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ANY-ANGLE PATH PLANNING by Alex Nash a Dissertation Presented ANY-ANGLE PATH PLANNING by Alex Nash A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) August 2012 Copyright 2012 Alex Nash Table of Contents List of Tables vii List of Figures ix Abstract xii Chapter 1: Introduction 1 1.1 PathPlanning .................................... 2 1.1.1 The Generate-Graph Problem . 3 1.1.2 Notation and Definitions . 4 1.1.3 TheFind-PathProblem. 5 1.2 Hypotheses and Contributions . ..... 12 1.2.1 Hypothesis1 ................................ 12 1.2.2 Hypothesis2 ................................ 12 1.2.2.1 Basic Theta*: Any-Angle Path Planning in Known 2D Envi- ronments............................. 14 1.2.2.2 Lazy Theta*: Any-Angle Path Planning in Known 3D Envi- ronments............................. 14 1.2.2.3 Incremental Phi*: Any-Angle Path Planning in Unknown 2D Environments .......................... 15 1.3 Dissertation Structure . ..... 17 Chapter 2: Path Planning 18 2.1 Navigation...................................... 18 2.2 PathPlanning .................................... 19 2.2.1 The Generate-Graph Problem . 20 2.2.1.1 Skeletonization Techniques . 21 2.2.1.2 Cell Decomposition Techniques . 27 2.2.1.3 Hierarchical Generate-Graph Techniques . 34 2.2.1.4 The Generate-Graph Problem in Unknown 2D Environments . 35 2.2.2 TheFind-PathProblem. 39 2.2.2.1 Single-Shot Find-Path Algorithms . 39 2.2.2.2 Incremental Find-Path Algorithms . 43 2.2.3 Post-Processing Techniques . 46 2.2.4 Existing Any-Angle Find-Path Algorithms . ...... 48 ii 2.3 Conclusions..................................... 50 Chapter 3: Path Length Analysis on Regular Grids (Contribution 1) 52 3.1 Introduction.................................... 52 3.2 2DRegularGrids .................................. 53 3.2.1 Existing Work: 2D Regular Grids . 54 3.2.2 Notation and Definitions . 55 3.2.3 GridGraphProperties . 57 3.2.4 PathLengthAnalysis. 59 3.2.4.1 Notation and Definitions . 61 3.2.4.2 Part 1: Unblocked Path . 61 3.2.4.3 Part 2: Two Edge Types . 65 3.2.4.4 Part 3: Scaling Method . 66 3.2.5 Geometric Relationships: 2D Regular Grids . ....... 67 3.2.5.1 Double Tri Graphs and Hex Graphs . 68 3.3 3DRegularGrids .................................. 70 3.3.1 Notation and Definitions . 70 3.3.2 GridGraphProperties . 71 3.3.3 PathLengthAnalysis. 71 3.3.3.1 Part 1: Unblocked Path . 73 3.3.3.2 Part 2: Lagrange Method . 77 3.3.4 Geometric Relationships: 3D Regular Grids . ....... 78 3.3.5 CubicGraphs................................ 78 3.3.6 Lagrange Method: 2D Regular Grids . 79 3.4 Conclusions..................................... 80 Chapter 4: Basic Theta*: Any-Angle Path Planning in Known 2D Environments (Con- tribution 2) 82 4.1 Introduction.................................... 83 4.2 NotationandDefinitions . 84 4.3 Existing Find-Path Algorithms in Known 2D Environments ............ 84 4.3.1 A*onOctileGraphs ............................ 87 4.3.2 A*withPost-Smoothing(A*PS) . 87 4.3.3 FieldD*(FD*)............................... 88 4.3.4 A*onVisibilityGraphs . 89 4.4 BasicTheta* .................................... 90 4.4.1 Operation of Basic Theta* . 91 4.4.2 Example Trace of Basic Theta* . 92 4.4.3 Properties of Basic Theta* . 93 4.4.3.1 Simplicity Property . 93 4.4.3.2 Generality Property . 93 4.4.3.3 Correctness and Completeness . 93 4.4.3.4 Optimality ............................ 95 4.4.3.5 HeadingChanges . 96 4.5 Angle-Propagation Theta* (AP Theta*) . ....... 97 iii 4.5.1 DefinitionofAngleRanges . 99 4.5.2 UpdatingAngleRanges . 100 4.5.3 ExampleTraceofAPTheta*. 102 4.5.4 PropertiesofAPTheta* . 102 4.6 ExperimentalResults . 110 4.7 Extensions of Basic Theta* and AP Theta* . 114 4.7.1 SingleSourcePaths. 114 4.7.2 Non-Uniform Traversal Costs . 116 4.8 Trading Off Runtime and Path Length: Exploiting h-Values............ 118 4.8.1 Weighted h-Values with w< 1 .......................119 4.8.2 TieBreaking ................................ 120 4.8.2.1 Tie Breaking and the Triangle Inequality . 121 4.8.3 Re-Expanding Vertices . 122 4.9 Trading Off Runtime and Path Length: Other Approaches . ........... 124 4.9.1 ThreePaths................................. 124 4.9.2 KeyVertices ................................ 125 4.9.3 Larger Branching Factors . 125 4.10Conclusions.................................... 126 Chapter 5: Lazy Theta*: Any-Angle Path Planning in Known 3D Environments (Con- tribution 3) 128 5.1 Introduction.................................... 129 5.2 NotationandDefinitions . 130 5.3 Find-Path Algorithms in Known 3D Environments . ........ 130 5.3.1 A*onTripleCubicGraphs. 130 5.3.2 A*withPost-Smoothing(A*PS) . 132 5.3.3 BasicTheta* ................................ 132 5.3.3.1 Example Trace of Basic Theta* . 134 5.4 LazyTheta*..................................... 134 5.4.1 OperationofLazyTheta* . 135 5.4.2 Example Trace of Lazy Theta* . 136 5.5 VariantsofLazyTheta*. 138 5.6 ExperimentalResults . 142 5.7 Trading Off Runtime and Path Length: Exploiting h-Values............ 144 5.7.1 Weighted h-Values with w> 1 .......................145 5.7.1.1 Environments in which Find-Path Algorithms Encounter Local Minima ............................. 146 5.7.1.2 Environments in which Find-Path Algorithms do not En- counter Local Minima . 150 5.8 Conclusions..................................... 154 Chapter 6: Incremental Phi*: Any-Angle Path Planning in Unknown 2D Environ- ments (Contribution 4) 155 6.1 Introduction.................................... 156 6.2 NotationandDefinitions . 157 iv 6.3 Find-Path Algorithms in Unknown 2D Environments . ......... 157 6.3.1 BasicTheta* ................................ 158 6.3.1.1 Example Trace of Basic Theta* . 159 6.4 Phi*.........................................161 6.4.1 DefinitionofAngleRanges . 162 6.4.2 UpdatingAngleRanges . 162 6.4.3 UsingAngleRanges ............................ 163 6.4.4 Tie Breaking and the Triangle Inequality . 164 6.4.5 ExampleTraceofPhi* ..... ..... ...... ..... ...... 164 6.5 IncrementalPhi* ................................. 165 6.5.1 Example Trace of Incremental Phi* . 167 6.5.2 MovingAgent ............................... 167 6.6 ExperimentalResults . 168 6.6.1 Phi*..................................... 169 6.6.2 IncrementalPhi* .............................. 170 6.7 Conclusions..................................... 171 Chapter 7: Conclusions 173 Bibliography 179 Appendix A CheckingLine-of-Sight . 186 Appendix B APTheta*Proofs ..................................... 188 Appendix C RegularGridProofs.................................. 194 C.1 Part1ofLemma1for2DRegularGrids . 194 C.2 Part1ofLemma2for3DRegularGrids . 199 C.2.1 NotationandDefinitions . 200 C.2.2 Category1 ................................. 201 C.2.2.1 Right Step (Q = R) .......................201 C.2.2.2 Back Step (Qb = B) .......................204 C.2.2.3 Up Step (Q b= U) ........................205 C.2.3 Category2 .................................b 207 C.2.3.1 Back Up Step (Q = BU) [Right Side] . 208 C.2.3.2 Back Right Stepb (Q = BR)[UpSide]. 209 C.2.3.3 Right Up Step (Q b= RU)[BackSide]. 210 b Appendix D Phi* and Incremental Phi* Proofs . 214 D.1 NotationandDefinitions . 214 D.2 Proofs........................................214 D.2.1 CoreProperties............................... 215 v D.2.2 Initialize .................................. 215 D.2.2.1 CoreProperties . 216 D.2.3 Procedures ComputeCost and UpdateVertex . 217 D.2.4 Procedure ComputeShortestPath . 219 D.2.4.1 HelperLemmata. 219 D.2.4.2 CoreProperties . 221 D.2.4.3 Completeness and Correctness . 225 D.2.4.4 Local Parent Path Property . 227 D.2.5 ProcedurePreProcess. 238 D.2.5.1 HelperLemmata. 239 D.2.5.2 CoreProperties . 240 Appendix E Impact of Any-Angle Path Planning . 248 E.1 Citations of Basic Theta* and its Variants . ......... 248 E.1.1 RoboticsConferences . 248 E.1.2 Artificial Intelligence Conferences . ....... 249 E.1.3 Aerospace Conferences . 249 E.1.4 VideoGameConferences . 249 E.1.5 Oceanography Conferences . 249 E.2 ExtensionsofBasicTheta* . 250 E.2.1 Faster Line-of-Sight Checks . 253 E.2.2 Non-Uniform Traversal Costs . 253 E.3 Experimental Comparisons with Basic Theta* . ......... 254 E.3.1 AcceleratedA*(AA*) . 254 E.3.2 BlockA* .................................. 256 vi List of Tables 1.1 SummaryofContribution1. 11 1.2 Summaries of Contributions 2-4 . ..... 11 2.1 Cell Decomposition Techniques . ..... 27 2.2 Find-PathAlgorithms. 38 3.1 Grid Graphs Constructed from 2D Regular Grids . ........ 54 3.2 Properties of Grid Graphs Constructed from 2D Regular Grids .......... 56 3.3 Path Length Analysis Results for Grid Graphs Constructed from 2D Regular Grids 60 3.4 Hexagonal Grid Step Sequence and Move Sequence for a Hex Graph . 62 3.5 Grid Graphs Constructed from 3D Regular Grids . ........ 68 3.6 Properties of Grid Graphs Constructed from 3D Regular Grids .......... 69 3.7 Path Length Analysis Results for Grid Graphs Constructed from 3D Regular Grids 69 3.8 Cubic Grid Step Sequence and Move Sequence for a Triple Cubic Graph . 69 4.1 PathLengths .................................... 106 4.2 Runtimes ...................................... 107 4.3 Number of Vertex Expansions . 108 4.4 NumberofHeadingChanges . 109 4.5 Find-Path Algorithms without Post-Processing Steps on Random 500 500 Grids × with 20 Percent Blocked Grid Cells . 114 4.6 Find-Path Algorithms on Random 1000 1000 Grids with Non-Uniform Traver- × salCosts....................................... 117 4.7 Tie Breaking: Random 500 500 Grids with 20 Percent Blocked Grid Cells . 123 × 4.8 Re-Expanding.
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