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)
Polygonal robot A translating at fix orientation
No rotation!
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
qinit
qgoal

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 Cfree at 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 qinit and 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 Cfree is 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?vs

19. Voronoi Diagram: Metrics

20. Voronoi Diagram (L2)
Note the curved edges

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