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5. Roadmaps

5. Roadmaps. Hyeokjae Kwon Sungmin Kim. 1. RoadMap Definition. 1. RoadMap Path Planning. 1. Visibility Graph methods. 1. The Visibility Graph in Action (1). 1. The Visibility Graph in Action (2). 1. The Visibility Graph in Action (3). 1. The Visibility Graph in Action (4).

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5. Roadmaps

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  1. 5. Roadmaps Hyeokjae Kwon Sungmin Kim

  2. 1. RoadMap Definition

  3. 1. RoadMap Path Planning

  4. 1. Visibility Graph methods

  5. 1. The Visibility Graph in Action (1)

  6. 1. The Visibility Graph in Action (2)

  7. 1. The Visibility Graph in Action (3)

  8. 1. The Visibility Graph in Action (4)

  9. 1. The Visibility Graph (Done)

  10. 1. Reduced Visibility Graphs Start Goal

  11. 1. The Sweepline Algorithm

  12. 1. Sweepline Algorithm Example

  13. 2. Generalized Voronoi Diagram

  14. 2. Two-Equidistant

  15. 2. Homotopy Classes Start Start Goal Goal

  16. 2. Sensor-Based Construction of the GVD

  17. 3. General Voronoi Graph

  18. 3. Retract-like Structure Connectivity

  19. 3. Retract-like Structure Connectivity

  20. The Rod-Hierarchical Generalized Voronoi Graph What is different? *a point robot ㅡ> a Rod Robot *Non-Euclidean *Sensor Based Approach *Workspace -> Configuration Space (However, we measure distance in the workspace, not configuration space.)

  21. Distance The nearest point rod

  22. Rod-GVG-edges (a1) Rod-GVG-edges: each of the clusters represents a set of configurations equidistant to three obstacles. (a2) The configurations of the rod that are equidistant to three obstacles in the workspace.

  23. R-edges (b1) R-edges: the rods are two-way equidistant and tangent to a planar point-GVG edge. (b2) The configurations of the rod that are tangent to the planar point-GVG in the workspace.

  24. rod-HGVG The rod-HGVG then comprises rod-GVG edges and R-edges (c1) Placements of the rod along the rod-HGVG. (c2) The entire rod-HGVG

  25. Silhouette Methods

  26. Silhouette Methods The silhouette approaches use extrema of a function defined on a codimension one hyperplane called a slice.

  27. Silhouette Methods • Canny's Roadmap Algorithm • Opportunistic Path Planner(OPP)

  28. Canny's Roadmap Algorithm Canny's Roadmap Algorithm is one of the classical motion planning techniques that uses critical points. critical points

  29. The Basic Ideas • Pick a sweeping surface • As sweeping happens, detect extremal points and critical points (= places where connectivity changes) • For each slice where a critical point occurs, repeat this process recursively • Use this as the roadmap

  30. How To Find Extrema In order to find the extrema on a manifold we will refer to the Lagrange Multiplier Theorem.

  31. Canny's Roadmap Algorithm Sweep direction Critical points The silhouette curves trace the boundary of the environment. Critical points occur when the slice is tangent to the roadmap

  32. Accessibility and Departability In order to access and depart the roadmap we treat the slices which contain qstart and qgoalas critical slices and run the algorithm the same way.

  33. Connectivity Changes at Critical Points

  34. Connectivity Changes at Critical Points Silhouette curves on the torus

  35. Connectivity Changes at Critical Points

  36. Connectivity Changes at Critical Points

  37. Building the Roadmap We can now findtheextrema necessary to build the silhouette curves. We can findthecritical points where linking is necessary We can run the algorithm recursively to construct the whole roadmap

  38. Illustrative Example Let S be the ellipsoid with a through hole. Pc is a hyperplane of codimension1 ( x = c ) which will be swept through S in the X direction.

  39. Illustrative Example This is not a roadmap, it’s not connected.

  40. Illustrative Example Find the critical points . The roadmap is the union of all silhouette curves.

  41. Opportunistic Path Planner

  42. Opportunistic Path Planner The Opportunistic Path Planner is similar to Canny’s Roadmap but differs in the following ways • Silhouette curves are now called freeways and are constructed slightly differently • Linking curves are now called bridges • It does not always construct the whole roadmap • The algorithm is not recursive

  43. The bridge curves are constructed in the vicinity of interesting critical point Bridge curves are also built when freeways terminate in the free space at bifurcation points A bridge curve is built leading away from a bifurcation point to another freeway curve. The union of bridge and freeway curves, sometimes termed a skeleton, forms the one-dimensional roadmap.

  44. Opportunistic Path Planner

  45. OPP method looks for connectivity changes in the slice in the free configuration space. We are assured that we only need to look for critical points to connect disconnected components of the roadmap. If the start and goal freeways are connected, then the algorithm terminates.

  46. Building the Roadmap (1) Start tracing a freeway curve from the start configuration, and also from the goal. (2) If the curves leading from start and goal are not connected enumerate a split point or join point and add a bridge curve near the point. Else stop. (3) Find all points on the bridge curve that lie on other freeways and trace from these freeways. Go to step 2.

  47. Reference *Algorithms for Sensor-Based Robotics: RoadMap Methods CS 336, G.D. Hager (loosely based on notes by Nancy Amato and HowieChoset) *Robot Motion Control and Planning http://www.cs.bilkent.edu.tr/~saranli/courses/cs548 *Principles of Robot Motion-Theory, Algorithms, and Implementation

  48. Reference *AnOpportunisticGlobalPathPlanner1 JohnF.Canny2andMingC.Lin3 * Robotic Motion Planning: Roadmap Methods http://voronoi.sbp.ri.cmu.edu/~choset

  49. Question & Answer

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