1 / 10

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.

edythe
Download Presentation

Roadmap Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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}

More Related