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Roadmap Methods

How do I get there?. Roadmap Methods. Visibility Graph Voronoid Diagram. The Roadmap Idea. Capture the connectivity of C free in a network of 1-D curves: the roadmap. Visibility Graph Method (VGM). No rotation!. Polygonal robot A translating at fix orientation.

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Roadmap Methods

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  1. How do I get there? Roadmap Methods • Visibility Graph • Voronoid Diagram

  2. The Roadmap Idea Capture the connectivity of Cfree in a network of 1-D curves: the roadmap

  3. Visibility Graph Method (VGM) No rotation! • Polygonal robot A translating at fix orientation • Polygonal obstacle in R2 • VGM: construct a semi-free path as a simple polygonal line connecting qinit to qgoal

  4. Main Proposition • CB a polygonal region of the plane There exists a semi-free path between qinit and qgoal   There exists a simple polygonal line lying in cl(Cfree) with end points qinit and qgoal and such that its vertices are certices of CB

  5. Example qgoal qinit

  6. Visibility Graph - Definition The visibility graph is the non-directed graph G specified as: • Nodes: qinit, qgoal and vertices of CB • Edges: 2 nodes connected if either the line segment joining them is an edge of CB, or it lies entirely in Cfreeat endpoints Algorithm of the visibility graph method: • Construct visibility graph G • Search G for a path from qinit to qgoal • If a path is found, return it; otherwise failure

  7. Constructing the VG: Naïve Approach • X, X’: qinit, qgoal or CB vertices • If X, X’ endpoints of same edge of CB, then the nodes are connected by a link • Otherwise X, X’ are connected by a link iff the line passing through them does not intersect CB • Complexity of algorithm O(n3)

  8. Constructing the VG: Improvement • Variation of sweep-line algorithm • For each X, compute the orientation i of every half-line from X to another point Xi. Sort these orientations. • Rotate half-line from X, from 0 to 2. Stop at each i. At each stop, update intersection with CB • Algorithm is O(n2logn)

  9. Retraction Approach • Def.: X a topological space, Y a subspace of X. A surjective map XY is a retraction iff it is continuous and its restriction to Y is the identity • Def.: the retraction  preserves connectivity iff for all xX, x and (x) are in the same path-connected component. • Proposition: Let :Cfree R, where R  Cfree is a network of 1D curves, be a CPR. There exists a free-path between qinitand qgoal iff there exists a path in R between  (qinit)and  (qgoal )

  10. Voronoid Diagram • Def.: let =Cfree. For any q in Cfree, let Clearance(q)=minp ||q-p|| Near(q)={p   / ||q-p||=clearance(q)} • The Voronoid diagram of Cfreeis the set: Vor(Cfree)={q  Cfree / card(near(q))>1}

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