1. Slide 1 Robot Path Planning William Regli Department of Computer Science (and Departments of ECE and MEM) Drexel University

2. Slide 2 Introduction to Motion Planning Applications Overview of the Problem Basics – Planning for Point Robot –Visibility Graphs –Roadmap –Cell Decomposition –Potential Field

3. Slide 3 Motion planning is the ability for an agent to compute its own motions in order to achieve certain goals. All autonomous robots and digital actors should eventually have this ability

4. Slide 4 Goals Compute motion strategies, e.g., –Geometric paths –Time-parameterized trajectories –Sequence of sensor-based motion commands Achieve high-level goals, e.g., –Go to the door and do not collide with obstacles –Assemble/disassemble the engine –Build a map of the hallway –Find and track the target (an intruder, a missing pet, etc.)

5. Slide 5 Fundamental Question Are two given points connected by a path?

6. Slide 6 Basic Problem  Statement: Compute a collision-free path for a rigid or articulated object among static obstacles  Inputs: Geometry of moving object and obstacles Kinematics of moving object (degrees of freedom) Initial and goal configurations (placements)  Output: Continuous collision-free path connecting the initial and goal configurations

7. Slide 7 Types of Path Constraints Local constraints –Collision-free paths Differential constraints –A car cannot move sideways –Have bound curvature Global constraints –Shortest or optimal paths Path Planning Motion Planning

8. Slide 8 Non-Holonomic Constraints v θ 3 degrees of freedom, but… only 2 control parameters Not every path in configuration space is valid...

9. Slide 9 On a Terrain Planar motion, but 3-dimensional obstacles.

10. Slide 10 Multiple Robots Robots translate and rotate Collisions between robots.

11. Slide 11 Moving Obstacles Add time dimension to C. Only time-monotone paths are allowed

12. Slide 12 Movable Obstacles Robot can change C free !

13. Slide 13 More Huge numbers of degrees of freedom. Group motions Deformable robots …

14. Slide 14 What is the number of DoF’s?  a polygon robot translating in the plane  a polygon robot translating and rotating  a spherical robot moving in space  a spatial robot translating and rotating  a snake robot in the plane with 3 links

15. Slide 15 Example: Rigid Objects

16. Slide 16 Piano Mover’s Problem 2D or 3D rigid models

17. Slide 17 piano.mpg Bench6.avi

18. Slide 18 Example: Articulated Robot

19. Slide 19 Is it easy?

20. Slide 20 Alpha.avi Alpha-another-path.avi

21. Slide 21 Hardness Results Several variants of the path planning problem have been proven to be PSPACE-hard. A complete algorithm may take exponential time. –A complete algorithm finds a path if one exists and reports no path exists otherwise. Examples –Planar linkages [Hopcroft et al., 1984] –Multiple rectangles [Hopcroft et al., 1984]

22. Slide 22 Hardness  The problem is hard when k is part of the input [Reif 79], [Hopcroft et al. 84], …  [Reif 79]: planning a free path for a robot made of an arbitrary number of polyhedral bodies connected together at some joint vertices, among a finite set of polyhedral obstacles, between any two given configurations, is a PSPACE-hard problem  Translating rectangles, planar linkages

23. Slide 23 Complete solutions, I the Piano Movers series [Schwartz-Sharir 83], cell decomposition: a doubly-exponential solution, O(nd) 3^k ) expected time assuming the robot complexity is constant, n is the complexity of the obstacles and d is the algebraic complexity of the problem

24. Slide 24 Complete solutions, II roadmap [Canny 87]: a singly exponential solution, n k (log n)d O(k^2) expected time

25. Slide 25 Tool: Configuration Space Difficulty –Number of degrees of freedom (dimension of configuration space) –Geometric complexity

26. Slide 26 Extensions of the Basic Problem More complex robots –Multiple robots –Movable objects –Nonholonomic & dynamic constraints –Physical models and deformable objects –Sensorless motions (exploiting task mechanics) –Uncertainty in control

27. Slide 27 Extensions of the Basic Problem More complex environments –Moving obstacles –Uncertainty in sensing More complex objectives –Optimal motion planning –Integration of planning and control –Assembly planning –Sensing the environment Model building Target finding, tracking

28. Slide 28 Practical Algorithms A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise. Most motion planning problems are hard, meaning that complete planners take exponential time in the number of degrees of freedom, moving objects, etc.

29. Slide 29 Practical Algorithms Theoretical algorithms strive for completeness and low worst-case complexity –Difficult to implement –Not robust Heuristic algorithms strive for efficiency in commonly encountered situations. –No performance guarantee Practical algorithms with performance guarantees –Weaker forms of completeness –Simplifying assumptions on the space: “exponential time” algorithms that work in practice

30. Slide 30 Problem Formulation for Point Robot Input –Robot represented as a point in the plane –Obstacles represented as polygons –Initial and goal positions Output –A collision-free path between the initial and goal positions

31. Slide 31 Motion planning: the basic problem Let B be a system (the robot) with k degrees of freedom moving in a known environment cluttered with obstacles. Given free start and goal placements for B decide whether there is a collision free motion for B from start to goal and if so plan such a motion.

32. Slide 32 Configuration space of a robot system with k degrees of freedom  the space of parametric representation of all possible robot configurations  C-obstacles: the expanded obstacles  the robot -> a point  k dimensional space  point in configuration space: free, forbidden, semi-free  path -> curve [Lozano-Peréz ’ 79]

33. Slide 33 Some Examples

34. Slide 34 Visibility Graphs

35. Slide 35 Framework

36. Slide 36 Visibility Graph Method Observation: If there is a collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices. Why? –Any collision-free path can be transformed into a polygonal path that bends only at the obstacle vertices. A polygonal path is a piecewise linear curve.

37. Slide 37 Visibility Graph A visibility graph is a graph such that –Nodes: q init, q goal, or an obstacle vertex. –Edges: An edge exists between nodes u and v if the line segment between u and v is an obstacle edge or it does not intersect the obstacles.

38. Slide 38 A Simple Algorithm for Building Visibility Graphs

39. Slide 39 Computational Efficiency Simple algorithm O(n 3 ) time More efficient algorithms –Rotational sweep O(n 2 log n) time –Optimal algorithm O(n 2 ) time –Output sensitive algorithms O(n 2 ) space

40. Slide 40 g s Issues With Visibility Graphs  Difficult to extend from point robots to rigid or articulated robots A L-shaped robot

41. Slide 41 Framework

42. Slide 42 Breadth-First Search

43. Slide 43 Breadth-First Search

44. Slide 44 Breadth-First Search

45. Slide 45 Breadth-First Search

46. Slide 46 Breadth-First Search

47. Slide 47 Breadth-First Search

48. Slide 48 Breadth-First Search

49. Slide 49 Breadth-First Search

50. Slide 50 Breadth-First Search

51. Slide 51 Breadth-First Search

52. Slide 52 Other Search Algorithms Depth-First Search Best-First Search, A*

53. Slide 53 Framework

54. Slide 54 Summary Discretize the space by constructing visibility graph Search the visibility graph with breadth- first search Q: How to perform the intersection test?

55. Slide 55 Summary Represent the connectivity of the configuration space in the visibility graph Running time O(n3) –Compute the visibility graph –Search the graph –An optimal O(n2) time algorithm exists. Space O(n2) Can we do better?

56. Slide 56 Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

57. Slide 57 Roadmap Visibility graph Shakey Project, SRI [Nilsson, 1969] Voronoi Diagram Introduced by computational geometry researchers. Generate paths that maximizes clearance. Applicable mostly to 2-D configuration spaces.

58. Slide 58 Voronoi Diagram Space O(n) Run time O(n log n)

59. Slide 59 Other Roadmap Methods Silhouette First complete general method that applies to spaces of any dimensions and is singly exponential in the number of dimensions [Canny 1987] Probabilistic roadmaps

60. Slide 60 Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

61. Slide 61 Cell-decomposition Methods Exact cell decomposition The free space F is represented by a collection of non-overlapping simple cells whose union is exactly F Examples of cells: trapezoids, triangles

62. Slide 62 Point robot www.seas.upenn.edu/~jwk/motionPlanning

63. Slide 63 Adjacency Graph Nodes: cells Edges: There is an edge between every pair of nodes whose corresponding cells are adjacent.

64. Slide 64 Trapezoidal Decomposition

65. Slide 65 Trapezoidal decomposition c 11 c1c1 c2c2 c4c4 c3c3 c6c6 c5c5 c8c8 c7c7 c 10 c9c9 c 12 c 13 c 14 c 15 www.seas.upenn.edu/~jwk/motionPlanning

66. Slide 66 Connectivity graph c1c1 c 10 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 c8c8 c9c9 c 11 c 12 c 13 c 14 c 15 c 11 c1c1 c2c2 c4c4 c3c3 c6c6 c5c5 c8c8 c7c7 c 10 c9c9 c 12 c 13 c 14 c 15 www.seas.upenn.edu/~jwk/motionPlanning

67. Slide 67 Computational Efficiency Running time O(n log n) by planar sweep Space O(n) Mostly for 2-D configuration spaces

68. Slide 68 Summary Discretize the space by constructing an adjacency graph of the cells Search the adjacency graph

69. Slide 69 Cell-decomposition Methods Exact cell decomposition Approximate cell decomposition –F is represented by a collection of non- overlapping cells whose union is contained in F. –Cells usually have simple, regular shapes, e.g., rectangles, squares. –Facilitate hierarchical space decomposition

70. Slide 70 Quadtree Decomposition

71. Slide 71 Octree Decomposition

72. Slide 72 Algorithm Outline

73. Slide 73 Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

74. Slide 74 Potential Fields Initially proposed for real-time collision avoidance [Khatib 1986]. Hundreds of papers published. A potential field is a scalar function over the free space. To navigate, the robot applies a force proportional to the negated gradient of the potential field. A navigation function is an ideal potential field that –has global minimum at the goal –has no local minima –grows to infinity near obstacles –is smooth

75. Slide 75 Attractive & Repulsive Fields

76. Slide 76 How Does It Work?

77. Slide 77 Algorithm Outline Place a regular grid G over the configuration space Compute the potential field over G Search G using a best-first algorithm with potential field as the heuristic function

78. Slide 78 Local Minima What can we do? –Escape from local minima by taking random walks –Build an ideal potential field – navigation function – that does not have local minima

79. Slide 79 Question Can such an ideal potential field be constructed efficiently in general?

80. Slide 80 Completeness A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise. –Is the visibility graph algorithm complete? Yes. –How about the exact cell decomposition algorithm and the potential field algorithm?

81. Slide 81 Why Complete Motion Planning? Probabilistic roadmap motion planning –Efficient –Work for complex problems with many DOF –Difficult for narrow passages –May not terminate when no path exists Complete motion planning –Always terminate –Not efficient –Not robust even for low DOF

82. Slide 82 Path Non-existence Problem Obstacle Goal Initial Robot

83. Slide 83 Main Challenge Obstacle Goal Initial Robot Exponential complexity: n DOF –Degree of freedom: DOF –Geometric complexity: n More difficult than finding a path –To check all possible paths

84. Slide 84 Approaches Exact Motion Planning –Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid planner

85. Slide 85 Configuration Space: 2D Translation Workspace Configuration Space x y Robot Start Goal Free Obstacle C- obstacle

86. Slide 86 Configuration Space Computation [Varadhan et al, ICRA 2006] 2 Translation + 1 Rotation 215 seconds Obstacle Robot x  y

87. Slide 87 Obstacles in C-space Workspace Configuration Space x y Robot Start Goal Free Obstacle C-obstacle A 2D Translating Robot

88. Slide 88 Obstacles in C-space A configuration q is collision-free, or free, if a moving object placed at q does not intersect any obstacles in the workspace. The free space F is the set of free configurations. A configuration space obstacle (C-obstacle) is the set of configurations where the moving object collides with workspace obstacles.

89. Slide 89 Disc in 2-D Workspace workspaceconfiguration space

90. Slide 90 Polygonal Robot Translating in 2-D Workspace workspace configuration space

91. Slide 91 Articulated Robot in 2-D Workspace workspace configuration space

92. Slide 92 Fundamental Question workspace configuration space Are two given points connected by a path?

93. Slide 93 Problem: Computing C-obstacles Input: –Polygonal moving object translating in 2-D workspace –Polygonal obstacles Output: configuration space obstacles represented as polygons

94. Slide 94 Minkowski Sum A B

95. Slide 95 Exercise A B A  B = ?

96. Slide 96 Minkowski Sum

97. Slide 97 Minkowski Sum

98. Slide 98 Minkowski Sum

99. Slide 99 Minkowski Sum The Minkowski sum of two sets A and B, denoted by A  B, is defined as A  B = { a+b | a  A, b  B } Similarly, the Minkowski difference is defined as A – B = { a–b | a  A, b  B } p q

100. Slide 100 Minkowski Sum of Non-convex Polyhedra

101. Slide 101 Configuration Space Obstacle If P is an obstacle in the workspace and M is a translating object. Then the C-space obstacle corresponding to P is P – M. P - M Obstacle P Robot M C- obstacle Classic result by Lozano-Perez and Wesley 1979

102. Slide 102 Minkowski Sum of Convex Polygons The Minkowski sum of two convex polygons A and B of m and n vertices respectively is a convex polygon A + B of m + n vertices. –The vertices of A + B are the “sums” of vertices of A and B.

103. Slide 103 Complexity of Minkowksi Sum 2D convex polygons: O(n+m) 2D non-convex polygons: O(n 2 m 2 ) –Decompose into convex polygons (e.g., triangles or trapezoids), compute the Minkowski sums, and take the union 3-D convex polyhedra: O(nm) 3-D non-convex polyhedra: O(n 3 m 3 )

104. Slide 104 Complexity of Computing C-obstacles 3D rigid robots with both translational and rotational DOF –6D C-space –Arrangement of non-linear surfaces –High combinatorial complexity Conclusion –Explicit computation of the boundary of C-obstacle is difficult and impractical for robots more than 3 DOFs

105. Slide 105 C-space Workspace Configuration Space x y Robot Initial Goal Free Obstacle C-obstacle A 2D Translating Robot

106. Slide 106 Computing C-Obstacles Difficult due to geometric and space complexity Practical solutions are only available for –Translating rigid robots: Minkowski sum –Robots with no more than 3 DOFs

107. Slide 107 Exact Motion Planning Approaches –Exact cell decomposition [Schwartz et al. 83] –Roadmap [Canny 88] –Criticality based method [Latombe 99] –Voronoi Diagram –Star-shaped roadmap [Varadhan et al. 06] Not practical –Due to free space computation –Limit for special and simple objects Ladders, sphere, convex shapes 3DOF

108. Slide 108 Approaches Exact Motion Planning –Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid Planner Combing ACD and PRM

109. Slide 109 Approximation Cell Decomposition (ACD) Not compute the free space exactly at once But compute it incrementally Relatively easy to implement –[Lozano-Pérez 83] –[Zhu et al. 91] –[Latombe 91] –[Zhang et al. 06]

110. Slide 110 fullmixed empty Approximation Cell Decomposition Full cell Empty cell Mixed cell –Mixed –Uncertain Configuration Space

111. Slide 111 Connectivity Graph G f : Free Connectivity GraphG: Connectivity Graph G f is a subgraph of G

112. Slide 112 Finding a Path by ACD Goal Initial G f : Free Connectivity Graph

113. Slide 113 Finding a Path by ACD L: Guiding Path First Graph Cut Algorithm –Guiding path in connectivity graph G –Only subdivide along this path –Update the graphs G and G f Described in Latombe’s book

114. Slide 114 First Graph Cut Algorithm Only subdivide along L L

115. Slide 115 Finding a Path by ACD

116. Slide 116 ACD for Path Non-existence C-space Goal Initial

117. Slide 117 Connectivity Graph Guiding Path ACD for Path Non-existence

118. Slide 118 ACD for Path Non-existence Connectivity graph is not connected No path! Sufficient condition for deciding path non-existence

119. Slide 119 Two-gear Example no path! Cells in C-obstacle Initial Goal Roadmap in F Vide o 3.356s

120. Slide 120 Cell Labeling Free Cell Query –Whether a cell completely lies in free space? C-obstacle Cell Query –Whether a cell completely lies in C-obstacle? fullmixed empty

121. Slide 121 Free Cell Query A Collision Detection Problem Does the cell lie inside free space? Do robot and obstacle separate at all configurations? Obstacle Workspace Configuration space ? Robot

122. Slide 122 Clearance Separation distance –A well studied geometric problem Determine a volume in C-space which are completely free d

123. Slide 123 C-obstacle Query Another Collision Detection Problem Does the cell lie inside C-obstacle? Do robot and obstacle intersect at all configurations? Obstacle Workspace Configuration space ? Robot

124. Slide 124 ‘Forbiddance’ ‘Forbiddance’: dual to clearance Penetration Depth –A geometric computation problem less investigated [Zhang et al. ACM SPM 2006] PD

125. Slide 125 Limitation of ACD Combinatorial complexity of cell decomposition Limited for low DOF problem –3-DOF robots

126. Slide 126 Approaches Exact Motion Planning –Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid Planner Combing ACD and PRM

127. Slide 127 Hybrid Planning Probabilistic roadmap motion planning + Efficient + Many DOFs -Narrow passages -Path non-existence Complete Motion Planning + Complete -Not efficient Can we combine them together?

128. Slide 128 Hybrid Approach for Complete Motion Planning Use Probabilistic Roadmap (PRM): –Capture the connectivity for mixed cells –Avoid substantial subdivision Use Approximation Cell Decomposition (ACD) –Completeness –Improve the sampling on narrow passages

129. Slide 129 Connectivity Graph G f : Free Connectivity GraphG: Connectivity Graph G f is a subgraph of G

130. Slide 130 Pseudo-free edges Pseudo free edge for two adjacent cells Goal Initial

131. Slide 131 Pseudo-free Connectivity Graph: G sf Goal Initial G sf = G f + Pseudo-edges

132. Slide 132 Algorithm G f G sf G

133. Slide 133 Results of Hybrid Planning

134. Slide 134 Results of Hybrid Planning

135. Slide 135 Results of Hybrid Planning 2.5 - 10 times speedup 3 DOF4 DOF timingcells #timingcells #timingcells # Hybrid34s50K16s48K102s164K ACD85s168K???? Speedup2.53.3≥10? ?

136. Slide 136 Summary Difficult for Exact Motion Planning –Due to the difficulty of free space configuration computation ACD is more practical –Explore the free space incrementally Hybrid Planning –Combine the completeness of ACD and efficiency of PRM

137. Slide 137 Outline Approximate cell decomposition Sampling-based motion planning

138. Slide 138 Approximate Cell Decomposition (ACD) Not compute the free space exactly at once But compute it incrementally Relatively easy to implement –[Lozano-Pérez 83] –[Zhu et al. 91] –[Latombe 91] –[Zhang et al. 06] Octree decomposition

139. Slide 139 fullmixed empty Approximate Cell Decomposition Full cell Empty cell Mixed cell –Mixed –Uncertain Cell labelling algorithms –[Zhang et al 06] Configuration Space

140. Slide 140 Finding a Path by ACD Goal Initial

141. Slide 141 Connectivity Graph G f : Free Connectivity GraphG: Connectivity Graph G f is a subgraph of G

142. Slide 142 Finding a Path by ACD Goal Initial G f : Free Connectivity Graph

143. Slide 143 Finding a Path by ACD L: Guiding Path First Graph Cut Algorithm –Guiding path in connectivity graph G –Only subdivide along this path –Update the graphs G and G f

144. Slide 144 First Graph Cut Algorithm Only subdivide the cells along L L : Guiding Path new G f

145. Slide 145 Finding a Path by ACD GfGf A channel Can be used for path smoothing.

146. Slide 146 ACD for Path Non-existence C-space Goal Initial

147. Slide 147 Connectivity Graph Guiding Path ACD for Path Non-existence

148. Slide 148 ACD for Path Non-existence Connectivity graph is not connected No path! A sufficient condition for deciding path non-existence

149. Slide 149 Live Demo –Gear-2DOF –Gear-3DOF

150. Slide 150 Five-gear Example Vide o Initial Goal roadmap in free space Total timing85s # of total cells168K

151. Slide 151 Two-gear Example no path! Cells in C-obstacle Initial Goal Roadmap in F Vide o 3.356s

152. Slide 152 Motion Planning Framework Continuous representation Discretization Graph search

153. Slide 153 Summary: Approximate Cell Decomposition Simple and easy to implement Efficient and practical for low DOF robots –Inefficient for 5 or more DOFs robot Resolution-complete –Find a path if there is one –Otherwise, report path non-existence –Up to some resolution of the cell

154. Slide 154 Outline Approximate cell decomposition Sampling-based motion planning

155. Slide 155 Motivation Geometric complexity Space dimensionality

156. Slide 156 Probabilistic Roadmap (PRM) free space q init q goal milestone [Kavraki, Svetska, Latombe,Overmars, 95] local path

157. Slide 157 Basic PRM Aalgorithm Input: geometry of the moving object & obstacles Output: roadmap G = (V, E) 1: V   and E  . 2: repeat 3: q  a configuration sampled uniformly at random from C. 4: if CLEAR(q)then 5: Add q to V. 6: N q  a set of nodes in V that are close to q. 6: for each q’  N q, in order of increasing d(q,q’) 7: if LINK(q’,q)then 8: Add an edge between q and q’ to E.

158. Slide 158 Two Geometric Primitives in C-space C LEAR( q ) Is configuration q collision free or not? L INK( q, q’) Is the straight-line path between q and q’ collision-free ?

159. Slide 159 Query Processing Connect q init and q goal to the roadmap Start at q init and q goal, perform a random walk, and try to connect with one of the milestones nearby Try multiple times

160. Slide 160 Two Tenets of PRM Planning  Checking sampled configurations and connections between samples for collision can be done efficiently.  Hierarchical collision checking [Hierarchical collision checking methods were developed independently from PRM, roughly at the same time]  A relatively small number of milestones and local paths are sufficient to capture the connectivity of the free space.  Exponential convergence in expansive free space (probabilistic completeness)

161. Slide 161 Why does it work? Intuition A small number of milestones almost “cover” the entire free space.

162. Slide 162 Two Tenets of PRM Planning  Checking sampled configurations and connections between samples for collision can be done efficiently.  Hierarchical collision checking [Hierarchical collision checking methods were developed independently from PRM, roughly at the same time]  A relatively small number of milestones and local paths are sufficient to capture the connectivity of the free space.  Exponential convergence in expansive free space (probabilistic completeness)

163. Slide 163 Narrow Passage Problem Narrow passages are difficult to be sampled due to their small volumes in C-space Narrow passage Alpha puzzle q init q goal Configuration Space

164. Slide 164 Difficulty Many small connected components

165. Slide 165 Strategies to Improve PRM Where to sample new milestones? –Sampling strategy Which milestones to connect? –Connection strategy Goal: –Minimize roadmap size to correctly answer motion-planning queries

166. Slide 166 Sampling Strategies

167. Slide 167 small visibility sets good visibility poor visibility Poor Visibility in Narrow Passages Non-uniform sampling strategies

168. Slide 168 But how to identify poor visibility regions? What is the source of information? –Robot and environment geometry How to exploit it? –Workspace-guided strategies –Dilation-based strategies –Filtering strategies –Adaptive strategies

169. Slide 169  Identify narrow passages in the workspace and map them into the configuration space Workspace-guided strategies

170. Slide 170 F O Dilation-based strategies During roadmap construction, allow milestones with small penetration Dilate the free space –[Hsu et al. 98, Saha et al. 05, Cheng et al. 06, Zhang et al. 07] A milestone with small penetration

171. Slide 171 Error If a path is returned, the answer is always correct. If no path is found, the answer may or may not be correct. We hope it is correct with high probability.

172. Slide 172 Weaker Completeness  Complete planner  Too slow  Heuristic planner  Too unreliable  Probabilistic completeness: If a solution path exists, then the probability that the planner will find one is a fast growing function that goes to 1 as the running time increases

173. Slide 173 Kinodynamic Planning

174. Slide 174 RRT for Kinodynamic Systems Rapidly-exploring Random Tree Randomly select a control input

175. Slide 175 More Examples Car pulling trailers (complicated kinematics -- no dynamics)

176. Slide 176 Summary: Sampling-based Motion Planning + Efficient in practice + Work for robots with many DOF (high- dimensional configuration spaces) + Has been applied for various motion planning problems (non-holonomic, kinodynamic planning etc.) -Narrow passages problems (one of the hot areas) -May not terminate when no path exists

177. Slide 177 Summary Configuration space Visibility graph Approximate cell decomposition –Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Sampling-based approach –High-dimensional Configuration Spaces –Capture the connectivity of the free space by sampling

178. Slide 178 References Books –J.C. Latombe. Robot Motion Planning, 1991. –S.M. LaValle, Planning Algorithms, 2006 Free book: http://msl.cs.uiuc.edu/planning/ –H. Choset et al. Principles of Robot Motion: Theory, Algorithms, and Implementations, 2005 Conferences –ICRA: IEEE International Conference on Robotics and Automation –IROS: IEEE/RSJ International Conference on Intelligent RObots and Systems –WAFR: Workshop on the Algorithmic Foundations of Robotics