3-D.Asegmentoftheexistingpathisexaminedwhichmakesasignicantchangeinheadingorpitchangleinproximitytoobstacles.Theconceptisto`pull'thetrajectorysegmenttowardstheobstaclesaroundwhichitisturninginordertoreducethedistancetraveledbythispath.Theorderincreasealgorithmispresentedwithexamplesinboth2-Dand3-D. 5-12 andisdescribedwiththefollowingsteps: 1. 5-12A ().Therststepistodeneaclosedpolygonwithpointsalongthetrajectorysegment,asshowninFigure 5-12B ().ObstacleverticesthatliewithinthepolygonareidentiedandshowninFigure 5-12B (). 2. 5-12C ().Theinitialandterminalpositionsareguaranteedtoalsobeverticesoftheconvexhullbecausetheywereextremeverticesoftheoriginalpolygondenition. 3. 115

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B C DFigure5-12. Stepsofthe2-Dorderincreasealgorithmwithinitialposition(X)andterminalposition(+)oftrajectorysegment:A)Theoriginalpath(),B)Polygon()andinternalobstaclevertices(),C)Convexhull(),andD)Newpath()comparedtooriginalpath(). Table5-7. Resultsoforderincreasealgorithm2-Dexample. OriginalPathNewPathChange TotalCongurations34+1PathLength16.4413.51-2.92(-17.77%) Theresultsofthis2-DexamplearepresentedinTable 5-7 .Thisexampledemonstratesthattheorderincreasealgorithmholdsbenetsinreducingthepathlengthof2-Dtrajectorysegmentsconsistingofturnsthroughenvironmentswithobstacles. 116

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Resultsoforderincreasealgorithm3-Dexample. OriginalPathNewPathChange TotalWaypoints34+1PathDistance18.4715.84-2.63(-14.25%) 5-13 anddescribedbythefollowingsteps: 1. 5-13A .TheprojectionofthesegmentontheX-Yplanedenesaclosedpolygonandobstacleverticeswhoseprojectionsliewithinthepolygonareidentied. 2. 5-13B withzerotilt.Thisplanewillcontainthewaypointsoftheeventualresultingpath.Obstacleedgesthatintersectthewaypointplanein3-Dspaceareidentied. 3. 2 ,alongwiththeinitialandnalsegmentpositions,deneaconvexhullwithinthewaypointplane.Theinitialandnalsegmentpositionsareverticesoftheconvexhull,alongwithcertainobstacleverticesasshowninFigure 5-13C 4. 5-13D alongwiththeoriginalsegment.Theresultsofthis3-DexamplearepresentedinTable 5-8 .Theexamplespresenteddemonstratethatstrategicnodereplacementand/oradditioncanimprovethepathlengthofcertaintrajectorysegmentsthatconsistofturnsand/orpitchingmaneuversinanenvironmentwithconstrainingobstacles.Thenalobstacleoftheexampleisvariedinheighttohighlightthe3-Dnatureoftheorderexpansionalgorithm.Thenalobstacleisshortenedsuchthatitliesentirely 117

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B C DFigure5-13. Stepsofthe3-Dorderincreasealgorithmwithinitialposition()andterminalposition()oftrajectorysegment:A)originaltrajectorysegment(),projectedpolygon(),andincludedobstaclevertexprojections(*),B)waypointplaneandobstaclevertexintersectionpoints(*),C)convexhull(),andD)newtrajectorysegment()andoriginaltrajectorysegment(). belowthezero-tiltwaypointplane.Theshortobstacleverticesarenotincludedintheconvexhulldenitionandtheresultingnewpathsegmenthasasingleintermediatewaypointandthentraversesabovetheshortenedobstacletoreachtherequiredterminalconguration.ThisresultingpathsegmentisshowninFigure 5-14A .Thenalobstacleheightisthenalteredsuchthatthetopsurfaceoftheobstacleintersectsthezero-tiltwaypointplane.Theintersectionpointsbetweenthetopedges 118

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BFigure5-14. Exampleoforderexpansionalgorithmwithobstaclesofdifferentheights,newpath(),originalpath(),andwaypoints():A)thirdobstacleshortenedbelowwaypointplane,resultinginpaththattravelsovershortenedobstacle,andB)thirdobstacleheightsetsuchthattopofobstacleintersectswaypointplane,resultinginwaypointalongtopedgeofobstacle. oftheobstacleandthewaypointplanearethenpossibleconvexhullvertices.Onesuchintersectionpointinthisexampleremainsasaconvexhullvertexandthereforeawaypointoftheresultingpathsegment,asshowninFigure 5-14B .ThewaypointplanedenedinStep 2 canbeofanytiltanglewhileincludingtheinitialandnalpositions.Differenttiltangleswillresultindifferentnewtrajectorysegments,potentiallygoingover,under,oraroundvariousobstacles.Examplesofthewaypointplanetilted30rightandleftareshowninFigures 5-15A and 5-15B ,respectively,withtheobstaclecongurationfromFigure 5-14B .Multiplewaypointplanetiltanglescanbebetestedforagiventrajectorysegmenttondthebestnewtrajectorysegment.Traversingoverorunderanobstaclemayprovideashorterpaththangoingaroundit,orviceversa,foragivenenvironmentandoriginaltrajectorysegment. 119

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BFigure5-15. Examplesoforderexpansionwithwaypointplanetiltedfromlevel,originalpath(),newpath(),andwaypoints():A)waypointplanetilted30rightandB)waypointplanetilted30left. withrelativelyfewcollecteddatapointsintheprocessofefcientglobaloptimization(EGO),whichisparticularlyusefulforexperimentalsystemsinwhicheachdatapointisexpansiveand/ortimeconsumingtocollect[ 123 125 ].Ineachiterationoftheoptimizationprocessamodelisconstructedbasedontheknowndatapoints.Themodelcanbeconstructedfromavarietyofmethodsincludingkriging,supportvectorregression,andradialbasisneuralnetwork.Thismodelprovidesanestimateofthecostfunctionthroughoutthedesignspace.Theexpectedmodelimprovementcanalsothenbecalculated,whichprovidesanestimateastowhatdatasetswouldbemostbenecialtoimprovetheaccuracyofthesurrogatemodel.Theexpectedimprovementisbasedonthestatisticaldistributionofdesignvariablesandtheirsubsequentcostfunctionvariations,andisweightedtowardstheestimatedoptimalcostfunctionlocation.Themathematicaldetailsofthevariousmethodscanbefoundinliterature[ 124 126 128 ].ImplementationofsuchmethodsinthisstudywasaccomplishedusingtheSURROGATESMatlabRToolbox[ 129 ].Surrogatemodelingcanpresentbenetswithintherandomizedtrajectoryplanningframework.TherandomizedsamplingprocessoftheRRTalgorithmpresents 120

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5-16 .Thex-andy-coordinatesoftherandomlysampledwaypoints,alongwiththeircorrespondingcostfunctionoftime,areinputintothesurrogatemodelingalgorithm.Thecostfunctionmodelisconstructedusingkriging[ 126 128 ],andispresentedinFigure 5-17 .TheexpectedimprovementmodelisalsoconstructedandpresentedinFigure 5-18 .InbothFigures 5-17 and 5-18 ,therandomlysampledwaypointlocationsareidentiedbyspheres.ThepredictedcostmodelinFigure 5-17 providesaconcavesurfacewiththeoptimalwaypointlocationalongthelineformedbyadirectpath.Thissimplepathcould 121

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Randomlysampledpathswithsingleintermediatewaypointinobstacle-freeenvironment. BFigure5-17. Surrogatemodeloftimecostfunctionforsinglewaypointlocationinobstacle-freeenvironment:A)2-DandB)3-D. beoptimizedbyothermethods,includinginspection,butthisexampledemonstratesthatsurrogatemodelingcanprovideintuitivelyexpectedresultsformotionplanningproblems.TheexpectedimprovementmodelinFigure 5-18 suggestsaddingwaypointstothedatasetinregionsnearthepredictedoptimalcostandwheretherearenodatapoints.Thisexampledemonstratesthebenetthatsurrogatemodelingcanbringtotherandomizedmotionplanningframework.Itcanbeseenthattherandomlysampleddataisnotuniformlydistributedthroughoutthedesignspace.Ittendstobeclustered 122

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BFigure5-18. Expectedimprovementmodelforsinglewaypointlocationinobstacle-freeenvironment:A)2-DandB)3-D. towardstheorigin,andthereisnoguaranteethatrandomsamplingalargernumberofpointswillremedythiscondition.Theexpectedimprovementmodel,however,providesguidancethatthemostadvantageouslocationstoaddfurtherdatapointsliesneartheterminalpointandalongthedirectpathbetweeninitialandterminalpositionswherethecostmodelpredictstheoptimalcosttolie.Theexpectedimprovementmodeldoesnotsuggestaddingfurtherdatapointselsewherebecauseiteitherhasahealthycollectionofpoints,asneartheorigin,oritviewsitasunlikelythattheregionimportanttominimizingthecostfunctionwillbeimproveduponbyaddingdatapointsthatwilllikelyhaveveryhighcostfunctions,asnear(10,0)and(0,10).Inclusionofsurrogateandexpectedmodelimprovementmodelsinarandomizedmotionplanningframeworkhasthepotentialtoreducethesizeoftherandomdatasetandmakefurtheriterationmorebenecialtotheproblemofminimizingcost,whichwouldalsolikelymaketheresultsmoreconsistentthanapurerandomapproach. 123

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BFigure5-19. Randomlysampledseedingdatafor3-Dobstacle-freeexample:A)completepathsandB)waypointlocationswithcolorindicatingpathlengthcostfunction. vectorofmotionpointsdirectlyatthenalpositionandtheturnrateandvelocityareunity,aswasthecaseinthe2-Dexample.Thepathsareconstructedwithasinglewaypointand3-D2-elementtrajectoryprimitives.Thecostfunctionisagaintotalpathlength.The50randomlysampledwaypointsandtheresultingpathsareshowninFigure 5-19 .Thesurrogatemodelisconstructedusingtherandomlysampled3-DwaypointlocationsshowninFigure 5-19 .The3-Dsurrogatemodelisrepresentedwithcontoursofpredictedcostwithinspeciedpercentagethresholdsoftheminimumpredictedcost.Theregionscorrespondingtothelowest1.5,5,and10%oftheminimumpredictedcostareshowninFigure 5-20 .RegionsofminimumpredictedcostofthesurrogatemodelarepresentedinFigure 5-20 .This3-Dexample,likethe2-Dobstacle-freeexample,producesintuitiveresultsthatthelowestcostpathswillbeconstructedwithawaypointlyingalongthestraightlinebetweentheinitialandnalpositions.TheexpectedimprovementmodelispresentedinFigure 5-21 ,showingregionswiththehighest2.5,5,and10%predictedimprovement.Thethreethresholdcontoursareconcentrated,makingthis3-Dregionanalogoustoaspikeina2-Dmodel.Thegreatest 124

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BFigure5-20. Surrogatemodelregionsofminimumpathlengthcostforintermediatewaypointlocationof3-Dobstacle-freeexample:A)isometricviewandB)topview. BFigure5-21. Regionsofmaximumpredictedmodelimprovementforintermediatewaypointlocationof3-Dobstacle-treeexample:A)isometricviewandB)topview. expectedimprovementispredictedtoliealongthelineofminimumcost,whichislogicalandagreeswiththe2-Dexample.The3-Dobstacle-freeexampleisextendedtoincludeconstraintsontheterminalheadingandightpathangleof90and0,respectively.Additionally,theinitialheadingandightpathangleconstraintsarebothalteredto0.Thismodicationshouldbias 125

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BFigure5-22. Isometricviewsofsurrogatemodelcontoursofminimumpathlengthforintermediatewaypointlocationwithinitialposition()andterminalposition(). theoptimalwaypointlocationawayfromthestraightlinebetweeninitialandterminalpositions.Waypointlocationsaresampledthroughoutthedesignspaceandthesurrogatemodelisconstructed.ThecontoursofminimumpathlengththresholdsarepresentedinFigures 5-22 and 5-23 .Figure 5-23 showsatopviewofthesurrogatemodelcontoursalongwiththe45linebetweentheinitialandterminalpositions.Itcanbeseenthatthesurrogatemodelcontoursarebiasedtotherightofthestraightline,asisexpectedduetotheheadingconstraints. 126

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Topviewofsurrogatemodelcontoursofminimumpathlengthforintermediatewaypointlocationwithinitialposition(),terminalposition(),andstraightlineconnectingthem(). 5.3.1 .Inthiscase,however,threeobstaclesareplacedwithinthedesignspace.ThesuccessfulpathsresultingfromtherandomizedseedingprocessarepresentedinFigure 5-24 .TherewereotherrandomlysampledwaypointpathsthatresultedininfeasibletrajectoriesthatarenotpresentedinFigure 5-24 .Thesurrogatemodelofpredictedcost,constructedinthisexamplewithsupportvectorregression(SVR),ispresentedinFigure 5-25 .TheexpectedimprovementmodelisalsoconstructedandpresentedinFigure 5-26 .Thesurrogatemodelofcostagainprovidesresultsthatareintuitivelyexpected.InFigure 5-25 theredregionsrepresentwaypointlocationsthatresultininfeasibletrajectories.Themodelpredictsthefourobviousregionsforasinglewaypointtobeplacedandcompleteafeasibletrajectorybetweentheinitialandterminalcongurations:thetwoextremecornersofthedesignspace,near(10,0)and(0,10),andthetwochannelswhichrunbetweeneachpairofobstacles. 127

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Randomlysampledpathswithsingleintermediatewaypointinenvironmentwithobstacles. BFigure5-25. Surrogatemodeloftimecostfunctionforsinglewaypointlocationinenvironmentwithobstacles:A)2-DandB)3-D. 5-27 .Theinitialandnalheadingsarealteredtobe0and90,respectively,relativetothex-axisandbothightpathanglesare0.Thelowest-costregionsofthe3-DsurrogatemodelarepresentedinFigure 5-28 ,againconstructedwithSVR. 128

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BFigure5-26. Expectedimprovementmodelforsinglewaypointlocationinenvironmentwithobstacles:A)2-DandB)3-D. Figure5-27. Randomlysampledwaypointlocationswithcolorindicatingresultingpathlengthcostfunction. Thesurrogatemodelprovidesresultswhichareagainintuitiveandagreewiththe2-Dexamplewithobstacles.Thebestregionsareidentiedastraversingthegapbetweenthecenterandrightobstacle.Thisregionissuperiorinpathdistancetothegapbetweentheleftandcenterobstaclesbecauseoftheinitialandnalheadingconstraints,althoughtherearesomerelativelylow-costregionsidentiedtotheleftofthedesignspace.Theexpectedimprovementmodelpredictsimprovementinthemodelbyaddingwaypointsinareasthatproducefeasibleandfairlylow-costpaths,asshownin 129

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BFigure5-28. 3-Dsurrogatemodelregionsofminimumpathlengthcostforintermediatewaypointlocationinenvironmentwithobstacles:A)isometricviewandB)topview. BFigure5-29. Regionsofmaximumpredictedmodelimprovementforintermediatewaypointlocationinenvironmentwithobstacles:A)isometricviewandB)topview. Figure 5-29 .Noregionsthatwouldresultinaninfeasibletrajectoryareidentiedformodelimprovement,indicatingthatthesurrogatemodelcapturesthenatureoftheobstaclecollisionsappropriately.The3-Dnatureofthesurrogatemodelisdemonstratedbyanexampleidenticaltothepreviousexamplebutwiththecenterobstacleshortened.Theresultingsurrogatemodelidentiesregionsforwaypointplacementthatresultinlow-costtrajectoriesboth 130

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BFigure5-30. Surrogatemodelregionsofminimumpathlengthcostwithcenterobstacleshortened:A)isometricviewandB)topview. aroundandoverthecenterobstacle,showninFigure 5-30 .Theminimumpredictedcoststillresultsfromawaypointplacementaroundthecenterobstacle,butthepathsthattraverseabovethecenterobstacleareclosetotheminimumpredictedcost. 130 ]. 131

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Reductionofintermediatecongurations:originalpath(),originalnodes(),reduced-orderpath(),andreduced-ordernodes(+). 5.3.3.12-DimensionalThelocalsurrogatemodelingapproachisdemonstratedin2-Dwiththereduced-orderpathfromtheorderreductionexampleinSection 5.1 .Theexampleenvironmentandthereduced-orderpathareshowninFigure 5-31 .Thereduced-orderpathconsistsoftwointermediatewaypointsthatconnecttheinitialandterminalcongurationswithtrajectoryprimitives.Alocaldesignregionisdenedaroundeachoftheseintermediatewaypoints.Thelocaldesignregioniscenteredaroundthecorrespondingwaypointandextends1unitinbothdirections.Eachlocaldesignregionissampledwithrandomwaypointlocations,alongwithwaypointslocatedalongtheboundaryofthedesignregion,tocreatethecollectionofseedingdata.Itisfoundthatincludingpointsontheboundaryofthedesignregionimprovestheaccuracyandconsistencyoftheresultingsurrogatemodels.Withanunsampledregionalongtheedgeofthedesignspace,ora`freeedge',thesurrogatemodelcansometimesdivergeinthatregionandinaccuratelypredicttheextrema.ThesurrogatemodelsofthetwolocaldesignregionsinthisexampleareshowninFigure 5-32 .Thecontoursofthesurrogatemodelsagreewiththeareasonewould 132

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Surrogatemodelswithinlocaldesignregionsforindividualwaypointimprovementwithinitialposition(),terminalposition(+),andpredictedoptimalwaypointlocations(). intuitivelythinkwouldbegoodorpoorwaypointlocationsbasedonobstacleproximity.Therstlocaldesignregioncontainsachannelthroughwhichthevehiclecouldtraversetosuccessfullyreachthenextwaypoint.Thesecondlocaldesignregionhasfeasibleareasonlyinthelowerportionoftheregionandtotherightoftheoverlappingobstacle.TheexpectedimprovementmodelisalsoconstructedfromthesurrogatemodelandisshowninFigure 5-33 .Thismodelpredictsthatgreatermodelimprovementcouldbeachievedthroughadditionalsamplingoftherstlocaldesignregionratherthanthesecondlocaldesignregion. 5.1 .The3-Denvironmentandreduced-orderpathareshowninFigure 5-34 .Alocaldesignregionisdenedaroundeachintermediatewaypointconstructingthereduced-orderpath.Thelocaldesignregionsarecenteredonthecorrespondingwaypointandspan1unitinallthreedirections.Seedingdataforeachlocaldesign 133

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Expectedimprovementmodelswithinlocaldesignregionsforindividualwaypointimprovementwithinitialposition()andterminalposition(+). BFigure5-34. Reductionofintermediatecongurationswithoriginalpath(),originalnodes(),reduced-orderpath(),andreduced-ordernodes(+):A)isometricviewandB)topview. 134

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Localsurrogatemodelseedingwaypoints. regionisthencompiledwithrandomsamplingalongwithpointslocatedattheextremesofthedesignregion.ThelocaldesignregionswiththeseedingwaypointsareshowninFigure 5-35 .Theseedingdataandtheresultingpathlengthsareutilizedtocalculatethesurrogatemodelineachlocaldesignregion.The3-Dlocalsurrogatesaredisplayedwithcontoursofvariousthresholdsrelativetothepredictedextremes.Thelocalsurrogatemodelcontoursofpredictedpathlengthcostwithin1.5,5,and10%oftheminimumpredictedcostareshowninFigure 5-36 .Anadditionaltechniqueisappliedtohelpvisualizethepredictedcostthroughouttheentirelocaldesignregions.Insteadofdisplayingthecontoursofminimumcost,asshowninFigure 5-36 ,thecontoursofmaximumpredictedcostaredisplayed.Contourscorrespondingtothehighest5,10,and20%ofthepredictedcostareshowninFigure 5-37 ,withobstaclesremovedforclarityinFigure 5-37A .Theexpectedimprovementmodeliscalculatedforeachlocaldesignregion.Itisfoundthatthethresholdsforvisualizationmustbeincreasedtoahigherpercentagetoyieldameaningfulvisualization.Contoursofthepredictedmaximum25,50,and 135

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BFigure5-36. Localsurrogatemodelregionsofminimumpredictedpathlengthcost:A)isometricviewandB)topview. BFigure5-37. Localsurrogatemodelregionsofmaximumpathlengthcost:A)isometricview(obstaclesremovedforclarity)andB)topview. 75%modelimprovementforthelocaldesignregionsareshowninFigure 5-38 .Theimplicationofhavingtoraisethethresholdssodrasticallyisthatthereisonesmallregionthatisidentiedasbeingpotentiallyextremelybenecialtomodelimprovement;however,theremainderofthedesignregionislikelywellsampledandhaslowuncertainty. 136

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BFigure5-38. Localregionsofmaximumpredictedmodelimprovement:A)isometricviewandB)topview. 5-39 .Thevehicleisconstrainedto2-Dplanarmotionandisutilizingaforward-lookingsensor.Thecostfunctionincorporatesbothsensingeffectivenessandthetotalpathtime.Weightsareincludedtoallowmission-specicbiastobeappliedtowardseither 137

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Environmentforcombinatorialcostfunctionsensingmission:initialposition( minimizingtotaltimeorsensingeffectiveness.TheweightsontimeandsensingeffectivenessareWtandWs,respectively.ThiscostfunctionisshowninEquation 5 5-40A and 5-40B fortargets1and2,respectively.Itcanbeseenthatincludingpathtimeasaportionofthecostresultsinfeasiblepathsproducingcostfunctionsbelowthenitemaximumduetothetimeportionofthecostfunction,despitelackofanysensingoftheappropriatetarget.Thiscouldbeusefulinmissionswheresensingatargetisdesiredbutonlyifitdoesnotsignicantlylengthentherequiredpathtimetoreachthegoal. 5-39 .Inthisinstancetheoptionisprovidedtouseasinglevehiclepathortwovehiclepathstominimizethecostfunction.Thecostfunctionincludesbothvehiclepathtimes, 138

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BFigure5-40. Surrogatemodelsofcombinatorialcostfunction:A)target1andB)target2. thoughifasinglevehiclepathisexaminedtheotherpathtimeisbydefaultzero.Thesensingeffectivenessofbothtargets,asdescribedinSection 5.3.4 ,areincludedinthecostfunction.Weightsareagainprovidedtobiastheresulttowardsminimizingtimeorsensoreffectiveness,althoughtheweightsremainequalinthisexample.ThecooperativecostfunctioninpresentedinEquation 5 5-41 alongwiththemosteffectivesensingLOSforeachtarget. 139

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Twopaths()andmosteffectivesensorLOS()forseparatevehiclesorseparatemissionstominimizethecostfunctionofsensingbothtargets. Theresultsindicatingtwopathsshouldbeusedisnotunexpected,astheconstraintsontheproblemensurethatapathcansenseonlyoneofthetwotargetsintheircurrentlocations.Theproblemisalteredbymovingthelocationoftarget2from(9.5,3.5)to(8,8).Thesameframeworkandcostfunctionareutilizedwiththenewtargetlocations.Thesurrogatemodelnowpredictsthatasinglevehiclepathwouldminimizethecostfunction.ThesinglepathispresentedinFigure 5-42 alongwiththemosteffectivesensingLOSforeachtarget. Figure5-42. Singlepath()andmosteffectivesensorLOS()estimatedtominimizesensingcostfunctionofbothtargets. 140

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5-41 .However,thetotalrequiredvehicletimeisapproximatelycutinhalfascomparedtoperformingtwovehiclepaths. 5.4.1OverviewAnexampleispresentedwhichdemonstratesutilizationofthepreviouslyintroducedtrajectoryplanningandimprovementmethods.Thisexampleconsistsofamissionwhichrequiresatargetbesensedwithmaximumeffectivenesswhilesimultaneouslyminimizingthedistanceandtimetotravelbetweentheinitialandnalpositions.TheenvironmentisshowninFigure 5-43 .Vehiclevelocityandturnrateareunity,theinitialpositionis(0,0)withaheadingof45,andthenalpositionis(20,20)withnoheadingrequirement.Thesensorismodeledasasimpleforward-lookingline-of-sightsensor,meaningobstaclesobstructtheviewandthusprohibiteffectivesensing.Thetotaleld-of-viewofthesensoris40,or20.Whenthetargetiswithintheeld-of-viewandunobstructed,thesensingeffectivenesscostfunctionistheEuclideandistancebetweenthesensorandthetarget.Theoverallsensingeffectivenesscostfunctionforatrajectoryistheminimuminstantaneoussensingeffectivenesscostfunctionobtainedatanypointalongthetrajectory.Assuch,theobjectiveistominimizetheoverallsensingeffectivenesscostfunctionwithtrajectoryplanning. 5-44 ,whereregionsAandBwillbetraversed.RegionAistheregion 141

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Environmentusedincombinatorialexample:initialposition(),nalposition(+),andsensingtarget(*). wherepathlengthistobeminimizedandregionBistheregioninwhichthesensingeffectivenesscostfunctionistobeminimized. 1. 3.3 2. 1 withasfewintermediatecongurationsaspossible,asdescribedinSection 5.1 3. 5.2 ,toattempttoaddandreplacewaypointstothealreadyreducedpathsoastocreateatrajectorywithalowerpathlength. 142

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Environmentusedincombinatorialexampledividedintoregions.RegionsAandBwillbeutilizedandwillprioritizepathlengthandsensing,respectively. TheresultingpathsthroughregionAfromeachofthestepsaboveisshowninFigure 5-45 .Thenatureofeachtechniqueisseeninthebehaviorofeachpath.Thetree-basedpathtendstowanderanddoesnottraversethemostdirectroute.Theorderreductionalgorithmsimpliesthetree-basedpathbutisconstrainedtoplacewaypointsontheoriginalpath.Theorderexpansionalgorithmliftsthisrequirementandplaceswaypointsadjacenttoobstaclevertices.TheresultingpathlengthandnumberofnodesofeachstepoftheregionAtrajectoryplanningareshowninTable 5-9 Table5-9. CombinatorialexampleperformanceresultsofeachstepoftheregionAtrajectoryplanningprocess. TrajectoryDistanceNo.ofNodes RDT25.7810.OrderReduction15.974.OrderExpansion15.525. 143

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ResultingpathsthroughregionAfromeachstepofthecombinatorialexamplewithinitialposition()andnalposition():randomdensetree(),orderreduction(),andorderexpansion(). minimizedbythetrajectorytraversingquadrantB.Thisstageusessurrogatemodelingtoimprovethetrajectorybasedonthesensingeffectivenesscostfunction.ThetrajectorythroughregionBislimitedtoasingleintermediatewaypointand2-elementtrajectoryprimitivesconnecttheregionBstartingconguration,theintermediatewaypoint,andthenalpositon.TheregionBdesignspaceisrandomlysampledwithintermediatewaypointlocationsuntil30successfulcollision-freepathsareconstructed,asshowninFigure 5-46 .Costfunctionsforanypathsthatareunsuccessfulinviewingthetargetorencounteranobstaclecollisionatanytimealongthepatharesettoanarbitrarilylargevalue.ThesensingeffectivenesscostfunctioniscalculatedforeachpathandthesurrogatemodelisconstructedextrapolatingthecostfunctionthroughoutregionB.ThemodelisconstructedwithsupportvectorregressionandcanbeseeninFigure 5-47 .ItisclearthatatrajectorymusttravelthroughthelowerrightregionofregionBtosensethetarget. 144

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Randomlysampledpathswithsingleintermediatewaypointforsensingtarget(*)inthelocalenvironmentofregionB. BFigure5-47. SurrogatemodelofsensingqualitycostfunctionthroughoutregionBinA)topviewandB)isometricview. 5.4.3 and 5.4.4 ,respectively,arecombinedtocreateacompletetrajectoryshowninFigure 5-48 .TheportionofthetrajectorythroughregionAattemptstominimizepathlengthwhiletheportionofthetrajectorythroughregionBattemptstominimizethesensingeffectivenesscostfunction.Thetrajectorythisapproachyieldsisnotoptimalineitherobjective,norisitoptimalinthecombinedobjectives.However,thistrajectoryreasonablysatisesbothcriteriawithrelativelylowcomputationalcost. 145

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Thetotaltrajectoryresultingfromthecombinatorialplanningexample()withinitialposition(),nalposition(),waypointnodes(),andtarget(*). 146

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4.4 .ThepathtimeoftheportionsofthetrajectorynotcriticaltothesensingobjectivearethenimprovedbytheorderreductionandorderincreasealgorithmsdescribedinSections 5.1 and 5.2 ,respectively.Theresultingpathrepresentsakinematically-feasibletrajectorywhicheffectivelysensestherequiredtargetsandnavigatesbetweenthestartandendpositionswithlowbutsuboptimalpathtime.TheenvironmentthroughwhichthetrajectoryisplannedispresentedinFigure 6-1 .Theoriginservesastheinitialandnalposition.Theinitialvectorofmotionisalongthex-axisandthenalvectorofmotionisalongthenegativex-axis.TheparameterspertainingtotheenvironmentandthetargetsarepresentedinTable 6-1 BFigure6-1. Theenvironmentforthesensor-basedtrajectoryplanningexamplewithtargets():A)isometricviewandB)topview. Thevehiclefeaturesadownward-lookingsensorwhosesensingeffectivenessmetricsarethosepresentedinSection 4.3.4 .ThesensorandvehicleparametersarepresentedinTable 6-2 147

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Environmentparametersforsensorplanningexample. ParameterValue InitialPosition(0.0,0.0,0.0)InitialDirection[1.0,0.0,0.0]FinalPosition(0.0,0.0,0.0)FinalDirection[-1.0,0.0,0.0]Target1Location(1.5,5.5,0.0)Target1NormalDirection[0.0,0.0,1.0]Target2Location(3.0,8.5,3.0)Target2NormalDirection[0.0,0.0,1.0]Target3Location(7.5,3.3,3.0)Target3NormalDirection[-1.0,0.0,0.0]Incmax(AllTargets)60(1.0472rad) Table6-2. Sensorandvehicleparametersforsensorplanningexample. ParameterValue Pan0(0rad)Tilt-90(1.5708rad)f0.1unitRmax3.0unitFOVmax45(0.7854rad/s)IPVmax0.15unit/sV1unit/s!1rad/s 4.4.1 .Thesensingeffectivenessofatrajectoryprimitiveiscalculatedasthebestinstantaneoussensingeffectivenessachievedatanypointalongthetrajectoryprimitive.Eachlocalsamplingregionissampleduntilatleast25trajectoryprimitivesarefoundwithnon-zeroinstantaneoussensingeffectiveness.ThesampledtrajectoryprimitivesaroundeachtargetaredisplayedinFigure 6-2A .Thetrajectoryprimitivewhichyieldsthehighestinstantaneoussensingeffectivenessforeachtargetisthenchosen.ThesechosensensingtrajectoryprimitivesarepresentedinFigure 6-2B andtheresultingsensingeffectivenessforeachtargetispresentedinTable 6-3 148

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BFigure6-2. Sensingtrajectoryprimitivesampling:A)sampledtrajectoryprimitivesandB)bestsensingtrajectoryprimitivesforeachtarget. Table6-3. Bestinstantaneoussensingqualityforexampletargets. TargetSensingQuality 10.962720.962730.9121 4.4.2.1 .Asmallsegmentofeachsensingtrajectoryprimitiveisidentiedduringwhichthemosteffectivesensingoccurs.Thesesegmentswillbepreservedinallfuturestepsinordertoguaranteethesensingqualityalreadyachieved.Itisthenattemptedtoimprovetheportionsofthetrajectorybetweenthepreservedsensingsegments.Theorderreductionalgorithmisappliedtoeachapplicableportion,asdescribedinSection 5.1 ,followedbytheorderincreasealgorithm,asdescribedinSection 5.2 6-3 presentsthetrajectorieswhichresultfromeachstepofthisprocess:randomdensetree,orderreduction,andorderincrease.Thepathlengthofeach 149

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6-4 alongwiththechangeinpathlengthcreatedbyeachstepandthetotalchangeinpathlength. BFigure6-3. A)TopviewandB)isometricview(obstaclesomittedforclarity)ofresultingpathsfromsensor-basedtrajectoryplanningexample:randomdensetreepath( ). Table6-4. Pathlengthsfromeachstepinthesensor-basedtrajectoryplanningexample. PathLengthChangeChange(%)TotalChangeTotalChange(%) RDT79.51N/AN/AN/AN/AReducedOrder50.03-29.48-37.07%-29.48-37.07%IncreasedOrder39.25-10.78-21.59%-40.26-50.64% ItcanbeseenfromFigure 6-3 thattheboldsensingsegmentsarepreservedineachstepoftheprocess.Theorderreductionalgorithmimproveseachpathbetweensensingsegmentsandshortensthetotaltrajectoryover37%ascomparedtotherandomdensetreetrajectory.Theorderincreasealgorithmcannotimproveuponseveralregionsofthepath(thisisdifculttoseeinFigure 6-3 asthepathsarecoincident).However,onetrajectoryportionbetweensensingsegmentsisimprovedsignicantlybytheorderincreasealgorithm.Thisimprovementyieldsanadditionalpathlengthimprovementof21.5%.ThenaltrajectoryislessthanhalfthepathlengthoftherandomtreetrajectoryandthehighsensingeffectivenessforeachtargetpresentedinTable 6-3 ispreserved. 150

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4 andthepathimprovementtechniquespresentedinChapter 5 .Thiscombinationofapproachesproducesrelativelyshorttrajectoriesthroughclutteredenvironmentswhichaccomplishthespeciedclose-proximitysensingrequirementswhileaccountingforvehiclemotionandsensorcharacteristics.Thisframeworkgenerallyrequireslesscomputationalresourcesforcomplicatedenvironmentsandhighlyconstrainedvehiclesthanthatwhichwouldberequiredbyoptimaltrajectorysolutions. 151

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4 ismodiedtoincludehighangle-of-attackightcharacteristics.Thepotentialadvantagesofincludinghighangle-of-attackightareexaminedbothintermsofsensingeffectivenessandreducingpathtime. 7-1 ,whereangle-of-attackisrepresentedby,pitchangleisrepresentedby,andtheightpathinclinationisrepresentedby. Figure7-1. Graphicalrepresentationofangle-of-attack. 152

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7 ,wherevbxandvbzarebody-framevelocitiesinthexandzdirectionsrespectively. 18 ].Highangle-of-attackightcouldalsoprovidegreaterobstacleavoidancecapabilitieswhenyinginurbanterrainsincetheturningradiusinhighangle-of-attackightisgreatlyreducedascomparedtothatofconventionalforwardight.Sensingcapabilities 153

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19 ]. 131 132 ].AnovelcongurationfortheX-29usedforward-sweptwingsandacanardtodemonstrateightattheseconditions[ 133 ].Ineachcase,theightdynamicswerefoundtobechallengingtomodelbecauseoftheinuenceofaerodynamics.Additionalissues,suchaswingrock,werealsonotedinvaryinglevelsatvirtuallyallangle-of-attackconditionsinthepost-stallregime.TheF/A-18HARVandtheX-29canbeseeninFigure 7-2 [ 134 135 ]. BFigure7-2. A)F/A-18HARV[ http://www.dfrc.nasa.gov/gallery/Photo/F-18HARV/ ,reprintedwithpermission]andB)X-29[ http://www.dfrc.nasa.gov/gallery/Photo/X-29/ ,reprintedwithpermission]. TheightdynamicsassociatedwithsmallUAVsarereceivingsignicantattentioninthecommunityasaresultoftheirmissionpotential.Severalsmall,man-portablevehiclesfeatureadeep-stallshortlandingmodeinwhichtheelevatordeectsandthethrottleisreducedwhiletheaircraftsteeplydescendsinanear-levelattitudeuntilit 154

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136 137 ].TheRQ-11BRaven,showninFigure 7-3 [ 138 ],isanexampleofsuchaUAVthatfeaturesadeep-stalllandingmode.TheightdynamicsofsmallUAVshavealsobeenstudiedinhoveringmodestoenableautonomouscontrol[ 139 141 ]. Figure7-3. RQ-11BRavenSmallUnmannedAircraftSystem[ http://www.avinc.com/media gallery2.asp?id=224 ,reprintedwithpermission]. SmallUAVsandremote-controlled(RC)aircraftarerapidlymaturinginightcapabilityforavarietyofmissionsincludingurbanoperations.Assuch,theabilitytooperateathighangle-of-attackconditionsisacriticalrequirementfortheseplatforms.Suchcapabilityisusuallyachievedthankstothecombinationofthrustgenerationandcontrolsurfaces.Theseaircraftoftenutilizefront-mountedpropellersandtractorpropulsionthatproducesairowoverthewingsandtail.Thelargecontrolsurfaces,whichconstitute50%ormoreofthetailareaandmaydeect45ormore,arethusabletousethispropwashtomaintaincontrolauthority.Flightathighangle-of-attackconditionsisoftencharacterizedbywingrock.Thisphenomenonisdescribedasuncommandedself-inducedoscillationsprimarilyabouttherollaxis.Someresearchhasindicatedthat,althoughdominatedbyrollmotions,theuncommandedwingrockmayactuallybealightly-dampeddutchrollmotion[ 142 144 ].Thesourceoftheuncommandedwingrockisnotcompletelyknownandseemstovarybyaircraftconguration.Someresearchhasledtothebeliefthatthewingrock 155

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145 146 ]oranaerodynamichysteresiswhichgeneratesthespring-likeforcesrequiredtodrivetheLCO[ 147 149 ].Thepresenceofsidesliphasbeenshowntohaveaneffectonwingrockathighangles-of-attack,bothasacauseandmitigator[ 148 150 151 ].Additionally,quiteafewstudieshavefoundtheuncommandedwingrockphenomenontobesomewhatunpredictableinnature,bothinmagnitudeandperiodicity[ 150 152 155 ].Someresearchhasdeterminedthephenomenontobecausedbyvorticesfromtheleadingedgesofthewings[ 142 156 157 ],whileotherresearchhasdeterminedittobecausedlargelyorentirelybyvorticesgeneratedfromslenderforebodiesimpingingupondownstreamairframecomponentssuchastheverticaltail[ 154 156 158 162 ].Thishasledtoresearchonthewingrockrelationshipwithandwithoutaverticaltailsurface[ 163 ]. 7.2.1AircraftTheightcharacteristicsoftheMiniShowTimearestudiedinahighangle-of-attackcondition.Thiselectric-poweredaircraft,showninFigure 7-4 ,isacommercially-availableoff-the-shelfplatformthatiscommonlyusedbytheremote-control(RC)community.Thisaircraftisconstructedfromalightweightbalsawoodstructurethatallowsthewingspantobelargeincomparisonwiththevehicleweight.Thespecicplatformhasaweightofapproximately820galongwiththecharacteristicsgiveninTable 7-1 [ 164 ].TheaircraftwasoutttedwithconventionalRCcomponentsforcontrolandpropulsion,whicharelistedinTable 7-2 Thisaircraftisusedinthecommunityforaerobaticsbecauseofitsexcellentagilityandoutstandingcharacteristicsathighangle-of-attackconditions.Inparticular,theaircraftishighlycontrollableathighangle-of-attackconditionsasaresultoflowwing-loadingandhighthrust-to-weightratioalongwithlargecontrolsurfaces.These 156

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MiniShowTimeaircraft.Photocourtesyofauthor. Table7-1. MiniShowTimespecications. ParameterValue Wingspan1090mmLength1065mmWingArea26.7dm2FlyingWeight820-850gWingLoading30.7-31.8g/dm2FlightSpeed0-20m/s Table7-2. MiniShowTimecomponents. ComponentManufacturerModel TransmitterSpektrumDX7ReceiverSpektrumAR6100eServos(4)JRDS285BatteryEliminatorCircuit(BEC)CastleCreationsCCBECElectronicSpeedControl(ESC)E-ite40-AmpBrushless(V2)MotorE-itePark480BLOutrunner,1020KvPropellerAPC12x6EBatteryThunderPower3S,11.1V,2100mAh

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7 ,whereSvrepresentsverticaltailarea,LvrepresentsthedistancefromtheCGoftheaircrafttotheaerodynamiccenteroftheverticaltail,Swrepresentswingarea,andbrepresentswingspan[ 165 ]. 7-3 .Therangeoftailsizesprovidesareasandverticaltailvolumecoefcients 158

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7-5 ,andatailmountedonthemodiedfuselageisshowninFigure 7-6 Table7-3. Verticaltailspecications. TailHeight(mm)Area(cm2)Vv BFigure7-5. Verticaltails:A)drawingandB)actual.Photocourtesyofauthor. 159

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Interchangeableverticaltailmountedonfuselage.Photocourtesyofauthor. 7-7 .Apairofsensorpackagesresultfromaninertialmeasurementunit(IMU)andglobalpositioningsystem(GPS).TheIMU,whichisaMEMSensenIMU,isaMEMS-basedunitwithtemperaturecompensationanddigitalI2Coutputof3-axisaccelerations,angularrates,andmagneticux.TheGPS,whichisanEagleTreeExpanderModule,noteslocation,groundspeed,course,andUTCtimestampatarateof5Hz.Anadditionalightdatarecorder(FDR),whichistheEagleTreeSystemsFDRPro,logsthesensoroutputsalongwithbarometricaltitudeandservocommands.Thissystemisabletoobtainmorethan15minutesofdataatarateof25Hz. 160

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B CFigure7-7. Airbornesensors:A)IMU,B)GPSreceiver,andC)ightdatarecorder.Photoscourtesyofauthor. Thesesensorsarerelativelysmall,asnotedinTable 7-4 [ 167 168 ],andineachcasetheweightisnearlynegligibleontheightdynamics. Table7-4. Sizeandmassofavionics. UnitSize(mm)Mass(g) IMU46.5x22.9x13.920GPS36.0x43.0x13.023FDR50.0x35.0x17.022 TheIMUismountedonaspecially-installedshelfwithinthefuselagetolieveryclosetothecenterofgravityalongallthreeaxes.TheGPSandFDR,alongwiththeconventionalRCreceiverandbatteryeliminatorcircuit(BEC),areinstalledunderthecanopy.ThemountedlocationofeachcanbeseeninFigure 7-8 .AnadapterwasassembledtoconnecttheIMU,whichusesaHiroseHR306-pinconnector,totheFDR,whichusesa4-wireplug.TheconnectionsequenceisshowninTable 7-5 [ 167 ].Thecompletewiringdiagramoftheexperimentalaircraft,includingconventionalRCcomponents,isshowninFigure 7-9 .TheBECmustbeprogrammedtooutputavoltageintherangeof5.4-7.0VtoproperlypowertheIMU,FDR,andRCsystemssimultaneously[ 167 168 ].Theseavionicshavebeendemonstratedashighlyaccuratewhencomparedtohigh-qualityavionics.Whencomparedtoahigh-qualityIMUwithlaser-ringgyros,the 161

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Avionicsmountedundercanopy:A)FDR,B)GPS,C)RCreceiver(underGPS),D)IMU,E)BEC.Photocourtesyofauthor. Table7-5. IMU/FDRwiringsequence. HirosePortNo.I2CFunctionFDRWire 1SDAYellow2VDDRed3NotUsedNA4NotUsedNA5GNDWhite6SCLBrown nIMUprovidedmeasurementsthatyieldedvelocityestimateswithinstandarddeviationsofapproximately0.2m=sonallaxesandattitudeestimateswithinstandarddeviationsofapproximately0.2inrollandpitchand0.35inheading[ 166 ].ThemanufacturerspecicationsaregiveninTable 7-6 foreachofthesensors[ 167 ]. Table7-6. TechnicalspecicationsofIMUsensors. SensorDynamicRangeDigitalSensitivityOffset/DriftNoise Gyro600o=s0.01831o=s1o=s0.56o=sAccelerometer5g1.5259e4g30mg4.87mgMagnetometer1.9Gauss5.79e5Gauss2700ppm=oC5.6e4Gauss

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WiringdiagramofMiniShowTimeaircraftfordatacollection.Photocourtesyofauthor. ThedataoutputfromthenIMUandrecordedbytheFDRisintheformofsigned16-bitintegers.Thegyro,accelerometer,andmagnetometerareconvertedtoappropriateunitsthroughtheuseofEquation 7 [ 167 ]. 21.5 32768!(7)BasedonthedynamicrangeofeachsensoraspresentedinTable 7-6 ,Equation 7 resultsinmultipliersoftherawsensoroutputandresultingunitsaspresentedinTable 7-7 163

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IMUrawoutputmultipliersandresultingunits. SensorMultiplierResultingUnits Gyro2.747e-2deg=sAccelerometer2.289e-4GMagnetometer8.698e-5Gauss 7-10 .Aseriesofightswasperformedtocollectdataduringdoubletmaneuversandsteadyhighangle-of-attackightwiththestock(normal)verticaltail.Eachightbeganbytakingoffandestablishingtheaircraftinhighangle-of-attackight.Acompletepassathighangle-of-attackwasrstperformedtoestablishthetrimpositionforallcontrol 164

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MiniShowTimeinuprighthighangle-of-attackight.Photocourtesyofauthor. surfaces.Eachsubsequentpassthenincludedthreedistinctactions:establishingthehighangle-of-attacktrimconditionatthedesiredheading,performingadoubletmaneuverwithacontrolsurface,andreestablishingthetrimconditionfortheremainderofthepasslength.Eachightinvolveddoubletsofvaryingsizebyasinglecontrolsurface.Thelargestdoubletforeachcontrolwasconstrainedeitherbymaximumcontroldeectionsorbythepilot'sabilitytoquicklyreestablishstraightandleveltrimight.Adoubletwasperformedwithmaximumdeectionandseveralwereperformedwithprogressivelysmallerdeections.Aminimumofthreepasseswithdoubletmaneuverswereperformedforeachight.Aseriesofightswasthenperformedtocollectdataduringsteadyhighangle-of-attackightwithtendifferentverticaltailcongurations:uprightandinvertedwitheachoftheveverticaltails.Invertedhighangle-of-attackwasperformedinasimilarfashiontouprightbutwithdown(positive)elevatordeectiontomaintainthepitchangle.Additionally,slightdifferencesinruddertrimexistedbetweenuprightandinvertedduetothelargeyawmomentfromp-factorinhighangle-of-attackight.TheMiniShowTimeininvertedhighangle-of-attackightisshowninFigure 7-11 165

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MiniShowTimeininvertedhighangle-of-attackight.Photocourtesyofauthor. Eachightconsistedofestablishingtheaircraftineitheruprightorinvertedhighangle-of-attackighttodetermineapproximatetrimconditions.Aminimumofvestraight,horizontalpassesinhighangle-of-attackightwereperformedwitheachtailconguration.Anassistantwithastopwatchrecordedthetimesatwhicheachhighangle-of-attackpassbeganandended,aswellaswhendoubletswereperformed.Aftereachight,thedatafromtheFDRwasdownloadedtoacomputerwiththeUSBcableandtheFDR'sbufferwascleared.Thetrimconditionsfortheaircraftremainedconsistentfromighttoight.Theighttestingprocedureinvolved14ightswhichspannedfourdays. 7.4.1ProcedureLeast-squaresregressionisacommonandeffectivemethodofttingdata.Itisparticularlyusefulformodelinganoverdeterminedsystemwithmeasurementsthatcontainrandomerrors.Residualsarethedifferencesbetweenthemeasuredand 166

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7 ,whereAisannxmmatrixofregressors,Xisanmx1vectorofunknowncoefcients,Lisannx1vectorofmeasurements,andVisannx1vectorofresiduals,wherenisthenumberofsamplesandmisthenumberofstatesintheleast-squaresmodel[ 169 ]. 7 [ 169 ]. 170 ].Thecompletemodelsalsoincludenonlinearquadratic,cubic,andquartictermsinbothangularratesandcontrolinputs.Themodelsarebasedsolelyondirectmeasurementsfromtheonboardavionics,asapost-processingINSsolutionwasnotavailableatthetimeofthisresearch.Knowledgeofaircraftattitudeandangles-of-attackandsideslipcouldbeincorporatedintosystemidenticationmodelswhentheyareavailable.Eachregressorintheleast-squaresmodelsisanaverageofthreedatapointsinordertoprovidelowpasslteringwithinthemodel.Alagofapproximately160msisaddedintothecontrolinputregressortermstocompensateforthespeedoftheservos.Thislagisintroducedbymakingthecontrolinputregressortermsfunctionsofcontrolinputsfromseveralpriorsamplingperiodsandnotthemostrecentcontrolinputs.A 167

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169 ].Thisisaccomplishedthroughbackwardelimination,inwhicherroneousorinsignicantlysmallindividualcomponentsineachcompletemodelareremovediteratively.Ateachiterationthemodelisusedtosimulatetheoutputwiththesameinputsasthemodelisbasedupon.Thecontributionofeachregressortermisplottedtoidentifyinsignicantorerroneouscontributionswhichcanthenberemoved.Iftheremovalofaparticularregressortermreducestheaccuracyofthemodelinpredictingthedoubletresponses,thetermisretained.Thenalreducedmodelsarepresentedinthefollowingsections. 168

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7.4.2.1LongitudinalAmodelofthelongitudinaldynamicsisgeneratedtorelatethepitchratetotheelevatorcommands.Suchamodelisestimatedtorelateasetofdoubletcommandstotheelevatorandtheresultingpitchrate.ThemeasurementsofpitchrateandthesimulatedvaluesfromthemodelareshowninFigure 7-12 alongwiththedoublets. BFigure7-12. A)ElevatorandB)pitchrateduringdoublets:measured()andsimulated(). ThemodelwithsimulatedresponseinFigure 7-12 isgiveninEquation 7 asadiscrete-timeequation,whereqispitchrateindeg=sandeiselevatordeectioninpercentage.Thismodeldependsuponalinearcombinationoflaggedvaluesforelevatoranglecorrespondingto5and7previoustimesteps.Thecurrentvalueofpitchratealsodependsuponlaggedvaluesofpitchratecorrespondingtoaveragevaluesfrom2,4and6previoustimesteps. 169

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30.2077e(k6)+e(k7)+e(k8) 3+1.1516q(k1)+q(k2)+q(k3) 30.3956q(k3)+q(k4)+q(k5) 30.0215q(k5)+q(k6)+q(k7) 3 ThecontributionsofeachtermfromEquation 7 totheresponseinFigure 7-12 isshowninFigure 7-13 .Thelargestcontributionresultedfromanegativepitchrateattimeofkresultingfromapositiveelevatordeectionattimeofk5;however,somehigher-orderdynamicsisalsopresentbecauseoftheneedtoalsoretainacontributionfromtheelevatordeectionattimeofk7.Thestatedynamicsareevidencedbythecontributionsfromseverallaggedvaluesofpitchrate.Apositivepitchrateattimeofk2generatesapositivecontributiontocurrentpitchratewhileapositivepitchrateateithertimeofk4ork6actuallygeneratesanegativecontributiontopitchrate.Suchdisparityispartlyduetoout-of-phasestatesfromashort-periodmodealthoughthedynamicsathighangle-of-attackdonotnecessarilyhavetraditionalmodes. Figure7-13. Individualcontributionstoresponsefromlongitudinalmodel:elevator(k-5)(o),elevator(k-7)(),pitchrate(k-2)(),pitchrate(k-4)(),pitchrate(k-6)(). 170

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7-12 ;however,theselongitudinaldynamicsareexpectedlydifculttomodelathighangle-of-attackconditions.Certainlytheaerodynamicsarenotnecessarilylinearnornite-dimensionalfunctionsofightcondition.Also,thelackofangle-of-attackmeasurementsdoesnotnecessarilylimitthedelity,giventhatatransferfunctionalwaysexistsbetweenaninputandanoutput,butthelackalmostcertainlylimitstheinterpretationoftheresultingmodel. 7-14 .Thedoubletsandresultingrollratesactuallyvarybyroughlyafactorof2betweentherstandthirdcommandsoarichsetofdataisavailableforthemodel. BFigure7-14. A)AileronandB)rollrateduringdoublets:measured()andsimulated(). ThemodelthatsimulatedtherollrateinresponsetothedoubletsinFigure 7-14 isgiveninEquation 7 ,wherearepresentsailerondeectioninpercentage.Thismodelsimplygeneratestherollrateattimeofkfromabiastermalongwithafneandquadratictermsassociatedwithaverageaileronangleattimesofk5. 171

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38.42e52a(k4)+2a(k5)+2a(k6) 3 Thismodelindicatesthelateraldynamicsaredominatedbyatraditionalmodeofrollconvergence.Theresponseisnearlylinearinaileronangle,asshowninFigure 7-15 ,sincethenonlinearcontributionisnegligible.Sucharesultissomewhatlogicalgiventhatanyeffectsofhighangle-of-attackconditionswouldinuencetheaerodynamicsoflongitudinalmotionmuchmorethanthelateralmotion. Figure7-15. Individualcontributionstoresponsefromlateralmodel:Aileron()andAileron2(). 7-16 .ThemodelthatsimulatedtheyawrateinresponsetothedoubletsinFigure 7-16 isgiveninEquation 7 ,wherearepresentsailerondeectioninpercentage,rrepresentsrudderdeectionin%,andprepresentsrollrateindeg=s.Thismodelrequiresmoretermstodescribethedynamicsthaneitherthelongitudinalmodelor 172

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BFigure7-16. A)RudderandB)yawrateduringdoublets:Measured()andSimulated(). lateralmodel.Essentially,thedirectionaldynamicsareestimatedasbeinganonlinearfunctionofboththerudderandtheaileronalongwithbeinganonlinearfunctionoftherollrate. 3+0.00592r(k4)+2r(k5)+2r(k6) 30.5073a(k4)+a(k5)+a(k6) 30.03132a(k4)+2a(k5)+2a(k6) 3+0.0259p(k1)+p(k2)+p(k3) 30.0009p2(k1)+p2(k2)+p2(k3) 3 ThecontributionsfromeachterminEquation 7 tothesimulatedresponseinFigure 7-16 areshowninFigure 7-17 .Theresponseisdominatedbythecontributionsfromtherudderwiththelineartermprovidingthesignicantportion.ThedirectionaldynamicsarechallengingtomodelasevidencedbytheinconsistentqualityofthetinFigure 7-16 despitethenonlineartermsinEquation 7 .Thenatureofthelinearcontributionsisnotconsistentgiventhatapositiveruddergenerates 173

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Individualcontributionstoresponsefromdirectionalmodel:rudder(o),rudder2(x),aileron(),aileron2(),rollrate(),rollrate2(). negativeyawrateandapositiveailerongeneratesnegativeyawrateasadverseyawbutapositiverollrategeneratesapositiveyawratetoimplysomeproverseyaw[ 171 ].Eventhenonlinearitiesareinconsistentsinceanyruddergeneratesasmallpositivecontributiontoyawratewhileanyaileronorrollrategeneratesasmallnegativecontributiontoyawrate.Theissueofighttestingmustbeconsideredwhentryingtoevaluatethequalityofthemodelandanyassociatedinconsistencies.Inparticular,theinuenceofgustscanbeextremeonthedirectionaldynamicswhenyingathighangle-of-attackconditionsbutofcourseanygustexcitationisnotproperlyrepresentedinthemodel. 7.4.3.1LongitudinalFlightdataassociatedwithsteady-statehighangle-of-attackightisshowninFigure 7-18 fortheelevatorcommandsandassociatedvaluesofpitchrate.Inthiscase,theelevatorcommandsinFigure 7-18 areafactorof5lessthantheelevatorcommandsinFigure 7-12 toindicatethepilotismerelymovingthecontrolsurfacestomaintainightconditionandthusnotintroducingsignicantenergy. 174

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BFigure7-18. A)ElevatorandB)pitchrateduringsteadyight:measured()andsimulated(). ThecontributiontopitchratethatisnotpredictedbythemodelinEquation 7 isshowninFigure 7-19 .Clearlythemodelisnotabletoreproducethecompleteresponseindicatingtheelevatorisnotabletoaccountfortheentiretyofthemeasuredpitchrate. Figure7-19. Uncommandedpitchrate. ThisuncommandedestimateofpitchratefromFigure 7-19 isrepresentedinthefrequencydomaininFigure 7-20 byapowerspectraldensity(PSD).Theenergyisconcentratedacrosslowfrequenciesbutapairofmodes,around3rad/sand6rad/s,isclearlyevident.Thesemodesmaycorrelatetoashort-periodmode;however,they 175

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Figure7-20. PSDofmeasuredpitchrate(),uncommandedpitchrate(),andelevatorinput(). Figure 7-20 indicatesthatbetweenapproximately2and4rad/stheuncommandedpitchrateislargerthanthemeasuredpitchrate.Thisseemsimproper,astheuncommandedpitchrateisextractedfromthemeasuredpitchrateandshouldthereforebeasubsetofthemeasuredpitchrate.However,phaseshiftsbetweenthemodel-predictedpitchrateandthemeasuredpitchratecanresultinuncommandedpitchratesestimatedathighervaluesthanthemeasuredpitchrate. 7-21 alongwithassociatedaileroncommands.ThedeectionsoftheaileronarenearlyanorderofmagnitudelessthanthesizeofthedoubletsinFigure 7-14 toindicatethepilotisprovidingonlyminimalexcitation.SomeunexplaineddrifttonegativedeectionisclearlyevidentinFigure 7-21 ;however,thisdriftisextremelylow-frequencyandthuscanbedirectlyeliminatedintheanalysis. 176

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BFigure7-21. A)AileronandB)rollrateduringsteadyight:measured()andsimulated(). Theuncommandedportionoftheresponse,asdeterminedbysubtractingthesimulatedresponsetotheaileronfromthemeasuredresponse,isshowninFigure 7-22 .Thisportionisquitelargeinmagnitudeandactuallyappearsquiteperiodic. Figure7-22. Uncommandedrollrate. TheperiodicitynotedinFigure 7-22 isquantiedbyaPSDonthatdatatoobtainthefrequency-domainrepresentationinFigure 7-23 .Thefrequency-domainrepresentationofthemeasuredrollrateandtheuncommandedrollrateareshownalongwiththeaileroncommand.Therollrateandtheaileronshowapeakat2.2rad/sindicating 177

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Figure7-23. PSDofmeasuredrollrate(),rncommandedrollrate(),andaileroninput(). 7-24 .ThisdataconsistsofsmallruddercommandstomaintaintheightconditionandtheassociatedyawrateswhichweremeasuredandsimulatedfromthemodelinEquation 7 .TheamountofyawratewhichcannotbepredictedbythemodelisgiveninFigure 7-25 .Theyawratealsohassomeclearperiodicityandasubstantialmagnitudeindicatingsomeelementofsteady-stateightisnotcapturedbythemodelgeneratedfromdoublets.Thefrequency-domainrepresentationoftheyawrate,bothmeasuredanduncommanded,alongwithruddercommandsindicatestheperiodicity.Thisdata,asshowninFigure 7-26 ,hasanoticeablemodenear2.2rad/s.Boththerudderand 178

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BFigure7-24. A)RudderandB)yawrateduringsteadyight:measured()andsimulated(). Figure7-25. Uncommandedyawrate. theyawratecontainthismode,asdidtheaileronandtherollrate,whichmaymeanitcorrelatestoaroll-yawcoupledmodethatthepilotisattemptingtodampoutandmaintaincondition.Figure 7-26 indicatesfrequencyregionswheretheuncommandedyawrateisoflargercorrelationthanthemeasuredyawrate,particularlybetweenapproximately2.5and5rad/sand6and10rad/s.Thisseemsimproper,astheuncommandedyawrateisextractedfromthemeasuredyawrateandshouldthereforebeasubsetofthe 179

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PSDofmeasuredyawrate(),uncommandedyawrate(),andrudderinput(). measuredyawrate.However,phaseshiftsbetweenthemodel-predictedyawrateandthemeasuredyawratecanresultinuncommandedyawratesestimatedathighervaluesthanthemeasuredyawrate. 7.5.1UprightTailAseriesofighttestsareperformedwiththeairplaneinanuprightorientationwithdifferentverticaltails[ 172 173 ].Rollrateswereexaminedastheprimaryindicatorofwingrockcharacteristics.TherollratemeasuredduringthesetestsisshowninFigure 7-27 alongwiththeassociatedailerondeections.Theailerondeectionisdramaticallysmallerthantherollrateandthusdifculttodistinguish.Clearlytherollrateshowssomeamountofperiodicityamongallthetails;consequently,wingrockappearsforanysizeofthesetails.Also,themagnitudeoftherollrateshowsvariationduringtheresponse;however,thismagnitudeisactuallysomewhatconsistentdespitevariationsinthetailsize.Afrequency-domainrepresentationoftherollratesandailerondeectionsfromFigure 7-27 iscomputedandshowninFigure 7-28 usingaPSD.Therollrateshowsaconsistentamountofenergyaround4rad/swhichcorrelateswiththeconsistent 180

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B C D EFigure7-27. TimeresponsesinuprightcongurationforA)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggertail:rollrate()indeg/sandailerondeection(

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7-28 representsuncommandedwingrock.Also,thispeakinenergyisactuallysomewhatbroadforeverytailandrangesfromapproximately2rad/sto6rad/sindicatingthewingrockisabroad-bandphenomenon. BFigure7-28. PSDinuprightcongurationofA)rollrateandB)ailerondeection:smallertail(),smalltail(),normaltail(),bigtail(),biggertail(x). Atime-frequencyrepresentationiscomputedfortherollratetoinvestigatethetemporalnatureofanyinstantaneousfrequenciesinthewingrock.TheserepresentationsareshowninFigure 7-29 ascomputedbywavelettransformsusingaMorletwavelet.Thewingrockisevidentbythehighcorrelationsshownaround4rad/s;however,thisrepresentationisnotablydifferentthanthefrequency-domaincharacterizationinFigure 7-28 .ThewingrockisshowntoactuallyhaveanarrowbandofenergywhenlocalizedintimeusingFigure 7-29 .Thebroad-bandnatureobservedinFigure 7-28 resultsfromthevariationsobservedinFigure 7-29 inthecentralfrequencyofthatnarrowband.Thewavelettransformsoftheailerondeection,asshowninFigure 7-30 ,donotshowsignicantcorrelationwiththewingrock.Ineachcase,theailerondeectionsarepredominatelyatlowerfrequencies. 182

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B C D EFigure7-29. Wavelettransformsofrollrateinuprightconguration:A)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggertail.

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B C D EFigure7-30. Wavelettransformsofailerondeectioninuprightconguration:A)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggerTail.

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7-31 alongwiththeassociatedailerondeections.Therollratesareapproximatelyanorderofmagnitudegreaterthantheailerondeectionsalthoughbothappeartohaveperiodicity.Ineachcase,thismagnitudeandperiodicityarerelativelyconsistentdespitevariationstothetail.Thefrequency-domainrepresentations,showninFigure 7-32 ,agreewiththetime-domainanalysisofFigure 7-31 .Theenergyoftherollrateisaboutanorderofmagnitudegreaterthantheaileronforeachtail.Also,therollrateshowsaminorbroad-bandpeaknear4rad/salthoughtheenergyisnotexcessive.Awavelettransformisappliedtothetime-domaindatatogeneratetherepresentation,showninFigure 7-33 ,inthetime-frequencydomain.Theseplotsshowsomecorrelationaround4rad/s;however,themagnitudeofcorrelationisnotexcessivelyhighincomparisontothelowerfrequencies.ThislackofexcessivecorrelationagreeswiththePSDsinFigure 7-32 andisevidentforeachtail.Asimilartime-frequencyrepresentationoftheailerondata,ascomputedthroughwavelettransformandshowninFigure 7-34 ,indicatesastrongcorrelationtotherollratesthatareshowninFigure 7-33 .Inparticular,mostofthepeaksincorrelationforrollratearematchedbyapeakinaileronatthesametimeandfrequency.Examplesofsuchinput/outputmatchingcanbeseenwiththesmallertailatapproximately2.5to3rad/sbetween50and60sandwiththebiggertailatapproximately3rad/sbetween10and20sandatapproximately3to4rad/sbetween45and60s. 7-27 and 7-31 respectively,areanalyzed.PeakaveragesandfrequenciesareextractedforeachverticaltailsizeandshowninFigure 7-35 185

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B C D EFigure7-31. TimeresponsesininvertedcongurationforA)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggertail:rollrate()andailerondeection(

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BFigure7-32. PSDininvertedcongurationofrollrate(left)andailerondeection(right):smallertail(),smalltail(),normaltail(),bigtail(),biggertail(x). Thetime-domainanalysisinFigure 7-35 indicatesthatwingrockfrequencyandmagnitudedoesnotdependuponthesizeoftheverticaltail.However,thewingrockmagnitudedoeschangebasedontheverticaltailconguration:theaveragewingrockpeakmagnitudeintheinvertedightconditionisapproximatelyhalfofthatintheuprightightconditionacrossalltailsizes.Theaveragewingrockfrequencyisnearlyconstantatapproximately4rad/sforalltailsizesinbothuprightandinvertedhighangle-of-attackight,whichagreeswiththepeaksseeninthePSDsinFigures 7-28 and 7-32 .ThedifferenceinmagnitudeofuncommandedwingrockbetweenuprightandinvertedightislikelymorepronouncedthanFigure 7-35 indicates.Figures 7-29 and 7-30 showlittlecorrelationbetweenaileroninputandrollratefrequenciesintheuprightconguration.Figures 7-33 and 7-34 indicate,however,thatastrongcorrelationexistsbetweenrollrateandaileroninputfrequenciesatessentiallyalltimeswhenrollrateoscillationsareobservedintheinvertedconguration.Thisindicatesthatuncommandedwingrockisvirtuallynonexistentwhenintheinvertedconguration.TheupperandlowerboundsofwingrockfrequencyareextractedfromFigures 7-28 and 7-29 andareshowninFigure 7-36 .Onlytheuprightcongurationisanalyzedfor 187

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B C D EFigure7-33. Wavelettransformsofrollrateininvertedconguration:A)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggertail.

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B C D EFigure7-34. Wavelettransformsofailerondeectionininvertedconguration:A)smallertail,B)smalltail,C)normaltail,D)bigtail,E)biggertail.

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Meanrollratepeakmagnitudes()andpeak-to-peakfrequencies(). suchboundsbecauseonlyuncommandedwingrockisofinterest.TheupperandlowerboundsasdeterminedbyboththePSDsandwaveletsarefairlyconsistentforalluprighttailsizes.Theseboundsindicatethatwingrockexistsacrossaspectrumoffrequencies,withbandwidthsofapproximately2rad/sforalltailsizes. Figure7-36. UpperandlowerboundsofwingrockfrequencyfromPSDs()andwavelets(). 190

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7-29 actuallyindicatethatthewingrockoscillationismuchnarrowerinbandwidthwhenlocalizedintimethanFigure 7-36 indicates.TheaveragebandwidthofthewingrockoscillationsatalltimesforeachtailareextractedfromthewaveletplotsandareshowninFigure 7-37 alongwiththebroadbandsextractedfromFigure 7-36 .ThebandwidthatmosttimesisnotablysmallerinFigure 7-37 thaninFigure 7-36 .Thisindicatesthatwingrockisactuallyafairlynarrow-bandphenomenon,butvariationsinfrequencyovertimewithinabroaderbandleadtotheappearanceofabroad-bandphenomenonwithmanyclassicalanalysistechniquessuchasthePSD. Figure7-37. Meanwingrockbandwidths:PSD(),waveletbroadlimits(),andwaveletnarrowlimits(). 3.3 3.4 ,isutilizedwiththeobjectiveofminimumpathtime.Highangle-of-attackightsignicantlyreducestheforwardvelocity.Thereforehighangle-of-attackighthasadisadvantageforplanningminimum-timepaths.However,thereducedvelocityproducessmallermaneuveringrequirementsandthereforepotentiallymoredirectpathsthroughaclutteredenvironment[ 174 ]. 191

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7.5 ,thesensingqualitymaysufferduetoblurringcausedbythewingrockrollrate. 192

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7-38 .Thevelocityforeachhighangle-of-attackprimitiverangesbetweenVmin,atmax,andVmax,intraditionalwing-borneight,andscalesproportionallythroughouttheangle-of-attackrange.Thetimedurationofeachprimitiveisscaledappropriatelybasedontheresultingvelocity,andineachlocalplanningstageoftheRRTgrowththetrajectoryprimitivewhichyieldstheshortesttimeandmaintainsobstacleavoidanceisselectedforthebranchaddition.Thechoiceofhighangle-of-attacktrajectoryprimitivesthenrepresentsatrade-offbetweenthereducedvelocityandtheincreasedmaneuverability,particularlythereducedturnradius.Theadditionofhighangle-of-attacktrajectoryprimitivesdoublestheprimitivelibraryfromwhichthepathplannercanchooseeachlocalsolution.Both2-elementand3-elementtrajectoryprimitivescanbeperformedinhighangle-of-attackight.The3-elementhighangle-of-attacktrajectoryprimitivesareaugmentedto4-elementprimitivesinthe3-Dsense,asdescribedinSection 2.3.2 .Theavailable2-elementtrajectoryprimitivesincludinghighangle-of-attackightareasfollows:

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Depictionofconditionalrulesestablishedforhighangle-of-attacktrajectoryprimitives:A)climbingandB)descending.

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7-39 BFigure7-39. Environmentforhighangle-of-attackMonteCarlosimulationswithinitialposition( Thetraditionalwing-bornevelocityandturnrate,Vmaxand!,respectively,arebothunity.Themaximumangle-of-attack,max,is45andthecorrespondingminimumvelocity,Vmin,is1 3V.Theturnrate!ismaintainedatunityresultinginaturnradiusproportionaltothevelocity.Theinitialpositionis(0,0,0)withavectorofmotionalongthex-axis([1,0,0])andthenalpositionis(10,10,10)withavectorofmotionof[0.7071,0.7071,0]. 195

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7-40A .Manysuchpathsfeaturedlengthyturningsegmentsduetotheminimumturnradiusbeinginsufcienttomakemoredirectturnstowardstherandomlysampledwaypointswhilemaintainingobstacleavoidance.AnexampleofapathfoundintheMonteCarlosimulationwithhighangle-of-attackprimitivesisshowninFigure 7-40B withthehighangle-of-attacksegmentshighlightedwithboldlinesegments.Typicallytheinclusionofhighangle-of-attackprimitivesisfoundtocreatemoredirectpathsthroughtheenvironment. BFigure7-40. Exampletrajectoriesfromhighangle-of-attackMonteCarlosimulations.A)Withouthighangle-of-attackprimitives(34.80s)andB)withhighangle-of-attackprimitives(27.57s):traditionalforwardight( )andhighangle-of-attackight( ).Obstaclesomittedforclarity. ThedistributionsofpathtimesfromtheMonteCarlosimulationsarepresentedinFigure 7-41 .ThestatisticalsummaryofthepathtimesispresentedinTable 7-8 .Theinclusionofhighangle-of-attackprimitivesinthisclutteredenvironmentreducesthemeanpathtimeby21%andthestandarddeviationby55%.Thesimulationwithouthighangle-of-attackprimitivesactuallygeneratedalowerminimumpathtimethanthesimulationwithhighangle-of-attackprimitives;however,therestofthedatastrongly 196

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PathtimeresultsofMonteCarlosimulationswithandwithouthighangle-of-attackprimitives(HA). MeanSt.DeviationMinimumMaximum WithoutHA35.42s10.23s19.41s64.13sWithHA27.84s4.60s21.49s39.01sChange-7.48s-5.63s+2.08s-25.12sChange(%)-21.12%-55.03%+10.72%-39.17% indicatesthattheopportunitytochoosehighangle-of-attackprimitivesallowsformoreconsistentlyshorterpathtimes. BFigure7-41. HistogramsofpathtimesfromMonteCarlosimulationswithmean( ):A)withouthighangle-of-attackprimitivesandB)withhighangle-of-attackprimitives. Therelativeperformanceofpathsplannedwithandwithouthighangle-of-attackprimitiveswilldependheavilyontheenvironmentandvehicleconstraints.Anenvironmentwithobstaclesmorewidelyspacedmaybetraversedmoredirectlywithtraditionalwing-borneprimitives,reducingtheadvantageofthelowervelocityhighangle-of-attackprimitives.Itwouldbeexpected,however,thatastheenvironmentismademoreclutteredrelativetotherequiredmaneuversizeofthevehicle,theadvantageofincludinghighangle-of-attackprimitiveswouldincrease. 197

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4.4 isutilizedwiththeinclusionofhighangle-of-attackightcapability.Highangle-of-attackprimitives,asdescribedinSection 7.6.1 ,areincludedinbothphasesofthesensorplanningframework:sensingtrajectoryprimitivesamplingandpathplanningbetweensensingprimitives.Thesensingtrajectoryprimitivesamplingtakesintoaccountthevehicleorientationwhileperformingthehighangle-of-attackprimitiveaswellasthereducedightspeed.Additionally,uncommandedwingrockasdescribedinSection 7.5 isaccountedforbyaugmentingtheimageplanevelocitysensoreffectivenessmetric.Thestagethatplanspathsbetweenthesensingtrajectoryprimitivesthenutilizeshighangle-of-attacktrajectoryprimitivesasdescribedinSection 7.6.2 .Theinclusionofhighangle-of-attackprimitivesinthisstageispurelyforthepotentialbenetofcreatingshortertimepathsthroughaclutteredenvironmentduetothetighterturnradiusaffordedinhighangle-of-attackight. 7-38 .Thebankangleofthevehicleisalwaysassumedtobezerowhileinhighangle-of-attackight,whichisbasedonpilotexperiencemaneuveringaircraftinhighangle-of-attackight.Itisdifculttomaintainthenecessaryliftinhighangle-of-attackightifasustainedbankangleisintroduced.Therefore,turningmaneuversinhighangle-of-attackightaregenerallyperformedbyyawingthevehiclewhilemaintainingawings-levelorientation.Nosideslipistakenintoaccountatthistime;thereforethevehicleheadingcorrespondstothedirectionofightatalltimes. 198

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7 7 ftan(FOVmax)(7)Aside-lookingsensorwillexperiencevaryingblurringacrosstheverticalaxisoftheimageplane,withthehighestblurringoccurringatthecenteroftheverticalaxiswhentherollrateduetowingrockishighest.Thefunctionutilizedtoaugmenttheimageplanevelocityforaside-lookingsensorispresentedinEquation 7 199

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B CFigure7-42. ImageplanevelocityaugmentationtoaccountforwingrockwithA)forward-lookingsensor,B)downward-lookingsensor,andC)side-lookingsensor. ftan(FOVmax)(7)Figure 7-42 presentsgraphicaldepictionsofthewingrockimageplanevelocityaugmentationfunctionsinEquations 7 7 200

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Targetparametersforhighangle-of-attackMonteCarlosensingsimulations. ParameterValue TargetLocation(5.0,5.0,5.0)TargetNormalDirection(Upward)[0.0,0.0,1.0]TargetNormalDirection(Side)[1.0,0.0,0.0]TargetNormalDirection(Downward)[0.0,0.0,-1.0]Incmax60(1.0472rad) Figure7-43. Environmentforhighangle-of-attackMonteCarlosensingsimulationswithtarget(4)andboundingboxaroundlocalsamplingregion(). non-zerooverallsensingeffectivenessforatleastoneofthethreetypesofprimitives:full-speed,highangle-of-attackwithoutwingrock,andhighangle-of-attackwithwingrock.Thesensingeffectivenessofthefull-speedtrajectoryprimitivesiscomparedtothatofthehighangle-of-attackprimitiveswithandwithoutwingrockeffects.ThetargetparametersutilizedintheMonteCarlosimulationsarepresentedinTable 7-9 .Thetargetlocationalongwiththelocalsamplingregionforallthehighangle-of-attackMonteCarlosensingsimulationsisshowninFigure 7-43 .Thevehicleparametersutilizedinthehighangle-of-attackMonteCarlosensingsimulationsforbothfull-speedandhighangle-of-attackprimitivesarepresentedinTable 7-10 .ThesensorparametersutilizedarealsopresentedinTable 7-11 .TheresultsfromtheMonteCarlosimulationforatargetwithanupwardnormalvectorarepresentedinTable 7-12 .Themeansensingeffectivenessofprimitiveswithnon-zeroeffectiveness,themaximumsensingeffectiveness,andthenumberof 201

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Vehicleparametersforhighangle-of-attackMonteCarlosensingsimulations. ParameterValue 3unit/spwr100/s(1.7452rad/s) Table7-11. Sensorparametersforhighangle-of-attackMonteCarlosensingsimulations. ParameterValue Pan(Forward)0(0rad)Pan(Downward)0(0rad)Pan(Side)90(1.5708rad)Tilt(Forward)0(0rad)Tilt(Downward)-90(-1.5708rad)Tilt(Side)0(0rad)f0.1unitRmax3.0unitFOVmax45(0.7854rad)IPVmax0.15unit/s primitiveswhichproducednon-zerosensingeffectivenessarepresentedforeachofthethreetypesofprimitiveswithvarioussensororientations.Thehighangle-of-attackprimitives(withandwithoutwingrock)provideahighermeansensingeffectivenessforallthreesensororientations.Thehighangle-of-attackprimitivesalsoprovideahighermaximumsensingeffectivenessforallthreesensororientations,thoughthewingrockpenaltyreducesthemaximumtobelowthatofthefull-speedprimitiveswithaforward-lookingsensor.Thenumberofprimitivesyieldinganon-zerosensingeffectivenessishigherforhighangle-of-attackprimitiveswithbothdownwardandsidewaysorientedsensors,indicatingthatitiseasiertondprimitiveswhichsensethetargetwhenathighangle-of-attack.Theoppositeistruefortheforwardsensororientation:itiseasiertondafull-speedprimitivewhichsensesthetarget.Therefore,itisseenthathighangle-of-attackprimitives,bothwithandwithoutwingrock,providean 202

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ResultsofMonteCarlosensingprimitivesimulationfortargetwithupwardnormalvector. Full-SpeedHighAngle-of-AttackHighAngle-of-Attack(NoWingRock)(WingRock) advantageoverfull-speedprimitivesintermsofsensingeffectivenessoreaseofndingeffectiveprimitives,orboth,withallsensororientations.TheresultsfromtheMonteCarlosimulationforatargetwithasidewaysnormalvectorarepresentedinTable 7-13 .Themeansensingeffectivenessofprimitiveswithnon-zeroeffectiveness,themaximumsensingeffectiveness,andthenumberofprimitiveswhichproducednon-zerosensingeffectivenessarepresentedforeachofthethreetypesofprimitiveswithvarioussensororientations.Thehighangle-of-attackprimitivesdonotperformaswellasfull-speedprimitiveswithadownwardsensorintermsofeithermeansensingeffectivenessornumberofprimitivesyieldingnon-zeroeffectiveness.Highangle-of-attackprimitivesprovideaslightadvantagewithaforwardsensorinallthreecategories.Highangle-of-attackprimitivesdonotprovideasignicantsensingeffectivenessimprovementoverfull-speedprimitiveswithasidewayssensor;however,asignicantimprovementisshownintermsofthenumberofprimitiveswithnon-zeroeffectiveness,indicatingthathighangle-of-attackprimitiveswhichsensethetargetareeasiertondwithsuchasensorandtargetorientation. 203

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ResultsofMonteCarlosensingprimitivesimulationfortargetwithsidewaysnormalvector. Full-SpeedHighAngle-of-AttackHighAngle-of-Attack(NoWingRock)(WingRock) TheresultsfromtheMonteCarlosimulationforatargetwithadownwardnormalvectorarepresentedinTable 7-14 .Themeansensingeffectivenessofprimitiveswithnon-zeroeffectiveness,themaximumsensingeffectiveness,andthenumberofprimitiveswhichproducednon-zerosensingeffectivenessarepresentedforeachofthethreetypesofprimitiveswithvarioussensororientations.Withadownwardtargetthehighangle-of-attackprimitivesareunabletoimproveuponthemeansensingeffectivenessorthenumberofprimitivesyieldingnon-zeroeffectivenessdeliveredbythefull-speedprimitiveswithanysensororientation.However,themaximumsensingeffectivenessproducedbyhighangle-of-attackprimitivesishigherthanfull-speedprimitivesforallsensororientations.Therefore,theinclusionofhighangle-of-attackprimitivesinthesamplingphaseofthesensorplanningframeworkmayprovidebenecialresults.Theinclusionofhighangle-of-attacktrajectoryprimitivescansignicantlyimprovethesensingeffectivenessorthespeedofndingeffectiveprimitives;however,theperformancevariessignicantlybysensorandtargetorientation.Theeffectofthewingrockisseenincomparingtheresultsfromhighangle-of-attackprimitiveswithand 204

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ResultsofMonteCarlosensingprimitivesimulationfortargetwithdownwardnormalvector. Full-SpeedHighAngle-of-AttackHighAngle-of-Attack(NoWingRock)(WingRock) withoutwingrock.Typicallyallthreecategories,meansensingeffectiveness,maximumsensingeffectiveness,andnumberofsensingprimitives,arehinderedslightlybytheinclusionoftheimageplanevelocityaugmentationdescribedinSection 7.6.3.2 .TheseMonteCarlosimulationshavenoobstaclesandthereforeprovideacomparisonofonlythesensingeffectivenessaffordedbythedifferentprimitives.Obstaclesincloseproximitytothetargetwillresultinsomeprimitivesbeinginfeasibleduetoobstaclecollision.Highangle-of-attackprimitiveshavesmallerturningradiiandthereforearelikelytocreatemoredirectpathsthatarecontainedinasmallerarea,typicallyresultinginfewerprimitiveswhichencounterobstaclecollisions.Therefore,theadvantagesofincludinghighangle-of-attackprimitivesinsensingnearobstaclesmaybegreaterthanindicatedbytheseMonteCarlosimulations. 7.6.2 demonstratesthattheinclusionofhighangle-of-attackprimitivescanproducetrajectorieswithshortertotal 205

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5.1 and 5.2 respectively,areagainutilizedtoattempttoimprovethepathsgeneratedbytherandomdensetrees.Ifneithermethodcanimproveuponthepathtimegeneratedbytherandomdensetreebetweensensingprimitivestheoriginaltree-basedpathisretained,includinghighangle-of-attacksegments. 7.6.3.2 .Thesensorisrigidlyxedtotheairframeandpointsdownward.Thepathsbetweensensingprimitivesaredeterminedwiththemultiple-goaltreegrowthmethod,describedinSection 4.4.2.1 .Highangle-of-attacktrajectoryprimitivesareincludedwithinthelibraryofpossibleprimitivesfortreegrowthofsuchpaths,providingapotentialadvantageintermsofpathtimeaspresentedinSection 7.6.2 .TheenvironmentandtargetsutilizedinthisexamplearepresentedinFigure 7-44 .Theenvironment,vehicle,andsensorparametersutilizedinthisexamplearepresentedinTables 7-15 7-16 ,and 7-17 ,respectively.Alocalregionaroundeachtargetisrandomlysampledforinitialandnalcongurationsoftrajectoryprimitives,asdescribedinSection 4.4.1 .Bothfull-speedandhighangle-of-attacktrajectoryprimitivesarecalculatedforeachsampledsetofcongurations,asdescribedinSection 7.6.3.3 .Atotalof100feasibletrajectoryprimitivesaresampled,includingbothfull-speedandhighangle-of-attackprimitives.ThesampledprimitivesforeachtargetareshowninFigure 7-45A .Thetrajectoryprimitiveswhichyieldthehighestsensingeffectivenessforeachtargetareselectedasthebasisfortheoveralltrajectory.ThesebestsensingtrajectoryprimitivesareshowninFigure 7-45B 206

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BFigure7-44. Environmentwithtargets(4)forhighangle-of-attacksensorplanningexample:A)topviewandB)isometricview. Table7-15. Environmentparametersforhighangle-of-attacksensorplanningexample. ParameterValue InitialPosition(0.0,0.0,0.0)InitialDirection[1.0,0.0,0.0]FinalPosition(0.0,0.0,0.0)FinalDirection[-1.0,0.0,0.0]Target1Location(1.5,5.5,0.0)Target1NormalDirection[0.0,0.0,1.0]Target2Location(3.0,8.5,3.0)Target2NormalDirection[0.0,0.0,1.0]Target3Location(7.5,3.3,3.0)Target3NormalDirection[-1.0,0.0,0.0]Incmax(AllTargets)60(1.0472rad) Table 7-18 presentstheresultingsensingeffectivenessvalueforeachtargetandindicateswhethertheprimitivecorrespondingtothattargetistobeperformedatfull-speedorhighangle-of-attack.Itisseenthatforthetwoupwardfacingtargetshigh Table7-16. Vehicleparametersforhighangle-of-attacksensorplanningexample. ParameterValue 3unit/spwr100/s(1.7452rad/s) 207

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Sensorparametersforhighangle-of-attacksensorplanningexample. ParameterValue Pan0(0rad)Tilt-90(1.5708rad)f0.1unitRmax3.0unitFOVmax45(0.7854rad)IPVmax0.15unit/s BFigure7-45. Sensingprimitiveselectionforhighangle-of-attacksensorplanningexample:A)sampledtrajectoryprimitiveswithfullspeedprimitives()andhighangle-of-attackprimitives( )andB)bestsensingtrajectoryprimitives. angle-of-attackprimitivesprovidethebestsensingeffectiveness,whichisinagreementwiththeMonteCarloresultsinTable 7-12 .Target3,whichfacessideways,isfoundtobebestsensedwithafull-speedprimitivewhichisalsoinagreementwiththeMonteCarloresultsinTable 7-13 Table7-18. Sensingqualityforselectedsensingtrajectoryprimitives. TargetHighAngle-of-Attack(HA)MaximumSensingorFull-Speed(FS)Effectiveness 1HA0.85612HA0.94813FS0.9850 RandomdensetreesaregrownasdescribedinSection 4.4.2.1 untilallsensingtrajectoryprimitivesareexecutedandthetrajectoryreachestheterminalconguration. 208

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7-46 BFigure7-46. Tree-basedsensingtrajectory:A)topviewandB)isometricview(obstaclesomittedforclarity)withfullspeedpathsegments(),highangle-of-attackpathsegments( ),sensingprimitivesperformedatfullspeed(),sensingprimitivesperformedathighangle-of-attack( Theorderreductionpathimprovementtechnique,describedinSection 5.1 ,isappliedtothetrajectorybetweensensingtrajectoryprimitives.Theorderreductionalgorithmimprovesuponthepathtimeoftherstandlastpathsegmentsbutisunabletoreducethepathtimeofthesecondandthirdpathsegments.Thereforethehighangle-of-attacksegmentsperformedinthesecondandthirdpathsegmentsarepreservedinthetotaltrajectory.Thetotaltrajectoryafterorderreductionyieldsapathtimeof48.46s,whichrepresentsanimprovementof15.27%overthetree-basedpathtime.ThetotaltrajectoryafterorderreductionispresentedinFigure 7-47 .Theorderincreasepathimprovementtechnique,describedinSection 5.2 ,isappliedtoeachtrajectorysegmentbetweensensingtrajectoryprimitives.Theorderincreasealgorithmprovidesslightimprovementstothepathtimesoftherstand 209

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BFigure7-47. Reduced-ordersensingtrajectory:A)topviewandB)isometricview(obstaclesomittedforclarity)withfullspeedpathsegments(),highangle-of-attackpathsegments( ),sensingprimitivesperformedatfullspeed(),sensingprimitivesperformedathighangle-of-attack( lastsegments,whichwerealreadyimproveduponbytheorderreductionalgorithm.However,theorderincreasealgorithmisunabletoimproveuponthetree-basedsecondandthirdpathsegments.Therefore,thehighangle-of-attacksegmentsinthesecondandthirdtrajectorysegmentsarepreservedinthetotaltrajectory.Thetotaltrajectoryaftertheorderincreasealgorithmisappliedyieldsapathtimeof48.06s,whichrepresentsamodestimprovementof0.82%overthereducedordertrajectorytime.ThetotaltrajectoryaftertheorderincreasealgorithmisappliedispresentedinFigure 7-48 .AsummaryofthepathtimeandimprovementofeachsegmentofthetrajectoryispresentedinTable 7-19 .AsummaryofthepathtimeandimprovementofthetotalsensingtrajectoryispresentedinTable 7-20 210

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BFigure7-48. Increased-ordersensingtrajectory:A)topviewandB)isometricview(obstaclesomittedforclarity)withfullspeedpathsegments(),highangle-of-attackpathsegments( ),sensingprimitivesperformedatfullspeed(),sensingprimitivesperformedathighangle-of-attack( Table7-19. Pathtimeandimprovementofeachsegmentofhighangle-of-attacksensingtrajectory. SensingStart!Target1!Target3!Target3!PrimitivesTarget1Target3Target2Finish Tree-Based9.51s6.40s11.08s12.25s17.94sReduced-Order9.51s6.25s11.08s12.25s9.36sImprovement(s)N/A0.15s0.00s0.00s8.58sImprovement(%)N/A2.34%0.00%0.00%47.83%Increased-Order9.51s5.88s11.08s12.25s9.33sImprovement(s)N/A0.37s0.00s0.00s0.03sImprovement(%)N/A5.92%0.00%0.00%0.32% effectivenessaswellasreducingtree-basedpathtimesthroughclutteredenvironments.Theuncommandedwingrockispenalizedaccordingtosensororientation,andthesensingeffectivenessofhighangle-of-attackightcanstillprovideimprovementsoverthatoffull-speedmaneuversforsometargetorientations. Table7-20. Pathtimeandimprovementoftotalhighangle-of-attacksensingtrajectory. Tree-BasedReduced-OrderIncreased-Order PathTime(s)57.19s48.46s48.06sImprovementN/A8.73s0.40sImprovement(%)N/A15.27%0.83% 211

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BaronJohnson'saviationpursuitsbeganwithhisrstaerobaticridesafewdayspriortobeingborninMemphis,TNin1983.AftermovingtoOcala,FL,BaronsoloedanL23SuperBlaniksailplaneattheageof14,andthensoloedaPiperJ-3Cubonhis16thbirthday.Baronthenpursuedhisprivatepilotlicense,whichhereceivedattheageof17,andhiscommercialpilotlicense,whichhereceivedattheageof18,alongwithinstrumentandmulti-engineratings.AftergraduatingfromBelleviewHighSchoolin2002,BaronattendedtheUniversityofFloridainGainesville,FL.HereceivedaBachelorofSciencein2007,aMasterofSciencein2009,andaPh.D.in2011,allinaerospaceengineering.Whilestudying,BaronhashadopportunitiestoworkonmanyexcitingUAVandMAVprogramswhileemployedbytheFlightControlLab,MicroAirVehicleLab,andFloridaCooperativeFish&WildlifeResearchUnit.TheFlightControlLabaffordedBarontheopportunitytoworkonmanyuniqueprojectsandalsolmseveraldocumentaries,includingonein3DwithSirDavidAttenborough.WhilememberandpilotfortheUFMicroAirVehicleTeam,theteamcaptured4consecutivevictoriesattheInternationalMicroAirVehicleCompetition.BaronhasalsobeenanactiveRCaviatorduringhisyears.HisprimarypassionsinRCareIMACScaleAerobaticsandF3CHelicopterAerobatics.BaronhashadtheopportunitytoymanyRCdemonstrationsandcompetitions,andhaswon5nationalchampionshiptitlesalongwithotherhighnishesatinternationalcompetitions.Baronalsoauthoredachildren'sbooktitledHistoryTakesAWildRide,forwhichhewasawardedthekeytothecityofMemphisandappearedonNBC'sTheTodayShow.Baronwillbetyingtheknotwithhisbeautifulanceesoonaftergraduationin2011andthey'llbemovingtoHuntsville,ALwhereBaronacceptedapositionintheUnmannedSystemsDepartmentofDynetics. 231