  A maximum-clearance roadmap tries to keep as far as possible from , as shown for the corridor in Figure 6.8. The resulting solution paths are sometimes preferred in mobile robotics applications because it is difficult to measure and control the precise position of a mobile robot. Traveling along the maximum-clearance roadmap reduces the chances of collisions due to these uncertainties. Other names for this roadmap are generalized Voronoi diagram and retraction method . It is considered as a generalization of the Voronoi diagram (recall from Section 5.2.2) from the case of points to the case of polygons. Each point along a roadmap edge is equidistant from two points on the boundary of . Each roadmap vertex corresponds to the intersection of two or more roadmap edges and is therefore equidistant from three or more points along the boundary of .

The retraction term comes from topology and provides a nice intuition about the method. A subspace is a deformation retract of a topological space if the following continuous homotopy, , can be defined as follows :

1. for all .
2. is a continuous function that maps every element of to some element of .
3. For all , for any .
The intuition is that is gradually thinned through the homotopy process, until a skeleton, , is obtained. An approximation to this shrinking process can be imagined by shaving off a thin layer around the whole boundary of . If this is repeated iteratively, the maximum-clearance roadmap is the only part that remains (assuming that the shaving always stops when thin slivers'' are obtained).

To construct the maximum-clearance roadmap, the concept of features from Section 5.3.3 is used again. Let the feature set refer to the set of all edges and vertices of . Candidate paths for the roadmap are produced by every pair of features. This leads to a naive time algorithm as follows. For every edge-edge feature pair, generate a line as shown in Figure 6.9a. For every vertex-vertex pair, generate a line as shown in Figure 6.9b. The maximum-clearance path between a point and a line is a parabola. Thus, for every edge-point pair, generate a parabolic curve as shown in Figure 6.9c. The portions of the paths that actually lie on the maximum-clearance roadmap are determined by intersecting the curves. Several algorithms exist that provide better asymptotic running times [616,626], but they are considerably more difficult to implement. The best-known algorithm runs in time in which is the number of roadmap curves .

Steven M LaValle 2020-08-14