Roadmap Methods

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 qinitand qgoal  There exists a simple polygonal line lying in cl(Cfree) with end points qinitand 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. The Visibility Graph in Action (Part 1) • First, draw lines of sight from the start and goal to all “visible” vertices and corners of the world. goal start

9. The Visibility Graph in Action (Part 2) • Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight. goal start

10. The Visibility Graph in Action (Part 3) • Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight. goal start

11. The Visibility Graph in Action (Part 4) • Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight. goal start

12. The Visibility Graph (Done) • Repeat until you’re done. goal start

13. 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)

14. 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 )

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

16. General Voronoid Graph A GVG is formed by paths equidistant from the two closest objects This generates a very safe roadmap which avoids obstacles as much as possible

17. General Voronoi Diagram

18. What about concave obstacles? L L vs

19. Voronoi Diagram: Metrics

20. Voronoi Diagram (L2) Note the curved edges

21. Voronoi Diagram (L1) Note the lack of curved edges