Visibility Roadmaps

A visibility roadmap is obtained by generating all line segments between pairs of obstacle region vertices. Any line segment that lies entirely in the free space is added to the roadmap. Figure 5.3 shows an example. When a path planning is given, the initial position and goal position are also treated as vertices (in other words, they are connected to other vertices, if possible). This generates a connectivity graph that can be searched for a solution. The naive approach to constructing this graph takes time O(n³). The sweep-line principle can be applied to yield a more efficient algorithm.

There are two important notes about visibility roadmaps:

• The shortest-path solutions found in the roadmap will actually be the shortest-path solutions for the original problem. In addition, the paths ``touch'' the obstacle region. This is not acceptable in terms of the original problem, and the resulting paths should be modified.
• Many edges can be removed from the visibility roadmap, and optimal solutions will still be obtained. Any edge can be removed if the following property holds: extend the edge in both direction by a small amount. If either end ``pokes'' into the obstacle region, then the edge can be removed. Figure .b shows the edges that remain after this removal of performed.

 

Figure 5.3:   a) A visibility graph is constructed by joining obstacle region vertices; b) It is generally possible to reduce the number of edges in the visibility roadmap.