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5. Roadmaps Hyeokjae Kwon Sungmin Kim

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1. RoadMap Definition

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1. RoadMap Path Planning

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1. Visibility Graph methods

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1. The Visibility Graph in Action (1)

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1. The Visibility Graph in Action (2)

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1. The Visibility Graph in Action (3)

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1. The Visibility Graph in Action (4)

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1. The Visibility Graph (Done)

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Start Goal 1. Reduced Visibility Graphs

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1. The Sweepline Algorithm

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1. Sweepline Algorithm Example

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2. Generalized Voronoi Diagram

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2. Two-Equidistant

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2. Homotopy Classes Start Goal Start Goal

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2. Sensor-Based Construction of the GVD

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3. General Voronoi Graph

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3. Retract-like Structure Connectivity

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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.)

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Distance rod The nearest point

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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.

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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.

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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

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Silhouette Methods

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

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

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

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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

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

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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

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Accessibility and Departability In order to access and depart the roadmap we treat the slices which contain q start and q goal as critical slices and run the algorithm the same way.

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Connectivity Changes at Critical Points

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Silhouette curves on the torus

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Connectivity Changes at Critical Points

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Building the Roadmap We can now find the extrema necessary to build the silhouette curves. We can find the critical points where linking is necessary We can run the algorithm recursively to construct the whole roadmap

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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.

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

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

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Opportunistic Path Planner

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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

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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.

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Opportunistic Path Planner

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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.

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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.

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Reference *Algorithms for Sensor-Based Robotics: RoadMap Methods CS 336, G.D. Hager (loosely based on notes by Nancy Amato and Howie Choset) *Robot Motion Control and Planning http://www.cs.bilkent.edu.tr/~saranli/courses/c s548 *Principles of Robot Motion-Theory, Algorithms, and Implementation

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

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Question & Answer

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