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

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

3. Roadmap Definition Let G be a graph that maps into C free. Let S be the set of all points reached by G. Then G is a roadmap if it satisfies: Accessibility: from any q  C free it is simple and efficient to compute a path  :[0,1]  C free s.t.  (0)=q, and  (1)  S. Connectivity Preserving: if there exists a path  :[0,1]  C free s.t.  (0)=q start and  (1)=q goal then there exists two points s 1 and s 2 in S that can be connected to q start and q goal and there exists a path  ’:[0,1]  C free s.t.  ’(0)=s 1 and  ’(1)=s 2

4. Visibility Graph Method (VGM) Polygonal robot A translating at fix orientation No rotation! Polygonal obstacle in R 2 VGM: construct a semi-free path as a simple polygonal line connecting q init to q goal

5. Main Proposition CB a polygonal region of the plane There exists a semi-free path between q init and q goal   There exists a simple polygonal line lying in cl(C free ) with end points q init and q goal and such that its vertices are vertices of CB

6. Example q init q goal

7. Visibility Graph - Definition The visibility graph is the non-directed graph G specified as: Nodes: q init, q goal 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 C free at endpoints Algorithm of the visibility graph method: 1.Construct visibility graph G 2.Search G for a path from q init to q goal 3.If a path is found, return it; otherwise failure

8. Constructing the VG: Naïve Approach X, X’: q init, q goal 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(n 3 )

9. Variation of sweep-line algorithm For each X, compute the orientation  i of every half-line from X to another point X i. 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(n 2 logn) Constructing the VG: Improvement

10. 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  :C free  R, where R  C free is a network of 1D curves, be a CPR. There exists a free-path between q init and q goal iff there exists a path in R between  (q init) and  (q goal )

11. Voronoid Diagram Def.: let  =  C free. For any q in C free, let Clearance(q)=min p   ||q-p|| Near(q)={p   / ||q-p||=clearance(q)} The Voronoid diagram of C free is the set: Vor(C free )={q  C free / card(near(q))>1}

12. Three possible cases